Subgroups — C₂×C₃₁×#3097 #21517272

Order 7936.

2170 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#536	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#520	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#3097	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#484	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#485	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#485	C₂	1	2	yes	maximal, complemented
7	C₂×C₃₁×#483	C₂	1	2	yes	maximal, complemented
8	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#419	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×C₃₁×#103	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#536	—	2	4	no	—
18	C₃₁×#536	C₂²	1	8	yes	—
19	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
20	C₂×C₃₁×#124	—	2	1	no	—
21	C₂×C₃₁×#123	—	2	2	no	—
22	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
23	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₃₁×#520	—	2	4	no	—
25	C₃₁×#520	C₂²	1	8	yes	—
26	C₂×C₄×Q₈×C₃₁	—	2	1	no	—
27	C₂×C₄×D₄×C₃₁	—	2	2	no	—
28	C₂×C₄×D₄×C₃₁	—	2	1	no	—
29	C₂×C₃₁×#117	C₂²	1	2	yes	—
30	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
31	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#485	C₂²	1	16	yes	complemented
35	C₃₁×#485	C₂²	1	16	yes	complemented
36	C₃₁×#484	C₂²	1	16	yes	complemented
37	C₃₁×#483	C₂²	1	16	yes	complemented
38	C₃₁×#428	C₂²	1	8	yes	complemented
39	C₃₁×#427	C₂²	1	8	yes	complemented
40	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
41	C₂×C₃₁×#107	C₂²	1	2	yes	—
42	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
43	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
44	C₂×C₃₁×#106	C₂²	1	2	yes	—
45	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
46	C₃₁×#419	C₂²	1	8	yes	complemented
47	C₂×C₃₁×#103	C₂²	1	2	yes	—
48	C₂×C₃₁×#103	C₂²	1	2	yes	—
49	C₂×C₃₁×#103	C₂²	1	2	yes	—
50	C₂×C₃₁×#103	C₂²	1	2	yes	—
51	C₂×C₃₁×#103	C₂²	1	8	yes	complemented
52	C₃₁×#408	C₂²	1	8	yes	—
53	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
55	C₂×#41×C₃₁	—	2	4	no	—
56	C₂×#41×C₃₁	—	2	4	no	—
57	C₂×#41×C₃₁	D₄	1	4	yes	—
58	C₂×#41×C₃₁	C₂³	1	4	yes	—
59	C₂²×Q₈×C₃₁	—	2	1	no	—
60	C₂²×Q₈×C₃₁	—	2	2	no	—
61	C₂²×Q₈×C₃₁	—	2	4	no	—
62	C₂²×Q₈×C₃₁	—	2	4	no	—
63	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
65	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
66	C₂²×D₄×C₃₁	D₄	1	2	yes	—
67	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂³×C₄×C₃₁	—	2	1	no	—
69	C₂³×C₄×C₃₁	—	2	2	no	—
70	C₂³×C₄×C₃₁	—	2	1	no	—
71	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
72	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×M₁₆×C₃₁	—	2	4	no	—
74	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
75	C₂²×C₈×C₃₁	—	2	4	no	—
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
78	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
80	C₃₁×C₄⋊Q₈	—	2	8	no	—
81	C₃₁×#124	—	2	8	no	—
82	C₃₁×#123	—	2	16	no	—
83	C₃₁×C₂²⋊Q₈	—	2	4	no	—
84	C₃₁×C₂²⋊Q₈	—	2	4	no	—
85	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
86	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
87	C₄×Q₈×C₃₁	—	2	8	no	—
88	C₄×D₄×C₃₁	—	2	16	no	—
89	C₄×D₄×C₃₁	—	2	8	no	—
90	C₃₁×#117	—	2	4	no	—
91	C₃₁×#117	—	2	2	no	—
92	C₃₁×#117	—	2	2	no	—
93	C₃₁×#117	C₂³	1	4	yes	—
94	C₃₁×#117	C₂³	1	4	yes	—
