Subgroups — C₃₁×#60718 #21517477

Order 7936.

1942 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3511	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3182	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3175	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3170	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3169	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3147	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#3141	C₂	1	2	yes	maximal, complemented
9	C₃₁×#3061	C₂	1	4	yes	maximal, complemented
10	C₃₁×#2951	C₂	1	4	yes	maximal, complemented
11	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	C₂²	1	4	yes	—
13	C₂×C₃₁×#131	C₂²	1	2	yes	—
14	C₃₁×#556	—	2	2	no	—
15	C₃₁×#555	—	2	2	no	—
16	C₃₁×#554	—	2	2	no	—
17	C₃₁×#553	—	2	4	no	—
18	C₃₁×#548	—	2	4	no	—
19	C₃₁×#547	—	2	2	no	—
20	C₃₁×#542	C₂²	1	8	yes	—
21	C₃₁×#537	C₂²	1	4	yes	—
22	C₃₁×#533	C₂²	1	4	yes	—
23	C₃₁×#532	C₂²	1	2	yes	—
24	C₃₁×#532	C₂²	1	4	yes	—
25	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
26	C₃₁×#520	C₂²	1	4	yes	—
27	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#500	C₂²	1	2	yes	—
29	C₃₁×#500	C₂²	1	4	yes	—
30	C₃₁×#500	C₂²	1	2	yes	—
31	C₃₁×#495	C₂²	1	4	yes	—
32	C₃₁×#494	C₂²	1	1	yes	characteristic
33	C₃₁×#493	C₂²	1	1	yes	characteristic
34	C₃₁×#492	C₂²	1	2	yes	—
35	C₃₁×#489	C₂²	1	8	yes	—
36	C₃₁×#487	C₂²	1	4	yes	—
37	C₃₁×#486	C₂²	1	4	yes	—
38	C₃₁×#468	C₂²	1	16	yes	—
39	C₃₁×#467	C₂²	1	8	yes	—
40	C₃₁×#466	C₂²	1	8	yes	—
41	C₃₁×#440	C₂²	1	16	yes	—
42	C₄×Q₁₆×C₃₁	C₂²	1	8	yes	—
43	C₄×D₈×C₃₁	C₂²	1	8	yes	—
44	C₂×C₃₁×#107	C₂²	1	4	yes	—
45	C₃₁×#417	C₂²	1	8	yes	—
46	C₃₁×#405	C₂²	1	4	yes	—
47	C₃₁×#405	C₂²	1	2	yes	—
48	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
49	C₄×M₁₆×C₃₁	C₂²	1	2	yes	—
50	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
51	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
52	C₂×#41×C₃₁	—	2	4	no	—
53	C₂×#41×C₃₁	C₂³	1	4	yes	—
54	C₂×#41×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×Q₈×C₃₁	—	2	2	no	—
56	C₃₁×#135	—	2	16	no	—
57	C₂×Q₁₆×C₃₁	C₂³	1	4	yes	—
58	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
59	C₂×D₈×C₃₁	C₂³	1	4	yes	—
60	C₃₁×#131	—	2	8	no	—
61	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
63	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
64	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
65	C₃₁×C₄⋊Q₈	—	2	2	no	—
66	C₃₁×C₄⋊Q₈	—	2	2	no	—
67	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
68	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
70	C₃₁×#127	C₂³	1	2	yes	—
71	C₃₁×#126	—	2	8	no	—
72	C₃₁×#125	—	2	4	no	—
73	C₃₁×#124	—	2	8	no	—
74	C₃₁×#124	—	2	4	no	—
75	C₃₁×#124	C₂³	1	4	yes	—
76	C₃₁×#123	—	2	8	no	—
77	C₃₁×C₂²⋊Q₈	—	2	4	no	—
78	C₃₁×C₂²⋊Q₈	—	2	4	no	—
79	C₃₁×C₂²⋊Q₈	—	2	8	no	—
80	C₃₁×C₂²⋊Q₈	C₂³	1	8	yes	—
81	C₃₁×#121	—	2	4	no	—
82	C₃₁×#121	—	2	8	no	—
83	C₃₁×#121	—	2	2	no	—
