Subgroups — C₃₁×#22799 #21479558

Order 7936.

952 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#432	C₂	1	2	yes	maximal, complemented
2	C₄²×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1954	C₂	1	2	yes	maximal
4	C₃₁×#1924	C₂	1	4	yes	maximal, complemented
5	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1797	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1770	C₂	1	2	yes	maximal, complemented
8	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
10	C₂×C₄×D₄×C₃₁	C₄	1	1	yes	characteristic
11	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄²×C₃₁	C₂²	1	2	yes	—
13	C₃₁×#432	—	2	8	no	—
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₂×C₃₁×#97	C₂²	1	4	yes	—
22	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
23	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₃₁	C₄	1	1	yes	characteristic
25	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
27	C₄×C₂²⋊C₄×C₃₁	C₄	1	2	yes	—
28	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
29	C₄³×C₃₁	C₂²	1	1	yes	characteristic
30	C₃₁×#336	C₂²	1	4	yes	—
31	C₃₁×#336	C₂²	1	2	yes	—
32	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂×C₄	1	2	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
37	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
38	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
39	C₄×D₄×C₃₁	—	2	4	no	—
40	C₄×D₄×C₃₁	—	2	4	no	—
41	C₄×D₄×C₃₁	—	2	2	no	—
42	C₄×D₄×C₃₁	—	2	2	no	—
43	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₃₁	—	2	4	no	—
50	C₂×C₄²×C₃₁	—	2	4	no	—
51	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
58	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
61	C₃₁×C₄⋊C₈	—	2	4	no	—
62	C₃₁×C₄⋊C₈	—	2	4	no	—
63	C₃₁×C₄⋊C₈	—	2	2	no	—
64	C₃₁×C₄⋊C₈	—	4	2	no	—
65	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
66	C₃₁×C₂²⋊C₈	—	2	8	no	—
67	C₃₁×C₂²⋊C₈	—	4	4	no	—
68	C₃₁×#97	—	2	8	no	—
69	C₄×C₈×C₃₁	—	2	4	no	—
70	C₃₁×#95	C₂×C₄	1	2	yes	—
71	C₃₁×#95	C₂×C₄	1	2	yes	—
72	C₃₁×#95	C₂×C₄	1	2	yes	—
73	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₃₁	—	2	2	no	—
75	C₂×D₄×C₃₁	—	2	2	no	—
76	C₂×D₄×C₃₁	M₁₆	1	4	yes	—
77	C₂²×C₄×C₃₁	—	2	4	no	—
78	C₂²×C₄×C₃₁	—	2	4	no	—
79	C₂²×C₄×C₃₁	—	2	4	no	—
80	C₂²×C₄×C₃₁	—	2	4	no	—
81	C₂²×C₄×C₃₁	—	2	4	no	—
82	C₂²×C₄×C₃₁	—	2	2	no	—
83	C₂²×C₄×C₃₁	—	2	2	no	—
84	C₂²×C₄×C₃₁	—	2	2	no	—
85	C₂²×C₄×C₃₁	—	2	2	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	4	2	no	—
89	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
91	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
93	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
99	C₂×C₈×C₃₁	—	2	2	no	—
100	C₂×C₈×C₃₁	—	2	2	no	—
101	C₂×C₈×C₃₁	—	2	4	no	—
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	4	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	4	no	—
107	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
108	C₄⋊C₄×C₃₁	—	2	2	no	—
109	C₄⋊C₄×C₃₁	—	2	2	no	—
110	C₄⋊C₄×C₃₁	—	2	2	no	—
111	C₂²⋊C₄×C₃₁	—	2	4	no	—
112	C₂²⋊C₄×C₃₁	—	2	4	no	—
113	C₂²⋊C₄×C₃₁	—	2	4	no	—
114	C₄²×C₃₁	—	2	4	no	—
115	C₄²×C₃₁	—	2	4	no	—
116	C₄²×C₃₁	—	2	2	no	—
117	C₄²×C₃₁	—	2	2	no	—
118	C₄²×C₃₁	—	2	2	no	—
119	C₄²×C₃₁	—	2	2	no	—
120	C₄²×C₃₁	—	2	1	no	—
121	C₄²×C₃₁	—	2	1	no	—
122	C₄²×C₃₁	—	4	1	no	—
123	C₄²×C₃₁	M₁₆	1	4	yes	—
124	C₄²×C₃₁	C₂×D₄	1	2	yes	—
125	C₄²×C₃₁	#41	1	2	yes	—
126	C₄²×C₃₁	#41	1	2	yes	—
127	C₄²×C₃₁	#41	1	2	yes	—
128	#22799	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
129	C₂³×C₃₁	—	2	2	no	—
130	C₂³×C₃₁	—	2	2	no	—
131	C₂³×C₃₁	—	2	2	no	—
132	C₂³×C₃₁	—	2	2	no	—
133	C₂³×C₃₁	—	2	2	no	—
134	C₂³×C₃₁	—	2	2	no	—
135	C₂³×C₃₁	C₂×M₁₆	1	4	yes	—
