Subgroups — C₂×C₃₁×#3099 #21517274

Order 7936.

2234 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#537	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#520	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3099	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₃₁×#481	C₂	1	2	yes	maximal, complemented
5	C₂×C₃₁×#485	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#482	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	C₂²×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×#417	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#416	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#408	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
14	C₂²×#41×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×#537	—	2	4	no	—
18	C₃₁×#537	C₂²	1	8	yes	—
19	C₂×C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₂×C₃₁×#121	—	2	1	no	—
22	C₂×C₃₁×#121	—	2	2	no	—
23	C₂×C₃₁×#121	C₂²	1	2	yes	—
24	C₃₁×#520	—	2	4	no	—
25	C₃₁×#520	C₂²	1	8	yes	complemented
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	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
34	C₃₁×#485	C₂²	1	16	yes	complemented
35	C₃₁×#483	C₂²	1	16	yes	complemented
36	C₃₁×#482	C₂²	1	16	yes	complemented
37	C₃₁×#481	C₂²	1	16	yes	complemented
38	C₃₁×#428	C₂²	1	8	yes	complemented
39	C₂×C₃₁×#107	C₂²	1	2	yes	—
40	C₂×C₃₁×#106	C₂²	1	2	yes	—
41	C₂×C₃₁×#106	C₂²	1	2	yes	—
42	C₂×C₃₁×#106	C₂²	1	2	yes	—
43	C₂×C₃₁×#106	C₂²	1	8	yes	complemented
44	C₃₁×#417	C₂²	1	8	yes	complemented
45	C₃₁×#416	C₂²	1	8	yes	complemented
46	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
47	C₂×C₃₁×#103	C₂²	1	2	yes	—
48	C₂×C₃₁×#103	C₂²	1	1	yes	characteristic
49	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
50	C₂×C₃₁×#102	C₂²	1	2	yes	—
51	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
52	C₃₁×#408	C₂²	1	8	yes	complemented
53	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
54	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
55	C₂×#41×C₃₁	—	2	2	no	—
56	C₂×#41×C₃₁	—	2	4	no	—
57	C₂×#41×C₃₁	—	2	4	no	—
58	C₂×#41×C₃₁	—	2	4	no	—
59	C₂×#41×C₃₁	—	2	4	no	—
60	C₂×#41×C₃₁	D₄	1	4	yes	—
61	C₂×#41×C₃₁	C₂³	1	4	yes	—
62	C₂×#41×C₃₁	C₂³	1	4	yes	—
63	C₂²×Q₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₃₁	—	2	1	no	—
66	C₂²×D₄×C₃₁	—	2	2	no	—
67	C₂²×D₄×C₃₁	D₄	1	2	yes	—
68	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₃₁	—	2	2	no	—
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	C₂³	1	2	yes	—
75	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
76	C₂×M₁₆×C₃₁	—	2	4	no	—
77	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
78	C₂²×C₈×C₃₁	—	2	4	no	—
79	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
82	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
83	C₃₁×C₂²⋊Q₈	—	2	8	no	—
84	C₃₁×C₂²⋊Q₈	—	2	4	no	—
85	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
86	C₃₁×#121	—	2	8	no	—
87	C₃₁×#121	—	2	16	no	—
88	C₃₁×#121	—	2	4	no	—
89	C₃₁×#121	C₂³	1	8	yes	—
90	C₄×Q₈×C₃₁	—	2	8	no	—
