Subgroups — C₃₁×#23031 #21479790

Order 7936.

1038 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#466	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₁×#442	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#427	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#2363	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1972	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1943	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1925	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1875	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1862	C₂	1	2	yes	maximal, complemented
10	C₃₁×#1776	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1752	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#121	C₂²	1	2	yes	—
15	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	—
16	C₂×C₃₁×#117	C₂²	1	2	yes	—
17	C₃₁×#466	C₄	1	8	yes	complemented
18	C₃₁×#442	—	2	4	no	—
19	C₃₁×#427	—	2	4	no	—
20	C₂×C₃₁×#107	C₂²	1	2	yes	—
21	C₂×C₃₁×#107	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×#106	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	2	yes	—
26	C₂×C₃₁×#102	C₂²	1	2	yes	—
27	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
28	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
29	C₂×C₃₁×#97	C₂²	1	1	yes	characteristic
30	C₃₁×#390	—	2	2	no	—
31	C₃₁×#390	C₂²	1	2	yes	—
32	C₃₁×#389	C₂²	1	2	yes	—
33	C₃₁×#388	—	2	2	no	—
34	C₃₁×#386	—	2	2	no	—
35	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
36	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
37	C₄×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	2	yes	—
40	C₃₁×#336	C₂²	1	1	yes	characteristic
41	C₃₁×#336	C₂²	1	1	yes	characteristic
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
44	C₂³×C₄×C₃₁	—	2	2	no	—
45	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₈×C₃₁	—	2	2	no	—
47	C₂×D₈×C₃₁	—	2	2	no	—
48	C₂×D₈×C₃₁	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₃₁	—	2	2	no	—
50	C₂²×C₈×C₃₁	—	2	2	no	—
51	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
52	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
53	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
55	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
56	C₃₁×#121	—	2	4	no	—
57	C₃₁×#121	C₂×C₄	1	8	yes	complemented
58	C₄×D₄×C₃₁	—	2	4	no	—
59	C₄×D₄×C₃₁	—	4	2	no	—
60	C₃₁×#117	—	2	4	no	—
61	C₃₁×#117	—	4	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
71	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
72	C₂×C₄²×C₃₁	—	2	2	no	—
73	C₂×C₄²×C₃₁	—	2	2	no	—
74	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	—	2	4	no	—
77	C₃₁×#107	C₂×C₄	1	4	yes	complemented
78	C₃₁×#106	—	2	2	no	—
79	C₃₁×#106	D₄	1	4	yes	complemented
80	C₃₁×#102	—	2	4	no	—
81	C₃₁×#102	—	4	2	no	—
82	C₃₁×#102	C₂×C₄	1	8	yes	complemented
83	C₃₁×C₂²⋊C₈	—	2	2	no	—
84	C₃₁×#97	—	2	2	no	—
85	C₃₁×#97	—	2	2	no	—
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₃₁×#95	D₄	1	2	yes	—
89	C₂⁴×C₃₁	—	2	2	no	—
90	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
91	C₂×D₄×C₃₁	—	2	8	no	—
92	C₂×D₄×C₃₁	—	2	2	no	—
