Subgroups — #5113

Order 192.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#235	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#926	C₂	1	1	yes	maximal, characteristic, complemented
4	#821	C₂	1	2	yes	maximal, complemented
5	#816	C₂	1	2	yes	maximal, complemented
6	#416	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
8	C₂×#67	—	2	1	no	—
9	#235	—	2	2	no	—
10	#235	C₄	1	4	yes	complemented
11	C₂×D₁₂	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₃	—	2	1	no	—
13	C₂×Dic₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈	—	2	2	no	—
15	C₂×C₃×C₈	C₂²	1	2	yes	—
16	#217	—	2	1	no	—
17	C₄⋊Dic₃	C₂²	1	2	yes	—
18	C₄×Dic₃	—	2	1	no	—
19	C₂×#41	—	3	1	no	—
20	C₂²×C₈	D₃	1	1	yes	characteristic, complemented
21	#117	—	3	1	no	—
22	#103	—	3	2	no	—
23	#102	—	3	2	no	—
24	C₂³×C₃	D₄	1	1	yes	characteristic
25	C₂²×D₃	—	2	1	no	—
26	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
27	C₂×C₃×C₄	D₄	1	1	yes	characteristic
28	C₂×C₃×C₄	D₄	1	2	yes	—
29	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
30	#67	—	2	4	no	—
31	C₂×Dic₃	—	2	1	no	—
32	C₂×Dic₃	—	2	2	no	—
33	D₁₂	—	2	1	no	—
34	D₁₂	C₂×C₄	1	2	yes	—
35	C₄×D₃	—	2	4	no	—
36	Dic₆	—	2	1	no	—
37	Dic₆	C₂×C₄	1	2	yes	—
38	C₃×C₈	—	2	2	no	—
39	#41	—	3	4	no	—
40	#41	—	6	2	no	—
41	C₂×Q₈	—	3	1	no	—
42	C₂×D₄	—	3	1	no	—
43	C₂×D₄	—	6	1	no	—
44	C₂²×C₄	—	6	1	no	—
45	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₂×C₈	—	2	2	no	—
47	C₂×C₈	C₂×D₃	1	2	yes	—
48	C₄⋊C₄	—	3	2	no	—
49	C₂²⋊C₄	—	6	1	no	—
50	C₄²	—	6	1	no	—
51	C₂²×C₃	—	2	2	no	—
52	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
53	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
54	C₂×D₃	—	2	2	no	—
55	C₂×D₃	—	4	2	no	—
56	C₃×C₄	C₂²⋊C₄	1	2	yes	—
57	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	Dic₃	—	2	2	no	—
60	Dic₃	—	4	2	no	—
61	C₂³	—	6	1	no	—
62	C₂³	D₁₂	1	1	yes	characteristic
63	Q₈	—	3	2	no	—
64	Q₈	—	6	1	no	—
65	D₄	—	3	2	no	—
66	D₄	—	6	4	no	—
67	D₄	—	6	1	no	—
68	C₂×C₄	—	6	1	no	—
69	C₂×C₄	—	6	4	no	—
70	C₂×C₄	—	6	2	no	—
71	C₂×C₄	C₄×D₃	1	2	yes	—
72	C₂×C₄	D₁₂	1	1	yes	characteristic
73	C₂×C₄	#67	1	2	yes	—
74	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₂×C₃	—	2	2	no	—
77	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₃	#135	1	2	yes	—
79	D₃	—	4	2	no	—
80	C₂²	—	2	2	no	—
81	C₂²	—	6	2	no	—
82	C₂²	—	12	2	no	—
83	C₂²	(C₂×D₃)⋊C₄	1	2	yes	—
84	C₂²	C₂×D₁₂	1	1	yes	characteristic
85	C₄	—	6	2	no	—
86	C₄	—	12	2	no	—
87	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
88	C₄	C₂×C₄×D₃	1	1	yes	characteristic
89	C₄	C₂×#67	1	1	yes	characteristic
90	C₃	#416	1	1	yes	characteristic, complemented, Sylow
91	C₂	—	2	2	no	—
92	C₂	—	12	2	no	—
93	C₂	#904	1	2	yes	—
94	C₂	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₂²×C₃×C₈	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×#235	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂²×C₈	D₃
hypocenter	C₃	#416
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×C₈	C₂	semidirect
3. #926	C₂	semidirect
4. #821	C₂	semidirect
5. #816	C₂	semidirect
10. #235	C₄	semidirect
20. C₂²×C₈	D₃	semidirect
90. C₃	#416	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
C₂²×C₃×C₈	2
#926	2
#821	2
#816	2
#416	3