Subgroups — C₃₁×#33296 #21490055

Order 7936.

678 conjugacy classes of subgroups.

The subgroup lattice has 180 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2862	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2859	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1993	C₂	1	4	yes	maximal
4	C₃₁×#1991	C₂	1	4	yes	maximal
5	C₃₁×#1836	C₂	1	4	yes	maximal
6	C₄×C₃₁×#97	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₃₁×#125	C₂²	1	2	yes	—
9	C₂×C₃₁×#97	C₂²	1	2	yes	—
10	C₂×C₃₁×#97	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#391	—	2	4	no	—
13	C₃₁×#389	—	2	4	no	—
14	C₃₁×#384	C₂²	1	2	yes	—
15	C₃₁×#382	C₂²	1	2	yes	—
16	C₃₁×#382	C₂²	1	2	yes	—
17	C₃₁×#382	C₂²	1	2	yes	—
18	C₄³×C₃₁	C₂²	1	1	yes	characteristic
19	C₃₁×#340	C₂²	1	16	yes	—
20	C₂²×Q₈×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
22	C₃₁×C₄⋊Q₈	—	2	8	no	—
23	C₃₁×#125	—	2	8	no	—
24	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
25	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
26	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
27	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
29	C₂×C₄²×C₃₁	D₄	1	4	yes	—
30	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
33	C₃₁×#97	—	2	4	no	—
34	C₃₁×#97	Q₈	1	8	yes	complemented
35	C₄×C₈×C₃₁	—	2	4	no	—
36	C₃₁×#95	—	2	4	no	—
37	C₃₁×#95	—	2	4	no	—
38	C₃₁×#95	—	2	4	no	—
39	C₂×Q₈×C₃₁	—	4	8	no	—
40	C₂×Q₈×C₃₁	—	4	4	no	—
41	C₂²×C₄×C₃₁	—	2	4	no	—
42	C₂²×C₄×C₃₁	—	2	4	no	—
43	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
46	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
47	C₂×C₈×C₃₁	—	2	4	no	—
48	C₂×C₈×C₃₁	—	2	4	no	—
49	C₂×C₈×C₃₁	C₂×Q₈	1	8	yes	—
50	C₄⋊C₄×C₃₁	—	2	8	no	—
51	C₄⋊C₄×C₃₁	—	2	8	no	—
52	C₄⋊C₄×C₃₁	—	4	8	no	—
53	C₄⋊C₄×C₃₁	—	4	8	no	—
54	C₄⋊C₄×C₃₁	—	4	4	no	—
55	C₄²×C₃₁	—	4	4	no	—
56	C₄²×C₃₁	C₂×Q₈	1	4	yes	—
57	C₄²×C₃₁	#41	1	4	yes	—
58	C₄²×C₃₁	#41	1	4	yes	—
59	#33296	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
61	Q₈×C₃₁	—	8	8	no	—
62	C₂×C₄×C₃₁	—	2	4	no	—
63	C₂×C₄×C₃₁	—	2	4	no	—
64	C₂×C₄×C₃₁	—	2	4	no	—
65	C₂×C₄×C₃₁	—	4	4	no	—
66	C₂×C₄×C₃₁	—	4	4	no	—
67	C₂×C₄×C₃₁	—	4	8	no	—
68	C₂×C₄×C₃₁	—	4	8	no	—
69	C₂×C₄×C₃₁	#124	1	4	yes	—
70	C₂×C₄×C₃₁	#124	1	4	yes	—
71	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
72	C₂×C₄×C₃₁	C₂²×Q₈	1	2	yes	—
73	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
74	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
75	C₈×C₃₁	—	2	8	no	—
76	#2862	C₂×C₃₁	1	1	yes	characteristic
77	#2859	C₂×C₃₁	1	1	yes	characteristic
78	#1993	C₂×C₃₁	1	4	yes	—
79	#1991	C₂×C₃₁	1	4	yes	—
80	#1836	C₂×C₃₁	1	4	yes	—
81	C₄×#97	C₂×C₃₁	1	1	yes	characteristic
82	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
83	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
84	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
85	C₂²×C₃₁	#575	1	4	yes	—
86	C₄×C₃₁	—	2	4	no	—
87	C₄×C₃₁	—	2	4	no	—
88	C₄×C₃₁	—	2	4	no	—
89	C₄×C₃₁	—	8	4	no	—
90	C₄×C₃₁	—	8	4	no	—
91	C₄×C₃₁	#533	1	4	yes	—
92	C₂×C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
93	C₂×#125	C₂²×C₃₁	1	2	yes	—
94	C₂×#97	C₂²×C₃₁	1	2	yes	—
95	C₂×#97	C₂²×C₃₁	1	2	yes	—
