Subgroups — C₃₁×#17316 #21474075

Order 7936.

890 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	C₈×#41×C₃₁	C₂	1	2	yes	maximal
2	C₄×C₃₁×#131	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1450	C₂	1	3	yes	maximal, complemented
4	C₈×M₁₆×C₃₁	C₂	1	6	yes	maximal, complemented
5	C₂×C₃₁×#322	C₂	1	3	yes	maximal, complemented
6	C₂×C₃₁×#131	C₄	1	2	yes	—
7	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
8	C₈×Q₈×C₃₁	C₄	1	4	yes	—
9	C₈×D₄×C₃₁	C₄	1	12	yes	—
10	C₃₁×#431	C₂²	1	6	yes	—
11	C₃₁×#405	C₄	1	6	yes	—
12	C₄×M₁₆×C₃₁	C₂²	1	3	yes	—
13	C₂×C₄×C₈×C₃₁	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₃₁	C₂²	1	6	yes	—
15	C₃₁×#322	C₂²	1	1	yes	characteristic
16	C₃₁×#322	C₂²	1	6	yes	—
17	C₃₁×#322	C₂²	1	3	yes	—
18	C₈²×C₃₁	C₂²	1	6	yes	—
19	C₂×#41×C₃₁	C₂×C₄	1	1	yes	characteristic
20	C₃₁×#131	C₈	1	16	yes	complemented
21	C₂×M₁₆×C₃₁	C₂×C₄	1	6	yes	—
22	C₂²×C₈×C₃₁	C₂×C₄	1	12	yes	—
23	C₂²×C₈×C₃₁	C₂×C₄	1	6	yes	—
24	C₄×Q₈×C₃₁	C₂×C₄	1	2	yes	—
25	C₄×D₄×C₃₁	C₂×C₄	1	6	yes	—
26	C₃₁×#117	C₂×C₄	1	3	yes	—
27	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
28	C₃₁×C₄⋊C₈	C₂×C₄	1	12	yes	—
29	C₃₁×C₂²⋊C₈	C₂×C₄	1	12	yes	—
30	C₃₁×#97	C₂×C₄	1	6	yes	—
31	C₄×C₈×C₃₁	C₂×C₄	1	6	yes	—
32	C₄×C₈×C₃₁	C₂×C₄	1	12	yes	—
33	C₄×C₈×C₃₁	C₂³	1	2	yes	—
34	C₄×C₈×C₃₁	C₂³	1	1	yes	characteristic
35	C₄×C₈×C₃₁	C₂³	1	3	yes	—
36	C₄×C₈×C₃₁	C₂³	1	6	yes	—
37	#41×C₃₁	C₂×C₈	1	8	yes	—
38	C₂×Q₈×C₃₁	C₄²	1	1	yes	characteristic
39	C₂×D₄×C₃₁	C₄²	1	3	yes	—
40	C₂²×C₄×C₃₁	C₂²×C₄	1	6	yes	—
41	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
42	M₁₆×C₃₁	C₂×C₈	1	24	yes	complemented
43	C₂×C₈×C₃₁	—	4	12	no	—
44	C₂×C₈×C₃₁	C₂×C₈	1	24	yes	complemented
45	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
46	C₂×C₈×C₃₁	C₂²×C₄	1	12	yes	—
47	C₂×C₈×C₃₁	C₂²×C₄	1	2	yes	—
48	C₂×C₈×C₃₁	C₂²×C₄	1	6	yes	—
49	C₄⋊C₄×C₃₁	C₄²	1	6	yes	—
50	C₂²⋊C₄×C₃₁	C₄²	1	6	yes	—
51	C₄²×C₃₁	C₂²×C₄	1	3	yes	—
52	C₄²×C₃₁	C₂²×C₄	1	6	yes	—
53	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
54	#17316	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₃₁	C₂×C₄²	1	3	yes	—
56	Q₈×C₃₁	C₄×C₈	1	4	yes	—
57	D₄×C₃₁	C₄×C₈	1	12	yes	—
58	C₂×C₄×C₃₁	—	2	12	no	—
59	C₂×C₄×C₃₁	C₂×C₄²	1	6	yes	—
60	C₂×C₄×C₃₁	C₂×C₄²	1	3	yes	—
61	C₂×C₄×C₃₁	C₂²×C₈	1	12	yes	—
62	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
63	C₂×C₄×C₃₁	C₂³×C₄	1	2	yes	—
64	C₈×C₃₁	—	2	4	no	—
65	C₈×C₃₁	—	2	12	no	—
66	C₈×C₃₁	C₂²×C₈	1	4	yes	—
67	C₈×C₃₁	C₂²×C₈	1	12	yes	—
68	C₈×#41	C₂×C₃₁	1	2	yes	—
69	C₄×#131	C₂×C₃₁	1	1	yes	characteristic
70	#1450	C₂×C₃₁	1	3	yes	complemented
71	C₈×M₁₆	C₂×C₃₁	1	6	yes	complemented
72	C₂×#322	C₂×C₃₁	1	3	yes	complemented
73	C₂²×C₃₁	—	2	6	no	—
74	C₂²×C₃₁	C₂×C₄×C₈	1	6	yes	—
