Subgroups — #5535

Order 192.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#117	C₂	1	1	yes	maximal, characteristic, complemented
2	#930	C₂	1	2	yes	maximal, complemented
3	#894	C₂	1	4	yes	maximal, complemented
4	#884	C₂	1	4	yes	maximal, complemented
5	#874	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₁₂	C₂	1	2	yes	maximal, complemented
7	#870	C₂	1	1	yes	maximal, characteristic, complemented
8	#540	—	3	1	no	maximal, Sylow
9	C₂²×C₃×C₄	C₂²	1	1	yes	characteristic
10	C₂×#67	C₂²	1	4	yes	—
11	C₂×D₁₂	C₂²	1	2	yes	—
12	C₂×D₁₂	C₂²	1	4	yes	—
13	C₂×C₄×D₃	C₂²	1	4	yes	—
14	C₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₃×C₂²⋊C₄	C₂²	1	2	yes	—
16	C₃×C₄²	C₂²	1	1	yes	characteristic
17	C₃×C₄²	C₂²	1	1	yes	characteristic
18	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
19	(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
20	C₄⋊Dic₃	C₂²	1	2	yes	—
21	Dic₃⋊C₄	C₂²	1	4	yes	—
22	#127	—	3	1	no	—
23	#125	—	3	1	no	—
24	#123	—	3	4	no	—
25	#121	—	3	2	no	—
26	#121	—	3	4	no	—
27	C₄×D₄	—	3	2	no	—
28	#117	D₃	1	1	yes	characteristic, complemented
29	C₂³×C₃	C₂³	1	1	yes	characteristic
30	C₂²×D₃	C₂³	1	4	yes	—
31	C₂×C₃×C₄	—	2	2	no	—
32	C₂×C₃×C₄	C₂³	1	2	yes	—
33	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
34	C₂×C₃×C₄	C₂³	1	2	yes	—
35	C₂×C₃×C₄	C₂³	1	1	yes	characteristic
36	#67	—	4	4	no	—
37	C₂×Dic₃	C₂³	1	4	yes	—
38	D₁₂	—	2	4	no	—
39	D₁₂	—	4	4	no	—
40	C₄×D₃	—	4	4	no	—
41	C₂×D₄	—	3	4	no	—
42	C₂×D₄	—	3	2	no	—
43	C₂×D₄	—	3	4	no	—
44	C₂²×C₄	—	3	4	no	—
45	C₂²×C₄	C₂×D₃	1	1	yes	characteristic
46	C₄⋊C₄	—	3	2	no	—
47	C₄⋊C₄	—	3	4	no	—
48	C₄⋊C₄	C₂×D₃	1	2	yes	—
49	C₂²⋊C₄	—	3	4	no	—
50	C₂²⋊C₄	—	3	4	no	—
51	C₂²⋊C₄	C₂×D₃	1	2	yes	—
52	C₄²	C₂×D₃	1	1	yes	characteristic
53	C₄²	C₂×D₃	1	1	yes	characteristic
54	C₂²×C₃	—	2	1	no	—
55	C₂²×C₃	—	2	2	no	—
56	C₂²×C₃	C₂⁴	1	1	yes	characteristic
57	C₂×D₃	—	2	4	no	—
58	C₂×D₃	—	2	4	no	—
59	C₂×D₃	—	2	4	no	—
60	C₃×C₄	—	2	1	no	—
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	—	2	2	no	—
63	C₃×C₄	#41	1	2	yes	—
64	Dic₃	—	2	4	no	—
65	C₂³	—	3	4	no	—
66	C₂³	C₂²×D₃	1	1	yes	characteristic
67	D₄	—	6	4	no	—
68	D₄	—	12	4	no	—
69	D₄	—	12	4	no	—
70	C₂×C₄	—	2	2	no	—
71	C₂×C₄	—	3	4	no	—
72	C₂×C₄	—	12	4	no	—
73	C₂×C₄	C₂²×D₃	1	2	yes	—
74	C₂×C₄	C₂²×D₃	1	2	yes	—
75	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂×C₃	—	4	1	no	—
78	C₂×C₃	C₂×#41	1	1	yes	characteristic
79	C₂×C₃	H(5,𝔽₂)	1	2	yes	—
80	D₃	—	4	4	no	—
81	C₂²	—	2	1	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	6	4	no	—
84	C₂²	—	6	4	no	—
85	C₂²	—	6	4	no	—
86	C₂²	C₂³×D₃	1	1	yes	characteristic
87	C₄	—	2	1	no	—
88	C₄	—	2	2	no	—
89	C₄	—	2	2	no	—
90	C₄	—	6	4	no	—
91	C₄	#235	1	2	yes	—
92	C₃	#540	1	1	yes	characteristic, complemented, Sylow
93	C₂	—	4	1	no	—
94	C₂	—	12	4	no	—
95	C₂	C₂×#235	1	1	yes	characteristic
96	C₂	#1009	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁴
Fitting	C₃×#117	C₂
hypercenter	#117	D₃
hypocenter	C₃	#540
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. C₃×#117	C₂	semidirect
2. #930	C₂	semidirect
3. #894	C₂	semidirect
4. #884	C₂	semidirect
5. #874	C₂	semidirect
6. C₄×D₁₂	C₂	semidirect
7. #870	C₂	semidirect
28. #117	D₃	semidirect
92. C₃	#540	semidirect

Maximal subgroups

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

subgroup	index
C₃×#117	2
#930	2
#894	2
#884	2
#874	2
C₄×D₁₂	2
#870	2
#540	3