Subgroups — C₃₁×#19981 #21476740

Order 7936.

658 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3362	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3298	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#3284	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1708	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1680	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1677	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1666	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1642	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#1635	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#1627	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈×SD₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₃₁×#135	—	2	1	no	—
17	Q₈²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#550	C₂²	1	1	yes	characteristic
19	D₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
20	C₃₁×#547	C₂²	1	1	yes	characteristic
21	C₃₁×#498	—	2	1	no	—
22	C₃₁×#498	C₂²	1	1	yes	characteristic
23	C₃₁×#492	C₂²	1	1	yes	characteristic
24	C₃₁×#487	C₂²	1	1	yes	characteristic
25	C₃₁×#487	C₂²	1	1	yes	characteristic
26	C₃₁×#486	—	2	1	no	—
27	C₃₁×#486	C₂²	1	1	yes	characteristic
28	C₃₁×#486	C₂²	1	1	yes	characteristic
29	C₃₁×#483	—	2	1	no	—
30	C₃₁×#476	—	2	1	no	—
31	C₃₁×#476	—	2	1	no	—
32	C₃₁×#476	C₂²	1	1	yes	characteristic
33	C₃₁×#476	C₂²	1	1	yes	characteristic
34	C₃₁×#475	—	2	1	no	—
35	C₃₁×#475	C₂²	1	1	yes	characteristic
36	C₃₁×#475	C₂²	1	1	yes	characteristic
37	C₃₁×#465	—	2	1	no	—
38	C₃₁×#465	—	2	1	no	—
39	C₃₁×#462	C₂²	1	1	yes	characteristic
40	C₃₁×#461	C₂²	1	1	yes	characteristic
41	C₃₁×#460	—	2	1	no	—
42	C₃₁×#460	C₂²	1	1	yes	characteristic
43	C₃₁×#459	C₂²	1	1	yes	characteristic
44	C₃₁×#452	—	2	1	no	—
45	C₃₁×#449	—	2	1	no	—
46	C₃₁×#448	—	2	1	no	—
47	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
48	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₄×SD₁₆×C₃₁	—	2	1	no	—
50	C₄×SD₁₆×C₃₁	—	2	1	no	—
51	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
52	C₄×SD₁₆×C₃₁	C₂²	1	1	yes	characteristic
53	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
54	C₈×D₄×C₃₁	C₂²	1	1	yes	characteristic
55	C₂×C₃₁×#103	—	2	1	no	—
56	C₃₁×#334	C₂²	1	1	yes	characteristic
57	C₃₁×#334	C₂²	1	1	yes	characteristic
58	C₃₁×#333	C₂²	1	1	yes	characteristic
59	C₃₁×#332	C₂²	1	1	yes	characteristic
60	C₃₁×#332	C₂²	1	1	yes	characteristic
61	C₃₁×#331	C₂²	1	1	yes	characteristic
62	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
63	C₃₁×Q₈⋊C₈	C₂²	1	1	yes	characteristic
64	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
65	C₃₁×D₄⋊C₈	C₂²	1	1	yes	characteristic
66	C₈²×C₃₁	C₂²	1	1	yes	characteristic
67	C₂×#41×C₃₁	—	2	1	no	—
68	C₂×#41×C₃₁	—	2	1	no	—
69	C₂²×Q₈×C₃₁	—	2	1	no	—
70	C₃₁×#135	—	4	4	no	—
71	C₂×Q₁₆×C₃₁	—	2	1	no	—
