Subgroups — C₂×C₃₁×#3196 #21517498

Order 7936.

2342 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#525	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3196	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#484	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₃₁×#481	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃₁×#485	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#470	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#468	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#452	C₂	1	2	yes	maximal, complemented
10	C₂×C₃₁×#451	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#448	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#407	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
17	C₂²×SD₁₆×C₃₁	C₂²	1	2	yes	complemented
18	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
19	C₃₁×#525	—	2	4	no	—
20	C₃₁×#525	C₂²	1	8	yes	—
21	C₂×C₃₁×#123	—	2	1	no	—
22	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
23	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
24	C₂×C₃₁×C₂²⋊Q₈	—	2	4	no	—
25	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	8	yes	complemented
30	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#121	C₂²	1	1	yes	characteristic
32	C₂×C₃₁×#120	—	2	1	no	—
33	C₂×C₄×D₄×C₃₁	—	2	1	no	—
34	C₂×C₄×D₄×C₃₁	—	2	1	no	—
35	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
37	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
39	C₃₁×#485	C₂²	1	16	yes	complemented
40	C₃₁×#484	C₂²	1	8	yes	complemented
41	C₃₁×#481	C₂²	1	8	yes	complemented
42	C₃₁×#470	C₂²	1	16	yes	complemented
43	C₃₁×#468	C₂²	1	16	yes	complemented
44	C₃₁×#452	C₂²	1	16	yes	complemented
45	C₃₁×#451	C₂²	1	8	yes	complemented
46	C₃₁×#448	C₂²	1	8	yes	complemented
47	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
48	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
49	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
50	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
51	C₂×C₃₁×#103	C₂²	1	2	yes	complemented
52	C₂×C₃₁×#102	C₂²	1	2	yes	complemented
53	C₃₁×#407	C₂²	1	8	yes	complemented
54	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
55	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
56	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
57	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
58	C₂×#41×C₃₁	—	2	2	no	—
59	C₂×#41×C₃₁	—	4	4	no	—
60	C₂×#41×C₃₁	C₂³	1	4	yes	—
61	C₂²×Q₈×C₃₁	—	2	2	no	—
62	C₂²×Q₈×C₃₁	—	2	1	no	—
63	C₂²×Q₈×C₃₁	—	4	4	no	—
64	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
67	C₂²×D₄×C₃₁	—	2	1	no	—
68	C₂²×D₄×C₃₁	—	2	1	no	—
69	C₂²×D₄×C₃₁	—	2	1	no	—
70	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	—	2	1	no	—
72	C₂³×C₄×C₃₁	—	2	1	no	—
73	C₂³×C₄×C₃₁	—	2	1	no	—
74	C₂³×C₄×C₃₁	—	4	4	no	—
75	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
76	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
77	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
78	C₂³×C₄×C₃₁	D₄	1	4	yes	—
79	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
