Subgroups — #83174

Order 384.

648 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#127	C₂	1	1	yes	maximal, characteristic
2	#5220	C₂	1	12	yes	maximal, complemented
3	#4937	C₂	1	2	yes	maximal, complemented
4	#2380	—	3	1	no	maximal, Sylow
5	C₂²×C₃×D₄	C₂²	1	6	yes	—
6	C₂³×Dic₃	C₂²	1	4	yes	—
7	C₃×#127	C₄	1	8	yes	complemented
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×#217	C₂²	1	12	yes	—
10	C₃⋊#95	C₂²	1	12	yes	—
11	C₂×#127	D₃	1	1	yes	characteristic
12	#386	—	3	12	no	—
13	#383	—	3	2	no	—
14	C₂⁴×C₃	C₂³	1	4	yes	—
15	C₂×C₃×D₄	—	4	6	no	—
16	C₂×C₃×D₄	—	4	6	no	—
17	C₂×C₃×D₄	C₂×C₄	1	24	yes	complemented
18	C₂²×C₃×C₄	C₂³	1	3	yes	—
19	C₂²×Dic₃	—	4	12	no	—
20	C₂²×Dic₃	C₂³	1	8	yes	—
21	C₃×C₄²	C₂×C₄	1	4	yes	complemented
22	#217	—	4	12	no	—
23	C₂²×D₄	C₂×D₃	1	6	yes	—
24	C₂³×C₄	—	3	4	no	—
25	#127	Dic₃	1	8	yes	complemented
26	C₂×C₂²⋊C₄	—	3	12	no	—
27	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
28	#95	—	3	12	no	—
29	C₂³×C₃	—	2	12	no	—
30	C₂³×C₃	—	2	12	no	—
31	C₂³×C₃	C₂²×C₄	1	8	yes	—
32	C₂³×C₃	C₂⁴	1	1	yes	characteristic
33	C₃×D₄	—	4	24	no	—
34	C₂×C₃×C₄	—	2	6	no	—
35	C₂×C₃×C₄	C₂²×C₄	1	6	yes	—
36	C₂×Dic₃	—	2	24	no	—
37	C₂×Dic₃	—	4	16	no	—
38	C₂⁴	C₂²×D₃	1	4	yes	—
39	C₂×D₄	—	4	6	no	—
40	C₂×D₄	—	4	6	no	—
41	C₂×D₄	C₂×Dic₃	1	24	yes	complemented
42	C₂²×C₄	—	3	8	no	—
43	C₂²×C₄	—	12	12	no	—
44	C₂²×C₄	C₂²×D₃	1	3	yes	—
45	C₂²⋊C₄	—	12	12	no	—
46	C₄²	C₂×Dic₃	1	4	yes	complemented
47	C₂²×C₃	—	2	24	no	—
48	C₂²×C₃	—	4	4	no	—
49	C₂²×C₃	—	4	12	no	—
50	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
51	C₂²×C₃	H(5,𝔽₂)	1	6	yes	—
52	C₃×C₄	—	2	6	no	—
53	Dic₃	—	4	8	no	—
54	C₂³	—	2	12	no	—
55	C₂³	—	2	12	no	—
56	C₂³	C₂²×Dic₃	1	8	yes	—
57	C₂³	C₂³×D₃	1	1	yes	characteristic
58	D₄	—	4	24	no	—
59	C₂×C₄	—	2	6	no	—
60	C₂×C₄	—	6	24	no	—
61	C₂×C₄	—	12	16	no	—
62	C₂×C₄	C₂²×Dic₃	1	6	yes	—
63	C₂×C₃	—	4	8	no	—
64	C₂×C₃	#518	1	3	yes	—
65	C₂×C₃	#560	1	4	yes	—
66	C₂²	—	2	24	no	—
67	C₂²	—	4	4	no	—
68	C₂²	—	4	12	no	—
69	C₂²	#1004	1	6	yes	—
70	C₂²	C₂³×Dic₃	1	1	yes	characteristic
71	C₄	—	2	6	no	—
72	C₄	—	12	8	no	—
73	C₃	#2380	1	1	yes	characteristic, complemented, Sylow
74	C₂	—	4	8	no	—
75	C₂	#5720	1	4	yes	—
76	C₂	#5798	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting, prime radical	C₂×C₃×#127	C₂
socle, cosocle	C₂³×C₃	C₂⁴
hypercenter	C₂×#127	D₃
hypocenter	C₃	#2380
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
2. #5220	C₂	semidirect
3. #4937	C₂	semidirect
7. C₃×#127	C₄	semidirect
17. C₂×C₃×D₄	C₂×C₄	semidirect
21. C₃×C₄²	C₂×C₄	semidirect
25. #127	Dic₃	semidirect
41. C₂×D₄	C₂×Dic₃	semidirect
46. C₄²	C₂×Dic₃	semidirect
73. C₃	#2380	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#127	2
#5220	2
#4937	2
#2380	3