Subgroups — C₂×C₃₁×#1446 #21470608

Order 7936.

4830 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#95	C₂	1	6	yes	maximal
3	C₃₁×#1446	C₂	1	8	yes	maximal, direct product factor
4	C₂³×Q₈×C₃₁	—	2	24	no	—
5	C₂³×Q₈×C₃₁	C₄	1	8	yes	complemented
6	C₂³×Q₈×C₃₁	C₂²	1	4	yes	—
7	C₂⁴×C₄×C₃₁	C₂²	1	4	yes	—
8	C₂⁴×C₄×C₃₁	C₂²	1	3	yes	—
9	C₂×C₃₁×#95	—	2	24	no	—
10	C₂×C₃₁×#95	C₂²	1	24	yes	—
11	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
12	C₂²×Q₈×C₃₁	—	2	128	no	—
13	C₂²×Q₈×C₃₁	—	2	96	no	—
14	C₂²×Q₈×C₃₁	—	2	48	no	—
15	C₂²×Q₈×C₃₁	C₂×C₄	1	16	yes	complemented
16	C₂³×C₄×C₃₁	—	2	24	no	—
17	C₂³×C₄×C₃₁	—	2	24	no	—
18	C₂³×C₄×C₃₁	—	2	24	no	—
19	C₂³×C₄×C₃₁	C₂×C₄	1	12	yes	—
20	C₂³×C₄×C₃₁	D₄	1	24	yes	—
21	C₂³×C₄×C₃₁	Q₈	1	16	yes	complemented
22	C₂³×C₄×C₃₁	C₂³	1	8	yes	—
23	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
24	C₃₁×#95	—	2	48	no	—
25	C₂⁴×C₃₁	C₂²×C₄	1	2	yes	—
26	C₂⁴×C₃₁	C₂×D₄	1	12	yes	—
27	C₂⁴×C₃₁	C₂×Q₈	1	8	yes	—
28	C₂⁴×C₃₁	#41	1	8	yes	—
29	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
30	C₂×Q₈×C₃₁	—	2	128	no	—
31	C₂×Q₈×C₃₁	—	2	256	no	—
32	C₂×Q₈×C₃₁	—	2	96	no	—
33	C₂²×C₄×C₃₁	—	2	48	no	—
34	C₂²×C₄×C₃₁	—	2	96	no	—
35	C₂²×C₄×C₃₁	—	2	24	no	—
36	C₂²×C₄×C₃₁	—	2	96	no	—
37	C₂²×C₄×C₃₁	—	2	24	no	—
38	C₂²×C₄×C₃₁	—	2	24	no	—
39	C₂²×C₄×C₃₁	—	4	16	no	—
40	C₂²×C₄×C₃₁	—	4	48	no	—
41	C₂²×C₄×C₃₁	C₂²⋊C₄	1	48	yes	—
42	C₂²×C₄×C₃₁	C₂²×C₄	1	12	yes	—
43	C₂²×C₄×C₃₁	C₂×D₄	1	24	yes	—
44	C₂²×C₄×C₃₁	C₂×Q₈	1	16	yes	complemented
45	C₂×#1446	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂³×C₃₁	C₂×C₂²⋊C₄	1	12	yes	—
47	C₂³×C₃₁	C₄×Q₈	1	16	yes	—
48	C₂³×C₃₁	#120	1	16	yes	—
49	C₂³×C₃₁	C₂²⋊Q₈	1	48	yes	—
50	C₂³×C₃₁	#124	1	24	yes	—
51	C₂³×C₃₁	C₄⋊Q₈	1	24	yes	—
52	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
53	C₂³×C₃₁	C₂²×D₄	1	6	yes	—
54	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
55	C₂³×C₃₁	C₂×#41	1	4	yes	—
56	Q₈×C₃₁	—	2	128	no	—
57	C₂×C₄×C₃₁	—	2	24	no	—
58	C₂×C₄×C₃₁	—	2	48	no	—
59	C₂×C₄×C₃₁	—	2	96	no	—
60	C₂×C₄×C₃₁	—	2	48	no	—
61	C₂×C₄×C₃₁	—	4	8	no	—
62	C₂×C₄×C₃₁	—	4	24	no	—
63	C₂×C₄×C₃₁	—	4	64	no	—
64	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	24	yes	—
65	C₂⁴×Q₈	C₂×C₃₁	1	1	yes	characteristic
66	C₂²×#95	C₂×C₃₁	1	6	yes	—
67	#1446	C₂×C₃₁	1	8	yes	direct product factor
68	C₂²×C₃₁	#379	1	8	yes	—
69	C₂²×C₃₁	#391	1	48	yes	—
70	C₂²×C₃₁	#393	1	32	yes	—
71	C₂²×C₃₁	#395	1	32	yes	—
72	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	3	yes	—
73	C₂²×C₃₁	C₂×C₄×Q₈	1	4	yes	—
74	C₂²×C₃₁	C₂×#120	1	4	yes	—
75	C₂²×C₃₁	C₂×C₂²⋊Q₈	1	12	yes	—
76	C₂²×C₃₁	C₂×#124	1	6	yes	—
77	C₂²×C₃₁	C₂×C₄⋊Q₈	1	6	yes	—
78	C₄×C₃₁	—	2	24	no	—
79	C₄×C₃₁	—	4	16	no	—
80	C₂³×Q₈	—	2	24	no	—
81	C₂³×Q₈	C₄×C₃₁	1	8	yes	complemented
82	C₂³×Q₈	C₂²×C₃₁	1	4	yes	—
83	C₂⁴×C₄	C₂²×C₃₁	1	4	yes	—
84	C₂⁴×C₄	C₂²×C₃₁	1	3	yes	—
85	C₂×#95	—	2	24	no	—
86	C₂×#95	C₂²×C₃₁	1	24	yes	—
87	C₂×C₃₁	#1446	1	16	yes	direct product factor
88	C₂×C₃₁	C₂×#379	1	1	yes	characteristic
89	C₂×C₃₁	C₂×#391	1	6	yes	—
90	C₂×C₃₁	C₂×#393	1	4	yes	—
91	C₂×C₃₁	C₂×#395	1	4	yes	—
