Subgroups — #26011880

Order 11072.

144 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583815	C₂	1	2	yes	maximal, complemented
2	C₂×D₁₃₈₄	C₂	1	2	yes	maximal, complemented
3	#19149676	C₂	1	1	yes	maximal, characteristic, complemented
4	#19149667	C₂	1	1	yes	maximal, characteristic, complemented
5	#97×C₁₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₃₄₆	C₂²	1	1	yes	characteristic
7	C₂×D₆₉₂	—	2	2	no	—
8	C₂×D₆₉₂	C₂²	1	1	yes	characteristic
9	C₂×D₆₉₂	C₂²	1	2	yes	—
10	#14583815	—	2	4	no	—
11	D₁₃₈₄	—	2	4	no	—
12	#14583803	—	2	2	no	—
13	C₂×C₈×C₁₇₃	C₂²	1	2	yes	—
14	C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
15	C₂²×D₁₇₃	—	2	2	no	—
16	C₂²×D₁₇₃	—	2	1	no	—
17	Dic₃₄₆	—	2	2	no	—
18	D₆₉₂	—	2	2	no	—
19	D₆₉₂	—	2	2	no	—
20	D₆₉₂	—	2	2	no	—
21	D₆₉₂	—	4	4	no	—
22	C₂×Dic₁₇₃	—	2	1	no	—
23	C₂×C₄×C₁₇₃	D₄	1	2	yes	—
24	C₂×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
25	C₈×C₁₇₃	D₄	1	4	yes	complemented
26	C₂²×C₁₇₃	C₂×D₄	1	1	yes	characteristic
27	C₂×D₁₇₃	—	4	1	no	—
28	C₂×D₁₇₃	—	4	2	no	—
29	C₂×D₁₇₃	—	4	4	no	—
30	C₂×D₁₇₃	—	4	2	no	—
31	C₄×C₁₇₃	—	2	2	no	—
32	C₄×C₁₇₃	C₂×D₄	1	2	yes	—
33	Dic₁₇₃	—	4	1	no	—
34	C₂×C₁₇₃	#127	1	1	yes	characteristic
35	C₂×C₁₇₃	AGL(1,ℤ/8)	1	2	yes	—
36	D₁₇₃	—	8	2	no	—
37	D₁₇₃	—	8	1	no	—
38	C₁₇₃	#496	1	1	yes	characteristic, complemented, Sylow
39	#496	—	173	1	no	maximal, Sylow
40	C₂×SD₁₆	—	173	2	no	—
41	C₂×D₈	—	173	2	no	—
42	#127	—	173	1	no	—
43	#124	—	173	1	no	—
44	#97	D₁₇₃	1	1	yes	characteristic, complemented
45	C₂×Q₈	—	173	1	no	—
46	C₂×D₄	—	346	2	no	—
47	C₂×D₄	—	173	1	no	—
48	C₂×D₄	—	173	2	no	—
49	SD₁₆	—	346	4	no	—
50	D₈	—	346	4	no	—
51	C₂×C₈	C₂×D₁₇₃	1	2	yes	—
52	C₂²⋊C₄	—	346	2	no	—
53	C₄²	C₂×D₁₇₃	1	1	yes	characteristic
54	C₂³	—	346	1	no	—
55	C₂³	—	346	2	no	—
56	Q₈	—	346	2	no	—
57	D₄	—	346	2	no	—
58	D₄	—	346	2	no	—
59	D₄	—	346	2	no	—
60	D₄	—	692	4	no	—
61	C₂×C₄	—	346	1	no	—
62	C₂×C₄	D₆₉₂	1	2	yes	—
63	C₂×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
64	C₈	D₆₉₂	1	4	yes	complemented
65	C₂²	—	692	1	no	—
66	C₂²	—	692	2	no	—
67	C₂²	—	692	4	no	—
68	C₂²	—	692	2	no	—
69	C₂²	C₂×D₆₉₂	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	692	1	no	—
72	C₄	C₂×D₆₉₂	1	2	yes	—
73	C₂	—	1384	2	no	—
74	C₂	—	1384	1	no	—
75	C₂	#19149676	1	1	yes	characteristic
76	C₂	#19149707	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₆₉₂
commutator, cosocle	C₂×C₄×C₁₇₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₇₃
Fitting	#97×C₁₇₃	C₂
socle	C₂²×C₁₇₃	C₂×D₄
hypercenter	#97	D₁₇₃
hypocenter	C₁₇₃	#496
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
1. C₂×#14583815	C₂	semidirect
2. C₂×D₁₃₈₄	C₂	semidirect
3. #19149676	C₂	semidirect
4. #19149667	C₂	semidirect
5. #97×C₁₇₃	C₂	semidirect
25. C₈×C₁₇₃	D₄	semidirect
38. C₁₇₃	#496	semidirect
44. #97	D₁₇₃	semidirect
64. C₈	D₆₉₂	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583815	2
C₂×D₁₃₈₄	2
#19149676	2
#19149667	2
#97×C₁₇₃	2
#496	173