Subgroups — #67539

Order 384.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#5071	C₂	1	1	yes	maximal, characteristic
2	C₃×#353	C₂	1	1	yes	maximal, characteristic
3	#4545	C₂	1	1	yes	maximal, characteristic
4	#1412	—	3	1	no	maximal, Sylow
5	C₃×#127	C₂²	1	1	yes	characteristic
6	#870	C₄	1	1	yes	characteristic, complemented
7	C₃×#99	—	2	1	no	—
8	#834	—	2	1	no	—
9	#832	—	2	1	no	—
10	#832	—	2	1	no	—
11	#803	C₄	1	1	yes	characteristic, complemented
12	#490	—	3	1	no	—
13	#353	—	3	1	no	—
14	#353	D₃	1	1	yes	characteristic
15	C₂×C₃×D₄	—	4	1	no	—
16	C₂×C₃×D₄	D₄	1	1	yes	characteristic
17	C₂×C₃×D₄	D₄	1	1	yes	characteristic
18	C₃×C₂²⋊C₄	—	4	1	no	—
19	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
20	#217	—	4	1	no	—
21	C₄⋊Dic₃	—	2	1	no	—
22	Dic₃⋊C₄	—	4	1	no	—
23	C₂×C₃⋊C₈	—	2	1	no	—
24	#127	C₂×D₃	1	1	yes	characteristic
25	#125	—	3	1	no	—
26	#102	—	6	1	no	—
27	#102	—	6	1	no	—
28	#99	—	2	1	no	—
29	#99	—	6	1	no	—
30	#97	—	3	1	no	—
31	C₂³×C₃	—	2	1	no	—
32	C₂³×C₃	—	2	1	no	—
33	C₃×D₄	—	4	1	no	—
34	C₃×D₄	—	4	1	no	—
35	C₃×D₄	—	8	2	no	—
36	C₂×C₃×C₄	—	2	1	no	—
37	C₂×C₃×C₄	—	8	1	no	—
38	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×Dic₃	—	4	1	no	—
40	C₂×Dic₃	—	8	1	no	—
41	C₃⋊C₈	—	4	1	no	—
42	C₂×D₄	—	4	1	no	—
43	C₂×D₄	D₁₂	1	1	yes	characteristic
44	C₂×D₄	#67	1	1	yes	characteristic
45	C₂×C₈	—	6	1	no	—
46	C₄⋊C₄	—	6	1	no	—
47	C₄⋊C₄	—	12	1	no	—
48	C₂²⋊C₄	—	4	1	no	—
49	C₂²⋊C₄	—	12	1	no	—
50	C₄²	C₄×D₃	1	1	yes	characteristic
51	C₂²×C₃	—	4	1	no	—
52	C₂²×C₃	—	4	1	no	—
53	C₂²×C₃	—	8	1	no	—
54	C₂²×C₃	—	8	1	no	—
55	C₂²×C₃	#99	1	1	yes	characteristic
56	C₃×C₄	—	2	1	no	—
57	C₃×C₄	—	4	1	no	—
58	C₃×C₄	—	16	1	no	—
59	Dic₃	—	8	1	no	—
60	Dic₃	—	16	1	no	—
61	C₂³	—	2	1	no	—
62	C₂³	—	2	1	no	—
63	D₄	—	4	1	no	—
64	D₄	—	4	1	no	—
65	D₄	—	8	2	no	—
66	C₂×C₄	—	2	1	no	—
67	C₂×C₄	—	8	1	no	—
68	C₂×C₄	—	12	1	no	—
69	C₂×C₄	—	24	1	no	—
70	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
71	C₈	—	12	1	no	—
72	C₂×C₃	—	2	1	no	—
73	C₂×C₃	—	8	1	no	—
74	C₂×C₃	—	8	1	no	—
75	C₂×C₃	#354	1	1	yes	characteristic
76	C₂²	—	4	1	no	—
77	C₂²	—	4	1	no	—
78	C₂²	—	8	1	no	—
79	C₂²	—	8	1	no	—
80	C₂²	#806	1	1	yes	characteristic
81	C₄	—	2	1	no	—
82	C₄	—	4	1	no	—
83	C₄	—	16	1	no	—
84	C₄	—	24	1	no	—
85	C₄	—	48	1	no	—
86	C₃	#1412	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	2	1	no	—
88	C₂	—	8	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#4473	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4473
commutator	C₃×C₄²	C₂×C₄
Frattini	#127	C₂×D₃
Fitting	C₃×#353	C₂
socle	C₂×C₃	#354
prime radical, cosocle	C₃×#127	C₂²
hypercenter	#353	D₃
hypocenter	C₃	#1412
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #353

Complements

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

N	H ≅ G/N	type
6. #870	C₄	semidirect
11. #803	C₄	semidirect
86. C₃	#1412	semidirect

Maximal subgroups

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

subgroup	index
#5071	2
C₃×#353	2
#4545	2
#1412	3