Subgroups — C₃₁×#61951 #21518710

Order 7936.

2168 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3498	C₂	1	1	yes	maximal, characteristic
2	Q₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3392	C₂	1	2	yes	maximal, complemented
4	C₃₁×#3362	C₂	1	2	yes	maximal, complemented
5	C₃₁×#3295	C₂	1	6	yes	maximal, complemented
6	D₄×SD₁₆×C₃₁	C₂	1	6	yes	maximal, complemented
7	C₃₁×#3246	C₂	1	3	yes	maximal, complemented
8	C₃₁×#3208	C₂	1	3	yes	maximal, complemented
9	C₃₁×#3103	C₂	1	3	yes	maximal, complemented
10	C₂×C₃₁×#460	C₂	1	3	yes	maximal, complemented
11	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×H(5,𝔽₂)	—	2	1	no	—
13	C₂²×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
14	Q₈²×C₃₁	C₂²	1	1	yes	characteristic, complemented
15	C₃₁×#557	—	2	1	no	—
16	C₃₁×#557	C₂²	1	1	yes	characteristic, complemented
17	C₃₁×#554	—	2	3	no	—
18	C₃₁×#554	C₂²	1	3	yes	complemented
19	C₃₁×#550	—	2	1	no	—
20	C₃₁×#550	C₂²	1	2	yes	—
21	D₄×Q₈×C₃₁	C₂²	1	3	yes	complemented
22	D₄²×C₃₁	—	2	3	no	—
23	D₄²×C₃₁	C₂²	1	6	yes	—
24	C₃₁×#535	—	2	3	no	—
25	C₃₁×#533	—	2	3	no	—
26	C₃₁×#533	C₂²	1	3	yes	complemented
27	C₃₁×#532	C₂²	1	3	yes	—
28	C₂×C₃₁×#127	C₂²	1	3	yes	—
29	C₄×#41×C₃₁	—	2	1	no	—
30	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	complemented
32	C₃₁×#492	C₂²	1	6	yes	complemented
33	C₃₁×#486	C₂²	1	6	yes	complemented
34	C₃₁×#481	C₂²	1	6	yes	complemented
35	C₃₁×#476	C₂²	1	6	yes	complemented
36	C₃₁×#475	C₂²	1	1	yes	characteristic, complemented
37	C₃₁×#475	C₂²	1	3	yes	complemented
38	C₃₁×#465	C₂²	1	12	yes	complemented
39	C₃₁×#461	C₂²	1	6	yes	complemented
40	C₃₁×#460	C₂²	1	3	yes	complemented
41	C₃₁×#460	C₂²	1	3	yes	complemented
42	C₃₁×#460	C₂²	1	6	yes	complemented
43	C₃₁×#460	C₂²	1	6	yes	complemented
44	C₃₁×#460	C₂²	1	6	yes	complemented
45	C₃₁×#450	C₂²	1	12	yes	complemented
46	C₃₁×#448	C₂²	1	6	yes	complemented
47	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
48	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
49	C₄×SD₁₆×C₃₁	C₂²	1	6	yes	complemented
50	C₈×D₄×C₃₁	C₂²	1	6	yes	complemented
51	C₃₁×#431	C₂²	1	3	yes	complemented
52	C₂×C₃₁×C₄⋊C₈	C₂²	1	3	yes	complemented
53	C₂×C₃₁×#102	C₂²	1	6	yes	complemented
54	C₃₁×H(5,𝔽₂)	—	4	8	no	—
55	C₂×#41×C₃₁	—	2	2	no	—
56	C₂×#41×C₃₁	—	2	3	no	—
57	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₃₁	C₂³	1	3	yes	—
59	C₂²×D₄×C₃₁	—	2	6	no	—
60	C₂²×D₄×C₃₁	—	2	3	no	—
61	C₂²×D₄×C₃₁	—	2	3	no	—
62	C₂²×D₄×C₃₁	C₂³	1	6	yes	—