95	C₃₁×#117	C₂³	1	8	yes	—
96	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
102	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₄²×C₃₁	—	2	2	no	—
115	C₂×C₄²×C₃₁	—	2	1	no	—
116	C₂×C₄²×C₃₁	—	2	1	no	—
117	C₃₁×#107	C₂³	1	4	yes	—
118	C₃₁×#107	C₂³	1	4	yes	—
119	C₃₁×#107	C₂³	1	8	yes	—
120	C₃₁×#106	C₂³	1	4	yes	—
121	C₃₁×#106	C₂³	1	4	yes	—
122	C₃₁×#106	C₂³	1	8	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#103	C₂³	1	8	yes	—
125	C₃₁×#103	C₂³	1	8	yes	—
126	C₃₁×#103	C₂³	1	8	yes	—
127	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
128	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
129	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
130	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
131	#41×C₃₁	—	2	16	no	—
132	#41×C₃₁	—	2	8	no	—
133	#41×C₃₁	—	2	8	no	—
134	C₂×Q₈×C₃₁	—	2	4	no	—
135	C₂×Q₈×C₃₁	—	2	2	no	—
136	C₂×Q₈×C₃₁	—	2	4	no	—
137	C₂×Q₈×C₃₁	—	2	4	no	—
138	C₂×Q₈×C₃₁	—	2	4	no	—
139	C₂×Q₈×C₃₁	—	2	8	no	—
140	C₂×Q₈×C₃₁	—	2	8	no	—
141	C₂×Q₈×C₃₁	—	4	4	no	—
142	C₂×Q₈×C₃₁	—	4	8	no	—
143	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
144	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
145	C₂×D₄×C₃₁	—	2	8	no	—
146	C₂×D₄×C₃₁	—	2	4	no	—
147	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
148	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₃₁	—	2	4	no	—
150	C₂²×C₄×C₃₁	—	2	4	no	—
151	C₂²×C₄×C₃₁	—	2	4	no	—
152	C₂²×C₄×C₃₁	—	2	1	no	—
153	C₂²×C₄×C₃₁	—	2	1	no	—
154	C₂²×C₄×C₃₁	—	2	1	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	1	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	8	no	—
159	C₂²×C₄×C₃₁	—	2	4	no	—
160	C₂²×C₄×C₃₁	—	4	4	no	—
161	C₂²×C₄×C₃₁	—	8	2	no	—
162	C₂²×C₄×C₃₁	—	8	4	no	—
163	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
164	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	#41	1	4	yes	—
168	C₂²×C₄×C₃₁	#41	1	4	yes	—
169	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
172	M₁₆×C₃₁	—	2	8	no	—
173	C₂×C₈×C₃₁	—	2	2	no	—
174	C₂×C₈×C₃₁	—	2	2	no	—
175	C₂×C₈×C₃₁	—	2	8	no	—
176	C₂×C₈×C₃₁	—	2	4	no	—
177	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
178	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
179	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
180	C₄⋊C₄×C₃₁	—	2	1	no	—
181	C₄⋊C₄×C₃₁	—	2	2	no	—
182	C₄⋊C₄×C₃₁	—	2	8	no	—
183	C₄⋊C₄×C₃₁	—	2	4	no	—
184	C₄⋊C₄×C₃₁	—	2	8	no	—
185	C₄⋊C₄×C₃₁	—	2	8	no	—
186	C₄⋊C₄×C₃₁	—	2	1	no	—
187	C₄⋊C₄×C₃₁	—	2	1	no	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
190	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
191	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
192	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
193	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
194	C₂²⋊C₄×C₃₁	—	2	8	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	8	no	—
198	C₂²⋊C₄×C₃₁	—	2	8	no	—
199	C₄²×C₃₁	—	2	8	no	—
200	C₄²×C₃₁	—	2	4	no	—
201	C₄²×C₃₁	—	2	4	no	—
202	C₂×#3097	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
203	C₂³×C₃₁	—	2	2	no	—
204	C₂³×C₃₁	—	2	2	no	—
205	C₂³×C₃₁	—	2	4	no	—
206	C₂³×C₃₁	—	2	4	no	—
207	C₂³×C₃₁	—	2	4	no	—
208	C₂³×C₃₁	#123	1	4	yes	—
209	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
210	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
211	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₃₁	—	2	8	no	—