93	C₂×D₄×C₃₁	—	2	2	no	—
94	C₂×D₄×C₃₁	—	2	2	no	—
95	C₂×D₄×C₃₁	—	2	2	no	—
96	C₂×D₄×C₃₁	—	2	2	no	—
97	C₂×D₄×C₃₁	—	8	2	no	—
98	C₂×D₄×C₃₁	—	8	2	no	—
99	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
100	C₂²×C₄×C₃₁	—	2	1	no	—
101	C₂²×C₄×C₃₁	—	2	1	no	—
102	C₂²×C₄×C₃₁	—	2	1	no	—
103	C₂²×C₄×C₃₁	—	2	1	no	—
104	C₂²×C₄×C₃₁	—	2	2	no	—
105	C₂²×C₄×C₃₁	—	2	2	no	—
106	C₂²×C₄×C₃₁	—	2	2	no	—
107	C₂²×C₄×C₃₁	—	4	4	no	—
108	C₂²×C₄×C₃₁	—	4	4	no	—
109	C₂²×C₄×C₃₁	—	8	2	no	—
110	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
111	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
112	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	#41	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
116	D₈×C₃₁	—	2	8	no	—
117	C₂×C₈×C₃₁	—	2	1	no	—
118	C₂×C₈×C₃₁	—	2	1	no	—
119	C₂×C₈×C₃₁	—	2	8	no	—
120	C₂×C₈×C₃₁	—	2	1	no	—
121	C₂×C₈×C₃₁	—	2	1	no	—
122	C₂×C₈×C₃₁	—	2	1	no	—
123	C₂×C₈×C₃₁	—	2	1	no	—
124	C₂×C₈×C₃₁	—	2	1	no	—
125	C₂×C₈×C₃₁	—	2	1	no	—
126	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
127	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
128	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
129	C₂×C₈×C₃₁	#41	1	2	yes	—
130	C₄⋊C₄×C₃₁	—	2	2	no	—
131	C₄⋊C₄×C₃₁	—	2	2	no	—
132	C₄⋊C₄×C₃₁	—	2	2	no	—
133	C₄⋊C₄×C₃₁	—	2	2	no	—
134	C₄⋊C₄×C₃₁	—	8	2	no	—
135	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
136	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
137	C₂²⋊C₄×C₃₁	—	2	8	no	—
138	C₂²⋊C₄×C₃₁	—	4	2	no	—
139	C₂²⋊C₄×C₃₁	—	4	4	no	—
140	C₂²⋊C₄×C₃₁	—	4	4	no	—
141	C₂²⋊C₄×C₃₁	—	4	4	no	—
142	C₂²⋊C₄×C₃₁	—	8	2	no	—
143	C₄²×C₃₁	—	2	1	no	—
144	C₄²×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	4	4	no	—
148	C₄²×C₃₁	—	8	2	no	—
149	#23031	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂³×C₃₁	—	2	1	no	—
151	C₂³×C₃₁	—	2	1	no	—
152	C₂³×C₃₁	—	2	2	no	—
153	C₂³×C₃₁	—	2	2	no	—
154	C₂³×C₃₁	—	2	4	no	—
155	C₂³×C₃₁	—	4	2	no	—
156	C₂³×C₃₁	—	4	2	no	—
157	C₂³×C₃₁	—	4	2	no	—
158	C₂³×C₃₁	—	4	2	no	—
159	C₂³×C₃₁	—	4	2	no	—
160	C₂³×C₃₁	—	4	2	no	—
161	C₂³×C₃₁	C₄×D₄	1	2	yes	—
162	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	2	4	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	8	8	no	—
168	C₂×C₄×C₃₁	—	2	1	no	—
169	C₂×C₄×C₃₁	—	2	1	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	—	2	4	no	—
175	C₂×C₄×C₃₁	—	2	2	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	2	no	—
181	C₂×C₄×C₃₁	—	2	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	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	8	8	no	—
190	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
191	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
192	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
195	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
196	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
197	C₈×C₃₁	—	2	2	no	—
198	C₈×C₃₁	—	4	1	no	—
199	C₈×C₃₁	—	4	1	no	—
200	C₈×C₃₁	C₄×D₄	1	4	yes	—
201	C₂×#466	C₂×C₃₁	1	1	yes	characteristic
202	C₂×#442	C₂×C₃₁	1	1	yes	characteristic, complemented
203	C₂×#427	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#2363	C₂×C₃₁	1	2	yes	complemented
