Subgroups — #5672

Order 192.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#124	C₂	1	1	yes	maximal, characteristic
2	#944	C₂	1	2	yes	maximal, complemented
3	#935	C₂	1	2	yes	maximal
4	#879	C₂	1	4	yes	maximal, complemented
5	#878	C₂	1	4	yes	maximal, complemented
6	#870	C₂	1	2	yes	maximal, complemented
7	#563	—	3	1	no	maximal, Sylow
8	C₂×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×D₄	C₂²	1	1	yes	characteristic
10	C₂²×Dic₃	C₂²	1	2	yes	—
11	C₂×Dic₆	C₂²	1	2	yes	—
12	C₃×C₂²⋊C₄	C₂²	1	4	yes	—
13	C₃×C₄²	C₂²	1	1	yes	characteristic
14	#217	C₂²	1	4	yes	—
15	#217	C₂²	1	2	yes	—
16	C₄⋊Dic₃	C₂²	1	2	yes	—
17	C₄⋊Dic₃	C₂²	1	2	yes	—
18	Dic₃⋊C₄	C₂²	1	4	yes	—
19	Dic₃⋊C₄	C₂²	1	4	yes	—
20	Dic₃⋊C₄	C₂²	1	4	yes	—
21	C₄×Dic₃	C₂²	1	2	yes	—
22	C₄⋊Q₈	—	3	2	no	—
23	#126	—	3	4	no	—
24	#125	—	3	2	no	—
25	#124	D₃	1	1	yes	characteristic
26	#123	—	3	2	no	—
27	C₂²⋊Q₈	—	3	4	no	—
28	C₂³×C₃	C₂³	1	2	yes	—
29	C₃×Q₈	—	4	1	no	—
30	C₃×D₄	—	4	1	no	—
31	C₂×C₃×C₄	C₂³	1	2	yes	—
32	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×C₃×C₄	C₂³	1	2	yes	—
34	C₂×Dic₃	—	4	2	no	—
35	C₂×Dic₃	C₂³	1	4	yes	—
36	C₂×Dic₃	C₂³	1	4	yes	—
37	Dic₆	—	4	2	no	—
38	C₂×Q₈	—	3	2	no	—
39	C₂×Q₈	C₂×D₃	1	1	yes	characteristic
40	C₂×D₄	C₂×D₃	1	1	yes	characteristic
41	C₂²×C₄	—	3	2	no	—
42	C₄⋊C₄	—	3	2	no	—
43	C₄⋊C₄	—	3	4	no	—
44	C₄⋊C₄	—	3	4	no	—
45	C₄⋊C₄	—	3	4	no	—
46	C₄⋊C₄	—	3	2	no	—
47	C₂²⋊C₄	—	3	4	no	—
48	C₂²⋊C₄	—	3	2	no	—
49	C₂²⋊C₄	C₂×D₃	1	4	yes	—
50	C₄²	—	3	2	no	—
51	C₄²	C₂×D₃	1	1	yes	characteristic
52	C₂²×C₃	—	2	2	no	—
53	C₂²×C₃	—	2	4	no	—
54	C₂²×C₃	C₂⁴	1	1	yes	characteristic
55	C₃×C₄	—	2	2	no	—
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	—	2	2	no	—
58	Dic₃	—	2	4	no	—
59	Dic₃	—	2	4	no	—
60	C₂³	C₂²×D₃	1	2	yes	—
61	Q₈	—	4	1	no	—
62	Q₈	—	12	2	no	—
63	D₄	—	4	1	no	—
64	C₂×C₄	—	3	4	no	—
65	C₂×C₄	—	3	4	no	—
66	C₂×C₄	—	12	2	no	—
67	C₂×C₄	C₂²×D₃	1	2	yes	—
68	C₂×C₄	C₂²×D₃	1	2	yes	—
69	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₃	—	4	2	no	—
71	C₂×C₃	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₃	2₋¹⁺⁴	1	2	yes	—
73	C₂²	—	2	2	no	—
74	C₂²	—	2	4	no	—
75	C₂²	C₂³×D₃	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	6	4	no	—
80	C₄	—	6	4	no	—
81	C₃	#563	1	1	yes	characteristic, complemented, Sylow
82	C₂	—	4	2	no	—
83	C₂	#1004	1	1	yes	characteristic
84	C₂	#1010	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#124	C₂
prime radical	C₂×C₃×D₄	C₂²
hypercenter	#124	D₃
hypocenter	C₃	#563
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #124

Complements

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

N	H ≅ G/N	type
2. #944	C₂	semidirect
4. #879	C₂	semidirect
5. #878	C₂	semidirect
6. #870	C₂	semidirect
81. C₃	#563	semidirect

Maximal subgroups

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

subgroup	index
C₃×#124	2
#944	2
#935	2
#879	2
#878	2
#870	2
#563	3