91	C₄×D₄×C₃₁	—	2	16	no	—
92	C₄×D₄×C₃₁	—	2	8	no	—
93	C₃₁×#117	—	2	4	no	—
94	C₃₁×#117	—	2	2	no	—
95	C₃₁×#117	C₂³	1	4	yes	—
96	C₃₁×#117	C₂³	1	8	yes	—
97	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
100	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
101	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
102	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
103	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
104	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
105	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
112	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
113	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
115	C₂×C₄²×C₃₁	—	2	2	no	—
116	C₂×C₄²×C₃₁	—	2	1	no	—
117	C₃₁×#107	C₂³	1	8	yes	—
118	C₃₁×#106	C₂³	1	8	yes	—
119	C₃₁×#106	C₂³	1	8	yes	—
120	C₃₁×#106	C₂³	1	8	yes	—
121	C₃₁×#103	C₂³	1	4	yes	—
122	C₃₁×#103	C₂³	1	4	yes	—
123	C₃₁×#103	C₂³	1	8	yes	—
124	C₃₁×#102	C₂³	1	4	yes	—
125	C₃₁×#102	C₂³	1	4	yes	—
126	C₃₁×#102	C₂³	1	8	yes	—
127	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
128	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
129	C₂⁴×C₃₁	—	2	1	no	—
130	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
131	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
132	#41×C₃₁	—	2	16	no	—
133	#41×C₃₁	—	2	8	no	—
134	#41×C₃₁	—	2	8	no	—
135	#41×C₃₁	—	2	8	no	—
136	#41×C₃₁	—	2	8	no	—
137	#41×C₃₁	—	4	8	no	—
138	C₂×Q₈×C₃₁	—	2	4	no	—
139	C₂×Q₈×C₃₁	—	2	1	no	—
140	C₂×Q₈×C₃₁	—	2	2	no	—
141	C₂×Q₈×C₃₁	—	2	2	no	—
142	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
143	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
144	C₂×D₄×C₃₁	—	2	8	no	—
145	C₂×D₄×C₃₁	—	2	4	no	—
146	C₂×D₄×C₃₁	—	2	8	no	—
147	C₂×D₄×C₃₁	—	2	4	no	—
148	C₂×D₄×C₃₁	—	2	1	no	—
149	C₂×D₄×C₃₁	—	2	2	no	—
150	C₂×D₄×C₃₁	—	2	2	no	—
151	C₂×D₄×C₃₁	—	4	4	no	—
152	C₂×D₄×C₃₁	—	8	4	no	—
153	C₂×D₄×C₃₁	C₂×D₄	1	8	yes	—
154	C₂×D₄×C₃₁	C₂×D₄	1	4	yes	—
155	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
156	C₂²×C₄×C₃₁	—	2	4	no	—
157	C₂²×C₄×C₃₁	—	2	4	no	—
158	C₂²×C₄×C₃₁	—	2	1	no	—
159	C₂²×C₄×C₃₁	—	2	2	no	—
160	C₂²×C₄×C₃₁	—	2	1	no	—
161	C₂²×C₄×C₃₁	—	2	2	no	—
162	C₂²×C₄×C₃₁	—	2	1	no	—
163	C₂²×C₄×C₃₁	—	2	8	no	—
164	C₂²×C₄×C₃₁	—	2	4	no	—
165	C₂²×C₄×C₃₁	—	2	4	no	—
166	C₂²×C₄×C₃₁	—	2	4	no	—
167	C₂²×C₄×C₃₁	—	4	4	no	—
168	C₂²×C₄×C₃₁	—	8	2	no	—
169	C₂²×C₄×C₃₁	—	8	4	no	—
170	C₂²×C₄×C₃₁	—	8	2	no	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
172	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
175	C₂²×C₄×C₃₁	#41	1	4	yes	—
176	C₂²×C₄×C₃₁	#41	1	4	yes	—
177	C₂²×C₄×C₃₁	C₂⁴	1	4	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
179	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
180	M₁₆×C₃₁	—	2	8	no	—
181	C₂×C₈×C₃₁	—	2	2	no	—
182	C₂×C₈×C₃₁	—	2	2	no	—
183	C₂×C₈×C₃₁	—	2	8	no	—
184	C₂×C₈×C₃₁	—	2	4	no	—
185	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
186	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