213	Q₈×C₃₁	—	2	4	no	—
214	Q₈×C₃₁	—	2	4	no	—
215	Q₈×C₃₁	—	4	4	no	—
216	Q₈×C₃₁	—	4	8	no	—
217	D₄×C₃₁	—	2	16	no	—
218	D₄×C₃₁	—	2	8	no	—
219	C₂×C₄×C₃₁	—	2	4	no	—
220	C₂×C₄×C₃₁	—	2	8	no	—
221	C₂×C₄×C₃₁	—	2	4	no	—
222	C₂×C₄×C₃₁	—	2	4	no	—
223	C₂×C₄×C₃₁	—	2	2	no	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	2	no	—
226	C₂×C₄×C₃₁	—	2	4	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	4	no	—
229	C₂×C₄×C₃₁	—	4	8	no	—
230	C₂×C₄×C₃₁	—	4	4	no	—
231	C₂×C₄×C₃₁	—	4	2	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	4	2	no	—
234	C₂×C₄×C₃₁	—	4	2	no	—
235	C₂×C₄×C₃₁	—	4	2	no	—
236	C₂×C₄×C₃₁	—	8	4	no	—
237	C₂×C₄×C₃₁	—	8	8	no	—
238	C₂×C₄×C₃₁	#123	1	4	yes	—
239	C₂×C₄×C₃₁	#123	1	8	yes	—
240	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
241	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
242	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
243	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
244	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
245	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
246	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
247	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
248	C₈×C₃₁	—	2	2	no	—
249	C₈×C₃₁	—	2	2	no	—
250	C₈×C₃₁	—	2	4	no	—
251	C₂×#536	C₂×C₃₁	1	1	yes	characteristic
252	C₂×#520	C₂×C₃₁	1	1	yes	characteristic
253	#3097	C₂×C₃₁	1	16	yes	direct product factor
254	C₂×#484	C₂×C₃₁	1	2	yes	complemented
255	C₂×#485	C₂×C₃₁	1	2	yes	complemented
256	C₂×#485	C₂×C₃₁	1	2	yes	complemented
257	C₂×#483	C₂×C₃₁	1	2	yes	complemented
258	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
259	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
260	C₂×#419	C₂×C₃₁	1	1	yes	characteristic, complemented
261	C₂²×#103	C₂×C₃₁	1	1	yes	characteristic, complemented
262	C₂×#408	C₂×C₃₁	1	1	yes	characteristic
263	C₂²×C₃₁	—	2	8	no	—
264	C₂²×C₃₁	—	2	2	no	—
265	C₂²×C₃₁	—	2	2	no	—
266	C₂²×C₃₁	—	2	4	no	—
267	C₂²×C₃₁	—	2	4	no	—
268	C₂²×C₃₁	—	2	4	no	—
269	C₂²×C₃₁	—	4	8	no	—
270	C₂²×C₃₁	C₂×#123	1	4	yes	—
271	C₂²×C₃₁	C₂×#123	1	2	yes	—
272	C₂²×C₃₁	#577	1	2	yes	—
273	C₂²×C₃₁	#578	1	2	yes	—
274	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	4	4	no	—
277	C₄×C₃₁	—	4	2	no	—
278	C₄×C₃₁	—	4	4	no	—
279	C₄×C₃₁	—	4	2	no	—
280	C₄×C₃₁	—	4	2	no	—
281	C₄×C₃₁	—	4	2	no	—
282	C₄×C₃₁	C₂×#123	1	2	yes	—
283	C₄×C₃₁	C₂×#123	1	4	yes	—
284	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
285	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
286	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
287	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
288	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
289	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
290	#536	—	2	4	no	—
291	#536	C₂²×C₃₁	1	8	yes	—
292	C₂×C₄⋊Q₈	—	2	1	no	—
293	C₂×#124	—	2	1	no	—
294	C₂×#123	—	2	2	no	—
295	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
296	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
297	#520	—	2	4	no	—
298	#520	C₂²×C₃₁	1	8	yes	—
299	C₂×C₄×Q₈	—	2	1	no	—
300	C₂×C₄×D₄	—	2	2	no	—
301	C₂×C₄×D₄	—	2	1	no	—
302	C₂×#117	C₂²×C₃₁	1	2	yes	—
303	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
304	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
305	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
