Subgroups — #26038789

Order 11456.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19153113	C₂	1	1	yes	maximal, characteristic
2	#19153037	C₂	1	4	yes	maximal
3	C₄×C₁₇₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Q₈×C₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₃₅₈	—	2	1	no	—
6	C₂×C₁₇₉⋊C₈	C₂²	1	2	yes	—
7	C₄⋊Dic₁₇₉	—	2	1	no	—
8	C₄⋊Dic₁₇₉	C₂²	1	2	yes	—
9	C₂×Q₈×C₁₇₉	C₂²	1	2	yes	—
10	C₄⋊C₄×C₁₇₉	—	2	1	no	—
11	C₄⋊C₄×C₁₇₉	—	2	1	no	—
12	C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
13	Dic₃₅₈	—	4	2	no	—
14	C₂×Dic₁₇₉	—	2	2	no	—
15	C₁₇₉⋊C₈	—	2	2	no	—
16	Q₈×C₁₇₉	—	2	4	no	—
17	C₂×C₄×C₁₇₉	—	2	2	no	—
18	C₂×C₄×C₁₇₉	D₄	1	1	yes	characteristic
19	C₂×C₄×C₁₇₉	D₄	1	1	yes	characteristic
20	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
21	C₂²×C₁₇₉	C₂×D₄	1	1	yes	characteristic
22	C₄×C₁₇₉	—	4	2	no	—
23	C₄×C₁₇₉	SD₁₆	1	2	yes	—
24	C₄×C₁₇₉	Q₁₆	1	2	yes	—
25	C₄×C₁₇₉	#41	1	2	yes	—
26	Dic₁₇₉	—	4	2	no	—
27	C₂×C₁₇₉	#124	1	1	yes	characteristic
28	C₂×C₁₇₉	C₂×SD₁₆	1	1	yes	characteristic
29	C₂×C₁₇₉	C₂×Q₁₆	1	1	yes	characteristic
30	C₁₇₉	#487	1	1	yes	characteristic, complemented, Sylow
31	#487	—	179	1	no	maximal, Sylow
32	C₄⋊Q₈	—	179	1	no	—
33	C₄⋊Q₈	D₁₇₉	1	1	yes	characteristic
34	#103	—	179	4	no	—
35	C₄×C₈	—	179	1	no	—
36	C₂×Q₈	—	358	1	no	—
37	C₂×Q₈	C₂×D₁₇₉	1	2	yes	—
38	C₂×C₈	—	179	2	no	—
39	C₄⋊C₄	—	2	1	no	—
40	C₄⋊C₄	—	2	1	no	—
41	C₄⋊C₄	—	358	1	no	—
42	C₄⋊C₄	—	179	2	no	—
43	C₄²	C₂×D₁₇₉	1	1	yes	characteristic
44	Q₈	—	2	4	no	—
45	Q₈	—	716	2	no	—
46	C₂×C₄	—	2	2	no	—
47	C₂×C₄	—	358	2	no	—
48	C₂×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
49	C₂×C₄	C₂²⋊D₁₇₉	1	1	yes	characteristic
50	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
51	C₈	—	358	2	no	—
52	C₂²	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
53	C₄	—	4	2	no	—
54	C₄	—	716	2	no	—
55	C₄	#15668246	1	2	yes	—
56	C₄	#15668249	1	2	yes	—
57	C₄	#15668258	1	2	yes	—
58	C₂	#19153100	1	1	yes	characteristic
59	C₂	C₂×#15668246	1	1	yes	characteristic
60	C₂	C₂×#15668249	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, prime radical	C₂²×C₁₇₉	C₂×D₄
hypercenter	C₄⋊Q₈	D₁₇₉
hypocenter	C₁₇₉	#487
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
30. C₁₇₉	#487	semidirect

Maximal subgroups

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

subgroup	index
#19153113	2
#19153037	2
C₄×C₁₇₉⋊C₈	2
C₄⋊Q₈×C₁₇₉	2
#487	179