Subgroups — C₃₁×#8785 #21465544

Order 7936.

382 conjugacy classes of subgroups.

The subgroup lattice has 156 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#1831	C₂	1	2	yes	maximal
2	C₄²×C₈×C₃₁	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1277	C₂	1	4	yes	maximal
4	C₂×C₃₁×C₄⋊C₈	C₂²	1	4	yes	—
5	C₂×C₄×C₈×C₃₁	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₃₁	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₃₁	C₂²	1	2	yes	—
9	C₄³×C₃₁	C₂²	1	1	yes	characteristic
10	C₃₁×#336	—	2	4	no	—
11	C₂²×C₈×C₃₁	—	2	4	no	—
12	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₃₁	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₃₁	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₃₁	D₄	1	2	yes	—
17	C₂×C₄²×C₃₁	Q₈	1	2	yes	—
18	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
19	C₃₁×C₄⋊C₈	—	2	8	no	—
20	C₄×C₈×C₃₁	—	2	4	no	—
21	C₄×C₈×C₃₁	—	2	2	no	—
22	C₄×C₈×C₃₁	—	2	4	no	—
23	C₄×C₈×C₃₁	C₈	1	4	yes	complemented
24	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
25	C₂²×C₄×C₃₁	C₄⋊C₄	1	2	yes	—
26	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₃₁	C₂×Q₈	1	1	yes	characteristic
29	C₂×C₈×C₃₁	—	2	2	no	—
30	C₂×C₈×C₃₁	—	2	2	no	—
31	C₂×C₈×C₃₁	—	2	2	no	—
32	C₂×C₈×C₃₁	—	2	2	no	—
33	C₂×C₈×C₃₁	—	4	4	no	—
34	C₂×C₈×C₃₁	—	4	8	no	—
35	C₂×C₈×C₃₁	C₂×C₈	1	4	yes	—
36	C₂×C₈×C₃₁	M₁₆	1	4	yes	—
37	C₄²×C₃₁	—	2	4	no	—
38	C₄²×C₃₁	—	2	2	no	—
39	C₄²×C₃₁	—	2	4	no	—
40	C₄²×C₃₁	C₂×C₈	1	2	yes	—
41	C₄²×C₃₁	M₁₆	1	2	yes	—
42	C₄²×C₃₁	#41	1	4	yes	—
43	#8785	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	C₂³×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₁	—	2	2	no	—
46	C₂×C₄×C₃₁	—	2	2	no	—
47	C₂×C₄×C₃₁	—	2	2	no	—
48	C₂×C₄×C₃₁	—	2	2	no	—
49	C₂×C₄×C₃₁	—	2	2	no	—
50	C₂×C₄×C₃₁	C₄⋊C₈	1	4	yes	—
51	C₂×C₄×C₃₁	#108	1	4	yes	—
52	C₂×C₄×C₃₁	#117	1	2	yes	—
53	C₂×C₄×C₃₁	#124	1	2	yes	—
54	C₂×C₄×C₃₁	#125	1	2	yes	—
55	C₂×C₄×C₃₁	C₂²×C₈	1	1	yes	characteristic
56	C₂×C₄×C₃₁	C₂×M₁₆	1	2	yes	—
57	C₂×C₄×C₃₁	C₂×M₁₆	1	1	yes	characteristic
58	C₂×C₄×C₃₁	#135	1	4	yes	—
59	C₈×C₃₁	—	2	4	no	—
60	C₈×C₃₁	—	2	4	no	—
61	C₈×C₃₁	—	8	4	no	—
62	#1831	C₂×C₃₁	1	2	yes	—
63	C₄²×C₈	C₂×C₃₁	1	1	yes	characteristic
64	#1277	C₂×C₃₁	1	4	yes	—
65	C₂²×C₃₁	#382	1	1	yes	characteristic
66	C₂²×C₃₁	C₂×C₄⋊C₈	1	1	yes	characteristic
67	C₂²×C₃₁	#423	1	1	yes	characteristic
68	C₂²×C₃₁	#427	1	2	yes	—
69	C₂²×C₃₁	C₂×#108	1	2	yes	—
70	C₄×C₃₁	—	2	4	no	—
71	C₄×C₃₁	—	2	4	no	—
72	C₄×C₃₁	—	2	2	no	—
73	C₄×C₃₁	#431	1	2	yes	—
74	C₄×C₃₁	#432	1	2	yes	—
75	C₄×C₃₁	#488	1	4	yes	—
76	C₂×C₄⋊C₈	C₂²×C₃₁	1	4	yes	—
77	C₂×C₄×C₈	C₄×C₃₁	1	2	yes	—
78	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