72	C₂×SD₁₆×C₃₁	—	2	1	no	—
73	C₂×SD₁₆×C₃₁	—	2	1	no	—
74	C₂×SD₁₆×C₃₁	—	2	1	no	—
75	C₂×SD₁₆×C₃₁	—	2	1	no	—
76	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
77	C₂×SD₁₆×C₃₁	D₄	1	1	yes	characteristic
78	C₂×D₈×C₃₁	—	2	1	no	—
79	C₂²×C₈×C₃₁	—	2	1	no	—
80	C₂²×C₈×C₃₁	—	2	1	no	—
81	C₃₁×C₄⋊Q₈	—	2	1	no	—
82	C₃₁×C₄⋊Q₈	—	2	1	no	—
83	C₃₁×C₄⋊Q₈	—	2	1	no	—
84	C₃₁×C₄⋊Q₈	—	4	1	no	—
85	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
86	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
87	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
88	C₃₁×#127	—	2	1	no	—
89	C₃₁×#127	C₂³	1	1	yes	characteristic
90	C₃₁×#123	—	4	1	no	—
91	C₃₁×C₂²⋊Q₈	—	2	1	no	—
92	C₃₁×C₂²⋊Q₈	—	2	1	no	—
93	C₃₁×C₂²⋊Q₈	—	4	1	no	—
94	C₃₁×#121	—	2	1	no	—
95	C₃₁×#121	—	2	1	no	—
96	C₃₁×#121	—	4	1	no	—
97	C₄×Q₈×C₃₁	—	2	1	no	—
98	C₄×Q₈×C₃₁	—	2	1	no	—
99	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
100	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
101	C₄×D₄×C₃₁	—	2	1	no	—
102	C₄×D₄×C₃₁	—	2	1	no	—
103	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
106	C₃₁×#106	—	2	1	no	—
107	C₃₁×#106	—	2	1	no	—
108	C₃₁×#106	—	2	1	no	—
109	C₃₁×#106	—	2	1	no	—
110	C₃₁×#106	—	2	1	no	—
111	C₃₁×#106	—	2	1	no	—
112	C₃₁×#106	D₄	1	1	yes	characteristic
113	C₃₁×#106	D₄	1	1	yes	characteristic
114	C₃₁×C₄⋊C₈	—	2	1	no	—
115	C₃₁×C₄⋊C₈	—	2	1	no	—
116	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
117	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
118	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
119	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
120	C₃₁×#103	—	2	1	no	—
121	C₃₁×#103	—	2	1	no	—
122	C₃₁×#103	—	2	1	no	—
123	C₃₁×#103	—	2	1	no	—
124	C₃₁×#103	—	2	1	no	—
125	C₃₁×#103	—	2	1	no	—
126	C₃₁×#103	—	4	1	no	—
127	C₃₁×#103	D₄	1	1	yes	characteristic
128	C₃₁×#103	D₄	1	1	yes	characteristic
129	C₃₁×#102	—	2	1	no	—
130	C₃₁×#102	—	2	1	no	—
131	C₃₁×#102	—	2	1	no	—
132	C₃₁×#102	—	2	1	no	—
133	C₃₁×#102	—	2	1	no	—
134	C₃₁×#102	—	2	1	no	—
135	C₃₁×#102	—	4	1	no	—
136	C₃₁×#102	D₄	1	1	yes	characteristic
137	C₃₁×#102	D₄	1	1	yes	characteristic
138	C₃₁×C₂²⋊C₈	—	2	1	no	—
139	C₃₁×C₂²⋊C₈	—	2	1	no	—
140	C₄×C₈×C₃₁	—	2	1	no	—
141	C₄×C₈×C₃₁	—	2	1	no	—
142	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
145	#41×C₃₁	—	4	2	no	—
146	#41×C₃₁	—	4	2	no	—
147	#41×C₃₁	—	8	1	no	—
148	#41×C₃₁	—	8	1	no	—
149	C₂×Q₈×C₃₁	—	2	1	no	—
150	C₂×Q₈×C₃₁	—	2	1	no	—
151	C₂×Q₈×C₃₁	—	2	1	no	—
152	C₂×Q₈×C₃₁	—	4	1	no	—
153	C₂×Q₈×C₃₁	—	4	1	no	—
154	C₂×Q₈×C₃₁	—	4	2	no	—
155	C₂×Q₈×C₃₁	—	8	1	no	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
158	C₂×D₄×C₃₁	—	2	1	no	—
159	C₂×D₄×C₃₁	—	2	1	no	—
160	C₂×D₄×C₃₁	—	2	1	no	—
161	C₂×D₄×C₃₁	—	4	1	no	—
162	C₂×D₄×C₃₁	—	4	1	no	—
163	C₂×D₄×C₃₁	—	4	1	no	—
164	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	—	2	1	no	—
167	C₂²×C₄×C₃₁	—	2	1	no	—
