Subgroups — #26038801

Order 11456.

100 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19153101	C₂	1	1	yes	maximal, characteristic
2	#19153036	C₂	1	2	yes	maximal, complemented
3	#19153033	C₂	1	2	yes	maximal
4	#19153024	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄⋊Q₈×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₅₈	C₂²	1	1	yes	characteristic
7	C₂×D₇₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₇₉⋊C₈	C₂²	1	2	yes	—
9	(C₂×D₁₇₉)⋊C₄	—	2	1	no	—
10	(C₂×D₁₇₉)⋊C₄	—	2	1	no	—
11	C₂×Q₈×C₁₇₉	—	2	1	no	—
12	C₄⋊C₄×C₁₇₉	—	2	1	no	—
13	C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
14	C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₇₉	—	2	1	no	—
16	Dic₃₅₈	—	2	2	no	—
17	D₇₁₆	—	2	2	no	—
18	C₂×Dic₁₇₉	—	2	1	no	—
19	C₁₇₉⋊C₈	—	2	2	no	—
20	Q₈×C₁₇₉	—	4	2	no	—
21	C₂×C₄×C₁₇₉	—	2	2	no	—
22	C₂×C₄×C₁₇₉	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₇₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
25	C₂²×C₁₇₉	C₂×D₄	1	1	yes	characteristic
26	C₂×D₁₇₉	—	4	1	no	—
27	C₂×D₁₇₉	—	4	1	no	—
28	C₂×D₁₇₉	—	4	1	no	—
29	C₄×C₁₇₉	—	2	1	no	—
30	C₄×C₁₇₉	—	2	1	no	—
31	C₄×C₁₇₉	—	4	2	no	—
32	C₄×C₁₇₉	#41	1	2	yes	—
33	Dic₁₇₉	—	4	1	no	—
34	C₂×C₁₇₉	#124	1	1	yes	characteristic
35	C₂×C₁₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
36	C₂×C₁₇₉	#137	1	1	yes	characteristic
37	D₁₇₉	—	8	1	no	—
38	C₁₇₉	#489	1	1	yes	characteristic, complemented, Sylow
39	#489	—	179	1	no	maximal, Sylow
40	C₄⋊Q₈	D₁₇₉	1	1	yes	characteristic, complemented
41	#124	—	179	1	no	—
42	#103	—	179	2	no	—
43	#102	—	179	2	no	—
44	#97	—	179	1	no	—
45	C₂×Q₈	—	2	1	no	—
46	C₂×Q₈	—	179	1	no	—
47	C₂×D₄	—	179	1	no	—
48	C₂×C₈	—	179	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
51	C₂²⋊C₄	—	358	1	no	—
52	C₂²⋊C₄	—	358	1	no	—
53	C₄²	C₂×D₁₇₉	1	1	yes	characteristic
54	C₂³	—	358	1	no	—
55	Q₈	—	4	2	no	—
56	Q₈	—	358	2	no	—
57	D₄	—	358	2	no	—
58	C₂×C₄	—	2	2	no	—
59	C₂×C₄	—	358	1	no	—
60	C₂×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
62	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
63	C₈	—	358	2	no	—
64	C₂²	—	716	1	no	—
65	C₂²	—	716	1	no	—
66	C₂²	—	716	1	no	—
67	C₂²	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
68	C₄	—	2	1	no	—
69	C₄	—	2	1	no	—
70	C₄	—	4	2	no	—
71	C₄	—	716	1	no	—
72	C₄	#15668260	1	2	yes	—
73	C₂	—	1432	1	no	—
74	C₂	#19153102	1	1	yes	characteristic
75	C₂	#19153139	1	1	yes	characteristic
76	C₂	#19153144	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₇₉
commutator, cosocle	C₂×C₄×C₁₇₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₇₉
Fitting	C₄⋊Q₈×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×D₄
prime radical	C₂×D₇₁₆	C₂²
hypercenter	C₄⋊Q₈	D₁₇₉
hypocenter	C₁₇₉	#489
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #19153036	C₂	semidirect
4. #19153024	C₂	semidirect
5. C₄⋊Q₈×C₁₇₉	C₂	semidirect
38. C₁₇₉	#489	semidirect
40. C₄⋊Q₈	D₁₇₉	semidirect

Maximal subgroups

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

subgroup	index
#19153101	2
#19153036	2
#19153033	2
#19153024	2
C₄⋊Q₈×C₁₇₉	2
#489	179