Subgroups — C₃₁×#3233 #16843166

Order 3968.

332 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#529	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#499	C₂	1	2	yes	maximal, complemented
4	C₃₁×#488	C₂	1	2	yes	maximal, complemented
5	C₃₁×#485	C₂	1	2	yes	maximal, complemented
6	C₃₁×#482	C₂	1	2	yes	maximal
7	C₃₁×#477	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#476	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#475	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#474	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₃₁×#126	—	2	1	no	—
15	C₃₁×#125	C₂²	1	2	yes	—
16	C₃₁×#125	C₂²	1	1	yes	characteristic
17	C₃₁×#124	C₂²	1	1	yes	characteristic
18	C₃₁×#123	—	2	1	no	—
19	C₃₁×C₂²⋊Q₈	—	2	1	no	—
20	C₃₁×C₂²⋊Q₈	—	2	1	no	—
21	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃₁×#121	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₃₁	—	2	1	no	—
24	C₄×Q₈×C₃₁	—	2	1	no	—
25	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₃₁	—	2	1	no	—
27	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
28	C₃₁×#117	—	2	1	no	—
29	C₃₁×#117	C₂²	1	2	yes	—
30	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
31	C₃₁×#107	C₂²	1	2	yes	—
32	C₃₁×#107	C₂²	1	2	yes	—
33	C₃₁×#106	C₂²	1	2	yes	—
34	C₃₁×#106	C₂²	1	2	yes	—
35	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#103	C₂²	1	2	yes	—
38	C₃₁×#103	C₂²	1	2	yes	—
39	C₃₁×#102	C₂²	1	2	yes	—
40	C₃₁×#102	C₂²	1	2	yes	—
41	C₃₁×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₄×C₈×C₃₁	C₂²	1	2	yes	—
43	C₂×Q₈×C₃₁	—	2	1	no	—
44	C₂×Q₈×C₃₁	—	2	1	no	—
45	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₃₁	—	2	1	no	—
47	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₃₁	—	2	1	no	—
49	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
52	C₂×C₈×C₃₁	C₂³	1	2	yes	—
53	C₂×C₈×C₃₁	C₂³	1	2	yes	—
54	C₄⋊C₄×C₃₁	—	2	1	no	—
55	C₄⋊C₄×C₃₁	—	2	1	no	—
56	C₄⋊C₄×C₃₁	—	2	1	no	—
57	C₄⋊C₄×C₃₁	—	2	2	no	—
58	C₄⋊C₄×C₃₁	—	2	2	no	—
59	C₄⋊C₄×C₃₁	—	2	1	no	—
60	C₄⋊C₄×C₃₁	—	2	1	no	—
61	C₄⋊C₄×C₃₁	—	2	1	no	—
62	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
63	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂²⋊C₄×C₃₁	—	2	2	no	—
67	C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂²⋊C₄×C₃₁	—	2	1	no	—
70	C₂²⋊C₄×C₃₁	—	2	1	no	—
71	C₂²⋊C₄×C₃₁	—	2	1	no	—
72	C₄²×C₃₁	—	2	1	no	—
73	C₄²×C₃₁	—	2	2	no	—
74	C₄²×C₃₁	D₄	1	1	yes	characteristic
75	C₄²×C₃₁	D₄	1	1	yes	characteristic
76	C₄²×C₃₁	C₂³	1	1	yes	characteristic
77	C₄²×C₃₁	C₂³	1	1	yes	characteristic
78	C₂³×C₃₁	—	2	1	no	—
79	C₂³×C₃₁	C₂×D₄	1	1	yes	characteristic
80	Q₈×C₃₁	—	2	1	no	—
81	Q₈×C₃₁	—	2	1	no	—
82	Q₈×C₃₁	—	4	2	no	—
83	Q₈×C₃₁	—	4	2	no	—
84	D₄×C₃₁	—	2	1	no	—
85	D₄×C₃₁	—	2	1	no	—
86	D₄×C₃₁	—	4	2	no	—
87	C₂×C₄×C₃₁	—	2	2	no	—
88	C₂×C₄×C₃₁	—	2	1	no	—
89	C₂×C₄×C₃₁	—	2	1	no	—
90	C₂×C₄×C₃₁	—	2	1	no	—
91	C₂×C₄×C₃₁	—	2	1	no	—
92	C₂×C₄×C₃₁	—	2	2	no	—
93	C₂×C₄×C₃₁	—	2	2	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	4	1	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
98	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
99	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
103	C₈×C₃₁	—	2	2	no	—
104	C₈×C₃₁	—	2	2	no	—
105	#3233	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₃₁	—	2	1	no	—
107	C₂²×C₃₁	—	2	1	no	—
108	C₂²×C₃₁	—	2	1	no	—
109	C₂²×C₃₁	—	4	1	no	—
110	C₂²×C₃₁	—	4	1	no	—
111	C₂²×C₃₁	—	4	1	no	—
112	C₂²×C₃₁	C₂²×D₄	1	1	yes	characteristic
113	C₄×C₃₁	—	2	1	no	—
114	C₄×C₃₁	—	2	1	no	—
115	C₄×C₃₁	—	2	1	no	—
116	C₄×C₃₁	—	2	1	no	—
117	C₄×C₃₁	—	4	1	no	—
118	C₄×C₃₁	—	4	1	no	—
119	C₄×C₃₁	—	4	2	no	—
120	C₄×C₃₁	—	4	1	no	—
121	C₄×C₃₁	—	4	2	no	—
122	C₄×C₃₁	#135	1	2	yes	—
123	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
124	C₄×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
125	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
126	#529	C₂×C₃₁	1	1	yes	characteristic
127	#499	C₂×C₃₁	1	2	yes	complemented
128	#488	C₂×C₃₁	1	2	yes	complemented
129	#485	C₂×C₃₁	1	2	yes	complemented
130	#482	C₂×C₃₁	1	2	yes	—
131	#477	C₂×C₃₁	1	1	yes	characteristic, complemented
132	#476	C₂×C₃₁	1	1	yes	characteristic, complemented