63	C₂×SD₁₆×C₃₁	—	4	12	no	—
64	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
65	C₂×SD₁₆×C₃₁	C₂³	1	6	yes	—
66	C₂×SD₁₆×C₃₁	C₂³	1	12	yes	—
67	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
68	C₃₁×C₄⋊Q₈	—	2	3	no	—
69	C₃₁×C₄⋊Q₈	—	2	3	no	—
70	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
71	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
72	C₃₁×C₄⋊Q₈	C₂³	1	3	yes	—
73	C₃₁×#127	—	2	3	no	—
74	C₃₁×#127	—	2	3	no	—
75	C₃₁×#127	C₂³	1	3	yes	—
76	C₃₁×#127	C₂³	1	6	yes	—
77	C₃₁×#125	—	2	3	no	—
78	C₃₁×#125	—	2	3	no	—
79	C₃₁×#125	—	2	3	no	—
80	C₃₁×#124	—	2	3	no	—
81	C₃₁×C₂²⋊Q₈	—	2	3	no	—
82	C₃₁×C₂²⋊Q₈	—	2	6	no	—
83	C₃₁×C₂²⋊Q₈	—	2	6	no	—
84	C₃₁×C₂²⋊Q₈	C₂³	1	6	yes	—
85	C₃₁×#121	—	2	3	no	—
86	C₃₁×#121	—	2	6	no	—
87	C₃₁×#121	—	2	6	no	—
88	C₃₁×#121	C₂³	1	6	yes	—
89	C₃₁×#120	—	2	12	no	—
90	C₄×Q₈×C₃₁	—	2	1	no	—
91	C₄×Q₈×C₃₁	—	2	1	no	—
92	C₄×Q₈×C₃₁	—	2	3	no	—
93	C₄×Q₈×C₃₁	—	2	3	no	—
94	C₄×Q₈×C₃₁	—	2	3	no	—
95	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
97	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
98	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
99	C₄×Q₈×C₃₁	C₂³	1	3	yes	—
100	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₃₁	—	2	3	no	—
102	C₄×D₄×C₃₁	—	2	3	no	—
103	C₄×D₄×C₃₁	—	2	3	no	—
104	C₄×D₄×C₃₁	D₄	1	3	yes	—
105	C₄×D₄×C₃₁	C₂³	1	6	yes	—
106	C₄×D₄×C₃₁	C₂³	1	3	yes	—
107	C₃₁×#117	—	2	3	no	—
108	C₃₁×#117	D₄	1	3	yes	—
109	C₂×C₄⋊C₄×C₃₁	—	2	3	no	—
110	C₂×C₄⋊C₄×C₃₁	C₂³	1	3	yes	—
111	C₂×C₄²×C₃₁	—	2	3	no	—
112	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
113	C₃₁×#106	C₂³	1	2	yes	—
114	C₃₁×#106	C₂³	1	6	yes	—
115	C₃₁×C₄⋊C₈	C₂³	1	3	yes	—
116	C₃₁×C₄⋊C₈	C₂³	1	6	yes	—
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×#103	C₂³	1	2	yes	—
119	C₃₁×#103	C₂³	1	6	yes	—
120	C₃₁×#102	C₂³	1	6	yes	—
121	C₃₁×#102	C₂³	1	12	yes	—
122	C₃₁×#102	C₂³	1	6	yes	—
123	C₃₁×C₂²⋊C₈	C₂³	1	6	yes	—
124	C₄×C₈×C₃₁	C₂³	1	6	yes	—
125	C₂⁴×C₃₁	—	2	6	no	—
126	#41×C₃₁	—	4	8	no	—
127	#41×C₃₁	—	4	12	no	—
128	#41×C₃₁	SD₁₆	1	8	yes	complemented
129	C₂×Q₈×C₃₁	—	2	3	no	—
130	C₂×Q₈×C₃₁	—	2	1	no	—
131	C₂×Q₈×C₃₁	—	2	3	no	—
132	C₂×Q₈×C₃₁	—	2	6	no	—
133	C₂×Q₈×C₃₁	—	4	3	no	—
134	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
135	C₂×Q₈×C₃₁	C₂⁴	1	3	yes	—
136	C₂×Q₈×C₃₁	C₂⁴	1	1	yes	characteristic
137	C₂×D₄×C₃₁	—	2	6	no	—
138	C₂×D₄×C₃₁	—	2	3	no	—
139	C₂×D₄×C₃₁	—	2	6	no	—
140	C₂×D₄×C₃₁	—	2	3	no	—
141	C₂×D₄×C₃₁	—	2	6	no	—
142	C₂×D₄×C₃₁	—	2	6	no	—
