Subgroups — C₃₁×#3505 #16843438

Order 3968.

1152 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#553	C₂	1	2	yes	maximal, complemented
2	C₃₁×#553	C₂	1	4	yes	maximal, complemented
3	C₃₁×#552	C₂	1	2	yes	maximal, complemented
4	C₃₁×#551	C₂	1	4	yes	maximal, complemented
5	C₃₁×#548	C₂	1	2	yes	maximal, complemented
6	C₃₁×#546	C₂	1	2	yes	maximal, complemented
7	C₃₁×#539	C₂	1	2	yes	maximal, complemented
8	C₃₁×#538	C₂	1	2	yes	maximal, complemented
9	C₃₁×#534	C₂	1	2	yes	maximal, complemented
10	C₃₁×#529	C₂	1	2	yes	maximal, complemented
11	C₃₁×#525	C₂	1	2	yes	maximal, complemented
12	C₂×C₃₁×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×C₃₁×#121	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#518	C₂	1	2	yes	maximal, complemented
15	C₂×C₄×D₄×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
17	C₂²×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
18	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic
19	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
21	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
23	C₃₁×#126	C₂²	1	4	yes	complemented
24	C₃₁×#126	C₂²	1	4	yes	complemented
25	C₃₁×#126	C₂²	1	4	yes	complemented
26	C₃₁×#125	C₂²	1	2	yes	complemented
27	C₃₁×#125	C₂²	1	4	yes	complemented
28	C₃₁×#124	C₂²	1	2	yes	complemented
29	C₃₁×#124	C₂²	1	4	yes	complemented
30	C₃₁×#123	C₂²	1	4	yes	—
31	C₃₁×#123	C₂²	1	4	yes	complemented
32	C₃₁×#123	C₂²	1	4	yes	complemented
33	C₃₁×#123	C₂²	1	4	yes	complemented
34	C₃₁×#123	C₂²	1	4	yes	complemented
35	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
36	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
37	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
38	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
39	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
40	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
41	C₃₁×#121	C₂²	1	2	yes	—
42	C₃₁×#121	C₂²	1	2	yes	—
43	C₃₁×#121	C₂²	1	2	yes	complemented
44	C₃₁×#121	C₂²	1	2	yes	complemented
45	C₃₁×#121	C₂²	1	2	yes	complemented
46	C₃₁×#121	C₂²	1	4	yes	complemented
47	C₃₁×#121	C₂²	1	4	yes	complemented
48	C₃₁×#121	C₂²	1	4	yes	complemented
49	C₃₁×#120	C₂²	1	4	yes	—
50	C₃₁×#120	C₂²	1	4	yes	complemented
51	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
52	C₄×D₄×C₃₁	C₂²	1	2	yes	—
53	C₄×D₄×C₃₁	C₂²	1	2	yes	—
54	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
55	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
56	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
57	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
58	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
59	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
60	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
61	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
62	C₃₁×#117	C₂²	1	4	yes	complemented
63	C₃₁×#117	C₂²	1	2	yes	complemented
64	C₃₁×#117	C₂²	1	4	yes	complemented
65	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
66	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
67	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
68	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
69	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
70	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
71	C₂×C₂²⋊C₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
72	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
73	C₂×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	complemented
74	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
75	C₂⁴×C₃₁	C₂³	1	2	yes	—
76	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