168	C₂²×C₄×C₃₁	—	4	1	no	—
169	C₂²×C₄×C₃₁	—	4	1	no	—
170	C₂²×C₄×C₃₁	—	4	1	no	—
171	Q₁₆×C₃₁	—	4	2	no	—
172	SD₁₆×C₃₁	—	2	2	no	—
173	SD₁₆×C₃₁	—	4	2	no	—
174	SD₁₆×C₃₁	—	4	2	no	—
175	SD₁₆×C₃₁	—	8	1	no	—
176	SD₁₆×C₃₁	—	8	1	no	—
177	SD₁₆×C₃₁	SD₁₆	1	4	yes	complemented
178	D₈×C₃₁	—	4	2	no	—
179	C₂×C₈×C₃₁	—	2	1	no	—
180	C₂×C₈×C₃₁	—	2	1	no	—
181	C₂×C₈×C₃₁	—	2	1	no	—
182	C₂×C₈×C₃₁	—	2	1	no	—
183	C₂×C₈×C₃₁	—	2	1	no	—
184	C₂×C₈×C₃₁	—	4	2	no	—
185	C₂×C₈×C₃₁	—	8	1	no	—
186	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
187	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
188	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
189	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
190	C₄⋊C₄×C₃₁	—	2	1	no	—
191	C₄⋊C₄×C₃₁	—	2	1	no	—
192	C₄⋊C₄×C₃₁	—	2	1	no	—
193	C₄⋊C₄×C₃₁	—	2	1	no	—
194	C₄⋊C₄×C₃₁	—	2	1	no	—
195	C₄⋊C₄×C₃₁	—	2	1	no	—
196	C₄⋊C₄×C₃₁	—	2	1	no	—
197	C₄⋊C₄×C₃₁	—	2	1	no	—
198	C₄⋊C₄×C₃₁	—	4	1	no	—
199	C₄⋊C₄×C₃₁	—	4	1	no	—
200	C₄⋊C₄×C₃₁	—	4	1	no	—
201	C₄⋊C₄×C₃₁	—	4	1	no	—
202	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
203	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
204	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
205	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
206	C₂²⋊C₄×C₃₁	—	2	1	no	—
207	C₂²⋊C₄×C₃₁	—	2	1	no	—
208	C₂²⋊C₄×C₃₁	—	4	1	no	—
209	C₂²⋊C₄×C₃₁	—	4	1	no	—
210	C₂²⋊C₄×C₃₁	—	4	1	no	—
211	C₄²×C₃₁	—	2	1	no	—
212	C₄²×C₃₁	—	2	1	no	—
213	C₄²×C₃₁	—	4	1	no	—
214	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
215	#19981	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
216	C₂³×C₃₁	—	2	1	no	—
217	C₂³×C₃₁	—	2	1	no	—
218	C₂³×C₃₁	—	4	1	no	—
219	Q₈×C₃₁	—	2	1	no	—
220	Q₈×C₃₁	—	2	1	no	—
221	Q₈×C₃₁	—	4	1	no	—
222	Q₈×C₃₁	—	4	1	no	—
223	Q₈×C₃₁	—	4	1	no	—
224	Q₈×C₃₁	—	4	1	no	—
225	Q₈×C₃₁	—	8	2	no	—
226	Q₈×C₃₁	—	8	1	no	—
227	Q₈×C₃₁	—	16	1	no	—
228	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
229	Q₈×C₃₁	#135	1	2	yes	complemented
230	D₄×C₃₁	—	2	1	no	—
231	D₄×C₃₁	—	2	1	no	—
232	D₄×C₃₁	—	2	1	no	—
233	D₄×C₃₁	—	4	1	no	—
234	D₄×C₃₁	—	4	1	no	—
235	D₄×C₃₁	—	4	1	no	—
236	D₄×C₃₁	—	4	1	no	—
237	D₄×C₃₁	—	8	2	no	—
238	D₄×C₃₁	—	8	1	no	—
239	D₄×C₃₁	—	8	2	no	—
240	D₄×C₃₁	—	16	1	no	—
241	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
242	C₂×C₄×C₃₁	—	2	1	no	—
243	C₂×C₄×C₃₁	—	2	1	no	—
244	C₂×C₄×C₃₁	—	2	1	no	—
245	C₂×C₄×C₃₁	—	2	1	no	—
246	C₂×C₄×C₃₁	—	2	1	no	—
247	C₂×C₄×C₃₁	—	2	1	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	—	4	1	no	—
250	C₂×C₄×C₃₁	—	4	1	no	—
251	C₂×C₄×C₃₁	—	4	1	no	—
252	C₂×C₄×C₃₁	—	4	1	no	—
253	C₂×C₄×C₃₁	—	4	1	no	—
254	C₂×C₄×C₃₁	—	4	1	no	—
255	C₂×C₄×C₃₁	—	8	2	no	—
256	C₂×C₄×C₃₁	—	8	2	no	—
257	C₂×C₄×C₃₁	—	8	2	no	—
258	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
260	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
261	C₈×C₃₁	—	2	1	no	—