80	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
83	C₂×Q₁₆×C₃₁	—	4	4	no	—
84	C₂×Q₁₆×C₃₁	C₂³	1	8	yes	complemented
85	C₂×SD₁₆×C₃₁	—	4	4	no	—
86	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	complemented
87	C₂×M₁₆×C₃₁	—	4	4	no	—
88	C₂×M₁₆×C₃₁	C₂³	1	8	yes	—
89	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
90	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
91	C₃₁×#123	—	2	8	no	—
92	C₃₁×C₂²⋊Q₈	—	2	4	no	—
93	C₃₁×C₂²⋊Q₈	—	2	4	no	—
94	C₃₁×C₂²⋊Q₈	—	2	8	no	—
95	C₃₁×C₂²⋊Q₈	—	2	8	no	—
96	C₃₁×C₂²⋊Q₈	—	2	2	no	—
97	C₃₁×C₂²⋊Q₈	—	2	2	no	—
98	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
99	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
100	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
101	C₃₁×C₂²⋊Q₈	C₂³	1	4	yes	—
102	C₃₁×#121	—	2	2	no	—
103	C₃₁×#121	—	2	2	no	—
104	C₃₁×#121	C₂³	1	4	yes	—
105	C₃₁×#121	C₂³	1	4	yes	—
106	C₃₁×#120	—	2	8	no	—
107	C₄×D₄×C₃₁	—	2	8	no	—
108	C₄×D₄×C₃₁	—	2	8	no	—
109	C₃₁×#117	—	2	2	no	—
110	C₃₁×#117	C₂³	1	4	yes	—
111	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
114	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
116	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
117	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
118	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
124	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
125	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
127	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
128	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
129	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
130	C₂×C₄²×C₃₁	—	2	1	no	—
131	C₃₁×#107	C₂³	1	8	yes	complemented
132	C₃₁×#106	C₂³	1	8	yes	complemented
133	C₃₁×#103	C₂³	1	8	yes	complemented
134	C₃₁×#103	C₂³	1	8	yes	complemented
135	C₃₁×#103	C₂³	1	8	yes	complemented
136	C₃₁×#102	C₂³	1	8	yes	complemented
137	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
138	C₃₁×C₂²⋊C₈	C₂³	1	4	yes	—
139	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
140	C₂⁴×C₃₁	—	2	4	no	—
141	C₂⁴×C₃₁	—	2	1	no	—
142	C₂⁴×C₃₁	—	4	2	no	—
143	C₂⁴×C₃₁	—	4	2	no	—
144	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
145	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
146	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
147	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
148	#41×C₃₁	—	4	8	no	—
149	#41×C₃₁	—	4	8	no	—
150	C₂×Q₈×C₃₁	—	2	4	no	—
151	C₂×Q₈×C₃₁	—	2	2	no	—
152	C₂×Q₈×C₃₁	—	2	1	no	—
153	C₂×Q₈×C₃₁	—	2	4	no	—
154	C₂×Q₈×C₃₁	—	2	4	no	—
155	C₂×Q₈×C₃₁	—	2	4	no	—
156	C₂×Q₈×C₃₁	—	4	8	no	—
157	C₂×Q₈×C₃₁	—	4	4	no	—
158	C₂×Q₈×C₃₁	—	4	8	no	—
159	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
160	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
161	C₂×D₄×C₃₁	—	2	4	no	—
162	C₂×D₄×C₃₁	—	2	4	no	—
163	C₂×D₄×C₃₁	—	2	4	no	—
164	C₂×D₄×C₃₁	—	2	1	no	—
165	C₂×D₄×C₃₁	—	2	4	no	—
166	C₂×D₄×C₃₁	—	4	4	no	—
167	C₂×D₄×C₃₁	—	4	4	no	—
168	C₂×D₄×C₃₁	—	4	4	no	—
169	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₃₁	—	2	4	no	—
171	C₂²×C₄×C₃₁	—	2	4	no	—
