Subgroups — C₃₁×#61373 #21518132

Order 7936.

1778 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3503	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3386	C₂	1	4	yes	maximal, complemented
3	C₃₁×#3385	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3383	C₂	1	2	yes	maximal, complemented
5	Q₈×Q₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3380	C₂	1	2	yes	maximal, complemented
7	Q₈×D₈×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₃₁×#3164	C₂	1	2	yes	maximal, complemented
9	C₂×C₃₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#3083	C₂	1	4	yes	maximal, complemented
11	C₃₁×#3079	C₂	1	2	yes	maximal, complemented
12	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#2951	C₂	1	2	yes	maximal
14	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic
15	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
16	Q₈²×C₃₁	C₂²	1	4	yes	—
17	C₃₁×#557	C₂²	1	4	yes	—
18	C₃₁×#554	—	2	4	no	—
19	C₃₁×#554	C₂²	1	4	yes	—
20	D₄×Q₈×C₃₁	—	2	4	no	—
21	D₄×Q₈×C₃₁	C₂²	1	4	yes	—
22	C₃₁×#544	—	2	4	no	—
23	C₃₁×#544	C₂²	1	4	yes	—
24	C₃₁×#533	—	2	2	no	—
25	C₃₁×#533	C₂²	1	1	yes	characteristic
26	C₂×C₃₁×C₄⋊Q₈	—	2	2	no	—
27	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
28	C₃₁×#520	C₂²	1	4	yes	—
29	C₄×#41×C₃₁	C₂²	1	2	yes	—
30	C₃₁×#501	C₂²	1	8	yes	—
31	C₃₁×#500	C₂²	1	4	yes	—
32	C₃₁×#500	C₂²	1	2	yes	—
33	C₃₁×#500	C₂²	1	2	yes	—
34	C₃₁×#500	C₂²	1	2	yes	—
35	C₃₁×#500	C₂²	1	2	yes	—
36	C₃₁×#499	C₂²	1	4	yes	—
37	C₃₁×#479	C₂²	1	8	yes	—
38	C₃₁×#478	C₂²	1	8	yes	—
39	C₃₁×#477	C₂²	1	4	yes	—
40	C₃₁×#477	C₂²	1	8	yes	—
41	C₃₁×#476	C₂²	1	4	yes	—
42	C₃₁×#475	C₂²	1	4	yes	—
43	C₃₁×#474	C₂²	1	4	yes	—
44	C₃₁×#474	C₂²	1	8	yes	—
45	C₃₁×#446	C₂²	1	2	yes	—
46	C₃₁×#446	C₂²	1	2	yes	—
47	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
48	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
49	C₃₁×#440	C₂²	1	8	yes	—
50	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
51	C₄×Q₁₆×C₃₁	C₂²	1	4	yes	—
52	C₄×Q₁₆×C₃₁	C₂²	1	1	yes	characteristic
53	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
54	C₄×SD₁₆×C₃₁	C₂²	1	2	yes	—
55	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
56	C₄×D₈×C₃₁	C₂²	1	4	yes	—
57	C₄×D₈×C₃₁	C₂²	1	1	yes	characteristic
58	C₃₁×#428	C₂²	1	4	yes	—
59	C₃₁×#427	C₂²	1	1	yes	characteristic
60	C₂×C₃₁×#107	C₂²	1	2	yes	—
61	C₂×C₃₁×#107	C₂²	1	2	yes	—
62	C₃₁×#424	C₂²	1	2	yes	—
63	C₃₁×#423	C₂²	1	1	yes	characteristic
64	C₃₁×#417	C₂²	1	4	yes	—
65	C₃₁×#416	C₂²	1	2	yes	—
66	C₃₁×#405	C₂²	1	2	yes	—
67	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
68	C₂×#41×C₃₁	C₂³	1	2	yes	—
69	C₂²×Q₈×C₃₁	—	2	4	no	—
70	C₂²×Q₈×C₃₁	—	2	2	no	—
71	C₃₁×#135	Q₈	1	8	yes	complemented
72	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
73	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
74	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
76	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×C₄⋊Q₈	—	2	4	no	—
79	C₃₁×C₄⋊Q₈	—	2	4	no	—
80	C₃₁×C₄⋊Q₈	—	2	4	no	—
