Subgroups — C₃₁×#33498 #21490257

Order 7936.

494 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2183	C₂	1	6	yes	maximal, complemented
3	Q₈×C₁₆×C₃₁	C₂	1	2	yes	maximal, complemented
4	C₃₁×#2163	C₂	1	3	yes	maximal
5	C₃₁×#2162	C₂	1	3	yes	maximal
6	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₃₁	C₄	1	3	yes	—
8	C₈×Q₈×C₃₁	C₄	1	1	yes	characteristic
9	C₈×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₈×Q₈×C₃₁	C₂²	1	3	yes	—
11	C₂×C₃₁×C₄⋊C₈	C₄	1	3	yes	—
12	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
13	C₃₁×C₄⋊C₁₆	C₂²	1	6	yes	—
14	C₃₁×C₄⋊C₁₆	C₂²	1	6	yes	—
15	C₃₁×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₃₁×C₂²⋊C₁₆	C₂²	1	3	yes	—
17	C₃₁×#346	C₂²	1	6	yes	—
18	C₄×C₁₆×C₃₁	C₂²	1	6	yes	—
19	C₂²×Q₈×C₃₁	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₃₁	C₂×C₄	1	3	yes	—
21	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
22	C₄×Q₈×C₃₁	C₈	1	3	yes	—
23	C₄×Q₈×C₃₁	C₈	1	1	yes	characteristic
24	C₄×Q₈×C₃₁	C₂×C₄	1	3	yes	—
25	C₄×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊C₄×C₃₁	C₈	1	3	yes	—
27	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
28	C₂×C₁₆×C₃₁	C₂³	1	2	yes	—
29	C₂×C₁₆×C₃₁	C₂³	1	6	yes	—
30	C₃₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
31	C₃₁×C₄⋊C₈	C₂×C₄	1	6	yes	—
32	C₃₁×C₄⋊C₈	C₂×C₄	1	3	yes	—
33	C₄×C₈×C₃₁	C₂×C₄	1	3	yes	—
34	C₄×C₈×C₃₁	C₂×C₄	1	3	yes	—
35	C₄×C₈×C₃₁	C₂³	1	3	yes	—
36	C₄×C₈×C₃₁	C₂³	1	3	yes	—
37	C₂×Q₈×C₃₁	—	2	4	no	—
38	C₂×Q₈×C₃₁	C₂×C₈	1	3	yes	—
39	C₂×Q₈×C₃₁	C₂×C₈	1	1	yes	characteristic
40	C₂²×C₄×C₃₁	C₂×C₈	1	3	yes	—
41	C₂²×C₄×C₃₁	C₂×C₈	1	3	yes	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₃₁	—	2	2	no	—
44	C₂×C₈×C₃₁	—	4	3	no	—
45	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
47	C₂×C₈×C₃₁	C₂²×C₄	1	3	yes	—
48	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
49	C₄⋊C₄×C₃₁	C₂×C₈	1	6	yes	—
50	C₄⋊C₄×C₃₁	C₂×C₈	1	3	yes	—
51	C₄⋊C₄×C₃₁	C₂×C₈	1	3	yes	—
52	C₄²×C₃₁	C₂×C₈	1	3	yes	—
53	C₄²×C₃₁	C₂×C₈	1	3	yes	—
54	C₄²×C₃₁	C₂²×C₄	1	3	yes	—
55	C₄²×C₃₁	C₂²×C₄	1	3	yes	—
56	C₁₆×C₃₁	—	2	2	no	—
57	C₁₆×C₃₁	—	2	6	no	—
58	#33498	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂³×C₃₁	C₂²×C₈	1	1	yes	characteristic
60	Q₈×C₃₁	—	2	6	no	—
61	Q₈×C₃₁	M₃₂	1	4	yes	—
62	C₂×C₄×C₃₁	—	2	6	no	—
63	C₂×C₄×C₃₁	—	2	1	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	6	no	—
66	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
67	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
68	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
69	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
70	C₂×C₄×C₃₁	C₂²×C₈	1	3	yes	—
71	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
72	C₈×C₃₁	—	2	1	no	—
73	C₈×C₃₁	—	2	3	no	—
74	C₈×C₃₁	—	2	3	no	—
75	C₈×C₃₁	2₋¹⁺⁴	1	2	yes	—
76	C₂×C₈×Q₈	C₂×C₃₁	1	1	yes	characteristic
77	#2183	C₂×C₃₁	1	6	yes	complemented
78	Q₈×C₁₆	C₂×C₃₁	1	2	yes	complemented
79	#2163	C₂×C₃₁	1	3	yes	—
80	#2162	C₂×C₃₁	1	3	yes	—
81	C₂²×C₃₁	—	2	1	no	—
82	C₂²×C₃₁	—	2	1	no	—
83	C₂²×C₃₁	#504	1	2	yes	—
84	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
85	C₄×C₃₁	—	2	3	no	—
86	C₄×C₃₁	—	2	3	no	—
87	C₄×C₃₁	—	2	3	no	—
88	C₄×C₃₁	—	2	1	no	—
89	C₄×C₃₁	C₂×M₃₂	1	6	yes	—
90	C₄×C₃₁	#519	1	1	yes	characteristic
91	C₄×C₃₁	#568	1	1	yes	characteristic
92	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
93	C₈×Q₈	C₄×C₃₁	1	3	yes	—
94	C₈×Q₈	C₄×C₃₁	1	1	yes	characteristic
