Subgroups — C₂×C₃₁×#3193 #21517495

Order 7936.

2666 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×#121	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3193	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#484	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#480	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#468	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×#448	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#450	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#407	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
11	C₂³×D₄×C₃₁	—	2	1	no	—
12	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
13	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
15	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	complemented
16	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
17	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
18	C₂×C₃₁×C₂²⋊Q₈	—	2	4	no	—
19	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
21	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	8	yes	complemented
22	C₂×C₃₁×#121	—	2	1	no	—
23	C₂×C₃₁×#121	—	2	4	no	—
24	C₂×C₃₁×#121	—	2	1	no	—
25	C₂×C₃₁×#121	C₂²	1	2	yes	—
26	C₂×C₃₁×#121	C₂²	1	8	yes	—
27	C₂×C₃₁×#121	C₂²	1	2	yes	—
28	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
31	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
32	C₂²×C₂²⋊C₄×C₃₁	—	2	1	no	—
33	C₃₁×#484	C₂²	1	16	yes	complemented
34	C₃₁×#480	C₂²	1	16	yes	complemented
35	C₃₁×#468	C₂²	1	32	yes	complemented
36	C₃₁×#450	C₂²	1	16	yes	complemented
37	C₃₁×#448	C₂²	1	16	yes	complemented
38	C₂×C₃₁×#107	C₂²	1	4	yes	complemented
39	C₂×C₃₁×#103	C₂²	1	4	yes	complemented
40	C₂×C₃₁×#102	C₂²	1	4	yes	complemented
41	C₃₁×#407	C₂²	1	8	yes	complemented
42	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
43	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	complemented
44	C₂⁵×C₃₁	—	2	1	no	—
45	C₂⁵×C₃₁	D₄	1	1	yes	characteristic
46	C₂²×Q₈×C₃₁	—	2	2	no	—
47	C₂²×Q₈×C₃₁	—	2	1	no	—
48	C₂²×Q₈×C₃₁	—	4	4	no	—
49	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
50	C₂²×Q₈×C₃₁	C₂³	1	4	yes	complemented
51	C₂²×D₄×C₃₁	—	2	8	no	—
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	—	2	2	no	—
54	C₂²×D₄×C₃₁	—	2	2	no	—
55	C₂²×D₄×C₃₁	—	2	1	no	—
56	C₂²×D₄×C₃₁	—	4	4	no	—
57	C₂²×D₄×C₃₁	—	4	8	no	—
58	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
59	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
60	C₂³×C₄×C₃₁	—	2	1	no	—
61	C₂³×C₄×C₃₁	—	2	1	no	—
62	C₂³×C₄×C₃₁	—	2	1	no	—
63	C₂³×C₄×C₃₁	—	4	4	no	—
64	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
66	C₂³×C₄×C₃₁	D₄	1	4	yes	—
67	C₂³×C₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
72	C₂×SD₁₆×C₃₁	—	4	8	no	—
73	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	complemented
74	C₂×M₁₆×C₃₁	—	4	4	no	—
75	C₂×M₁₆×C₃₁	C₂³	1	8	yes	complemented
76	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
77	C₃₁×C₂²⋊Q₈	—	2	4	no	—
78	C₃₁×C₂²⋊Q₈	—	2	8	no	—
79	C₃₁×C₂²⋊Q₈	—	2	8	no	—
80	C₃₁×C₂²⋊Q₈	—	2	4	no	—
81	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
82	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	complemented
83	C₃₁×#121	—	2	4	no	—
84	C₃₁×#121	—	2	4	no	—
85	C₃₁×#121	—	2	8	no	—
86	C₃₁×#121	—	2	8	no	—
87	C₃₁×#121	C₂³	1	8	yes	—
88	C₃₁×#121	C₂³	1	8	yes	—
89	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
91	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
