Subgroups — C₃₁×#13645 #21470404

Order 7936.

204 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₃₁×C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic
4	C₃₁×C₄⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×C₂²⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#1397	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
10	C₈×D₄×C₃₁	C₄	1	1	yes	characteristic
11	C₃₁×M₆₄	—	2	2	no	—
12	C₂×C₃₁×C₃₂	—	2	2	no	—
13	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×C₃₂	C₂²	1	1	yes	characteristic
17	C₃₁×C₄⋊C₁₆	C₄	1	1	yes	characteristic
18	C₃₁×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
19	C₃₁×C₂²⋊C₁₆	C₄	1	1	yes	characteristic
20	C₄×C₁₆×C₃₁	C₂²	1	1	yes	characteristic
21	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
23	C₄×D₄×C₃₁	C₈	1	1	yes	characteristic
24	C₂×C₁₆×C₃₁	—	2	1	no	—
25	C₂×C₁₆×C₃₁	—	2	1	no	—
26	C₂×C₁₆×C₃₁	—	2	1	no	—
27	C₂×C₁₆×C₃₁	—	2	1	no	—
28	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₃₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₁₆×C₃₁	C₂³	1	1	yes	characteristic
32	C₃₁×C₄⋊C₈	C₈	1	1	yes	characteristic
33	C₃₁×C₂²⋊C₈	C₈	1	1	yes	characteristic
34	C₃₁×C₂²⋊C₈	C₈	1	1	yes	characteristic
35	C₄×C₈×C₃₁	C₂×C₄	1	1	yes	characteristic
36	C₃₁×C₃₂	—	2	1	no	—
37	C₃₁×C₃₂	—	2	1	no	—
38	C₃₁×C₃₂	—	2	1	no	—
39	C₃₁×C₃₂	D₄	1	2	yes	complemented
40	C₂×D₄×C₃₁	C₁₆	1	1	yes	characteristic
41	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
42	C₂²×C₄×C₃₁	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₃₁	—	2	1	no	—
44	C₂×C₈×C₃₁	—	2	1	no	—
45	C₂×C₈×C₃₁	—	2	1	no	—
46	C₂×C₈×C₃₁	—	2	1	no	—
47	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
48	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
49	C₂×C₈×C₃₁	C₂×C₈	1	1	yes	characteristic
50	C₂×C₈×C₃₁	C₂²×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₃₁	C₁₆	1	1	yes	characteristic
52	C₂²⋊C₄×C₃₁	C₁₆	1	1	yes	characteristic
53	C₂²⋊C₄×C₃₁	C₁₆	1	1	yes	characteristic
54	C₄²×C₃₁	C₂×C₈	1	1	yes	characteristic
55	C₁₆×C₃₁	—	2	1	no	—
56	C₁₆×C₃₁	—	2	1	no	—
57	C₁₆×C₃₁	—	2	1	no	—
58	C₁₆×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₁₆×C₃₁	#41	1	1	yes	characteristic
60	#13645	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₃₁	C₂×C₁₆	1	1	yes	characteristic
62	C₂³×C₃₁	C₂×C₁₆	1	1	yes	characteristic
63	D₄×C₃₁	—	2	2	no	—
64	C₂×C₄×C₃₁	—	2	1	no	—
65	C₂×C₄×C₃₁	—	2	1	no	—
66	C₂×C₄×C₃₁	—	2	1	no	—
67	C₂×C₄×C₃₁	—	2	1	no	—
68	C₂×C₄×C₃₁	C₂×C₁₆	1	1	yes	characteristic
69	C₂×C₄×C₃₁	C₂×C₁₆	1	1	yes	characteristic
70	C₂×C₄×C₃₁	C₂×C₁₆	1	1	yes	characteristic
71	C₂×C₄×C₃₁	C₂×C₁₆	1	1	yes	characteristic
72	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