143	C₂×D₄×C₃₁	—	4	6	no	—
144	C₂×D₄×C₃₁	—	4	6	no	—
145	C₂×D₄×C₃₁	—	4	24	no	—
146	C₂×D₄×C₃₁	—	4	12	no	—
147	C₂×D₄×C₃₁	—	8	6	no	—
148	C₂×D₄×C₃₁	C₂×D₄	1	3	yes	—
149	C₂×D₄×C₃₁	C₂⁴	1	6	yes	—
150	C₂²×C₄×C₃₁	—	2	3	no	—
151	C₂²×C₄×C₃₁	—	2	3	no	—
152	C₂²×C₄×C₃₁	—	2	6	no	—
153	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
154	C₂²×C₄×C₃₁	C₂×D₄	1	3	yes	—
155	C₂²×C₄×C₃₁	C₂⁴	1	3	yes	—
156	SD₁₆×C₃₁	—	2	12	no	—
157	SD₁₆×C₃₁	—	4	12	no	—
158	SD₁₆×C₃₁	—	4	6	no	—
159	C₂×C₈×C₃₁	—	4	6	no	—
160	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
161	C₂×C₈×C₃₁	C₂⁴	1	6	yes	—
162	C₄⋊C₄×C₃₁	—	2	1	no	—
163	C₄⋊C₄×C₃₁	—	2	3	no	—
164	C₄⋊C₄×C₃₁	—	2	3	no	—
165	C₄⋊C₄×C₃₁	—	2	3	no	—
166	C₄⋊C₄×C₃₁	—	2	3	no	—
167	C₄⋊C₄×C₃₁	—	2	3	no	—
168	C₄⋊C₄×C₃₁	—	2	3	no	—
169	C₄⋊C₄×C₃₁	—	2	1	no	—
170	C₄⋊C₄×C₃₁	—	2	3	no	—
171	C₄⋊C₄×C₃₁	—	2	6	no	—
172	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
173	C₄⋊C₄×C₃₁	C₂×D₄	1	3	yes	—
174	C₄⋊C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₄⋊C₄×C₃₁	C₂⁴	1	3	yes	—
176	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂²⋊C₄×C₃₁	—	2	6	no	—
178	C₂²⋊C₄×C₃₁	—	2	6	no	—
179	C₂²⋊C₄×C₃₁	—	2	6	no	—
180	C₂²⋊C₄×C₃₁	C₂×D₄	1	6	yes	—
181	C₄²×C₃₁	—	2	1	no	—
182	C₄²×C₃₁	—	2	3	no	—
183	C₄²×C₃₁	—	2	3	no	—
184	C₄²×C₃₁	—	2	3	no	—
185	C₄²×C₃₁	C₂×D₄	1	3	yes	—
186	C₄²×C₃₁	C₂×D₄	1	3	yes	—
187	C₄²×C₃₁	C₂⁴	1	3	yes	—
188	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
189	#61951	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
190	C₂³×C₃₁	—	2	6	no	—
191	C₂³×C₃₁	—	4	12	no	—
192	C₂³×C₃₁	—	8	6	no	—
193	C₂³×C₃₁	—	8	6	no	—
194	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
195	Q₈×C₃₁	—	2	1	no	—
196	Q₈×C₃₁	—	2	3	no	—
197	Q₈×C₃₁	—	2	3	no	—
198	Q₈×C₃₁	—	4	6	no	—
199	Q₈×C₃₁	—	4	2	no	—
200	Q₈×C₃₁	—	8	3	no	—
201	Q₈×C₃₁	C₂×SD₁₆	1	4	yes	complemented
202	Q₈×C₃₁	H(5,𝔽₂)	1	2	yes	complemented
203	D₄×C₃₁	—	2	6	no	—
204	D₄×C₃₁	—	2	6	no	—
205	D₄×C₃₁	—	4	6	no	—
206	D₄×C₃₁	—	4	12	no	—
207	D₄×C₃₁	—	4	12	no	—
208	D₄×C₃₁	—	4	6	no	—
209	D₄×C₃₁	—	8	6	no	—
210	D₄×C₃₁	—	8	6	no	—
211	D₄×C₃₁	C₂×SD₁₆	1	12	yes	complemented
212	C₂×C₄×C₃₁	—	2	1	no	—
213	C₂×C₄×C₃₁	—	2	3	no	—
214	C₂×C₄×C₃₁	—	2	3	no	—
215	C₂×C₄×C₃₁	—	2	1	no	—
216	C₂×C₄×C₃₁	—	2	3	no	—
217	C₂×C₄×C₃₁	—	2	3	no	—
218	C₂×C₄×C₃₁	—	2	2	no	—
219	C₂×C₄×C₃₁	—	4	3	no	—
220	C₂×C₄×C₃₁	—	4	6	no	—
221	C₂×C₄×C₃₁	—	4	12	no	—
222	C₂×C₄×C₃₁	—	8	3	no	—
223	C₂×C₄×C₃₁	C₂×SD₁₆	1	12	yes	complemented