205	#1972	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#1943	C₂×C₃₁	1	1	yes	characteristic, complemented
207	#1925	C₂×C₃₁	1	2	yes	complemented
208	#1875	C₂×C₃₁	1	2	yes	complemented
209	#1862	C₂×C₃₁	1	2	yes	complemented
210	#1776	C₂×C₃₁	1	1	yes	characteristic, complemented
211	#1752	C₂×C₃₁	1	1	yes	characteristic, complemented
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	2	no	—
215	C₂²×C₃₁	—	4	2	no	—
216	C₂²×C₃₁	—	4	2	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	4	4	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	8	2	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	8	2	no	—
223	C₂²×C₃₁	—	8	2	no	—
224	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
225	C₂²×C₃₁	D₄²	1	1	yes	characteristic
226	C₂²×C₃₁	#547	1	1	yes	characteristic
227	C₂²×C₃₁	#575	1	2	yes	—
228	C₂²×C₃₁	#576	1	2	yes	—
229	C₄×C₃₁	—	2	2	no	—
230	C₄×C₃₁	—	4	2	no	—
231	C₄×C₃₁	—	4	4	no	—
232	C₄×C₃₁	—	4	2	no	—
233	C₄×C₃₁	—	4	4	no	—
234	C₄×C₃₁	—	8	2	no	—
235	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
236	C₄×C₃₁	#520	1	1	yes	characteristic
237	C₄×C₃₁	#548	1	1	yes	characteristic
238	C₄×C₃₁	#553	1	1	yes	characteristic
239	C₂²×D₈	C₂²×C₃₁	1	1	yes	characteristic
240	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
241	C₂×#121	C₂²×C₃₁	1	2	yes	—
242	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	—
243	C₂×#117	C₂²×C₃₁	1	2	yes	—
244	#466	C₄×C₃₁	1	8	yes	complemented
245	#442	—	2	4	no	—
246	#427	—	2	4	no	—
247	C₂×#107	C₂²×C₃₁	1	2	yes	—
248	C₂×#107	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#106	C₂²×C₃₁	1	1	yes	characteristic
251	C₂×#102	C₂²×C₃₁	1	2	yes	—
252	C₂×#102	C₂²×C₃₁	1	2	yes	—
253	C₂×#102	C₂²×C₃₁	1	2	yes	—
254	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
255	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
256	C₂×#97	C₂²×C₃₁	1	1	yes	characteristic
257	#390	—	2	2	no	—
258	#390	C₂²×C₃₁	1	2	yes	—
259	#389	C₂²×C₃₁	1	2	yes	—
260	#388	—	2	2	no	—
261	#386	—	2	2	no	—
262	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
263	C₄×C₂²⋊C₄	—	2	2	no	—
264	C₄×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
265	#340	C₂²×C₃₁	1	2	yes	—
266	#340	C₂²×C₃₁	1	2	yes	—
267	#336	C₂²×C₃₁	1	1	yes	characteristic
268	#336	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₃₁	—	4	2	no	—
270	C₂×C₃₁	—	8	4	no	—
271	C₂×C₃₁	#2361	1	1	yes	characteristic
272	C₂×C₃₁	#2943	1	1	yes	characteristic
273	C₂×C₃₁	#2948	1	1	yes	characteristic
274	C₂×C₃₁	#3268	1	2	yes	—
275	C₂×C₃₁	#3347	1	2	yes	—
276	C₂²×D₄	—	2	2	no	—
277	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
278	C₂³×C₄	—	2	2	no	—
279	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
280	C₂×D₈	—	2	2	no	—
281	C₂×D₈	—	2	2	no	—
282	C₂×D₈	C₂×C₄×C₃₁	1	4	yes	—
283	C₂²×C₈	—	2	2	no	—
284	C₂²×C₈	—	2	2	no	—
285	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
286	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
287	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
288	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
290	#121	—	2	4	no	—
