Subgroups — C₃₁×#33172 #21489931

Order 7936.

1142 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#494	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#467	C₂	1	4	yes	maximal
3	C₂²×C₄×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2857	C₂	1	2	yes	maximal
5	C₃₁×#1969	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1956	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₁₆×C₃₁	C₂²	1	4	yes	—
9	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#494	—	2	4	no	—
14	C₃₁×#467	—	2	16	no	—
15	C₂×C₃₁×#107	C₂²	1	4	yes	—
16	C₂×C₃₁×#103	C₂²	1	8	yes	—
17	C₂×C₄×C₈×C₃₁	—	2	4	no	—
18	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
21	C₃₁×#391	C₂²	1	4	yes	—
22	C₃₁×#389	—	2	4	no	—
23	C₃₁×#384	C₂²	1	2	yes	—
24	C₃₁×#380	—	2	4	no	—
25	C₂²×Q₈×C₃₁	C₂³	1	4	yes	—
26	C₂³×C₄×C₃₁	D₄	1	2	yes	—
27	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
28	C₂×Q₁₆×C₃₁	—	2	8	no	—
29	C₂×Q₁₆×C₃₁	—	2	8	no	—
30	C₂×Q₁₆×C₃₁	—	2	8	no	—
31	C₂²×C₈×C₃₁	—	2	4	no	—
32	C₂²×C₈×C₃₁	—	2	4	no	—
33	C₂²×C₈×C₃₁	D₄	1	8	yes	—
34	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
35	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
36	C₃₁×C₄⋊Q₈	—	2	4	no	—
37	C₃₁×C₄⋊Q₈	—	2	4	no	—
38	C₃₁×C₂²⋊Q₈	—	2	16	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
40	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
41	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
44	C₂×C₄²×C₃₁	—	2	2	no	—
45	C₂×C₄²×C₃₁	—	2	2	no	—
46	C₂×C₄²×C₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
49	C₃₁×#107	—	2	8	no	—
50	C₃₁×#103	—	2	16	no	—
51	C₄×C₈×C₃₁	—	2	4	no	—
52	C₄×C₈×C₃₁	—	2	2	no	—
53	C₄×C₈×C₃₁	D₄	1	4	yes	—
54	C₃₁×#95	—	2	8	no	—
55	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
56	C₂×Q₈×C₃₁	—	2	8	no	—
57	C₂×Q₈×C₃₁	—	2	4	no	—
58	C₂×Q₈×C₃₁	—	2	4	no	—
59	C₂×Q₈×C₃₁	—	2	8	no	—
60	C₂²×C₄×C₃₁	—	2	4	no	—
61	C₂²×C₄×C₃₁	—	2	4	no	—
62	C₂²×C₄×C₃₁	—	2	1	no	—
63	C₂²×C₄×C₃₁	—	2	1	no	—
64	C₂²×C₄×C₃₁	—	2	2	no	—
65	C₂²×C₄×C₃₁	—	2	4	no	—
66	C₂²×C₄×C₃₁	—	2	4	no	—
67	C₂²×C₄×C₃₁	Q₁₆	1	8	yes	complemented
68	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
70	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
73	Q₁₆×C₃₁	—	4	16	no	—
74	C₂×C₈×C₃₁	—	2	2	no	—
75	C₂×C₈×C₃₁	—	2	2	no	—
76	C₂×C₈×C₃₁	—	2	16	no	—
77	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
79	C₂×C₈×C₃₁	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₃₁	#41	1	4	yes	—
81	C₄⋊C₄×C₃₁	—	2	8	no	—
82	C₄⋊C₄×C₃₁	—	4	8	no	—
83	C₄⋊C₄×C₃₁	—	4	8	no	—
84	C₄⋊C₄×C₃₁	—	4	8	no	—
85	C₂²⋊C₄×C₃₁	—	4	8	no	—
86	C₄²×C₃₁	—	2	4	no	—
87	C₄²×C₃₁	—	2	1	no	—
88	C₄²×C₃₁	—	2	1	no	—
89	C₄²×C₃₁	C₂×D₄	1	2	yes	—
90	C₄²×C₃₁	#41	1	2	yes	—
91	#33172	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	1	no	—
93	C₂³×C₃₁	—	2	1	no	—
94	C₂³×C₃₁	—	2	2	no	—
95	C₂³×C₃₁	#127	1	2	yes	—
96	C₂³×C₃₁	C₂×Q₁₆	1	4	yes	—
97	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₃₁	—	4	8	no	—
99	Q₈×C₃₁	—	4	8	no	—
100	C₂×C₄×C₃₁	—	2	1	no	—
101	C₂×C₄×C₃₁	—	2	2	no	—
102	C₂×C₄×C₃₁	—	2	1	no	—
103	C₂×C₄×C₃₁	—	2	2	no	—
104	C₂×C₄×C₃₁	—	2	4	no	—