224	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
226	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
227	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₃₁	C₂²×D₄	1	3	yes	—
229	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
230	C₈×C₃₁	—	2	6	no	—
231	C₈×C₃₁	—	2	2	no	—
232	#3498	C₂×C₃₁	1	1	yes	characteristic
233	Q₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#3392	C₂×C₃₁	1	2	yes	complemented
235	#3362	C₂×C₃₁	1	2	yes	complemented
236	#3295	C₂×C₃₁	1	6	yes	complemented
237	D₄×SD₁₆	C₂×C₃₁	1	6	yes	complemented
238	#3246	C₂×C₃₁	1	3	yes	complemented
239	#3208	C₂×C₃₁	1	3	yes	complemented
240	#3103	C₂×C₃₁	1	3	yes	complemented
241	C₂×#460	C₂×C₃₁	1	3	yes	complemented
242	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
243	C₂²×C₃₁	—	2	3	no	—
244	C₂²×C₃₁	—	2	3	no	—
245	C₂²×C₃₁	—	4	6	no	—
246	C₂²×C₃₁	—	4	6	no	—
247	C₂²×C₃₁	—	4	6	no	—
248	C₂²×C₃₁	—	16	12	no	—
249	C₂²×C₃₁	C₂²×SD₁₆	1	6	yes	—
250	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
251	C₄×C₃₁	—	2	1	no	—
252	C₄×C₃₁	—	2	3	no	—
253	C₄×C₃₁	—	2	3	no	—
254	C₄×C₃₁	—	2	2	no	—
255	C₄×C₃₁	—	4	1	no	—
256	C₄×C₃₁	—	4	3	no	—
257	C₄×C₃₁	—	4	3	no	—
258	C₄×C₃₁	—	4	2	no	—
259	C₄×C₃₁	C₂²×SD₁₆	1	6	yes	—
260	C₄×C₃₁	C₂²×SD₁₆	1	2	yes	—
261	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
262	C₄×C₃₁	#585	1	1	yes	characteristic
263	C₂×H(5,𝔽₂)	—	2	1	no	—
264	C₂²×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
265	Q₈²	C₂²×C₃₁	1	1	yes	characteristic, complemented
266	#557	—	2	1	no	—
267	#557	C₂²×C₃₁	1	1	yes	characteristic, complemented
268	#554	—	2	3	no	—
269	#554	C₂²×C₃₁	1	3	yes	complemented
270	#550	—	2	1	no	—
271	#550	C₂²×C₃₁	1	2	yes	—
272	D₄×Q₈	C₂²×C₃₁	1	3	yes	complemented
273	D₄²	—	2	3	no	—
274	D₄²	C₂²×C₃₁	1	6	yes	—
275	#535	—	2	3	no	—
276	#533	—	2	3	no	—
277	#533	C₂²×C₃₁	1	3	yes	complemented
278	#532	C₂²×C₃₁	1	3	yes	—
279	C₂×#127	C₂²×C₃₁	1	3	yes	—
280	C₄×#41	—	2	1	no	—
281	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
282	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	complemented
283	#492	C₂²×C₃₁	1	6	yes	complemented
284	#486	C₂²×C₃₁	1	6	yes	complemented
285	#481	C₂²×C₃₁	1	6	yes	complemented
286	#476	C₂²×C₃₁	1	6	yes	complemented
287	#475	C₂²×C₃₁	1	1	yes	characteristic, complemented
288	#475	C₂²×C₃₁	1	3	yes	complemented
289	#465	C₂²×C₃₁	1	12	yes	complemented
290	#461	C₂²×C₃₁	1	6	yes	complemented
291	#460	C₂²×C₃₁	1	3	yes	complemented
292	#460	C₂²×C₃₁	1	3	yes	complemented