291	#121	C₂×C₄×C₃₁	1	8	yes	complemented
292	C₄×D₄	—	2	4	no	—
293	C₄×D₄	—	4	2	no	—
294	#117	—	2	4	no	—
295	#117	—	4	2	no	—
296	C₂×C₄⋊C₄	—	2	2	no	—
297	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
298	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
299	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
300	C₂×C₂²⋊C₄	—	2	2	no	—
301	C₂×C₂²⋊C₄	—	2	2	no	—
302	C₂×C₂²⋊C₄	—	2	2	no	—
303	C₂×C₂²⋊C₄	—	2	2	no	—
304	C₂×C₂²⋊C₄	—	2	2	no	—
305	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	—	2	2	no	—
308	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
310	#107	—	2	4	no	—
311	#107	C₂×C₄×C₃₁	1	4	yes	complemented
312	#106	—	2	2	no	—
313	#106	D₄×C₃₁	1	4	yes	complemented
314	#102	—	2	4	no	—
315	#102	—	4	2	no	—
316	#102	C₂×C₄×C₃₁	1	8	yes	complemented
317	C₂²⋊C₈	—	2	2	no	—
318	#97	—	2	2	no	—
319	#97	—	2	2	no	—
320	#95	—	2	2	no	—
321	#95	—	2	2	no	—
322	#95	D₄×C₃₁	1	2	yes	—
323	C₃₁	#23031	1	1	yes	characteristic, direct product factor, Sylow
324	C₂⁴	—	2	2	no	—
325	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂×D₄	—	2	8	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	2	2	no	—
331	C₂×D₄	—	2	2	no	—
332	C₂×D₄	—	8	2	no	—
333	C₂×D₄	—	8	2	no	—
334	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
335	C₂²×C₄	—	2	1	no	—
336	C₂²×C₄	—	2	1	no	—
337	C₂²×C₄	—	2	1	no	—
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	2	no	—
340	C₂²×C₄	—	2	2	no	—
341	C₂²×C₄	—	2	2	no	—
342	C₂²×C₄	—	4	4	no	—
343	C₂²×C₄	—	4	4	no	—
344	C₂²×C₄	—	8	2	no	—
345	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
346	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
347	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
348	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
349	C₂²×C₄	#41×C₃₁	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
351	D₈	—	2	8	no	—
352	C₂×C₈	—	2	1	no	—
353	C₂×C₈	—	2	1	no	—
354	C₂×C₈	—	2	8	no	—
355	C₂×C₈	—	2	1	no	—
356	C₂×C₈	—	2	1	no	—
357	C₂×C₈	—	2	1	no	—
358	C₂×C₈	—	2	1	no	—
359	C₂×C₈	—	2	1	no	—
360	C₂×C₈	—	2	1	no	—
361	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
363	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
364	C₂×C₈	#41×C₃₁	1	2	yes	—
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	—	8	2	no	—
370	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
371	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
372	C₂²⋊C₄	—	2	8	no	—
373	C₂²⋊C₄	—	4	2	no	—
374	C₂²⋊C₄	—	4	4	no	—
375	C₂²⋊C₄	—	4	4	no	—
376	C₂²⋊C₄	—	4	4	no	—
377	C₂²⋊C₄	—	8	2	no	—
378	C₄²	—	2	1	no	—
379	C₄²	—	2	1	no	—
380	C₄²	—	2	1	no	—
381	C₄²	—	2	1	no	—
382	C₄²	—	4	4	no	—
383	C₄²	—	8	2	no	—
384	C₂³	—	2	1	no	—
385	C₂³	—	2	1	no	—
386	C₂³	—	2	2	no	—
387	C₂³	—	2	2	no	—
388	C₂³	—	2	4	no	—
389	C₂³	—	4	2	no	—
390	C₂³	—	4	2	no	—
391	C₂³	—	4	2	no	—
392	C₂³	—	4	2	no	—
393	C₂³	—	4	2	no	—
394	C₂³	—	4	2	no	—
395	C₂³	C₄×D₄×C₃₁	1	2	yes	—
396	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
397	D₄	—	2	4	no	—
398	D₄	—	2	4	no	—
399	D₄	—	4	2	no	—
400	D₄	—	4	2	no	—
401	D₄	—	8	8	no	—
402	C₂×C₄	—	2	1	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	4	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	2	4	no	—
