Subgroups — #26038757

Order 11456.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19153087	C₂	1	1	yes	maximal, characteristic
2	#19153044	C₂	1	2	yes	maximal, complemented
3	#19153033	C₂	1	2	yes	maximal, complemented
4	C₂²⋊(C₁₇₉⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₇₁₆	C₂²	1	1	yes	characteristic
7	C₂×C₂²⋊D₁₇₉	—	2	1	no	—
8	C₂×C₁₇₉⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₁₇₉	C₂²	1	1	yes	characteristic
10	(C₂×D₁₇₉)⋊C₄	—	2	1	no	—
11	C₂²×C₄×C₁₇₉	—	2	1	no	—
12	C₂²×C₄×C₁₇₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₇₉	—	2	1	no	—
14	C₄⋊C₄×C₁₇₉	C₂²	1	2	yes	—
15	C₂²×D₁₇₉	—	2	1	no	—
16	D₇₁₆	—	2	2	no	—
17	C₂²⋊D₁₇₉	—	4	2	no	—
18	C₂×Dic₁₇₉	—	2	1	no	—
19	C₁₇₉⋊C₈	—	2	2	no	—
20	C₂³×C₁₇₉	D₄	1	1	yes	characteristic
21	C₂×C₄×C₁₇₉	—	2	2	no	—
22	C₂×C₄×C₁₇₉	—	2	2	no	—
23	C₂×C₄×C₁₇₉	—	4	2	no	—
24	C₂×C₄×C₁₇₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
26	C₂²×C₁₇₉	—	2	1	no	—
27	C₂²×C₁₇₉	—	2	1	no	—
28	C₂²×C₁₇₉	SD₁₆	1	2	yes	—
29	C₂²×C₁₇₉	C₂×D₄	1	1	yes	characteristic
30	C₂×D₁₇₉	—	4	1	no	—
31	C₂×D₁₇₉	—	4	1	no	—
32	C₂×D₁₇₉	—	4	1	no	—
33	C₄×C₁₇₉	—	2	1	no	—
34	C₄×C₁₇₉	—	4	2	no	—
35	C₄×C₁₇₉	#41	1	2	yes	—
36	Dic₁₇₉	—	4	1	no	—
37	C₂×C₁₇₉	—	2	2	no	—
38	C₂×C₁₇₉	#123	1	1	yes	characteristic
39	C₂×C₁₇₉	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₁₇₉	AGL(1,ℤ/8)	1	1	yes	characteristic
41	D₁₇₉	—	8	1	no	—
42	C₁₇₉	#481	1	1	yes	characteristic, complemented, Sylow
43	#481	—	179	1	no	maximal, Sylow
44	#121	—	179	1	no	—
45	C₂×C₄⋊C₄	D₁₇₉	1	1	yes	characteristic, complemented
46	#106	—	179	2	no	—
47	#102	—	179	2	no	—
48	C₂²⋊C₈	—	179	1	no	—
49	C₂×D₄	—	358	1	no	—
50	C₂×D₄	—	179	1	no	—
51	C₂²×C₄	—	2	1	no	—
52	C₂²×C₄	C₂×D₁₇₉	1	1	yes	characteristic
53	C₂×C₈	—	179	2	no	—
54	C₄⋊C₄	—	2	1	no	—
55	C₄⋊C₄	—	179	1	no	—
56	C₄⋊C₄	C₂×D₁₇₉	1	2	yes	—
57	C₂²⋊C₄	—	358	1	no	—
58	C₂³	—	358	1	no	—
59	C₂³	C₂²⋊D₁₇₉	1	1	yes	characteristic
60	D₄	—	358	2	no	—
61	D₄	—	716	2	no	—
62	C₂×C₄	—	2	2	no	—
63	C₂×C₄	—	2	2	no	—
64	C₂×C₄	—	4	2	no	—
65	C₂×C₄	—	358	1	no	—
66	C₂×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
67	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
68	C₈	—	358	2	no	—
69	C₂²	—	2	1	no	—
70	C₂²	—	2	1	no	—
71	C₂²	—	716	1	no	—
72	C₂²	—	716	1	no	—
73	C₂²	—	716	1	no	—
74	C₂²	#15668246	1	2	yes	—
75	C₂²	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	716	1	no	—
79	C₄	#15668259	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	—	1432	1	no	—
82	C₂	#19153096	1	1	yes	characteristic
83	C₂	C₂×#15668246	1	1	yes	characteristic
84	C₂	#19153139	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₇₉
commutator, cosocle	C₂×C₄×C₁₇₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₇₉
Fitting	C₂×C₄⋊C₄×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×D₄
prime radical	#19153087	C₂
hypercenter	C₂×C₄⋊C₄	D₁₇₉
hypocenter	C₁₇₉	#481
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #19153044	C₂	semidirect
3. #19153033	C₂	semidirect
4. C₂²⋊(C₁₇₉⋊C₈)	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₇₉	C₂	semidirect
42. C₁₇₉	#481	semidirect
45. C₂×C₄⋊C₄	D₁₇₉	semidirect

Maximal subgroups

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

subgroup	index
#19153087	2
#19153044	2
#19153033	2
C₂²⋊(C₁₇₉⋊C₈)	2
C₂×C₄⋊C₄×C₁₇₉	2
#481	179