Subgroups — C₃₁×#22096 #21478855

Order 7936.

726 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3091	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2929	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#424	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#1946	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1938	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1922	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1871	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#1866	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1855	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1826	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#131	C₂²	1	1	yes	characteristic
13	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#525	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
17	C₃₁×#485	C₄	1	2	yes	complemented
18	C₃₁×#484	C₄	1	1	yes	characteristic, complemented
19	C₃₁×#483	C₄	1	1	yes	characteristic, complemented
20	C₃₁×#482	C₄	1	2	yes	complemented
21	C₃₁×#481	C₄	1	1	yes	characteristic, complemented
22	C₃₁×#480	C₄	1	1	yes	characteristic, complemented
23	C₃₁×#435	—	2	1	no	—
24	C₃₁×#435	—	2	1	no	—
25	C₈×D₄×C₃₁	—	2	1	no	—
26	C₈×D₄×C₃₁	—	2	1	no	—
27	C₂×C₃₁×#108	C₂²	1	2	yes	—
28	C₃₁×#427	C₂²	1	2	yes	—
29	C₃₁×#424	—	2	1	no	—
30	C₃₁×#424	—	2	1	no	—
31	C₃₁×#424	C₂²	1	2	yes	—
32	C₃₁×#424	C₂²	1	2	yes	—
33	C₃₁×#424	C₂²	1	1	yes	characteristic
34	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
35	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
36	C₂×C₃₁×#104	C₂²	1	2	yes	—
37	C₃₁×#416	C₂²	1	2	yes	—
38	C₃₁×#408	C₂²	1	1	yes	characteristic
39	C₃₁×#407	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×#405	C₂²	1	1	yes	characteristic
42	C₂×C₃₁×#97	C₂²	1	2	yes	—
43	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
44	C₃₁×#341	C₂²	1	2	yes	—
45	C₃₁×#339	C₂²	1	2	yes	—
46	C₃₁×#339	C₂²	1	2	yes	—
47	C₃₁×#339	C₂²	1	2	yes	—
48	C₃₁×#336	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#131	—	4	2	no	—
52	C₂×M₁₆×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂×M₁₆×C₃₁	—	4	2	no	—
55	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×M₁₆×C₃₁	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	4	2	no	—
61	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₃₁	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×#123	—	2	1	no	—
69	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
70	C₃₁×C₂²⋊Q₈	C₂×C₄	1	1	yes	characteristic
71	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
72	C₃₁×#121	C₂×C₄	1	1	yes	characteristic
73	C₃₁×#120	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₃₁×#117	—	2	1	no	—
77	C₃₁×#117	—	2	1	no	—
78	C₃₁×#117	C₂×C₄	1	2	yes	—
79	C₃₁×#117	C₂³	1	1	yes	characteristic
80	C₃₁×#117	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
83	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
85	C₂×C₄²×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
87	C₃₁×#108	—	4	2	no	—
88	C₃₁×#107	C₂×C₄	1	2	yes	—
89	C₃₁×#107	C₂×C₄	1	2	yes	—
90	C₃₁×#106	C₂×C₄	1	2	yes	—
91	C₃₁×#106	C₂×C₄	1	2	yes	—
92	C₃₁×C₄⋊C₈	—	2	2	no	—
93	C₃₁×C₄⋊C₈	—	2	2	no	—
94	C₃₁×C₄⋊C₈	—	2	2	no	—
95	C₃₁×C₄⋊C₈	—	2	2	no	—
96	C₃₁×C₄⋊C₈	—	2	1	no	—
97	C₃₁×C₄⋊C₈	—	2	1	no	—
98	C₃₁×#104	—	4	2	no	—
99	C₃₁×#103	C₂×C₄	1	2	yes	—
100	C₃₁×#103	C₂×C₄	1	2	yes	—
101	C₃₁×#102	C₂×C₄	1	2	yes	—