77	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
78	#41×C₃₁	—	4	4	no	—
79	C₂×Q₈×C₃₁	—	4	2	no	—
80	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
81	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
82	C₂×D₄×C₃₁	—	2	4	no	—
83	C₂×D₄×C₃₁	—	2	8	no	—
84	C₂×D₄×C₃₁	—	4	2	no	—
85	C₂×D₄×C₃₁	C₂³	1	2	yes	—
86	C₂×D₄×C₃₁	C₂³	1	4	yes	—
87	C₂×D₄×C₃₁	C₂³	1	2	yes	—
88	C₂×D₄×C₃₁	C₂³	1	4	yes	—
89	C₂×D₄×C₃₁	C₂³	1	2	yes	—
90	C₂×D₄×C₃₁	C₂³	1	4	yes	—
91	C₂×D₄×C₃₁	C₂³	1	4	yes	—
92	C₂×D₄×C₃₁	C₂³	1	2	yes	—
93	C₂²×C₄×C₃₁	—	4	2	no	—
94	C₂²×C₄×C₃₁	—	4	2	no	—
95	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
98	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
100	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
103	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
104	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
105	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
106	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
107	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
108	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
109	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
111	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
115	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
116	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
117	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
118	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
119	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
121	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
123	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
127	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
131	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
134	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
135	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
137	C₄²×C₃₁	C₂³	1	2	yes	—
138	C₄²×C₃₁	C₂³	1	2	yes	—
139	C₄²×C₃₁	C₂³	1	4	yes	—
140	C₄²×C₃₁	C₂³	1	4	yes	—
141	C₂³×C₃₁	—	2	2	no	—
142	C₂³×C₃₁	—	2	4	no	—
143	C₂³×C₃₁	—	4	1	no	—
144	C₂³×C₃₁	—	4	1	no	—
145	C₂³×C₃₁	—	4	1	no	—
146	C₂³×C₃₁	—	4	1	no	—
147	C₂³×C₃₁	—	4	2	no	—
148	C₂³×C₃₁	—	4	2	no	—
149	C₂³×C₃₁	#41	1	4	yes	complemented
150	C₂³×C₃₁	C₂⁴	1	2	yes	—
151	C₂³×C₃₁	C₂⁴	1	4	yes	—
152	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
153	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
154	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
155	Q₈×C₃₁	—	2	4	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	2	8	no	—
159	D₄×C₃₁	—	2	4	no	—
160	D₄×C₃₁	—	2	8	no	—
161	D₄×C₃₁	—	2	4	no	—
162	D₄×C₃₁	—	4	2	no	—
163	D₄×C₃₁	—	4	4	no	—
164	D₄×C₃₁	—	4	4	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	2	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	2	2	no	—
169	C₂×C₄×C₃₁	—	2	2	no	—
170	C₂×C₄×C₃₁	—	2	4	no	—
171	C₂×C₄×C₃₁	—	4	1	no	—
172	C₂×C₄×C₃₁	—	4	2	no	—
173	C₂×C₄×C₃₁	—	4	1	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	4	no	—
178	C₂×C₄×C₃₁	—	4	2	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	4	no	—
181	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
182	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
183	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
184	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
186	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
187	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
188	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
191	#3505	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
192	C₂²×C₃₁	—	2	1	no	—
193	C₂²×C₃₁	—	2	1	no	—
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	—	2	1	no	—