105	C₂×C₄×C₃₁	—	2	4	no	—
106	C₂×C₄×C₃₁	—	2	16	no	—
107	C₂×C₄×C₃₁	—	4	4	no	—
108	C₂×C₄×C₃₁	—	4	4	no	—
109	C₂×C₄×C₃₁	—	4	4	no	—
110	C₂×C₄×C₃₁	—	4	4	no	—
111	C₂×C₄×C₃₁	—	4	8	no	—
112	C₂×C₄×C₃₁	#121	1	4	yes	—
113	C₂×C₄×C₃₁	#127	1	2	yes	—
114	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
115	C₂×C₄×C₃₁	C₂×Q₁₆	1	4	yes	—
116	C₂×C₄×C₃₁	#135	1	4	yes	—
117	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
119	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
120	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
121	C₈×C₃₁	—	2	4	no	—
122	C₈×C₃₁	#121	1	8	yes	—
123	C₂×#494	C₂×C₃₁	1	1	yes	characteristic, complemented
124	C₂×#467	C₂×C₃₁	1	4	yes	—
125	C₂²×C₄×C₈	C₂×C₃₁	1	1	yes	characteristic
126	#2857	C₂×C₃₁	1	2	yes	—
127	#1969	C₂×C₃₁	1	4	yes	complemented
128	#1956	C₂×C₃₁	1	2	yes	complemented
129	#1845	C₂×C₃₁	1	1	yes	characteristic, complemented
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	—	2	4	no	—
133	C₂²×C₃₁	—	2	4	no	—
134	C₂²×C₃₁	#494	1	4	yes	complemented
135	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
136	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
137	C₂²×C₃₁	#532	1	1	yes	characteristic
138	C₂²×C₃₁	C₂²×Q₁₆	1	2	yes	—
139	C₂²×C₃₁	C₂×#135	1	2	yes	—
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	4	no	—
142	C₄×C₃₁	—	8	4	no	—
143	C₄×C₃₁	—	8	4	no	—
144	C₄×C₃₁	#467	1	8	yes	complemented
145	C₄×C₃₁	C₂×#121	1	2	yes	—
146	C₄×C₃₁	#532	1	2	yes	—
147	C₂²×Q₁₆	C₂²×C₃₁	1	4	yes	—
148	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
150	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
151	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
152	#494	—	2	4	no	—
153	#467	—	2	16	no	—
154	C₂×#107	C₂²×C₃₁	1	4	yes	—
155	C₂×#103	C₂²×C₃₁	1	8	yes	—
156	C₂×C₄×C₈	—	2	4	no	—
157	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
160	#391	C₂²×C₃₁	1	4	yes	—
161	#389	—	2	4	no	—
162	#384	C₂²×C₃₁	1	2	yes	—
163	#380	—	2	4	no	—
164	C₂×C₃₁	—	2	4	no	—
165	C₂×C₃₁	#2856	1	1	yes	characteristic
166	C₂×C₃₁	C₂×#467	1	2	yes	—
167	C₂×C₃₁	C₂×#494	1	1	yes	characteristic
168	C₂×C₃₁	#3147	1	1	yes	characteristic
169	C₂×C₃₁	#3170	1	2	yes	—
170	C₂²×Q₈	C₂³×C₃₁	1	4	yes	—
171	C₂³×C₄	D₄×C₃₁	1	2	yes	—
172	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
173	C₂×Q₁₆	—	2	8	no	—
174	C₂×Q₁₆	—	2	8	no	—
175	C₂×Q₁₆	—	2	8	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	D₄×C₃₁	1	8	yes	—
179	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
180	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
181	C₄⋊Q₈	—	2	4	no	—
182	C₄⋊Q₈	—	2	4	no	—
183	C₂²⋊Q₈	—	2	16	no	—
184	C₂×C₄⋊C₄	—	2	4	no	—
185	C₂×C₄⋊C₄	—	2	4	no	—
186	C₂×C₄⋊C₄	—	2	4	no	—
187	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
188	C₂×C₂²⋊C₄	—	2	4	no	—
189	C₂×C₄²	—	2	2	no	—
190	C₂×C₄²	—	2	2	no	—
191	C₂×C₄²	D₄×C₃₁	1	2	yes	—
192	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
194	#107	—	2	8	no	—
195	#103	—	2	16	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	D₄×C₃₁	1	4	yes	—
199	#95	—	2	8	no	—
