Subgroups — C₃₁×#22798 #21479557

Order 7936.

808 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#433	C₂	1	2	yes	maximal, complemented
2	C₃₁×#2293	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1955	C₂	1	2	yes	maximal
4	C₃₁×#1924	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1923	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1798	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1771	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1767	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂×C₃₁×#117	C₂²	1	2	yes	—
12	C₃₁×#433	—	2	8	no	—
13	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
21	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
23	C₂×C₃₁×#97	C₂²	1	2	yes	—
24	C₂×C₃₁×#97	C₂²	1	2	yes	—
25	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#390	C₄	1	1	yes	characteristic
27	C₃₁×#390	C₄	1	1	yes	characteristic
28	C₃₁×#389	C₄	1	1	yes	characteristic
29	C₃₁×#389	C₄	1	1	yes	characteristic
30	C₃₁×#388	C₄	1	2	yes	—
31	C₃₁×#388	C₄	1	2	yes	—
32	C₄×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
34	C₄³×C₃₁	C₂²	1	1	yes	characteristic
35	C₃₁×#336	C₂²	1	2	yes	—
36	C₃₁×#336	C₂²	1	2	yes	—
37	C₃₁×#336	C₂²	1	2	yes	—
38	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic
39	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
40	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
41	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
42	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
43	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
44	C₄×D₄×C₃₁	—	2	2	no	—
45	C₄×D₄×C₃₁	—	2	2	no	—
46	C₃₁×#117	—	2	4	no	—
47	C₃₁×#117	—	2	4	no	—
48	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
54	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
55	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
56	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₃₁	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
66	C₃₁×C₄⋊C₈	—	2	4	no	—
67	C₃₁×C₄⋊C₈	—	2	4	no	—
68	C₃₁×C₄⋊C₈	—	2	2	no	—
69	C₃₁×C₄⋊C₈	—	4	1	no	—
70	C₃₁×C₄⋊C₈	—	4	1	no	—
71	C₃₁×C₄⋊C₈	D₄	1	4	yes	complemented
72	C₃₁×C₂²⋊C₈	—	2	4	no	—
73	C₃₁×C₂²⋊C₈	—	2	4	no	—
74	C₃₁×C₂²⋊C₈	—	4	2	no	—
75	C₃₁×C₂²⋊C₈	—	4	2	no	—
76	C₃₁×#97	—	2	4	no	—
77	C₃₁×#97	—	2	4	no	—
78	C₄×C₈×C₃₁	—	2	4	no	—
79	C₃₁×#95	C₂×C₄	1	2	yes	—
80	C₃₁×#95	C₂×C₄	1	2	yes	—
81	C₃₁×#95	C₂×C₄	1	2	yes	—
82	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
83	C₂×D₄×C₃₁	—	2	2	no	—
84	C₂×D₄×C₃₁	—	4	1	no	—
85	C₂×D₄×C₃₁	—	4	1	no	—
86	C₂²×C₄×C₃₁	—	2	2	no	—
87	C₂²×C₄×C₃₁	—	2	2	no	—
88	C₂²×C₄×C₃₁	—	2	2	no	—
89	C₂²×C₄×C₃₁	—	2	2	no	—
90	C₂²×C₄×C₃₁	—	2	2	no	—
91	C₂²×C₄×C₃₁	—	2	2	no	—
92	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
101	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
102	C₂×C₈×C₃₁	—	2	2	no	—
103	C₂×C₈×C₃₁	—	2	2	no	—
104	C₂×C₈×C₃₁	—	2	2	no	—
105	C₂×C₈×C₃₁	—	2	2	no	—
106	C₂×C₈×C₃₁	—	2	2	no	—
107	C₂×C₈×C₃₁	—	2	2	no	—
108	C₂×C₈×C₃₁	—	2	2	no	—
109	C₂×C₈×C₃₁	—	2	2	no	—
110	C₂×C₈×C₃₁	—	2	2	no	—
111	C₂×C₈×C₃₁	—	2	2	no	—
112	C₂×C₈×C₃₁	—	2	2	no	—
113	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
114	C₄⋊C₄×C₃₁	—	2	2	no	—
115	C₄⋊C₄×C₃₁	—	2	4	no	—
116	C₄⋊C₄×C₃₁	—	2	4	no	—
117	C₄⋊C₄×C₃₁	—	4	1	no	—
118	C₄⋊C₄×C₃₁	—	4	1	no	—
119	C₂²⋊C₄×C₃₁	—	2	4	no	—
120	C₂²⋊C₄×C₃₁	—	2	4	no	—
121	C₂²⋊C₄×C₃₁	—	2	4	no	—
122	C₂²⋊C₄×C₃₁	—	2	4	no	—
123	C₂²⋊C₄×C₃₁	—	4	2	no	—
124	C₄²×C₃₁	—	2	2	no	—
125	C₄²×C₃₁	—	2	2	no	—
126	C₄²×C₃₁	—	2	2	no	—