293	#460	C₂²×C₃₁	1	6	yes	complemented
294	#460	C₂²×C₃₁	1	6	yes	complemented
295	#460	C₂²×C₃₁	1	6	yes	complemented
296	#450	C₂²×C₃₁	1	12	yes	complemented
297	#448	C₂²×C₃₁	1	6	yes	complemented
298	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
299	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
300	C₄×SD₁₆	C₂²×C₃₁	1	6	yes	complemented
301	C₈×D₄	C₂²×C₃₁	1	6	yes	complemented
302	#431	C₂²×C₃₁	1	3	yes	complemented
303	C₂×C₄⋊C₈	C₂²×C₃₁	1	3	yes	complemented
304	C₂×#102	C₂²×C₃₁	1	6	yes	complemented
305	C₂×C₃₁	—	2	6	no	—
306	C₂×C₃₁	—	8	6	no	—
307	C₂×C₃₁	#3488	1	1	yes	characteristic
308	C₂×C₃₁	C₂³×SD₁₆	1	1	yes	characteristic
309	C₂×C₃₁	#3585	1	1	yes	characteristic
310	H(5,𝔽₂)	—	4	8	no	—
311	C₂×#41	—	2	2	no	—
312	C₂×#41	—	2	3	no	—
313	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
314	C₂²×Q₈	C₂³×C₃₁	1	3	yes	—
315	C₂²×D₄	—	2	6	no	—
316	C₂²×D₄	—	2	3	no	—
317	C₂²×D₄	—	2	3	no	—
318	C₂²×D₄	C₂³×C₃₁	1	6	yes	—
319	C₂×SD₁₆	—	4	12	no	—
320	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
321	C₂×SD₁₆	C₂³×C₃₁	1	12	yes	—
322	C₂×SD₁₆	C₂³×C₃₁	1	6	yes	—
323	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
324	C₄⋊Q₈	—	2	3	no	—
325	C₄⋊Q₈	—	2	3	no	—
326	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
327	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₄⋊Q₈	C₂³×C₃₁	1	3	yes	—
329	#127	—	2	3	no	—
330	#127	—	2	3	no	—
331	#127	C₂³×C₃₁	1	3	yes	—
332	#127	C₂³×C₃₁	1	6	yes	—
333	#125	—	2	3	no	—
334	#125	—	2	3	no	—
335	#125	—	2	3	no	—
336	#124	—	2	3	no	—
337	C₂²⋊Q₈	—	2	3	no	—
338	C₂²⋊Q₈	—	2	6	no	—
339	C₂²⋊Q₈	—	2	6	no	—
340	C₂²⋊Q₈	C₂³×C₃₁	1	6	yes	—
341	#121	—	2	3	no	—
342	#121	—	2	6	no	—
343	#121	—	2	6	no	—
344	#121	C₂³×C₃₁	1	6	yes	—
345	#120	—	2	12	no	—
346	C₄×Q₈	—	2	1	no	—
347	C₄×Q₈	—	2	1	no	—
348	C₄×Q₈	—	2	3	no	—
349	C₄×Q₈	—	2	3	no	—
350	C₄×Q₈	—	2	3	no	—
351	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
352	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
353	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
354	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
355	C₄×Q₈	C₂³×C₃₁	1	3	yes	—
356	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
357	C₄×D₄	—	2	3	no	—
358	C₄×D₄	—	2	3	no	—
359	C₄×D₄	—	2	3	no	—
360	C₄×D₄	D₄×C₃₁	1	3	yes	—
361	C₄×D₄	C₂³×C₃₁	1	6	yes	—
362	C₄×D₄	C₂³×C₃₁	1	3	yes	—
363	#117	—	2	3	no	—
364	#117	D₄×C₃₁	1	3	yes	—
365	C₂×C₄⋊C₄	—	2	3	no	—
366	C₂×C₄⋊C₄	C₂³×C₃₁	1	3	yes	—
367	C₂×C₄²	—	2	3	no	—