102	C₃₁×#102	C₂×C₄	1	2	yes	—
103	C₃₁×C₂²⋊C₈	—	2	1	no	—
104	C₃₁×C₂²⋊C₈	—	2	1	no	—
105	C₃₁×C₂²⋊C₈	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	1	no	—
107	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
108	C₃₁×C₂²⋊C₈	C₂×C₄	1	1	yes	characteristic
109	C₃₁×C₂²⋊C₈	C₂×C₄	1	2	yes	—
110	C₃₁×#97	—	2	1	no	—
111	C₃₁×#97	—	4	2	no	—
112	C₄×C₈×C₃₁	—	2	1	no	—
113	C₃₁×#95	—	2	1	no	—
114	C₂⁴×C₃₁	#41	1	1	yes	characteristic
115	#41×C₃₁	—	2	2	no	—
116	#41×C₃₁	—	2	2	no	—
117	C₂×Q₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
118	C₂×D₄×C₃₁	—	2	1	no	—
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	2	no	—
124	C₂²×C₄×C₃₁	—	2	1	no	—
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	2	no	—
128	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
129	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
133	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
134	M₁₆×C₃₁	—	2	2	no	—
135	M₁₆×C₃₁	—	2	2	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	—	2	2	no	—
138	M₁₆×C₃₁	—	4	2	no	—
139	M₁₆×C₃₁	—	4	2	no	—
140	C₂×C₈×C₃₁	—	2	2	no	—
141	C₂×C₈×C₃₁	—	2	2	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	2	2	no	—
144	C₂×C₈×C₃₁	—	2	1	no	—
145	C₂×C₈×C₃₁	—	2	1	no	—
146	C₂×C₈×C₃₁	—	2	2	no	—
147	C₂×C₈×C₃₁	—	4	1	no	—
148	C₂×C₈×C₃₁	—	4	1	no	—
149	C₂×C₈×C₃₁	—	4	1	no	—
150	C₂×C₈×C₃₁	—	4	1	no	—
151	C₂×C₈×C₃₁	—	4	2	no	—
152	C₂×C₈×C₃₁	—	4	2	no	—
153	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
154	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
155	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
159	C₄⋊C₄×C₃₁	—	2	1	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	2	1	no	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
164	C₄⋊C₄×C₃₁	C₂²×C₄	1	2	yes	—
165	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
166	C₄⋊C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
167	C₂²⋊C₄×C₃₁	—	2	2	no	—
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	2	1	no	—
170	C₂²⋊C₄×C₃₁	—	2	2	no	—
171	C₂²⋊C₄×C₃₁	—	2	1	no	—
172	C₂²⋊C₄×C₃₁	—	2	1	no	—
173	C₂²⋊C₄×C₃₁	—	2	1	no	—
174	C₂²⋊C₄×C₃₁	—	4	2	no	—
175	C₄²×C₃₁	—	2	2	no	—
176	C₄²×C₃₁	—	2	1	no	—
177	C₄²×C₃₁	—	2	2	no	—
178	C₄²×C₃₁	—	2	2	no	—
179	C₄²×C₃₁	—	2	2	no	—
180	#22096	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
181	C₂³×C₃₁	—	2	2	no	—
182	C₂³×C₃₁	—	2	2	no	—
183	C₂³×C₃₁	—	2	1	no	—
184	C₂³×C₃₁	—	4	1	no	—
185	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
186	C₂³×C₃₁	C₄×D₄	1	1	yes	characteristic
187	C₂³×C₃₁	C₂×#41	1	1	yes	characteristic
188	Q₈×C₃₁	—	2	2	no	—
189	D₄×C₃₁	—	2	2	no	—
190	D₄×C₃₁	—	4	2	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	2	no	—
193	C₂×C₄×C₃₁	—	2	2	no	—
194	C₂×C₄×C₃₁	—	2	2	no	—
195	C₂×C₄×C₃₁	—	2	2	no	—
196	C₂×C₄×C₃₁	—	2	1	no	—
197	C₂×C₄×C₃₁	—	2	1	no	—
198	C₂×C₄×C₃₁	—	2	1	no	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	2	no	—
201	C₂×C₄×C₃₁	—	2	2	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	2	1	no	—
204	C₂×C₄×C₃₁	—	2	1	no	—
205	C₂×C₄×C₃₁	—	4	2	no	—
206	C₂×C₄×C₃₁	—	4	2	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	2	no	—
212	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₄×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
216	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
217	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
218	C₈×C₃₁	—	2	2	no	—
219	C₈×C₃₁	—	2	2	no	—