200	C₃₁	#33172	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
202	C₂×Q₈	—	2	8	no	—
203	C₂×Q₈	—	2	4	no	—
204	C₂×Q₈	—	2	4	no	—
205	C₂×Q₈	—	2	8	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	Q₁₆×C₃₁	1	8	yes	complemented
214	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
216	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
219	Q₁₆	—	4	16	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	16	no	—
223	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
224	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
225	C₂×C₈	C₂×D₄×C₃₁	1	4	yes	—
226	C₂×C₈	#41×C₃₁	1	4	yes	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	4	8	no	—
229	C₄⋊C₄	—	4	8	no	—
230	C₄⋊C₄	—	4	8	no	—
231	C₂²⋊C₄	—	4	8	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	C₂×D₄×C₃₁	1	2	yes	—
236	C₄²	#41×C₃₁	1	2	yes	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	C₃₁×#127	1	2	yes	—
241	C₂³	C₂×Q₁₆×C₃₁	1	4	yes	—
242	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
243	Q₈	—	4	8	no	—
244	Q₈	—	4	8	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	16	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	C₃₁×#121	1	4	yes	—
258	C₂×C₄	C₃₁×#127	1	2	yes	—
259	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
260	C₂×C₄	C₂×Q₁₆×C₃₁	1	4	yes	—
261	C₂×C₄	C₃₁×#135	1	4	yes	—
262	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
263	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
264	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
265	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	C₃₁×#121	1	8	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	C₃₁×#494	1	4	yes	complemented
273	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
274	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
275	C₂²	C₃₁×#532	1	1	yes	characteristic
276	C₂²	C₂²×Q₁₆×C₃₁	1	2	yes	—
277	C₂²	C₂×C₃₁×#135	1	2	yes	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	8	4	no	—
281	C₄	—	8	4	no	—
282	C₄	C₃₁×#467	1	8	yes	complemented
283	C₄	C₂×C₃₁×#121	1	2	yes	—
284	C₄	C₃₁×#532	1	2	yes	—
285	C₂	—	2	4	no	—
286	C₂	C₃₁×#2856	1	1	yes	characteristic
287	C₂	C₂×C₃₁×#467	1	2	yes	—
288	C₂	C₂×C₃₁×#494	1	1	yes	characteristic
289	C₂	C₃₁×#3147	1	1	yes	characteristic
290	C₂	C₃₁×#3170	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂⁴×C₃₁	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#494	C₂	semidirect
5. C₃₁×#1969	C₂	semidirect
6. C₃₁×#1956	C₂	semidirect
7. C₃₁×#1845	C₂	semidirect
67. C₂²×C₄×C₃₁	Q₁₆	semidirect
91. #33172	C₃₁	direct (N × H)
123. C₂×#494	C₂×C₃₁	semidirect
127. #1969	C₂×C₃₁	semidirect
128. #1956	C₂×C₃₁	semidirect
129. #1845	C₂×C₃₁	semidirect
134. C₂²×C₃₁	#494	semidirect
144. C₄×C₃₁	#467	semidirect
200. C₃₁	#33172	direct (N × H)
213. C₂²×C₄	Q₁₆×C₃₁	semidirect
272. C₂²	C₃₁×#494	semidirect
282. C₄	C₃₁×#467	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#494	2
C₂×C₃₁×#467	2
C₂²×C₄×C₈×C₃₁	2
C₃₁×#2857	2
C₃₁×#1969	2
C₃₁×#1956	2
C₃₁×#1845	2
#33172	31

Subgroups — C31×#33172 #21489931

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33172 #21489931