368	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
369	#106	C₂³×C₃₁	1	2	yes	—
370	#106	C₂³×C₃₁	1	6	yes	—
371	C₄⋊C₈	C₂³×C₃₁	1	3	yes	—
372	C₄⋊C₈	C₂³×C₃₁	1	6	yes	—
373	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
374	#103	C₂³×C₃₁	1	2	yes	—
375	#103	C₂³×C₃₁	1	6	yes	—
376	#102	C₂³×C₃₁	1	6	yes	—
377	#102	C₂³×C₃₁	1	12	yes	—
378	#102	C₂³×C₃₁	1	6	yes	—
379	C₂²⋊C₈	C₂³×C₃₁	1	6	yes	—
380	C₄×C₈	C₂³×C₃₁	1	6	yes	—
381	C₃₁	#61951	1	1	yes	characteristic, direct product factor, Sylow
382	C₂⁴	—	2	6	no	—
383	#41	—	4	8	no	—
384	#41	—	4	12	no	—
385	#41	SD₁₆×C₃₁	1	8	yes	complemented
386	C₂×Q₈	—	2	3	no	—
387	C₂×Q₈	—	2	1	no	—
388	C₂×Q₈	—	2	3	no	—
389	C₂×Q₈	—	2	6	no	—
390	C₂×Q₈	—	4	3	no	—
391	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
392	C₂×Q₈	C₂⁴×C₃₁	1	3	yes	—
393	C₂×Q₈	C₂⁴×C₃₁	1	1	yes	characteristic
394	C₂×D₄	—	2	6	no	—
395	C₂×D₄	—	2	3	no	—
396	C₂×D₄	—	2	6	no	—
397	C₂×D₄	—	2	3	no	—
398	C₂×D₄	—	2	6	no	—
399	C₂×D₄	—	2	6	no	—
400	C₂×D₄	—	4	6	no	—
401	C₂×D₄	—	4	6	no	—
402	C₂×D₄	—	4	24	no	—
403	C₂×D₄	—	4	12	no	—
404	C₂×D₄	—	8	6	no	—
405	C₂×D₄	C₂×D₄×C₃₁	1	3	yes	—
406	C₂×D₄	C₂⁴×C₃₁	1	6	yes	—
407	C₂²×C₄	—	2	3	no	—
408	C₂²×C₄	—	2	3	no	—
409	C₂²×C₄	—	2	6	no	—
410	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
411	C₂²×C₄	C₂×D₄×C₃₁	1	3	yes	—
412	C₂²×C₄	C₂⁴×C₃₁	1	3	yes	—
413	SD₁₆	—	2	12	no	—
414	SD₁₆	—	4	12	no	—
415	SD₁₆	—	4	6	no	—
416	C₂×C₈	—	4	6	no	—
417	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
418	C₂×C₈	C₂⁴×C₃₁	1	6	yes	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	3	no	—
421	C₄⋊C₄	—	2	3	no	—
422	C₄⋊C₄	—	2	3	no	—
423	C₄⋊C₄	—	2	3	no	—
424	C₄⋊C₄	—	2	3	no	—
425	C₄⋊C₄	—	2	3	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	3	no	—
428	C₄⋊C₄	—	2	6	no	—
429	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
430	C₄⋊C₄	C₂×D₄×C₃₁	1	3	yes	—
431	C₄⋊C₄	C₂⁴×C₃₁	1	1	yes	characteristic
432	C₄⋊C₄	C₂⁴×C₃₁	1	3	yes	—
433	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
434	C₂²⋊C₄	—	2	6	no	—
435	C₂²⋊C₄	—	2	6	no	—
436	C₂²⋊C₄	—	2	6	no	—
437	C₂²⋊C₄	C₂×D₄×C₃₁	1	6	yes	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	3	no	—
440	C₄²	—	2	3	no	—
441	C₄²	—	2	3	no	—
442	C₄²	C₂×D₄×C₃₁	1	3	yes	—
443	C₄²	C₂×D₄×C₃₁	1	3	yes	—
444	C₄²	C₂⁴×C₃₁	1	3	yes	—
445	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
446	C₂³	—	2	6	no	—
447	C₂³	—	4	12	no	—
448	C₂³	—	8	6	no	—
449	C₂³	—	8	6	no	—
450	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