133	#475	C₂×C₃₁	1	1	yes	characteristic, complemented
134	#474	C₂×C₃₁	1	1	yes	characteristic, complemented
135	#431	C₂×C₃₁	1	1	yes	characteristic, complemented
136	C₂×C₃₁	—	4	1	no	—
137	C₂×C₃₁	—	8	1	no	—
138	C₂×C₃₁	#536	1	1	yes	characteristic
139	C₂×C₃₁	C₂×#135	1	1	yes	characteristic
140	C₂×C₃₁	#575	1	1	yes	characteristic
141	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
142	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
143	#126	—	2	1	no	—
144	#125	C₂²×C₃₁	1	1	yes	characteristic
145	#125	C₂²×C₃₁	1	2	yes	—
146	#124	C₂²×C₃₁	1	1	yes	characteristic
147	#123	—	2	1	no	—
148	C₂²⋊Q₈	—	2	1	no	—
149	C₂²⋊Q₈	—	2	1	no	—
150	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
151	#121	C₂²×C₃₁	1	1	yes	characteristic
152	C₄×Q₈	—	2	1	no	—
153	C₄×Q₈	—	2	1	no	—
154	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
155	C₄×D₄	—	2	1	no	—
156	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
157	#117	—	2	1	no	—
158	#117	C₂²×C₃₁	1	2	yes	—
159	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
160	#107	C₂²×C₃₁	1	2	yes	—
161	#107	C₂²×C₃₁	1	2	yes	—
162	#106	C₂²×C₃₁	1	2	yes	—
163	#106	C₂²×C₃₁	1	2	yes	—
164	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
165	C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
166	#103	C₂²×C₃₁	1	2	yes	—
167	#103	C₂²×C₃₁	1	2	yes	—
168	#102	C₂²×C₃₁	1	2	yes	—
169	#102	C₂²×C₃₁	1	2	yes	—
170	C₂²⋊C₈	C₂²×C₃₁	1	2	yes	—
171	C₄×C₈	C₂²×C₃₁	1	2	yes	—
172	C₃₁	#3233	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
180	C₂²×C₄	D₄×C₃₁	1	1	yes	characteristic
181	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
182	C₂×C₈	C₂³×C₃₁	1	2	yes	—
183	C₂×C₈	C₂³×C₃₁	1	2	yes	—
184	C₄⋊C₄	—	2	1	no	—
185	C₄⋊C₄	—	2	1	no	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	2	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
193	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
194	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
195	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
196	C₂²⋊C₄	—	2	2	no	—
197	C₂²⋊C₄	—	2	1	no	—
198	C₂²⋊C₄	—	2	1	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	2	2	no	—
204	C₄²	D₄×C₃₁	1	1	yes	characteristic
205	C₄²	D₄×C₃₁	1	1	yes	characteristic
206	C₄²	C₂³×C₃₁	1	1	yes	characteristic
207	C₄²	C₂³×C₃₁	1	1	yes	characteristic
208	C₂³	—	2	1	no	—
209	C₂³	C₂×D₄×C₃₁	1	1	yes	characteristic
210	Q₈	—	2	1	no	—
211	Q₈	—	2	1	no	—
212	Q₈	—	4	2	no	—
213	Q₈	—	4	2	no	—
214	D₄	—	2	1	no	—
215	D₄	—	2	1	no	—
216	D₄	—	4	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	4	1	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
232	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	C₂²×D₄×C₃₁	1	1	yes	characteristic
242	C₄	—	2	1	no	—
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	4	1	no	—
247	C₄	—	4	1	no	—
248	C₄	—	4	2	no	—
249	C₄	—	4	1	no	—
250	C₄	—	4	2	no	—
251	C₄	C₃₁×#135	1	2	yes	—
252	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
253	C₄	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
254	C₂	—	4	1	no	—
255	C₂	—	8	1	no	—
256	C₂	C₃₁×#536	1	1	yes	characteristic
257	C₂	C₂×C₃₁×#135	1	1	yes	characteristic
258	C₂	C₃₁×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₁
prime radical	C₃₁×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#533	C₂	semidirect
3. C₃₁×#499	C₂	semidirect
4. C₃₁×#488	C₂	semidirect
5. C₃₁×#485	C₂	semidirect
7. C₃₁×#477	C₂	semidirect
8. C₃₁×#476	C₂	semidirect
9. C₃₁×#475	C₂	semidirect
10. C₃₁×#474	C₂	semidirect
11. C₃₁×#431	C₂	semidirect
105. #3233	C₃₁	direct (N × H)
125. #533	C₂×C₃₁	semidirect
127. #499	C₂×C₃₁	semidirect
128. #488	C₂×C₃₁	semidirect
129. #485	C₂×C₃₁	semidirect
131. #477	C₂×C₃₁	semidirect
132. #476	C₂×C₃₁	semidirect
133. #475	C₂×C₃₁	semidirect
134. #474	C₂×C₃₁	semidirect
135. #431	C₂×C₃₁	semidirect
172. C₃₁	#3233	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃₁×#533	2
C₃₁×#529	2
C₃₁×#499	2
C₃₁×#488	2
C₃₁×#485	2
C₃₁×#482	2
C₃₁×#477	2
C₃₁×#476	2
C₃₁×#475	2
C₃₁×#474	2
C₃₁×#431	2
#3233	31

Subgroups — C31×#3233 #16843166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3233 #16843166