187	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
188	C₄⋊C₄×C₃₁	—	2	1	no	—
189	C₄⋊C₄×C₃₁	—	2	2	no	—
190	C₄⋊C₄×C₃₁	—	2	8	no	—
191	C₄⋊C₄×C₃₁	—	2	4	no	—
192	C₄⋊C₄×C₃₁	—	2	4	no	—
193	C₄⋊C₄×C₃₁	—	2	2	no	—
194	C₄⋊C₄×C₃₁	—	2	8	no	—
195	C₄⋊C₄×C₃₁	—	2	1	no	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
197	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
198	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
199	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	—	2	4	no	—
202	C₂²⋊C₄×C₃₁	—	2	8	no	—
203	C₂²⋊C₄×C₃₁	—	2	8	no	—
204	C₄²×C₃₁	—	2	8	no	—
205	C₄²×C₃₁	—	2	4	no	—
206	C₂×#3099	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	2	2	no	—
209	C₂³×C₃₁	—	2	4	no	—
210	C₂³×C₃₁	—	2	4	no	—
211	C₂³×C₃₁	—	2	4	no	—
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	—	4	2	no	—
214	C₂³×C₃₁	—	4	2	no	—
215	C₂³×C₃₁	—	4	2	no	—
216	C₂³×C₃₁	#123	1	4	yes	—
217	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
218	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
219	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
220	Q₈×C₃₁	—	2	8	no	—
221	Q₈×C₃₁	—	2	2	no	—
222	Q₈×C₃₁	—	2	2	no	—
223	Q₈×C₃₁	—	4	2	no	—
224	D₄×C₃₁	—	2	16	no	—
225	D₄×C₃₁	—	2	8	no	—
226	D₄×C₃₁	—	2	2	no	—
227	D₄×C₃₁	—	2	2	no	—
228	D₄×C₃₁	—	4	8	no	—
229	D₄×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	8	8	no	—
231	C₂×C₄×C₃₁	—	2	4	no	—
232	C₂×C₄×C₃₁	—	2	8	no	—
233	C₂×C₄×C₃₁	—	2	4	no	—
234	C₂×C₄×C₃₁	—	2	4	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	2	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	4	2	no	—
241	C₂×C₄×C₃₁	—	4	2	no	—
242	C₂×C₄×C₃₁	—	4	4	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	4	no	—
245	C₂×C₄×C₃₁	—	4	8	no	—
246	C₂×C₄×C₃₁	—	8	4	no	—
247	C₂×C₄×C₃₁	—	8	8	no	—
248	C₂×C₄×C₃₁	—	8	4	no	—
249	C₂×C₄×C₃₁	#123	1	4	yes	—
250	C₂×C₄×C₃₁	#123	1	8	yes	—
251	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
252	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
253	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
254	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
255	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
256	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
257	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
258	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
259	C₈×C₃₁	—	2	2	no	—
260	C₈×C₃₁	—	2	2	no	—
261	C₈×C₃₁	—	2	4	no	—
262	C₂×#537	C₂×C₃₁	1	1	yes	characteristic
263	C₂×#520	C₂×C₃₁	1	1	yes	characteristic, complemented
264	#3099	C₂×C₃₁	1	16	yes	direct product factor
265	C₂×#481	C₂×C₃₁	1	2	yes	complemented
266	C₂×#485	C₂×C₃₁	1	2	yes	complemented
267	C₂×#482	C₂×C₃₁	1	2	yes	complemented
268	C₂×#483	C₂×C₃₁	1	2	yes	complemented
269	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
270	C₂²×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
271	C₂×#417	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂×#416	C₂×C₃₁	1	1	yes	characteristic, complemented
273	C₂×#408	C₂×C₃₁	1	1	yes	characteristic, complemented
