Subgroups — #5233

Order 192.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#41	C₂	1	1	yes	maximal, characteristic
2	#926	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#835	C₂	1	2	yes	maximal, complemented
5	#832	C₂	1	2	yes	maximal, complemented
6	#416	—	3	1	no	maximal, Sylow
7	C₃×#41	—	2	2	no	—
8	C₃×#41	C₄	1	4	yes	complemented
9	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₃×D₄	—	2	1	no	—
11	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄	—	2	1	no	—
13	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
14	#217	—	2	1	no	—
15	C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₄×Dic₃	—	2	1	no	—
17	C₂×C₃⋊C₈	—	2	2	no	—
18	C₂×C₃⋊C₈	C₂²	1	2	yes	—
19	C₂×#41	D₃	1	1	yes	characteristic
20	C₂²×C₈	—	3	1	no	—
21	#117	—	3	1	no	—
22	#103	—	3	2	no	—
23	#102	—	3	2	no	—
24	C₂³×C₃	—	2	1	no	—
25	C₂³×C₃	D₄	1	1	yes	characteristic
26	C₃×Q₈	—	2	1	no	—
27	C₃×Q₈	C₂×C₄	1	2	yes	—
28	C₃×D₄	—	2	4	no	—
29	C₃×D₄	—	2	1	no	—
30	C₃×D₄	C₂×C₄	1	2	yes	—
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	—	2	4	no	—
33	C₂×C₃×C₄	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₄	D₄	1	2	yes	—
35	C₂×C₃×C₄	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂×Dic₃	—	2	2	no	—
38	C₃⋊C₈	—	2	2	no	—
39	#41	—	2	2	no	—
40	#41	Dic₃	1	4	yes	complemented
41	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
42	C₂×D₄	—	2	1	no	—
43	C₂×D₄	C₂×D₃	1	1	yes	characteristic
44	C₂²×C₄	—	2	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	—	3	2	no	—
49	C₂²⋊C₄	—	6	1	no	—
50	C₄²	—	6	1	no	—
51	C₂²×C₃	—	2	2	no	—
52	C₂²×C₃	—	2	2	no	—
53	C₂²×C₃	—	4	2	no	—
54	C₂²×C₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₃	C₂×D₄	1	1	yes	characteristic
56	C₃×C₄	—	2	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	—	4	2	no	—
61	C₂³	—	2	1	no	—
62	C₂³	#67	1	1	yes	characteristic
63	Q₈	—	2	1	no	—
64	Q₈	C₂×Dic₃	1	2	yes	—
65	D₄	—	2	4	no	—
66	D₄	—	2	1	no	—
67	D₄	C₂×Dic₃	1	2	yes	—
68	C₂×C₄	—	2	1	no	—
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	6	2	no	—
71	C₂×C₄	C₂×Dic₃	1	2	yes	—
72	C₂×C₄	#67	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	—	4	2	no	—
78	C₂×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₃	#135	1	2	yes	—
80	C₂²	—	2	2	no	—
81	C₂²	—	2	2	no	—
82	C₂²	—	4	2	no	—
83	C₂²	#217	1	2	yes	—
84	C₂²	C₂×#67	1	1	yes	characteristic
85	C₄	—	2	2	no	—
86	C₄	—	12	2	no	—
87	C₄	#217	1	2	yes	—
88	C₄	C₂²×Dic₃	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	—	4	2	no	—
93	C₂	#950	1	2	yes	—
94	C₂	C₂×#217	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#67
commutator	C₃×C₄	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃
Fitting, prime radical	C₂×C₃×#41	C₂
socle	C₂²×C₃	C₂×D₄
cosocle	C₂×C₃×C₄	C₂³
hypercenter	C₂×#41	D₃
hypocenter	C₃	#416
LCS	G ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
2. #926	C₂	semidirect
3. C₂²×C₃⋊C₈	C₂	semidirect
4. #835	C₂	semidirect
5. #832	C₂	semidirect
8. C₃×#41	C₄	semidirect
40. #41	Dic₃	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×C₃×#41	2
#926	2
C₂²×C₃⋊C₈	2
#835	2
#832	2
#416	3