411	C₂×C₄	—	2	2	no	—
412	C₂×C₄	—	2	2	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	4	2	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	C₄×D₄×C₃₁	1	2	yes	—
425	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
426	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
427	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
428	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
429	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
430	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
431	C₈	—	2	2	no	—
432	C₈	—	4	1	no	—
433	C₈	—	4	1	no	—
434	C₈	C₄×D₄×C₃₁	1	4	yes	—
435	C₂²	—	2	2	no	—
436	C₂²	—	2	2	no	—
437	C₂²	—	2	2	no	—
438	C₂²	—	4	2	no	—
439	C₂²	—	4	2	no	—
440	C₂²	—	4	4	no	—
441	C₂²	—	4	4	no	—
442	C₂²	—	4	4	no	—
443	C₂²	—	8	2	no	—
444	C₂²	—	8	2	no	—
445	C₂²	—	8	2	no	—
446	C₂²	—	8	2	no	—
447	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
448	C₂²	D₄²×C₃₁	1	1	yes	characteristic
449	C₂²	C₃₁×#547	1	1	yes	characteristic
450	C₂²	C₃₁×#575	1	2	yes	—
451	C₂²	C₃₁×#576	1	2	yes	—
452	C₄	—	2	2	no	—
453	C₄	—	4	2	no	—
454	C₄	—	4	4	no	—
455	C₄	—	4	2	no	—
456	C₄	—	4	4	no	—
457	C₄	—	8	2	no	—
458	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
459	C₄	C₃₁×#520	1	1	yes	characteristic
460	C₄	C₃₁×#548	1	1	yes	characteristic
461	C₄	C₃₁×#553	1	1	yes	characteristic
462	C₂	—	4	2	no	—
463	C₂	—	8	4	no	—
464	C₂	C₃₁×#2361	1	1	yes	characteristic
465	C₂	C₃₁×#2943	1	1	yes	characteristic
466	C₂	C₃₁×#2948	1	1	yes	characteristic
467	C₂	C₃₁×#3268	1	2	yes	—
468	C₂	C₃₁×#3347	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂×C₃₁×#466	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄×C₂²⋊C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₃₁×#442	C₂	semidirect
3. C₂×C₃₁×#427	C₂	semidirect
4. C₃₁×#2363	C₂	semidirect
5. C₃₁×#1972	C₂	semidirect
6. C₃₁×#1943	C₂	semidirect
7. C₃₁×#1925	C₂	semidirect
8. C₃₁×#1875	C₂	semidirect
9. C₃₁×#1862	C₂	semidirect
10. C₃₁×#1776	C₂	semidirect
11. C₃₁×#1752	C₂	semidirect
17. C₃₁×#466	C₄	semidirect
51. C₂²×C₈×C₃₁	C₂×C₄	semidirect
57. C₃₁×#121	C₂×C₄	semidirect
77. C₃₁×#107	C₂×C₄	semidirect
79. C₃₁×#106	D₄	semidirect
82. C₃₁×#102	C₂×C₄	semidirect
149. #23031	C₃₁	direct (N × H)
202. C₂×#442	C₂×C₃₁	semidirect
203. C₂×#427	C₂×C₃₁	semidirect
204. #2363	C₂×C₃₁	semidirect
205. #1972	C₂×C₃₁	semidirect
206. #1943	C₂×C₃₁	semidirect
207. #1925	C₂×C₃₁	semidirect
208. #1875	C₂×C₃₁	semidirect
209. #1862	C₂×C₃₁	semidirect
210. #1776	C₂×C₃₁	semidirect
211. #1752	C₂×C₃₁	semidirect
244. #466	C₄×C₃₁	semidirect
285. C₂²×C₈	C₂×C₄×C₃₁	semidirect
291. #121	C₂×C₄×C₃₁	semidirect
311. #107	C₂×C₄×C₃₁	semidirect
313. #106	D₄×C₃₁	semidirect
316. #102	C₂×C₄×C₃₁	semidirect
323. C₃₁	#23031	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#466	2
C₂×C₃₁×#442	2
C₂×C₃₁×#427	2
C₃₁×#2363	2
C₃₁×#1972	2
C₃₁×#1943	2
C₃₁×#1925	2
C₃₁×#1875	2
C₃₁×#1862	2
C₃₁×#1776	2
C₃₁×#1752	2
#23031	31

Subgroups — C31×#23031 #21479790

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#23031 #21479790