220	C₈×C₃₁	—	2	1	no	—
221	C₈×C₃₁	—	2	1	no	—
222	C₈×C₃₁	—	2	1	no	—
223	C₈×C₃₁	—	2	1	no	—
224	C₈×C₃₁	—	4	1	no	—
225	C₈×C₃₁	—	4	2	no	—
226	C₈×C₃₁	—	4	1	no	—
227	#3091	C₂×C₃₁	1	1	yes	characteristic
228	#2929	C₂×C₃₁	1	1	yes	characteristic
229	C₂×#424	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#1946	C₂×C₃₁	1	2	yes	complemented
231	#1938	C₂×C₃₁	1	2	yes	complemented
232	#1922	C₂×C₃₁	1	2	yes	complemented
233	#1871	C₂×C₃₁	1	1	yes	characteristic
234	#1866	C₂×C₃₁	1	2	yes	complemented
235	#1855	C₂×C₃₁	1	1	yes	characteristic, complemented
236	#1826	C₂×C₃₁	1	1	yes	characteristic, complemented
237	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
238	C₂²×C₃₁	—	2	1	no	—
239	C₂²×C₃₁	—	2	1	no	—
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	2	1	no	—
242	C₂²×C₃₁	—	2	1	no	—
243	C₂²×C₃₁	—	2	1	no	—
244	C₂²×C₃₁	—	2	1	no	—
245	C₂²×C₃₁	—	4	4	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	—	4	1	no	—
248	C₂²×C₃₁	—	4	1	no	—
249	C₂²×C₃₁	#444	1	2	yes	—
250	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
251	C₂²×C₃₁	#539	1	2	yes	—
252	C₄×C₃₁	—	2	1	no	—
253	C₄×C₃₁	—	2	1	no	—
254	C₄×C₃₁	—	4	1	no	—
255	C₄×C₃₁	—	4	2	no	—
256	C₄×C₃₁	—	4	1	no	—
257	C₄×C₃₁	—	4	2	no	—
258	C₄×C₃₁	—	4	1	no	—
259	C₄×C₃₁	—	8	2	no	—
260	C₄×C₃₁	#520	1	2	yes	—
261	C₄×C₃₁	#534	1	1	yes	characteristic
262	C₄×C₃₁	#536	1	1	yes	characteristic
263	C₂×#131	C₂²×C₃₁	1	1	yes	characteristic
264	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
265	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
266	#525	C₂²×C₃₁	1	1	yes	characteristic
267	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
268	#485	C₄×C₃₁	1	2	yes	complemented
269	#484	C₄×C₃₁	1	1	yes	characteristic, complemented
270	#483	C₄×C₃₁	1	1	yes	characteristic, complemented
271	#482	C₄×C₃₁	1	2	yes	complemented
272	#481	C₄×C₃₁	1	1	yes	characteristic, complemented
273	#480	C₄×C₃₁	1	1	yes	characteristic, complemented
274	#435	—	2	1	no	—
275	#435	—	2	1	no	—
276	C₈×D₄	—	2	1	no	—
277	C₈×D₄	—	2	1	no	—
278	C₂×#108	C₂²×C₃₁	1	2	yes	—
279	#427	C₂²×C₃₁	1	2	yes	—
280	#424	—	2	1	no	—
281	#424	—	2	1	no	—
282	#424	C₂²×C₃₁	1	2	yes	—
283	#424	C₂²×C₃₁	1	2	yes	—
284	#424	C₂²×C₃₁	1	1	yes	characteristic
285	C₂×C₄⋊C₈	—	2	1	no	—
286	C₂×C₄⋊C₈	—	2	1	no	—
287	C₂×#104	C₂²×C₃₁	1	2	yes	—
288	#416	C₂²×C₃₁	1	2	yes	—
289	#408	C₂²×C₃₁	1	1	yes	characteristic
290	#407	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
292	#405	C₂²×C₃₁	1	1	yes	characteristic
293	C₂×#97	C₂²×C₃₁	1	2	yes	—
294	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
295	#341	C₂²×C₃₁	1	2	yes	—
296	#339	C₂²×C₃₁	1	2	yes	—
297	#339	C₂²×C₃₁	1	2	yes	—
298	#339	C₂²×C₃₁	1	2	yes	—
299	#336	C₂²×C₃₁	1	2	yes	—
300	C₂×C₃₁	—	2	2	no	—
301	C₂×C₃₁	—	2	2	no	—
302	C₂×C₃₁	—	4	1	no	—
303	C₂×C₃₁	—	8	1	no	—
304	C₂×C₃₁	#2322	1	1	yes	characteristic
305	C₂×C₃₁	C₂×#444	1	1	yes	characteristic
306	C₂×C₃₁	#2955	1	1	yes	characteristic
307	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
308	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
309	#131	—	4	2	no	—
310	C₂×M₁₆	—	2	1	no	—
311	C₂×M₁₆	—	2	1	no	—
312	C₂×M₁₆	—	4	2	no	—
313	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
314	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
315	C₂×M₁₆	C₂³×C₃₁	1	1	yes	characteristic
316	C₂²×C₈	—	2	1	no	—
317	C₂²×C₈	—	2	1	no	—
318	C₂²×C₈	—	4	2	no	—
