Subgroups — C₃₁×#33957 #21490716

Order 7936.

444 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3383	C₂	1	2	yes	maximal
2	C₃₁×#2255	C₂	1	2	yes	maximal
3	C₃₁×#2253	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2229	C₂	1	2	yes	maximal
5	C₃₁×#2229	C₂	1	2	yes	maximal
6	C₃₁×#2227	C₂	1	2	yes	maximal
7	C₃₁×#2183	C₂	1	1	yes	maximal, characteristic
8	C₃₁×#2176	C₂	1	2	yes	maximal
9	C₄×C₃₁×Q₃₂	C₂	1	1	yes	maximal, characteristic
10	C₃₁×#557	—	2	2	no	—
11	C₂×C₃₁×Q₃₂	C₂²	1	1	yes	characteristic
12	C₃₁×#500	C₂²	1	2	yes	—
13	C₃₁×#499	C₂²	1	2	yes	—
14	C₃₁×#499	C₂²	1	2	yes	—
15	C₃₁×#479	—	2	2	no	—
16	C₃₁×#479	—	2	2	no	—
17	C₃₁×#475	—	2	2	no	—
18	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
19	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
20	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
21	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	—
22	C₃₁×#367	C₂²	1	2	yes	—
23	C₃₁×#367	C₂²	1	2	yes	—
24	C₃₁×#366	C₂²	1	1	yes	characteristic
25	C₃₁×#366	C₂²	1	1	yes	characteristic
26	C₃₁×#366	C₂²	1	2	yes	—
27	C₃₁×#366	C₂²	1	1	yes	characteristic
28	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
29	C₃₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₃₁×#358	C₂²	1	2	yes	—
31	C₃₁×#358	C₂²	1	2	yes	—
32	C₃₁×#358	C₂²	1	2	yes	—
33	C₃₁×#346	C₂²	1	2	yes	—
34	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
35	C₂×Q₁₆×C₃₁	C₂³	1	2	yes	—
36	C₃₁×C₄⋊Q₈	—	2	2	no	—
37	C₃₁×C₄⋊Q₈	—	4	2	no	—
38	C₃₁×#125	—	2	2	no	—
39	C₃₁×#125	—	2	2	no	—
40	C₃₁×#125	—	4	2	no	—
41	C₃₁×#125	—	4	2	no	—
42	C₄×Q₈×C₃₁	—	2	2	no	—
43	C₄×Q₈×C₃₁	—	2	2	no	—
44	C₄×Q₈×C₃₁	—	2	2	no	—
45	C₄×Q₈×C₃₁	—	4	2	no	—
46	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
47	C₃₁×Q₃₂	Q₈	1	4	yes	complemented
48	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
49	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
50	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
51	C₃₁×#107	—	2	2	no	—
52	C₃₁×#107	C₂³	1	2	yes	—
53	C₃₁×#107	C₂³	1	2	yes	—
54	C₃₁×#107	C₂³	1	2	yes	—
55	C₃₁×#106	—	2	2	no	—
56	C₃₁×#106	—	2	2	no	—
57	C₃₁×C₄⋊C₈	D₄	1	2	yes	—
58	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
59	C₃₁×#103	—	2	2	no	—
60	C₃₁×#103	—	2	2	no	—
61	C₃₁×#103	—	2	2	no	—
62	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₃₁	C₂³	1	2	yes	—
64	C₂×Q₈×C₃₁	—	2	2	no	—
65	C₂×Q₈×C₃₁	—	4	2	no	—
66	C₂×Q₈×C₃₁	D₈	1	1	yes	characteristic
67	Q₁₆×C₃₁	—	2	2	no	—
68	Q₁₆×C₃₁	C₂×Q₈	1	4	yes	—
69	C₂×C₈×C₃₁	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
72	C₄⋊C₄×C₃₁	—	2	2	no	—
73	C₄⋊C₄×C₃₁	—	2	2	no	—
74	C₄⋊C₄×C₃₁	—	2	2	no	—
75	C₄⋊C₄×C₃₁	—	4	2	no	—
76	C₄⋊C₄×C₃₁	—	4	2	no	—
77	C₄⋊C₄×C₃₁	—	4	2	no	—
78	C₄⋊C₄×C₃₁	—	4	2	no	—
79	C₄⋊C₄×C₃₁	—	4	2	no	—
80	C₄⋊C₄×C₃₁	—	4	2	no	—
81	C₄⋊C₄×C₃₁	—	4	2	no	—
82	C₄⋊C₄×C₃₁	—	4	2	no	—
83	C₄⋊C₄×C₃₁	D₈	1	2	yes	—