172	C₂²×C₄×C₃₁	—	2	4	no	—
173	C₂²×C₄×C₃₁	—	2	4	no	—
174	C₂²×C₄×C₃₁	—	2	4	no	—
175	C₂²×C₄×C₃₁	—	2	1	no	—
176	C₂²×C₄×C₃₁	—	2	1	no	—
177	C₂²×C₄×C₃₁	—	2	2	no	—
178	C₂²×C₄×C₃₁	—	2	1	no	—
179	C₂²×C₄×C₃₁	—	2	2	no	—
180	C₂²×C₄×C₃₁	—	2	4	no	—
181	C₂²×C₄×C₃₁	—	4	8	no	—
182	C₂²×C₄×C₃₁	—	4	8	no	—
183	C₂²×C₄×C₃₁	—	4	2	no	—
184	C₂²×C₄×C₃₁	—	4	2	no	—
185	C₂²×C₄×C₃₁	—	4	2	no	—
186	C₂²×C₄×C₃₁	—	4	4	no	—
187	C₂²×C₄×C₃₁	—	4	4	no	—
188	C₂²×C₄×C₃₁	—	4	4	no	—
189	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
190	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
191	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
192	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
196	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
197	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
198	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
201	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
202	Q₁₆×C₃₁	—	4	8	no	—
203	SD₁₆×C₃₁	—	4	8	no	—
204	M₁₆×C₃₁	—	4	8	no	—
205	C₂×C₈×C₃₁	—	2	4	no	—
206	C₂×C₈×C₃₁	—	2	4	no	—
207	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
208	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
209	C₄⋊C₄×C₃₁	—	2	8	no	—
210	C₄⋊C₄×C₃₁	—	2	8	no	—
211	C₄⋊C₄×C₃₁	—	2	4	no	—
212	C₄⋊C₄×C₃₁	—	2	2	no	—
213	C₄⋊C₄×C₃₁	—	2	4	no	—
214	C₄⋊C₄×C₃₁	—	2	4	no	—
215	C₄⋊C₄×C₃₁	—	2	1	no	—
216	C₄⋊C₄×C₃₁	—	2	1	no	—
217	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
218	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
219	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
220	C₂²⋊C₄×C₃₁	—	2	8	no	—
221	C₂²⋊C₄×C₃₁	—	2	8	no	—
222	C₂²⋊C₄×C₃₁	—	2	4	no	—
223	C₂²⋊C₄×C₃₁	—	2	4	no	—
224	C₂²⋊C₄×C₃₁	—	2	4	no	—
225	C₂²⋊C₄×C₃₁	—	2	4	no	—
226	C₂²⋊C₄×C₃₁	—	2	4	no	—
227	C₄²×C₃₁	—	2	4	no	—
228	C₂×#3196	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
229	C₂³×C₃₁	—	2	2	no	—
230	C₂³×C₃₁	—	2	2	no	—
231	C₂³×C₃₁	—	2	2	no	—
232	C₂³×C₃₁	—	2	2	no	—
233	C₂³×C₃₁	—	2	8	no	—
234	C₂³×C₃₁	—	2	2	no	—
235	C₂³×C₃₁	—	2	2	no	—
236	C₂³×C₃₁	—	2	2	no	—
237	C₂³×C₃₁	—	2	2	no	—
238	C₂³×C₃₁	—	2	2	no	—
239	C₂³×C₃₁	—	4	4	no	—
240	C₂³×C₃₁	—	4	4	no	—
241	C₂³×C₃₁	—	4	16	no	—
242	C₂³×C₃₁	—	4	2	no	—
243	C₂³×C₃₁	—	4	2	no	—
244	C₂³×C₃₁	—	4	2	no	—
245	C₂³×C₃₁	#137	1	4	yes	—
246	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
247	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
248	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
249	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
250	Q₈×C₃₁	—	2	4	no	—
251	Q₈×C₃₁	—	2	4	no	—
252	Q₈×C₃₁	—	2	4	no	—
253	Q₈×C₃₁	—	4	8	no	—
254	Q₈×C₃₁	—	4	4	no	—
255	Q₈×C₃₁	—	4	2	no	—
256	D₄×C₃₁	—	2	4	no	—
257	D₄×C₃₁	—	4	8	no	—
258	D₄×C₃₁	—	4	8	no	—
259	D₄×C₃₁	—	4	8	no	—
260	D₄×C₃₁	—	4	2	no	—
261	C₂×C₄×C₃₁	—	2	2	no	—
262	C₂×C₄×C₃₁	—	2	2	no	—
263	C₂×C₄×C₃₁	—	2	2	no	—
264	C₂×C₄×C₃₁	—	2	4	no	—
265	C₂×C₄×C₃₁	—	2	4	no	—
266	C₂×C₄×C₃₁	—	2	4	no	—
267	C₂×C₄×C₃₁	—	2	4	no	—
268	C₂×C₄×C₃₁	—	2	4	no	—
269	C₂×C₄×C₃₁	—	2	4	no	—
270	C₂×C₄×C₃₁	—	2	2	no	—
271	C₂×C₄×C₃₁	—	2	2	no	—
272	C₂×C₄×C₃₁	—	2	4	no	—