73	C₈×C₃₁	—	2	1	no	—
74	C₈×C₃₁	—	2	1	no	—
75	C₈×C₃₁	—	2	1	no	—
76	C₈×C₃₁	C₄×D₄	1	1	yes	characteristic
77	C₈×C₃₁	#131	1	1	yes	characteristic
78	C₂×M₆₄	C₂×C₃₁	1	1	yes	characteristic, complemented
79	C₂²×C₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
80	D₄×C₁₆	C₂×C₃₁	1	1	yes	characteristic
81	C₄⋊C₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
82	C₂²⋊C₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
83	C₂²⋊C₃₂	C₂×C₃₁	1	1	yes	characteristic, complemented
84	#1397	C₂×C₃₁	1	1	yes	characteristic, complemented
85	C₂²×C₃₁	—	2	1	no	—
86	C₂²×C₃₁	—	2	1	no	—
87	C₂²×C₃₁	—	2	1	no	—
88	C₂²×C₃₁	—	2	1	no	—
89	C₂²×C₃₁	—	2	1	no	—
90	C₂²×C₃₁	M₆₄	1	2	yes	complemented
91	C₂²×C₃₁	C₂²×C₁₆	1	1	yes	characteristic
92	C₄×C₃₁	—	2	1	no	—
93	C₄×C₃₁	—	2	1	no	—
94	C₄×C₃₁	—	2	1	no	—
95	C₄×C₃₁	—	2	1	no	—
96	C₄×C₃₁	C₈×D₄	1	1	yes	characteristic
97	C₄×C₃₁	#504	1	1	yes	characteristic
98	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
99	C₂²×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
100	C₈×D₄	C₄×C₃₁	1	1	yes	characteristic
101	M₆₄	—	2	2	no	—
102	C₂×C₃₂	—	2	2	no	—
103	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
104	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
105	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
106	C₂×C₃₂	C₂²×C₃₁	1	1	yes	characteristic
107	C₄⋊C₁₆	C₄×C₃₁	1	1	yes	characteristic
108	C₂²⋊C₁₆	C₄×C₃₁	1	1	yes	characteristic
109	C₂²⋊C₁₆	C₄×C₃₁	1	1	yes	characteristic
110	C₄×C₁₆	C₂²×C₃₁	1	1	yes	characteristic
111	C₂×C₃₁	—	2	2	no	—
112	C₂×C₃₁	—	4	1	no	—
113	C₂×C₃₁	D₄×C₁₆	1	1	yes	characteristic
114	C₂×C₃₁	C₂×M₆₄	1	1	yes	characteristic
115	C₂×C₃₁	#2258	1	1	yes	characteristic
116	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
117	C₂²×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
118	C₄×D₄	C₈×C₃₁	1	1	yes	characteristic
119	C₂×C₁₆	—	2	1	no	—
120	C₂×C₁₆	—	2	1	no	—
121	C₂×C₁₆	—	2	1	no	—
122	C₂×C₁₆	—	2	1	no	—
123	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
124	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
125	C₂×C₁₆	C₂×C₄×C₃₁	1	1	yes	characteristic
126	C₂×C₁₆	C₂³×C₃₁	1	1	yes	characteristic
127	C₄⋊C₈	C₈×C₃₁	1	1	yes	characteristic
128	C₂²⋊C₈	C₈×C₃₁	1	1	yes	characteristic
129	C₂²⋊C₈	C₈×C₃₁	1	1	yes	characteristic
130	C₄×C₈	C₂×C₄×C₃₁	1	1	yes	characteristic
131	C₃₂	—	2	1	no	—
132	C₃₂	—	2	1	no	—
133	C₃₂	—	2	1	no	—
134	C₃₂	D₄×C₃₁	1	2	yes	complemented
135	C₃₁	#13645	1	1	yes	characteristic, direct product factor, Sylow
136	C₂×D₄	C₁₆×C₃₁	1	1	yes	characteristic
137	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
138	C₂²×C₄	C₂×C₈×C₃₁	1	1	yes	characteristic