262	C₈×C₃₁	—	2	1	no	—
263	C₈×C₃₁	—	2	1	no	—
264	C₈×C₃₁	—	4	1	no	—
265	C₈×C₃₁	—	4	1	no	—
266	C₈×C₃₁	—	4	1	no	—
267	C₈×C₃₁	—	4	1	no	—
268	C₈×C₃₁	—	4	1	no	—
269	C₈×C₃₁	C₂×SD₁₆	1	2	yes	—
270	Q₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
271	#3362	C₂×C₃₁	1	1	yes	characteristic, complemented
272	#3298	C₂×C₃₁	1	1	yes	characteristic, complemented
273	#3284	C₂×C₃₁	1	1	yes	characteristic
274	#1708	C₂×C₃₁	1	1	yes	characteristic, complemented
275	#1680	C₂×C₃₁	1	1	yes	characteristic, complemented
276	#1677	C₂×C₃₁	1	1	yes	characteristic, complemented
277	#1671	C₂×C₃₁	1	1	yes	characteristic, complemented
278	#1666	C₂×C₃₁	1	1	yes	characteristic, complemented
279	#1645	C₂×C₃₁	1	1	yes	characteristic, complemented
280	#1642	C₂×C₃₁	1	1	yes	characteristic, complemented
281	#1635	C₂×C₃₁	1	1	yes	characteristic, complemented
282	#1627	C₂×C₃₁	1	1	yes	characteristic, complemented
283	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
284	C₈×SD₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
285	C₂²×C₃₁	—	2	2	no	—
286	C₂²×C₃₁	—	4	1	no	—
287	C₂²×C₃₁	—	4	1	no	—
288	C₂²×C₃₁	—	4	1	no	—
289	C₂²×C₃₁	—	4	1	no	—
290	C₂²×C₃₁	—	4	1	no	—
291	C₂²×C₃₁	—	8	1	no	—
292	C₂²×C₃₁	—	8	1	no	—
293	C₂²×C₃₁	—	8	1	no	—
294	C₂²×C₃₁	D₄²	1	1	yes	characteristic
295	C₄×C₃₁	—	2	2	no	—
296	C₄×C₃₁	—	2	2	no	—
297	C₄×C₃₁	—	2	1	no	—
298	C₄×C₃₁	—	4	1	no	—
299	C₄×C₃₁	—	4	1	no	—
300	C₄×C₃₁	—	4	1	no	—
301	C₄×C₃₁	—	4	1	no	—
302	C₄×C₃₁	—	8	1	no	—
303	C₄×C₃₁	—	8	1	no	—
304	C₄×C₃₁	—	8	1	no	—
305	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
306	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
307	C₄×C₃₁	#577	1	1	yes	characteristic
308	C₄×C₃₁	#577	1	1	yes	characteristic
309	C₂×#135	—	2	1	no	—
310	Q₈²	C₂²×C₃₁	1	1	yes	characteristic
311	#550	C₂²×C₃₁	1	1	yes	characteristic
312	D₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
313	#547	C₂²×C₃₁	1	1	yes	characteristic
314	#498	—	2	1	no	—
315	#498	C₂²×C₃₁	1	1	yes	characteristic
316	#492	C₂²×C₃₁	1	1	yes	characteristic
317	#487	C₂²×C₃₁	1	1	yes	characteristic
318	#487	C₂²×C₃₁	1	1	yes	characteristic
319	#486	—	2	1	no	—
320	#486	C₂²×C₃₁	1	1	yes	characteristic
321	#486	C₂²×C₃₁	1	1	yes	characteristic
322	#483	—	2	1	no	—
323	#476	—	2	1	no	—
324	#476	—	2	1	no	—
325	#476	C₂²×C₃₁	1	1	yes	characteristic
326	#476	C₂²×C₃₁	1	1	yes	characteristic
327	#475	—	2	1	no	—
328	#475	C₂²×C₃₁	1	1	yes	characteristic
329	#475	C₂²×C₃₁	1	1	yes	characteristic
330	#465	—	2	1	no	—
331	#465	—	2	1	no	—
332	#462	C₂²×C₃₁	1	1	yes	characteristic
333	#461	C₂²×C₃₁	1	1	yes	characteristic
334	#460	—	2	1	no	—
335	#460	C₂²×C₃₁	1	1	yes	characteristic
336	#459	C₂²×C₃₁	1	1	yes	characteristic
337	#452	—	2	1	no	—
338	#449	—	2	1	no	—
339	#448	—	2	1	no	—
340	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
341	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