92	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
93	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
95	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
99	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	complemented
100	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
102	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
104	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
106	C₃₁×#107	C₂³	1	16	yes	complemented
107	C₃₁×#103	C₂³	1	16	yes	complemented
108	C₃₁×#102	C₂³	1	16	yes	complemented
109	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
110	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	complemented
111	C₂⁴×C₃₁	—	2	4	no	—
112	C₂⁴×C₃₁	—	2	4	no	—
113	C₂⁴×C₃₁	—	2	4	no	—
114	C₂⁴×C₃₁	—	2	2	no	—
115	C₂⁴×C₃₁	—	4	2	no	—
116	C₂⁴×C₃₁	—	4	4	no	—
117	C₂⁴×C₃₁	—	4	4	no	—
118	C₂⁴×C₃₁	—	4	4	no	—
119	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂⁴×C₃₁	C₂×D₄	1	4	yes	—
121	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
122	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Q₈×C₃₁	—	2	2	no	—
124	C₂×Q₈×C₃₁	—	2	4	no	—
125	C₂×Q₈×C₃₁	—	2	8	no	—
126	C₂×Q₈×C₃₁	—	4	8	no	—
127	C₂×Q₈×C₃₁	—	4	4	no	—
128	C₂×Q₈×C₃₁	—	4	8	no	—
129	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
130	C₂×D₄×C₃₁	—	2	8	no	—
131	C₂×D₄×C₃₁	—	2	8	no	—
132	C₂×D₄×C₃₁	—	2	4	no	—
133	C₂×D₄×C₃₁	—	2	2	no	—
134	C₂×D₄×C₃₁	—	2	8	no	—
135	C₂×D₄×C₃₁	—	4	8	no	—
136	C₂×D₄×C₃₁	—	4	4	no	—
137	C₂×D₄×C₃₁	—	4	8	no	—
138	C₂×D₄×C₃₁	—	4	32	no	—
139	C₂×D₄×C₃₁	—	4	8	no	—
140	C₂×D₄×C₃₁	—	4	8	no	—
141	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
142	C₂²×C₄×C₃₁	—	2	4	no	—
143	C₂²×C₄×C₃₁	—	2	4	no	—
144	C₂²×C₄×C₃₁	—	2	4	no	—
145	C₂²×C₄×C₃₁	—	2	4	no	—
146	C₂²×C₄×C₃₁	—	2	4	no	—
147	C₂²×C₄×C₃₁	—	2	4	no	—
148	C₂²×C₄×C₃₁	—	2	2	no	—
149	C₂²×C₄×C₃₁	—	2	2	no	—
150	C₂²×C₄×C₃₁	—	2	2	no	—
151	C₂²×C₄×C₃₁	—	4	8	no	—
152	C₂²×C₄×C₃₁	—	4	8	no	—
153	C₂²×C₄×C₃₁	—	4	2	no	—
154	C₂²×C₄×C₃₁	—	4	2	no	—
155	C₂²×C₄×C₃₁	—	4	2	no	—
156	C₂²×C₄×C₃₁	—	4	4	no	—
157	C₂²×C₄×C₃₁	—	4	4	no	—
158	C₂²×C₄×C₃₁	—	4	4	no	—
159	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
160	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
161	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
162	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	SD₁₆×C₃₁	—	4	16	no	—
173	M₁₆×C₃₁	—	4	8	no	—
174	C₂×C₈×C₃₁	—	2	8	no	—
175	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
176	C₄⋊C₄×C₃₁	—	2	8	no	—
177	C₄⋊C₄×C₃₁	—	2	8	no	—
178	C₄⋊C₄×C₃₁	—	2	4	no	—
179	C₄⋊C₄×C₃₁	—	2	2	no	—
180	C₄⋊C₄×C₃₁	—	2	2	no	—
181	C₄⋊C₄×C₃₁	—	2	4	no	—
182	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
183	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
184	C₂²⋊C₄×C₃₁	—	2	8	no	—
185	C₂²⋊C₄×C₃₁	—	2	8	no	—
186	C₂²⋊C₄×C₃₁	—	2	8	no	—
187	C₂²⋊C₄×C₃₁	—	2	8	no	—
188	C₂×#3193	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
189	C₂³×C₃₁	—	2	2	no	—
190	C₂³×C₃₁	—	2	2	no	—
191	C₂³×C₃₁	—	2	2	no	—
192	C₂³×C₃₁	—	2	2	no	—
193	C₂³×C₃₁	—	2	2	no	—
194	C₂³×C₃₁	—	2	8	no	—
195	C₂³×C₃₁	—	2	2	no	—
196	C₂³×C₃₁	—	2	8	no	—
197	C₂³×C₃₁	—	2	2	no	—
198	C₂³×C₃₁	—	2	4	no	—
199	C₂³×C₃₁	—	4	4	no	—
200	C₂³×C₃₁	—	4	16	no	—
201	C₂³×C₃₁	—	4	8	no	—
202	C₂³×C₃₁	—	4	8	no	—
203	C₂³×C₃₁	—	4	4	no	—
204	C₂³×C₃₁	—	4	4	no	—
205	C₂³×C₃₁	—	4	8	no	—
206	C₂³×C₃₁	—	4	4	no	—
207	C₂³×C₃₁	—	8	16	no	—
208	C₂³×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