196	C₂²×C₃₁	—	2	1	no	—
197	C₂²×C₃₁	—	2	1	no	—
198	C₂²×C₃₁	—	2	2	no	—
199	C₂²×C₃₁	—	2	4	no	—
200	C₂²×C₃₁	—	2	2	no	—
201	C₂²×C₃₁	—	2	4	no	—
202	C₂²×C₃₁	—	2	4	no	—
203	C₂²×C₃₁	—	2	2	no	—
204	C₂²×C₃₁	—	4	4	no	—
205	C₂²×C₃₁	—	4	4	no	—
206	C₂²×C₃₁	—	8	4	no	—
207	C₂²×C₃₁	C₂×#41	1	2	yes	—
208	C₂²×C₃₁	C₂×#41	1	2	yes	—
209	C₂²×C₃₁	C₂×#41	1	2	yes	—
210	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
211	C₄×C₃₁	—	2	4	no	—
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	2	1	no	—
214	C₄×C₃₁	—	2	2	no	—
215	C₄×C₃₁	—	2	1	no	—
216	C₄×C₃₁	—	2	1	no	—
217	C₄×C₃₁	—	2	4	no	—
218	C₄×C₃₁	—	2	2	no	—
219	C₄×C₃₁	—	2	4	no	—
220	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
221	#553	C₂×C₃₁	1	2	yes	complemented
222	#553	C₂×C₃₁	1	4	yes	complemented
223	#552	C₂×C₃₁	1	2	yes	complemented
224	#551	C₂×C₃₁	1	4	yes	complemented
225	#548	C₂×C₃₁	1	2	yes	complemented
226	#546	C₂×C₃₁	1	2	yes	complemented
227	#539	C₂×C₃₁	1	2	yes	complemented
228	#538	C₂×C₃₁	1	2	yes	complemented
229	#534	C₂×C₃₁	1	2	yes	complemented
230	#529	C₂×C₃₁	1	2	yes	complemented
231	#525	C₂×C₃₁	1	2	yes	complemented
232	C₂×C₂²⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
233	C₂×#121	C₂×C₃₁	1	1	yes	characteristic, complemented
234	#518	C₂×C₃₁	1	2	yes	complemented
235	C₂×C₄×D₄	C₂×C₃₁	1	1	yes	characteristic, complemented
236	C₂×C₃₁	—	2	2	no	—
237	C₂×C₃₁	—	2	2	no	—
238	C₂×C₃₁	—	2	2	no	—
239	C₂×C₃₁	—	4	2	no	—
240	C₂×C₃₁	—	4	4	no	—
241	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
242	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
243	C₂×C₃₁	#585	1	1	yes	characteristic
244	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
245	C₂²×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
246	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic
247	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
248	C₂²×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
249	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic
250	C₂³×C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
251	#126	C₂²×C₃₁	1	4	yes	complemented
252	#126	C₂²×C₃₁	1	4	yes	complemented
253	#126	C₂²×C₃₁	1	4	yes	complemented
254	#125	C₂²×C₃₁	1	2	yes	complemented
255	#125	C₂²×C₃₁	1	4	yes	complemented
256	#124	C₂²×C₃₁	1	2	yes	complemented
257	#124	C₂²×C₃₁	1	4	yes	complemented
258	#123	C₂²×C₃₁	1	4	yes	—
259	#123	C₂²×C₃₁	1	4	yes	complemented
260	#123	C₂²×C₃₁	1	4	yes	complemented
261	#123	C₂²×C₃₁	1	4	yes	complemented
262	#123	C₂²×C₃₁	1	4	yes	complemented
263	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
264	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
265	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
266	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
267	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
268	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
269	#121	C₂²×C₃₁	1	2	yes	—
270	#121	C₂²×C₃₁	1	2	yes	—
271	#121	C₂²×C₃₁	1	2	yes	complemented
272	#121	C₂²×C₃₁	1	2	yes	complemented
273	#121	C₂²×C₃₁	1	2	yes	complemented
274	#121	C₂²×C₃₁	1	4	yes	complemented
275	#121	C₂²×C₃₁	1	4	yes	complemented
276	#121	C₂²×C₃₁	1	4	yes	complemented
277	#120	C₂²×C₃₁	1	4	yes	—
278	#120	C₂²×C₃₁	1	4	yes	complemented
279	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
280	C₄×D₄	C₂²×C₃₁	1	2	yes	—
281	C₄×D₄	C₂²×C₃₁	1	2	yes	—
282	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
283	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
284	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
285	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
