Subgroups — #37510902

Order 22912.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×Dic₃₅₈	C₂	1	1	yes	maximal, characteristic
2	Dic₁₇₉⋊C₁₆	C₂	1	2	yes	maximal
3	C₄×C₁₇₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₆×Dic₁₇₉	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₃₅₈	C₄	1	1	yes	characteristic
7	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₄⋊(C₁₇₉⋊C₈)	C₄	1	1	yes	characteristic
10	Dic₁₇₉⋊C₈	C₄	1	2	yes	—
11	C₈×Dic₁₇₉	C₂²	1	2	yes	—
12	C₂×C₁₆×C₁₇₉	C₂²	1	2	yes	—
13	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₅₈	C₈	1	1	yes	characteristic
15	C₂×C₁₇₉⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₁₇₉	C₈	1	1	yes	characteristic
17	Dic₁₇₉⋊C₄	C₈	1	2	yes	—
18	C₄×Dic₁₇₉	C₂×C₄	1	2	yes	—
19	C₁₇₉⋊C₁₆	—	2	1	no	—
20	C₁₇₉⋊C₁₆	Q₈	1	2	yes	complemented
21	C₂×C₈×C₁₇₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
23	C₄²×C₁₇₉	C₂×C₄	1	1	yes	characteristic
24	C₁₆×C₁₇₉	—	2	2	no	—
25	Dic₃₅₈	C₁₆	1	4	yes	complemented
26	C₂×Dic₁₇₉	C₂×C₈	1	2	yes	—
27	C₂×Dic₁₇₉	C₂×C₈	1	2	yes	—
28	C₁₇₉⋊C₈	—	2	2	no	—
29	C₂×C₄×C₁₇₉	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₇₉	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₁₇₉	—	2	1	no	—
33	C₈×C₁₇₉	C₂×Q₈	1	1	yes	characteristic
34	C₈×C₁₇₉	#41	1	1	yes	characteristic
35	C₂²×C₁₇₉	C₂²×C₈	1	1	yes	characteristic
36	C₄×C₁₇₉	—	2	1	no	—
37	C₄×C₁₇₉	C₂×C₁₆	1	2	yes	—
38	C₄×C₁₇₉	C₄×Q₈	1	1	yes	characteristic
39	C₄×C₁₇₉	#131	1	1	yes	characteristic
40	Dic₁₇₉	—	2	2	no	—
41	Dic₁₇₉	C₂×C₁₆	1	4	yes	—
42	C₂×C₁₇₉	C₈×Q₈	1	1	yes	characteristic
43	C₂×C₁₇₉	C₂²×C₁₆	1	1	yes	characteristic
44	C₂×C₁₇₉	#504	1	1	yes	characteristic
45	C₁₇₉	Q₈×C₁₆	1	1	yes	characteristic, complemented, Sylow
46	Q₈×C₁₆	—	179	1	no	maximal, Sylow
47	C₈×Q₈	—	179	1	no	—
48	C₄⋊C₁₆	—	179	2	no	—
49	C₄⋊C₁₆	D₁₇₉	1	1	yes	characteristic
50	C₄×C₁₆	—	179	2	no	—
51	C₄×C₁₆	—	179	1	no	—
52	C₄×Q₈	—	179	1	no	—
53	C₂×C₁₆	—	179	1	no	—
54	C₂×C₁₆	—	179	1	no	—
55	C₂×C₁₆	C₂×D₁₇₉	1	2	yes	—
56	C₄⋊C₈	—	179	2	no	—
57	C₄⋊C₈	—	179	1	no	—
58	C₄×C₈	—	179	2	no	—
59	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
60	C₂×Q₈	—	179	1	no	—
61	C₂×C₈	—	179	2	no	—
62	C₂×C₈	C₄×D₁₇₉	1	1	yes	characteristic
63	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
64	C₄⋊C₄	—	179	2	no	—
65	C₄⋊C₄	—	179	1	no	—
66	C₄²	—	179	2	no	—
67	C₄²	C₄×D₁₇₉	1	1	yes	characteristic
68	C₁₆	—	2	2	no	—
69	C₁₆	—	358	1	no	—
70	C₁₆	—	179	2	no	—
71	Q₈	—	179	4	no	—
72	C₂×C₄	—	179	2	no	—
73	C₂×C₄	—	179	2	no	—
74	C₂×C₄	C₈×D₁₇₉	1	1	yes	characteristic
75	C₂×C₄	C₈×D₁₇₉	1	1	yes	characteristic
76	C₂×C₄	C₂×C₄×D₁₇₉	1	1	yes	characteristic
77	C₈	—	2	1	no	—
78	C₈	—	358	2	no	—
79	C₈	Q₈×D₁₇₉	1	1	yes	characteristic
80	C₈	#15668258	1	1	yes	characteristic
81	C₂²	C₂×C₈×D₁₇₉	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	358	2	no	—
84	C₄	—	179	4	no	—
85	C₄	C₁₆×D₁₇₉	1	2	yes	—
86	C₄	#19153078	1	1	yes	characteristic
87	C₄	#19153121	1	1	yes	characteristic
88	C₂	#26038563	1	1	yes	characteristic
89	C₂	C₂×C₁₆×D₁₇₉	1	1	yes	characteristic
90	C₂	#26038857	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₁₇₉
commutator	C₂×C₁₇₉	C₂²×C₁₆
Fitting	C₄⋊C₁₆×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	C₂²×C₈
cosocle	C₂×C₈×C₁₇₉	C₂³
hypercenter	C₄⋊C₁₆	D₁₇₉
hypocenter	C₁₇₉	Q₈×C₁₆
LCS	G ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
20. C₁₇₉⋊C₁₆	Q₈	semidirect
25. Dic₃₅₈	C₁₆	semidirect
45. C₁₇₉	Q₈×C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₈×Dic₃₅₈	2
Dic₁₇₉⋊C₁₆	2
C₄×C₁₇₉⋊C₁₆	2
C₁₆×Dic₁₇₉	2
C₄⋊C₁₆×C₁₇₉	2
Q₈×C₁₆	179