209	C₂³×C₃₁	#137	1	4	yes	—
210	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
211	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
212	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
213	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	Q₈×C₃₁	—	2	8	no	—
215	Q₈×C₃₁	—	4	8	no	—
216	Q₈×C₃₁	—	4	4	no	—
217	D₄×C₃₁	—	2	8	no	—
218	D₄×C₃₁	—	4	8	no	—
219	D₄×C₃₁	—	4	16	no	—
220	D₄×C₃₁	—	4	16	no	—
221	D₄×C₃₁	—	4	4	no	—
222	C₂×C₄×C₃₁	—	2	2	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	4	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	4	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	2	4	no	—
230	C₂×C₄×C₃₁	—	2	4	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	4	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	4	4	no	—
235	C₂×C₄×C₃₁	—	4	4	no	—
236	C₂×C₄×C₃₁	—	4	4	no	—
237	C₂×C₄×C₃₁	—	4	8	no	—
238	C₂×C₄×C₃₁	—	4	8	no	—
239	C₂×C₄×C₃₁	—	4	4	no	—
240	C₂×C₄×C₃₁	—	4	8	no	—
241	C₂×C₄×C₃₁	—	4	4	no	—
242	C₂×C₄×C₃₁	—	4	4	no	—
243	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
247	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
248	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
249	C₈×C₃₁	—	2	8	no	—
250	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
251	C₂²×#121	C₂×C₃₁	1	1	yes	characteristic
252	#3193	C₂×C₃₁	1	16	yes	direct product factor
253	C₂×#484	C₂×C₃₁	1	2	yes	complemented
254	C₂×#480	C₂×C₃₁	1	2	yes	complemented
255	C₂×#468	C₂×C₃₁	1	4	yes	complemented
256	C₂×#448	C₂×C₃₁	1	2	yes	complemented
257	C₂×#450	C₂×C₃₁	1	2	yes	complemented
258	C₂×#407	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂²×C₃₁	—	2	2	no	—
260	C₂²×C₃₁	—	2	2	no	—
261	C₂²×C₃₁	—	2	8	no	—
262	C₂²×C₃₁	—	2	8	no	—
263	C₂²×C₃₁	—	2	2	no	—
264	C₂²×C₃₁	—	2	2	no	—
265	C₂²×C₃₁	—	2	2	no	—
266	C₂²×C₃₁	—	2	2	no	—
267	C₂²×C₃₁	—	2	2	no	—
268	C₂²×C₃₁	—	4	16	no	—
269	C₂²×C₃₁	—	4	4	no	—
270	C₂²×C₃₁	—	4	4	no	—
271	C₂²×C₃₁	—	4	4	no	—
272	C₂²×C₃₁	—	4	4	no	—
273	C₂²×C₃₁	—	8	16	no	—
274	C₂²×C₃₁	—	8	8	no	—
275	C₂²×C₃₁	#534	1	2	yes	—
276	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
277	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
278	C₂²×C₃₁	C₂×#137	1	4	yes	—
279	C₂²×C₃₁	C₂×#137	1	2	yes	—
280	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
281	C₄×C₃₁	—	2	2	no	—
282	C₄×C₃₁	—	2	2	no	—
283	C₄×C₃₁	—	2	2	no	—
284	C₄×C₃₁	—	4	4	no	—
285	C₄×C₃₁	—	4	4	no	—
286	C₄×C₃₁	—	4	4	no	—
287	C₄×C₃₁	#534	1	2	yes	—
288	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
289	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
290	C₂³×D₄	—	2	1	no	—
291	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
292	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
293	C₂²×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
294	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	complemented
295	C₂×C₂²⋊Q₈	—	2	1	no	—
296	C₂×C₂²⋊Q₈	—	2	1	no	—
297	C₂×C₂²⋊Q₈	—	2	4	no	—
298	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
299	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
300	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	8	yes	complemented
301	C₂×#121	—	2	1	no	—
302	C₂×#121	—	2	1	no	—
303	C₂×#121	—	2	4	no	—
304	C₂×#121	C₂²×C₃₁	1	2	yes	—
305	C₂×#121	C₂²×C₃₁	1	8	yes	—
306	C₂×#121	C₂²×C₃₁	1	2	yes	—
307	C₂²×C₄⋊C₄	—	2	1	no	—
308	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