306	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
307	#485	C₂²×C₃₁	1	16	yes	complemented
308	#485	C₂²×C₃₁	1	16	yes	complemented
309	#484	C₂²×C₃₁	1	16	yes	complemented
310	#483	C₂²×C₃₁	1	16	yes	complemented
311	#428	C₂²×C₃₁	1	8	yes	complemented
312	#427	C₂²×C₃₁	1	8	yes	complemented
313	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
314	C₂×#107	C₂²×C₃₁	1	2	yes	—
315	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
316	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
317	C₂×#106	C₂²×C₃₁	1	2	yes	—
318	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
319	#419	C₂²×C₃₁	1	8	yes	complemented
320	C₂×#103	C₂²×C₃₁	1	2	yes	—
321	C₂×#103	C₂²×C₃₁	1	2	yes	—
322	C₂×#103	C₂²×C₃₁	1	2	yes	—
323	C₂×#103	C₂²×C₃₁	1	2	yes	—
324	C₂×#103	C₂²×C₃₁	1	8	yes	complemented
325	#408	C₂²×C₃₁	1	8	yes	—
326	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
327	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
328	C₂×C₃₁	—	2	4	no	—
329	C₂×C₃₁	—	4	4	no	—
330	C₂×C₃₁	#3097	1	4	yes	direct product factor
331	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
332	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
333	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
334	C₂×#41	—	2	4	no	—
335	C₂×#41	—	2	4	no	—
336	C₂×#41	D₄×C₃₁	1	4	yes	—
337	C₂×#41	C₂³×C₃₁	1	4	yes	—
338	C₂²×Q₈	—	2	1	no	—
339	C₂²×Q₈	—	2	2	no	—
340	C₂²×Q₈	—	2	4	no	—
341	C₂²×Q₈	—	2	4	no	—
342	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
343	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
344	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
345	C₂²×D₄	D₄×C₃₁	1	2	yes	—
346	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
347	C₂³×C₄	—	2	1	no	—
348	C₂³×C₄	—	2	2	no	—
349	C₂³×C₄	—	2	1	no	—
350	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
351	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂×M₁₆	—	2	4	no	—
353	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
354	C₂²×C₈	—	2	4	no	—
355	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
356	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
358	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
359	C₄⋊Q₈	—	2	8	no	—
360	#124	—	2	8	no	—
361	#123	—	2	16	no	—
362	C₂²⋊Q₈	—	2	4	no	—
363	C₂²⋊Q₈	—	2	4	no	—
364	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
365	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
366	C₄×Q₈	—	2	8	no	—
367	C₄×D₄	—	2	16	no	—
368	C₄×D₄	—	2	8	no	—
369	#117	—	2	4	no	—
370	#117	—	2	2	no	—
371	#117	—	2	2	no	—
372	#117	C₂³×C₃₁	1	4	yes	—
373	#117	C₂³×C₃₁	1	8	yes	—
374	#117	C₂³×C₃₁	1	4	yes	—
375	C₂×C₄⋊C₄	—	2	2	no	—
376	C₂×C₄⋊C₄	—	2	1	no	—
377	C₂×C₄⋊C₄	—	2	4	no	—
378	C₂×C₄⋊C₄	—	2	2	no	—
379	C₂×C₄⋊C₄	—	2	2	no	—
380	C₂×C₄⋊C₄	—	2	2	no	—
381	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
384	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
385	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
386	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
388	C₂×C₂²⋊C₄	—	2	2	no	—
389	C₂×C₂²⋊C₄	—	2	1	no	—
390	C₂×C₂²⋊C₄	—	2	1	no	—
391	C₂×C₂²⋊C₄	—	2	2	no	—
392	C₂×C₂²⋊C₄	—	2	2	no	—
393	C₂×C₄²	—	2	2	no	—
394	C₂×C₄²	—	2	1	no	—
395	C₂×C₄²	—	2	1	no	—
396	#107	C₂³×C₃₁	1	4	yes	—
397	#107	C₂³×C₃₁	1	4	yes	—
398	#107	C₂³×C₃₁	1	8	yes	—
399	#106	C₂³×C₃₁	1	4	yes	—
