Subgroups — #37510872

Order 22912.

148 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	C₄×Dic₇₁₆	C₂	1	2	yes	maximal
3	#26038267	C₂	1	1	yes	maximal, characteristic
4	#26038244	C₂	1	2	yes	maximal
5	#346×C₁₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₁₆	C₂²	1	2	yes	—
7	C₄×Dic₃₅₈	—	2	2	no	—
8	Dic₁₄₃₂	C₄	1	4	yes	complemented
9	#19153047	C₂²	1	2	yes	—
10	#19153038	—	2	2	no	—
11	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₇₉	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₁₇₉	C₂²	1	1	yes	characteristic
14	C₂×Dic₃₅₈	—	2	2	no	—
15	Dic₇₁₆	—	2	2	no	—
16	Dic₇₁₆	C₂×C₄	1	4	yes	—
17	C₄⋊Dic₁₇₉	—	2	2	no	—
18	Dic₁₇₉⋊C₄	—	4	2	no	—
19	C₄×Dic₁₇₉	—	4	2	no	—
20	C₂×C₈×C₁₇₉	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₇₉	C₂³	1	1	yes	characteristic
22	C₄²×C₁₇₉	D₄	1	1	yes	characteristic
23	C₁₆×C₁₇₉	—	2	1	no	—
24	C₁₆×C₁₇₉	C₂×C₄	1	2	yes	—
25	Dic₃₅₈	—	2	4	no	—
26	Dic₃₅₈	—	4	2	no	—
27	C₂×Dic₁₇₉	—	4	2	no	—
28	C₂×Dic₁₇₉	—	4	2	no	—
29	C₂×C₄×C₁₇₉	D₈	1	2	yes	—
30	C₂×C₄×C₁₇₉	C₂×D₄	1	1	yes	characteristic
31	C₈×C₁₇₉	—	2	1	no	—
32	C₈×C₁₇₉	C₂²×C₄	1	1	yes	characteristic
33	C₈×C₁₇₉	#41	1	1	yes	characteristic
34	C₂²×C₁₇₉	C₂×D₈	1	1	yes	characteristic
35	C₄×C₁₇₉	—	2	2	no	—
36	C₄×C₁₇₉	C₄×D₄	1	1	yes	characteristic
37	C₄×C₁₇₉	#135	1	1	yes	characteristic
38	Dic₁₇₉	—	4	4	no	—
39	Dic₁₇₉	—	8	2	no	—
40	C₂×C₁₇₉	C₄×D₈	1	1	yes	characteristic
41	C₂×C₁₇₉	#510	1	2	yes	—
42	C₁₇₉	#2176	1	1	yes	characteristic, complemented, Sylow
43	#2176	—	179	1	no	maximal, Sylow
44	C₂×Q₃₂	—	179	1	no	—
45	C₄×Q₁₆	—	179	2	no	—
46	#367	—	179	1	no	—
47	#358	—	179	2	no	—
48	#346	D₁₇₉	1	1	yes	characteristic
49	C₂×Q₁₆	—	179	2	no	—
50	C₄×Q₈	—	358	2	no	—
51	Q₃₂	—	179	4	no	—
52	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
53	C₂×C₁₆	C₂×D₁₇₉	1	1	yes	characteristic
54	#107	—	179	2	no	—
55	#103	—	358	2	no	—
56	C₄×C₈	C₂×D₁₇₉	1	1	yes	characteristic
57	C₂×Q₈	—	358	2	no	—
58	Q₁₆	—	358	2	no	—
59	Q₁₆	—	179	4	no	—
60	C₂×C₈	D₇₁₆	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₇₉	1	1	yes	characteristic
62	C₄⋊C₄	—	358	2	no	—
63	C₄⋊C₄	—	716	2	no	—
64	C₄²	—	716	2	no	—
65	C₄²	D₇₁₆	1	1	yes	characteristic
66	C₁₆	—	2	1	no	—
67	C₁₆	C₄×D₁₇₉	1	2	yes	—
68	Q₈	—	358	4	no	—
69	Q₈	—	716	2	no	—
70	C₂×C₄	—	716	2	no	—
71	C₂×C₄	—	716	2	no	—
72	C₂×C₄	D₁₄₃₂	1	2	yes	—
73	C₂×C₄	C₂×D₇₁₆	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	C₂×C₄×D₁₇₉	1	1	yes	characteristic
76	C₈	#15668259	1	1	yes	characteristic
77	C₂²	C₂×D₁₄₃₂	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	1432	2	no	—
80	C₄	—	716	4	no	—
81	C₄	C₄×D₇₁₆	1	1	yes	characteristic
82	C₄	#19153135	1	1	yes	characteristic
83	C₂	C₄×D₁₄₃₂	1	1	yes	characteristic
84	C₂	#26038882	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₄₃₂
commutator	C₈×C₁₇₉	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₁₇₉
Fitting	#346×C₁₇₉	C₂
socle, prime radical	C₂²×C₁₇₉	C₂×D₈
cosocle	C₂×C₈×C₁₇₉	C₂³
hypercenter	#346	D₁₇₉
hypocenter	C₁₇₉	#2176
LCS	G ▷ C₈×C₁₇₉ ▷ C₄×C₁₇₉ ▷ C₂×C₁₇₉ ▷ C₁₇₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
8. Dic₁₄₃₂	C₄	semidirect
42. C₁₇₉	#2176	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
C₄×Dic₇₁₆	2
#26038267	2
#26038244	2
#346×C₁₇₉	2
#2176	179