309	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
310	C₂²×C₂²⋊C₄	—	2	1	no	—
311	C₂²×C₂²⋊C₄	—	2	1	no	—
312	#484	C₂²×C₃₁	1	16	yes	complemented
313	#480	C₂²×C₃₁	1	16	yes	complemented
314	#468	C₂²×C₃₁	1	32	yes	complemented
315	#450	C₂²×C₃₁	1	16	yes	complemented
316	#448	C₂²×C₃₁	1	16	yes	complemented
317	C₂×#107	C₂²×C₃₁	1	4	yes	complemented
318	C₂×#103	C₂²×C₃₁	1	4	yes	complemented
319	C₂×#102	C₂²×C₃₁	1	4	yes	complemented
320	#407	C₂²×C₃₁	1	8	yes	complemented
321	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
322	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	complemented
323	C₂×C₃₁	—	2	4	no	—
324	C₂×C₃₁	—	2	4	no	—
325	C₂×C₃₁	—	4	2	no	—
326	C₂×C₃₁	—	8	4	no	—
327	C₂×C₃₁	#3193	1	4	yes	direct product factor
328	C₂×C₃₁	C₂×#534	1	1	yes	characteristic
329	C₂×C₃₁	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
330	C₂×C₃₁	C₂²×#137	1	1	yes	characteristic
331	C₂⁵	—	2	1	no	—
332	C₂⁵	D₄×C₃₁	1	1	yes	characteristic
333	C₂²×Q₈	—	2	1	no	—
334	C₂²×Q₈	—	2	2	no	—
335	C₂²×Q₈	—	4	4	no	—
336	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
337	C₂²×Q₈	C₂³×C₃₁	1	4	yes	complemented
338	C₂²×D₄	—	2	2	no	—
339	C₂²×D₄	—	2	8	no	—
340	C₂²×D₄	—	2	2	no	—
341	C₂²×D₄	—	2	1	no	—
342	C₂²×D₄	—	2	2	no	—
343	C₂²×D₄	—	4	4	no	—
344	C₂²×D₄	—	4	8	no	—
345	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
346	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	—	2	1	no	—
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	—	4	4	no	—
351	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
352	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
353	C₂³×C₄	D₄×C₃₁	1	4	yes	—
354	C₂³×C₄	D₄×C₃₁	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
356	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
358	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
359	C₂×SD₁₆	—	4	8	no	—
360	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	complemented
361	C₂×M₁₆	—	4	4	no	—
362	C₂×M₁₆	C₂³×C₃₁	1	8	yes	complemented
363	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
364	C₂²⋊Q₈	—	2	4	no	—
365	C₂²⋊Q₈	—	2	8	no	—
366	C₂²⋊Q₈	—	2	8	no	—
367	C₂²⋊Q₈	—	2	4	no	—
368	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
369	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	complemented
370	#121	—	2	4	no	—
371	#121	—	2	4	no	—
372	#121	—	2	8	no	—
373	#121	—	2	8	no	—
374	#121	C₂³×C₃₁	1	8	yes	—
375	#121	C₂³×C₃₁	1	8	yes	—
376	C₂×C₄⋊C₄	—	2	2	no	—
377	C₂×C₄⋊C₄	—	2	8	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	1	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	—	2	2	no	—
382	C₂×C₄⋊C₄	—	2	1	no	—
383	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
385	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	complemented
387	C₂×C₂²⋊C₄	—	2	2	no	—
388	C₂×C₂²⋊C₄	—	2	8	no	—
389	C₂×C₂²⋊C₄	—	2	2	no	—
390	C₂×C₂²⋊C₄	—	2	2	no	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	8	no	—
393	#107	C₂³×C₃₁	1	16	yes	complemented
394	#103	C₂³×C₃₁	1	16	yes	complemented
395	#102	C₂³×C₃₁	1	16	yes	complemented
396	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
397	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	complemented
398	C₃₁	C₂×#3193	1	1	yes	characteristic, direct product factor, Sylow
399	C₂⁴	—	2	4	no	—
400	C₂⁴	—	2	4	no	—
401	C₂⁴	—	2	4	no	—
402	C₂⁴	—	2	2	no	—
403	C₂⁴	—	4	2	no	—
404	C₂⁴	—	4	4	no	—
405	C₂⁴	—	4	4	no	—
406	C₂⁴	—	4	4	no	—
407	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