274	C₂²×C₃₁	—	2	8	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	2	no	—
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	2	4	no	—
279	C₂²×C₃₁	—	2	4	no	—
280	C₂²×C₃₁	—	4	8	no	—
281	C₂²×C₃₁	—	4	2	no	—
282	C₂²×C₃₁	—	4	2	no	—
283	C₂²×C₃₁	—	4	2	no	—
284	C₂²×C₃₁	—	8	4	no	—
285	C₂²×C₃₁	C₂×#123	1	4	yes	—
286	C₂²×C₃₁	C₂×#123	1	2	yes	—
287	C₂²×C₃₁	#577	1	2	yes	—
288	C₂²×C₃₁	#577	1	2	yes	—
289	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
290	C₄×C₃₁	—	2	4	no	—
291	C₄×C₃₁	—	4	2	no	—
292	C₄×C₃₁	—	4	2	no	—
293	C₄×C₃₁	—	4	4	no	—
294	C₄×C₃₁	—	4	2	no	—
295	C₄×C₃₁	—	4	4	no	—
296	C₄×C₃₁	C₂×#123	1	2	yes	—
297	C₄×C₃₁	C₂×#123	1	4	yes	—
298	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
299	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
300	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
301	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
302	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
303	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
304	#537	—	2	4	no	—
305	#537	C₂²×C₃₁	1	8	yes	—
306	C₂×C₂²⋊Q₈	—	2	1	no	—
307	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
308	C₂×#121	—	2	1	no	—
309	C₂×#121	—	2	2	no	—
310	C₂×#121	C₂²×C₃₁	1	2	yes	—
311	#520	—	2	4	no	—
312	#520	C₂²×C₃₁	1	8	yes	complemented
313	C₂×C₄×Q₈	—	2	1	no	—
314	C₂×C₄×D₄	—	2	2	no	—
315	C₂×C₄×D₄	—	2	1	no	—
316	C₂×#117	C₂²×C₃₁	1	2	yes	—
317	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
318	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
319	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
320	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
321	#485	C₂²×C₃₁	1	16	yes	complemented
322	#483	C₂²×C₃₁	1	16	yes	complemented
323	#482	C₂²×C₃₁	1	16	yes	complemented
324	#481	C₂²×C₃₁	1	16	yes	complemented
325	#428	C₂²×C₃₁	1	8	yes	complemented
326	C₂×#107	C₂²×C₃₁	1	2	yes	—
327	C₂×#106	C₂²×C₃₁	1	2	yes	—
328	C₂×#106	C₂²×C₃₁	1	2	yes	—
329	C₂×#106	C₂²×C₃₁	1	2	yes	—
330	C₂×#106	C₂²×C₃₁	1	8	yes	complemented
331	#417	C₂²×C₃₁	1	8	yes	complemented
332	#416	C₂²×C₃₁	1	8	yes	complemented
333	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
334	C₂×#103	C₂²×C₃₁	1	2	yes	—
335	C₂×#103	C₂²×C₃₁	1	1	yes	characteristic
336	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×#102	C₂²×C₃₁	1	2	yes	—
338	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
339	#408	C₂²×C₃₁	1	8	yes	complemented
340	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
341	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
342	C₂×C₃₁	—	2	4	no	—
343	C₂×C₃₁	—	4	4	no	—
344	C₂×C₃₁	—	8	2	no	—
345	C₂×C₃₁	#3099	1	4	yes	direct product factor
346	C₂×C₃₁	C₂²×#123	1	1	yes	characteristic
347	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
348	C₂×C₃₁	C₂×#577	1	1	yes	characteristic
349	C₂×#41	—	2	2	no	—
350	C₂×#41	—	2	4	no	—
351	C₂×#41	—	2	4	no	—
352	C₂×#41	—	2	4	no	—
353	C₂×#41	—	2	4	no	—
354	C₂×#41	D₄×C₃₁	1	4	yes	—
355	C₂×#41	C₂³×C₃₁	1	4	yes	—
356	C₂×#41	C₂³×C₃₁	1	4	yes	—