286	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
287	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
288	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
289	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
290	#117	C₂²×C₃₁	1	4	yes	complemented
291	#117	C₂²×C₃₁	1	2	yes	complemented
292	#117	C₂²×C₃₁	1	4	yes	complemented
293	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
294	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
295	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
296	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
297	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
298	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
299	C₂×C₂²⋊C₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
300	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
301	C₂×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	complemented
302	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
303	C₃₁	#3505	1	1	yes	characteristic, direct product factor, Sylow
304	C₂⁴	C₂³×C₃₁	1	2	yes	—
305	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
306	C₂⁴	C₂³×C₃₁	1	1	yes	characteristic
307	#41	—	4	4	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
310	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
311	C₂×D₄	—	2	4	no	—
312	C₂×D₄	—	2	8	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	C₂³×C₃₁	1	2	yes	—
315	C₂×D₄	C₂³×C₃₁	1	4	yes	—
316	C₂×D₄	C₂³×C₃₁	1	2	yes	—
317	C₂×D₄	C₂³×C₃₁	1	4	yes	—
318	C₂×D₄	C₂³×C₃₁	1	2	yes	—
319	C₂×D₄	C₂³×C₃₁	1	4	yes	—
320	C₂×D₄	C₂³×C₃₁	1	4	yes	—
321	C₂×D₄	C₂³×C₃₁	1	2	yes	—
322	C₂²×C₄	—	4	2	no	—
323	C₂²×C₄	—	4	2	no	—
324	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
326	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
327	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
328	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
329	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
330	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
331	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
332	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
333	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
334	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
337	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
338	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
340	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
344	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
345	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
347	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
348	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
349	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
350	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
351	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
352	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
353	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
354	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
355	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
356	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
357	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
358	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
359	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
360	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
361	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
362	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
363	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
364	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