95	C₈×Q₈	C₂²×C₃₁	1	1	yes	characteristic
96	C₈×Q₈	C₂²×C₃₁	1	3	yes	—
97	C₂×C₄⋊C₈	C₄×C₃₁	1	3	yes	—
98	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
99	C₄⋊C₁₆	C₂²×C₃₁	1	6	yes	—
100	C₄⋊C₁₆	C₂²×C₃₁	1	6	yes	—
101	C₂²⋊C₁₆	C₂²×C₃₁	1	1	yes	characteristic
102	C₂²⋊C₁₆	C₂²×C₃₁	1	3	yes	—
103	#346	C₂²×C₃₁	1	6	yes	—
104	C₄×C₁₆	C₂²×C₃₁	1	6	yes	—
105	C₂×C₃₁	—	2	2	no	—
106	C₂×C₃₁	#2982	1	1	yes	characteristic
107	C₂×C₃₁	C₂²×M₃₂	1	1	yes	characteristic
108	C₂×C₃₁	C₂×#504	1	1	yes	characteristic
109	C₂²×Q₈	C₈×C₃₁	1	1	yes	characteristic
110	C₂²×C₈	C₂×C₄×C₃₁	1	3	yes	—
111	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
112	C₄×Q₈	C₈×C₃₁	1	3	yes	—
113	C₄×Q₈	C₈×C₃₁	1	1	yes	characteristic
114	C₄×Q₈	C₂×C₄×C₃₁	1	3	yes	—
115	C₄×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
116	C₂×C₄⋊C₄	C₈×C₃₁	1	3	yes	—
117	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
118	C₂×C₁₆	C₂³×C₃₁	1	2	yes	—
119	C₂×C₁₆	C₂³×C₃₁	1	6	yes	—
120	C₄⋊C₈	C₂×C₄×C₃₁	1	3	yes	—
121	C₄⋊C₈	C₂×C₄×C₃₁	1	6	yes	—
122	C₄⋊C₈	C₂×C₄×C₃₁	1	3	yes	—
123	C₄×C₈	C₂×C₄×C₃₁	1	3	yes	—
124	C₄×C₈	C₂×C₄×C₃₁	1	3	yes	—
125	C₄×C₈	C₂³×C₃₁	1	3	yes	—
126	C₄×C₈	C₂³×C₃₁	1	3	yes	—
127	C₃₁	#33498	1	1	yes	characteristic, direct product factor, Sylow
128	C₂×Q₈	—	2	4	no	—
129	C₂×Q₈	C₂×C₈×C₃₁	1	3	yes	—
130	C₂×Q₈	C₂×C₈×C₃₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₈×C₃₁	1	3	yes	—
132	C₂²×C₄	C₂×C₈×C₃₁	1	3	yes	—
133	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	4	3	no	—
136	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
137	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
138	C₂×C₈	C₂²×C₄×C₃₁	1	3	yes	—
139	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
140	C₄⋊C₄	C₂×C₈×C₃₁	1	6	yes	—
141	C₄⋊C₄	C₂×C₈×C₃₁	1	3	yes	—
142	C₄⋊C₄	C₂×C₈×C₃₁	1	3	yes	—
143	C₄²	C₂×C₈×C₃₁	1	3	yes	—
144	C₄²	C₂×C₈×C₃₁	1	3	yes	—
145	C₄²	C₂²×C₄×C₃₁	1	3	yes	—
146	C₄²	C₂²×C₄×C₃₁	1	3	yes	—
147	C₁₆	—	2	2	no	—
148	C₁₆	—	2	6	no	—
149	C₂³	C₂²×C₈×C₃₁	1	1	yes	characteristic
150	Q₈	—	2	6	no	—
151	Q₈	C₃₁×M₃₂	1	4	yes	—
152	C₂×C₄	—	2	6	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	2	1	no	—
155	C₂×C₄	—	2	1	no	—
156	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
157	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
158	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
159	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
160	C₂×C₄	C₂²×C₈×C₃₁	1	3	yes	—
161	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
162	C₈	—	2	1	no	—
163	C₈	—	2	3	no	—
164	C₈	—	2	3	no	—
165	C₈	C₃₁×2₋¹⁺⁴	1	2	yes	—
166	C₂²	—	2	1	no	—
167	C₂²	—	2	1	no	—
168	C₂²	C₃₁×#504	1	2	yes	—
169	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
170	C₄	—	2	3	no	—
171	C₄	—	2	3	no	—
172	C₄	—	2	3	no	—
173	C₄	—	2	1	no	—
174	C₄	C₂×C₃₁×M₃₂	1	6	yes	—
175	C₄	C₃₁×#519	1	1	yes	characteristic
176	C₄	C₃₁×#568	1	1	yes	characteristic
177	C₂	—	2	2	no	—
178	C₂	C₃₁×#2982	1	1	yes	characteristic
179	C₂	C₂²×C₃₁×M₃₂	1	1	yes	characteristic
180	C₂	C₂×C₃₁×#504	1	1	yes	characteristic

	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
58. #33498	C₃₁	direct (N × H)
77. #2183	C₂×C₃₁	semidirect
78. Q₈×C₁₆	C₂×C₃₁	semidirect
127. C₃₁	#33498	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₈×Q₈×C₃₁	2
C₃₁×#2183	2
Q₈×C₁₆×C₃₁	2
C₃₁×#2163	2
C₃₁×#2162	2
#33498	31

Subgroups — C31×#33498 #21490257

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#33498 #21490257