81	C₃₁×C₄⋊Q₈	—	2	8	no	—
82	C₃₁×C₄⋊Q₈	—	2	2	no	—
83	C₃₁×C₄⋊Q₈	—	2	2	no	—
84	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
85	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
86	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
87	C₃₁×C₄⋊Q₈	C₂³	1	8	yes	—
88	C₃₁×C₄⋊Q₈	C₂³	1	4	yes	—
89	C₃₁×#125	—	2	4	no	—
90	C₃₁×#125	—	2	8	no	—
91	C₃₁×#125	D₄	1	2	yes	—
92	C₃₁×#125	C₂³	1	8	yes	—
93	C₃₁×C₂²⋊Q₈	—	2	4	no	—
94	C₃₁×C₂²⋊Q₈	—	2	4	no	—
95	C₃₁×C₂²⋊Q₈	—	2	8	no	—
96	C₃₁×C₂²⋊Q₈	—	2	8	no	—
97	C₃₁×C₂²⋊Q₈	—	2	8	no	—
98	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
99	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
100	C₄×Q₈×C₃₁	—	2	4	no	—
101	C₄×Q₈×C₃₁	—	2	4	no	—
102	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
103	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
104	C₄×Q₈×C₃₁	C₂³	1	8	yes	—
105	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
106	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
107	C₄×D₄×C₃₁	—	2	2	no	—
108	C₄×D₄×C₃₁	—	2	2	no	—
109	C₄×D₄×C₃₁	—	2	8	no	—
110	C₄×D₄×C₃₁	C₂³	1	2	yes	—
111	C₄×D₄×C₃₁	C₂³	1	2	yes	—
112	C₄×D₄×C₃₁	C₂³	1	8	yes	—
113	C₃₁×#117	—	2	4	no	—
114	C₃₁×#117	—	2	2	no	—
115	C₃₁×#117	C₂³	1	2	yes	—
116	C₃₁×#117	C₂³	1	2	yes	—
117	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
118	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
119	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
120	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₂×C₄²×C₃₁	—	2	2	no	—
122	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
123	C₃₁×#107	C₂³	1	4	yes	—
124	C₃₁×#107	C₂³	1	4	yes	—
125	C₃₁×#107	C₂³	1	4	yes	—
126	C₃₁×#107	C₂³	1	4	yes	—
127	C₃₁×#107	C₂³	1	1	yes	characteristic
128	C₃₁×#107	C₂³	1	1	yes	characteristic
129	C₃₁×#107	C₂³	1	8	yes	—
130	C₃₁×#106	C₂³	1	2	yes	—
131	C₃₁×#106	C₂³	1	8	yes	—
132	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
133	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
134	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
135	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
136	C₃₁×#103	C₂³	1	2	yes	—
137	C₃₁×#103	C₂³	1	2	yes	—
138	C₃₁×#103	C₂³	1	8	yes	—
139	C₃₁×#102	C₂³	1	2	yes	—
140	C₃₁×#102	C₂³	1	2	yes	—
141	C₃₁×#102	C₂³	1	8	yes	—
142	C₃₁×#97	C₂³	1	4	yes	—
143	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
145	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
146	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
147	C₄×C₈×C₃₁	C₂³	1	4	yes	—
148	#41×C₃₁	—	2	4	no	—
149	#41×C₃₁	C₂×Q₈	1	8	yes	complemented
150	C₂×Q₈×C₃₁	—	2	4	no	—
151	C₂×Q₈×C₃₁	—	2	8	no	—
152	C₂×Q₈×C₃₁	—	2	4	no	—
153	C₂×Q₈×C₃₁	—	8	4	no	—
154	C₂×Q₈×C₃₁	—	8	8	no	—
155	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
157	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
158	C₂×D₄×C₃₁	—	2	2	no	—
159	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	2	8	no	—
165	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
168	Q₁₆×C₃₁	C₂×Q₈	1	4	yes	complemented