319	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₈	D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
324	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
326	#123	—	2	1	no	—
327	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
328	C₂²⋊Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
329	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
330	#121	C₂×C₄×C₃₁	1	1	yes	characteristic
331	#120	—	2	1	no	—
332	C₄×D₄	—	2	1	no	—
333	C₄×D₄	—	2	1	no	—
334	#117	—	2	1	no	—
335	#117	—	2	1	no	—
336	#117	C₂×C₄×C₃₁	1	2	yes	—
337	#117	C₂³×C₃₁	1	2	yes	—
338	#117	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
341	C₂×C₂²⋊C₄	—	2	1	no	—
342	C₂×C₂²⋊C₄	—	2	1	no	—
343	C₂×C₄²	—	2	1	no	—
344	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
345	#108	—	4	2	no	—
346	#107	C₂×C₄×C₃₁	1	2	yes	—
347	#107	C₂×C₄×C₃₁	1	2	yes	—
348	#106	C₂×C₄×C₃₁	1	2	yes	—
349	#106	C₂×C₄×C₃₁	1	2	yes	—
350	C₄⋊C₈	—	2	2	no	—
351	C₄⋊C₈	—	2	2	no	—
352	C₄⋊C₈	—	2	2	no	—
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	—	2	1	no	—
356	#104	—	4	2	no	—
357	#103	C₂×C₄×C₃₁	1	2	yes	—
358	#103	C₂×C₄×C₃₁	1	2	yes	—
359	#102	C₂×C₄×C₃₁	1	2	yes	—
360	#102	C₂×C₄×C₃₁	1	2	yes	—
361	C₂²⋊C₈	—	2	1	no	—
362	C₂²⋊C₈	—	2	1	no	—
363	C₂²⋊C₈	—	2	1	no	—
364	C₂²⋊C₈	—	2	1	no	—
365	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
366	C₂²⋊C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
367	C₂²⋊C₈	C₂×C₄×C₃₁	1	2	yes	—
368	#97	—	2	1	no	—
369	#97	—	4	2	no	—
370	C₄×C₈	—	2	1	no	—
371	#95	—	2	1	no	—
372	C₃₁	#22096	1	1	yes	characteristic, direct product factor, Sylow
373	C₂⁴	#41×C₃₁	1	1	yes	characteristic
374	#41	—	2	2	no	—
375	#41	—	2	2	no	—
376	C₂×Q₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
377	C₂×D₄	—	2	1	no	—
378	C₂×D₄	—	2	1	no	—
379	C₂×D₄	—	2	1	no	—
380	C₂×D₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
381	C₂²×C₄	—	2	2	no	—
382	C₂²×C₄	—	2	2	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	2	1	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
389	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
391	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
392	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
393	M₁₆	—	2	2	no	—
394	M₁₆	—	2	2	no	—
395	M₁₆	—	2	2	no	—
396	M₁₆	—	2	2	no	—
397	M₁₆	—	4	2	no	—
398	M₁₆	—	4	2	no	—
399	C₂×C₈	—	2	2	no	—
400	C₂×C₈	—	2	2	no	—
401	C₂×C₈	—	2	2	no	—
402	C₂×C₈	—	2	2	no	—
403	C₂×C₈	—	2	1	no	—
404	C₂×C₈	—	2	1	no	—
405	C₂×C₈	—	2	2	no	—
406	C₂×C₈	—	4	1	no	—
407	C₂×C₈	—	4	1	no	—
408	C₂×C₈	—	4	1	no	—
409	C₂×C₈	—	4	1	no	—
410	C₂×C₈	—	4	2	no	—
411	C₂×C₈	—	4	2	no	—
412	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
413	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
414	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
415	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
416	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
417	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	1	no	—
422	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
423	C₄⋊C₄	C₂²×C₄×C₃₁	1	2	yes	—
424	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
425	C₄⋊C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
426	C₂²⋊C₄	—	2	2	no	—
427	C₂²⋊C₄	—	2	1	no	—
428	C₂²⋊C₄	—	2	1	no	—
429	C₂²⋊C₄	—	2	2	no	—
430	C₂²⋊C₄	—	2	1	no	—
431	C₂²⋊C₄	—	2	1	no	—
432	C₂²⋊C₄	—	2	1	no	—
433	C₂²⋊C₄	—	4	2	no	—
434	C₄²	—	2	2	no	—
435	C₄²	—	2	1	no	—