408	C₂⁴	C₂×D₄×C₃₁	1	4	yes	—
409	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
410	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	—	2	4	no	—
413	C₂×Q₈	—	2	8	no	—
414	C₂×Q₈	—	4	8	no	—
415	C₂×Q₈	—	4	4	no	—
416	C₂×Q₈	—	4	8	no	—
417	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
418	C₂×D₄	—	2	8	no	—
419	C₂×D₄	—	2	8	no	—
420	C₂×D₄	—	2	4	no	—
421	C₂×D₄	—	2	2	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	4	32	no	—
427	C₂×D₄	—	4	8	no	—
428	C₂×D₄	—	4	8	no	—
429	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	2	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	4	8	no	—
440	C₂²×C₄	—	4	8	no	—
441	C₂²×C₄	—	4	2	no	—
442	C₂²×C₄	—	4	2	no	—
443	C₂²×C₄	—	4	2	no	—
444	C₂²×C₄	—	4	4	no	—
445	C₂²×C₄	—	4	4	no	—
446	C₂²×C₄	—	4	4	no	—
447	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
450	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
451	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
453	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
454	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
455	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
460	SD₁₆	—	4	16	no	—
461	M₁₆	—	4	8	no	—
462	C₂×C₈	—	2	8	no	—
463	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	2	no	—
468	C₄⋊C₄	—	2	2	no	—
469	C₄⋊C₄	—	2	4	no	—
470	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
471	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
472	C₂²⋊C₄	—	2	8	no	—
473	C₂²⋊C₄	—	2	8	no	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂³	—	2	2	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	2	2	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	8	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	8	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	4	4	no	—
487	C₂³	—	4	16	no	—
488	C₂³	—	4	8	no	—
489	C₂³	—	4	8	no	—
490	C₂³	—	4	4	no	—
491	C₂³	—	4	4	no	—
492	C₂³	—	4	8	no	—
493	C₂³	—	4	4	no	—
494	C₂³	—	8	16	no	—
495	C₂³	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
496	C₂³	C₃₁×#137	1	4	yes	—
497	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
498	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
499	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
500	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
501	Q₈	—	2	8	no	—
502	Q₈	—	4	8	no	—
503	Q₈	—	4	4	no	—
504	D₄	—	2	8	no	—
505	D₄	—	4	8	no	—
506	D₄	—	4	16	no	—
507	D₄	—	4	16	no	—
508	D₄	—	4	4	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	4	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	4	4	no	—
522	C₂×C₄	—	4	4	no	—
523	C₂×C₄	—	4	4	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	4	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	—	4	4	no	—
529	C₂×C₄	—	4	4	no	—
530	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
532	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
534	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
535	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
536	C₈	—	2	8	no	—
537	C₂²	—	2	2	no	—
538	C₂²	—	2	2	no	—
539	C₂²	—	2	8	no	—
540	C₂²	—	2	8	no	—
541	C₂²	—	2	2	no	—
542	C₂²	—	2	2	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	4	16	no	—
547	C₂²	—	4	4	no	—
548	C₂²	—	4	4	no	—
549	C₂²	—	4	4	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	8	16	no	—
552	C₂²	—	8	8	no	—
553	C₂²	C₃₁×#534	1	2	yes	—
554	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