169	SD₁₆×C₃₁	C₂×Q₈	1	8	yes	complemented
170	D₈×C₃₁	C₂×Q₈	1	4	yes	complemented
171	M₁₆×C₃₁	—	2	4	no	—
172	C₂×C₈×C₃₁	—	2	2	no	—
173	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
174	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
177	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
178	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₄⋊C₄×C₃₁	—	2	4	no	—
180	C₄⋊C₄×C₃₁	—	2	4	no	—
181	C₄⋊C₄×C₃₁	—	2	8	no	—
182	C₄⋊C₄×C₃₁	—	2	4	no	—
183	C₄⋊C₄×C₃₁	—	2	8	no	—
184	C₄⋊C₄×C₃₁	—	2	2	no	—
185	C₄⋊C₄×C₃₁	—	2	2	no	—
186	C₄⋊C₄×C₃₁	—	2	8	no	—
187	C₄⋊C₄×C₃₁	—	2	8	no	—
188	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
189	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
190	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
191	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
192	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
193	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
194	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
195	C₄⋊C₄×C₃₁	C₂⁴	1	8	yes	—
196	C₄⋊C₄×C₃₁	C₂⁴	1	2	yes	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	2	no	—
199	C₂²⋊C₄×C₃₁	—	2	2	no	—
200	C₂²⋊C₄×C₃₁	—	2	8	no	—
201	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
202	C₄²×C₃₁	—	2	2	no	—
203	C₄²×C₃₁	—	2	2	no	—
204	C₄²×C₃₁	—	2	2	no	—
205	C₄²×C₃₁	—	2	8	no	—
206	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
207	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
208	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
209	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
210	C₄²×C₃₁	C₂⁴	1	4	yes	—
211	#61373	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
212	C₂³×C₃₁	—	2	2	no	—
213	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
214	Q₈×C₃₁	—	2	4	no	—
215	Q₈×C₃₁	—	2	4	no	—
216	Q₈×C₃₁	—	2	2	no	—
217	Q₈×C₃₁	—	4	8	no	—
218	Q₈×C₃₁	—	8	4	no	—
219	Q₈×C₃₁	—	8	8	no	—
220	Q₈×C₃₁	C₂²×Q₈	1	4	yes	—
221	D₄×C₃₁	—	2	8	no	—
222	D₄×C₃₁	—	2	2	no	—
223	D₄×C₃₁	C₂²×Q₈	1	4	yes	—
224	C₂×C₄×C₃₁	—	2	2	no	—
225	C₂×C₄×C₃₁	—	2	8	no	—
226	C₂×C₄×C₃₁	—	2	2	no	—
227	C₂×C₄×C₃₁	—	2	2	no	—
228	C₂×C₄×C₃₁	—	2	2	no	—
229	C₂×C₄×C₃₁	—	2	2	no	—
230	C₂×C₄×C₃₁	—	2	8	no	—
231	C₂×C₄×C₃₁	—	4	4	no	—
232	C₂×C₄×C₃₁	—	4	4	no	—
233	C₂×C₄×C₃₁	—	8	8	no	—
234	C₂×C₄×C₃₁	—	8	4	no	—
235	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
236	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
237	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
238	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
240	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
242	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
243	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
244	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
245	C₈×C₃₁	—	2	1	no	—
246	C₈×C₃₁	—	2	1	no	—
247	C₈×C₃₁	—	2	4	no	—
248	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
249	C₈×C₃₁	C₂²×Q₈	1	2	yes	—
250	#3503	C₂×C₃₁	1	2	yes	complemented
251	#3386	C₂×C₃₁	1	4	yes	complemented