451	Q₈	—	2	1	no	—
452	Q₈	—	2	3	no	—
453	Q₈	—	2	3	no	—
454	Q₈	—	4	6	no	—
455	Q₈	—	4	2	no	—
456	Q₈	—	8	3	no	—
457	Q₈	C₂×SD₁₆×C₃₁	1	4	yes	complemented
458	Q₈	C₃₁×H(5,𝔽₂)	1	2	yes	complemented
459	D₄	—	2	6	no	—
460	D₄	—	2	6	no	—
461	D₄	—	4	6	no	—
462	D₄	—	4	12	no	—
463	D₄	—	4	12	no	—
464	D₄	—	4	6	no	—
465	D₄	—	8	6	no	—
466	D₄	—	8	6	no	—
467	D₄	C₂×SD₁₆×C₃₁	1	12	yes	complemented
468	C₂×C₄	—	2	1	no	—
469	C₂×C₄	—	2	3	no	—
470	C₂×C₄	—	2	3	no	—
471	C₂×C₄	—	2	1	no	—
472	C₂×C₄	—	2	3	no	—
473	C₂×C₄	—	2	3	no	—
474	C₂×C₄	—	2	2	no	—
475	C₂×C₄	—	4	3	no	—
476	C₂×C₄	—	4	6	no	—
477	C₂×C₄	—	4	12	no	—
478	C₂×C₄	—	8	3	no	—
479	C₂×C₄	C₂×SD₁₆×C₃₁	1	12	yes	complemented
480	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
481	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
482	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
483	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
484	C₂×C₄	C₂²×D₄×C₃₁	1	3	yes	—
485	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
486	C₈	—	2	6	no	—
487	C₈	—	2	2	no	—
488	C₂²	—	2	3	no	—
489	C₂²	—	2	3	no	—
490	C₂²	—	4	6	no	—
491	C₂²	—	4	6	no	—
492	C₂²	—	4	6	no	—
493	C₂²	—	16	12	no	—
494	C₂²	C₂²×SD₁₆×C₃₁	1	6	yes	—
495	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
496	C₄	—	2	3	no	—
497	C₄	—	2	2	no	—
498	C₄	—	2	1	no	—
499	C₄	—	2	3	no	—
500	C₄	—	4	3	no	—
501	C₄	—	4	2	no	—
502	C₄	—	4	1	no	—
503	C₄	—	4	3	no	—
504	C₄	C₂²×SD₁₆×C₃₁	1	6	yes	—
505	C₄	C₂²×SD₁₆×C₃₁	1	2	yes	—
506	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
507	C₄	C₃₁×#585	1	1	yes	characteristic
508	C₂	—	2	6	no	—
509	C₂	—	8	6	no	—
510	C₂	C₃₁×#3488	1	1	yes	characteristic
511	C₂	C₂³×SD₁₆×C₃₁	1	1	yes	characteristic
512	C₂	C₃₁×#3585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₃₁×#3498	C₂
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
2. Q₈×#41×C₃₁	C₂	semidirect
3. C₃₁×#3392	C₂	semidirect
4. C₃₁×#3362	C₂	semidirect
5. C₃₁×#3295	C₂	semidirect
6. D₄×SD₁₆×C₃₁	C₂	semidirect
7. C₃₁×#3246	C₂	semidirect
8. C₃₁×#3208	C₂	semidirect
9. C₃₁×#3103	C₂	semidirect
10. C₂×C₃₁×#460	C₂	semidirect
11. C₃₁×#2988	C₂	semidirect
13. C₂²×SD₁₆×C₃₁	C₂²	semidirect
14. Q₈²×C₃₁	C₂²	semidirect
16. C₃₁×#557	C₂²	semidirect
18. C₃₁×#554	C₂²	semidirect
21. D₄×Q₈×C₃₁	C₂²	semidirect
26. C₃₁×#533	C₂²	semidirect
31. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
32. C₃₁×#492	C₂²	semidirect
33. C₃₁×#486	C₂²	semidirect
34. C₃₁×#481	C₂²	semidirect