357	C₂²×Q₈	D₄×C₃₁	1	1	yes	characteristic
358	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
359	C₂²×D₄	—	2	1	no	—
360	C₂²×D₄	—	2	2	no	—
361	C₂²×D₄	D₄×C₃₁	1	2	yes	—
362	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
363	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
364	C₂³×C₄	—	2	2	no	—
365	C₂³×C₄	—	2	1	no	—
366	C₂³×C₄	—	2	1	no	—
367	C₂³×C₄	—	2	1	no	—
368	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
369	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
370	C₂×M₁₆	—	2	4	no	—
371	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
372	C₂²×C₈	—	2	4	no	—
373	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
374	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
375	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
376	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
377	C₂²⋊Q₈	—	2	8	no	—
378	C₂²⋊Q₈	—	2	4	no	—
379	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
380	#121	—	2	8	no	—
381	#121	—	2	16	no	—
382	#121	—	2	4	no	—
383	#121	C₂³×C₃₁	1	8	yes	—
384	C₄×Q₈	—	2	8	no	—
385	C₄×D₄	—	2	16	no	—
386	C₄×D₄	—	2	8	no	—
387	#117	—	2	4	no	—
388	#117	—	2	2	no	—
389	#117	C₂³×C₃₁	1	4	yes	—
390	#117	C₂³×C₃₁	1	8	yes	—
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	—	2	1	no	—
393	C₂×C₄⋊C₄	—	2	4	no	—
394	C₂×C₄⋊C₄	—	2	2	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	2	no	—
397	C₂×C₄⋊C₄	—	2	2	no	—
398	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
399	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
400	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
401	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
402	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
403	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
404	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
405	C₂×C₂²⋊C₄	—	2	2	no	—
406	C₂×C₂²⋊C₄	—	2	1	no	—
407	C₂×C₂²⋊C₄	—	2	2	no	—
408	C₂×C₂²⋊C₄	—	2	2	no	—
409	C₂×C₄²	—	2	2	no	—
410	C₂×C₄²	—	2	1	no	—
411	#107	C₂³×C₃₁	1	8	yes	—
412	#106	C₂³×C₃₁	1	8	yes	—
413	#106	C₂³×C₃₁	1	8	yes	—
414	#106	C₂³×C₃₁	1	8	yes	—
415	#103	C₂³×C₃₁	1	4	yes	—
416	#103	C₂³×C₃₁	1	4	yes	—
417	#103	C₂³×C₃₁	1	8	yes	—
418	#102	C₂³×C₃₁	1	4	yes	—
419	#102	C₂³×C₃₁	1	4	yes	—
420	#102	C₂³×C₃₁	1	8	yes	—
421	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
422	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
423	C₃₁	C₂×#3099	1	1	yes	characteristic, direct product factor, Sylow
424	C₂⁴	—	2	1	no	—
425	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
426	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
427	#41	—	2	16	no	—
428	#41	—	2	8	no	—
429	#41	—	2	8	no	—
430	#41	—	2	8	no	—
431	#41	—	2	8	no	—
432	#41	—	4	8	no	—
433	C₂×Q₈	—	2	4	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
438	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
439	C₂×D₄	—	2	8	no	—
440	C₂×D₄	—	2	4	no	—
441	C₂×D₄	—	2	8	no	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	1	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	4	4	no	—
