Subgroups — C₃₁×#22706 #21479465

Order 7936.

938 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2352	C₂	1	2	yes	maximal
3	C₃₁×#1949	C₂	1	2	yes	maximal, complemented
4	C₃₁×#1854	C₂	1	2	yes	maximal, complemented
5	C₃₁×#1853	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1835	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1808	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1807	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₃₁×#106	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1735	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#340	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₂³×C₈×C₃₁	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#117	C₂²	1	2	yes	—
16	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
17	C₈×D₄×C₃₁	C₄	1	8	yes	complemented
18	C₂×C₃₁×#107	C₂²	1	2	yes	—
19	C₂×C₃₁×#106	C₂²	1	2	yes	—
20	C₂×C₃₁×#106	C₂²	1	2	yes	—
21	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
25	C₃₁×#388	—	2	2	no	—
26	C₃₁×#387	—	2	2	no	—
27	C₃₁×#386	—	2	2	no	—
28	C₃₁×#382	C₂²	1	2	yes	—
29	C₃₁×#380	C₂²	1	2	yes	—
30	C₃₁×#380	C₂²	1	2	yes	—
31	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
32	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
33	C₃₁×#340	C₂²	1	1	yes	characteristic
34	C₃₁×#340	C₂²	1	1	yes	characteristic
35	C₃₁×#340	C₂²	1	1	yes	characteristic
36	C₃₁×#340	C₂²	1	2	yes	—
37	C₃₁×#340	C₂²	1	2	yes	—
38	C₃₁×#340	C₂²	1	2	yes	—
39	C₃₁×#340	C₂²	1	1	yes	characteristic
40	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
41	C₂³×C₄×C₃₁	—	2	2	no	—
42	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
43	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₃₁	—	4	2	no	—
45	C₂²×C₈×C₃₁	—	4	2	no	—
46	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	—
47	C₂²×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
48	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
52	C₄×D₄×C₃₁	C₂×C₄	1	4	yes	—
53	C₄×D₄×C₃₁	Q₈	1	4	yes	—
54	C₃₁×#117	—	2	4	no	—
55	C₃₁×#117	—	4	2	no	—
56	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
57	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₂×C₄⋊C₄×C₃₁	—	4	2	no	—
59	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
61	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
62	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
65	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
66	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
68	C₂×C₂²⋊C₄×C₃₁	D₄	1	1	yes	characteristic
69	C₂×C₄²×C₃₁	—	2	2	no	—
70	C₂×C₄²×C₃₁	—	2	2	no	—
71	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
72	C₃₁×#107	—	4	2	no	—
73	C₃₁×#106	—	2	4	no	—
74	C₃₁×#106	—	4	2	no	—
75	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
76	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	—
77	C₃₁×C₂²⋊C₈	C₂×C₄	1	4	yes	complemented
78	C₄×C₈×C₃₁	C₂×C₄	1	4	yes	complemented
79	C₃₁×#95	—	2	2	no	—
80	C₃₁×#95	—	2	2	no	—
81	C₃₁×#95	—	2	2	no	—
82	C₃₁×#95	—	2	2	no	—
83	C₃₁×#95	—	2	2	no	—
84	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
85	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
86	C₂×D₄×C₃₁	—	2	2	no	—
87	C₂×D₄×C₃₁	—	2	2	no	—
88	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
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	—	2	2	no	—
93	C₂²×C₄×C₃₁	—	2	2	no	—
94	C₂²×C₄×C₃₁	—	2	2	no	—
95	C₂²×C₄×C₃₁	—	2	2	no	—
96	C₂²×C₄×C₃₁	—	2	2	no	—
97	C₂²×C₄×C₃₁	—	4	4	no	—
98	C₂²×C₄×C₃₁	—	4	4	no	—
99	C₂²×C₄×C₃₁	—	8	2	no	—
100	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
107	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
108	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
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	—	4	4	no	—
114	C₂×C₈×C₃₁	—	4	4	no	—
115	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
