Subgroups — C₃₁×#14353 #21471112

Order 7936.

2070 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#380	C₂	1	6	yes	maximal
2	C₂²×C₃₁×#95	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1444	C₂	1	8	yes	maximal, complemented
4	C₂⁴×C₄×C₃₁	C₂²	1	3	yes	—
5	C₂²×C₄⋊C₄×C₃₁	C₂²	1	12	yes	—
6	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
7	C₃₁×#380	—	2	24	no	—
8	C₂×C₃₁×#95	C₄	1	8	yes	complemented
9	C₂×C₃₁×#95	C₂²	1	12	yes	—
10	C₂×C₃₁×#95	C₂²	1	4	yes	—
11	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
12	C₂³×C₄×C₃₁	—	4	12	no	—
13	C₂³×C₄×C₃₁	C₂×C₄	1	12	yes	—
14	C₂³×C₄×C₃₁	D₄	1	12	yes	—
15	C₂³×C₄×C₃₁	Q₈	1	12	yes	—
16	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
17	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
18	C₂×C₄⋊C₄×C₃₁	—	2	24	no	—
19	C₂×C₄⋊C₄×C₃₁	—	4	24	no	—
20	C₂×C₂²⋊C₄×C₃₁	—	2	24	no	—
21	C₃₁×#95	—	2	24	no	—
22	C₃₁×#95	C₂×C₄	1	16	yes	—
23	C₂⁴×C₃₁	—	2	6	no	—
24	C₂⁴×C₃₁	—	2	2	no	—
25	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
26	C₂⁴×C₃₁	C₂×D₄	1	6	yes	—
27	C₂⁴×C₃₁	C₂×Q₈	1	6	yes	—
28	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
29	C₂²×C₄×C₃₁	—	2	24	no	—
30	C₂²×C₄×C₃₁	—	2	24	no	—
31	C₂²×C₄×C₃₁	—	2	24	no	—
32	C₂²×C₄×C₃₁	—	2	8	no	—
33	C₂²×C₄×C₃₁	—	4	24	no	—
34	C₂²×C₄×C₃₁	—	4	24	no	—
35	C₂²×C₄×C₃₁	—	4	24	no	—
36	C₂²×C₄×C₃₁	C₄⋊C₄	1	48	yes	complemented
37	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
38	C₂²×C₄×C₃₁	C₂×D₄	1	12	yes	—
39	C₂²×C₄×C₃₁	C₂×Q₈	1	12	yes	—
40	C₄⋊C₄×C₃₁	—	4	48	no	—
41	C₂²⋊C₄×C₃₁	—	4	16	no	—
42	#14353	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
43	C₂³×C₃₁	—	2	6	no	—
44	C₂³×C₃₁	—	2	2	no	—
45	C₂³×C₃₁	—	2	12	no	—
46	C₂³×C₃₁	—	2	12	no	—
47	C₂³×C₃₁	—	2	12	no	—
48	C₂³×C₃₁	—	2	12	no	—
49	C₂³×C₃₁	C₂×C₄⋊C₄	1	12	yes	—
50	C₂³×C₃₁	#127	1	4	yes	—
51	C₂³×C₃₁	C₄⋊Q₈	1	12	yes	—
52	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
53	C₂³×C₃₁	C₂²×D₄	1	3	yes	—
54	C₂³×C₃₁	C₂²×Q₈	1	3	yes	—
55	C₂³×C₃₁	H(5,𝔽₂)	1	6	yes	—
56	C₂³×C₃₁	2₋¹⁺⁴	1	2	yes	—
57	C₂×C₄×C₃₁	—	2	24	no	—
58	C₂×C₄×C₃₁	—	2	24	no	—
59	C₂×C₄×C₃₁	—	2	24	no	—
60	C₂×C₄×C₃₁	—	4	48	no	—
61	C₂×C₄×C₃₁	—	4	8	no	—
62	C₂×C₄×C₃₁	—	4	24	no	—
63	C₂×C₄×C₃₁	—	4	24	no	—
64	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	24	yes	—
65	C₂×#380	C₂×C₃₁	1	6	yes	—
66	C₂²×#95	C₂×C₃₁	1	1	yes	characteristic
67	#1444	C₂×C₃₁	1	8	yes	complemented
68	C₂²×C₃₁	—	2	12	no	—
69	C₂²×C₃₁	—	2	12	no	—
70	C₂²×C₃₁	—	2	8	no	—
71	C₂²×C₃₁	—	2	24	no	—
72	C₂²×C₃₁	#384	1	8	yes	—
73	C₂²×C₃₁	C₂²×C₄⋊C₄	1	3	yes	—
74	C₂²×C₃₁	#518	1	3	yes	—
75	C₂²×C₃₁	#519	1	1	yes	characteristic
76	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
77	C₂²×C₃₁	C₂×C₄⋊Q₈	1	3	yes	—
78	C₂²×C₃₁	#534	1	6	yes	—
79	C₂²×C₃₁	#537	1	6	yes	—
80	C₂²×C₃₁	#543	1	6	yes	—
81	C₂²×C₃₁	#544	1	6	yes	—
82	C₄×C₃₁	—	2	24	no	—
83	C₄×C₃₁	—	8	8	no	—
84	C₂⁴×C₄	C₂²×C₃₁	1	3	yes	—
85	C₂²×C₄⋊C₄	C₂²×C₃₁	1	12	yes	—
86	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
87	#380	—	2	24	no	—
88	C₂×#95	C₄×C₃₁	1	8	yes	complemented
89	C₂×#95	C₂²×C₃₁	1	12	yes	—
90	C₂×#95	C₂²×C₃₁	1	4	yes	—
91	C₂×C₃₁	—	2	8	no	—