342	C₄×SD₁₆	—	2	1	no	—
343	C₄×SD₁₆	—	2	1	no	—
344	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
345	C₄×SD₁₆	C₂²×C₃₁	1	1	yes	characteristic
346	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
347	C₈×D₄	C₂²×C₃₁	1	1	yes	characteristic
348	C₂×#103	—	2	1	no	—
349	#334	C₂²×C₃₁	1	1	yes	characteristic
350	#334	C₂²×C₃₁	1	1	yes	characteristic
351	#333	C₂²×C₃₁	1	1	yes	characteristic
352	#332	C₂²×C₃₁	1	1	yes	characteristic
353	#332	C₂²×C₃₁	1	1	yes	characteristic
354	#331	C₂²×C₃₁	1	1	yes	characteristic
355	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
356	Q₈⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
357	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
358	D₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
359	C₈²	C₂²×C₃₁	1	1	yes	characteristic
360	C₂×C₃₁	—	4	2	no	—
361	C₂×C₃₁	—	8	1	no	—
362	C₂×C₃₁	—	16	1	no	—
363	C₂×C₃₁	D₄×SD₁₆	1	1	yes	characteristic
364	C₂×C₃₁	#3282	1	1	yes	characteristic
365	C₂×C₃₁	#3290	1	1	yes	characteristic
366	C₂×#41	—	2	1	no	—
367	C₂×#41	—	2	1	no	—
368	C₂²×Q₈	—	2	1	no	—
369	#135	—	4	4	no	—
370	C₂×Q₁₆	—	2	1	no	—
371	C₂×SD₁₆	—	2	1	no	—
372	C₂×SD₁₆	—	2	1	no	—
373	C₂×SD₁₆	—	2	1	no	—
374	C₂×SD₁₆	—	2	1	no	—
375	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
376	C₂×SD₁₆	D₄×C₃₁	1	1	yes	characteristic
377	C₂×D₈	—	2	1	no	—
378	C₂²×C₈	—	2	1	no	—
379	C₂²×C₈	—	2	1	no	—
380	C₄⋊Q₈	—	2	1	no	—
381	C₄⋊Q₈	—	2	1	no	—
382	C₄⋊Q₈	—	2	1	no	—
383	C₄⋊Q₈	—	4	1	no	—
384	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
385	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
386	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
387	#127	—	2	1	no	—
388	#127	C₂³×C₃₁	1	1	yes	characteristic
389	#123	—	4	1	no	—
390	C₂²⋊Q₈	—	2	1	no	—
391	C₂²⋊Q₈	—	2	1	no	—
392	C₂²⋊Q₈	—	4	1	no	—
393	#121	—	2	1	no	—
394	#121	—	2	1	no	—
395	#121	—	4	1	no	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
399	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
400	C₄×D₄	—	2	1	no	—
401	C₄×D₄	—	2	1	no	—
402	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
403	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
404	C₂×C₄⋊C₄	—	2	1	no	—
405	#106	—	2	1	no	—
406	#106	—	2	1	no	—
407	#106	—	2	1	no	—
408	#106	—	2	1	no	—
409	#106	—	2	1	no	—
410	#106	—	2	1	no	—
411	#106	D₄×C₃₁	1	1	yes	characteristic
412	#106	D₄×C₃₁	1	1	yes	characteristic
413	C₄⋊C₈	—	2	1	no	—
414	C₄⋊C₈	—	2	1	no	—
415	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
416	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
417	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
418	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
419	#103	—	2	1	no	—
420	#103	—	2	1	no	—
421	#103	—	2	1	no	—
422	#103	—	2	1	no	—
423	#103	—	2	1	no	—
424	#103	—	2	1	no	—
425	#103	—	4	1	no	—
426	#103	D₄×C₃₁	1	1	yes	characteristic