116	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
117	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₃₁	—	2	2	no	—
120	C₄⋊C₄×C₃₁	—	2	2	no	—
121	C₄⋊C₄×C₃₁	—	2	2	no	—
122	C₄⋊C₄×C₃₁	—	2	2	no	—
123	C₄⋊C₄×C₃₁	—	2	2	no	—
124	C₄⋊C₄×C₃₁	—	2	2	no	—
125	C₄⋊C₄×C₃₁	—	2	2	no	—
126	C₄⋊C₄×C₃₁	—	4	4	no	—
127	C₄⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
128	C₄⋊C₄×C₃₁	#41	1	4	yes	—
129	C₂²⋊C₄×C₃₁	—	4	4	no	—
130	C₂²⋊C₄×C₃₁	—	4	4	no	—
131	C₂²⋊C₄×C₃₁	—	8	2	no	—
132	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
133	C₂²⋊C₄×C₃₁	C₄⋊C₄	1	4	yes	—
134	C₄²×C₃₁	—	4	4	no	—
135	C₄²×C₃₁	—	8	2	no	—
136	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
137	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
138	#22706	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
139	C₂³×C₃₁	—	2	2	no	—
140	C₂³×C₃₁	—	2	2	no	—
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	2	1	no	—
143	C₂³×C₃₁	—	2	1	no	—
144	C₂³×C₃₁	—	2	2	no	—
145	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
146	C₂³×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
147	C₂³×C₃₁	#135	1	2	yes	—
148	C₂³×C₃₁	#135	1	2	yes	—
149	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
150	D₄×C₃₁	—	2	8	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	1	no	—
155	C₂×C₄×C₃₁	—	2	2	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	4	no	—
161	C₂×C₄×C₃₁	—	2	4	no	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	4	2	no	—
164	C₂×C₄×C₃₁	—	4	2	no	—
165	C₂×C₄×C₃₁	—	4	2	no	—
166	C₂×C₄×C₃₁	—	4	2	no	—
167	C₂×C₄×C₃₁	—	4	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	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	2	no	—
174	C₂×C₄×C₃₁	—	8	8	no	—
175	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
176	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
177	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
178	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
179	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
182	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
183	C₈×C₃₁	—	2	2	no	—
184	C₈×C₃₁	—	2	2	no	—
185	C₈×C₃₁	—	2	2	no	—
186	C₈×C₃₁	—	2	2	no	—
187	C₂×C₈×D₄	C₂×C₃₁	1	1	yes	characteristic
188	#2352	C₂×C₃₁	1	2	yes	—
189	#1949	C₂×C₃₁	1	2	yes	complemented
190	#1854	C₂×C₃₁	1	2	yes	complemented
191	#1853	C₂×C₃₁	1	2	yes	complemented
192	#1835	C₂×C₃₁	1	1	yes	characteristic, complemented
193	#1808	C₂×C₃₁	1	1	yes	characteristic, complemented
194	#1807	C₂×C₃₁	1	1	yes	characteristic, complemented
195	C₄×#106	C₂×C₃₁	1	1	yes	characteristic, complemented
196	#1735	C₂×C₃₁	1	1	yes	characteristic, complemented
197	C₂×#340	C₂×C₃₁	1	1	yes	characteristic, complemented
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	2	2	no	—
200	C₂²×C₃₁	—	2	4	no	—
201	C₂²×C₃₁	—	2	4	no	—
202	C₂²×C₃₁	—	2	2	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	4	2	no	—
207	C₂²×C₃₁	#427	1	4	yes	—
208	C₂²×C₃₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
209	C₂²×C₃₁	#546	1	2	yes	—
210	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
211	C₂²×C₃₁	C₂×#135	1	1	yes	characteristic
212	C₂²×C₃₁	#577	1	2	yes	—
213	C₄×C₃₁	—	2	2	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	2	2	no	—
216	C₄×C₃₁	—	2	2	no	—
217	C₄×C₃₁	—	4	4	no	—
218	C₄×C₃₁	—	8	2	no	—
219	C₄×C₃₁	—	8	2	no	—
220	C₄×C₃₁	—	8	2	no	—
221	C₄×C₃₁	C₄×#41	1	1	yes	characteristic
222	C₄×C₃₁	#518	1	1	yes	characteristic
223	C₄×C₃₁	#554	1	2	yes	—
224	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
225	C₂³×C₈	C₂²×C₃₁	1	1	yes	characteristic
226	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
227	C₂×#117	C₂²×C₃₁	1	2	yes	—
228	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
229	C₈×D₄	C₄×C₃₁	1	8	yes	complemented
230	C₂×#107	C₂²×C₃₁	1	2	yes	—
231	C₂×#106	C₂²×C₃₁	1	2	yes	—
232	C₂×#106	C₂²×C₃₁	1	2	yes	—
233	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