365	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
366	C₄²	C₂³×C₃₁	1	2	yes	—
367	C₄²	C₂³×C₃₁	1	2	yes	—
368	C₄²	C₂³×C₃₁	1	4	yes	—
369	C₄²	C₂³×C₃₁	1	4	yes	—
370	C₂³	—	2	2	no	—
371	C₂³	—	2	4	no	—
372	C₂³	—	4	1	no	—
373	C₂³	—	4	1	no	—
374	C₂³	—	4	1	no	—
375	C₂³	—	4	1	no	—
376	C₂³	—	4	2	no	—
377	C₂³	—	4	2	no	—
378	C₂³	#41×C₃₁	1	4	yes	complemented
379	C₂³	C₂⁴×C₃₁	1	2	yes	—
380	C₂³	C₂⁴×C₃₁	1	4	yes	—
381	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
382	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
383	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
384	Q₈	—	2	4	no	—
385	Q₈	—	4	2	no	—
386	D₄	—	2	4	no	—
387	D₄	—	2	8	no	—
388	D₄	—	2	4	no	—
389	D₄	—	2	8	no	—
390	D₄	—	2	4	no	—
391	D₄	—	4	2	no	—
392	D₄	—	4	4	no	—
393	D₄	—	4	4	no	—
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	2	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	4	no	—
400	C₂×C₄	—	4	1	no	—
401	C₂×C₄	—	4	2	no	—
402	C₂×C₄	—	4	1	no	—
403	C₂×C₄	—	4	1	no	—
404	C₂×C₄	—	4	1	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	4	no	—
407	C₂×C₄	—	4	2	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	4	no	—
410	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
411	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
412	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
413	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
414	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
415	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
416	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
417	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
418	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
419	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
420	C₂²	—	2	1	no	—
421	C₂²	—	2	1	no	—
422	C₂²	—	2	1	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	—	2	2	no	—
427	C₂²	—	2	4	no	—
428	C₂²	—	2	2	no	—
429	C₂²	—	2	4	no	—
430	C₂²	—	2	4	no	—
431	C₂²	—	2	2	no	—
432	C₂²	—	4	4	no	—
433	C₂²	—	4	4	no	—
434	C₂²	—	8	4	no	—
435	C₂²	C₂×#41×C₃₁	1	2	yes	—
436	C₂²	C₂×#41×C₃₁	1	2	yes	—
437	C₂²	C₂×#41×C₃₁	1	2	yes	—
438	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
439	C₄	—	2	4	no	—
440	C₄	—	2	2	no	—
441	C₄	—	2	1	no	—
442	C₄	—	2	2	no	—
443	C₄	—	2	1	no	—
444	C₄	—	2	1	no	—
445	C₄	—	2	4	no	—
446	C₄	—	2	2	no	—
447	C₄	—	2	4	no	—
448	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
449	C₂	—	2	2	no	—
450	C₂	—	2	2	no	—
451	C₂	—	2	2	no	—
452	C₂	—	4	2	no	—
453	C₂	—	4	4	no	—
454	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
455	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
456	C₂	C₃₁×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#553	C₂	semidirect
2. C₃₁×#553	C₂	semidirect
3. C₃₁×#552	C₂	semidirect
4. C₃₁×#551	C₂	semidirect
5. C₃₁×#548	C₂	semidirect
6. C₃₁×#546	C₂	semidirect
7. C₃₁×#539	C₂	semidirect
8. C₃₁×#538	C₂	semidirect
9. C₃₁×#534	C₂	semidirect
10. C₃₁×#529	C₂	semidirect
11. C₃₁×#525	C₂	semidirect
12. C₂×C₃₁×C₂²⋊Q₈	C₂	semidirect
13. C₂×C₃₁×#121	C₂	semidirect
14. C₃₁×#518	C₂	semidirect
15. C₂×C₄×D₄×C₃₁	C₂	semidirect
16. C₂×#41×C₃₁	C₂²	semidirect
17. C₂²×Q₈×C₃₁	C₂²	semidirect
20. C₂²×D₄×C₃₁	C₂²	semidirect
22. C₂³×C₄×C₃₁	C₂²	semidirect
23. C₃₁×#126	C₂²	semidirect
24. C₃₁×#126	C₂²	semidirect
25. C₃₁×#126	C₂²	semidirect
26. C₃₁×#125	C₂²	semidirect
27. C₃₁×#125	C₂²	semidirect
28. C₃₁×#124	C₂²	semidirect
29. C₃₁×#124	C₂²	semidirect
31. C₃₁×#123	C₂²	semidirect
32. C₃₁×#123	C₂²	semidirect
33. C₃₁×#123	C₂²	semidirect
34. C₃₁×#123	C₂²	semidirect
35. C₃₁×C₂²⋊Q₈	C₂²	semidirect
36. C₃₁×C₂²⋊Q₈	C₂²	semidirect
37. C₃₁×C₂²⋊Q₈	C₂²	semidirect
38. C₃₁×C₂²⋊Q₈	C₂²	semidirect
