Subgroups — #15689676

Order 3008.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13094023	C₂	1	2	yes	maximal, complemented
2	#13094015	C₂	1	2	yes	maximal, complemented
3	#13094001	C₂	1	4	yes	maximal
4	C₄×Dic₉₄	C₂	1	4	yes	maximal, complemented
5	#13093983	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Dic₄₇	C₂	1	1	yes	maximal, characteristic
7	#117×C₄₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₄×C₄₇	C₂²	1	1	yes	characteristic
9	C₂²×Dic₄₇	C₂²	1	2	yes	—
10	C₂×Dic₉₄	C₂²	1	4	yes	—
11	C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
12	C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
13	C₄²×C₄₇	C₂²	1	2	yes	—
14	C₂²⋊Dic₄₇	C₂²	1	2	yes	—
15	C₄⋊Dic₄₇	C₂²	1	2	yes	—
16	C₄⋊Dic₄₇	C₂²	1	4	yes	—
17	Dic₄₇⋊C₄	C₂²	1	4	yes	—
18	Dic₄₇⋊C₄	C₂²	1	4	yes	—
19	C₄×Dic₄₇	C₂²	1	2	yes	—
20	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
21	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
22	C₄×Dic₄₇	C₂²	1	2	yes	—
23	C₂³×C₄₇	C₂³	1	1	yes	characteristic
24	C₂×C₄×C₄₇	Q₈	1	4	yes	complemented
25	C₂×C₄×C₄₇	C₂³	1	4	yes	—
26	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₄₇	—	2	4	no	—
29	C₂×Dic₄₇	C₂³	1	2	yes	—
30	C₂×Dic₄₇	C₂³	1	4	yes	—
31	C₂×Dic₄₇	C₂³	1	2	yes	—
32	Dic₉₄	—	2	8	no	—
33	C₂²×C₄₇	—	2	2	no	—
34	C₂²×C₄₇	C₂×Q₈	1	2	yes	—
35	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
36	C₄×C₄₇	—	2	4	no	—
37	C₄×C₄₇	C₂×Q₈	1	2	yes	—
38	C₄×C₄₇	C₂×Q₈	1	2	yes	—
39	Dic₄₇	—	2	4	no	—
40	Dic₄₇	—	2	2	no	—
41	Dic₄₇	#41	1	4	yes	—
42	C₂×C₄₇	—	2	2	no	—
43	C₂×C₄₇	C₂²×Q₈	1	1	yes	characteristic
44	C₂×C₄₇	C₂×#41	1	2	yes	—
45	#533	—	47	1	no	maximal, Sylow
46	C₄₇	#533	1	1	yes	characteristic, complemented, Sylow
47	C₄⋊Q₈	—	47	2	no	—
48	#125	—	47	2	no	—
49	C₂²⋊Q₈	—	47	4	no	—
50	C₄×Q₈	—	47	4	no	—
51	#117	—	47	1	no	—
52	#117	D₄₇	1	1	yes	characteristic
53	C₂×C₄²	—	47	1	no	—
54	C₂×Q₈	—	47	4	no	—
55	C₂²×C₄	—	47	2	no	—
56	C₂²×C₄	C₂×D₄₇	1	1	yes	characteristic
57	C₄⋊C₄	—	47	4	no	—
58	C₄⋊C₄	—	47	4	no	—
59	C₄⋊C₄	—	47	2	no	—
60	C₄⋊C₄	—	47	4	no	—
61	C₄⋊C₄	C₂×D₄₇	1	2	yes	—
62	C₂²⋊C₄	—	47	2	no	—
63	C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
64	C₄²	—	47	1	no	—
65	C₄²	—	47	1	no	—
66	C₄²	—	47	2	no	—
67	C₄²	—	47	2	no	—
68	C₄²	C₂×D₄₇	1	2	yes	—
69	C₂³	C₂²×D₄₇	1	1	yes	characteristic
70	Q₈	—	94	8	no	—
71	C₂×C₄	—	47	2	no	—
72	C₂×C₄	—	47	2	no	—
73	C₂×C₄	—	47	4	no	—
74	C₂×C₄	—	94	4	no	—
75	C₂×C₄	Dic₉₄	1	4	yes	complemented
76	C₂×C₄	C₂²×D₄₇	1	4	yes	—
77	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	C₂×Dic₉₄	1	2	yes	—
81	C₂²	C₂³×D₄₇	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	47	4	no	—
84	C₄	—	94	2	no	—
85	C₄	—	94	4	no	—
86	C₄	C₂×Dic₉₄	1	2	yes	—
87	C₄	C₂×Dic₉₄	1	2	yes	—
88	C₂	—	2	2	no	—
89	C₂	C₂²×Dic₉₄	1	1	yes	characteristic
90	C₂	#41×D₄₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Frattini	C₂²	C₂³×D₄₇
Fitting	#117×C₄₇	C₂
prime radical	C₂³×C₄₇	C₂³
hypercenter	#117	D₄₇
hypocenter	C₄₇	#533
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
1. #13094023	C₂	semidirect
2. #13094015	C₂	semidirect
4. C₄×Dic₉₄	C₂	semidirect
24. C₂×C₄×C₄₇	Q₈	semidirect
46. C₄₇	#533	semidirect
75. C₂×C₄	Dic₉₄	semidirect

Maximal subgroups

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

subgroup	index
#13094023	2
#13094015	2
#13094001	2
C₄×Dic₉₄	2
#13093983	2
C₂×C₄×Dic₄₇	2
#117×C₄₇	2
#533	47