92	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
93	C₂²×Q₈	—	2	128	no	—
94	C₂²×Q₈	—	2	96	no	—
95	C₂²×Q₈	—	2	48	no	—
96	C₂²×Q₈	C₂×C₄×C₃₁	1	16	yes	complemented
97	C₂³×C₄	—	2	24	no	—
98	C₂³×C₄	—	2	24	no	—
99	C₂³×C₄	—	2	24	no	—
100	C₂³×C₄	C₂×C₄×C₃₁	1	12	yes	—
101	C₂³×C₄	D₄×C₃₁	1	24	yes	—
102	C₂³×C₄	Q₈×C₃₁	1	16	yes	complemented
103	C₂³×C₄	C₂³×C₃₁	1	8	yes	—
104	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
105	#95	—	2	48	no	—
106	C₃₁	C₂×#1446	1	1	yes	characteristic, direct product factor, Sylow
107	C₂⁴	C₂²×C₄×C₃₁	1	2	yes	—
108	C₂⁴	C₂×D₄×C₃₁	1	12	yes	—
109	C₂⁴	C₂×Q₈×C₃₁	1	8	yes	—
110	C₂⁴	#41×C₃₁	1	8	yes	—
111	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
112	C₂×Q₈	—	2	128	no	—
113	C₂×Q₈	—	2	256	no	—
114	C₂×Q₈	—	2	96	no	—
115	C₂²×C₄	—	2	48	no	—
116	C₂²×C₄	—	2	96	no	—
117	C₂²×C₄	—	2	24	no	—
118	C₂²×C₄	—	2	96	no	—
119	C₂²×C₄	—	2	24	no	—
120	C₂²×C₄	—	2	24	no	—
121	C₂²×C₄	—	4	16	no	—
122	C₂²×C₄	—	4	48	no	—
123	C₂²×C₄	C₂²⋊C₄×C₃₁	1	48	yes	—
124	C₂²×C₄	C₂²×C₄×C₃₁	1	12	yes	—
125	C₂²×C₄	C₂×D₄×C₃₁	1	24	yes	—
126	C₂²×C₄	C₂×Q₈×C₃₁	1	16	yes	complemented
127	C₂³	C₂×C₂²⋊C₄×C₃₁	1	12	yes	—
128	C₂³	C₄×Q₈×C₃₁	1	16	yes	—
129	C₂³	C₃₁×#120	1	16	yes	—
130	C₂³	C₃₁×C₂²⋊Q₈	1	48	yes	—
131	C₂³	C₃₁×#124	1	24	yes	—
132	C₂³	C₃₁×C₄⋊Q₈	1	24	yes	—
133	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
134	C₂³	C₂²×D₄×C₃₁	1	6	yes	—
135	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
136	C₂³	C₂×#41×C₃₁	1	4	yes	—
137	Q₈	—	2	128	no	—
138	C₂×C₄	—	2	24	no	—
139	C₂×C₄	—	2	48	no	—
140	C₂×C₄	—	2	96	no	—
141	C₂×C₄	—	2	48	no	—
142	C₂×C₄	—	4	8	no	—
143	C₂×C₄	—	4	24	no	—
144	C₂×C₄	—	4	64	no	—
145	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	24	yes	—
146	C₂²	C₃₁×#379	1	8	yes	—
147	C₂²	C₃₁×#391	1	48	yes	—
148	C₂²	C₃₁×#393	1	32	yes	—
149	C₂²	C₃₁×#395	1	32	yes	—
150	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	3	yes	—
151	C₂²	C₂×C₄×Q₈×C₃₁	1	4	yes	—
152	C₂²	C₂×C₃₁×#120	1	4	yes	—
153	C₂²	C₂×C₃₁×C₂²⋊Q₈	1	12	yes	—
154	C₂²	C₂×C₃₁×#124	1	6	yes	—
155	C₂²	C₂×C₃₁×C₄⋊Q₈	1	6	yes	—
156	C₄	—	2	24	no	—
157	C₄	—	4	16	no	—
158	C₂	C₃₁×#1446	1	16	yes	direct product factor
159	C₂	C₂×C₃₁×#379	1	1	yes	characteristic
160	C₂	C₂×C₃₁×#391	1	6	yes	—
161	C₂	C₂×C₃₁×#393	1	4	yes	—
162	C₂	C₂×C₃₁×#395	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₃₁×#1446	C₂	direct (N × H)
5. C₂³×Q₈×C₃₁	C₄	semidirect
15. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
21. C₂³×C₄×C₃₁	Q₈	semidirect
44. C₂²×C₄×C₃₁	C₂×Q₈	semidirect
45. C₂×#1446	C₃₁	direct (N × H)
67. #1446	C₂×C₃₁	direct (N × H)
81. C₂³×Q₈	C₄×C₃₁	semidirect
87. C₂×C₃₁	#1446	direct (N × H)
96. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
102. C₂³×C₄	Q₈×C₃₁	semidirect
106. C₃₁	C₂×#1446	direct (N × H)
126. C₂²×C₄	C₂×Q₈×C₃₁	semidirect
158. C₂	C₃₁×#1446	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₃₁	2
C₂²×C₃₁×#95	2
C₃₁×#1446	2
C₂×#1446	31

Subgroups — C2×C31×#1446 #21470608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1446 #21470608