Subgroups — #5002

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	C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#810	C₂	1	2	yes	maximal, complemented
5	#809	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	C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₂²⋊C₄	—	2	1	no	—
16	C₃×C₄²	—	2	1	no	—
17	C₂×C₃⋊C₈	—	2	2	no	—
18	C₂×C₃⋊C₈	C₂²	1	2	yes	—
19	C₂×#41	—	3	1	no	—
20	C₂²×C₈	—	3	1	no	—
21	#117	D₃	1	1	yes	characteristic, complemented
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	—	2	2	no	—
27	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
28	C₂×C₃×C₄	D₄	1	2	yes	—
29	C₂×C₃×C₄	D₄	1	1	yes	characteristic
30	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
31	#67	—	2	4	no	—
32	C₂×Dic₃	—	2	1	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	—	3	2	no	—
47	C₂×C₈	—	6	2	no	—
48	C₄⋊C₄	C₂×D₃	1	2	yes	—
49	C₂²⋊C₄	—	2	1	no	—
50	C₄²	—	2	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	—	4	2	no	—
57	C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	Dic₃	—	2	2	no	—
61	C₂³	—	6	1	no	—
62	C₂³	#67	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	—	2	2	no	—
69	C₂×C₄	—	6	1	no	—
70	C₂×C₄	—	6	4	no	—
71	C₂×C₄	C₄×D₃	1	2	yes	—
72	C₂×C₄	D₁₂	1	2	yes	—
73	C₂×C₄	#67	1	1	yes	characteristic
74	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₈	—	6	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×#67	1	1	yes	characteristic
85	C₄	—	4	2	no	—
86	C₄	—	6	2	no	—
87	C₄	(C₂×D₃)⋊C₄	1	2	yes	—
88	C₄	C₂×C₄×D₃	1	1	yes	characteristic
89	C₄	C₂×D₁₂	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	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
94	C₂	#950	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#67
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting	C₃×#117	C₂
socle	C₂²×C₃	C₂×D₄
prime radical	C₂×#235	C₂
cosocle	C₂×C₃×C₄	C₂³
hypercenter	#117	D₃
hypocenter	C₃	#416
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₃×#117	C₂	semidirect
3. C₂²×C₃⋊C₈	C₂	semidirect
4. #810	C₂	semidirect
5. #809	C₂	semidirect
10. #235	C₄	semidirect
21. #117	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
C₃×#117	2
C₂²×C₃⋊C₈	2
#810	2
#809	2
#416	3