Subgroups — C₃₁×#33500 #21490259

Order 7936.

460 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2183	C₂	1	4	yes	maximal, complemented
3	Q₈×C₁₆×C₃₁	C₂	1	4	yes	maximal, complemented
4	C₃₁×#2164	C₂	1	4	yes	maximal
5	C₃₁×#2162	C₂	1	2	yes	maximal
6	C₃₁×#533	C₄	1	1	yes	characteristic
7	C₃₁×#446	C₄	1	4	yes	—
8	C₈×Q₈×C₃₁	C₂²	1	4	yes	—
9	C₃₁×#424	C₄	1	2	yes	—
10	C₃₁×#423	C₄	1	1	yes	characteristic
11	C₃₁×#405	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
13	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
14	C₃₁×C₄⋊C₁₆	C₂²	1	2	yes	—
15	C₃₁×C₄⋊C₁₆	C₂²	1	8	yes	—
16	C₃₁×C₂²⋊C₁₆	C₂²	1	4	yes	—
17	C₃₁×#346	C₂²	1	4	yes	—
18	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
19	C₄×C₁₆×C₃₁	C₂²	1	4	yes	—
20	C₂×M₁₆×C₃₁	C₂×C₄	1	2	yes	—
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
23	C₃₁×C₄⋊Q₈	C₈	1	2	yes	—
24	C₃₁×#125	C₈	1	2	yes	—
25	C₃₁×C₂²⋊Q₈	C₈	1	4	yes	—
26	C₄×Q₈×C₃₁	C₂×C₄	1	4	yes	—
27	C₃₁×#117	C₂×C₄	1	2	yes	—
28	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
30	C₂×C₁₆×C₃₁	C₂³	1	4	yes	—
31	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	—
32	C₃₁×C₄⋊C₈	C₂×C₄	1	8	yes	—
33	C₃₁×#97	C₂×C₄	1	4	yes	—
34	C₄×C₈×C₃₁	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₃₁	C₂³	1	2	yes	—
36	C₄×C₈×C₃₁	C₂³	1	4	yes	—
37	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
38	C₂²×C₄×C₃₁	C₂×C₈	1	2	yes	—
39	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
40	M₁₆×C₃₁	—	4	2	no	—
41	C₂×C₈×C₃₁	—	2	2	no	—
42	C₂×C₈×C₃₁	—	4	1	no	—
43	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
48	C₄⋊C₄×C₃₁	C₂×C₈	1	8	yes	—
49	C₄⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
50	C₂²⋊C₄×C₃₁	C₂×C₈	1	4	yes	—
51	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
52	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
53	C₄²×C₃₁	C₂²×C₄	1	2	yes	—
54	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
55	C₁₆×C₃₁	—	2	4	no	—
56	C₁₆×C₃₁	—	2	4	no	—
57	#33500	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
59	Q₈×C₃₁	—	2	8	no	—
60	C₂×C₄×C₃₁	—	2	1	no	—
61	C₂×C₄×C₃₁	—	2	1	no	—
62	C₂×C₄×C₃₁	—	2	4	no	—
63	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
64	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
65	C₂×C₄×C₃₁	C₂²×C₈	1	8	yes	—
66	C₂×C₄×C₃₁	C₂²×C₈	1	2	yes	—
67	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
68	C₈×C₃₁	—	2	1	no	—
69	C₈×C₃₁	—	2	2	no	—
70	C₈×C₃₁	—	2	4	no	—
71	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
72	#2960	C₂×C₃₁	1	1	yes	characteristic
73	#2183	C₂×C₃₁	1	4	yes	complemented
74	Q₈×C₁₆	C₂×C₃₁	1	4	yes	complemented
75	#2164	C₂×C₃₁	1	4	yes	—
76	#2162	C₂×C₃₁	1	2	yes	—
77	C₂²×C₃₁	—	2	2	no	—
78	C₂²×C₃₁	—	2	1	no	—
79	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
80	C₄×C₃₁	—	2	2	no	—
81	C₄×C₃₁	—	2	8	no	—
82	C₄×C₃₁	—	2	1	no	—
83	C₄×C₃₁	#504	1	4	yes	—
84	C₄×C₃₁	#519	1	1	yes	characteristic
85	C₄×C₃₁	#568	1	1	yes	characteristic
86	#533	C₄×C₃₁	1	1	yes	characteristic
87	#446	C₄×C₃₁	1	4	yes	—
88	C₈×Q₈	C₂²×C₃₁	1	4	yes	—
89	#424	C₄×C₃₁	1	2	yes	—
90	#423	C₄×C₃₁	1	1	yes	characteristic
91	#405	C₂²×C₃₁	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
93	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
94	C₄⋊C₁₆	C₂²×C₃₁	1	2	yes	—
95	C₄⋊C₁₆	C₂²×C₃₁	1	8	yes	—
