Subgroups — #37507141

Order 22912.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26038855	C₂	1	1	yes	maximal, characteristic
2	#26038273	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₂×C₁₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19153119	C₄	1	1	yes	characteristic
5	#19153053	C₄	1	1	yes	characteristic
6	#19153052	—	2	1	no	—
7	C₁₆×D₁₇₉	—	2	1	no	—
8	C₁₇₉⋊C₃₂	—	2	1	no	—
9	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₇₉	—	2	1	no	—
11	#15668259	C₈	1	1	yes	characteristic
12	#15668243	C₈	1	1	yes	characteristic
13	#15668242	—	2	1	no	—
14	C₈×D₁₇₉	—	2	1	no	—
15	C₁₇₉⋊C₁₆	—	2	1	no	—
16	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
17	C₁₆×C₁₇₉	D₄	1	1	yes	characteristic
18	C₁₆×C₁₇₉	D₄	1	1	yes	characteristic
19	Dic₃₅₈	C₁₆	1	1	yes	characteristic
20	D₇₁₆	C₁₆	1	1	yes	characteristic
21	C₂²⋊D₁₇₉	—	2	1	no	—
22	C₄×D₁₇₉	—	2	1	no	—
23	C₁₇₉⋊C₈	—	2	1	no	—
24	C₂×C₄×C₁₇₉	C₂×C₈	1	1	yes	characteristic
25	C₈×C₁₇₉	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₇₉	M₁₆	1	1	yes	characteristic
27	C₂²×C₁₇₉	M₃₂	1	1	yes	characteristic
28	C₂×D₁₇₉	—	2	1	no	—
29	C₄×C₁₇₉	C₂²⋊C₈	1	1	yes	characteristic
30	C₄×C₁₇₉	C₂×C₁₆	1	1	yes	characteristic
31	Dic₁₇₉	—	2	1	no	—
32	C₂×C₁₇₉	—	2	1	no	—
33	C₂×C₁₇₉	C₂²⋊C₁₆	1	1	yes	characteristic
34	D₁₇₉	—	4	1	no	—
35	C₁₇₉	#1401	1	1	yes	characteristic, complemented, Sylow
36	#1401	—	179	1	no	maximal, Sylow
37	#504	—	179	1	no	—
38	M₆₄	—	179	1	no	—
39	C₂×C₃₂	D₁₇₉	1	1	yes	characteristic, complemented
40	#131	—	179	1	no	—
41	M₃₂	—	358	1	no	—
42	M₃₂	—	179	1	no	—
43	C₂×C₁₆	—	358	1	no	—
44	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
45	C₃₂	—	2	1	no	—
46	C₃₂	—	358	1	no	—
47	#41	—	179	1	no	—
48	M₁₆	—	358	1	no	—
49	M₁₆	—	179	1	no	—
50	C₂×C₈	—	358	1	no	—
51	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
52	C₁₆	—	358	1	no	—
53	C₁₆	C₂²⋊D₁₇₉	1	1	yes	characteristic
54	C₁₆	D₇₁₆	1	1	yes	characteristic
55	Q₈	—	179	1	no	—
56	D₄	—	358	1	no	—
57	D₄	—	179	1	no	—
58	C₂×C₄	—	358	1	no	—
59	C₂×C₄	C₈×D₁₇₉	1	1	yes	characteristic
60	C₈	—	358	1	no	—
61	C₈	(C₂×D₁₇₉)⋊C₄	1	1	yes	characteristic
62	C₈	#15668242	1	1	yes	characteristic
63	C₂²	—	358	1	no	—
64	C₂²	#19153052	1	1	yes	characteristic
65	C₄	—	358	1	no	—
66	C₄	(C₂×D₁₇₉)⋊C₈	1	1	yes	characteristic
67	C₄	C₁₆×D₁₇₉	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	716	1	no	—
70	C₂	(C₂×D₁₇₉)⋊C₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	D₇₁₆
commutator	C₄×C₁₇₉	C₂×C₁₆
Frattini	C₂×C₁₆	C₂×D₁₇₉
Fitting	C₂×C₃₂×C₁₇₉	C₂
socle	C₂×C₁₇₉	C₂²⋊C₁₆
prime radical	#15668259	C₈
cosocle	C₂×C₁₆×C₁₇₉	C₂²
hypercenter	C₂×C₃₂	D₁₇₉
hypocenter	C₁₇₉	#1401
LCS	G ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ C₂×C₃₂

Complements

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

N	H ≅ G/N	type
2. #26038273	C₂	semidirect
3. C₂×C₃₂×C₁₇₉	C₂	semidirect
35. C₁₇₉	#1401	semidirect
39. C₂×C₃₂	D₁₇₉	semidirect

Maximal subgroups

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

subgroup	index
#26038855	2
#26038273	2
C₂×C₃₂×C₁₇₉	2
#1401	179