Subgroups — #26038828

Order 11456.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#15668248	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₄₃₂	C₂	1	2	yes	maximal, complemented
3	#19153110	C₂	1	1	yes	maximal, characteristic, complemented
4	#19153101	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₅₈	C₂²	1	1	yes	characteristic
7	C₂×D₇₁₆	—	2	2	no	—
8	C₂×D₇₁₆	C₂²	1	1	yes	characteristic
9	C₂×D₇₁₆	C₂²	1	2	yes	—
10	#15668248	—	2	4	no	—
11	D₁₄₃₂	—	2	4	no	—
12	(C₂×D₁₇₉)⋊C₄	—	2	2	no	—
13	C₂×C₈×C₁₇₉	C₂²	1	2	yes	—
14	C₄²×C₁₇₉	C₂²	1	1	yes	characteristic
15	C₂²×D₁₇₉	—	2	1	no	—
16	C₂²×D₁₇₉	—	2	2	no	—
17	Dic₃₅₈	—	2	2	no	—
18	D₇₁₆	—	2	2	no	—
19	D₇₁₆	—	2	2	no	—
20	D₇₁₆	—	2	2	no	—
21	D₇₁₆	—	4	4	no	—
22	C₂×Dic₁₇₉	—	2	1	no	—
23	C₂×C₄×C₁₇₉	D₄	1	2	yes	—
24	C₂×C₄×C₁₇₉	C₂³	1	1	yes	characteristic
25	C₈×C₁₇₉	D₄	1	4	yes	complemented
26	C₂²×C₁₇₉	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₇₉	—	4	1	no	—
28	C₂×D₁₇₉	—	4	2	no	—
29	C₂×D₁₇₉	—	4	4	no	—
30	C₂×D₁₇₉	—	4	2	no	—
31	C₄×C₁₇₉	—	2	2	no	—
32	C₄×C₁₇₉	C₂×D₄	1	2	yes	—
33	Dic₁₇₉	—	4	1	no	—
34	C₂×C₁₇₉	#127	1	1	yes	characteristic
35	C₂×C₁₇₉	AGL(1,ℤ/8)	1	2	yes	—
36	D₁₇₉	—	8	2	no	—
37	D₁₇₉	—	8	1	no	—
38	C₁₇₉	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	179	1	no	maximal, Sylow
40	C₂×SD₁₆	—	179	2	no	—
41	C₂×D₈	—	179	2	no	—
42	#127	—	179	1	no	—
43	#124	—	179	1	no	—
44	#97	D₁₇₉	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	179	1	no	—
46	C₂×D₄	—	358	2	no	—
47	C₂×D₄	—	179	1	no	—
48	C₂×D₄	—	179	2	no	—
49	SD₁₆	—	358	4	no	—
50	D₈	—	358	4	no	—
51	C₂×C₈	C₂×D₁₇₉	1	2	yes	—
52	C₂²⋊C₄	—	358	2	no	—
53	C₄²	C₂×D₁₇₉	1	1	yes	characteristic
54	C₂³	—	358	1	no	—
55	C₂³	—	358	2	no	—
56	Q₈	—	358	2	no	—
57	D₄	—	358	2	no	—
58	D₄	—	358	2	no	—
59	D₄	—	358	2	no	—
60	D₄	—	716	4	no	—
61	C₂×C₄	—	358	1	no	—
62	C₂×C₄	D₇₁₆	1	2	yes	—
63	C₂×C₄	C₂²×D₁₇₉	1	1	yes	characteristic
64	C₈	D₇₁₆	1	4	yes	complemented
65	C₂²	—	716	1	no	—
66	C₂²	—	716	2	no	—
67	C₂²	—	716	4	no	—
68	C₂²	—	716	2	no	—
69	C₂²	C₂×D₇₁₆	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	716	1	no	—
72	C₄	C₂×D₇₁₆	1	2	yes	—
73	C₂	—	1432	2	no	—
74	C₂	—	1432	1	no	—
75	C₂	#19153110	1	1	yes	characteristic
76	C₂	#19153141	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₁₆
commutator, cosocle	C₂×C₄×C₁₇₉	C₂³
Frattini	C₂×C₄	C₂²×D₁₇₉
Fitting	#97×C₁₇₉	C₂
socle	C₂²×C₁₇₉	C₂×D₄
hypercenter	#97	D₁₇₉
hypocenter	C₁₇₉	#496
LCS	G ▷ C₂×C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×#15668248	C₂	semidirect
2. C₂×D₁₄₃₂	C₂	semidirect
3. #19153110	C₂	semidirect
4. #19153101	C₂	semidirect
5. #97×C₁₇₉	C₂	semidirect
25. C₈×C₁₇₉	D₄	semidirect
38. C₁₇₉	#496	semidirect
44. #97	D₁₇₉	semidirect
64. C₈	D₇₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#15668248	2
C₂×D₁₄₃₂	2
#19153110	2
#19153101	2
#97×C₁₇₉	2
#496	179