Subgroups — C₃₁×#8805 #21465564

Order 7936.

462 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×#1842	C₂	1	2	yes	maximal
2	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#334	C₂	1	4	yes	maximal
4	C₂×C₃₁×C₄⋊C₈	—	2	4	no	—
5	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
10	C₄³×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#334	—	2	8	no	—
12	C₂²×C₈×C₃₁	—	2	4	no	—
13	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
14	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄²×C₃₁	D₄	1	2	yes	—
18	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
19	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
20	C₃₁×C₄⋊C₈	—	2	8	no	—
21	C₃₁×C₄⋊C₈	—	4	8	no	—
22	C₄×C₈×C₃₁	—	2	4	no	—
23	C₄×C₈×C₃₁	—	2	2	no	—
24	C₄×C₈×C₃₁	C₈	1	4	yes	complemented
25	C₄×C₈×C₃₁	D₄	1	4	yes	—
26	C₄×C₈×C₃₁	Q₈	1	4	yes	—
27	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
28	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
29	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₈×C₃₁	—	2	2	no	—
33	C₂×C₈×C₃₁	—	2	2	no	—
34	C₂×C₈×C₃₁	—	4	4	no	—
35	C₂×C₈×C₃₁	—	4	8	no	—
36	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
37	C₂×C₈×C₃₁	C₄⋊C₄	1	4	yes	—
38	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
39	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
40	C₄²×C₃₁	—	2	4	no	—
41	C₄²×C₃₁	—	2	2	no	—
42	C₄²×C₃₁	C₂×C₈	1	2	yes	—
43	C₄²×C₃₁	M₁₆	1	2	yes	—
44	C₄²×C₃₁	SD₁₆	1	8	yes	—
45	C₄²×C₃₁	C₂×D₄	1	2	yes	—
46	C₄²×C₃₁	C₂×Q₈	1	2	yes	—
47	#8805	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₃₁	—	2	2	no	—
50	C₂×C₄×C₃₁	—	2	2	no	—
51	C₂×C₄×C₃₁	—	2	2	no	—
52	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
53	C₂×C₄×C₃₁	#106	1	4	yes	—
54	C₂×C₄×C₃₁	#106	1	4	yes	—
55	C₂×C₄×C₃₁	#108	1	4	yes	—
56	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₃₁	#127	1	1	yes	characteristic
58	C₂×C₄×C₃₁	C₄⋊Q₈	1	2	yes	—
59	C₂×C₄×C₃₁	C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
61	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
62	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂×SD₁₆	1	4	yes	—
64	C₈×C₃₁	—	2	4	no	—
65	C₈×C₃₁	—	8	4	no	—
66	C₈×C₃₁	C₄⋊C₈	1	8	yes	complemented
67	#1842	C₂×C₃₁	1	2	yes	—
68	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
69	C₂×#334	C₂×C₃₁	1	4	yes	—
70	C₂²×C₃₁	#384	1	1	yes	characteristic
71	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
72	C₂²×C₃₁	#423	1	1	yes	characteristic
73	C₂²×C₃₁	C₂×#106	1	2	yes	—
74	C₂²×C₃₁	C₂×#108	1	2	yes	—
75	C₄×C₃₁	—	2	4	no	—
76	C₄×C₃₁	—	2	2	no	—
77	C₄×C₃₁	#334	1	8	yes	complemented
78	C₄×C₃₁	C₂×C₄⋊C₈	1	2	yes	—
79	C₄×C₃₁	#423	1	2	yes	—
80	C₄×C₃₁	#492	1	2	yes	—
81	C₄×C₃₁	#498	1	2	yes	—
82	C₂×C₄⋊C₈	—	2	4	no	—
83	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
84	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
85	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
86	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
87	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
88	C₄³	C₂²×C₃₁	1	1	yes	characteristic
89	#334	—	2	8	no	—
90	C₂×C₃₁	C₂×#334	1	2	yes	—
91	C₂×C₃₁	#1566	1	2	yes	—
92	C₂×C₃₁	#1842	1	1	yes	characteristic
93	C₂×C₃₁	#1844	1	1	yes	characteristic