400	#106	C₂³×C₃₁	1	4	yes	—
401	#106	C₂³×C₃₁	1	8	yes	—
402	#103	C₂³×C₃₁	1	8	yes	—
403	#103	C₂³×C₃₁	1	8	yes	—
404	#103	C₂³×C₃₁	1	8	yes	—
405	#103	C₂³×C₃₁	1	8	yes	—
406	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
407	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
408	C₃₁	C₂×#3097	1	1	yes	characteristic, direct product factor, Sylow
409	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
410	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
411	#41	—	2	16	no	—
412	#41	—	2	8	no	—
413	#41	—	2	8	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	—	2	2	no	—
416	C₂×Q₈	—	2	4	no	—
417	C₂×Q₈	—	2	4	no	—
418	C₂×Q₈	—	2	4	no	—
419	C₂×Q₈	—	2	8	no	—
420	C₂×Q₈	—	2	8	no	—
421	C₂×Q₈	—	4	4	no	—
422	C₂×Q₈	—	4	8	no	—
423	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
424	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
425	C₂×D₄	—	2	8	no	—
426	C₂×D₄	—	2	4	no	—
427	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
428	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
429	C₂²×C₄	—	2	4	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	1	no	—
432	C₂²×C₄	—	2	1	no	—
433	C₂²×C₄	—	2	1	no	—
434	C₂²×C₄	—	2	2	no	—
435	C₂²×C₄	—	2	1	no	—
436	C₂²×C₄	—	2	2	no	—
437	C₂²×C₄	—	2	4	no	—
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	2	4	no	—
440	C₂²×C₄	—	4	4	no	—
441	C₂²×C₄	—	8	2	no	—
442	C₂²×C₄	—	8	4	no	—
443	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
445	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
447	C₂²×C₄	#41×C₃₁	1	4	yes	—
448	C₂²×C₄	#41×C₃₁	1	4	yes	—
449	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
450	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
452	M₁₆	—	2	8	no	—
453	C₂×C₈	—	2	2	no	—
454	C₂×C₈	—	2	2	no	—
455	C₂×C₈	—	2	8	no	—
456	C₂×C₈	—	2	4	no	—
457	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
458	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
459	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
460	C₄⋊C₄	—	2	1	no	—
461	C₄⋊C₄	—	2	2	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	—	2	4	no	—
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	8	no	—
466	C₄⋊C₄	—	2	1	no	—
467	C₄⋊C₄	—	2	1	no	—
468	C₄⋊C₄	—	2	1	no	—
469	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
470	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
471	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
472	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
473	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
474	C₂²⋊C₄	—	2	8	no	—
475	C₂²⋊C₄	—	2	4	no	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	8	no	—
478	C₂²⋊C₄	—	2	8	no	—
479	C₄²	—	2	8	no	—
480	C₄²	—	2	4	no	—
481	C₄²	—	2	4	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	2	no	—
484	C₂³	—	2	4	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	2	4	no	—
487	C₂³	C₃₁×#123	1	4	yes	—
488	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
489	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
490	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
491	Q₈	—	2	8	no	—
492	Q₈	—	2	4	no	—
493	Q₈	—	2	4	no	—
494	Q₈	—	4	4	no	—
495	Q₈	—	4	8	no	—
496	D₄	—	2	16	no	—
497	D₄	—	2	8	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	2	4	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	2	no	—
505	C₂×C₄	—	2	4	no	—
506	C₂×C₄	—	2	2	no	—