127	C₄²×C₃₁	—	2	1	no	—
128	C₄²×C₃₁	—	2	1	no	—
129	C₄²×C₃₁	—	4	1	no	—
130	C₄²×C₃₁	—	4	1	no	—
131	C₄²×C₃₁	—	4	1	no	—
132	C₄²×C₃₁	C₂×D₄	1	2	yes	—
133	C₄²×C₃₁	#41	1	2	yes	—
134	C₄²×C₃₁	#41	1	2	yes	—
135	C₄²×C₃₁	#41	1	2	yes	—
136	#22798	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂³×C₃₁	—	2	2	no	—
138	C₂³×C₃₁	—	2	2	no	—
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	2	2	no	—
143	C₂³×C₃₁	—	2	2	no	—
144	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
145	D₄×C₃₁	—	4	4	no	—
146	C₂×C₄×C₃₁	—	2	1	no	—
147	C₂×C₄×C₃₁	—	2	1	no	—
148	C₂×C₄×C₃₁	—	2	1	no	—
149	C₂×C₄×C₃₁	—	2	1	no	—
150	C₂×C₄×C₃₁	—	2	1	no	—
151	C₂×C₄×C₃₁	—	2	1	no	—
152	C₂×C₄×C₃₁	—	2	1	no	—
153	C₂×C₄×C₃₁	—	2	1	no	—
154	C₂×C₄×C₃₁	—	2	1	no	—
155	C₂×C₄×C₃₁	—	2	1	no	—
156	C₂×C₄×C₃₁	—	2	2	no	—
157	C₂×C₄×C₃₁	—	2	2	no	—
158	C₂×C₄×C₃₁	—	2	2	no	—
159	C₂×C₄×C₃₁	—	2	2	no	—
160	C₂×C₄×C₃₁	—	2	1	no	—
161	C₂×C₄×C₃₁	—	2	2	no	—
162	C₂×C₄×C₃₁	—	2	2	no	—
163	C₂×C₄×C₃₁	—	2	1	no	—
164	C₂×C₄×C₃₁	—	2	1	no	—
165	C₂×C₄×C₃₁	—	2	2	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	2	no	—
170	C₂×C₄×C₃₁	—	4	2	no	—
171	C₂×C₄×C₃₁	—	4	2	no	—
172	C₂×C₄×C₃₁	#117	1	2	yes	—
173	C₂×C₄×C₃₁	#117	1	2	yes	—
174	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₃₁	#131	1	2	yes	—
177	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
178	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
181	C₂×C₄×C₃₁	H(5,𝔽₂)	1	1	yes	characteristic
182	C₂×C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
183	C₈×C₃₁	—	2	4	no	—
184	C₈×C₃₁	—	4	2	no	—
185	C₈×C₃₁	—	4	2	no	—
186	C₈×C₃₁	—	4	2	no	—
187	C₂×#433	C₂×C₃₁	1	2	yes	complemented
188	#2293	C₂×C₃₁	1	1	yes	characteristic
189	#1955	C₂×C₃₁	1	2	yes	—
190	#1924	C₂×C₃₁	1	2	yes	complemented
191	#1923	C₂×C₃₁	1	2	yes	complemented
192	#1842	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1798	C₂×C₃₁	1	2	yes	complemented
194	#1771	C₂×C₃₁	1	2	yes	complemented
195	#1767	C₂×C₃₁	1	1	yes	characteristic, complemented
196	C₂²×C₃₁	—	4	2	no	—
197	C₂²×C₃₁	—	4	2	no	—
198	C₂²×C₃₁	—	4	2	no	—
199	C₂²×C₃₁	—	4	2	no	—
200	C₂²×C₃₁	—	4	2	no	—
201	C₂²×C₃₁	—	4	2	no	—
202	C₂²×C₃₁	—	4	2	no	—
203	C₂²×C₃₁	C₂×#117	1	1	yes	characteristic
204	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
205	C₂²×C₃₁	#520	1	1	yes	characteristic
206	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
207	C₂²×C₃₁	#568	1	1	yes	characteristic
208	C₂²×C₃₁	#568	1	1	yes	characteristic
209	C₂²×C₃₁	#568	1	1	yes	characteristic
210	C₄×C₃₁	—	2	2	no	—
211	C₄×C₃₁	—	2	2	no	—
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	4	2	no	—
216	C₄×C₃₁	—	4	2	no	—
217	C₄×C₃₁	—	4	2	no	—
218	C₄×C₃₁	—	4	1	no	—
219	C₄×C₃₁	#547	1	1	yes	characteristic
220	C₄×C₃₁	#550	1	1	yes	characteristic
221	C₄×C₃₁	#555	1	1	yes	characteristic
222	C₄×C₃₁	#556	1	1	yes	characteristic
223	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
224	C₂×#117	C₂²×C₃₁	1	2	yes	—
225	#433	—	2	8	no	—
226	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
228	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
229	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
231	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
232	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
233	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
234	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
235	C₂×C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
236	C₂×#97	C₂²×C₃₁	1	2	yes	—
237	C₂×#97	C₂²×C₃₁	1	2	yes	—
