Subgroups — C₃₁×#26040 #21482799

Order 7936.

914 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#474	C₂	1	2	yes	maximal
2	C₃₁×#2551	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₄⋊C₄×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2086	C₂	1	2	yes	maximal, complemented
5	C₃₁×#2057	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1947	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1939	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1862	C₂	1	2	yes	maximal
9	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1802	C₂	1	1	yes	maximal, characteristic
11	C₄×C₃₁×#102	C₂	1	1	yes	maximal, characteristic
12	C₂×C₃₁×C₄⋊Q₈	—	2	1	no	—
13	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
14	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₃₁	—	2	1	no	—
17	C₃₁×#474	—	2	8	no	—
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	—
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
25	C₂×C₃₁×#102	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
27	C₃₁×#395	—	2	2	no	—
28	C₃₁×#389	—	2	1	no	—
29	C₃₁×#389	C₂²	1	2	yes	—
30	C₃₁×#389	C₂²	1	2	yes	—
31	C₃₁×#385	—	2	2	no	—
32	C₃₁×#384	C₂²	1	1	yes	characteristic
33	C₃₁×#384	C₂²	1	1	yes	characteristic
34	C₃₁×#380	—	2	1	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
37	C₃₁×#340	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₃₁×#340	C₂²	1	2	yes	—
41	C₃₁×#336	C₂²	1	2	yes	—
42	C₂²×Q₈×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂³×C₄×C₃₁	—	2	1	no	—
45	C₂³×C₄×C₃₁	—	2	2	no	—
46	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
47	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×C₄⋊Q₈	—	2	4	no	—
50	C₃₁×C₄⋊Q₈	—	4	2	no	—
51	C₃₁×C₄⋊Q₈	—	4	4	no	—
52	C₄×D₄×C₃₁	—	2	4	no	—
53	C₄×D₄×C₃₁	—	2	2	no	—
54	C₄×D₄×C₃₁	—	4	2	no	—
55	C₄×D₄×C₃₁	Q₈	1	4	yes	complemented
56	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
59	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
60	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	—	4	4	no	—
64	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
68	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
69	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
71	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	—	2	4	no	—
77	C₃₁×#107	—	2	4	no	—
78	C₃₁×#107	—	4	2	no	—
79	C₃₁×C₄⋊C₈	—	2	4	no	—
80	C₃₁×C₄⋊C₈	—	2	2	no	—
81	C₃₁×#102	—	2	4	no	—
82	C₃₁×#102	D₄	1	4	yes	—
83	C₃₁×#102	Q₈	1	4	yes	complemented
84	C₄×C₈×C₃₁	—	2	2	no	—
85	C₃₁×#95	—	2	1	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	D₄	1	2	yes	—
89	C₂⁴×C₃₁	—	2	1	no	—
90	C₂×Q₈×C₃₁	—	4	4	no	—
91	C₂×Q₈×C₃₁	—	4	4	no	—
92	C₂×Q₈×C₃₁	—	8	2	no	—
93	C₂×D₄×C₃₁	—	2	1	no	—
94	C₂×D₄×C₃₁	—	2	1	no	—
95	C₂×D₄×C₃₁	—	2	1	no	—
96	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
97	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
98	C₂×D₄×C₃₁	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₃₁	—	2	1	no	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	4	no	—