79	C₂×C₄×C₈	C₄×C₃₁	1	1	yes	characteristic
80	C₂×C₄×C₈	C₂²×C₃₁	1	2	yes	—
81	C₄³	C₂²×C₃₁	1	1	yes	characteristic
82	#336	—	2	4	no	—
83	C₂×C₃₁	#1564	1	2	yes	—
84	C₂×C₃₁	#1568	1	2	yes	—
85	C₂×C₃₁	#1831	1	1	yes	characteristic
86	C₂×C₃₁	#1835	1	1	yes	characteristic
87	C₂×C₃₁	#1837	1	1	yes	characteristic
88	C₂²×C₈	—	2	4	no	—
89	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
90	C₂²×C₈	C₂×C₄×C₃₁	1	2	yes	—
91	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
92	C₂×C₄²	C₂×C₄×C₃₁	1	1	yes	characteristic
93	C₂×C₄²	D₄×C₃₁	1	2	yes	—
94	C₂×C₄²	Q₈×C₃₁	1	2	yes	—
95	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
96	C₄⋊C₈	—	2	8	no	—
97	C₄×C₈	—	2	4	no	—
98	C₄×C₈	—	2	2	no	—
99	C₄×C₈	—	2	4	no	—
100	C₄×C₈	C₈×C₃₁	1	4	yes	complemented
101	C₃₁	#8785	1	1	yes	characteristic, direct product factor, Sylow
102	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
103	C₂²×C₄	C₄⋊C₄×C₃₁	1	2	yes	—
104	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
105	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
106	C₂²×C₄	C₂×Q₈×C₃₁	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	4	4	no	—
112	C₂×C₈	—	4	8	no	—
113	C₂×C₈	C₂×C₈×C₃₁	1	4	yes	—
114	C₂×C₈	M₁₆×C₃₁	1	4	yes	—
115	C₄²	—	2	4	no	—
116	C₄²	—	2	2	no	—
117	C₄²	—	2	4	no	—
118	C₄²	C₂×C₈×C₃₁	1	2	yes	—
119	C₄²	M₁₆×C₃₁	1	2	yes	—
120	C₄²	#41×C₃₁	1	4	yes	—
121	C₂³	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	C₃₁×C₄⋊C₈	1	4	yes	—
128	C₂×C₄	C₃₁×#108	1	4	yes	—
129	C₂×C₄	C₃₁×#117	1	2	yes	—
130	C₂×C₄	C₃₁×#124	1	2	yes	—
131	C₂×C₄	C₃₁×#125	1	2	yes	—
132	C₂×C₄	C₂²×C₈×C₃₁	1	1	yes	characteristic
133	C₂×C₄	C₂×M₁₆×C₃₁	1	2	yes	—
134	C₂×C₄	C₂×M₁₆×C₃₁	1	1	yes	characteristic
135	C₂×C₄	C₃₁×#135	1	4	yes	—
136	C₈	—	2	4	no	—
137	C₈	—	2	4	no	—
138	C₈	—	8	4	no	—
139	C₂²	C₃₁×#382	1	1	yes	characteristic
140	C₂²	C₂×C₃₁×C₄⋊C₈	1	1	yes	characteristic
141	C₂²	C₃₁×#423	1	1	yes	characteristic
142	C₂²	C₃₁×#427	1	2	yes	—
143	C₂²	C₂×C₃₁×#108	1	2	yes	—
144	C₄	—	2	4	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	2	no	—
147	C₄	C₃₁×#431	1	2	yes	—
148	C₄	C₃₁×#432	1	2	yes	—
149	C₄	C₃₁×#488	1	4	yes	—
150	C₂	C₃₁×#1564	1	2	yes	—
151	C₂	C₃₁×#1568	1	2	yes	—
152	C₂	C₃₁×#1831	1	1	yes	characteristic
153	C₂	C₃₁×#1835	1	1	yes	characteristic
154	C₂	C₃₁×#1837	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
23. C₄×C₈×C₃₁	C₈	semidirect
43. #8785	C₃₁	direct (N × H)
100. C₄×C₈	C₈×C₃₁	semidirect
101. C₃₁	#8785	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×#1831	2
C₄²×C₈×C₃₁	2
C₃₁×#1277	2
#8785	31

Subgroups — C31×#8785 #21465544

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8785 #21465544