507	C₂×C₄	—	2	4	no	—
508	C₂×C₄	—	4	8	no	—
509	C₂×C₄	—	4	4	no	—
510	C₂×C₄	—	4	2	no	—
511	C₂×C₄	—	4	4	no	—
512	C₂×C₄	—	4	2	no	—
513	C₂×C₄	—	4	2	no	—
514	C₂×C₄	—	4	2	no	—
515	C₂×C₄	—	8	4	no	—
516	C₂×C₄	—	8	8	no	—
517	C₂×C₄	C₃₁×#123	1	4	yes	—
518	C₂×C₄	C₃₁×#123	1	8	yes	—
519	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
520	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
521	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
522	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
523	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
524	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
525	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
526	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
527	C₈	—	2	2	no	—
528	C₈	—	2	2	no	—
529	C₈	—	2	4	no	—
530	C₂²	—	2	8	no	—
531	C₂²	—	2	2	no	—
532	C₂²	—	2	2	no	—
533	C₂²	—	2	4	no	—
534	C₂²	—	2	4	no	—
535	C₂²	—	2	4	no	—
536	C₂²	—	4	8	no	—
537	C₂²	C₂×C₃₁×#123	1	4	yes	—
538	C₂²	C₂×C₃₁×#123	1	2	yes	—
539	C₂²	C₃₁×#577	1	2	yes	—
540	C₂²	C₃₁×#578	1	2	yes	—
541	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
542	C₄	—	2	4	no	—
543	C₄	—	4	4	no	—
544	C₄	—	4	4	no	—
545	C₄	—	4	2	no	—
546	C₄	—	4	2	no	—
547	C₄	—	4	2	no	—
548	C₄	—	4	2	no	—
549	C₄	C₂×C₃₁×#123	1	2	yes	—
550	C₄	C₂×C₃₁×#123	1	4	yes	—
551	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
552	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
553	C₂	—	2	4	no	—
554	C₂	—	4	4	no	—
555	C₂	C₃₁×#3097	1	4	yes	direct product factor
556	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
557	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
558	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂²×#41×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×#41×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#3097	C₂	direct (N × H)
4. C₂×C₃₁×#484	C₂	semidirect
5. C₂×C₃₁×#485	C₂	semidirect
6. C₂×C₃₁×#485	C₂	semidirect
7. C₂×C₃₁×#483	C₂	semidirect
8. C₂×C₃₁×#428	C₂	semidirect
9. C₂×C₃₁×#427	C₂	semidirect
10. C₂×C₃₁×#419	C₂	semidirect
11. C₂²×C₃₁×#103	C₂	semidirect
34. C₃₁×#485	C₂²	semidirect
35. C₃₁×#485	C₂²	semidirect
36. C₃₁×#484	C₂²	semidirect
37. C₃₁×#483	C₂²	semidirect
38. C₃₁×#428	C₂²	semidirect
39. C₃₁×#427	C₂²	semidirect
46. C₃₁×#419	C₂²	semidirect
51. C₂×C₃₁×#103	C₂²	semidirect
202. C₂×#3097	C₃₁	direct (N × H)
253. #3097	C₂×C₃₁	direct (N × H)
254. C₂×#484	C₂×C₃₁	semidirect
255. C₂×#485	C₂×C₃₁	semidirect
256. C₂×#485	C₂×C₃₁	semidirect
257. C₂×#483	C₂×C₃₁	semidirect
258. C₂×#428	C₂×C₃₁	semidirect
259. C₂×#427	C₂×C₃₁	semidirect
260. C₂×#419	C₂×C₃₁	semidirect
261. C₂²×#103	C₂×C₃₁	semidirect
307. #485	C₂²×C₃₁	semidirect
308. #485	C₂²×C₃₁	semidirect
309. #484	C₂²×C₃₁	semidirect
310. #483	C₂²×C₃₁	semidirect
311. #428	C₂²×C₃₁	semidirect
312. #427	C₂²×C₃₁	semidirect
319. #419	C₂²×C₃₁	semidirect
324. C₂×#103	C₂²×C₃₁	semidirect
330. C₂×C₃₁	#3097	direct (N × H)
408. C₃₁	C₂×#3097	direct (N × H)
555. C₂	C₃₁×#3097	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×#536	2
C₂×C₃₁×#520	2
C₃₁×#3097	2
C₂×C₃₁×#484	2
C₂×C₃₁×#485	2
C₂×C₃₁×#483	2
C₂×C₃₁×#428	2
C₂×C₃₁×#427	2
C₂×C₃₁×#419	2
C₂²×C₃₁×#103	2
C₂×C₃₁×#408	2
C₂×#3097	31

Subgroups — C2×C31×#3097 #21517272

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3097 #21517272