39. C₃₁×C₂²⋊Q₈	C₂²	semidirect
40. C₃₁×C₂²⋊Q₈	C₂²	semidirect
43. C₃₁×#121	C₂²	semidirect
44. C₃₁×#121	C₂²	semidirect
45. C₃₁×#121	C₂²	semidirect
46. C₃₁×#121	C₂²	semidirect
47. C₃₁×#121	C₂²	semidirect
48. C₃₁×#121	C₂²	semidirect
50. C₃₁×#120	C₂²	semidirect
51. C₄×Q₈×C₃₁	C₂²	semidirect
54. C₄×D₄×C₃₁	C₂²	semidirect
55. C₄×D₄×C₃₁	C₂²	semidirect
56. C₄×D₄×C₃₁	C₂²	semidirect
57. C₄×D₄×C₃₁	C₂²	semidirect
58. C₄×D₄×C₃₁	C₂²	semidirect
59. C₄×D₄×C₃₁	C₂²	semidirect
60. C₄×D₄×C₃₁	C₂²	semidirect
61. C₄×D₄×C₃₁	C₂²	semidirect
62. C₃₁×#117	C₂²	semidirect
63. C₃₁×#117	C₂²	semidirect
64. C₃₁×#117	C₂²	semidirect
65. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
66. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
67. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
68. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
71. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
72. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
73. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
74. C₂×C₄²×C₃₁	C₂²	semidirect
149. C₂³×C₃₁	#41	semidirect
191. #3505	C₃₁	direct (N × H)
221. #553	C₂×C₃₁	semidirect
222. #553	C₂×C₃₁	semidirect
223. #552	C₂×C₃₁	semidirect
224. #551	C₂×C₃₁	semidirect
225. #548	C₂×C₃₁	semidirect
226. #546	C₂×C₃₁	semidirect
227. #539	C₂×C₃₁	semidirect
228. #538	C₂×C₃₁	semidirect
229. #534	C₂×C₃₁	semidirect
230. #529	C₂×C₃₁	semidirect
231. #525	C₂×C₃₁	semidirect
232. C₂×C₂²⋊Q₈	C₂×C₃₁	semidirect
233. C₂×#121	C₂×C₃₁	semidirect
234. #518	C₂×C₃₁	semidirect
235. C₂×C₄×D₄	C₂×C₃₁	semidirect
244. C₂×#41	C₂²×C₃₁	semidirect
245. C₂²×Q₈	C₂²×C₃₁	semidirect
248. C₂²×D₄	C₂²×C₃₁	semidirect
250. C₂³×C₄	C₂²×C₃₁	semidirect
251. #126	C₂²×C₃₁	semidirect
252. #126	C₂²×C₃₁	semidirect
253. #126	C₂²×C₃₁	semidirect
254. #125	C₂²×C₃₁	semidirect
255. #125	C₂²×C₃₁	semidirect
256. #124	C₂²×C₃₁	semidirect
257. #124	C₂²×C₃₁	semidirect
259. #123	C₂²×C₃₁	semidirect
260. #123	C₂²×C₃₁	semidirect
261. #123	C₂²×C₃₁	semidirect
262. #123	C₂²×C₃₁	semidirect
263. C₂²⋊Q₈	C₂²×C₃₁	semidirect
264. C₂²⋊Q₈	C₂²×C₃₁	semidirect
265. C₂²⋊Q₈	C₂²×C₃₁	semidirect
266. C₂²⋊Q₈	C₂²×C₃₁	semidirect
267. C₂²⋊Q₈	C₂²×C₃₁	semidirect
268. C₂²⋊Q₈	C₂²×C₃₁	semidirect
271. #121	C₂²×C₃₁	semidirect
272. #121	C₂²×C₃₁	semidirect
273. #121	C₂²×C₃₁	semidirect
274. #121	C₂²×C₃₁	semidirect
275. #121	C₂²×C₃₁	semidirect
276. #121	C₂²×C₃₁	semidirect
278. #120	C₂²×C₃₁	semidirect
279. C₄×Q₈	C₂²×C₃₁	semidirect
282. C₄×D₄	C₂²×C₃₁	semidirect
283. C₄×D₄	C₂²×C₃₁	semidirect
284. C₄×D₄	C₂²×C₃₁	semidirect
285. C₄×D₄	C₂²×C₃₁	semidirect
286. C₄×D₄	C₂²×C₃₁	semidirect
287. C₄×D₄	C₂²×C₃₁	semidirect
288. C₄×D₄	C₂²×C₃₁	semidirect
289. C₄×D₄	C₂²×C₃₁	semidirect
290. #117	C₂²×C₃₁	semidirect
291. #117	C₂²×C₃₁	semidirect
292. #117	C₂²×C₃₁	semidirect
293. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
294. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
295. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
296. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
299. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
300. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
301. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
302. C₂×C₄²	C₂²×C₃₁	semidirect
303. C₃₁	#3505	direct (N × H)
378. C₂³	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#553	2
C₃₁×#552	2
C₃₁×#551	2
C₃₁×#548	2
C₃₁×#546	2
C₃₁×#539	2
C₃₁×#538	2
C₃₁×#534	2
C₃₁×#529	2
C₃₁×#525	2
C₂×C₃₁×C₂²⋊Q₈	2
C₂×C₃₁×#121	2
C₃₁×#518	2
C₂×C₄×D₄×C₃₁	2
#3505	31

Subgroups — C31×#3505 #16843438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3505 #16843438