84	C₄⋊C₄×C₃₁	D₈	1	1	yes	characteristic
85	C₄²×C₃₁	—	4	2	no	—
86	C₄²×C₃₁	—	4	2	no	—
87	C₄²×C₃₁	—	4	2	no	—
88	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₃₁	C₂×D₄	1	2	yes	—
90	C₁₆×C₃₁	—	2	1	no	—
91	C₁₆×C₃₁	—	2	2	no	—
92	C₁₆×C₃₁	C₂×Q₈	1	2	yes	—
93	#33957	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
94	Q₈×C₃₁	—	2	4	no	—
95	Q₈×C₃₁	—	4	1	no	—
96	Q₈×C₃₁	—	4	2	no	—
97	Q₈×C₃₁	—	8	4	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	4	2	no	—
102	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
104	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
105	C₂×C₄×C₃₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
107	C₈×C₃₁	—	2	1	no	—
108	C₈×C₃₁	—	2	2	no	—
109	C₈×C₃₁	C₂²×Q₈	1	1	yes	characteristic
110	C₈×C₃₁	2₋¹⁺⁴	1	1	yes	characteristic
111	#3383	C₂×C₃₁	1	2	yes	—
112	#2255	C₂×C₃₁	1	2	yes	—
113	#2253	C₂×C₃₁	1	1	yes	characteristic
114	#2229	C₂×C₃₁	1	2	yes	—
115	#2229	C₂×C₃₁	1	2	yes	—
116	#2227	C₂×C₃₁	1	2	yes	—
117	#2183	C₂×C₃₁	1	1	yes	characteristic
118	#2176	C₂×C₃₁	1	2	yes	—
119	C₄×Q₃₂	C₂×C₃₁	1	1	yes	characteristic
120	C₂²×C₃₁	C₂²×D₈	1	1	yes	characteristic
121	C₄×C₃₁	—	2	1	no	—
122	C₄×C₃₁	—	2	2	no	—
123	C₄×C₃₁	—	2	1	no	—
124	C₄×C₃₁	—	2	2	no	—
125	C₄×C₃₁	—	4	4	no	—
126	C₄×C₃₁	—	8	2	no	—
127	C₄×C₃₁	—	8	2	no	—
128	C₄×C₃₁	—	8	2	no	—
129	C₄×C₃₁	D₄×Q₈	1	1	yes	characteristic
130	C₄×C₃₁	#578	1	1	yes	characteristic
131	#557	—	2	2	no	—
132	C₂×Q₃₂	C₂²×C₃₁	1	1	yes	characteristic
133	#500	C₂²×C₃₁	1	2	yes	—
134	#499	C₂²×C₃₁	1	2	yes	—
135	#499	C₂²×C₃₁	1	2	yes	—
136	#479	—	2	2	no	—
137	#479	—	2	2	no	—
138	#475	—	2	2	no	—
139	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
140	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
141	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
142	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	—
143	#367	C₂²×C₃₁	1	2	yes	—
144	#367	C₂²×C₃₁	1	2	yes	—
145	#366	C₂²×C₃₁	1	1	yes	characteristic
146	#366	C₂²×C₃₁	1	1	yes	characteristic
147	#366	C₂²×C₃₁	1	2	yes	—
148	#366	C₂²×C₃₁	1	1	yes	characteristic
149	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
150	C₄⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
151	#358	C₂²×C₃₁	1	2	yes	—
152	#358	C₂²×C₃₁	1	2	yes	—
153	#358	C₂²×C₃₁	1	2	yes	—
154	#346	C₂²×C₃₁	1	2	yes	—
155	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
156	C₂×C₃₁	Q₈×D₈	1	1	yes	characteristic
157	C₂×C₃₁	#3414	1	1	yes	characteristic
158	C₂×C₃₁	#3417	1	1	yes	characteristic
159	C₂×Q₁₆	C₂³×C₃₁	1	2	yes	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₄⋊Q₈	—	4	2	no	—
162	#125	—	2	2	no	—
163	#125	—	2	2	no	—
164	#125	—	4	2	no	—
165	#125	—	4	2	no	—
166	C₄×Q₈	—	2	2	no	—
167	C₄×Q₈	—	2	2	no	—
168	C₄×Q₈	—	2	2	no	—
169	C₄×Q₈	—	4	2	no	—
170	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