136	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
137	D₄×C₃₁	—	2	8	no	—
138	C₂×C₄×C₃₁	—	2	1	no	—
139	C₂×C₄×C₃₁	—	2	1	no	—
140	C₂×C₄×C₃₁	—	2	1	no	—
141	C₂×C₄×C₃₁	—	2	1	no	—
142	C₂×C₄×C₃₁	—	2	1	no	—
143	C₂×C₄×C₃₁	—	2	1	no	—
144	C₂×C₄×C₃₁	—	2	2	no	—
145	C₂×C₄×C₃₁	—	2	2	no	—
146	C₂×C₄×C₃₁	—	2	2	no	—
147	C₂×C₄×C₃₁	—	2	2	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	2	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	2	no	—
152	C₂×C₄×C₃₁	—	2	2	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	2	no	—
155	C₂×C₄×C₃₁	—	2	2	no	—
156	C₂×C₄×C₃₁	—	4	8	no	—
157	C₂×C₄×C₃₁	—	4	2	no	—
158	C₂×C₄×C₃₁	—	4	2	no	—
159	C₂×C₄×C₃₁	—	4	2	no	—
160	C₂×C₄×C₃₁	—	4	2	no	—
161	C₂×C₄×C₃₁	—	4	8	no	—
162	C₂×C₄×C₃₁	#117	1	2	yes	—
163	C₂×C₄×C₃₁	#117	1	2	yes	—
164	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
167	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
168	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
169	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
170	C₂×C₄×C₃₁	#131	1	2	yes	—
171	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
172	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
176	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
177	C₈×C₃₁	—	2	4	no	—
178	C₈×C₃₁	—	4	4	no	—
179	C₈×C₃₁	—	4	2	no	—
180	C₂×#432	C₂×C₃₁	1	2	yes	complemented
181	C₄²×D₄	C₂×C₃₁	1	1	yes	characteristic
182	#1954	C₂×C₃₁	1	2	yes	—
183	#1924	C₂×C₃₁	1	4	yes	complemented
184	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
185	#1797	C₂×C₃₁	1	2	yes	complemented
186	#1770	C₂×C₃₁	1	2	yes	complemented
187	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic, complemented
188	C₂²×C₃₁	—	2	4	no	—
189	C₂²×C₃₁	—	2	4	no	—
190	C₂²×C₃₁	—	2	4	no	—
191	C₂²×C₃₁	—	4	2	no	—
192	C₂²×C₃₁	—	4	2	no	—
193	C₂²×C₃₁	—	4	2	no	—
194	C₂²×C₃₁	—	4	2	no	—
195	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
196	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
197	C₂²×C₃₁	#520	1	1	yes	characteristic
198	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
199	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
200	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
201	C₂²×C₃₁	#568	1	1	yes	characteristic
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	2	no	—
205	C₄×C₃₁	—	4	2	no	—
206	C₄×C₃₁	—	4	2	no	—
207	C₄×C₃₁	—	4	2	no	—
208	C₄×C₃₁	—	4	1	no	—
209	C₄×C₃₁	#432	1	4	yes	complemented
210	C₄×C₃₁	#436	1	4	yes	—
211	C₄×C₃₁	#547	1	1	yes	characteristic
212	C₄×C₃₁	#550	1	1	yes	characteristic
213	C₄×C₃₁	#555	1	1	yes	characteristic
214	C₄×C₃₁	#556	1	1	yes	characteristic
215	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
216	C₂×C₄×D₄	C₄×C₃₁	1	1	yes	characteristic
217	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
218	C₂²×C₄²	C₂²×C₃₁	1	2	yes	—
219	#432	—	2	8	no	—
220	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
221	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
222	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
223	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
224	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
226	C₂×C₂²⋊C₈	C₂²×C₃₁	1	4	yes	—
227	C₂×#97	C₂²×C₃₁	1	4	yes	—
228	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
229	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
230	C₄×C₄⋊C₄	C₄×C₃₁	1	1	yes	characteristic
231	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
232	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
233	C₄×C₂²⋊C₄	C₄×C₃₁	1	2	yes	—
234	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
235	C₄³	C₂²×C₃₁	1	1	yes	characteristic
