Subgroups — #69806

Order 384.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#97	C₂	1	1	yes	maximal, characteristic
2	#4932	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	4	yes	maximal
5	#1746	—	3	1	no	maximal, Sylow
6	C₂²×C₃×C₈	C₂²	1	2	yes	—
7	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₃	C₂²	1	2	yes	—
9	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
10	C₃×#97	C₄	1	4	yes	complemented
11	C₃⋊#95	C₄	1	4	yes	—
12	C₄×C₃⋊C₈	C₄	1	4	yes	complemented
13	C₂×#97	D₃	1	1	yes	characteristic
14	C₂×C₄×C₈	—	3	1	no	—
15	#376	—	3	1	no	—
16	#336	—	3	4	no	—
17	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
19	C₂²×Dic₃	C₂×C₄	1	4	yes	—
20	C₂×C₃×C₈	—	2	4	no	—
21	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
22	C₃×C₄²	C₂×C₄	1	2	yes	—
23	C₃×C₄²	C₂×C₄	1	2	yes	—
24	C₄×Dic₃	—	2	4	no	—
25	C₂×C₃⋊C₈	—	2	4	no	—
26	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₈	—	3	2	no	—
28	C₂²×C₈	C₂×D₃	1	2	yes	—
29	C₂×C₄²	—	3	2	no	—
30	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
31	#97	Dic₃	1	4	yes	complemented
32	C₄×C₈	—	3	4	no	—
33	#95	—	3	4	no	—
34	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃×C₄	—	2	4	no	—
36	C₂×C₃×C₄	C₄²	1	4	yes	—
37	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄	#41	1	4	yes	—
40	C₂×Dic₃	—	2	4	no	—
41	C₂×Dic₃	—	2	8	no	—
42	C₃×C₈	—	2	4	no	—
43	C₃⋊C₈	—	2	4	no	—
44	C₂²×C₄	—	3	4	no	—
45	C₂²×C₄	C₄×D₃	1	2	yes	—
46	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
47	C₂×C₈	—	2	4	no	—
48	C₂×C₈	—	3	4	no	—
49	C₂×C₈	—	6	4	no	—
50	C₂×C₈	C₂×Dic₃	1	4	yes	—
51	C₄²	—	6	4	no	—
52	C₄²	C₄×D₃	1	2	yes	—
53	C₄²	C₂×Dic₃	1	2	yes	—
54	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
55	C₂²×C₃	#117	1	2	yes	—
56	C₂²×C₃	#131	1	4	yes	—
57	C₃×C₄	—	2	4	no	—
58	C₃×C₄	#117	1	2	yes	—
59	C₃×C₄	#117	1	2	yes	—
60	Dic₃	—	4	4	no	—
61	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
62	C₂×C₄	—	2	4	no	—
63	C₂×C₄	—	6	4	no	—
64	C₂×C₄	—	6	8	no	—
65	C₂×C₄	C₄×Dic₃	1	4	yes	—
66	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
67	C₂×C₄	#235	1	4	yes	—
68	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
69	C₈	—	2	4	no	—
70	C₈	—	6	4	no	—
71	C₂×C₃	#376	1	1	yes	characteristic
72	C₂×C₃	#405	1	2	yes	—
73	C₂×C₃	#433	1	4	yes	—
74	C₂²	#872	1	2	yes	—
75	C₂²	#907	1	4	yes	—
76	C₂²	C₂×C₄×Dic₃	1	1	yes	characteristic
77	C₄	—	2	4	no	—
78	C₄	—	12	4	no	—
79	C₄	#872	1	2	yes	—
80	C₄	#926	1	2	yes	—
81	C₃	#1746	1	1	yes	characteristic, complemented, Sylow
82	C₂	#4709	1	4	yes	—
83	C₂	#4932	1	1	yes	characteristic
84	C₂	#5122	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₂×C₃×#97	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#97	D₃
hypocenter	C₃	#1746
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#97

Complements

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

N	H ≅ G/N	type
10. C₃×#97	C₄	semidirect
12. C₄×C₃⋊C₈	C₄	semidirect
31. #97	Dic₃	semidirect
81. C₃	#1746	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×#97	2
#4932	2
C₂×C₄×C₃⋊C₈	2
#4550	2
#1746	3