94	C₂×C₃₁	#1848	1	1	yes	characteristic
95	C₂²×C₈	—	2	4	no	—
96	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
97	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
98	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
99	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
100	C₂×C₄²	D₄×C₃₁	1	2	yes	—
101	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
102	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
103	C₄⋊C₈	—	2	8	no	—
104	C₄⋊C₈	—	4	8	no	—
105	C₄×C₈	—	2	4	no	—
106	C₄×C₈	—	2	2	no	—
107	C₄×C₈	C₈×C₃₁	1	4	yes	complemented
108	C₄×C₈	D₄×C₃₁	1	4	yes	—
109	C₄×C₈	Q₈×C₃₁	1	4	yes	—
110	C₃₁	#8805	1	1	yes	characteristic, direct product factor, Sylow
111	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
112	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
113	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
114	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
115	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	4	4	no	—
119	C₂×C₈	—	4	8	no	—
120	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
121	C₂×C₈	C₄⋊C₄×C₃₁	1	4	yes	—
122	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
123	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
124	C₄²	—	2	4	no	—
125	C₄²	—	2	2	no	—
126	C₄²	C₂×C₈×C₃₁	1	2	yes	—
127	C₄²	M₁₆×C₃₁	1	2	yes	—
128	C₄²	SD₁₆×C₃₁	1	8	yes	—
129	C₄²	C₂×D₄×C₃₁	1	2	yes	—
130	C₄²	C₂×Q₈×C₃₁	1	2	yes	—
131	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
136	C₂×C₄	C₃₁×#106	1	4	yes	—
137	C₂×C₄	C₃₁×#106	1	4	yes	—
138	C₂×C₄	C₃₁×#108	1	4	yes	—
139	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	2	yes	—
140	C₂×C₄	C₃₁×#127	1	1	yes	characteristic
141	C₂×C₄	C₃₁×C₄⋊Q₈	1	2	yes	—
142	C₂×C₄	C₃₁×C₄⋊Q₈	1	1	yes	characteristic
143	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
144	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
145	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
146	C₂×C₄	C₂×SD₁₆×C₃₁	1	4	yes	—
147	C₈	—	2	4	no	—
148	C₈	—	8	4	no	—
149	C₈	C₃₁×C₄⋊C₈	1	8	yes	complemented
150	C₂²	C₃₁×#384	1	1	yes	characteristic
151	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
152	C₂²	C₃₁×#423	1	1	yes	characteristic
153	C₂²	C₂×C₃₁×#106	1	2	yes	—
154	C₂²	C₂×C₃₁×#108	1	2	yes	—
155	C₄	—	2	4	no	—
156	C₄	—	2	2	no	—
157	C₄	C₃₁×#334	1	8	yes	complemented
158	C₄	C₂×C₃₁×C₄⋊C₈	1	2	yes	—
159	C₄	C₃₁×#423	1	2	yes	—
160	C₄	C₃₁×#492	1	2	yes	—
161	C₄	C₃₁×#498	1	2	yes	—
162	C₂	C₂×C₃₁×#334	1	2	yes	—
163	C₂	C₃₁×#1566	1	2	yes	—
164	C₂	C₃₁×#1842	1	1	yes	characteristic
165	C₂	C₃₁×#1844	1	1	yes	characteristic
166	C₂	C₃₁×#1848	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
socle, prime radical	C₂³×C₃₁	C₂×C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ C₄³×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₄×C₈×C₃₁	C₈	semidirect
47. #8805	C₃₁	direct (N × H)
66. C₈×C₃₁	C₄⋊C₈	semidirect
77. C₄×C₃₁	#334	semidirect
107. C₄×C₈	C₈×C₃₁	semidirect
110. C₃₁	#8805	direct (N × H)
149. C₈	C₃₁×C₄⋊C₈	semidirect
157. C₄	C₃₁×#334	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#1842	2
C₄²×C₈×C₃₁	2
C₂×C₃₁×#334	2
#8805	31

Subgroups — C31×#8805 #21465564

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8805 #21465564