75	C₂²×C₃₁	C₂²×C₄²	1	1	yes	characteristic
76	C₄×C₃₁	—	2	6	no	—
77	C₄×C₃₁	C₂×C₄×C₈	1	6	yes	—
78	C₄×C₃₁	C₂³×C₈	1	2	yes	—
79	C₄×C₃₁	#568	1	4	yes	—
80	C₂×#131	C₄×C₃₁	1	2	yes	—
81	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
82	C₈×Q₈	C₄×C₃₁	1	4	yes	—
83	C₈×D₄	C₄×C₃₁	1	12	yes	—
84	#431	C₂²×C₃₁	1	6	yes	—
85	#405	C₄×C₃₁	1	6	yes	—
86	C₄×M₁₆	C₂²×C₃₁	1	3	yes	—
87	C₂×C₄×C₈	C₂²×C₃₁	1	3	yes	—
88	C₂×C₄×C₈	C₂²×C₃₁	1	6	yes	—
89	#322	C₂²×C₃₁	1	1	yes	characteristic
90	#322	C₂²×C₃₁	1	6	yes	—
91	#322	C₂²×C₃₁	1	3	yes	—
92	C₈²	C₂²×C₃₁	1	6	yes	—
93	C₂×C₃₁	—	2	6	no	—
94	C₂×C₃₁	C₂²×C₄×C₈	1	1	yes	characteristic
95	C₂×C₃₁	#2875	1	2	yes	—
96	C₂×#41	C₂×C₄×C₃₁	1	1	yes	characteristic
97	#131	C₈×C₃₁	1	16	yes	complemented
98	C₂×M₁₆	C₂×C₄×C₃₁	1	6	yes	—
99	C₂²×C₈	C₂×C₄×C₃₁	1	12	yes	—
100	C₂²×C₈	C₂×C₄×C₃₁	1	6	yes	—
101	C₄×Q₈	C₂×C₄×C₃₁	1	2	yes	—
102	C₄×D₄	C₂×C₄×C₃₁	1	6	yes	—
103	#117	C₂×C₄×C₃₁	1	3	yes	—
104	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
105	C₄⋊C₈	C₂×C₄×C₃₁	1	12	yes	—
106	C₂²⋊C₈	C₂×C₄×C₃₁	1	12	yes	—
107	#97	C₂×C₄×C₃₁	1	6	yes	—
108	C₄×C₈	C₂×C₄×C₃₁	1	6	yes	—
109	C₄×C₈	C₂×C₄×C₃₁	1	12	yes	—
110	C₄×C₈	C₂³×C₃₁	1	3	yes	—
111	C₄×C₈	C₂³×C₃₁	1	6	yes	—
112	C₄×C₈	C₂³×C₃₁	1	2	yes	—
113	C₄×C₈	C₂³×C₃₁	1	1	yes	characteristic
114	C₃₁	#17316	1	1	yes	characteristic, direct product factor, Sylow
115	#41	C₂×C₈×C₃₁	1	8	yes	—
116	C₂×Q₈	C₄²×C₃₁	1	1	yes	characteristic
117	C₂×D₄	C₄²×C₃₁	1	3	yes	—
118	C₂²×C₄	C₂²×C₄×C₃₁	1	6	yes	—
119	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
120	M₁₆	C₂×C₈×C₃₁	1	24	yes	complemented
121	C₂×C₈	—	4	12	no	—
122	C₂×C₈	C₂×C₈×C₃₁	1	24	yes	complemented
123	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
124	C₂×C₈	C₂²×C₄×C₃₁	1	12	yes	—
125	C₂×C₈	C₂²×C₄×C₃₁	1	2	yes	—
126	C₂×C₈	C₂²×C₄×C₃₁	1	6	yes	—
127	C₄⋊C₄	C₄²×C₃₁	1	6	yes	—
128	C₂²⋊C₄	C₄²×C₃₁	1	6	yes	—
129	C₄²	C₂²×C₄×C₃₁	1	3	yes	—
130	C₄²	C₂²×C₄×C₃₁	1	6	yes	—
131	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
132	C₂³	C₂×C₄²×C₃₁	1	3	yes	—
133	Q₈	C₄×C₈×C₃₁	1	4	yes	—
134	D₄	C₄×C₈×C₃₁	1	12	yes	—
135	C₂×C₄	—	2	12	no	—
136	C₂×C₄	C₂×C₄²×C₃₁	1	6	yes	—
137	C₂×C₄	C₂×C₄²×C₃₁	1	3	yes	—
138	C₂×C₄	C₂²×C₈×C₃₁	1	12	yes	—
139	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
140	C₂×C₄	C₂³×C₄×C₃₁	1	2	yes	—
141	C₈	—	2	4	no	—
142	C₈	—	2	12	no	—
143	C₈	C₂²×C₈×C₃₁	1	4	yes	—
144	C₈	C₂²×C₈×C₃₁	1	12	yes	—
145	C₂²	—	2	6	no	—
146	C₂²	C₂×C₄×C₈×C₃₁	1	6	yes	—
147	C₂²	C₂²×C₄²×C₃₁	1	1	yes	characteristic
148	C₄	—	2	6	no	—
149	C₄	C₂×C₄×C₈×C₃₁	1	6	yes	—
150	C₄	C₂³×C₈×C₃₁	1	2	yes	—
151	C₄	C₃₁×#568	1	4	yes	—
152	C₂	—	2	6	no	—
153	C₂	C₂²×C₄×C₈×C₃₁	1	1	yes	characteristic
154	C₂	C₃₁×#2875	1	2	yes	—