273	C₂×C₄×C₃₁	—	2	2	no	—
274	C₂×C₄×C₃₁	—	2	4	no	—
275	C₂×C₄×C₃₁	—	4	4	no	—
276	C₂×C₄×C₃₁	—	4	4	no	—
277	C₂×C₄×C₃₁	—	4	4	no	—
278	C₂×C₄×C₃₁	—	4	8	no	—
279	C₂×C₄×C₃₁	—	4	8	no	—
280	C₂×C₄×C₃₁	—	4	8	no	—
281	C₂×C₄×C₃₁	—	4	2	no	—
282	C₂×C₄×C₃₁	—	4	4	no	—
283	C₂×C₄×C₃₁	—	4	2	no	—
284	C₂×C₄×C₃₁	—	4	4	no	—
285	C₂×C₄×C₃₁	—	4	2	no	—
286	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
291	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
292	C₈×C₃₁	—	2	4	no	—
293	C₈×C₃₁	—	2	4	no	—
294	C₂×#525	C₂×C₃₁	1	1	yes	characteristic
295	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
296	#3196	C₂×C₃₁	1	16	yes	direct product factor
297	C₂×#484	C₂×C₃₁	1	1	yes	characteristic, complemented
298	C₂×#481	C₂×C₃₁	1	1	yes	characteristic, complemented
299	C₂×#485	C₂×C₃₁	1	2	yes	complemented
300	C₂×#470	C₂×C₃₁	1	2	yes	complemented
301	C₂×#468	C₂×C₃₁	1	2	yes	complemented
302	C₂×#452	C₂×C₃₁	1	2	yes	complemented
303	C₂×#451	C₂×C₃₁	1	1	yes	characteristic, complemented
304	C₂×#448	C₂×C₃₁	1	1	yes	characteristic, complemented
305	C₂×#407	C₂×C₃₁	1	1	yes	characteristic, complemented
306	C₂²×C₃₁	—	2	2	no	—
307	C₂²×C₃₁	—	2	8	no	—
308	C₂²×C₃₁	—	2	2	no	—
309	C₂²×C₃₁	—	2	2	no	—
310	C₂²×C₃₁	—	2	2	no	—
311	C₂²×C₃₁	—	2	2	no	—
312	C₂²×C₃₁	—	2	2	no	—
313	C₂²×C₃₁	—	2	2	no	—
314	C₂²×C₃₁	—	2	2	no	—
315	C₂²×C₃₁	—	4	16	no	—
316	C₂²×C₃₁	—	4	4	no	—
317	C₂²×C₃₁	—	4	4	no	—
318	C₂²×C₃₁	—	4	2	no	—
319	C₂²×C₃₁	—	4	2	no	—
320	C₂²×C₃₁	—	4	2	no	—
321	C₂²×C₃₁	—	8	4	no	—
322	C₂²×C₃₁	#534	1	2	yes	—
323	C₂²×C₃₁	C₂×#137	1	4	yes	—
324	C₂²×C₃₁	C₂×#137	1	2	yes	—
325	C₂²×C₃₁	#575	1	2	yes	—
326	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
327	C₄×C₃₁	—	2	2	no	—
328	C₄×C₃₁	—	2	2	no	—
329	C₄×C₃₁	—	2	2	no	—
330	C₄×C₃₁	—	4	2	no	—
331	C₄×C₃₁	—	4	4	no	—
332	C₄×C₃₁	—	4	2	no	—
333	C₄×C₃₁	—	4	4	no	—
334	C₄×C₃₁	—	4	2	no	—
335	C₄×C₃₁	#534	1	2	yes	—
336	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
337	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
338	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
339	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
340	C₂²×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
341	C₂²×SD₁₆	C₂²×C₃₁	1	2	yes	complemented
342	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
343	#525	—	2	4	no	—
344	#525	C₂²×C₃₁	1	8	yes	—
345	C₂×#123	—	2	1	no	—
346	C₂×C₂²⋊Q₈	—	2	1	no	—
347	C₂×C₂²⋊Q₈	—	2	1	no	—
348	C₂×C₂²⋊Q₈	—	2	4	no	—
349	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
350	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
351	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
352	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
353	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
354	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
355	C₂×#121	C₂²×C₃₁	1	1	yes	characteristic
356	C₂×#120	—	2	1	no	—
357	C₂×C₄×D₄	—	2	1	no	—
358	C₂×C₄×D₄	—	2	1	no	—
359	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
360	C₂²×C₄⋊C₄	—	2	1	no	—
361	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