96	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
97	#391	—	2	4	no	—
98	#389	—	2	4	no	—
99	#384	C₂²×C₃₁	1	2	yes	—
100	#382	C₂²×C₃₁	1	2	yes	—
101	#382	C₂²×C₃₁	1	2	yes	—
102	#382	C₂²×C₃₁	1	2	yes	—
103	C₄³	C₂²×C₃₁	1	1	yes	characteristic
104	#340	C₂²×C₃₁	1	16	yes	—
105	C₂×C₃₁	#2862	1	1	yes	characteristic
106	C₂×C₃₁	#3135	1	2	yes	—
107	C₂×C₃₁	#3140	1	2	yes	—
108	C₂×C₃₁	#3166	1	2	yes	—
109	C₂²×Q₈	—	2	2	no	—
110	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
111	C₄⋊Q₈	—	2	8	no	—
112	#125	—	2	8	no	—
113	C₂×C₄⋊C₄	—	2	4	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	4	no	—
116	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
117	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
118	C₂×C₄²	D₄×C₃₁	1	4	yes	—
119	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
120	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
121	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
122	#97	—	2	4	no	—
123	#97	Q₈×C₃₁	1	8	yes	complemented
124	C₄×C₈	—	2	4	no	—
125	#95	—	2	4	no	—
126	#95	—	2	4	no	—
127	#95	—	2	4	no	—
128	C₃₁	#33296	1	1	yes	characteristic, direct product factor, Sylow
129	C₂×Q₈	—	4	8	no	—
130	C₂×Q₈	—	4	4	no	—
131	C₂²×C₄	—	2	4	no	—
132	C₂²×C₄	—	2	4	no	—
133	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
134	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
135	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
136	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
137	C₂×C₈	—	2	4	no	—
138	C₂×C₈	—	2	4	no	—
139	C₂×C₈	C₂×Q₈×C₃₁	1	8	yes	—
140	C₄⋊C₄	—	2	8	no	—
141	C₄⋊C₄	—	2	8	no	—
142	C₄⋊C₄	—	4	8	no	—
143	C₄⋊C₄	—	4	8	no	—
144	C₄⋊C₄	—	4	4	no	—
145	C₄²	—	4	4	no	—
146	C₄²	C₂×Q₈×C₃₁	1	4	yes	—
147	C₄²	#41×C₃₁	1	4	yes	—
148	C₄²	#41×C₃₁	1	4	yes	—
149	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
150	Q₈	—	8	8	no	—
151	C₂×C₄	—	2	4	no	—
152	C₂×C₄	—	2	4	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	4	4	no	—
155	C₂×C₄	—	4	4	no	—
156	C₂×C₄	—	4	8	no	—
157	C₂×C₄	—	4	8	no	—
158	C₂×C₄	C₃₁×#124	1	4	yes	—
159	C₂×C₄	C₃₁×#124	1	4	yes	—
160	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
161	C₂×C₄	C₂²×Q₈×C₃₁	1	2	yes	—
162	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
163	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
164	C₈	—	2	8	no	—
165	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
166	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
167	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
168	C₂²	C₃₁×#575	1	4	yes	—
169	C₄	—	2	4	no	—
170	C₄	—	2	4	no	—
171	C₄	—	2	4	no	—
172	C₄	—	8	4	no	—
173	C₄	—	8	4	no	—
174	C₄	C₃₁×#533	1	4	yes	—
175	C₂	C₃₁×#2862	1	1	yes	characteristic
176	C₂	C₃₁×#3135	1	2	yes	—
177	C₂	C₃₁×#3140	1	2	yes	—
178	C₂	C₃₁×#3166	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
34. C₃₁×#97	Q₈	semidirect
59. #33296	C₃₁	direct (N × H)
123. #97	Q₈×C₃₁	semidirect
128. C₃₁	#33296	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×#2862	2
C₃₁×#2859	2
C₃₁×#1993	2
C₃₁×#1991	2
C₃₁×#1836	2
C₄×C₃₁×#97	2
#33296	31

Subgroups — C31×#33296 #21490055

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33296 #21490055