139	C₂×C₈	—	2	1	no	—
140	C₂×C₈	—	2	1	no	—
141	C₂×C₈	—	2	1	no	—
142	C₂×C₈	—	2	1	no	—
143	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
144	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
145	C₂×C₈	C₂×C₈×C₃₁	1	1	yes	characteristic
146	C₂×C₈	C₂²×C₄×C₃₁	1	1	yes	characteristic
147	C₄⋊C₄	C₁₆×C₃₁	1	1	yes	characteristic
148	C₂²⋊C₄	C₁₆×C₃₁	1	1	yes	characteristic
149	C₂²⋊C₄	C₁₆×C₃₁	1	1	yes	characteristic
150	C₄²	C₂×C₈×C₃₁	1	1	yes	characteristic
151	C₁₆	—	2	1	no	—
152	C₁₆	—	2	1	no	—
153	C₁₆	—	2	1	no	—
154	C₁₆	C₂×D₄×C₃₁	1	1	yes	characteristic
155	C₁₆	#41×C₃₁	1	1	yes	characteristic
156	C₂³	C₂×C₁₆×C₃₁	1	1	yes	characteristic
157	C₂³	C₂×C₁₆×C₃₁	1	1	yes	characteristic
158	D₄	—	2	2	no	—
159	C₂×C₄	—	2	1	no	—
160	C₂×C₄	—	2	1	no	—
161	C₂×C₄	—	2	1	no	—
162	C₂×C₄	—	2	1	no	—
163	C₂×C₄	C₂×C₁₆×C₃₁	1	1	yes	characteristic
164	C₂×C₄	C₂×C₁₆×C₃₁	1	1	yes	characteristic
165	C₂×C₄	C₂×C₁₆×C₃₁	1	1	yes	characteristic
166	C₂×C₄	C₂×C₁₆×C₃₁	1	1	yes	characteristic
167	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
168	C₈	—	2	1	no	—
169	C₈	—	2	1	no	—
170	C₈	—	2	1	no	—
171	C₈	C₄×D₄×C₃₁	1	1	yes	characteristic
172	C₈	C₃₁×#131	1	1	yes	characteristic
173	C₂²	—	2	1	no	—
174	C₂²	—	2	1	no	—
175	C₂²	—	2	1	no	—
176	C₂²	—	2	1	no	—
177	C₂²	—	2	1	no	—
178	C₂²	C₃₁×M₆₄	1	2	yes	complemented
179	C₂²	C₂²×C₁₆×C₃₁	1	1	yes	characteristic
180	C₄	—	2	1	no	—
181	C₄	—	2	1	no	—
182	C₄	—	2	1	no	—
183	C₄	—	2	1	no	—
184	C₄	C₈×D₄×C₃₁	1	1	yes	characteristic
185	C₄	C₃₁×#504	1	1	yes	characteristic
186	C₂	—	2	2	no	—
187	C₂	—	4	1	no	—
188	C₂	D₄×C₁₆×C₃₁	1	1	yes	characteristic
189	C₂	C₂×C₃₁×M₆₄	1	1	yes	characteristic
190	C₂	C₃₁×#2258	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆×C₃₁	C₂³
commutator	C₂²	C₂²×C₁₆×C₃₁
Frattini, cosocle	C₂×C₁₆	C₂³×C₃₁
socle	C₂²×C₃₁	C₂²×C₁₆
prime radical	C₂×D₄×C₃₁	C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×M₆₄	C₂	semidirect
2. C₂²×C₃₁×C₃₂	C₂	semidirect
4. C₃₁×C₄⋊C₃₂	C₂	semidirect
5. C₃₁×C₂²⋊C₃₂	C₂	semidirect
6. C₃₁×C₂²⋊C₃₂	C₂	semidirect
7. C₃₁×#1397	C₂	semidirect
39. C₃₁×C₃₂	D₄	semidirect
60. #13645	C₃₁	direct (N × H)
78. C₂×M₆₄	C₂×C₃₁	semidirect
79. C₂²×C₃₂	C₂×C₃₁	semidirect
81. C₄⋊C₃₂	C₂×C₃₁	semidirect
82. C₂²⋊C₃₂	C₂×C₃₁	semidirect
83. C₂²⋊C₃₂	C₂×C₃₁	semidirect
84. #1397	C₂×C₃₁	semidirect
90. C₂²×C₃₁	M₆₄	semidirect
134. C₃₂	D₄×C₃₁	semidirect
135. C₃₁	#13645	direct (N × H)
178. C₂²	C₃₁×M₆₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×M₆₄	2
C₂²×C₃₁×C₃₂	2
D₄×C₁₆×C₃₁	2
C₃₁×C₄⋊C₃₂	2
C₃₁×C₂²⋊C₃₂	2
C₃₁×#1397	2
#13645	31

Subgroups — C31×#13645 #21470404

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#13645 #21470404