362	C₂²×C₂²⋊C₄	—	2	1	no	—
363	#485	C₂²×C₃₁	1	16	yes	complemented
364	#484	C₂²×C₃₁	1	8	yes	complemented
365	#481	C₂²×C₃₁	1	8	yes	complemented
366	#470	C₂²×C₃₁	1	16	yes	complemented
367	#468	C₂²×C₃₁	1	16	yes	complemented
368	#452	C₂²×C₃₁	1	16	yes	complemented
369	#451	C₂²×C₃₁	1	8	yes	complemented
370	#448	C₂²×C₃₁	1	8	yes	complemented
371	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
372	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
373	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
374	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
375	C₂×#103	C₂²×C₃₁	1	2	yes	complemented
376	C₂×#102	C₂²×C₃₁	1	2	yes	complemented
377	#407	C₂²×C₃₁	1	8	yes	complemented
378	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
379	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
380	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
381	C₂×C₃₁	—	2	4	no	—
382	C₂×C₃₁	—	4	2	no	—
383	C₂×C₃₁	—	4	2	no	—
384	C₂×C₃₁	—	8	2	no	—
385	C₂×C₃₁	#3196	1	4	yes	direct product factor
386	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
387	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
388	C₂×C₃₁	C₂×#575	1	1	yes	characteristic
389	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
390	C₂×#41	—	2	2	no	—
391	C₂×#41	—	4	4	no	—
392	C₂×#41	C₂³×C₃₁	1	4	yes	—
393	C₂²×Q₈	—	2	1	no	—
394	C₂²×Q₈	—	2	2	no	—
395	C₂²×Q₈	—	4	4	no	—
396	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
397	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
398	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
399	C₂²×D₄	—	2	1	no	—
400	C₂²×D₄	—	2	1	no	—
401	C₂²×D₄	—	2	1	no	—
402	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
403	C₂³×C₄	—	2	1	no	—
404	C₂³×C₄	—	2	1	no	—
405	C₂³×C₄	—	2	1	no	—
406	C₂³×C₄	—	4	4	no	—
407	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
408	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
409	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
410	C₂³×C₄	D₄×C₃₁	1	4	yes	—
411	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
412	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
413	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
414	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
415	C₂×Q₁₆	—	4	4	no	—
416	C₂×Q₁₆	C₂³×C₃₁	1	8	yes	complemented
417	C₂×SD₁₆	—	4	4	no	—
418	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	complemented
419	C₂×M₁₆	—	4	4	no	—
420	C₂×M₁₆	C₂³×C₃₁	1	8	yes	—
421	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
422	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
423	#123	—	2	8	no	—
424	C₂²⋊Q₈	—	2	4	no	—
425	C₂²⋊Q₈	—	2	4	no	—
426	C₂²⋊Q₈	—	2	8	no	—
427	C₂²⋊Q₈	—	2	8	no	—
428	C₂²⋊Q₈	—	2	2	no	—
429	C₂²⋊Q₈	—	2	2	no	—
430	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
431	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
432	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
433	C₂²⋊Q₈	C₂³×C₃₁	1	4	yes	—
434	#121	—	2	2	no	—
435	#121	—	2	2	no	—
436	#121	C₂³×C₃₁	1	4	yes	—
437	#121	C₂³×C₃₁	1	4	yes	—
438	#120	—	2	8	no	—
439	C₄×D₄	—	2	8	no	—