236	#336	C₂²×C₃₁	1	4	yes	—
237	#336	C₂²×C₃₁	1	2	yes	—
238	C₂×C₃₁	—	4	4	no	—
239	C₂×C₃₁	#2354	1	1	yes	characteristic
240	C₂×C₃₁	C₂×#432	1	1	yes	characteristic
241	C₂×C₃₁	#2925	1	1	yes	characteristic
242	C₂×C₃₁	C₂×#436	1	1	yes	characteristic
243	C₂×C₃₁	D₄×M₁₆	1	1	yes	characteristic
244	C₂×C₃₁	#2989	1	1	yes	characteristic
245	C₂×C₃₁	#2990	1	1	yes	characteristic
246	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
247	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
248	C₂³×C₄	C₂×C₄×C₃₁	1	2	yes	—
249	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
250	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
251	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
252	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
253	C₄×D₄	—	2	4	no	—
254	C₄×D₄	—	2	4	no	—
255	C₄×D₄	—	2	2	no	—
256	C₄×D₄	—	2	2	no	—
257	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
258	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
259	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
260	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
261	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
262	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
263	C₂×C₄²	—	2	4	no	—
264	C₂×C₄²	—	2	4	no	—
265	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
266	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
267	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
268	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
269	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
270	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
272	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
273	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
274	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
275	C₄⋊C₈	—	2	4	no	—
276	C₄⋊C₈	—	2	4	no	—
277	C₄⋊C₈	—	2	2	no	—
278	C₄⋊C₈	—	4	2	no	—
279	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
280	C₂²⋊C₈	—	2	8	no	—
281	C₂²⋊C₈	—	4	4	no	—
282	#97	—	2	8	no	—
283	C₄×C₈	—	2	4	no	—
284	#95	C₂×C₄×C₃₁	1	2	yes	—
285	#95	C₂×C₄×C₃₁	1	2	yes	—
286	#95	C₂×C₄×C₃₁	1	2	yes	—
287	C₃₁	#22799	1	1	yes	characteristic, direct product factor, Sylow
288	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
289	C₂×D₄	—	2	2	no	—
290	C₂×D₄	—	2	2	no	—
291	C₂×D₄	M₁₆×C₃₁	1	4	yes	—
292	C₂²×C₄	—	2	4	no	—
293	C₂²×C₄	—	2	4	no	—
294	C₂²×C₄	—	2	4	no	—
295	C₂²×C₄	—	2	4	no	—
296	C₂²×C₄	—	2	4	no	—
297	C₂²×C₄	—	2	2	no	—
298	C₂²×C₄	—	2	2	no	—
299	C₂²×C₄	—	2	2	no	—
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	4	2	no	—
304	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
305	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
306	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
307	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
308	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
309	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
310	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
312	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
313	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
314	C₂×C₈	—	2	2	no	—
315	C₂×C₈	—	2	2	no	—
316	C₂×C₈	—	2	4	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	4	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	—	2	2	no	—
321	C₂×C₈	—	2	4	no	—
322	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₂²⋊C₄	—	2	4	no	—
327	C₂²⋊C₄	—	2	4	no	—
328	C₂²⋊C₄	—	2	4	no	—
329	C₄²	—	2	4	no	—
330	C₄²	—	2	4	no	—
331	C₄²	—	2	2	no	—
332	C₄²	—	2	2	no	—
333	C₄²	—	2	2	no	—
334	C₄²	—	2	2	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	4	1	no	—
