Subgroups — #38734710

Order 24896.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27159379	C₂	1	4	yes	maximal
2	#27159376	C₂	1	1	yes	maximal, characteristic, complemented
3	#27159374	C₂	1	4	yes	maximal, complemented
4	#27159360	C₂	1	2	yes	maximal
5	Q₈×Dic₃₈₉	C₂	1	2	yes	maximal, complemented
6	#27159341	C₂	1	1	yes	maximal, characteristic
7	C₄⋊Q₈×C₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₈₉	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₃₈₉	C₂²	1	2	yes	—
10	C₄⋊Dic₃₈₉	C₂²	1	4	yes	—
11	#20261945	C₂²	1	2	yes	—
12	#20261945	C₂²	1	4	yes	—
13	#20261945	C₂²	1	4	yes	—
14	#20261943	C₂²	1	2	yes	—
15	#20261943	C₂²	1	4	yes	—
16	C₄×Dic₃₈₉	C₂²	1	1	yes	characteristic
17	C₄×Dic₃₈₉	C₂²	1	4	yes	—
18	C₂×Q₈×C₃₈₉	C₂²	1	2	yes	—
19	C₄⋊C₄×C₃₈₉	C₂²	1	4	yes	—
20	C₄²×C₃₈₉	C₂²	1	1	yes	characteristic
21	C₂²×D₃₈₉	C₂³	1	1	yes	characteristic
22	C₂×Dic₃₈₉	C₂³	1	2	yes	—
23	C₂×Dic₃₈₉	C₂³	1	4	yes	—
24	C₂×Dic₃₈₉	C₂³	1	1	yes	characteristic
25	C₄×D₃₈₉	—	2	2	no	—
26	Q₈×C₃₈₉	—	2	4	no	—
27	C₂×C₄×C₃₈₉	C₂³	1	2	yes	—
28	C₂×C₄×C₃₈₉	C₂³	1	4	yes	—
29	C₂×C₄×C₃₈₉	C₂³	1	1	yes	characteristic
30	C₂²×C₃₈₉	C₂⁴	1	1	yes	characteristic
31	C₂×D₃₈₉	—	2	1	no	—
32	C₂×D₃₈₉	—	2	2	no	—
33	C₄×C₃₈₉	—	2	4	no	—
34	C₄×C₃₈₉	—	2	2	no	—
35	C₄×C₃₈₉	#41	1	2	yes	—
36	Dic₃₈₉	—	2	1	no	—
37	Dic₃₈₉	—	2	2	no	—
38	Dic₃₈₉	—	2	4	no	—
39	C₂×C₃₈₉	C₂×#41	1	1	yes	characteristic
40	C₂×C₃₈₉	2₋¹⁺⁴	1	2	yes	—
41	D₃₈₉	—	4	1	no	—
42	C₃₈₉	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	389	1	no	maximal, Sylow
44	C₄⋊Q₈	D₃₈₉	1	1	yes	characteristic, complemented
45	#126	—	389	4	no	—
46	#125	—	389	4	no	—
47	#125	—	389	1	no	—
48	C₂²⋊Q₈	—	389	2	no	—
49	C₄×Q₈	—	389	2	no	—
50	#117	—	389	1	no	—
51	C₂×Q₈	C₂×D₃₈₉	1	2	yes	—
52	C₂²×C₄	—	389	1	no	—
53	C₄⋊C₄	—	389	2	no	—
54	C₄⋊C₄	—	389	4	no	—
55	C₄⋊C₄	—	389	4	no	—
56	C₄⋊C₄	—	389	2	no	—
57	C₄⋊C₄	—	389	4	no	—
58	C₄⋊C₄	C₂×D₃₈₉	1	4	yes	—
59	C₂²⋊C₄	—	389	2	no	—
60	C₂²⋊C₄	—	389	4	no	—
61	C₄²	—	389	1	no	—
62	C₄²	—	389	4	no	—
63	C₄²	C₂×D₃₈₉	1	1	yes	characteristic
64	C₂³	—	389	1	no	—
65	Q₈	—	2	4	no	—
66	C₂×C₄	—	778	2	no	—
67	C₂×C₄	—	389	2	no	—
68	C₂×C₄	—	389	4	no	—
69	C₂×C₄	—	389	1	no	—
70	C₂×C₄	C₂²×D₃₈₉	1	4	yes	—
71	C₂×C₄	C₂²×D₃₈₉	1	2	yes	—
72	C₂×C₄	C₂²×D₃₈₉	1	1	yes	characteristic
73	C₂²	—	778	1	no	—
74	C₂²	—	778	2	no	—
75	C₂²	C₂³×D₃₈₉	1	1	yes	characteristic
76	C₄	—	2	4	no	—
77	C₄	—	2	2	no	—
78	C₄	—	778	1	no	—
79	C₄	—	778	2	no	—
80	C₄	—	778	4	no	—
81	C₄	#20261965	1	2	yes	—
82	C₂	—	1556	1	no	—
83	C₂	C₂×#20261965	1	1	yes	characteristic
84	C₂	#27159432	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₈₉
commutator, socle, cosocle	C₂²×C₃₈₉	C₂⁴
Fitting	C₄⋊Q₈×C₃₈₉	C₂
prime radical	C₂²×D₃₈₉	C₂³
hypercenter	C₄⋊Q₈	D₃₈₉
hypocenter	C₃₈₉	#541
LCS	G ▷ C₂²×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #27159376	C₂	semidirect
3. #27159374	C₂	semidirect
5. Q₈×Dic₃₈₉	C₂	semidirect
7. C₄⋊Q₈×C₃₈₉	C₂	semidirect
42. C₃₈₉	#541	semidirect
44. C₄⋊Q₈	D₃₈₉	semidirect

Maximal subgroups

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

subgroup	index
#27159379	2
#27159376	2
#27159374	2
#27159360	2
Q₈×Dic₃₈₉	2
#27159341	2
C₄⋊Q₈×C₃₈₉	2
#541	389