440	C₄×D₄	—	2	8	no	—
441	#117	—	2	2	no	—
442	#117	C₂³×C₃₁	1	4	yes	—
443	C₂×C₄⋊C₄	—	2	2	no	—
444	C₂×C₄⋊C₄	—	2	2	no	—
445	C₂×C₄⋊C₄	—	2	8	no	—
446	C₂×C₄⋊C₄	—	2	1	no	—
447	C₂×C₄⋊C₄	—	2	2	no	—
448	C₂×C₄⋊C₄	—	2	1	no	—
449	C₂×C₄⋊C₄	—	2	1	no	—
450	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
451	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
452	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
453	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
454	C₂×C₂²⋊C₄	—	2	2	no	—
455	C₂×C₂²⋊C₄	—	2	2	no	—
456	C₂×C₂²⋊C₄	—	2	8	no	—
457	C₂×C₂²⋊C₄	—	2	1	no	—
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	1	no	—
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₄²	—	2	1	no	—
463	#107	C₂³×C₃₁	1	8	yes	complemented
464	#106	C₂³×C₃₁	1	8	yes	complemented
465	#103	C₂³×C₃₁	1	8	yes	complemented
466	#103	C₂³×C₃₁	1	8	yes	complemented
467	#103	C₂³×C₃₁	1	8	yes	complemented
468	#102	C₂³×C₃₁	1	8	yes	complemented
469	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
470	C₂²⋊C₈	C₂³×C₃₁	1	4	yes	—
471	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
472	C₃₁	C₂×#3196	1	1	yes	characteristic, direct product factor, Sylow
473	C₂⁴	—	2	4	no	—
474	C₂⁴	—	2	1	no	—
475	C₂⁴	—	4	2	no	—
476	C₂⁴	—	4	2	no	—
477	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
480	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
481	#41	—	4	8	no	—
482	#41	—	4	8	no	—
483	C₂×Q₈	—	2	4	no	—
484	C₂×Q₈	—	2	2	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	4	no	—
487	C₂×Q₈	—	2	4	no	—
488	C₂×Q₈	—	2	4	no	—
489	C₂×Q₈	—	4	8	no	—
490	C₂×Q₈	—	4	4	no	—
491	C₂×Q₈	—	4	8	no	—
492	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
493	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
494	C₂×D₄	—	2	4	no	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	2	1	no	—
498	C₂×D₄	—	2	4	no	—
499	C₂×D₄	—	4	4	no	—
500	C₂×D₄	—	4	4	no	—
501	C₂×D₄	—	4	4	no	—
502	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	4	no	—
505	C₂²×C₄	—	2	4	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	2	1	no	—
508	C₂²×C₄	—	2	4	no	—
509	C₂²×C₄	—	2	2	no	—
510	C₂²×C₄	—	2	1	no	—
511	C₂²×C₄	—	2	4	no	—
512	C₂²×C₄	—	2	2	no	—
513	C₂²×C₄	—	2	4	no	—
514	C₂²×C₄	—	4	8	no	—
515	C₂²×C₄	—	4	8	no	—
516	C₂²×C₄	—	4	2	no	—
517	C₂²×C₄	—	4	2	no	—
518	C₂²×C₄	—	4	2	no	—
519	C₂²×C₄	—	4	4	no	—
520	C₂²×C₄	—	4	4	no	—
521	C₂²×C₄	—	4	4	no	—
522	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
526	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
527	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
528	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
529	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
530	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
531	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
532	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
533	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
534	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
535	Q₁₆	—	4	8	no	—
536	SD₁₆	—	4	8	no	—
537	M₁₆	—	4	8	no	—
538	C₂×C₈	—	2	4	no	—
