Subgroups — #37511251

Order 22912.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₄₃₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#19153059	C₂	1	2	yes	maximal
3	#26038817	C₂	1	2	yes	maximal
4	C₁₆×Dic₁₇₉	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₆	C₂²	1	2	yes	—
7	C₂×#15668249	—	2	2	no	—
8	#19153111	—	2	2	no	—
9	Dic₁₄₃₂	—	2	2	no	—
10	#19153059	—	2	4	no	—
11	C₂×C₁₇₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₇₉	C₂²	1	1	yes	characteristic
13	C₂×Q₁₆×C₁₇₉	C₂²	1	2	yes	—
14	Q₃₂×C₁₇₉	—	2	2	no	—
15	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₅₈	—	2	2	no	—
17	Dic₇₁₆	—	2	2	no	—
18	Dic₇₁₆	—	2	2	no	—
19	#15668249	—	4	4	no	—
20	C₂×C₁₇₉⋊C₈	D₄	1	1	yes	characteristic
21	Dic₁₇₉⋊C₄	—	4	4	no	—
22	C₄×Dic₁₇₉	D₄	1	1	yes	characteristic
23	C₁₇₉⋊C₁₆	D₄	1	2	yes	—
24	C₂×Q₈×C₁₇₉	—	2	2	no	—
25	Q₁₆×C₁₇₉	—	2	2	no	—
26	Q₁₆×C₁₇₉	—	2	2	no	—
27	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
28	C₁₆×C₁₇₉	D₄	1	2	yes	—
29	Dic₃₅₈	—	4	2	no	—
30	Dic₃₅₈	—	4	2	no	—
31	C₂×Dic₁₇₉	—	4	2	no	—
32	C₂×Dic₁₇₉	D₈	1	2	yes	—
33	C₁₇₉⋊C₈	D₈	1	2	yes	—
34	Q₈×C₁₇₉	—	4	2	no	—
35	Q₈×C₁₇₉	—	4	2	no	—
36	C₂×C₄×C₁₇₉	—	4	2	no	—
37	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
38	C₈×C₁₇₉	C₂×D₄	1	1	yes	characteristic
39	C₈×C₁₇₉	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₁₇₉	C₂×D₈	1	1	yes	characteristic
41	C₄×C₁₇₉	—	8	2	no	—
42	C₄×C₁₇₉	#127	1	1	yes	characteristic
43	C₄×C₁₇₉	C₂×D₈	1	1	yes	characteristic
44	Dic₁₇₉	—	8	2	no	—
45	Dic₁₇₉	Q₃₂	1	4	yes	complemented
46	C₂×C₁₇₉	#493	1	1	yes	characteristic
47	C₂×C₁₇₉	C₂×Q₃₂	1	2	yes	—
48	C₁₇₉	#2247	1	1	yes	characteristic, complemented, Sylow
49	#2247	—	179	1	no	maximal, Sylow
50	C₂×Q₃₂	—	179	2	no	—
51	C₂×Q₃₂	—	179	1	no	—
52	C₂×Q₃₂	D₁₇₉	1	1	yes	characteristic
53	#494	—	179	2	no	—
54	C₄×C₁₆	—	179	1	no	—
55	C₂×Q₁₆	—	358	2	no	—
56	C₂×Q₁₆	—	179	2	no	—
57	C₂×Q₁₆	C₂×D₁₇₉	1	2	yes	—
58	C₄⋊Q₈	—	358	2	no	—
59	Q₃₂	—	2	2	no	—
60	Q₃₂	—	358	4	no	—
61	Q₃₂	—	358	2	no	—
62	C₂×C₁₆	—	179	1	no	—
63	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
64	C₄×C₈	—	179	1	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	—	358	2	no	—
67	Q₁₆	—	2	2	no	—
68	Q₁₆	—	2	2	no	—
69	Q₁₆	—	358	2	no	—
70	Q₁₆	—	358	2	no	—
71	Q₁₆	—	716	4	no	—
72	C₂×C₈	—	179	1	no	—
73	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
74	C₄⋊C₄	—	716	4	no	—
75	C₄²	—	179	1	no	—
76	C₁₆	—	179	2	no	—
77	C₁₆	C₂²⋊D₁₇₉	1	2	yes	—
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	Q₈	—	716	2	no	—
81	Q₈	—	716	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	179	2	no	—
84	C₂×C₄	—	716	2	no	—
85	C₂×C₄	D₄×D₁₇₉	1	1	yes	characteristic
86	C₈	—	179	2	no	—
87	C₈	D₄×D₁₇₉	1	1	yes	characteristic
88	C₈	C₂×C₂²⋊D₁₇₉	1	1	yes	characteristic
89	C₂²	D₈×D₁₇₉	1	1	yes	characteristic
90	C₄	—	8	2	no	—
91	C₄	—	1432	2	no	—
92	C₄	—	179	4	no	—
93	C₄	#19153109	1	1	yes	characteristic
94	C₄	D₈×D₁₇₉	1	1	yes	characteristic
95	C₂	#26038814	1	1	yes	characteristic
96	C₂	Q₃₂×D₁₇₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₁₇₉
commutator, cosocle	C₂×C₈×C₁₇₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₇₉
Fitting	C₂×Q₃₂×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	C₂×D₈
hypercenter	C₂×Q₃₂	D₁₇₉
hypocenter	C₁₇₉	#2247
LCS	G ▷ C₂×C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

Complements

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

N	H ≅ G/N	type
45. Dic₁₇₉	Q₃₂	semidirect
48. C₁₇₉	#2247	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₄₃₂	2
C₂×#19153059	2
#26038817	2
C₁₆×Dic₁₇₉	2
C₂×Q₃₂×C₁₇₉	2
#2247	179