Subgroups — #67521

Order 384.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4703	C₂	1	1	yes	maximal, characteristic
2	C₃×#347	C₂	1	1	yes	maximal, characteristic
3	#4466	C₂	1	1	yes	maximal, characteristic
4	#1356	—	3	1	no	maximal, Sylow
5	#870	C₄	1	1	yes	characteristic
6	#869	C₄	1	1	yes	characteristic
7	C₃×M₃₂	—	2	1	no	—
8	C₃×#97	C₂²	1	1	yes	characteristic
9	#816	—	2	1	no	—
10	#816	—	2	1	no	—
11	#812	—	2	1	no	—
12	#491	—	3	1	no	—
13	#347	—	3	1	no	—
14	#347	D₃	1	1	yes	characteristic
15	C₂×Dic₆	—	2	1	no	—
16	C₂×C₃×C₈	D₄	1	1	yes	characteristic
17	C₂×C₃×C₈	D₄	1	1	yes	characteristic
18	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
19	C₄⋊Dic₃	—	2	1	no	—
20	C₄⋊Dic₃	—	4	1	no	—
21	Dic₃⋊C₄	—	4	1	no	—
22	C₃×C₁₆	—	2	2	no	—
23	C₃⋊C₁₆	—	2	2	no	—
24	C₄⋊Q₈	—	3	1	no	—
25	#125	—	3	1	no	—
26	M₃₂	—	2	1	no	—
27	M₃₂	—	6	1	no	—
28	#103	—	6	1	no	—
29	#103	—	6	1	no	—
30	#97	C₂×D₃	1	1	yes	characteristic
31	C₂×C₃×C₄	—	2	1	no	—
32	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×Dic₃	—	4	1	no	—
34	C₂×Dic₃	—	4	1	no	—
35	Dic₆	—	4	1	no	—
36	Dic₆	—	4	1	no	—
37	C₃×C₈	—	2	1	no	—
38	C₃×C₈	—	2	1	no	—
39	C₂×Q₈	—	6	1	no	—
40	C₂×C₈	D₁₂	1	1	yes	characteristic
41	C₂×C₈	#67	1	1	yes	characteristic
42	C₄⋊C₄	—	6	1	no	—
43	C₄⋊C₄	—	12	1	no	—
44	C₄⋊C₄	—	12	1	no	—
45	C₄²	C₄×D₃	1	1	yes	characteristic
46	C₁₆	—	2	2	no	—
47	C₁₆	—	6	2	no	—
48	C₂²×C₃	#100	1	1	yes	characteristic
49	C₃×C₄	—	4	1	no	—
50	C₃×C₄	#104	1	1	yes	characteristic
51	C₃×C₄	#104	1	1	yes	characteristic
52	Dic₃	—	8	1	no	—
53	Dic₃	—	8	1	no	—
54	Q₈	—	12	1	no	—
55	Q₈	—	12	1	no	—
56	C₂×C₄	—	2	1	no	—
57	C₂×C₄	—	12	1	no	—
58	C₂×C₄	—	12	1	no	—
59	C₂×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
60	C₈	—	2	1	no	—
61	C₈	—	2	1	no	—
62	C₂×C₃	—	2	1	no	—
63	C₂×C₃	#329	1	1	yes	characteristic
64	C₂²	#823	1	1	yes	characteristic
65	C₄	—	4	1	no	—
66	C₄	—	24	1	no	—
67	C₄	—	24	1	no	—
68	C₄	#805	1	1	yes	characteristic
69	C₄	#825	1	1	yes	characteristic
70	C₃	#1356	1	1	yes	characteristic, complemented, Sylow
71	C₂	—	2	1	no	—
72	C₂	#4464	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#4464
commutator	C₃×C₄²	C₂×C₄
Frattini	#97	C₂×D₃
Fitting	C₃×#347	C₂
socle	C₂×C₃	#329
prime radical	C₂²×C₃	#100
cosocle	C₃×#97	C₂²
hypercenter	#347	D₃
hypocenter	C₃	#1356
LCS	G ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #97 ◁ #347

Complements

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

N	H ≅ G/N	type
70. C₃	#1356	semidirect

Maximal subgroups

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

subgroup	index
#4703	2
C₃×#347	2
#4466	2
#1356	3