92	C₂×C₃₁	C₂×#384	1	1	yes	characteristic
93	C₂×C₃₁	#2315	1	3	yes	—
94	C₂×C₃₁	#2316	1	3	yes	—
95	C₂×C₃₁	#2656	1	2	yes	—
96	C₂×C₃₁	#2659	1	6	yes	—
97	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
98	C₂³×C₄	—	4	12	no	—
99	C₂³×C₄	C₂×C₄×C₃₁	1	12	yes	—
100	C₂³×C₄	D₄×C₃₁	1	12	yes	—
101	C₂³×C₄	Q₈×C₃₁	1	12	yes	—
102	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
103	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
104	C₂×C₄⋊C₄	—	2	24	no	—
105	C₂×C₄⋊C₄	—	4	24	no	—
106	C₂×C₂²⋊C₄	—	2	24	no	—
107	#95	—	2	24	no	—
108	#95	C₂×C₄×C₃₁	1	16	yes	—
109	C₃₁	#14353	1	1	yes	characteristic, direct product factor, Sylow
110	C₂⁴	—	2	6	no	—
111	C₂⁴	—	2	2	no	—
112	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
113	C₂⁴	C₂×D₄×C₃₁	1	6	yes	—
114	C₂⁴	C₂×Q₈×C₃₁	1	6	yes	—
115	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
116	C₂²×C₄	—	2	24	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	—	2	24	no	—
119	C₂²×C₄	—	2	8	no	—
120	C₂²×C₄	—	4	24	no	—
121	C₂²×C₄	—	4	24	no	—
122	C₂²×C₄	—	4	24	no	—
123	C₂²×C₄	C₄⋊C₄×C₃₁	1	48	yes	complemented
124	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
125	C₂²×C₄	C₂×D₄×C₃₁	1	12	yes	—
126	C₂²×C₄	C₂×Q₈×C₃₁	1	12	yes	—
127	C₄⋊C₄	—	4	48	no	—
128	C₂²⋊C₄	—	4	16	no	—
129	C₂³	—	2	6	no	—
130	C₂³	—	2	2	no	—
131	C₂³	—	2	12	no	—
132	C₂³	—	2	12	no	—
133	C₂³	—	2	12	no	—
134	C₂³	—	2	12	no	—
135	C₂³	C₂×C₄⋊C₄×C₃₁	1	12	yes	—
136	C₂³	C₃₁×#127	1	4	yes	—
137	C₂³	C₃₁×C₄⋊Q₈	1	12	yes	—
138	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
139	C₂³	C₂²×D₄×C₃₁	1	3	yes	—
140	C₂³	C₂²×Q₈×C₃₁	1	3	yes	—
141	C₂³	C₃₁×H(5,𝔽₂)	1	6	yes	—
142	C₂³	C₃₁×2₋¹⁺⁴	1	2	yes	—
143	C₂×C₄	—	2	24	no	—
144	C₂×C₄	—	2	24	no	—
145	C₂×C₄	—	2	24	no	—
146	C₂×C₄	—	4	48	no	—
147	C₂×C₄	—	4	8	no	—
148	C₂×C₄	—	4	24	no	—
149	C₂×C₄	—	4	24	no	—
150	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	24	yes	—
151	C₂²	—	2	12	no	—
152	C₂²	—	2	12	no	—
153	C₂²	—	2	8	no	—
154	C₂²	—	2	24	no	—
155	C₂²	C₃₁×#384	1	8	yes	—
156	C₂²	C₂²×C₄⋊C₄×C₃₁	1	3	yes	—
157	C₂²	C₃₁×#518	1	3	yes	—
158	C₂²	C₃₁×#519	1	1	yes	characteristic
159	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
160	C₂²	C₂×C₃₁×C₄⋊Q₈	1	3	yes	—
161	C₂²	C₃₁×#534	1	6	yes	—
162	C₂²	C₃₁×#537	1	6	yes	—
163	C₂²	C₃₁×#543	1	6	yes	—
164	C₂²	C₃₁×#544	1	6	yes	—
165	C₄	—	2	24	no	—
166	C₄	—	8	8	no	—
167	C₂	—	2	8	no	—
168	C₂	C₂×C₃₁×#384	1	1	yes	characteristic
169	C₂	C₃₁×#2315	1	3	yes	—
170	C₂	C₃₁×#2316	1	3	yes	—
171	C₂	C₃₁×#2656	1	2	yes	—
172	C₂	C₃₁×#2659	1	6	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator	C₂³	C₂³×C₄×C₃₁
Frattini, cosocle	C₂⁴	C₂⁴×C₃₁
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1444	C₂	semidirect
8. C₂×C₃₁×#95	C₄	semidirect
36. C₂²×C₄×C₃₁	C₄⋊C₄	semidirect
42. #14353	C₃₁	direct (N × H)
67. #1444	C₂×C₃₁	semidirect
88. C₂×#95	C₄×C₃₁	semidirect
109. C₃₁	#14353	direct (N × H)
123. C₂²×C₄	C₄⋊C₄×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#380	2
C₂²×C₃₁×#95	2
C₃₁×#1444	2
#14353	31

Subgroups — C31×#14353 #21471112

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#14353 #21471112