234	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
237	#388	—	2	2	no	—
238	#387	—	2	2	no	—
239	#386	—	2	2	no	—
240	#382	C₂²×C₃₁	1	2	yes	—
241	#380	C₂²×C₃₁	1	2	yes	—
242	#380	C₂²×C₃₁	1	2	yes	—
243	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
244	C₄×C₂²⋊C₄	—	2	2	no	—
245	#340	C₂²×C₃₁	1	1	yes	characteristic
246	#340	C₂²×C₃₁	1	1	yes	characteristic
247	#340	C₂²×C₃₁	1	1	yes	characteristic
248	#340	C₂²×C₃₁	1	2	yes	—
249	#340	C₂²×C₃₁	1	2	yes	—
250	#340	C₂²×C₃₁	1	2	yes	—
251	#340	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₃₁	—	2	4	no	—
253	C₂×C₃₁	—	4	2	no	—
254	C₂×C₃₁	#2349	1	1	yes	characteristic
255	C₂×C₃₁	C₂×#427	1	1	yes	characteristic
256	C₂×C₃₁	#2912	1	1	yes	characteristic
257	C₂×C₃₁	#3296	1	2	yes	—
258	C₂×C₃₁	#3301	1	2	yes	—
259	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
260	C₂³×C₄	—	2	2	no	—
261	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
262	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
263	C₂²×C₈	—	4	2	no	—
264	C₂²×C₈	—	4	2	no	—
265	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	—
266	C₂²×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
267	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
268	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
269	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
270	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
271	C₄×D₄	C₂×C₄×C₃₁	1	4	yes	—
272	C₄×D₄	Q₈×C₃₁	1	4	yes	—
273	#117	—	2	4	no	—
274	#117	—	4	2	no	—
275	C₂×C₄⋊C₄	—	2	4	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	4	2	no	—
278	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
279	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
280	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
282	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
283	C₂×C₂²⋊C₄	—	2	2	no	—
284	C₂×C₂²⋊C₄	—	2	2	no	—
285	C₂×C₂²⋊C₄	—	2	2	no	—
286	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
287	C₂×C₂²⋊C₄	D₄×C₃₁	1	1	yes	characteristic
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
291	#107	—	4	2	no	—
292	#106	—	2	4	no	—
293	#106	—	4	2	no	—
294	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
295	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
296	C₂²⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
297	C₄×C₈	C₂×C₄×C₃₁	1	4	yes	complemented
298	#95	—	2	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	2	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	C₃₁	#22706	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
305	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
306	C₂×D₄	—	2	2	no	—
307	C₂×D₄	—	2	2	no	—
308	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
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	—	2	2	no	—
315	C₂²×C₄	—	2	2	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	4	4	no	—
318	C₂²×C₄	—	4	4	no	—
319	C₂²×C₄	—	8	2	no	—
320	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
321	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
323	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
327	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
328	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
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	—	4	4	no	—
334	C₂×C₈	—	4	4	no	—
335	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
336	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
337	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
338	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	2	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	2	2	no	—
344	C₄⋊C₄	—	2	2	no	—
345	C₄⋊C₄	—	2	2	no	—
346	C₄⋊C₄	—	4	4	no	—
347	C₄⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
348	C₄⋊C₄	#41×C₃₁	1	4	yes	—
349	C₂²⋊C₄	—	4	4	no	—
350	C₂²⋊C₄	—	4	4	no	—
351	C₂²⋊C₄	—	8	2	no	—
