Subgroups — C₃₁×#9691 #21466450

Order 7936.

570 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2868	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1842	C₂	1	1	yes	maximal, characteristic
3	C₄×C₃₁×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#333	C₂	1	4	yes	maximal
5	C₂×C₃₁×C₄⋊Q₈	C₄	1	4	yes	—
6	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₃₁×C₄⋊C₈	—	2	2	no	—
8	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	—	2	1	no	—
11	C₃₁×#384	—	2	1	no	—
12	C₃₁×#384	—	2	1	no	—
13	C₃₁×#384	—	2	2	no	—
14	C₄³×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#336	—	2	1	no	—
16	C₃₁×#333	—	2	8	no	—
17	C₂²×Q₈×C₃₁	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₃₁	—	2	2	no	—
19	C₂²×C₈×C₃₁	—	2	2	no	—
20	C₃₁×C₄⋊Q₈	—	2	4	no	—
21	C₃₁×C₄⋊Q₈	—	2	4	no	—
22	C₃₁×C₄⋊Q₈	—	4	8	no	—
23	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
29	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
30	C₂×C₄²×C₃₁	D₄	1	2	yes	—
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	—	2	4	no	—
33	C₃₁×C₄⋊C₈	—	2	2	no	—
34	C₃₁×C₄⋊C₈	—	4	4	no	—
35	C₃₁×C₄⋊C₈	D₄	1	4	yes	—
36	C₄×C₈×C₃₁	—	4	2	no	—
37	C₂×Q₈×C₃₁	—	2	8	no	—
38	C₂×Q₈×C₃₁	—	4	8	no	—
39	C₂²×C₄×C₃₁	—	2	4	no	—
40	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
41	C₂²×C₄×C₃₁	C₂²⋊C₄	1	2	yes	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₈×C₃₁	—	2	4	no	—
46	C₂×C₈×C₃₁	—	4	2	no	—
47	C₂×C₈×C₃₁	—	4	2	no	—
48	C₂×C₈×C₃₁	—	4	2	no	—
49	C₂×C₈×C₃₁	—	4	4	no	—
50	C₄⋊C₄×C₃₁	—	4	8	no	—
51	C₄⋊C₄×C₃₁	—	4	8	no	—
52	C₄⋊C₄×C₃₁	—	4	4	no	—
53	C₄⋊C₄×C₃₁	—	4	2	no	—
54	C₄⋊C₄×C₃₁	—	4	2	no	—
55	C₄²×C₃₁	—	2	4	no	—
56	C₄²×C₃₁	—	2	2	no	—
57	C₄²×C₃₁	SD₁₆	1	2	yes	—
58	C₄²×C₃₁	SD₁₆	1	4	yes	—
59	C₄²×C₃₁	Q₁₆	1	4	yes	—
60	C₄²×C₃₁	Q₁₆	1	2	yes	—
61	C₄²×C₃₁	C₂×D₄	1	2	yes	—
62	C₄²×C₃₁	#41	1	2	yes	—
63	#9691	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	Q₈×C₃₁	—	8	8	no	—
66	C₂×C₄×C₃₁	—	2	2	no	—
67	C₂×C₄×C₃₁	—	2	2	no	—
68	C₂×C₄×C₃₁	—	2	2	no	—
69	C₂×C₄×C₃₁	—	4	4	no	—
70	C₂×C₄×C₃₁	—	4	8	no	—
71	C₂×C₄×C₃₁	#100	1	4	yes	—
72	C₂×C₄×C₃₁	#103	1	4	yes	—
73	C₂×C₄×C₃₁	#103	1	4	yes	—
74	C₂×C₄×C₃₁	#103	1	4	yes	—
75	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
76	C₂×C₄×C₃₁	#121	1	2	yes	—
77	C₂×C₄×C₃₁	#124	1	1	yes	characteristic
78	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
79	C₂×C₄×C₃₁	C₂×SD₁₆	1	1	yes	characteristic
80	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
81	C₂×C₄×C₃₁	C₂×Q₁₆	1	1	yes	characteristic
82	C₂×C₄×C₃₁	C₂×Q₁₆	1	2	yes	—
83	C₂×C₄×C₃₁	#137	1	2	yes	—
84	C₈×C₃₁	—	4	4	no	—
85	C₈×C₃₁	—	8	2	no	—
86	#2868	C₂×C₃₁	1	1	yes	characteristic
87	#1842	C₂×C₃₁	1	1	yes	characteristic
88	C₄×C₄⋊C₈	C₂×C₃₁	1	1	yes	characteristic
89	C₂×#333	C₂×C₃₁	1	4	yes	—
90	C₂²×C₃₁	#390	1	1	yes	characteristic
91	C₂²×C₃₁	C₂×#100	1	2	yes	—
92	C₂²×C₃₁	C₂×#103	1	1	yes	characteristic
93	C₂²×C₃₁	C₂×#103	1	2	yes	—
94	C₂²×C₃₁	#419	1	1	yes	characteristic
95	C₄×C₃₁	—	2	4	no	—
96	C₄×C₃₁	—	2	2	no	—
97	C₄×C₃₁	—	8	4	no	—
98	C₄×C₃₁	#333	1	8	yes	complemented
99	C₄×C₃₁	#461	1	2	yes	—
100	C₄×C₃₁	#462	1	2	yes	—
101	C₄×C₃₁	#487	1	2	yes	—
102	C₄×C₃₁	#492	1	1	yes	characteristic
103	C₄×C₃₁	#494	1	1	yes	characteristic
104	C₂×C₄⋊Q₈	C₄×C₃₁	1	4	yes	—
105	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
106	C₂×C₄⋊C₈	—	2	2	no	—
107	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
108	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
109	C₂×C₄×C₈	—	2	1	no	—
110	#384	—	2	1	no	—
111	#384	—	2	1	no	—