238	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
239	#390	C₄×C₃₁	1	1	yes	characteristic
240	#390	C₄×C₃₁	1	1	yes	characteristic
241	#389	C₄×C₃₁	1	1	yes	characteristic
242	#389	C₄×C₃₁	1	1	yes	characteristic
243	#388	C₄×C₃₁	1	2	yes	—
244	#388	C₄×C₃₁	1	2	yes	—
245	C₄×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
246	C₄×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
247	C₄³	C₂²×C₃₁	1	1	yes	characteristic
248	#336	C₂²×C₃₁	1	2	yes	—
249	#336	C₂²×C₃₁	1	2	yes	—
250	#336	C₂²×C₃₁	1	2	yes	—
251	C₂×C₃₁	—	8	2	no	—
252	C₂×C₃₁	#2354	1	1	yes	characteristic
253	C₂×C₃₁	#2921	1	1	yes	characteristic
254	C₂×C₃₁	#2925	1	1	yes	characteristic
255	C₂×C₃₁	#2932	1	1	yes	characteristic
256	C₂×C₃₁	#2935	1	1	yes	characteristic
257	C₂×C₃₁	#2992	1	1	yes	characteristic
258	C₂×C₃₁	#2995	1	1	yes	characteristic
259	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic
260	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
261	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
262	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
263	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
264	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
265	C₄×D₄	—	2	2	no	—
266	C₄×D₄	—	2	2	no	—
267	#117	—	2	4	no	—
268	#117	—	2	4	no	—
269	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
270	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
271	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
272	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	1	yes	characteristic
274	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
275	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
276	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
277	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
278	C₂×C₂²⋊C₄	C₂×C₄×C₃₁	1	2	yes	—
279	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
280	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
281	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
282	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
283	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
284	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
285	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
286	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
287	C₄⋊C₈	—	2	4	no	—
288	C₄⋊C₈	—	2	4	no	—
289	C₄⋊C₈	—	2	2	no	—
290	C₄⋊C₈	—	4	1	no	—
291	C₄⋊C₈	—	4	1	no	—
292	C₄⋊C₈	D₄×C₃₁	1	4	yes	complemented
293	C₂²⋊C₈	—	2	4	no	—
294	C₂²⋊C₈	—	2	4	no	—
295	C₂²⋊C₈	—	4	2	no	—
296	C₂²⋊C₈	—	4	2	no	—
297	#97	—	2	4	no	—
298	#97	—	2	4	no	—
299	C₄×C₈	—	2	4	no	—
300	#95	C₂×C₄×C₃₁	1	2	yes	—
301	#95	C₂×C₄×C₃₁	1	2	yes	—
302	#95	C₂×C₄×C₃₁	1	2	yes	—
303	C₃₁	#22798	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
305	C₂×D₄	—	2	2	no	—
306	C₂×D₄	—	4	1	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂²×C₄	—	2	2	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	2	no	—
311	C₂²×C₄	—	2	2	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
316	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
317	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
318	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
320	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
321	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
322	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	—	2	2	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	2	no	—
329	C₂×C₈	—	2	2	no	—