	subgroup	G/H
center	C₄²×C₃₁	C₂⁴
commutator	C₂	C₂²×C₄×C₈×C₃₁
Frattini, cosocle	C₄²	C₂⁴×C₃₁
socle	C₂²×C₃₁	C₂²×C₄²
prime radical	C₂×#41×C₃₁	C₂×C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1450	C₂	semidirect
4. C₈×M₁₆×C₃₁	C₂	semidirect
5. C₂×C₃₁×#322	C₂	semidirect
20. C₃₁×#131	C₈	semidirect
42. M₁₆×C₃₁	C₂×C₈	semidirect
44. C₂×C₈×C₃₁	C₂×C₈	semidirect
54. #17316	C₃₁	direct (N × H)
70. #1450	C₂×C₃₁	semidirect
71. C₈×M₁₆	C₂×C₃₁	semidirect
72. C₂×#322	C₂×C₃₁	semidirect
97. #131	C₈×C₃₁	semidirect
114. C₃₁	#17316	direct (N × H)
120. M₁₆	C₂×C₈×C₃₁	semidirect
122. C₂×C₈	C₂×C₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₈×#41×C₃₁	2
C₄×C₃₁×#131	2
C₃₁×#1450	2
C₈×M₁₆×C₃₁	2
C₂×C₃₁×#322	2
#17316	31

Subgroups — C31×#17316 #21474075

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#17316 #21474075