Subgroups — #14583057

Order 2752.

148 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₈₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₇₂	C₂	1	2	yes	maximal, complemented
3	#13063304	C₂	1	1	yes	maximal, characteristic
4	#13063250	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄×C₄₃	C₂²	1	1	yes	characteristic
7	C₂²×Dic₄₃	—	2	1	no	—
8	C₂×Dic₈₆	—	2	1	no	—
9	C₂×Dic₈₆	—	2	4	no	—
10	C₂×Dic₈₆	C₂²	1	1	yes	characteristic
11	C₂×Dic₈₆	C₂²	1	2	yes	—
12	C₂×C₈×C₄₃	C₂²	1	2	yes	—
13	C₂²⋊Dic₄₃	—	2	1	no	—
14	C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
15	Dic₄₃⋊C₄	—	2	1	no	—
16	Dic₁₇₂	—	2	4	no	—
17	C₂³×C₄₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₄₃	—	2	2	no	—
19	C₂×C₄×C₄₃	D₄	1	1	yes	characteristic
20	C₂×C₄×C₄₃	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₃	—	2	4	no	—
22	C₂×Dic₄₃	—	2	2	no	—
23	C₂×Dic₄₃	—	2	1	no	—
24	C₂×Dic₄₃	—	2	1	no	—
25	Dic₈₆	—	2	4	no	—
26	Dic₈₆	—	2	2	no	—
27	Dic₈₆	—	2	2	no	—
28	Dic₈₆	D₄	1	4	yes	—
29	C₈×C₄₃	—	2	2	no	—
30	C₂²×C₄₃	—	2	1	no	—
31	C₂²×C₄₃	—	2	1	no	—
32	C₂²×C₄₃	Q₁₆	1	2	yes	complemented
33	C₂²×C₄₃	C₂×D₄	1	1	yes	characteristic
34	C₄×C₄₃	—	2	1	no	—
35	C₄×C₄₃	C₂×D₄	1	2	yes	—
36	Dic₄₃	—	2	4	no	—
37	Dic₄₃	—	4	1	no	—
38	Dic₄₃	—	4	1	no	—
39	C₂×C₄₃	—	2	2	no	—
40	C₂×C₄₃	#120	1	1	yes	characteristic
41	C₂×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
42	C₂×C₄₃	#137	1	1	yes	characteristic
43	#451	—	43	1	no	maximal, Sylow
44	C₄₃	#451	1	1	yes	characteristic, complemented, Sylow
45	C₂²×Q₈	—	43	1	no	—
46	C₂×Q₁₆	—	43	2	no	—
47	C₂²⋊Q₈	—	43	1	no	—
48	#103	—	43	2	no	—
49	C₂²⋊C₈	D₄₃	1	1	yes	characteristic
50	C₂×Q₈	—	43	2	no	—
51	C₂×Q₈	—	43	1	no	—
52	C₂×Q₈	—	86	4	no	—
53	C₂×Q₈	—	86	1	no	—
54	C₂²×C₄	—	86	1	no	—
55	C₂²×C₄	C₂×D₄₃	1	1	yes	characteristic
56	Q₁₆	—	86	4	no	—
57	C₂×C₈	C₂×D₄₃	1	2	yes	—
58	C₄⋊C₄	—	43	1	no	—
59	C₄⋊C₄	—	86	1	no	—
60	C₂²⋊C₄	—	86	1	no	—
61	C₂³	D₁₇₂	1	1	yes	characteristic
62	Q₈	—	43	4	no	—
63	Q₈	—	86	4	no	—
64	Q₈	—	86	2	no	—
65	Q₈	—	86	2	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	86	4	no	—
68	C₂×C₄	—	86	2	no	—
69	C₂×C₄	—	86	1	no	—
70	C₂×C₄	—	86	1	no	—
71	C₂×C₄	D₁₇₂	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₄₃	1	1	yes	characteristic
73	C₈	—	2	2	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	2	1	no	—
76	C₂²	Dic₁₇₂	1	2	yes	complemented
77	C₂²	C₂×D₁₇₂	1	1	yes	characteristic
78	C₄	—	2	1	no	—
79	C₄	—	86	4	no	—
80	C₄	—	172	1	no	—
81	C₄	—	172	1	no	—
82	C₄	D₄×D₄₃	1	2	yes	—
83	C₂	—	2	2	no	—
84	C₂	#13063295	1	1	yes	characteristic
85	C₂	C₂×Dic₁₇₂	1	1	yes	characteristic
86	C₂	#13063358	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₇₂
commutator, cosocle	C₂×C₄×C₄₃	C₂³
Frattini	C₂×C₄	C₂²×D₄₃
Fitting	C₂²⋊C₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂×D₄
prime radical	C₂³×C₄₃	D₄
hypercenter	C₂²⋊C₈	D₄₃
hypocenter	C₄₃	#451
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₁₇₂	C₂	semidirect
4. #13063250	C₂	semidirect
32. C₂²×C₄₃	Q₁₆	semidirect
44. C₄₃	#451	semidirect
76. C₂²	Dic₁₇₂	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₈₆	2
C₂×Dic₁₇₂	2
#13063304	2
#13063250	2
C₂²⋊C₈×C₄₃	2
#451	43