447	C₂×D₄	—	8	4	no	—
448	C₂×D₄	C₂×D₄×C₃₁	1	8	yes	—
449	C₂×D₄	C₂×D₄×C₃₁	1	4	yes	—
450	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
451	C₂²×C₄	—	2	4	no	—
452	C₂²×C₄	—	2	4	no	—
453	C₂²×C₄	—	2	1	no	—
454	C₂²×C₄	—	2	2	no	—
455	C₂²×C₄	—	2	1	no	—
456	C₂²×C₄	—	2	2	no	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	8	no	—
459	C₂²×C₄	—	2	4	no	—
460	C₂²×C₄	—	2	4	no	—
461	C₂²×C₄	—	2	4	no	—
462	C₂²×C₄	—	4	4	no	—
463	C₂²×C₄	—	8	2	no	—
464	C₂²×C₄	—	8	4	no	—
465	C₂²×C₄	—	8	2	no	—
466	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
468	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
470	C₂²×C₄	#41×C₃₁	1	4	yes	—
471	C₂²×C₄	#41×C₃₁	1	4	yes	—
472	C₂²×C₄	C₂⁴×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
475	M₁₆	—	2	8	no	—
476	C₂×C₈	—	2	2	no	—
477	C₂×C₈	—	2	2	no	—
478	C₂×C₈	—	2	8	no	—
479	C₂×C₈	—	2	4	no	—
480	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
481	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
482	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
483	C₄⋊C₄	—	2	1	no	—
484	C₄⋊C₄	—	2	2	no	—
485	C₄⋊C₄	—	2	8	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	4	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	8	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
492	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
493	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
494	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
495	C₂²⋊C₄	—	2	8	no	—
496	C₂²⋊C₄	—	2	4	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	8	no	—
499	C₄²	—	2	8	no	—
500	C₄²	—	2	4	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	2	no	—
503	C₂³	—	2	4	no	—
504	C₂³	—	2	4	no	—
505	C₂³	—	2	4	no	—
506	C₂³	—	2	2	no	—
507	C₂³	—	4	2	no	—
508	C₂³	—	4	2	no	—
509	C₂³	—	4	2	no	—
510	C₂³	C₃₁×#123	1	4	yes	—
511	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
512	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
513	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
514	Q₈	—	2	8	no	—
515	Q₈	—	2	2	no	—
516	Q₈	—	2	2	no	—
517	Q₈	—	4	2	no	—
518	D₄	—	2	16	no	—
519	D₄	—	2	8	no	—
520	D₄	—	2	2	no	—
521	D₄	—	2	2	no	—
522	D₄	—	4	8	no	—
523	D₄	—	4	2	no	—
524	D₄	—	8	8	no	—
525	C₂×C₄	—	2	4	no	—
526	C₂×C₄	—	2	8	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	4	no	—
529	C₂×C₄	—	2	2	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	4	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	4	2	no	—
535	C₂×C₄	—	4	2	no	—
536	C₂×C₄	—	4	4	no	—
537	C₂×C₄	—	4	2	no	—
538	C₂×C₄	—	4	4	no	—
539	C₂×C₄	—	4	8	no	—
540	C₂×C₄	—	8	4	no	—
541	C₂×C₄	—	8	8	no	—
542	C₂×C₄	—	8	4	no	—
543	C₂×C₄	C₃₁×#123	1	4	yes	—
544	C₂×C₄	C₃₁×#123	1	8	yes	—
545	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
546	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
547	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
548	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
549	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
