Subgroups — #20243300

Order 6016.

612 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	6	yes	maximal, complemented
2	C₂²×#41×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄₇⋊C₈	C₂²	1	4	yes	—
4	C₂²⋊(C₄₇⋊C₈)	—	2	12	no	—
5	C₂×#41×C₄₇	—	2	12	no	—
6	C₂²×Q₈×C₄₇	C₄	1	1	yes	characteristic
7	C₂²×D₄×C₄₇	C₄	1	3	yes	—
8	C₂³×C₄×C₄₇	C₂²	1	3	yes	—
9	C₂⁴×C₄₇	C₂×C₄	1	3	yes	—
10	#41×C₄₇	—	2	32	no	—
11	C₂×Q₈×C₄₇	—	2	6	no	—
12	C₂×D₄×C₄₇	—	2	12	no	—
13	C₂×D₄×C₄₇	—	2	6	no	—
14	C₂²×C₄×C₄₇	—	2	12	no	—
15	C₂²×C₄×C₄₇	C₂×C₄	1	3	yes	—
16	C₂²×C₄×C₄₇	D₄	1	12	yes	—
17	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
18	C₂×C₄₇⋊C₈	—	2	12	no	—
19	C₂³×C₄₇	—	2	6	no	—
20	C₂³×C₄₇	—	2	12	no	—
21	C₂³×C₄₇	C₂²⋊C₄	1	6	yes	—
22	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
23	Q₈×C₄₇	—	2	8	no	—
24	D₄×C₄₇	—	2	24	no	—
25	C₂×C₄×C₄₇	—	2	6	no	—
26	C₂×C₄×C₄₇	—	2	24	no	—
27	C₂×C₄×C₄₇	C₂²⋊C₄	1	6	yes	—
28	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
29	C₄₇⋊C₈	—	4	4	no	—
30	C₂²×C₄₇	—	2	6	no	—
31	C₂²×C₄₇	—	2	12	no	—
32	C₂²×C₄₇	—	4	12	no	—
33	C₂²×C₄₇	C₂×C₂²⋊C₄	1	3	yes	—
34	C₂²×C₄₇	#131	1	4	yes	—
35	C₄×C₄₇	—	2	6	no	—
36	C₄×C₄₇	#120	1	4	yes	—
37	#1780	—	47	1	no	maximal, Sylow
38	C₂×C₄₇	—	4	6	no	—
39	C₂×C₄₇	#379	1	1	yes	characteristic
40	C₂×C₄₇	#408	1	6	yes	—
41	C₂²×#41	D₄₇	1	1	yes	characteristic
42	C₂×C₂²⋊C₈	—	47	6	no	—
43	C₄₇	#1780	1	1	yes	characteristic, complemented, Sylow
44	C₂×#41	—	2	12	no	—
45	C₂²×Q₈	Dic₄₇	1	1	yes	characteristic
46	C₂²×D₄	Dic₄₇	1	3	yes	—
47	C₂³×C₄	C₂×D₄₇	1	3	yes	—
48	C₂²×C₈	—	47	4	no	—
49	C₂²⋊C₈	—	94	12	no	—
50	C₂⁴	C₂×Dic₄₇	1	3	yes	—
51	#41	—	2	32	no	—
52	C₂×Q₈	—	2	6	no	—
53	C₂×D₄	—	2	12	no	—
54	C₂×D₄	—	2	6	no	—
55	C₂²×C₄	—	2	12	no	—
56	C₂²×C₄	C₂×Dic₄₇	1	3	yes	—
57	C₂²×C₄	C₂²⋊D₄₇	1	12	yes	—
58	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
59	C₂×C₈	—	94	12	no	—
60	C₂³	—	2	6	no	—
61	C₂³	—	2	12	no	—
62	C₂³	C₂²⋊Dic₄₇	1	6	yes	—
63	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
64	Q₈	—	2	8	no	—
65	D₄	—	2	24	no	—
66	C₂×C₄	—	2	6	no	—
67	C₂×C₄	—	2	24	no	—
68	C₂×C₄	C₂²⋊Dic₄₇	1	6	yes	—
69	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
70	C₈	—	188	4	no	—
71	C₂²	—	2	6	no	—
72	C₂²	—	2	12	no	—
73	C₂²	—	4	12	no	—
74	C₂²	C₂×C₂²⋊Dic₄₇	1	3	yes	—
75	C₂²	#13094031	1	4	yes	—
76	C₄	—	2	6	no	—
77	C₄	#13093993	1	4	yes	—
78	C₂	—	4	6	no	—
79	C₂	#15688979	1	1	yes	characteristic
80	C₂	#15689090	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂²×C₄
Fitting	C₂²×#41×C₄₇	C₂
hypercenter	C₂²×#41	D₄₇
hypocenter	C₄₇	#1780
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×C₂²⋊(C₄₇⋊C₈)	C₂	semidirect
43. C₄₇	#1780	semidirect