103	C₂²×C₄×C₃₁	—	2	2	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	4	4	no	—
106	C₂²×C₄×C₃₁	—	4	2	no	—
107	C₂²×C₄×C₃₁	—	4	2	no	—
108	C₂²×C₄×C₃₁	—	4	4	no	—
109	C₂²×C₄×C₃₁	—	8	2	no	—
110	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
115	C₂×C₈×C₃₁	—	2	1	no	—
116	C₂×C₈×C₃₁	—	2	1	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	2	no	—
120	C₂×C₈×C₃₁	—	2	2	no	—
121	C₂×C₈×C₃₁	—	2	2	no	—
122	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
123	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	1	no	—
126	C₄⋊C₄×C₃₁	—	2	1	no	—
127	C₄⋊C₄×C₃₁	—	2	1	no	—
128	C₄⋊C₄×C₃₁	—	2	2	no	—
129	C₄⋊C₄×C₃₁	—	2	2	no	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	4	4	no	—
133	C₄⋊C₄×C₃₁	—	4	4	no	—
134	C₄⋊C₄×C₃₁	—	4	2	no	—
135	C₄⋊C₄×C₃₁	—	4	4	no	—
136	C₄⋊C₄×C₃₁	—	4	2	no	—
137	C₄⋊C₄×C₃₁	—	8	1	no	—
138	C₄⋊C₄×C₃₁	—	8	1	no	—
139	C₄⋊C₄×C₃₁	D₈	1	4	yes	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
141	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
142	C₄⋊C₄×C₃₁	C₂×Q₈	1	2	yes	—
143	C₂²⋊C₄×C₃₁	—	2	4	no	—
144	C₂²⋊C₄×C₃₁	—	4	4	no	—
145	C₄²×C₃₁	—	2	2	no	—
146	C₄²×C₃₁	—	2	2	no	—
147	C₄²×C₃₁	—	4	2	no	—
148	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
149	C₄²×C₃₁	#41	1	2	yes	—
150	#26040	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
151	C₂³×C₃₁	—	2	2	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	4	1	no	—
155	C₂³×C₃₁	—	4	1	no	—
156	C₂³×C₃₁	—	4	1	no	—
157	C₂³×C₃₁	—	4	1	no	—
158	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
159	Q₈×C₃₁	—	8	8	no	—
160	D₄×C₃₁	—	2	2	no	—
161	D₄×C₃₁	—	2	2	no	—
162	D₄×C₃₁	—	2	2	no	—
163	D₄×C₃₁	C₄⋊Q₈	1	4	yes	complemented
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	1	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	1	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	2	no	—
171	C₂×C₄×C₃₁	—	2	2	no	—
172	C₂×C₄×C₃₁	—	2	2	no	—
173	C₂×C₄×C₃₁	—	2	2	no	—
174	C₂×C₄×C₃₁	—	4	1	no	—
175	C₂×C₄×C₃₁	—	4	1	no	—
176	C₂×C₄×C₃₁	—	4	1	no	—
177	C₂×C₄×C₃₁	—	4	1	no	—
178	C₂×C₄×C₃₁	—	4	1	no	—
179	C₂×C₄×C₃₁	—	4	1	no	—
180	C₂×C₄×C₃₁	—	4	8	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	4	2	no	—
183	C₂×C₄×C₃₁	—	4	2	no	—
184	C₂×C₄×C₃₁	—	4	2	no	—
185	C₂×C₄×C₃₁	—	4	2	no	—
186	C₂×C₄×C₃₁	—	4	2	no	—
187	C₂×C₄×C₃₁	—	8	8	no	—
188	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
189	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
190	C₂×C₄×C₃₁	C₂×D₈	1	4	yes	—
191	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
192	C₂×C₄×C₃₁	#137	1	2	yes	—
193	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
197	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
199	C₈×C₃₁	—	2	2	no	—
200	C₈×C₃₁	—	2	2	no	—
201	C₈×C₃₁	—	4	2	no	—
202	C₂×#474	C₂×C₃₁	1	2	yes	—
203	#2551	C₂×C₃₁	1	1	yes	characteristic, complemented
204	D₄×C₄⋊C₄	C₂×C₃₁	1	1	yes	characteristic
205	#2086	C₂×C₃₁	1	2	yes	complemented
206	#2057	C₂×C₃₁	1	2	yes	complemented
207	#1947	C₂×C₃₁	1	1	yes	characteristic, complemented
208	#1939	C₂×C₃₁	1	1	yes	characteristic, complemented
209	#1862	C₂×C₃₁	1	2	yes	—