427	#103	D₄×C₃₁	1	1	yes	characteristic
428	#102	—	2	1	no	—
429	#102	—	2	1	no	—
430	#102	—	2	1	no	—
431	#102	—	2	1	no	—
432	#102	—	2	1	no	—
433	#102	—	2	1	no	—
434	#102	—	4	1	no	—
435	#102	D₄×C₃₁	1	1	yes	characteristic
436	#102	D₄×C₃₁	1	1	yes	characteristic
437	C₂²⋊C₈	—	2	1	no	—
438	C₂²⋊C₈	—	2	1	no	—
439	C₄×C₈	—	2	1	no	—
440	C₄×C₈	—	2	1	no	—
441	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
442	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
443	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
444	C₃₁	#19981	1	1	yes	characteristic, direct product factor, Sylow
445	#41	—	4	2	no	—
446	#41	—	4	2	no	—
447	#41	—	8	1	no	—
448	#41	—	8	1	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	4	1	no	—
453	C₂×Q₈	—	4	1	no	—
454	C₂×Q₈	—	4	2	no	—
455	C₂×Q₈	—	8	1	no	—
456	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
457	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	—	2	1	no	—
460	C₂×D₄	—	2	1	no	—
461	C₂×D₄	—	4	1	no	—
462	C₂×D₄	—	4	1	no	—
463	C₂×D₄	—	4	1	no	—
464	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
465	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	—	2	1	no	—
468	C₂²×C₄	—	4	1	no	—
469	C₂²×C₄	—	4	1	no	—
470	C₂²×C₄	—	4	1	no	—
471	Q₁₆	—	4	2	no	—
472	SD₁₆	—	2	2	no	—
473	SD₁₆	—	4	2	no	—
474	SD₁₆	—	4	2	no	—
475	SD₁₆	—	8	1	no	—
476	SD₁₆	—	8	1	no	—
477	SD₁₆	SD₁₆×C₃₁	1	4	yes	complemented
478	D₈	—	4	2	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	—	2	1	no	—
484	C₂×C₈	—	4	2	no	—
485	C₂×C₈	—	8	1	no	—
486	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
487	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
488	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
489	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	2	1	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	4	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	—	4	1	no	—
501	C₄⋊C₄	—	4	1	no	—
502	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
503	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
504	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
505	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
506	C₂²⋊C₄	—	2	1	no	—
507	C₂²⋊C₄	—	2	1	no	—
508	C₂²⋊C₄	—	4	1	no	—
509	C₂²⋊C₄	—	4	1	no	—
510	C₂²⋊C₄	—	4	1	no	—
511	C₄²	—	2	1	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	4	1	no	—
514	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
515	C₂³	—	2	1	no	—
516	C₂³	—	2	1	no	—
517	C₂³	—	4	1	no	—
518	Q₈	—	2	1	no	—
519	Q₈	—	2	1	no	—
520	Q₈	—	4	1	no	—
521	Q₈	—	4	1	no	—
522	Q₈	—	4	1	no	—
523	Q₈	—	4	1	no	—
524	Q₈	—	8	2	no	—
525	Q₈	—	8	1	no	—
526	Q₈	—	16	1	no	—
527	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
528	Q₈	C₃₁×#135	1	2	yes	complemented
529	D₄	—	2	1	no	—
530	D₄	—	2	1	no	—
531	D₄	—	2	1	no	—
532	D₄	—	4	1	no	—
533	D₄	—	4	1	no	—
534	D₄	—	4	1	no	—