338	C₄²	M₁₆×C₃₁	1	4	yes	—
339	C₄²	C₂×D₄×C₃₁	1	2	yes	—
340	C₄²	#41×C₃₁	1	2	yes	—
341	C₄²	#41×C₃₁	1	2	yes	—
342	C₄²	#41×C₃₁	1	2	yes	—
343	C₂³	—	2	2	no	—
344	C₂³	—	2	2	no	—
345	C₂³	—	2	2	no	—
346	C₂³	—	2	2	no	—
347	C₂³	—	2	2	no	—
348	C₂³	—	2	2	no	—
349	C₂³	C₂×M₁₆×C₃₁	1	4	yes	—
350	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
351	D₄	—	2	8	no	—
352	C₂×C₄	—	2	1	no	—
353	C₂×C₄	—	2	1	no	—
354	C₂×C₄	—	2	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	2	no	—
359	C₂×C₄	—	2	2	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	1	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	4	8	no	—
371	C₂×C₄	—	4	8	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	C₃₁×#117	1	2	yes	—
377	C₂×C₄	C₃₁×#117	1	2	yes	—
378	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
379	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
380	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
381	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
382	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
383	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
384	C₂×C₄	C₃₁×#131	1	2	yes	—
385	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
386	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
387	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
388	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
389	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
390	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
391	C₈	—	2	4	no	—
392	C₈	—	4	4	no	—
393	C₈	—	4	2	no	—
394	C₂²	—	2	4	no	—
395	C₂²	—	2	4	no	—
396	C₂²	—	2	4	no	—
397	C₂²	—	4	2	no	—
398	C₂²	—	4	2	no	—
399	C₂²	—	4	2	no	—
400	C₂²	—	4	2	no	—
401	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
402	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
403	C₂²	C₃₁×#520	1	1	yes	characteristic
404	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
405	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
406	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
407	C₂²	C₃₁×#568	1	1	yes	characteristic
408	C₄	—	2	2	no	—
409	C₄	—	2	2	no	—
410	C₄	—	2	2	no	—
411	C₄	—	4	2	no	—
412	C₄	—	4	2	no	—
413	C₄	—	4	2	no	—
414	C₄	—	4	1	no	—
415	C₄	C₃₁×#432	1	4	yes	complemented
416	C₄	C₃₁×#436	1	4	yes	—
417	C₄	C₃₁×#547	1	1	yes	characteristic
418	C₄	C₃₁×#550	1	1	yes	characteristic
419	C₄	C₃₁×#555	1	1	yes	characteristic
420	C₄	C₃₁×#556	1	1	yes	characteristic
421	C₂	—	4	4	no	—
422	C₂	C₃₁×#2354	1	1	yes	characteristic
423	C₂	C₂×C₃₁×#432	1	1	yes	characteristic
424	C₂	C₃₁×#2925	1	1	yes	characteristic
425	C₂	C₂×C₃₁×#436	1	1	yes	characteristic
426	C₂	D₄×M₁₆×C₃₁	1	1	yes	characteristic
427	C₂	C₃₁×#2989	1	1	yes	characteristic
428	C₂	C₃₁×#2990	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#432	C₂	semidirect
4. C₃₁×#1924	C₂	semidirect
5. C₃₁×#1842	C₂	semidirect
6. C₃₁×#1797	C₂	semidirect
7. C₃₁×#1770	C₂	semidirect
8. C₄×C₃₁×C₄⋊C₈	C₂	semidirect
65. C₃₁×C₄⋊C₈	D₄	semidirect
128. #22799	C₃₁	direct (N × H)
180. C₂×#432	C₂×C₃₁	semidirect
183. #1924	C₂×C₃₁	semidirect
184. #1842	C₂×C₃₁	semidirect
185. #1797	C₂×C₃₁	semidirect
186. #1770	C₂×C₃₁	semidirect
187. C₄×C₄⋊C₈	C₂×C₃₁	semidirect
209. C₄×C₃₁	#432	semidirect
279. C₄⋊C₈	D₄×C₃₁	semidirect
287. C₃₁	#22799	direct (N × H)
415. C₄	C₃₁×#432	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#432	2
C₄²×D₄×C₃₁	2
C₃₁×#1954	2
C₃₁×#1924	2
C₃₁×#1842	2
C₃₁×#1797	2
C₃₁×#1770	2
C₄×C₃₁×C₄⋊C₈	2
#22799	31

Subgroups — C31×#22799 #21479558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22799 #21479558