84	C₃₁×#121	—	2	2	no	—
85	C₃₁×#121	—	2	4	no	—
86	C₃₁×#121	C₂³	1	8	yes	—
87	C₄×Q₈×C₃₁	—	2	4	no	—
88	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
89	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
90	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
91	C₄×D₄×C₃₁	—	2	8	no	—
92	C₄×D₄×C₃₁	—	2	2	no	—
93	C₄×D₄×C₃₁	—	2	4	no	—
94	C₄×D₄×C₃₁	—	2	2	no	—
95	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
96	C₄×D₄×C₃₁	D₄	1	2	yes	—
97	C₄×D₄×C₃₁	D₄	1	2	yes	—
98	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
99	C₄×D₄×C₃₁	C₂³	1	8	yes	—
100	C₃₁×#117	—	2	4	no	—
101	C₃₁×#117	—	2	4	no	—
102	C₃₁×#117	C₂³	1	1	yes	characteristic
103	C₃₁×#117	C₂³	1	2	yes	—
104	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
105	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
106	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
107	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
109	C₃₁×#107	C₂³	1	8	yes	—
110	C₃₁×#107	C₂³	1	8	yes	—
111	C₃₁×#103	C₂³	1	16	yes	—
112	C₃₁×#102	C₂³	1	16	yes	—
113	C₃₁×#97	C₂³	1	4	yes	—
114	C₃₁×#97	C₂³	1	2	yes	—
115	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
116	C₄×C₈×C₃₁	C₂³	1	2	yes	—
117	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
118	C₄×C₈×C₃₁	C₂³	1	4	yes	—
119	C₄×C₈×C₃₁	C₂³	1	2	yes	—
120	#41×C₃₁	—	2	4	no	—
121	#41×C₃₁	—	2	16	no	—
122	#41×C₃₁	—	8	8	no	—
123	C₂×Q₈×C₃₁	—	2	4	no	—
124	C₂×Q₈×C₃₁	—	2	4	no	—
125	C₂×Q₈×C₃₁	—	8	4	no	—
126	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
128	C₂×D₄×C₃₁	—	2	4	no	—
129	C₂×D₄×C₃₁	—	2	8	no	—
130	C₂×D₄×C₃₁	—	2	2	no	—
131	C₂×D₄×C₃₁	—	2	2	no	—
132	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
134	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
135	C₂²×C₄×C₃₁	—	2	4	no	—
136	C₂²×C₄×C₃₁	—	2	4	no	—
137	C₂²×C₄×C₃₁	—	2	8	no	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
144	Q₁₆×C₃₁	—	2	8	no	—
145	SD₁₆×C₃₁	—	2	16	no	—
146	D₈×C₃₁	—	2	8	no	—
147	M₁₆×C₃₁	—	2	4	no	—
148	M₁₆×C₃₁	—	2	8	no	—
149	C₂×C₈×C₃₁	—	2	8	no	—
150	C₂×C₈×C₃₁	#41	1	8	yes	complemented
151	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
152	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
153	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
154	C₄⋊C₄×C₃₁	—	2	4	no	—
155	C₄⋊C₄×C₃₁	—	2	4	no	—
156	C₄⋊C₄×C₃₁	—	2	8	no	—
157	C₄⋊C₄×C₃₁	—	2	4	no	—
158	C₄⋊C₄×C₃₁	—	2	2	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	8	no	—
161	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
162	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
163	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
164	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
165	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
166	C₂²⋊C₄×C₃₁	—	2	8	no	—
167	C₂²⋊C₄×C₃₁	—	2	8	no	—
168	C₂²⋊C₄×C₃₁	—	2	8	no	—
169	C₂²⋊C₄×C₃₁	—	2	4	no	—
170	C₂²⋊C₄×C₃₁	—	2	4	no	—
171	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