252	#3385	C₂×C₃₁	1	4	yes	complemented
253	#3383	C₂×C₃₁	1	2	yes	complemented
254	Q₈×Q₁₆	C₂×C₃₁	1	2	yes	complemented
255	#3380	C₂×C₃₁	1	2	yes	complemented
256	Q₈×D₈	C₂×C₃₁	1	2	yes	complemented
257	#3164	C₂×C₃₁	1	2	yes	complemented
258	C₂×#500	C₂×C₃₁	1	1	yes	characteristic, complemented
259	#3083	C₂×C₃₁	1	4	yes	complemented
260	#3079	C₂×C₃₁	1	2	yes	complemented
261	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
262	#2951	C₂×C₃₁	1	2	yes	—
263	C₄×#135	C₂×C₃₁	1	1	yes	characteristic
264	C₂²×C₃₁	—	2	1	no	—
265	C₂²×C₃₁	—	2	1	no	—
266	C₂²×C₃₁	—	2	4	no	—
267	C₂²×C₃₁	—	4	2	no	—
268	C₂²×C₃₁	—	4	2	no	—
269	C₂²×C₃₁	D₄×Q₈	1	2	yes	—
270	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
271	C₄×C₃₁	—	2	4	no	—
272	C₄×C₃₁	—	2	1	no	—
273	C₄×C₃₁	—	2	4	no	—
274	C₄×C₃₁	—	2	1	no	—
275	C₄×C₃₁	—	2	4	no	—
276	C₄×C₃₁	—	4	2	no	—
277	C₄×C₃₁	—	4	8	no	—
278	C₄×C₃₁	—	4	2	no	—
279	C₄×C₃₁	D₄×Q₈	1	2	yes	—
280	C₄×C₃₁	#576	1	2	yes	—
281	C₄×C₃₁	#578	1	2	yes	—
282	C₄×C₃₁	C₂³×Q₈	1	1	yes	characteristic
283	C₄×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
284	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
285	Q₈²	C₂²×C₃₁	1	4	yes	—
286	#557	C₂²×C₃₁	1	4	yes	—
287	#554	—	2	4	no	—
288	#554	C₂²×C₃₁	1	4	yes	—
289	D₄×Q₈	—	2	4	no	—
290	D₄×Q₈	C₂²×C₃₁	1	4	yes	—
291	#544	—	2	4	no	—
292	#544	C₂²×C₃₁	1	4	yes	—
293	#533	—	2	2	no	—
294	#533	C₂²×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊Q₈	—	2	2	no	—
296	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
297	#520	C₂²×C₃₁	1	4	yes	—
298	C₄×#41	C₂²×C₃₁	1	2	yes	—
299	#501	C₂²×C₃₁	1	8	yes	—
300	#500	C₂²×C₃₁	1	4	yes	—
301	#500	C₂²×C₃₁	1	2	yes	—
302	#500	C₂²×C₃₁	1	2	yes	—
303	#500	C₂²×C₃₁	1	2	yes	—
304	#500	C₂²×C₃₁	1	2	yes	—
305	#499	C₂²×C₃₁	1	4	yes	—
306	#479	C₂²×C₃₁	1	8	yes	—
307	#478	C₂²×C₃₁	1	8	yes	—
308	#477	C₂²×C₃₁	1	4	yes	—
309	#477	C₂²×C₃₁	1	8	yes	—
310	#476	C₂²×C₃₁	1	4	yes	—
311	#475	C₂²×C₃₁	1	4	yes	—
312	#474	C₂²×C₃₁	1	4	yes	—
313	#474	C₂²×C₃₁	1	8	yes	—
314	#446	C₂²×C₃₁	1	2	yes	—
315	#446	C₂²×C₃₁	1	2	yes	—
316	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
317	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
318	#440	C₂²×C₃₁	1	8	yes	—
319	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
320	C₄×Q₁₆	C₂²×C₃₁	1	4	yes	—
321	C₄×Q₁₆	C₂²×C₃₁	1	1	yes	characteristic
322	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
323	C₄×SD₁₆	C₂²×C₃₁	1	2	yes	—
324	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
325	C₄×D₈	C₂²×C₃₁	1	4	yes	—
326	C₄×D₈	C₂²×C₃₁	1	1	yes	characteristic
327	#428	C₂²×C₃₁	1	4	yes	—
328	#427	C₂²×C₃₁	1	1	yes	characteristic
329	C₂×#107	C₂²×C₃₁	1	2	yes	—
330	C₂×#107	C₂²×C₃₁	1	2	yes	—
331	#424	C₂²×C₃₁	1	2	yes	—
332	#423	C₂²×C₃₁	1	1	yes	characteristic
333	#417	C₂²×C₃₁	1	4	yes	—
334	#416	C₂²×C₃₁	1	2	yes	—
335	#405	C₂²×C₃₁	1	2	yes	—
336	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
337	C₂×C₃₁	—	2	2	no	—
338	C₂×C₃₁	—	4	4	no	—
339	C₂×C₃₁	C₂×D₄×Q₈	1	1	yes	characteristic
