Subgroups — #26038722

Order 11456.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19153112	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
3	#19153045	C₂	1	1	yes	maximal, characteristic
4	#19153044	C₂	1	1	yes	maximal, characteristic
5	Dic₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#19153037	C₂	1	1	yes	maximal, characteristic
7	#103×C₁₇₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₃₅₈	—	2	1	no	—
9	C₂×C₁₇₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄⋊Dic₁₇₉	—	2	1	no	—
11	C₄⋊Dic₁₇₉	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₇₉	C₂²	1	1	yes	characteristic
13	Dic₁₇₉⋊C₄	—	2	1	no	—
14	C₄×Dic₁₇₉	—	2	1	no	—
15	C₄×Dic₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₁₇₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₁₇₉	C₂²	1	1	yes	characteristic
19	Dic₃₅₈	—	4	2	no	—
20	C₂×Dic₁₇₉	—	2	1	no	—
21	C₂×Dic₁₇₉	—	2	1	no	—
22	C₂×Dic₁₇₉	D₄	1	1	yes	characteristic
23	C₂×Dic₁₇₉	D₄	1	1	yes	characteristic
24	C₁₇₉⋊C₈	—	2	1	no	—
25	Q₈×C₁₇₉	—	2	1	no	—
26	Q₈×C₁₇₉	Q₈	1	2	yes	complemented
27	C₂×C₄×C₁₇₉	—	2	1	no	—
28	C₂×C₄×C₁₇₉	—	2	1	no	—
29	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
30	C₈×C₁₇₉	—	2	1	no	—
31	C₂²×C₁₇₉	C₂×D₄	1	1	yes	characteristic
32	C₄×C₁₇₉	—	2	2	no	—
33	C₄×C₁₇₉	—	4	1	no	—
34	C₄×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
35	C₄×C₁₇₉	#41	1	1	yes	characteristic
36	Dic₁₇₉	—	2	1	no	—
37	Dic₁₇₉	—	4	1	no	—
38	Dic₁₇₉	—	4	1	no	—
39	Dic₁₇₉	SD₁₆	1	2	yes	—
40	C₂×C₁₇₉	C₂²⋊Q₈	1	1	yes	characteristic
41	C₂×C₁₇₉	C₂×SD₁₆	1	1	yes	characteristic
42	C₂×C₁₇₉	#137	1	1	yes	characteristic
43	C₁₇₉	#475	1	1	yes	characteristic, complemented, Sylow
44	#475	—	179	1	no	maximal, Sylow
45	C₄⋊Q₈	—	179	1	no	—
46	C₄×Q₈	—	179	1	no	—
47	#106	—	179	1	no	—
48	#106	—	179	1	no	—
49	C₄⋊C₈	—	179	1	no	—
50	#103	—	179	1	no	—
51	#103	D₁₇₉	1	1	yes	characteristic
52	C₂×Q₈	—	358	1	no	—
53	C₂×Q₈	C₂×D₁₇₉	1	1	yes	characteristic
54	C₂×C₈	—	179	1	no	—
55	C₂×C₈	C₂×D₁₇₉	1	1	yes	characteristic
56	C₄⋊C₄	—	358	1	no	—
57	C₄⋊C₄	—	358	1	no	—
58	C₄⋊C₄	—	179	1	no	—
59	C₄⋊C₄	—	179	1	no	—
60	C₄⋊C₄	C₂×D₁₇₉	1	1	yes	characteristic
61	C₄²	—	358	1	no	—
62	C₄²	—	179	1	no	—
63	Q₈	—	2	1	no	—
64	Q₈	—	716	2	no	—
65	Q₈	Dic₃₅₈	1	2	yes	complemented
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	2	1	no	—
68	C₂×C₄	—	358	1	no	—
69	C₂×C₄	—	358	1	no	—
70	C₂×C₄	—	179	1	no	—
71	C₂×C₄	—	179	1	no	—
72	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
73	C₈	—	2	1	no	—
74	C₈	—	358	1	no	—
75	C₂²	D₄×D₁₇₉	1	1	yes	characteristic
76	C₄	—	2	2	no	—
77	C₄	—	4	1	no	—
78	C₄	—	358	1	no	—
79	C₄	—	179	2	no	—
80	C₄	—	716	1	no	—
81	C₄	—	716	1	no	—
82	C₄	C₂×Dic₃₅₈	1	1	yes	characteristic
83	C₄	#15668258	1	1	yes	characteristic
84	C₂	#19153090	1	1	yes	characteristic
85	C₂	SD₁₆×D₁₇₉	1	1	yes	characteristic
86	C₂	#19153145	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₇₉
commutator, cosocle	C₂×C₄×C₁₇₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₇₉
Fitting	#103×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	C₂×D₄
hypercenter	#103	D₁₇₉
hypocenter	C₁₇₉	#475
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #103

Complements

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

N	H ≅ G/N	type
26. Q₈×C₁₇₉	Q₈	semidirect
43. C₁₇₉	#475	semidirect
65. Q₈	Dic₃₅₈	semidirect

Maximal subgroups

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

subgroup	index
#19153112	2
Q₈×Dic₁₇₉	2
#19153045	2
#19153044	2
Dic₁₇₉⋊C₈	2
#19153037	2
#103×C₁₇₉	2
#475	179