172	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
173	C₂²⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
174	C₄²×C₃₁	—	2	8	no	—
175	C₄²×C₃₁	C₂×D₄	1	2	yes	—
176	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
177	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
178	C₄²×C₃₁	C₂×D₄	1	2	yes	—
179	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
180	C₄²×C₃₁	C₂⁴	1	2	yes	—
181	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
182	#60718	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
183	C₂³×C₃₁	—	2	4	no	—
184	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
185	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
186	Q₈×C₃₁	—	2	2	no	—
187	Q₈×C₃₁	—	2	8	no	—
188	Q₈×C₃₁	—	4	8	no	—
189	Q₈×C₃₁	—	8	4	no	—
190	D₄×C₃₁	—	2	4	no	—
191	D₄×C₃₁	—	2	2	no	—
192	D₄×C₃₁	—	2	8	no	—
193	D₄×C₃₁	—	4	8	no	—
194	D₄×C₃₁	—	4	4	no	—
195	D₄×C₃₁	—	4	4	no	—
196	D₄×C₃₁	—	8	8	no	—
197	C₂×C₄×C₃₁	—	2	2	no	—
198	C₂×C₄×C₃₁	—	2	4	no	—
199	C₂×C₄×C₃₁	—	2	2	no	—
200	C₂×C₄×C₃₁	—	2	4	no	—
201	C₂×C₄×C₃₁	—	2	4	no	—
202	C₂×C₄×C₃₁	—	2	4	no	—
203	C₂×C₄×C₃₁	—	2	8	no	—
204	C₂×C₄×C₃₁	—	4	8	no	—
205	C₂×C₄×C₃₁	—	8	8	no	—
206	C₂×C₄×C₃₁	—	8	4	no	—
207	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
208	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
209	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
210	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
212	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
213	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₄×C₃₁	C₂×#41	1	4	yes	—
216	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
217	C₈×C₃₁	—	2	4	no	—
218	C₈×C₃₁	C₂×#41	1	4	yes	—
219	C₈×C₃₁	C₂×#41	1	4	yes	—
220	#3511	C₂×C₃₁	1	2	yes	complemented
221	#3182	C₂×C₃₁	1	4	yes	complemented
222	#3175	C₂×C₃₁	1	8	yes	complemented
223	#3170	C₂×C₃₁	1	2	yes	complemented
224	#3169	C₂×C₃₁	1	2	yes	complemented
225	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
226	#3147	C₂×C₃₁	1	1	yes	characteristic, complemented
227	#3141	C₂×C₃₁	1	2	yes	complemented
228	#3061	C₂×C₃₁	1	4	yes	complemented
229	#2951	C₂×C₃₁	1	4	yes	complemented
230	C₄×#131	C₂×C₃₁	1	1	yes	characteristic, complemented
231	C₂²×C₃₁	—	2	1	no	—
232	C₂²×C₃₁	—	2	1	no	—
233	C₂²×C₃₁	—	2	2	no	—
234	C₂²×C₃₁	—	2	2	no	—
235	C₂²×C₃₁	—	2	2	no	—
236	C₂²×C₃₁	—	4	4	no	—
237	C₂²×C₃₁	—	4	4	no	—
238	C₂²×C₃₁	—	4	4	no	—
239	C₂²×C₃₁	#532	1	2	yes	—
240	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
241	C₄×C₃₁	—	2	2	no	—
242	C₄×C₃₁	—	2	2	no	—
243	C₄×C₃₁	—	2	1	no	—
244	C₄×C₃₁	—	2	2	no	—
245	C₄×C₃₁	—	2	1	no	—
246	C₄×C₃₁	—	4	4	no	—
247	C₄×C₃₁	—	4	8	no	—
248	C₄×C₃₁	#532	1	2	yes	—
249	C₄×C₃₁	#576	1	2	yes	—
250	C₄×C₃₁	#578	1	2	yes	—
251	C₄×C₃₁	C₂²×#41	1	2	yes	—
252	C₂×#135	C₂²×C₃₁	1	4	yes	—
253	C₂×#131	C₂²×C₃₁	1	2	yes	—
254	#556	—	2	2	no	—
255	#555	—	2	2	no	—
256	#554	—	2	2	no	—
257	#553	—	2	4	no	—