436	C₄²	—	2	2	no	—
437	C₄²	—	2	2	no	—
438	C₄²	—	2	2	no	—
439	C₂³	—	2	2	no	—
440	C₂³	—	2	2	no	—
441	C₂³	—	2	1	no	—
442	C₂³	—	4	1	no	—
443	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
444	C₂³	C₄×D₄×C₃₁	1	1	yes	characteristic
445	C₂³	C₂×#41×C₃₁	1	1	yes	characteristic
446	Q₈	—	2	2	no	—
447	D₄	—	2	2	no	—
448	D₄	—	4	2	no	—
449	D₄	—	4	2	no	—
450	D₄	—	4	2	no	—
451	C₂×C₄	—	2	2	no	—
452	C₂×C₄	—	2	2	no	—
453	C₂×C₄	—	2	2	no	—
454	C₂×C₄	—	2	1	no	—
455	C₂×C₄	—	2	1	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	2	no	—
459	C₂×C₄	—	2	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	4	2	no	—
464	C₂×C₄	—	4	2	no	—
465	C₂×C₄	—	4	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	4	2	no	—
470	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
471	C₂×C₄	C₄×D₄×C₃₁	1	1	yes	characteristic
472	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
473	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
474	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
475	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
476	C₈	—	2	2	no	—
477	C₈	—	2	2	no	—
478	C₈	—	2	1	no	—
479	C₈	—	2	1	no	—
480	C₈	—	2	1	no	—
481	C₈	—	2	1	no	—
482	C₈	—	4	1	no	—
483	C₈	—	4	2	no	—
484	C₈	—	4	1	no	—
485	C₂²	—	2	1	no	—
486	C₂²	—	2	1	no	—
487	C₂²	—	2	1	no	—
488	C₂²	—	2	1	no	—
489	C₂²	—	2	1	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	4	4	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	C₃₁×#444	1	2	yes	—
497	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
498	C₂²	C₃₁×#539	1	2	yes	—
499	C₄	—	2	1	no	—
500	C₄	—	2	1	no	—
501	C₄	—	4	1	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	1	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	1	no	—
506	C₄	—	8	2	no	—
507	C₄	C₃₁×#520	1	2	yes	—
508	C₄	C₃₁×#534	1	1	yes	characteristic
509	C₄	C₃₁×#536	1	1	yes	characteristic
510	C₂	—	2	2	no	—
511	C₂	—	2	2	no	—
512	C₂	—	4	1	no	—
513	C₂	—	8	1	no	—
514	C₂	C₃₁×#2322	1	1	yes	characteristic
515	C₂	C₂×C₃₁×#444	1	1	yes	characteristic
516	C₂	C₃₁×#2955	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂×C₄×D₄
prime radical	C₃₁×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁ ◁ C₂³×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₃₁×#424	C₂	semidirect
4. C₃₁×#1946	C₂	semidirect
5. C₃₁×#1938	C₂	semidirect
6. C₃₁×#1922	C₂	semidirect
8. C₃₁×#1866	C₂	semidirect
9. C₃₁×#1855	C₂	semidirect
10. C₃₁×#1826	C₂	semidirect
11. C₃₁×#1752	C₂	semidirect
17. C₃₁×#485	C₄	semidirect
18. C₃₁×#484	C₄	semidirect
19. C₃₁×#483	C₄	semidirect
20. C₃₁×#482	C₄	semidirect
21. C₃₁×#481	C₄	semidirect
22. C₃₁×#480	C₄	semidirect
180. #22096	C₃₁	direct (N × H)
229. C₂×#424	C₂×C₃₁	semidirect
230. #1946	C₂×C₃₁	semidirect
231. #1938	C₂×C₃₁	semidirect
232. #1922	C₂×C₃₁	semidirect
234. #1866	C₂×C₃₁	semidirect
235. #1855	C₂×C₃₁	semidirect
236. #1826	C₂×C₃₁	semidirect
237. #1752	C₂×C₃₁	semidirect
268. #485	C₄×C₃₁	semidirect
269. #484	C₄×C₃₁	semidirect
270. #483	C₄×C₃₁	semidirect
271. #482	C₄×C₃₁	semidirect
272. #481	C₄×C₃₁	semidirect
273. #480	C₄×C₃₁	semidirect
372. C₃₁	#22096	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#3091	2
C₃₁×#2929	2
C₂×C₃₁×#424	2
C₃₁×#1946	2
C₃₁×#1938	2
C₃₁×#1922	2
C₃₁×#1871	2
C₃₁×#1866	2
C₃₁×#1855	2
C₃₁×#1826	2
C₃₁×#1752	2
#22096	31

Subgroups — C31×#22096 #21478855

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22096 #21478855