352	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
353	C₂²⋊C₄	C₄⋊C₄×C₃₁	1	4	yes	—
354	C₄²	—	4	4	no	—
355	C₄²	—	8	2	no	—
356	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
357	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
358	C₂³	—	2	2	no	—
359	C₂³	—	2	2	no	—
360	C₂³	—	2	2	no	—
361	C₂³	—	2	1	no	—
362	C₂³	—	2	1	no	—
363	C₂³	—	2	2	no	—
364	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
365	C₂³	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
366	C₂³	C₃₁×#135	1	2	yes	—
367	C₂³	C₃₁×#135	1	2	yes	—
368	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
369	D₄	—	2	8	no	—
370	C₂×C₄	—	2	2	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	2	1	no	—
373	C₂×C₄	—	2	1	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	4	no	—
381	C₂×C₄	—	2	4	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	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	—	8	8	no	—
394	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
395	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
396	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
397	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
398	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
399	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
400	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
401	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
402	C₈	—	2	2	no	—
403	C₈	—	2	2	no	—
404	C₈	—	2	2	no	—
405	C₈	—	2	2	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	2	no	—
408	C₂²	—	2	4	no	—
409	C₂²	—	2	4	no	—
410	C₂²	—	2	2	no	—
411	C₂²	—	2	2	no	—
412	C₂²	—	2	2	no	—
413	C₂²	—	4	2	no	—
414	C₂²	—	4	2	no	—
415	C₂²	C₃₁×#427	1	4	yes	—
416	C₂²	C₂²×C₄⋊C₄×C₃₁	1	1	yes	characteristic
417	C₂²	C₃₁×#546	1	2	yes	—
418	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
419	C₂²	C₂×C₃₁×#135	1	1	yes	characteristic
420	C₂²	C₃₁×#577	1	2	yes	—
421	C₄	—	2	2	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	2	no	—
424	C₄	—	2	2	no	—
425	C₄	—	4	4	no	—
426	C₄	—	8	2	no	—
427	C₄	—	8	2	no	—
428	C₄	—	8	2	no	—
429	C₄	C₄×#41×C₃₁	1	1	yes	characteristic
430	C₄	C₃₁×#518	1	1	yes	characteristic
431	C₄	C₃₁×#554	1	2	yes	—
432	C₂	—	2	4	no	—
433	C₂	—	4	2	no	—
434	C₂	C₃₁×#2349	1	1	yes	characteristic
435	C₂	C₂×C₃₁×#427	1	1	yes	characteristic
436	C₂	C₃₁×#2912	1	1	yes	characteristic
437	C₂	C₃₁×#3296	1	2	yes	—
438	C₂	C₃₁×#3301	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	C₂²×D₄×C₃₁	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄×D₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1949	C₂	semidirect
4. C₃₁×#1854	C₂	semidirect
5. C₃₁×#1853	C₂	semidirect
6. C₃₁×#1835	C₂	semidirect
7. C₃₁×#1808	C₂	semidirect
8. C₃₁×#1807	C₂	semidirect
9. C₄×C₃₁×#106	C₂	semidirect
10. C₃₁×#1735	C₂	semidirect
11. C₂×C₃₁×#340	C₂	semidirect
17. C₈×D₄×C₃₁	C₄	semidirect
47. C₂²×C₈×C₃₁	C₂×C₄	semidirect
75. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
77. C₃₁×C₂²⋊C₈	C₂×C₄	semidirect
78. C₄×C₈×C₃₁	C₂×C₄	semidirect
138. #22706	C₃₁	direct (N × H)
189. #1949	C₂×C₃₁	semidirect
190. #1854	C₂×C₃₁	semidirect
191. #1853	C₂×C₃₁	semidirect
192. #1835	C₂×C₃₁	semidirect
193. #1808	C₂×C₃₁	semidirect
194. #1807	C₂×C₃₁	semidirect
195. C₄×#106	C₂×C₃₁	semidirect
196. #1735	C₂×C₃₁	semidirect
197. C₂×#340	C₂×C₃₁	semidirect
229. C₈×D₄	C₄×C₃₁	semidirect
266. C₂²×C₈	C₂×C₄×C₃₁	semidirect
294. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
296. C₂²⋊C₈	C₂×C₄×C₃₁	semidirect
297. C₄×C₈	C₂×C₄×C₃₁	semidirect
303. C₃₁	#22706	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×C₈×D₄×C₃₁	2
C₃₁×#2352	2
C₃₁×#1949	2
C₃₁×#1854	2
C₃₁×#1853	2
C₃₁×#1835	2
C₃₁×#1808	2
C₃₁×#1807	2
C₄×C₃₁×#106	2
C₃₁×#1735	2
C₂×C₃₁×#340	2
#22706	31

Subgroups — C31×#22706 #21479465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#22706 #21479465