Subgroups — C₃₁×#33257 #21490016

Order 7936.

646 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2864	C₂	1	2	yes	maximal
2	C₃₁×#1942	C₂	1	4	yes	maximal
3	C₃₁×#1941	C₂	1	4	yes	maximal
4	C₃₁×#1846	C₂	1	2	yes	maximal
5	C₃₁×#1835	C₂	1	2	yes	maximal
6	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂×C₃₁×#107	C₂²	1	4	yes	—
8	C₂×C₃₁×#106	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×#389	C₂²	1	8	yes	—
14	C₃₁×#387	—	2	8	no	—
15	C₃₁×#382	C₂²	1	4	yes	—
16	C₄³×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#340	C₂²	1	8	yes	—
18	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
19	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
22	C₂×C₄⋊C₄×C₃₁	C₂³	1	8	yes	—
23	C₂×C₄²×C₃₁	D₄	1	4	yes	—
24	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
25	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
26	C₃₁×#107	—	2	8	no	—
27	C₃₁×#106	—	2	8	no	—
28	C₄×C₈×C₃₁	—	2	4	no	—
29	C₄×C₈×C₃₁	—	2	4	no	—
30	C₄×C₈×C₃₁	Q₈	1	4	yes	—
31	C₄×C₈×C₃₁	Q₈	1	4	yes	—
32	C₃₁×#95	—	2	8	no	—
33	C₃₁×#95	—	2	8	no	—
34	C₂²×C₄×C₃₁	—	2	8	no	—
35	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
37	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
38	C₂×C₈×C₃₁	—	2	2	no	—
39	C₂×C₈×C₃₁	—	2	2	no	—
40	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	—
41	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
42	C₂×C₈×C₃₁	C₂×Q₈	1	2	yes	—
43	C₂×C₈×C₃₁	#41	1	4	yes	—
44	C₂×C₈×C₃₁	#41	1	4	yes	—
45	C₄⋊C₄×C₃₁	—	2	8	no	—
46	C₄⋊C₄×C₃₁	—	2	8	no	—
47	C₄⋊C₄×C₃₁	—	4	16	no	—
48	C₄²×C₃₁	—	2	8	no	—
49	C₄²×C₃₁	—	2	2	no	—
50	C₄²×C₃₁	—	2	2	no	—
51	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
52	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
53	#33257	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
55	C₂×C₄×C₃₁	—	2	4	no	—
56	C₂×C₄×C₃₁	—	2	4	no	—
57	C₂×C₄×C₃₁	—	2	4	no	—
58	C₂×C₄×C₃₁	—	4	8	no	—
59	C₂×C₄×C₃₁	—	4	16	no	—
60	C₂×C₄×C₃₁	C₄⋊Q₈	1	4	yes	—
61	C₂×C₄×C₃₁	#135	1	8	yes	—
62	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
65	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
66	C₈×C₃₁	—	2	4	no	—
67	C₈×C₃₁	#125	1	4	yes	—
68	C₈×C₃₁	#125	1	4	yes	—
69	#2864	C₂×C₃₁	1	2	yes	—
70	#1942	C₂×C₃₁	1	4	yes	—
71	#1941	C₂×C₃₁	1	4	yes	—
72	#1846	C₂×C₃₁	1	2	yes	—
73	#1835	C₂×C₃₁	1	2	yes	—
74	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
75	C₂²×C₃₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
76	C₂²×C₃₁	#532	1	2	yes	—
77	C₂²×C₃₁	C₂×#135	1	4	yes	—
78	C₄×C₃₁	—	2	4	no	—
79	C₄×C₃₁	—	2	8	no	—
80	C₄×C₃₁	—	8	8	no	—
81	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
82	C₄×C₃₁	C₂×#125	1	1	yes	characteristic
83	C₄×C₃₁	#533	1	2	yes	—
84	C₂×#107	C₂²×C₃₁	1	4	yes	—
85	C₂×#106	C₂²×C₃₁	1	4	yes	—
86	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
87	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
88	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
90	#389	C₂²×C₃₁	1	8	yes	—
91	#387	—	2	8	no	—
92	#382	C₂²×C₃₁	1	4	yes	—
93	C₄³	C₂²×C₃₁	1	1	yes	characteristic
94	#340	C₂²×C₃₁	1	8	yes	—
95	C₂×C₃₁	#2859	1	1	yes	characteristic
96	C₂×C₃₁	#3160	1	2	yes	—
97	C₂×C₃₁	#3168	1	4	yes	—
98	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
99	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
100	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	2	8	no	—
102	C₂×C₄⋊C₄	C₂³×C₃₁	1	8	yes	—
103	C₂×C₄²	D₄×C₃₁	1	4	yes	—
104	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
105	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
106	#107	—	2	8	no	—
107	#106	—	2	8	no	—
108	C₄×C₈	—	2	4	no	—
109	C₄×C₈	—	2	4	no	—
110	C₄×C₈	Q₈×C₃₁	1	4	yes	—
111	C₄×C₈	Q₈×C₃₁	1	4	yes	—
112	#95	—	2	8	no	—
113	#95	—	2	8	no	—
114	C₃₁	#33257	1	1	yes	characteristic, direct product factor, Sylow
115	C₂²×C₄	—	2	8	no	—
116	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
117	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
118	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	—
122	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
123	C₂×C₈	C₂×Q₈×C₃₁	1	2	yes	—
124	C₂×C₈	#41×C₃₁	1	4	yes	—
125	C₂×C₈	#41×C₃₁	1	4	yes	—
126	C₄⋊C₄	—	2	8	no	—
127	C₄⋊C₄	—	2	8	no	—
128	C₄⋊C₄	—	4	16	no	—
129	C₄²	—	2	8	no	—
130	C₄²	—	2	2	no	—
131	C₄²	—	2	2	no	—
132	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
133	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
134	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	4	8	no	—
139	C₂×C₄	—	4	16	no	—
140	C₂×C₄	C₃₁×C₄⋊Q₈	1	4	yes	—
141	C₂×C₄	C₃₁×#135	1	8	yes	—
142	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
143	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
144	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
145	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
146	C₈	—	2	4	no	—
147	C₈	C₃₁×#125	1	4	yes	—
148	C₈	C₃₁×#125	1	4	yes	—
149	C₂²	C₂×C₃₁×C₄⋊Q₈	1	1	yes	characteristic
150	C₂²	C₃₁×#532	1	2	yes	—
151	C₂²	C₂×C₃₁×#135	1	4	yes	—
152	C₄	—	2	4	no	—
153	C₄	—	2	8	no	—
154	C₄	—	8	8	no	—
155	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
156	C₄	C₂×C₃₁×#125	1	1	yes	characteristic
157	C₄	C₃₁×#533	1	2	yes	—
158	C₂	C₃₁×#2859	1	1	yes	characteristic
159	C₂	C₃₁×#3160	1	2	yes	—
160	C₂	C₃₁×#3168	1	4	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
53. #33257	C₃₁	direct (N × H)
114. C₃₁	#33257	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×#2864	2
C₃₁×#1942	2
C₃₁×#1941	2
C₃₁×#1846	2
C₃₁×#1835	2
C₄²×C₈×C₃₁	2
#33257	31

Subgroups — C31×#33257 #21490016

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33257 #21490016