171	Q₃₂	Q₈×C₃₁	1	4	yes	complemented
172	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
173	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
174	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
175	#107	—	2	2	no	—
176	#107	C₂³×C₃₁	1	2	yes	—
177	#107	C₂³×C₃₁	1	2	yes	—
178	#107	C₂³×C₃₁	1	2	yes	—
179	#106	—	2	2	no	—
180	#106	—	2	2	no	—
181	C₄⋊C₈	D₄×C₃₁	1	2	yes	—
182	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
183	#103	—	2	2	no	—
184	#103	—	2	2	no	—
185	#103	—	2	2	no	—
186	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
187	C₄×C₈	C₂³×C₃₁	1	2	yes	—
188	C₃₁	#33957	1	1	yes	characteristic, direct product factor, Sylow
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×Q₈	D₈×C₃₁	1	1	yes	characteristic
192	Q₁₆	—	2	2	no	—
193	Q₁₆	C₂×Q₈×C₃₁	1	4	yes	—
194	C₂×C₈	C₂×D₄×C₃₁	1	1	yes	characteristic
195	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
196	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	4	2	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	2	no	—
203	C₄⋊C₄	—	4	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₄⋊C₄	D₈×C₃₁	1	2	yes	—
209	C₄⋊C₄	D₈×C₃₁	1	1	yes	characteristic
210	C₄²	—	4	2	no	—
211	C₄²	—	4	2	no	—
212	C₄²	—	4	2	no	—
213	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
214	C₄²	C₂×D₄×C₃₁	1	2	yes	—
215	C₁₆	—	2	1	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	C₂×Q₈×C₃₁	1	2	yes	—
218	Q₈	—	2	4	no	—
219	Q₈	—	4	1	no	—
220	Q₈	—	4	2	no	—
221	Q₈	—	8	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
227	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
228	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
229	C₂×C₄	C₂×D₈×C₃₁	1	1	yes	characteristic
230	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
231	C₈	—	2	1	no	—
232	C₈	—	2	2	no	—
233	C₈	C₂²×Q₈×C₃₁	1	1	yes	characteristic
234	C₈	C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
235	C₂²	C₂²×D₈×C₃₁	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	2	1	no	—
238	C₄	—	2	1	no	—
239	C₄	—	2	2	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	—	8	2	no	—
243	C₄	—	8	2	no	—
244	C₄	D₄×Q₈×C₃₁	1	1	yes	characteristic
245	C₄	C₃₁×#578	1	1	yes	characteristic
246	C₂	Q₈×D₈×C₃₁	1	1	yes	characteristic
247	C₂	C₃₁×#3414	1	1	yes	characteristic
248	C₂	C₃₁×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₃₁	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₄×Q₈×C₃₁ ◁ C₈×Q₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
47. C₃₁×Q₃₂	Q₈	semidirect
93. #33957	C₃₁	direct (N × H)
171. Q₃₂	Q₈×C₃₁	semidirect
188. C₃₁	#33957	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×#3383	2
C₃₁×#2255	2
C₃₁×#2253	2
C₃₁×#2229	2
C₃₁×#2227	2
C₃₁×#2183	2
C₃₁×#2176	2
C₄×C₃₁×Q₃₂	2
#33957	31

Subgroups — C31×#33957 #21490716

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33957 #21490716