258	#548	—	2	4	no	—
259	#547	—	2	2	no	—
260	#542	C₂²×C₃₁	1	8	yes	—
261	#537	C₂²×C₃₁	1	4	yes	—
262	#533	C₂²×C₃₁	1	4	yes	—
263	#532	C₂²×C₃₁	1	2	yes	—
264	#532	C₂²×C₃₁	1	4	yes	—
265	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
266	#520	C₂²×C₃₁	1	4	yes	—
267	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
268	#500	C₂²×C₃₁	1	2	yes	—
269	#500	C₂²×C₃₁	1	4	yes	—
270	#500	C₂²×C₃₁	1	2	yes	—
271	#495	C₂²×C₃₁	1	4	yes	—
272	#494	C₂²×C₃₁	1	1	yes	characteristic
273	#493	C₂²×C₃₁	1	1	yes	characteristic
274	#492	C₂²×C₃₁	1	2	yes	—
275	#489	C₂²×C₃₁	1	8	yes	—
276	#487	C₂²×C₃₁	1	4	yes	—
277	#486	C₂²×C₃₁	1	4	yes	—
278	#468	C₂²×C₃₁	1	16	yes	—
279	#467	C₂²×C₃₁	1	8	yes	—
280	#466	C₂²×C₃₁	1	8	yes	—
281	#440	C₂²×C₃₁	1	16	yes	—
282	C₄×Q₁₆	C₂²×C₃₁	1	8	yes	—
283	C₄×D₈	C₂²×C₃₁	1	8	yes	—
284	C₂×#107	C₂²×C₃₁	1	4	yes	—
285	#417	C₂²×C₃₁	1	8	yes	—
286	#405	C₂²×C₃₁	1	4	yes	—
287	#405	C₂²×C₃₁	1	2	yes	—
288	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
289	C₄×M₁₆	C₂²×C₃₁	1	2	yes	—
290	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
292	C₂×C₃₁	—	2	2	no	—
293	C₂×C₃₁	—	4	2	no	—
294	C₂×C₃₁	—	8	4	no	—
295	C₂×C₃₁	C₂×#532	1	1	yes	characteristic
296	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
297	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
298	C₂×#41	—	2	4	no	—
299	C₂×#41	C₂³×C₃₁	1	4	yes	—
300	C₂×#41	C₂³×C₃₁	1	1	yes	characteristic
301	C₂²×Q₈	—	2	2	no	—
302	#135	—	2	16	no	—
303	C₂×Q₁₆	C₂³×C₃₁	1	4	yes	—
304	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
305	C₂×D₈	C₂³×C₃₁	1	4	yes	—
306	#131	—	2	8	no	—
307	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
308	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
309	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
310	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
311	C₄⋊Q₈	—	2	2	no	—
312	C₄⋊Q₈	—	2	2	no	—
313	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
314	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
315	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
316	#127	C₂³×C₃₁	1	2	yes	—
317	#126	—	2	8	no	—
318	#125	—	2	4	no	—
319	#124	—	2	8	no	—
320	#124	—	2	4	no	—
321	#124	C₂³×C₃₁	1	4	yes	—
322	#123	—	2	8	no	—
323	C₂²⋊Q₈	—	2	4	no	—
324	C₂²⋊Q₈	—	2	4	no	—
325	C₂²⋊Q₈	—	2	8	no	—
326	C₂²⋊Q₈	C₂³×C₃₁	1	8	yes	—
327	#121	—	2	4	no	—
328	#121	—	2	8	no	—
329	#121	—	2	2	no	—
330	#121	—	2	2	no	—
331	#121	—	2	4	no	—
332	#121	C₂³×C₃₁	1	8	yes	—
333	C₄×Q₈	—	2	4	no	—
334	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
335	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
336	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
337	C₄×D₄	—	2	8	no	—
338	C₄×D₄	—	2	2	no	—
339	C₄×D₄	—	2	4	no	—
340	C₄×D₄	—	2	2	no	—
341	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
342	C₄×D₄	D₄×C₃₁	1	2	yes	—
343	C₄×D₄	D₄×C₃₁	1	2	yes	—
344	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
