Subgroups — #5510

Order 192.

294 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#235	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#930	C₂	1	4	yes	maximal, complemented
4	#897	C₂	1	4	yes	maximal, complemented
5	#895	C₂	1	4	yes	maximal, complemented
6	#537	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	2	yes	—
8	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
9	C₂×#67	C₂²	1	4	yes	—
10	#235	—	2	8	no	—
11	C₂×D₁₂	C₂²	1	2	yes	—
12	C₂×D₁₂	C₂²	1	4	yes	—
13	C₂×C₄×D₃	C₂²	1	4	yes	—
14	C₂×Dic₆	C₂²	1	2	yes	—
15	C₃×C₄⋊C₄	C₂²	1	4	yes	—
16	(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
17	C₄⋊Dic₃	C₂²	1	4	yes	—
18	C₂×#41	—	3	2	no	—
19	C₂²⋊Q₈	—	3	4	no	—
20	#121	—	3	4	no	—
21	#121	—	3	4	no	—
22	C₂×C₄⋊C₄	D₃	1	1	yes	characteristic, complemented
23	C₂³×C₃	C₂³	1	1	yes	characteristic
24	C₂²×D₃	C₂³	1	4	yes	—
25	C₂×C₃×C₄	—	4	2	no	—
26	C₂×C₃×C₄	D₄	1	4	yes	complemented
27	C₂×C₃×C₄	C₂³	1	4	yes	—
28	C₂×C₃×C₄	C₂³	1	2	yes	—
29	#67	—	2	8	no	—
30	C₂×Dic₃	C₂³	1	4	yes	—
31	D₁₂	—	2	4	no	—
32	D₁₂	—	4	4	no	—
33	C₄×D₃	—	2	8	no	—
34	Dic₆	—	2	4	no	—
35	#41	—	6	8	no	—
36	C₂×Q₈	—	3	2	no	—
37	C₂×D₄	—	3	4	no	—
38	C₂×D₄	—	3	2	no	—
39	C₂×D₄	—	3	4	no	—
40	C₂²×C₄	—	3	4	no	—
41	C₂²×C₄	C₂×D₃	1	2	yes	—
42	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
43	C₄⋊C₄	—	3	4	no	—
44	C₄⋊C₄	C₂×D₃	1	4	yes	—
45	C₂²⋊C₄	—	3	8	no	—
46	C₂²×C₃	—	2	1	no	—
47	C₂²×C₃	—	2	1	no	—
48	C₂²×C₃	C₂×D₄	1	2	yes	—
49	C₂²×C₃	C₂⁴	1	1	yes	characteristic
50	C₂×D₃	—	2	4	no	—
51	C₂×D₃	—	2	4	no	—
52	C₂×D₃	—	2	4	no	—
53	C₃×C₄	—	2	4	no	—
54	C₃×C₄	C₂×D₄	1	4	yes	—
55	Dic₃	—	2	4	no	—
56	C₂³	—	3	4	no	—
57	C₂³	C₂²×D₃	1	1	yes	characteristic
58	Q₈	—	6	4	no	—
59	D₄	—	6	8	no	—
60	D₄	—	6	4	no	—
61	D₄	—	12	4	no	—
62	C₂×C₄	—	3	4	no	—
63	C₂×C₄	—	4	2	no	—
64	C₂×C₄	—	6	8	no	—
65	C₂×C₄	D₁₂	1	4	yes	complemented
66	C₂×C₄	C₂²×D₃	1	4	yes	—
67	C₂×C₄	C₂²×D₃	1	2	yes	—
68	C₂×C₃	—	2	2	no	—
69	C₂×C₃	C₂²×D₄	1	1	yes	characteristic
70	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
71	C₂×C₃	2₋¹⁺⁴	1	1	yes	characteristic
72	D₃	—	4	4	no	—
73	C₂²	—	2	1	no	—
74	C₂²	—	2	1	no	—
75	C₂²	—	6	4	no	—
76	C₂²	—	6	4	no	—
77	C₂²	—	6	4	no	—
78	C₂²	C₂×D₁₂	1	2	yes	—
79	C₂²	C₂³×D₃	1	1	yes	characteristic
80	C₄	—	2	4	no	—
81	C₄	—	6	4	no	—
82	C₄	C₂×D₁₂	1	4	yes	—
83	C₃	#537	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	2	2	no	—
85	C₂	—	12	4	no	—
86	C₂	C₂²×D₁₂	1	1	yes	characteristic
87	C₂	#1004	1	1	yes	characteristic
88	C₂	#1007	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₂×C₃×C₄⋊C₄	C₂
hypercenter	C₂×C₄⋊C₄	D₃
hypocenter	C₃	#537
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×#235	C₂	semidirect
2. C₂×C₃×C₄⋊C₄	C₂	semidirect
3. #930	C₂	semidirect
4. #897	C₂	semidirect
5. #895	C₂	semidirect
22. C₂×C₄⋊C₄	D₃	semidirect
26. C₂×C₃×C₄	D₄	semidirect
65. C₂×C₄	D₁₂	semidirect
83. C₃	#537	semidirect

Maximal subgroups

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

subgroup	index
C₂×#235	2
C₂×C₃×C₄⋊C₄	2
#930	2
#897	2
#895	2
#537	3