539	C₂×C₈	—	2	4	no	—
540	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
541	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
542	C₄⋊C₄	—	2	8	no	—
543	C₄⋊C₄	—	2	8	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	2	no	—
546	C₄⋊C₄	—	2	4	no	—
547	C₄⋊C₄	—	2	4	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
551	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
552	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
553	C₂²⋊C₄	—	2	8	no	—
554	C₂²⋊C₄	—	2	8	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	4	no	—
560	C₄²	—	2	4	no	—
561	C₂³	—	2	2	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	8	no	—
566	C₂³	—	2	2	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	2	2	no	—
570	C₂³	—	2	2	no	—
571	C₂³	—	4	4	no	—
572	C₂³	—	4	4	no	—
573	C₂³	—	4	16	no	—
574	C₂³	—	4	2	no	—
575	C₂³	—	4	2	no	—
576	C₂³	—	4	2	no	—
577	C₂³	C₃₁×#137	1	4	yes	—
578	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
579	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
580	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
581	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
582	Q₈	—	2	4	no	—
583	Q₈	—	2	4	no	—
584	Q₈	—	2	4	no	—
585	Q₈	—	4	8	no	—
586	Q₈	—	4	4	no	—
587	Q₈	—	4	2	no	—
588	D₄	—	2	4	no	—
589	D₄	—	4	8	no	—
590	D₄	—	4	8	no	—
591	D₄	—	4	8	no	—
592	D₄	—	4	2	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	4	no	—
599	C₂×C₄	—	2	4	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	4	4	no	—
608	C₂×C₄	—	4	4	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	4	8	no	—
611	C₂×C₄	—	4	8	no	—
612	C₂×C₄	—	4	8	no	—
613	C₂×C₄	—	4	2	no	—
614	C₂×C₄	—	4	4	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	4	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
619	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
620	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
621	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
622	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
623	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
624	C₈	—	2	4	no	—
625	C₈	—	2	4	no	—
626	C₂²	—	2	2	no	—
627	C₂²	—	2	8	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	2	2	no	—
633	C₂²	—	2	2	no	—
634	C₂²	—	2	2	no	—
635	C₂²	—	4	16	no	—
636	C₂²	—	4	4	no	—
637	C₂²	—	4	4	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	4	2	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	8	4	no	—
642	C₂²	C₃₁×#534	1	2	yes	—
643	C₂²	C₂×C₃₁×#137	1	4	yes	—
644	C₂²	C₂×C₃₁×#137	1	2	yes	—
645	C₂²	C₃₁×#575	1	2	yes	—
646	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
647	C₄	—	2	2	no	—
648	C₄	—	2	2	no	—
649	C₄	—	2	2	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	4	no	—
652	C₄	—	4	2	no	—
653	C₄	—	4	4	no	—
654	C₄	—	4	2	no	—
655	C₄	C₃₁×#534	1	2	yes	—
656	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
657	C₂	—	2	4	no	—
658	C₂	—	4	2	no	—
659	C₂	—	4	2	no	—
660	C₂	—	8	2	no	—
661	C₂	C₃₁×#3196	1	4	yes	direct product factor
662	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