330	C₂×C₈	—	2	2	no	—
331	C₂×C₈	—	2	2	no	—
332	C₂×C₈	—	2	2	no	—
333	C₂×C₈	—	2	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	4	1	no	—
340	C₄⋊C₄	—	4	1	no	—
341	C₂²⋊C₄	—	2	4	no	—
342	C₂²⋊C₄	—	2	4	no	—
343	C₂²⋊C₄	—	2	4	no	—
344	C₂²⋊C₄	—	2	4	no	—
345	C₂²⋊C₄	—	4	2	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	2	1	no	—
350	C₄²	—	2	1	no	—
351	C₄²	—	4	1	no	—
352	C₄²	—	4	1	no	—
353	C₄²	—	4	1	no	—
354	C₄²	C₂×D₄×C₃₁	1	2	yes	—
355	C₄²	#41×C₃₁	1	2	yes	—
356	C₄²	#41×C₃₁	1	2	yes	—
357	C₄²	#41×C₃₁	1	2	yes	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	2	no	—
362	C₂³	—	2	2	no	—
363	C₂³	—	2	2	no	—
364	C₂³	—	2	2	no	—
365	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
366	D₄	—	4	4	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	1	no	—
375	C₂×C₄	—	2	1	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	2	2	no	—
383	C₂×C₄	—	2	2	no	—
384	C₂×C₄	—	2	1	no	—
385	C₂×C₄	—	2	1	no	—
386	C₂×C₄	—	2	2	no	—
387	C₂×C₄	—	2	2	no	—
388	C₂×C₄	—	2	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	2	no	—
391	C₂×C₄	—	4	2	no	—
392	C₂×C₄	—	4	2	no	—
393	C₂×C₄	C₃₁×#117	1	2	yes	—
394	C₂×C₄	C₃₁×#117	1	2	yes	—
395	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
396	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
397	C₂×C₄	C₃₁×#131	1	2	yes	—
398	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
401	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
403	C₂×C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
404	C₈	—	2	4	no	—
405	C₈	—	4	2	no	—
406	C₈	—	4	2	no	—
407	C₈	—	4	2	no	—
408	C₂²	—	4	2	no	—
409	C₂²	—	4	2	no	—
410	C₂²	—	4	2	no	—
411	C₂²	—	4	2	no	—
412	C₂²	—	4	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	C₂×C₃₁×#117	1	1	yes	characteristic
416	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
417	C₂²	C₃₁×#520	1	1	yes	characteristic
418	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
419	C₂²	C₃₁×#568	1	1	yes	characteristic
420	C₂²	C₃₁×#568	1	1	yes	characteristic
421	C₂²	C₃₁×#568	1	1	yes	characteristic
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	—	2	2	no	—
426	C₄	—	2	2	no	—
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	1	no	—
431	C₄	C₃₁×#547	1	1	yes	characteristic
432	C₄	C₃₁×#550	1	1	yes	characteristic
433	C₄	C₃₁×#555	1	1	yes	characteristic
434	C₄	C₃₁×#556	1	1	yes	characteristic
435	C₂	—	8	2	no	—
436	C₂	C₃₁×#2354	1	1	yes	characteristic
437	C₂	C₃₁×#2921	1	1	yes	characteristic
438	C₂	C₃₁×#2925	1	1	yes	characteristic
439	C₂	C₃₁×#2932	1	1	yes	characteristic
440	C₂	C₃₁×#2935	1	1	yes	characteristic
441	C₂	C₃₁×#2992	1	1	yes	characteristic
442	C₂	C₃₁×#2995	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×#433	C₂	semidirect
4. C₃₁×#1924	C₂	semidirect
5. C₃₁×#1923	C₂	semidirect
6. C₃₁×#1842	C₂	semidirect
7. C₃₁×#1798	C₂	semidirect
8. C₃₁×#1771	C₂	semidirect
9. C₃₁×#1767	C₂	semidirect
71. C₃₁×C₄⋊C₈	D₄	semidirect
136. #22798	C₃₁	direct (N × H)
187. C₂×#433	C₂×C₃₁	semidirect
190. #1924	C₂×C₃₁	semidirect
191. #1923	C₂×C₃₁	semidirect
192. #1842	C₂×C₃₁	semidirect
193. #1798	C₂×C₃₁	semidirect
194. #1771	C₂×C₃₁	semidirect
195. #1767	C₂×C₃₁	semidirect
292. C₄⋊C₈	D₄×C₃₁	semidirect
303. C₃₁	#22798	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×#433	2
C₃₁×#2293	2
C₃₁×#1955	2
C₃₁×#1924	2
C₃₁×#1923	2
C₃₁×#1842	2
C₃₁×#1798	2
C₃₁×#1771	2
C₃₁×#1767	2
#22798	31

Subgroups — C31×#22798 #21479557

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22798 #21479557