340	C₂×C₃₁	C₂×#576	1	1	yes	characteristic
341	C₂×C₃₁	C₂×#578	1	1	yes	characteristic
342	C₂×#41	C₂³×C₃₁	1	2	yes	—
343	C₂²×Q₈	—	2	4	no	—
344	C₂²×Q₈	—	2	2	no	—
345	#135	Q₈×C₃₁	1	8	yes	complemented
346	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
347	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
348	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
349	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
350	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
351	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
352	C₄⋊Q₈	—	2	4	no	—
353	C₄⋊Q₈	—	2	4	no	—
354	C₄⋊Q₈	—	2	4	no	—
355	C₄⋊Q₈	—	2	8	no	—
356	C₄⋊Q₈	—	2	2	no	—
357	C₄⋊Q₈	—	2	2	no	—
358	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
359	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
360	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
361	C₄⋊Q₈	C₂³×C₃₁	1	4	yes	—
362	C₄⋊Q₈	C₂³×C₃₁	1	8	yes	—
363	#125	—	2	4	no	—
364	#125	—	2	8	no	—
365	#125	D₄×C₃₁	1	2	yes	—
366	#125	C₂³×C₃₁	1	8	yes	—
367	C₂²⋊Q₈	—	2	4	no	—
368	C₂²⋊Q₈	—	2	4	no	—
369	C₂²⋊Q₈	—	2	8	no	—
370	C₂²⋊Q₈	—	2	8	no	—
371	C₂²⋊Q₈	—	2	8	no	—
372	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
373	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
374	C₄×Q₈	—	2	4	no	—
375	C₄×Q₈	—	2	4	no	—
376	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
377	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
378	C₄×Q₈	C₂³×C₃₁	1	8	yes	—
379	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
380	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
381	C₄×D₄	—	2	2	no	—
382	C₄×D₄	—	2	2	no	—
383	C₄×D₄	—	2	8	no	—
384	C₄×D₄	C₂³×C₃₁	1	2	yes	—
385	C₄×D₄	C₂³×C₃₁	1	2	yes	—
386	C₄×D₄	C₂³×C₃₁	1	8	yes	—
387	#117	—	2	4	no	—
388	#117	—	2	2	no	—
389	#117	C₂³×C₃₁	1	2	yes	—
390	#117	C₂³×C₃₁	1	2	yes	—
391	C₂×C₄⋊C₄	—	2	2	no	—
392	C₂×C₄⋊C₄	—	2	4	no	—
393	C₂×C₄⋊C₄	—	2	4	no	—
394	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
395	C₂×C₄²	—	2	2	no	—
396	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
397	#107	C₂³×C₃₁	1	4	yes	—
398	#107	C₂³×C₃₁	1	4	yes	—
399	#107	C₂³×C₃₁	1	4	yes	—
400	#107	C₂³×C₃₁	1	4	yes	—
401	#107	C₂³×C₃₁	1	1	yes	characteristic
402	#107	C₂³×C₃₁	1	1	yes	characteristic
403	#107	C₂³×C₃₁	1	8	yes	—
404	#106	C₂³×C₃₁	1	2	yes	—
405	#106	C₂³×C₃₁	1	8	yes	—
406	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
407	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
408	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
409	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
410	#103	C₂³×C₃₁	1	2	yes	—
411	#103	C₂³×C₃₁	1	2	yes	—
412	#103	C₂³×C₃₁	1	8	yes	—
413	#102	C₂³×C₃₁	1	2	yes	—
414	#102	C₂³×C₃₁	1	2	yes	—
415	#102	C₂³×C₃₁	1	8	yes	—
416	#97	C₂³×C₃₁	1	4	yes	—
417	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
418	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
419	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
420	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