345	C₄×D₄	C₂³×C₃₁	1	8	yes	—
346	#117	—	2	4	no	—
347	#117	—	2	4	no	—
348	#117	C₂³×C₃₁	1	1	yes	characteristic
349	#117	C₂³×C₃₁	1	2	yes	—
350	C₂×C₄⋊C₄	—	2	2	no	—
351	C₂×C₄⋊C₄	—	2	4	no	—
352	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
354	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
355	#107	C₂³×C₃₁	1	8	yes	—
356	#107	C₂³×C₃₁	1	8	yes	—
357	#103	C₂³×C₃₁	1	16	yes	—
358	#102	C₂³×C₃₁	1	16	yes	—
359	#97	C₂³×C₃₁	1	4	yes	—
360	#97	C₂³×C₃₁	1	2	yes	—
361	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
362	C₄×C₈	C₂³×C₃₁	1	2	yes	—
363	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
364	C₄×C₈	C₂³×C₃₁	1	4	yes	—
365	C₄×C₈	C₂³×C₃₁	1	2	yes	—
366	C₃₁	#60718	1	1	yes	characteristic, direct product factor, Sylow
367	#41	—	2	4	no	—
368	#41	—	2	16	no	—
369	#41	—	8	8	no	—
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	4	no	—
372	C₂×Q₈	—	8	4	no	—
373	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
375	C₂×D₄	—	2	4	no	—
376	C₂×D₄	—	2	8	no	—
377	C₂×D₄	—	2	2	no	—
378	C₂×D₄	—	2	2	no	—
379	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
380	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
381	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	8	no	—
385	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
386	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
388	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
389	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
390	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
391	Q₁₆	—	2	8	no	—
392	SD₁₆	—	2	16	no	—
393	D₈	—	2	8	no	—
394	M₁₆	—	2	4	no	—
395	M₁₆	—	2	8	no	—
396	C₂×C₈	—	2	8	no	—
397	C₂×C₈	#41×C₃₁	1	8	yes	complemented
398	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
399	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
400	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
401	C₄⋊C₄	—	2	4	no	—
402	C₄⋊C₄	—	2	4	no	—
403	C₄⋊C₄	—	2	8	no	—
404	C₄⋊C₄	—	2	4	no	—
405	C₄⋊C₄	—	2	2	no	—
406	C₄⋊C₄	—	2	2	no	—
407	C₄⋊C₄	—	2	8	no	—
408	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
409	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
410	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
411	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
412	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
413	C₂²⋊C₄	—	2	8	no	—
414	C₂²⋊C₄	—	2	8	no	—
415	C₂²⋊C₄	—	2	8	no	—
416	C₂²⋊C₄	—	2	4	no	—
417	C₂²⋊C₄	—	2	4	no	—
418	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
419	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
420	C₂²⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
421	C₄²	—	2	8	no	—
422	C₄²	C₂×D₄×C₃₁	1	2	yes	—
423	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
424	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
425	C₄²	C₂×D₄×C₃₁	1	2	yes	—
426	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
427	C₄²	C₂⁴×C₃₁	1	2	yes	—