555	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
556	C₂²	C₂×C₃₁×#137	1	4	yes	—
557	C₂²	C₂×C₃₁×#137	1	2	yes	—
558	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
559	C₄	—	2	2	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	2	no	—
562	C₄	—	4	4	no	—
563	C₄	—	4	4	no	—
564	C₄	—	4	4	no	—
565	C₄	C₃₁×#534	1	2	yes	—
566	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
567	C₂	—	2	4	no	—
568	C₂	—	2	4	no	—
569	C₂	—	4	2	no	—
570	C₂	—	8	4	no	—
571	C₂	C₃₁×#3193	1	4	yes	direct product factor
572	C₂	C₂×C₃₁×#534	1	1	yes	characteristic
573	C₂	C₂²×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
574	C₂	C₂²×C₃₁×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×C₃₁×#121	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
1. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
3. C₃₁×#3193	C₂	direct (N × H)
4. C₂×C₃₁×#484	C₂	semidirect
5. C₂×C₃₁×#480	C₂	semidirect
6. C₂×C₃₁×#468	C₂	semidirect
7. C₂×C₃₁×#448	C₂	semidirect
8. C₂×C₃₁×#450	C₂	semidirect
9. C₂×C₃₁×#407	C₂	semidirect
10. C₂³×Q₈×C₃₁	C₂²	semidirect
14. C₂²×SD₁₆×C₃₁	C₂²	semidirect
15. C₂²×M₁₆×C₃₁	C₂²	semidirect
20. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
21. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
30. C₂²×C₄⋊C₄×C₃₁	C₂²	semidirect
33. C₃₁×#484	C₂²	semidirect
34. C₃₁×#480	C₂²	semidirect
35. C₃₁×#468	C₂²	semidirect
36. C₃₁×#450	C₂²	semidirect
37. C₃₁×#448	C₂²	semidirect
38. C₂×C₃₁×#107	C₂²	semidirect
39. C₂×C₃₁×#103	C₂²	semidirect
40. C₂×C₃₁×#102	C₂²	semidirect
41. C₃₁×#407	C₂²	semidirect
43. C₂×C₃₁×C₂²⋊C₈	C₂²	semidirect
50. C₂²×Q₈×C₃₁	C₂³	semidirect
73. C₂×SD₁₆×C₃₁	C₂³	semidirect
75. C₂×M₁₆×C₃₁	C₂³	semidirect
82. C₃₁×C₂²⋊Q₈	C₂³	semidirect
99. C₂×C₄⋊C₄×C₃₁	C₂³	semidirect
106. C₃₁×#107	C₂³	semidirect
107. C₃₁×#103	C₂³	semidirect
108. C₃₁×#102	C₂³	semidirect
110. C₃₁×C₂²⋊C₈	C₂³	semidirect
188. C₂×#3193	C₃₁	direct (N × H)
250. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
252. #3193	C₂×C₃₁	direct (N × H)
253. C₂×#484	C₂×C₃₁	semidirect
254. C₂×#480	C₂×C₃₁	semidirect
255. C₂×#468	C₂×C₃₁	semidirect
256. C₂×#448	C₂×C₃₁	semidirect
257. C₂×#450	C₂×C₃₁	semidirect
258. C₂×#407	C₂×C₃₁	semidirect
289. C₂³×Q₈	C₂²×C₃₁	semidirect
293. C₂²×SD₁₆	C₂²×C₃₁	semidirect
294. C₂²×M₁₆	C₂²×C₃₁	semidirect
299. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
300. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
309. C₂²×C₄⋊C₄	C₂²×C₃₁	semidirect
312. #484	C₂²×C₃₁	semidirect
313. #480	C₂²×C₃₁	semidirect
314. #468	C₂²×C₃₁	semidirect
315. #450	C₂²×C₃₁	semidirect
316. #448	C₂²×C₃₁	semidirect
317. C₂×#107	C₂²×C₃₁	semidirect
318. C₂×#103	C₂²×C₃₁	semidirect
319. C₂×#102	C₂²×C₃₁	semidirect
320. #407	C₂²×C₃₁	semidirect
322. C₂×C₂²⋊C₈	C₂²×C₃₁	semidirect
327. C₂×C₃₁	#3193	direct (N × H)
337. C₂²×Q₈	C₂³×C₃₁	semidirect
360. C₂×SD₁₆	C₂³×C₃₁	semidirect
362. C₂×M₁₆	C₂³×C₃₁	semidirect
369. C₂²⋊Q₈	C₂³×C₃₁	semidirect
386. C₂×C₄⋊C₄	C₂³×C₃₁	semidirect
393. #107	C₂³×C₃₁	semidirect
394. #103	C₂³×C₃₁	semidirect
395. #102	C₂³×C₃₁	semidirect
397. C₂²⋊C₈	C₂³×C₃₁	semidirect
398. C₃₁	C₂×#3193	direct (N × H)
571. C₂	C₃₁×#3193	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₃₁×#121	2
C₃₁×#3193	2
C₂×C₃₁×#484	2
C₂×C₃₁×#480	2
C₂×C₃₁×#468	2
C₂×C₃₁×#448	2
C₂×C₃₁×#450	2
C₂×C₃₁×#407	2
C₂×#3193	31

Subgroups — C2×C31×#3193 #21517495

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3193 #21517495