210	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1802	C₂×C₃₁	1	1	yes	characteristic
212	C₄×#102	C₂×C₃₁	1	1	yes	characteristic
213	C₂²×C₃₁	—	2	4	no	—
214	C₂²×C₃₁	—	4	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	2	no	—
218	C₂²×C₃₁	—	4	2	no	—
219	C₂²×C₃₁	—	4	2	no	—
220	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
221	C₂²×C₃₁	D₄²	1	1	yes	characteristic
222	C₂²×C₃₁	D₄×Q₈	1	1	yes	characteristic
223	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
224	C₂²×C₃₁	C₂×#137	1	1	yes	characteristic
225	C₂²×C₃₁	#576	1	1	yes	characteristic
226	C₂²×C₃₁	#578	1	1	yes	characteristic
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	4	no	—
229	C₄×C₃₁	—	4	2	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	2	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	8	2	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	#474	1	4	yes	complemented
236	C₄×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
237	C₄×C₃₁	#544	1	1	yes	characteristic
238	C₄×C₃₁	#547	1	1	yes	characteristic
239	C₄×C₃₁	#554	1	1	yes	characteristic
240	C₂×C₄⋊Q₈	—	2	1	no	—
241	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
242	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
243	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
244	C₂²×C₄⋊C₄	—	2	1	no	—
245	#474	—	2	8	no	—
246	C₂×#107	C₂²×C₃₁	1	2	yes	—
247	C₂×#107	C₂²×C₃₁	1	2	yes	—
248	C₂×#107	C₂²×C₃₁	1	2	yes	—
249	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
251	C₂×#102	C₂²×C₃₁	1	2	yes	—
252	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
253	C₂×#102	C₂²×C₃₁	1	1	yes	characteristic
254	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
255	#395	—	2	2	no	—
256	#389	—	2	1	no	—
257	#389	C₂²×C₃₁	1	2	yes	—
258	#389	C₂²×C₃₁	1	2	yes	—
259	#385	—	2	2	no	—
260	#384	C₂²×C₃₁	1	1	yes	characteristic
261	#384	C₂²×C₃₁	1	1	yes	characteristic
262	#380	—	2	1	no	—
263	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
264	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
265	#340	C₂²×C₃₁	1	1	yes	characteristic
266	#340	C₂²×C₃₁	1	2	yes	—
267	#340	C₂²×C₃₁	1	1	yes	characteristic
268	#340	C₂²×C₃₁	1	2	yes	—
269	#336	C₂²×C₃₁	1	2	yes	—
270	C₂×C₃₁	—	4	4	no	—
271	C₂×C₃₁	#2440	1	1	yes	characteristic
272	C₂×C₃₁	C₂×#474	1	1	yes	characteristic
273	C₂×C₃₁	#3083	1	1	yes	characteristic
274	C₂×C₃₁	#3276	1	1	yes	characteristic
275	C₂×C₃₁	D₄×D₈	1	1	yes	characteristic
276	C₂×C₃₁	Q₈×D₈	1	1	yes	characteristic
277	C₂×C₃₁	#3385	1	1	yes	characteristic
278	C₂²×Q₈	—	2	2	no	—
279	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂³×C₄	—	2	1	no	—
281	C₂³×C₄	—	2	2	no	—
282	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
283	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
284	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
285	C₄⋊Q₈	—	2	4	no	—
286	C₄⋊Q₈	—	4	2	no	—
287	C₄⋊Q₈	—	4	4	no	—
288	C₄×D₄	—	2	4	no	—
289	C₄×D₄	—	2	2	no	—
290	C₄×D₄	—	4	2	no	—
291	C₄×D₄	Q₈×C₃₁	1	4	yes	complemented
292	C₂×C₄⋊C₄	—	2	1	no	—
293	C₂×C₄⋊C₄	—	2	2	no	—