421	C₄×C₈	C₂³×C₃₁	1	4	yes	—
422	C₃₁	#61373	1	1	yes	characteristic, direct product factor, Sylow
423	#41	—	2	4	no	—
424	#41	C₂×Q₈×C₃₁	1	8	yes	complemented
425	C₂×Q₈	—	2	4	no	—
426	C₂×Q₈	—	2	8	no	—
427	C₂×Q₈	—	2	4	no	—
428	C₂×Q₈	—	8	4	no	—
429	C₂×Q₈	—	8	8	no	—
430	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
431	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
432	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
433	C₂×D₄	—	2	2	no	—
434	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	4	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	2	no	—
439	C₂²×C₄	—	2	8	no	—
440	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
441	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
443	Q₁₆	C₂×Q₈×C₃₁	1	4	yes	complemented
444	SD₁₆	C₂×Q₈×C₃₁	1	8	yes	complemented
445	D₈	C₂×Q₈×C₃₁	1	4	yes	complemented
446	M₁₆	—	2	4	no	—
447	C₂×C₈	—	2	2	no	—
448	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
449	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
450	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
451	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
452	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
453	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
454	C₄⋊C₄	—	2	4	no	—
455	C₄⋊C₄	—	2	4	no	—
456	C₄⋊C₄	—	2	8	no	—
457	C₄⋊C₄	—	2	4	no	—
458	C₄⋊C₄	—	2	8	no	—
459	C₄⋊C₄	—	2	2	no	—
460	C₄⋊C₄	—	2	2	no	—
461	C₄⋊C₄	—	2	8	no	—
462	C₄⋊C₄	—	2	8	no	—
463	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
464	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
465	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
466	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
467	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
468	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
469	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
470	C₄⋊C₄	C₂⁴×C₃₁	1	8	yes	—
471	C₄⋊C₄	C₂⁴×C₃₁	1	2	yes	—
472	C₂²⋊C₄	—	2	2	no	—
473	C₂²⋊C₄	—	2	2	no	—
474	C₂²⋊C₄	—	2	2	no	—
475	C₂²⋊C₄	—	2	8	no	—
476	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
477	C₄²	—	2	2	no	—
478	C₄²	—	2	2	no	—
479	C₄²	—	2	2	no	—
480	C₄²	—	2	8	no	—
481	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
482	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
483	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
484	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
485	C₄²	C₂⁴×C₃₁	1	4	yes	—
486	C₂³	—	2	2	no	—
487	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
488	Q₈	—	2	4	no	—
489	Q₈	—	2	4	no	—
490	Q₈	—	2	2	no	—
491	Q₈	—	4	8	no	—
492	Q₈	—	8	4	no	—
493	Q₈	—	8	8	no	—
494	Q₈	C₂²×Q₈×C₃₁	1	4	yes	—
495	D₄	—	2	8	no	—
496	D₄	—	2	2	no	—
497	D₄	C₂²×Q₈×C₃₁	1	4	yes	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	8	no	—
500	C₂×C₄	—	2	2	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	2	no	—
503	C₂×C₄	—	2	2	no	—
504	C₂×C₄	—	2	8	no	—
505	C₂×C₄	—	4	4	no	—
506	C₂×C₄	—	4	4	no	—
507	C₂×C₄	—	8	8	no	—
508	C₂×C₄	—	8	4	no	—
