Subgroups — #38734310

Order 24896.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Dic₁₅₅₆	C₂	1	2	yes	maximal
2	#27159384	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₇₇₈	C₂	1	1	yes	maximal, characteristic
4	#27159309	C₂	1	2	yes	maximal
5	C₄⋊C₈×C₃₈₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₇₈	C₂²	1	2	yes	—
7	C₂×Dic₇₇₈	C₂²	1	1	yes	characteristic
8	Dic₁₅₅₆	—	2	4	no	—
9	C₄⋊Dic₃₈₉	—	2	1	no	—
10	C₄⋊Dic₃₈₉	—	2	1	no	—
11	C₄⋊Dic₃₈₉	C₂²	1	1	yes	characteristic
12	#20261945	—	2	1	no	—
13	C₄×Dic₃₈₉	—	2	1	no	—
14	C₂×C₈×C₃₈₉	C₂²	1	2	yes	—
15	C₄²×C₃₈₉	C₂²	1	1	yes	characteristic
16	Dic₇₇₈	—	2	1	no	—
17	Dic₇₇₈	—	2	2	no	—
18	Dic₇₇₈	—	2	2	no	—
19	Dic₇₇₈	D₄	1	2	yes	—
20	C₂×Dic₃₈₉	—	2	1	no	—
21	C₂×Dic₃₈₉	—	2	1	no	—
22	C₂×Dic₃₈₉	—	2	2	no	—
23	C₂×C₄×C₃₈₉	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₈₉	D₄	1	1	yes	characteristic
25	C₂×C₄×C₃₈₉	C₂³	1	1	yes	characteristic
26	C₈×C₃₈₉	—	2	2	no	—
27	C₂²×C₃₈₉	C₂×D₄	1	1	yes	characteristic
28	C₄×C₃₈₉	—	2	1	no	—
29	C₄×C₃₈₉	Q₁₆	1	2	yes	complemented
30	C₄×C₃₈₉	C₂×D₄	1	1	yes	characteristic
31	C₄×C₃₈₉	#41	1	1	yes	characteristic
32	Dic₃₈₉	—	2	2	no	—
33	Dic₃₈₉	—	4	2	no	—
34	Dic₃₈₉	—	4	1	no	—
35	C₂×C₃₈₉	#121	1	1	yes	characteristic
36	C₂×C₃₈₉	C₂×Q₁₆	1	1	yes	characteristic
37	C₂×C₃₈₉	#137	1	1	yes	characteristic
38	C₃₈₉	#462	1	1	yes	characteristic, complemented, Sylow
39	#462	—	389	1	no	maximal, Sylow
40	C₂×Q₁₆	—	389	2	no	—
41	C₄⋊Q₈	—	389	1	no	—
42	C₄×Q₈	—	389	1	no	—
43	C₄⋊C₈	D₃₈₉	1	1	yes	characteristic
44	#103	—	389	2	no	—
45	C₂×Q₈	—	389	2	no	—
46	C₂×Q₈	—	389	1	no	—
47	Q₁₆	—	778	4	no	—
48	C₂×C₈	C₂×D₃₈₉	1	2	yes	—
49	C₄⋊C₄	—	778	1	no	—
50	C₄⋊C₄	—	778	1	no	—
51	C₄⋊C₄	—	778	1	no	—
52	C₄⋊C₄	—	389	1	no	—
53	C₄²	—	778	1	no	—
54	C₄²	C₂×D₃₈₉	1	1	yes	characteristic
55	Q₈	—	778	1	no	—
56	Q₈	—	778	2	no	—
57	Q₈	—	778	2	no	—
58	Q₈	—	389	2	no	—
59	C₂×C₄	—	778	1	no	—
60	C₂×C₄	—	778	1	no	—
61	C₂×C₄	—	778	2	no	—
62	C₂×C₄	D₁₅₅₆	1	1	yes	characteristic
63	C₂×C₄	D₁₅₅₆	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₃₈₉	1	1	yes	characteristic
65	C₈	—	2	2	no	—
66	C₂²	C₂×D₁₅₅₆	1	1	yes	characteristic
67	C₄	—	2	1	no	—
68	C₄	—	778	2	no	—
69	C₄	—	1556	2	no	—
70	C₄	—	1556	1	no	—
71	C₄	Dic₁₅₅₆	1	2	yes	complemented
72	C₄	D₄×D₃₈₉	1	1	yes	characteristic
73	C₄	#20261967	1	1	yes	characteristic
74	C₂	#27159359	1	1	yes	characteristic
75	C₂	C₂×Dic₁₅₅₆	1	1	yes	characteristic
76	C₂	#27159417	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₅₅₆
commutator, cosocle	C₂×C₄×C₃₈₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₈₉
Fitting	C₄⋊C₈×C₃₈₉	C₂
socle, prime radical	C₂²×C₃₈₉	C₂×D₄
hypercenter	C₄⋊C₈	D₃₈₉
hypocenter	C₃₈₉	#462
LCS	G ▷ C₂×C₄×C₃₈₉ ▷ C₂×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
29. C₄×C₃₈₉	Q₁₆	semidirect
38. C₃₈₉	#462	semidirect
71. C₄	Dic₁₅₅₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×Dic₁₅₅₆	2
#27159384	2
C₄×Dic₇₇₈	2
#27159309	2
C₄⋊C₈×C₃₈₉	2
#462	389