550	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
551	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
552	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
553	C₈	—	2	2	no	—
554	C₈	—	2	2	no	—
555	C₈	—	2	4	no	—
556	C₂²	—	2	8	no	—
557	C₂²	—	2	2	no	—
558	C₂²	—	2	2	no	—
559	C₂²	—	2	4	no	—
560	C₂²	—	2	4	no	—
561	C₂²	—	2	4	no	—
562	C₂²	—	4	8	no	—
563	C₂²	—	4	2	no	—
564	C₂²	—	4	2	no	—
565	C₂²	—	4	2	no	—
566	C₂²	—	8	4	no	—
567	C₂²	C₂×C₃₁×#123	1	4	yes	—
568	C₂²	C₂×C₃₁×#123	1	2	yes	—
569	C₂²	C₃₁×#577	1	2	yes	—
570	C₂²	C₃₁×#577	1	2	yes	—
571	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
572	C₄	—	2	4	no	—
573	C₄	—	4	2	no	—
574	C₄	—	4	4	no	—
575	C₄	—	4	2	no	—
576	C₄	—	4	4	no	—
577	C₄	—	4	2	no	—
578	C₄	C₂×C₃₁×#123	1	2	yes	—
579	C₄	C₂×C₃₁×#123	1	4	yes	—
580	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
581	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
582	C₂	—	2	4	no	—
583	C₂	—	4	4	no	—
584	C₂	—	8	2	no	—
585	C₂	C₃₁×#3099	1	4	yes	direct product factor
586	C₂	C₂²×C₃₁×#123	1	1	yes	characteristic
587	C₂	C₂×C₃₁×#577	1	1	yes	characteristic
588	C₂	C₂×C₃₁×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#537	C₂
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
2. C₂×C₃₁×#520	C₂	semidirect
3. C₃₁×#3099	C₂	direct (N × H)
4. C₂×C₃₁×#481	C₂	semidirect
5. C₂×C₃₁×#485	C₂	semidirect
6. C₂×C₃₁×#482	C₂	semidirect
7. C₂×C₃₁×#483	C₂	semidirect
8. C₂×C₃₁×#428	C₂	semidirect
9. C₂²×C₃₁×#106	C₂	semidirect
10. C₂×C₃₁×#417	C₂	semidirect
11. C₂×C₃₁×#416	C₂	semidirect
12. C₂×C₃₁×#408	C₂	semidirect
25. C₃₁×#520	C₂²	semidirect
34. C₃₁×#485	C₂²	semidirect
35. C₃₁×#483	C₂²	semidirect
36. C₃₁×#482	C₂²	semidirect
37. C₃₁×#481	C₂²	semidirect
38. C₃₁×#428	C₂²	semidirect
43. C₂×C₃₁×#106	C₂²	semidirect
44. C₃₁×#417	C₂²	semidirect
45. C₃₁×#416	C₂²	semidirect
52. C₃₁×#408	C₂²	semidirect
206. C₂×#3099	C₃₁	direct (N × H)
263. C₂×#520	C₂×C₃₁	semidirect
264. #3099	C₂×C₃₁	direct (N × H)
265. C₂×#481	C₂×C₃₁	semidirect
266. C₂×#485	C₂×C₃₁	semidirect
267. C₂×#482	C₂×C₃₁	semidirect
268. C₂×#483	C₂×C₃₁	semidirect
269. C₂×#428	C₂×C₃₁	semidirect
270. C₂²×#106	C₂×C₃₁	semidirect
271. C₂×#417	C₂×C₃₁	semidirect
272. C₂×#416	C₂×C₃₁	semidirect
273. C₂×#408	C₂×C₃₁	semidirect
312. #520	C₂²×C₃₁	semidirect
321. #485	C₂²×C₃₁	semidirect
322. #483	C₂²×C₃₁	semidirect
323. #482	C₂²×C₃₁	semidirect
324. #481	C₂²×C₃₁	semidirect
325. #428	C₂²×C₃₁	semidirect
330. C₂×#106	C₂²×C₃₁	semidirect
331. #417	C₂²×C₃₁	semidirect
332. #416	C₂²×C₃₁	semidirect
339. #408	C₂²×C₃₁	semidirect
345. C₂×C₃₁	#3099	direct (N × H)
423. C₃₁	C₂×#3099	direct (N × H)
585. C₂	C₃₁×#3099	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×#537	2
C₂×C₃₁×#520	2
C₃₁×#3099	2
C₂×C₃₁×#481	2
C₂×C₃₁×#485	2
C₂×C₃₁×#482	2
C₂×C₃₁×#483	2
C₂×C₃₁×#428	2
C₂²×C₃₁×#106	2
C₂×C₃₁×#417	2
C₂×C₃₁×#416	2
C₂×C₃₁×#408	2
C₂×#3099	31

Subgroups — C2×C31×#3099 #21517274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#3099 #21517274