294	C₂×C₄⋊C₄	—	2	1	no	—
295	C₂×C₄⋊C₄	—	2	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	—	2	1	no	—
298	C₂×C₄⋊C₄	—	2	1	no	—
299	C₂×C₄⋊C₄	—	4	4	no	—
300	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
301	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
302	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
303	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
304	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
305	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×C₂²⋊C₄	—	2	1	no	—
308	C₂×C₂²⋊C₄	—	2	2	no	—
309	C₂×C₄²	—	2	1	no	—
310	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
311	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
312	#107	—	2	4	no	—
313	#107	—	2	4	no	—
314	#107	—	4	2	no	—
315	C₄⋊C₈	—	2	4	no	—
316	C₄⋊C₈	—	2	2	no	—
317	#102	—	2	4	no	—
318	#102	D₄×C₃₁	1	4	yes	—
319	#102	Q₈×C₃₁	1	4	yes	complemented
320	C₄×C₈	—	2	2	no	—
321	#95	—	2	1	no	—
322	#95	—	2	2	no	—
323	#95	—	2	2	no	—
324	#95	D₄×C₃₁	1	2	yes	—
325	C₃₁	#26040	1	1	yes	characteristic, direct product factor, Sylow
326	C₂⁴	—	2	1	no	—
327	C₂×Q₈	—	4	4	no	—
328	C₂×Q₈	—	4	4	no	—
329	C₂×Q₈	—	8	2	no	—
330	C₂×D₄	—	2	1	no	—
331	C₂×D₄	—	2	1	no	—
332	C₂×D₄	—	2	1	no	—
333	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
334	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
335	C₂×D₄	C₂×Q₈×C₃₁	1	2	yes	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	4	4	no	—
343	C₂²×C₄	—	4	2	no	—
344	C₂²×C₄	—	4	2	no	—
345	C₂²×C₄	—	4	4	no	—
346	C₂²×C₄	—	8	2	no	—
347	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	1	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	2	no	—
357	C₂×C₈	—	2	2	no	—
358	C₂×C₈	—	2	2	no	—
359	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
360	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	2	1	no	—
363	C₄⋊C₄	—	2	1	no	—
364	C₄⋊C₄	—	2	1	no	—
365	C₄⋊C₄	—	2	2	no	—
366	C₄⋊C₄	—	2	2	no	—
367	C₄⋊C₄	—	2	2	no	—
368	C₄⋊C₄	—	2	2	no	—
369	C₄⋊C₄	—	4	4	no	—
370	C₄⋊C₄	—	4	4	no	—
371	C₄⋊C₄	—	4	2	no	—
372	C₄⋊C₄	—	4	4	no	—
373	C₄⋊C₄	—	4	2	no	—
374	C₄⋊C₄	—	8	1	no	—
375	C₄⋊C₄	—	8	1	no	—
376	C₄⋊C₄	D₈×C₃₁	1	4	yes	—
377	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
378	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
379	C₄⋊C₄	C₂×Q₈×C₃₁	1	2	yes	—
380	C₂²⋊C₄	—	2	4	no	—
381	C₂²⋊C₄	—	4	4	no	—
382	C₄²	—	2	2	no	—
383	C₄²	—	2	2	no	—
384	C₄²	—	4	2	no	—
385	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
386	C₄²	#41×C₃₁	1	2	yes	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	2	no	—
389	C₂³	—	2	2	no	—
390	C₂³	—	4	1	no	—
391	C₂³	—	4	1	no	—
392	C₂³	—	4	1	no	—
393	C₂³	—	4	1	no	—
394	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
395	Q₈	—	8	8	no	—
396	D₄	—	2	2	no	—
397	D₄	—	2	2	no	—
398	D₄	—	2	2	no	—
399	D₄	C₃₁×C₄⋊Q₈	1	4	yes	complemented
400	C₂×C₄	—	2	2	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	2	2	no	—