96	C₂²⋊C₁₆	C₂²×C₃₁	1	4	yes	—
97	#346	C₂²×C₃₁	1	4	yes	—
98	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
99	C₄×C₁₆	C₂²×C₃₁	1	4	yes	—
100	C₂×C₃₁	—	4	1	no	—
101	C₂×C₃₁	#2982	1	1	yes	characteristic
102	C₂×C₃₁	C₂×#504	1	2	yes	—
103	C₂×M₁₆	C₂×C₄×C₃₁	1	2	yes	—
104	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
105	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
106	C₄⋊Q₈	C₈×C₃₁	1	2	yes	—
107	#125	C₈×C₃₁	1	2	yes	—
108	C₂²⋊Q₈	C₈×C₃₁	1	4	yes	—
109	C₄×Q₈	C₂×C₄×C₃₁	1	4	yes	—
110	#117	C₂×C₄×C₃₁	1	2	yes	—
111	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
112	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
113	C₂×C₁₆	C₂³×C₃₁	1	4	yes	—
114	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	—
115	C₄⋊C₈	C₂×C₄×C₃₁	1	8	yes	—
116	#97	C₂×C₄×C₃₁	1	4	yes	—
117	C₄×C₈	C₂×C₄×C₃₁	1	2	yes	—
118	C₄×C₈	C₂³×C₃₁	1	2	yes	—
119	C₄×C₈	C₂³×C₃₁	1	4	yes	—
120	C₃₁	#33500	1	1	yes	characteristic, direct product factor, Sylow
121	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
122	C₂²×C₄	C₂×C₈×C₃₁	1	2	yes	—
123	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
124	M₁₆	—	4	2	no	—
125	C₂×C₈	—	2	2	no	—
126	C₂×C₈	—	4	1	no	—
127	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
128	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
129	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
130	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
131	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
132	C₄⋊C₄	C₂×C₈×C₃₁	1	8	yes	—
133	C₄⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
134	C₂²⋊C₄	C₂×C₈×C₃₁	1	4	yes	—
135	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
136	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
137	C₄²	C₂²×C₄×C₃₁	1	2	yes	—
138	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
139	C₁₆	—	2	4	no	—
140	C₁₆	—	2	4	no	—
141	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
142	Q₈	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	2	1	no	—
146	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
147	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
148	C₂×C₄	C₂²×C₈×C₃₁	1	8	yes	—
149	C₂×C₄	C₂²×C₈×C₃₁	1	2	yes	—
150	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
151	C₈	—	2	1	no	—
152	C₈	—	2	2	no	—
153	C₈	—	2	4	no	—
154	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
155	C₂²	—	2	2	no	—
156	C₂²	—	2	1	no	—
157	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
158	C₄	—	2	2	no	—
159	C₄	—	2	8	no	—
160	C₄	—	2	1	no	—
161	C₄	C₃₁×#504	1	4	yes	—
162	C₄	C₃₁×#519	1	1	yes	characteristic
163	C₄	C₃₁×#568	1	1	yes	characteristic
164	C₂	—	4	1	no	—
165	C₂	C₃₁×#2982	1	1	yes	characteristic
166	C₂	C₂×C₃₁×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂×C₈	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂³×C₈
prime radical	C₂³×C₃₁	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#2183	C₂	semidirect
3. Q₈×C₁₆×C₃₁	C₂	semidirect
57. #33500	C₃₁	direct (N × H)
73. #2183	C₂×C₃₁	semidirect
74. Q₈×C₁₆	C₂×C₃₁	semidirect
120. C₃₁	#33500	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×#2960	2
C₃₁×#2183	2
Q₈×C₁₆×C₃₁	2
C₃₁×#2164	2
C₃₁×#2162	2
#33500	31

Subgroups — C31×#33500 #21490259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33500 #21490259