Subgroups — #38734703

Order 24896.

240 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27159376	C₂	1	1	yes	maximal, characteristic, complemented
2	#27159365	C₂	1	4	yes	maximal, complemented
3	#27159356	C₂	1	4	yes	maximal, complemented
4	#27159355	C₂	1	2	yes	maximal, complemented
5	D₄×Dic₃₈₉	C₂	1	2	yes	maximal, complemented
6	#27159341	C₂	1	1	yes	maximal, characteristic, complemented
7	#127×C₃₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊D₃₈₉	C₂²	1	4	yes	—
9	C₂²×Dic₃₈₉	C₂²	1	4	yes	—
10	C₂×C₄×D₃₈₉	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₈₉	C₂²	1	2	yes	—
12	#20261945	C₂²	1	2	yes	—
13	#20261945	C₂²	1	4	yes	—
14	C₂²⋊Dic₃₈₉	C₂²	1	4	yes	—
15	C₂²⋊Dic₃₈₉	C₂²	1	4	yes	—
16	#20261943	C₂²	1	2	yes	—
17	C₄×Dic₃₈₉	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₃₈₉	C₂²	1	2	yes	—
19	C₂×D₄×C₃₈₉	C₂²	1	4	yes	—
20	C₄²×C₃₈₉	C₂²	1	1	yes	characteristic
21	C₂²×D₃₈₉	C₂³	1	1	yes	characteristic
22	C₂²⋊D₃₈₉	—	4	4	no	—
23	C₂×Dic₃₈₉	—	4	4	no	—
24	C₂×Dic₃₈₉	C₂³	1	2	yes	—
25	C₂×Dic₃₈₉	C₂³	1	1	yes	characteristic
26	C₂×Dic₃₈₉	C₂³	1	4	yes	—
27	C₄×D₃₈₉	—	2	2	no	—
28	C₂³×C₃₈₉	C₂³	1	4	yes	—
29	D₄×C₃₈₉	—	2	4	no	—
30	D₄×C₃₈₉	—	4	4	no	—
31	C₂×C₄×C₃₈₉	C₂³	1	2	yes	—
32	C₂×C₄×C₃₈₉	C₂³	1	1	yes	characteristic
33	C₂²×C₃₈₉	—	2	4	no	—
34	C₂²×C₃₈₉	—	2	4	no	—
35	C₂²×C₃₈₉	—	2	4	no	—
36	C₂²×C₃₈₉	C₂⁴	1	1	yes	characteristic
37	C₂×D₃₈₉	—	2	1	no	—
38	C₂×D₃₈₉	—	2	2	no	—
39	C₄×C₃₈₉	—	2	2	no	—
40	C₄×C₃₈₉	#41	1	2	yes	—
41	Dic₃₈₉	—	2	1	no	—
42	Dic₃₈₉	—	2	4	no	—
43	Dic₃₈₉	—	2	2	no	—
44	C₂×C₃₈₉	—	4	4	no	—
45	C₂×C₃₈₉	C₂×#41	1	1	yes	characteristic
46	C₂×C₃₈₉	H(5,𝔽₂)	1	2	yes	—
47	D₃₈₉	—	4	1	no	—
48	C₃₈₉	#540	1	1	yes	characteristic, complemented, Sylow
49	#540	—	389	1	no	maximal, Sylow
50	#127	D₃₈₉	1	1	yes	characteristic, complemented
51	#125	—	389	1	no	—
52	#123	—	389	4	no	—
53	#121	—	389	2	no	—
54	#121	—	389	4	no	—
55	C₄×D₄	—	389	2	no	—
56	#117	—	389	1	no	—
57	C₂×D₄	—	389	4	no	—
58	C₂×D₄	C₂×D₃₈₉	1	2	yes	—
59	C₂×D₄	C₂×D₃₈₉	1	4	yes	—
60	C₂²×C₄	—	389	1	no	—
61	C₂²×C₄	—	389	4	no	—
62	C₄⋊C₄	—	389	2	no	—
63	C₄⋊C₄	—	389	4	no	—
64	C₄⋊C₄	—	389	2	no	—
65	C₂²⋊C₄	—	389	4	no	—
66	C₂²⋊C₄	—	389	4	no	—
67	C₂²⋊C₄	—	389	2	no	—
68	C₄²	—	389	1	no	—
69	C₄²	C₂×D₃₈₉	1	1	yes	characteristic
70	C₂³	—	389	1	no	—
71	C₂³	C₂²×D₃₈₉	1	4	yes	—
72	D₄	—	2	4	no	—
73	D₄	—	4	4	no	—
74	D₄	—	1556	4	no	—
75	C₂×C₄	—	778	2	no	—
76	C₂×C₄	—	1556	4	no	—
77	C₂×C₄	—	389	2	no	—
78	C₂×C₄	—	389	1	no	—
79	C₂×C₄	—	389	4	no	—
80	C₂×C₄	C₂²×D₃₈₉	1	2	yes	—
81	C₂×C₄	C₂²×D₃₈₉	1	1	yes	characteristic
82	C₂²	—	2	4	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	778	1	no	—
86	C₂²	—	778	2	no	—
87	C₂²	C₂³×D₃₈₉	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	—	778	1	no	—
90	C₄	—	778	2	no	—
91	C₄	—	778	4	no	—
92	C₄	#20261965	1	2	yes	—
93	C₂	—	4	4	no	—
94	C₂	—	1556	1	no	—
95	C₂	C₂×#20261965	1	1	yes	characteristic
96	C₂	#27159429	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₈₉
commutator, socle, cosocle	C₂²×C₃₈₉	C₂⁴
Fitting	#127×C₃₈₉	C₂
hypercenter	#127	D₃₈₉
hypocenter	C₃₈₉	#540
LCS	G ▷ C₂²×C₃₈₉ ▷ C₃₈₉
UCS	C₁ ◁ C₂² ◁ #127

Complements

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

N	H ≅ G/N	type
1. #27159376	C₂	semidirect
2. #27159365	C₂	semidirect
3. #27159356	C₂	semidirect
4. #27159355	C₂	semidirect
5. D₄×Dic₃₈₉	C₂	semidirect
6. #27159341	C₂	semidirect
7. #127×C₃₈₉	C₂	semidirect
48. C₃₈₉	#540	semidirect
50. #127	D₃₈₉	semidirect

Maximal subgroups

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

subgroup	index
#27159376	2
#27159365	2
#27159356	2
#27159355	2
D₄×Dic₃₈₉	2
#27159341	2
#127×C₃₈₉	2
#540	389