35. C₃₁×#476	C₂²	semidirect
36. C₃₁×#475	C₂²	semidirect
37. C₃₁×#475	C₂²	semidirect
38. C₃₁×#465	C₂²	semidirect
39. C₃₁×#461	C₂²	semidirect
40. C₃₁×#460	C₂²	semidirect
41. C₃₁×#460	C₂²	semidirect
42. C₃₁×#460	C₂²	semidirect
43. C₃₁×#460	C₂²	semidirect
44. C₃₁×#460	C₂²	semidirect
45. C₃₁×#450	C₂²	semidirect
46. C₃₁×#448	C₂²	semidirect
47. C₈×Q₈×C₃₁	C₂²	semidirect
48. C₄×SD₁₆×C₃₁	C₂²	semidirect
49. C₄×SD₁₆×C₃₁	C₂²	semidirect
50. C₈×D₄×C₃₁	C₂²	semidirect
51. C₃₁×#431	C₂²	semidirect
52. C₂×C₃₁×C₄⋊C₈	C₂²	semidirect
53. C₂×C₃₁×#102	C₂²	semidirect
128. #41×C₃₁	SD₁₆	semidirect
189. #61951	C₃₁	direct (N × H)
201. Q₈×C₃₁	C₂×SD₁₆	semidirect
202. Q₈×C₃₁	H(5,𝔽₂)	semidirect
211. D₄×C₃₁	C₂×SD₁₆	semidirect
223. C₂×C₄×C₃₁	C₂×SD₁₆	semidirect
233. Q₈×#41	C₂×C₃₁	semidirect
234. #3392	C₂×C₃₁	semidirect
235. #3362	C₂×C₃₁	semidirect
236. #3295	C₂×C₃₁	semidirect
237. D₄×SD₁₆	C₂×C₃₁	semidirect
238. #3246	C₂×C₃₁	semidirect
239. #3208	C₂×C₃₁	semidirect
240. #3103	C₂×C₃₁	semidirect
241. C₂×#460	C₂×C₃₁	semidirect
242. #2988	C₂×C₃₁	semidirect
264. C₂²×SD₁₆	C₂²×C₃₁	semidirect
265. Q₈²	C₂²×C₃₁	semidirect
267. #557	C₂²×C₃₁	semidirect
269. #554	C₂²×C₃₁	semidirect
272. D₄×Q₈	C₂²×C₃₁	semidirect
277. #533	C₂²×C₃₁	semidirect
282. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
283. #492	C₂²×C₃₁	semidirect
284. #486	C₂²×C₃₁	semidirect
285. #481	C₂²×C₃₁	semidirect
286. #476	C₂²×C₃₁	semidirect
287. #475	C₂²×C₃₁	semidirect
288. #475	C₂²×C₃₁	semidirect
289. #465	C₂²×C₃₁	semidirect
290. #461	C₂²×C₃₁	semidirect
291. #460	C₂²×C₃₁	semidirect
292. #460	C₂²×C₃₁	semidirect
293. #460	C₂²×C₃₁	semidirect
294. #460	C₂²×C₃₁	semidirect
295. #460	C₂²×C₃₁	semidirect
296. #450	C₂²×C₃₁	semidirect
297. #448	C₂²×C₃₁	semidirect
298. C₈×Q₈	C₂²×C₃₁	semidirect
299. C₄×SD₁₆	C₂²×C₃₁	semidirect
300. C₄×SD₁₆	C₂²×C₃₁	semidirect
301. C₈×D₄	C₂²×C₃₁	semidirect
302. #431	C₂²×C₃₁	semidirect
303. C₂×C₄⋊C₈	C₂²×C₃₁	semidirect
304. C₂×#102	C₂²×C₃₁	semidirect
381. C₃₁	#61951	direct (N × H)
385. #41	SD₁₆×C₃₁	semidirect
457. Q₈	C₂×SD₁₆×C₃₁	semidirect
458. Q₈	C₃₁×H(5,𝔽₂)	semidirect
467. D₄	C₂×SD₁₆×C₃₁	semidirect
479. C₂×C₄	C₂×SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3498	2
Q₈×#41×C₃₁	2
C₃₁×#3392	2
C₃₁×#3362	2
C₃₁×#3295	2
D₄×SD₁₆×C₃₁	2
C₃₁×#3246	2
C₃₁×#3208	2
C₃₁×#3103	2
C₂×C₃₁×#460	2
C₃₁×#2988	2
#61951	31

Subgroups — C31×#61951 #21518710

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61951 #21518710