Subgroups — #15689325

Order 3008.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10625336	C₂	1	2	yes	maximal
2	C₄×D₁₈₈	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#13093948	C₂	1	2	yes	maximal, complemented
5	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₇	C₂²	1	2	yes	—
7	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₄₇	—	2	1	no	—
9	C₄⋊C₄×C₄₇	—	2	1	no	—
10	C₄⋊C₄×C₄₇	—	2	1	no	—
11	C₄²×C₄₇	C₂²	1	1	yes	characteristic
12	#10625336	—	2	4	no	—
13	(C₂×D₄₇)⋊C₄	—	2	1	no	—
14	C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄₇⋊C₈	C₂²	1	2	yes	—
16	C₂²×D₄₇	—	2	1	no	—
17	Q₈×C₄₇	—	2	2	no	—
18	Q₈×C₄₇	—	2	2	no	—
19	C₂×C₄×C₄₇	—	2	2	no	—
20	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂×Dic₄₇	—	2	1	no	—
24	D₁₈₈	—	2	1	no	—
25	D₁₈₈	D₄	1	2	yes	—
26	C₄×D₄₇	—	4	2	no	—
27	C₄₇⋊C₈	—	2	2	no	—
28	C₂²×C₄₇	C₂×D₄	1	1	yes	characteristic
29	C₂×D₄₇	—	2	2	no	—
30	C₂×D₄₇	—	4	1	no	—
31	C₂×D₄₇	—	4	1	no	—
32	C₄×C₄₇	—	2	1	no	—
33	C₄×C₄₇	—	4	2	no	—
34	C₄×C₄₇	SD₁₆	1	2	yes	complemented
35	C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₄×C₄₇	#41	1	1	yes	characteristic
37	Dic₄₇	—	4	1	no	—
38	C₂×C₄₇	#121	1	1	yes	characteristic
39	C₂×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₄₇	#137	1	1	yes	characteristic
41	D₄₇	—	4	2	no	—
42	#461	—	47	1	no	maximal, Sylow
43	C₄₇	#461	1	1	yes	characteristic, complemented, Sylow
44	C₂×SD₁₆	—	47	2	no	—
45	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
46	C₄×D₄	—	47	1	no	—
47	C₄⋊C₈	—	47	1	no	—
48	#103	—	47	2	no	—
49	C₂×Q₈	C₂×D₄₇	1	2	yes	—
50	C₂×D₄	—	47	1	no	—
51	C₂²×C₄	—	94	1	no	—
52	SD₁₆	—	94	4	no	—
53	C₂×C₈	—	47	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	2	1	no	—
56	C₄⋊C₄	—	47	1	no	—
57	C₂²⋊C₄	—	94	1	no	—
58	C₄²	C₂×D₄₇	1	1	yes	characteristic
59	C₂³	—	94	1	no	—
60	Q₈	—	2	2	no	—
61	Q₈	—	2	2	no	—
62	D₄	—	47	2	no	—
63	D₄	—	94	1	no	—
64	C₂×C₄	—	2	2	no	—
65	C₂×C₄	—	94	1	no	—
66	C₂×C₄	—	188	2	no	—
67	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
70	C₈	—	94	2	no	—
71	C₂²	—	94	2	no	—
72	C₂²	—	188	1	no	—
73	C₂²	—	188	1	no	—
74	C₂²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	2	no	—
77	C₄	—	188	1	no	—
78	C₄	#10625336	1	2	yes	complemented
79	C₄	D₄×D₄₇	1	1	yes	characteristic
80	C₄	#10625352	1	1	yes	characteristic
81	C₂	—	188	2	no	—
82	C₂	#13093995	1	1	yes	characteristic
83	C₂	C₂×#10625336	1	1	yes	characteristic
84	C₂	#13094055	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₇
commutator, cosocle	C₂×C₄×C₄₇	C₂³
Frattini	C₂×C₄	C₂²×D₄₇
Fitting	C₄⋊Q₈×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₄
prime radical	C₂×D₁₈₈	C₂²
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#461
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
3. C₄⋊(C₄₇⋊C₈)	C₂	semidirect
4. #13093948	C₂	semidirect
5. C₄⋊Q₈×C₄₇	C₂	semidirect
34. C₄×C₄₇	SD₁₆	semidirect
43. C₄₇	#461	semidirect
45. C₄⋊Q₈	D₄₇	semidirect
78. C₄	#10625336	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10625336	2
C₄×D₁₈₈	2
C₄⋊(C₄₇⋊C₈)	2
#13093948	2
C₄⋊Q₈×C₄₇	2
#461	47