663	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic
664	C₂	C₂×C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#525	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂⁴×C₄×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#3196	C₂	direct (N × H)
4. C₂×C₃₁×#484	C₂	semidirect
5. C₂×C₃₁×#481	C₂	semidirect
6. C₂×C₃₁×#485	C₂	semidirect
7. C₂×C₃₁×#470	C₂	semidirect
8. C₂×C₃₁×#468	C₂	semidirect
9. C₂×C₃₁×#452	C₂	semidirect
10. C₂×C₃₁×#451	C₂	semidirect
11. C₂×C₃₁×#448	C₂	semidirect
12. C₂×C₃₁×#407	C₂	semidirect
16. C₂²×Q₁₆×C₃₁	C₂²	semidirect
17. C₂²×SD₁₆×C₃₁	C₂²	semidirect
29. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
39. C₃₁×#485	C₂²	semidirect
40. C₃₁×#484	C₂²	semidirect
41. C₃₁×#481	C₂²	semidirect
42. C₃₁×#470	C₂²	semidirect
43. C₃₁×#468	C₂²	semidirect
44. C₃₁×#452	C₂²	semidirect
45. C₃₁×#451	C₂²	semidirect
46. C₃₁×#448	C₂²	semidirect
47. C₂×C₃₁×#107	C₂²	semidirect
48. C₂×C₃₁×#106	C₂²	semidirect
49. C₂×C₃₁×#103	C₂²	semidirect
50. C₂×C₃₁×#103	C₂²	semidirect
51. C₂×C₃₁×#103	C₂²	semidirect
52. C₂×C₃₁×#102	C₂²	semidirect
53. C₃₁×#407	C₂²	semidirect
84. C₂×Q₁₆×C₃₁	C₂³	semidirect
86. C₂×SD₁₆×C₃₁	C₂³	semidirect
131. C₃₁×#107	C₂³	semidirect
132. C₃₁×#106	C₂³	semidirect
133. C₃₁×#103	C₂³	semidirect
134. C₃₁×#103	C₂³	semidirect
135. C₃₁×#103	C₂³	semidirect
136. C₃₁×#102	C₂³	semidirect
228. C₂×#3196	C₃₁	direct (N × H)
295. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
296. #3196	C₂×C₃₁	direct (N × H)
297. C₂×#484	C₂×C₃₁	semidirect
298. C₂×#481	C₂×C₃₁	semidirect
299. C₂×#485	C₂×C₃₁	semidirect
300. C₂×#470	C₂×C₃₁	semidirect
301. C₂×#468	C₂×C₃₁	semidirect
302. C₂×#452	C₂×C₃₁	semidirect
303. C₂×#451	C₂×C₃₁	semidirect
304. C₂×#448	C₂×C₃₁	semidirect
305. C₂×#407	C₂×C₃₁	semidirect
340. C₂²×Q₁₆	C₂²×C₃₁	semidirect
341. C₂²×SD₁₆	C₂²×C₃₁	semidirect
353. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
363. #485	C₂²×C₃₁	semidirect
364. #484	C₂²×C₃₁	semidirect
365. #481	C₂²×C₃₁	semidirect
366. #470	C₂²×C₃₁	semidirect
367. #468	C₂²×C₃₁	semidirect
368. #452	C₂²×C₃₁	semidirect
369. #451	C₂²×C₃₁	semidirect
370. #448	C₂²×C₃₁	semidirect
371. C₂×#107	C₂²×C₃₁	semidirect
372. C₂×#106	C₂²×C₃₁	semidirect
373. C₂×#103	C₂²×C₃₁	semidirect
374. C₂×#103	C₂²×C₃₁	semidirect
375. C₂×#103	C₂²×C₃₁	semidirect
376. C₂×#102	C₂²×C₃₁	semidirect
377. #407	C₂²×C₃₁	semidirect
385. C₂×C₃₁	#3196	direct (N × H)
416. C₂×Q₁₆	C₂³×C₃₁	semidirect
418. C₂×SD₁₆	C₂³×C₃₁	semidirect
463. #107	C₂³×C₃₁	semidirect
464. #106	C₂³×C₃₁	semidirect
465. #103	C₂³×C₃₁	semidirect
466. #103	C₂³×C₃₁	semidirect
467. #103	C₂³×C₃₁	semidirect
468. #102	C₂³×C₃₁	semidirect
472. C₃₁	C₂×#3196	direct (N × H)
661. C₂	C₃₁×#3196	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃₁×#525	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₃₁×#3196	2
C₂×C₃₁×#484	2
C₂×C₃₁×#481	2
C₂×C₃₁×#485	2
C₂×C₃₁×#470	2
C₂×C₃₁×#468	2
C₂×C₃₁×#452	2
C₂×C₃₁×#451	2
C₂×C₃₁×#448	2
C₂×C₃₁×#407	2
C₂×#3196	31

Subgroups — C2×C31×#3196 #21517498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3196 #21517498