509	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
510	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
511	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
512	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
513	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
514	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
515	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
516	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
517	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
518	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
519	C₈	—	2	1	no	—
520	C₈	—	2	1	no	—
521	C₈	—	2	4	no	—
522	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
523	C₈	C₂²×Q₈×C₃₁	1	2	yes	—
524	C₂²	—	2	1	no	—
525	C₂²	—	2	1	no	—
526	C₂²	—	2	4	no	—
527	C₂²	—	4	2	no	—
528	C₂²	—	4	2	no	—
529	C₂²	D₄×Q₈×C₃₁	1	2	yes	—
530	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
531	C₄	—	2	4	no	—
532	C₄	—	2	1	no	—
533	C₄	—	2	4	no	—
534	C₄	—	2	1	no	—
535	C₄	—	2	4	no	—
536	C₄	—	4	2	no	—
537	C₄	—	4	8	no	—
538	C₄	—	4	2	no	—
539	C₄	D₄×Q₈×C₃₁	1	2	yes	—
540	C₄	C₃₁×#576	1	2	yes	—
541	C₄	C₃₁×#578	1	2	yes	—
542	C₄	C₂³×Q₈×C₃₁	1	1	yes	characteristic
543	C₄	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
544	C₂	—	2	2	no	—
545	C₂	—	4	4	no	—
546	C₂	C₂×D₄×Q₈×C₃₁	1	1	yes	characteristic
547	C₂	C₂×C₃₁×#576	1	1	yes	characteristic
548	C₂	C₂×C₃₁×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
prime radical	C₂×C₃₁×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3503	C₂	semidirect
2. C₃₁×#3386	C₂	semidirect
3. C₃₁×#3385	C₂	semidirect
4. C₃₁×#3383	C₂	semidirect
5. Q₈×Q₁₆×C₃₁	C₂	semidirect
6. C₃₁×#3380	C₂	semidirect
7. Q₈×D₈×C₃₁	C₂	semidirect
8. C₃₁×#3164	C₂	semidirect
9. C₂×C₃₁×#500	C₂	semidirect
10. C₃₁×#3083	C₂	semidirect
11. C₃₁×#3079	C₂	semidirect
12. C₃₁×#2960	C₂	semidirect
71. C₃₁×#135	Q₈	semidirect
149. #41×C₃₁	C₂×Q₈	semidirect
168. Q₁₆×C₃₁	C₂×Q₈	semidirect
169. SD₁₆×C₃₁	C₂×Q₈	semidirect
170. D₈×C₃₁	C₂×Q₈	semidirect
173. C₂×C₈×C₃₁	C₂×Q₈	semidirect
211. #61373	C₃₁	direct (N × H)
250. #3503	C₂×C₃₁	semidirect
251. #3386	C₂×C₃₁	semidirect
252. #3385	C₂×C₃₁	semidirect
253. #3383	C₂×C₃₁	semidirect
254. Q₈×Q₁₆	C₂×C₃₁	semidirect
255. #3380	C₂×C₃₁	semidirect
256. Q₈×D₈	C₂×C₃₁	semidirect
257. #3164	C₂×C₃₁	semidirect
258. C₂×#500	C₂×C₃₁	semidirect
259. #3083	C₂×C₃₁	semidirect
260. #3079	C₂×C₃₁	semidirect
261. #2960	C₂×C₃₁	semidirect
345. #135	Q₈×C₃₁	semidirect
422. C₃₁	#61373	direct (N × H)
424. #41	C₂×Q₈×C₃₁	semidirect
443. Q₁₆	C₂×Q₈×C₃₁	semidirect
444. SD₁₆	C₂×Q₈×C₃₁	semidirect
445. D₈	C₂×Q₈×C₃₁	semidirect
448. C₂×C₈	C₂×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3503	2
C₃₁×#3386	2
C₃₁×#3385	2
C₃₁×#3383	2
Q₈×Q₁₆×C₃₁	2
C₃₁×#3380	2
Q₈×D₈×C₃₁	2
C₃₁×#3164	2
C₂×C₃₁×#500	2
C₃₁×#3083	2
C₃₁×#3079	2
C₃₁×#2960	2
C₃₁×#2951	2
C₄×C₃₁×#135	2
#61373	31

Subgroups — C31×#61373 #21518132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61373 #21518132