Subgroups — #26012456

Order 11072.

130 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583836	C₂	1	1	yes	maximal, characteristic
2	C₂×D₁₇₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19149734	C₂	1	2	yes	maximal, complemented
4	C₄×#13063443	C₂	1	2	yes	maximal, complemented
5	#19149637	C₂	1	1	yes	maximal, characteristic
6	#14583836	C₄	1	4	yes	complemented
7	D₁₇₃⋊C₈	C₄	1	4	yes	complemented
8	C₂×#13063443	C₂²	1	1	yes	characteristic
9	C₂×#13063443	C₂²	1	2	yes	—
10	C₂×#13063443	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₇₃	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₁₇₃	C₄	1	1	yes	characteristic
13	#14583803	C₄	1	2	yes	—
14	C₄×Dic₁₇₃	C₂²	1	2	yes	—
15	C₄⋊C₄×C₁₇₃	C₄	1	1	yes	characteristic
16	#13063443	C₂×C₄	1	2	yes	—
17	#13063443	C₂×C₄	1	2	yes	—
18	#13063443	C₂×C₄	1	4	yes	—
19	C₂²×D₁₇₃	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₁₇₃	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₇₃	C₂³	1	1	yes	characteristic
22	C₄×D₁₇₃	C₂×C₄	1	2	yes	—
23	C₄×D₁₇₃	C₂×C₄	1	2	yes	—
24	C₂×C₄×C₁₇₃	C₂×C₄	1	2	yes	—
25	C₂×C₄×C₁₇₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₁₇₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₁₇₃	—	2	1	no	—
28	C₂×D₁₇₃	—	2	1	no	—
29	C₂×D₁₇₃	C₄²	1	2	yes	—
30	C₄×C₁₇₃	—	2	2	no	—
31	C₄×C₁₇₃	C₄²	1	2	yes	—
32	Dic₁₇₃	—	2	2	no	—
33	Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
34	Dic₁₇₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₁₇₃	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₁₇₃	#131	1	1	yes	characteristic
37	C₂×C₁₇₃	#131	1	1	yes	characteristic
38	D₁₇₃	—	2	2	no	—
39	C₁₇₃	#405	1	1	yes	characteristic, complemented, Sylow
40	#405	—	173	1	no	maximal, Sylow
41	C₂×M₁₆	—	173	1	no	—
42	C₂²×C₈	—	173	1	no	—
43	#117	—	173	1	no	—
44	#97	—	173	2	no	—
45	C₄×C₈	—	173	2	no	—
46	C₂²×C₄	—	173	1	no	—
47	M₁₆	—	173	4	no	—
48	C₂×C₈	—	173	1	no	—
49	C₂×C₈	—	173	2	no	—
50	C₂×C₈	—	173	1	no	—
51	C₂×C₈	—	173	4	no	—
52	C₄⋊C₄	—	173	1	no	—
53	C₄⋊C₄	#10621924	1	1	yes	characteristic
54	C₂²⋊C₄	—	173	2	no	—
55	C₄²	—	173	2	no	—
56	C₂³	—	173	1	no	—
57	C₂×C₄	—	173	2	no	—
58	C₂×C₄	—	173	1	no	—
59	C₂×C₄	—	173	2	no	—
60	C₂×C₄	—	173	2	no	—
61	C₂×C₄	C₂×#10621924	1	2	yes	—
62	C₂×C₄	C₂×#10621924	1	1	yes	characteristic
63	C₈	—	173	2	no	—
64	C₈	—	173	2	no	—
65	C₈	—	173	4	no	—
66	C₂²	—	346	1	no	—
67	C₂²	—	346	1	no	—
68	C₂²	—	173	2	no	—
69	C₂²	C₂²×#10621924	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	346	2	no	—
72	C₄	—	173	1	no	—
73	C₄	—	173	1	no	—
74	C₄	C₄×#10621924	1	2	yes	—
75	C₂	—	346	2	no	—
76	C₂	C₂×C₄×#10621924	1	1	yes	characteristic
77	C₂	#19149766	1	1	yes	characteristic
78	C₂	#19149767	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×#10621924
commutator	C₂×C₁₇₃	C₂×C₄²
Fitting	C₄⋊C₄×C₁₇₃	C₄
socle	C₂²×C₁₇₃	C₂²×C₄
prime radical	C₂²×D₁₇₃	C₂×C₄
cosocle	C₂×Dic₁₇₃	C₂³
hypercenter	C₄⋊C₄	#10621924
hypocenter	C₁₇₃	#405
LCS	G ▷ C₂×C₁₇₃ ▷ C₁₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #19149734	C₂	semidirect
4. C₄×#13063443	C₂	semidirect
6. #14583836	C₄	semidirect
7. D₁₇₃⋊C₈	C₄	semidirect
39. C₁₇₃	#405	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583836	2
C₂×D₁₇₃⋊C₈	2
#19149734	2
C₄×#13063443	2
#19149637	2
#405	173