Subgroups — #26011701

Order 11072.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583816	C₂	1	2	yes	maximal
2	#19149679	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#13063436	C₂	1	1	yes	maximal, characteristic
4	#19149602	C₂	1	2	yes	maximal
5	C₄×Q₈×C₁₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₄₆	C₂²	1	2	yes	—
7	#14583816	—	2	4	no	—
8	C₂×#13063436	C₂²	1	2	yes	—
9	C₄⋊Dic₁₇₃	—	2	1	no	—
10	C₄⋊Dic₁₇₃	—	2	1	no	—
11	C₂×Q₈×C₁₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₇₃	—	2	1	no	—
13	C₄⋊C₄×C₁₇₃	C₂²	1	1	yes	characteristic
14	C₄²×C₁₇₃	—	2	1	no	—
15	C₄²×C₁₇₃	C₂²	1	1	yes	characteristic
16	Dic₃₄₆	—	2	2	no	—
17	Dic₃₄₆	—	2	2	no	—
18	C₂×Dic₁₇₃	—	2	2	no	—
19	#13063436	—	2	2	no	—
20	Q₈×C₁₇₃	—	2	1	no	—
21	Q₈×C₁₇₃	D₄	1	2	yes	—
22	C₂×C₄×C₁₇₃	—	2	1	no	—
23	C₂×C₄×C₁₇₃	—	2	1	no	—
24	C₂×C₄×C₁₇₃	D₄	1	1	yes	characteristic
25	C₂×C₄×C₁₇₃	D₄	1	1	yes	characteristic
26	C₂×C₄×C₁₇₃	C₂³	1	1	yes	characteristic
27	C₂²×C₁₇₃	C₂×D₄	1	1	yes	characteristic
28	C₄×C₁₇₃	—	2	2	no	—
29	C₄×C₁₇₃	—	2	1	no	—
30	C₄×C₁₇₃	—	4	1	no	—
31	C₄×C₁₇₃	Q₁₆	1	2	yes	complemented
32	C₄×C₁₇₃	C₂×D₄	1	1	yes	characteristic
33	C₄×C₁₇₃	#41	1	1	yes	characteristic
34	Dic₁₇₃	—	4	2	no	—
35	C₂×C₁₇₃	#121	1	1	yes	characteristic
36	C₂×C₁₇₃	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₁₇₃	#137	1	1	yes	characteristic
38	C₁₇₃	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	173	1	no	maximal, Sylow
40	C₂×Q₁₆	—	173	2	no	—
41	C₄⋊Q₈	—	173	1	no	—
42	C₄×Q₈	D₁₇₃	1	1	yes	characteristic
43	C₄⋊C₈	—	173	1	no	—
44	#103	—	173	2	no	—
45	C₂×Q₈	—	173	2	no	—
46	C₂×Q₈	C₂×D₁₇₃	1	1	yes	characteristic
47	Q₁₆	—	346	4	no	—
48	C₂×C₈	—	173	2	no	—
49	C₄⋊C₄	—	2	1	no	—
50	C₄⋊C₄	—	346	1	no	—
51	C₄⋊C₄	—	346	1	no	—
52	C₄⋊C₄	C₂×D₁₇₃	1	1	yes	characteristic
53	C₄²	—	2	1	no	—
54	C₄²	C₂×D₁₇₃	1	1	yes	characteristic
55	Q₈	—	2	1	no	—
56	Q₈	—	346	2	no	—
57	Q₈	—	346	2	no	—
58	Q₈	D₆₉₂	1	2	yes	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	2	1	no	—
61	C₂×C₄	—	346	2	no	—
62	C₂×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
63	C₂×C₄	C₂²⋊D₁₇₃	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₁₇₃	1	1	yes	characteristic
65	C₈	—	346	2	no	—
66	C₂²	C₂×C₂²⋊D₁₇₃	1	1	yes	characteristic
67	C₄	—	2	2	no	—
68	C₄	—	2	1	no	—
69	C₄	—	4	1	no	—
70	C₄	—	692	2	no	—
71	C₄	#14583816	1	2	yes	complemented
72	C₄	C₂×D₆₉₂	1	1	yes	characteristic
73	C₄	#14583826	1	1	yes	characteristic
74	C₂	#19149653	1	1	yes	characteristic
75	C₂	C₂×#14583816	1	1	yes	characteristic
76	C₂	#19149709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₁₇₃
commutator, cosocle	C₂×C₄×C₁₇₃	C₂³
Frattini	C₂×C₄	C₂²×D₁₇₃
Fitting	C₄×Q₈×C₁₇₃	C₂
socle, prime radical	C₂²×C₁₇₃	C₂×D₄
hypercenter	C₄×Q₈	D₁₇₃
hypocenter	C₁₇₃	#462
LCS	G ▷ C₂×C₄×C₁₇₃ ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
31. C₄×C₁₇₃	Q₁₆	semidirect
38. C₁₇₃	#462	semidirect
71. C₄	#14583816	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583816	2
#19149679	2
C₄⋊#13063436	2
#19149602	2
C₄×Q₈×C₁₇₃	2
#462	173