428	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
429	C₂³	—	2	4	no	—
430	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
431	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
432	Q₈	—	2	2	no	—
433	Q₈	—	2	8	no	—
434	Q₈	—	4	8	no	—
435	Q₈	—	8	4	no	—
436	D₄	—	2	4	no	—
437	D₄	—	2	2	no	—
438	D₄	—	2	8	no	—
439	D₄	—	4	8	no	—
440	D₄	—	4	4	no	—
441	D₄	—	4	4	no	—
442	D₄	—	8	8	no	—
443	C₂×C₄	—	2	2	no	—
444	C₂×C₄	—	2	4	no	—
445	C₂×C₄	—	2	2	no	—
446	C₂×C₄	—	2	4	no	—
447	C₂×C₄	—	2	4	no	—
448	C₂×C₄	—	2	4	no	—
449	C₂×C₄	—	2	8	no	—
450	C₂×C₄	—	4	8	no	—
451	C₂×C₄	—	8	8	no	—
452	C₂×C₄	—	8	4	no	—
453	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
454	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
455	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
456	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
457	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
458	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
460	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₃₁	1	4	yes	—
462	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
463	C₈	—	2	4	no	—
464	C₈	C₂×#41×C₃₁	1	4	yes	—
465	C₈	C₂×#41×C₃₁	1	4	yes	—
466	C₂²	—	2	1	no	—
467	C₂²	—	2	1	no	—
468	C₂²	—	2	2	no	—
469	C₂²	—	2	2	no	—
470	C₂²	—	2	2	no	—
471	C₂²	—	4	4	no	—
472	C₂²	—	4	4	no	—
473	C₂²	—	4	4	no	—
474	C₂²	C₃₁×#532	1	2	yes	—
475	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	1	no	—
479	C₄	—	2	2	no	—
480	C₄	—	2	1	no	—
481	C₄	—	4	4	no	—
482	C₄	—	4	8	no	—
483	C₄	C₃₁×#532	1	2	yes	—
484	C₄	C₃₁×#576	1	2	yes	—
485	C₄	C₃₁×#578	1	2	yes	—
486	C₄	C₂²×#41×C₃₁	1	2	yes	—
487	C₂	—	2	2	no	—
488	C₂	—	4	2	no	—
489	C₂	—	8	4	no	—
490	C₂	C₂×C₃₁×#532	1	1	yes	characteristic
491	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
492	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3511	C₂	semidirect
2. C₃₁×#3182	C₂	semidirect
3. C₃₁×#3175	C₂	semidirect
4. C₃₁×#3170	C₂	semidirect
5. C₃₁×#3169	C₂	semidirect
6. C₂×C₃₁×#500	C₂	semidirect
7. C₃₁×#3147	C₂	semidirect
8. C₃₁×#3141	C₂	semidirect
9. C₃₁×#3061	C₂	semidirect
10. C₃₁×#2951	C₂	semidirect
11. C₄×C₃₁×#131	C₂	semidirect
150. C₂×C₈×C₃₁	#41	semidirect
182. #60718	C₃₁	direct (N × H)
220. #3511	C₂×C₃₁	semidirect
221. #3182	C₂×C₃₁	semidirect
222. #3175	C₂×C₃₁	semidirect
223. #3170	C₂×C₃₁	semidirect
224. #3169	C₂×C₃₁	semidirect
225. C₂×#500	C₂×C₃₁	semidirect
226. #3147	C₂×C₃₁	semidirect
227. #3141	C₂×C₃₁	semidirect
228. #3061	C₂×C₃₁	semidirect
229. #2951	C₂×C₃₁	semidirect
230. C₄×#131	C₂×C₃₁	semidirect
366. C₃₁	#60718	direct (N × H)
397. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3511	2
C₃₁×#3182	2
C₃₁×#3175	2
C₃₁×#3170	2
C₃₁×#3169	2
C₂×C₃₁×#500	2
C₃₁×#3147	2
C₃₁×#3141	2
C₃₁×#3061	2
C₃₁×#2951	2
C₄×C₃₁×#131	2
#60718	31

Subgroups — C31×#60718 #21517477

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60718 #21517477