405	C₂×C₄	—	2	2	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	2	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	4	1	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	1	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	8	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	—	4	2	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	2	no	—
422	C₂×C₄	—	4	2	no	—
423	C₂×C₄	—	8	8	no	—
424	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
425	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
426	C₂×C₄	C₂×D₈×C₃₁	1	4	yes	—
427	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
428	C₂×C₄	C₃₁×#137	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
431	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
432	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
433	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
434	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
435	C₈	—	2	2	no	—
436	C₈	—	2	2	no	—
437	C₈	—	4	2	no	—
438	C₂²	—	2	4	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	2	no	—
441	C₂²	—	4	2	no	—
442	C₂²	—	4	2	no	—
443	C₂²	—	4	2	no	—
444	C₂²	—	4	2	no	—
445	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	1	yes	characteristic
446	C₂²	D₄²×C₃₁	1	1	yes	characteristic
447	C₂²	D₄×Q₈×C₃₁	1	1	yes	characteristic
448	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
449	C₂²	C₂×C₃₁×#137	1	1	yes	characteristic
450	C₂²	C₃₁×#576	1	1	yes	characteristic
451	C₂²	C₃₁×#578	1	1	yes	characteristic
452	C₄	—	2	2	no	—
453	C₄	—	2	4	no	—
454	C₄	—	4	2	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	2	no	—
457	C₄	—	4	2	no	—
458	C₄	—	8	2	no	—
459	C₄	—	8	2	no	—
460	C₄	C₃₁×#474	1	4	yes	complemented
461	C₄	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
462	C₄	C₃₁×#544	1	1	yes	characteristic
463	C₄	C₃₁×#547	1	1	yes	characteristic
464	C₄	C₃₁×#554	1	1	yes	characteristic
465	C₂	—	4	4	no	—
466	C₂	C₃₁×#2440	1	1	yes	characteristic
467	C₂	C₂×C₃₁×#474	1	1	yes	characteristic
468	C₂	C₃₁×#3083	1	1	yes	characteristic
469	C₂	C₃₁×#3276	1	1	yes	characteristic
470	C₂	D₄×D₈×C₃₁	1	1	yes	characteristic
471	C₂	Q₈×D₈×C₃₁	1	1	yes	characteristic
472	C₂	C₃₁×#3385	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2551	C₂	semidirect
4. C₃₁×#2086	C₂	semidirect
5. C₃₁×#2057	C₂	semidirect
6. C₃₁×#1947	C₂	semidirect
7. C₃₁×#1939	C₂	semidirect
9. C₃₁×#1845	C₂	semidirect
55. C₄×D₄×C₃₁	Q₈	semidirect
83. C₃₁×#102	Q₈	semidirect
150. #26040	C₃₁	direct (N × H)
163. D₄×C₃₁	C₄⋊Q₈	semidirect
203. #2551	C₂×C₃₁	semidirect
205. #2086	C₂×C₃₁	semidirect
206. #2057	C₂×C₃₁	semidirect
207. #1947	C₂×C₃₁	semidirect
208. #1939	C₂×C₃₁	semidirect
210. #1845	C₂×C₃₁	semidirect
235. C₄×C₃₁	#474	semidirect
291. C₄×D₄	Q₈×C₃₁	semidirect
319. #102	Q₈×C₃₁	semidirect
325. C₃₁	#26040	direct (N × H)
399. D₄	C₃₁×C₄⋊Q₈	semidirect
460. C₄	C₃₁×#474	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#474	2
C₃₁×#2551	2
D₄×C₄⋊C₄×C₃₁	2
C₃₁×#2086	2
C₃₁×#2057	2
C₃₁×#1947	2
C₃₁×#1939	2
C₃₁×#1862	2
C₃₁×#1845	2
C₃₁×#1802	2
C₄×C₃₁×#102	2
#26040	31

Subgroups — C31×#26040 #21482799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#26040 #21482799