535	D₄	—	4	1	no	—
536	D₄	—	8	2	no	—
537	D₄	—	8	1	no	—
538	D₄	—	8	2	no	—
539	D₄	—	16	1	no	—
540	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	1	no	—
543	C₂×C₄	—	2	1	no	—
544	C₂×C₄	—	2	1	no	—
545	C₂×C₄	—	2	1	no	—
546	C₂×C₄	—	2	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	8	2	no	—
555	C₂×C₄	—	8	2	no	—
556	C₂×C₄	—	8	2	no	—
557	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
560	C₈	—	2	1	no	—
561	C₈	—	2	1	no	—
562	C₈	—	2	1	no	—
563	C₈	—	4	1	no	—
564	C₈	—	4	1	no	—
565	C₈	—	4	1	no	—
566	C₈	—	4	1	no	—
567	C₈	—	4	1	no	—
568	C₈	C₂×SD₁₆×C₃₁	1	2	yes	—
569	C₂²	—	2	2	no	—
570	C₂²	—	4	1	no	—
571	C₂²	—	4	1	no	—
572	C₂²	—	4	1	no	—
573	C₂²	—	4	1	no	—
574	C₂²	—	4	1	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	—	8	1	no	—
578	C₂²	D₄²×C₃₁	1	1	yes	characteristic
579	C₄	—	2	2	no	—
580	C₄	—	2	2	no	—
581	C₄	—	2	1	no	—
582	C₄	—	4	1	no	—
583	C₄	—	4	1	no	—
584	C₄	—	4	1	no	—
585	C₄	—	4	1	no	—
586	C₄	—	8	1	no	—
587	C₄	—	8	1	no	—
588	C₄	—	8	1	no	—
589	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
590	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
591	C₄	C₃₁×#577	1	1	yes	characteristic
592	C₄	C₃₁×#577	1	1	yes	characteristic
593	C₂	—	4	2	no	—
594	C₂	—	8	1	no	—
595	C₂	—	16	1	no	—
596	C₂	D₄×SD₁₆×C₃₁	1	1	yes	characteristic
597	C₂	C₃₁×#3282	1	1	yes	characteristic
598	C₂	C₃₁×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3284	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×SD₁₆×C₃₁	C₂	semidirect
2. C₃₁×#3362	C₂	semidirect
3. C₃₁×#3298	C₂	semidirect
5. C₃₁×#1708	C₂	semidirect
6. C₃₁×#1680	C₂	semidirect
7. C₃₁×#1677	C₂	semidirect
8. C₃₁×#1671	C₂	semidirect
9. C₃₁×#1666	C₂	semidirect
10. C₃₁×#1645	C₂	semidirect
11. C₃₁×#1642	C₂	semidirect
12. C₃₁×#1635	C₂	semidirect
13. C₃₁×#1627	C₂	semidirect
14. C₈×SD₁₆×C₃₁	C₂	semidirect
15. C₈×SD₁₆×C₃₁	C₂	semidirect
177. SD₁₆×C₃₁	SD₁₆	semidirect
215. #19981	C₃₁	direct (N × H)
229. Q₈×C₃₁	#135	semidirect
270. Q₈×SD₁₆	C₂×C₃₁	semidirect
271. #3362	C₂×C₃₁	semidirect
272. #3298	C₂×C₃₁	semidirect
274. #1708	C₂×C₃₁	semidirect
275. #1680	C₂×C₃₁	semidirect
276. #1677	C₂×C₃₁	semidirect
277. #1671	C₂×C₃₁	semidirect
278. #1666	C₂×C₃₁	semidirect
279. #1645	C₂×C₃₁	semidirect
280. #1642	C₂×C₃₁	semidirect
281. #1635	C₂×C₃₁	semidirect
282. #1627	C₂×C₃₁	semidirect
283. C₈×SD₁₆	C₂×C₃₁	semidirect
284. C₈×SD₁₆	C₂×C₃₁	semidirect
444. C₃₁	#19981	direct (N × H)
477. SD₁₆	SD₁₆×C₃₁	semidirect
528. Q₈	C₃₁×#135	semidirect

Maximal subgroups

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

subgroup	index
Q₈×SD₁₆×C₃₁	2
C₃₁×#3362	2
C₃₁×#3298	2
C₃₁×#3284	2
C₃₁×#1708	2
C₃₁×#1680	2
C₃₁×#1677	2
C₃₁×#1671	2
C₃₁×#1666	2
C₃₁×#1645	2
C₃₁×#1642	2
C₃₁×#1635	2
C₃₁×#1627	2
C₈×SD₁₆×C₃₁	2
#19981	31

Subgroups — C31×#19981 #21476740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19981 #21476740