112	#384	—	2	2	no	—
113	C₄³	C₂²×C₃₁	1	1	yes	characteristic
114	#336	—	2	1	no	—
115	#333	—	2	8	no	—
116	C₂×C₃₁	C₂×#333	1	2	yes	—
117	C₂×C₃₁	#1548	1	2	yes	—
118	C₂×C₃₁	#1956	1	1	yes	characteristic
119	C₂×C₃₁	#1965	1	1	yes	characteristic
120	C₂×C₃₁	#1981	1	1	yes	characteristic
121	C₂²×Q₈	C₂×C₄×C₃₁	1	4	yes	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	2	no	—
124	C₄⋊Q₈	—	2	4	no	—
125	C₄⋊Q₈	—	2	4	no	—
126	C₄⋊Q₈	—	4	8	no	—
127	C₂×C₄⋊C₄	—	2	4	no	—
128	C₂×C₄⋊C₄	—	2	4	no	—
129	C₂×C₄⋊C₄	—	2	2	no	—
130	C₂×C₄⋊C₄	—	2	2	no	—
131	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	—
132	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
133	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
134	C₂×C₄²	D₄×C₃₁	1	2	yes	—
135	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
136	C₄⋊C₈	—	2	4	no	—
137	C₄⋊C₈	—	2	2	no	—
138	C₄⋊C₈	—	4	4	no	—
139	C₄⋊C₈	D₄×C₃₁	1	4	yes	—
140	C₄×C₈	—	4	2	no	—
141	C₃₁	#9691	1	1	yes	characteristic, direct product factor, Sylow
142	C₂×Q₈	—	2	8	no	—
143	C₂×Q₈	—	4	8	no	—
144	C₂²×C₄	—	2	4	no	—
145	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
146	C₂²×C₄	C₂²⋊C₄×C₃₁	1	2	yes	—
147	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
148	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
149	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
150	C₂×C₈	—	2	4	no	—
151	C₂×C₈	—	4	2	no	—
152	C₂×C₈	—	4	2	no	—
153	C₂×C₈	—	4	2	no	—
154	C₂×C₈	—	4	4	no	—
155	C₄⋊C₄	—	4	8	no	—
156	C₄⋊C₄	—	4	8	no	—
157	C₄⋊C₄	—	4	4	no	—
158	C₄⋊C₄	—	4	2	no	—
159	C₄⋊C₄	—	4	2	no	—
160	C₄²	—	2	4	no	—
161	C₄²	—	2	2	no	—
162	C₄²	SD₁₆×C₃₁	1	2	yes	—
163	C₄²	SD₁₆×C₃₁	1	4	yes	—
164	C₄²	Q₁₆×C₃₁	1	4	yes	—
165	C₄²	Q₁₆×C₃₁	1	2	yes	—
166	C₄²	C₂×D₄×C₃₁	1	2	yes	—
167	C₄²	#41×C₃₁	1	2	yes	—
168	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
169	Q₈	—	8	8	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	4	8	no	—
174	C₂×C₄	—	4	4	no	—
175	C₂×C₄	C₃₁×#100	1	4	yes	—
176	C₂×C₄	C₃₁×#103	1	4	yes	—
177	C₂×C₄	C₃₁×#103	1	4	yes	—
178	C₂×C₄	C₃₁×#103	1	4	yes	—
179	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
180	C₂×C₄	C₃₁×#121	1	2	yes	—
181	C₂×C₄	C₃₁×#124	1	1	yes	characteristic
182	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
183	C₂×C₄	C₂×SD₁₆×C₃₁	1	1	yes	characteristic
184	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
185	C₂×C₄	C₂×Q₁₆×C₃₁	1	1	yes	characteristic
186	C₂×C₄	C₂×Q₁₆×C₃₁	1	2	yes	—
187	C₂×C₄	C₃₁×#137	1	2	yes	—
188	C₈	—	4	4	no	—
189	C₈	—	8	2	no	—
190	C₂²	C₃₁×#390	1	1	yes	characteristic
191	C₂²	C₂×C₃₁×#100	1	2	yes	—
192	C₂²	C₂×C₃₁×#103	1	1	yes	characteristic
193	C₂²	C₂×C₃₁×#103	1	2	yes	—
194	C₂²	C₃₁×#419	1	1	yes	characteristic
195	C₄	—	2	4	no	—
196	C₄	—	2	2	no	—
197	C₄	—	8	4	no	—
198	C₄	C₃₁×#333	1	8	yes	complemented
199	C₄	C₃₁×#461	1	2	yes	—
200	C₄	C₃₁×#462	1	2	yes	—
201	C₄	C₃₁×#487	1	2	yes	—
202	C₄	C₃₁×#492	1	1	yes	characteristic
203	C₄	C₃₁×#494	1	1	yes	characteristic
204	C₂	C₂×C₃₁×#333	1	2	yes	—
205	C₂	C₃₁×#1548	1	2	yes	—
206	C₂	C₃₁×#1956	1	1	yes	characteristic
207	C₂	C₃₁×#1965	1	1	yes	characteristic
208	C₂	C₃₁×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
63. #9691	C₃₁	direct (N × H)
98. C₄×C₃₁	#333	semidirect
141. C₃₁	#9691	direct (N × H)
198. C₄	C₃₁×#333	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#2868	2
C₃₁×#1842	2
C₄×C₃₁×C₄⋊C₈	2
C₂×C₃₁×#333	2
#9691	31

Subgroups — C31×#9691 #21466450

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#9691 #21466450