Subgroups — #27131313

Order 12128.

78 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊D₃₇₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₃₇₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊Dic₃₇₉	C₂	1	1	yes	maximal, characteristic, complemented
4	Dic₃₇₉⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊Dic₃₇₉	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃₇₉)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²⋊C₄×C₃₇₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₃₇₉	C₂²	1	1	yes	characteristic
9	C₂²⋊D₃₇₉	—	2	2	no	—
10	C₂×Dic₃₇₉	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₇₉	C₂²	1	1	yes	characteristic
12	C₂×Dic₃₇₉	C₂²	1	1	yes	characteristic
13	C₄×D₃₇₉	—	2	2	no	—
14	C₂³×C₃₇₉	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₃₇₉	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₃₇₉	C₂²	1	1	yes	characteristic
17	C₂²×C₃₇₉	—	2	1	no	—
18	C₂²×C₃₇₉	—	2	1	no	—
19	C₂²×C₃₇₉	—	2	1	no	—
20	C₂²×C₃₇₉	C₂³	1	1	yes	characteristic
21	C₂×D₃₇₉	—	2	1	no	—
22	C₂×D₃₇₉	—	2	1	no	—
23	C₂×D₃₇₉	D₄	1	2	yes	—
24	C₄×C₃₇₉	—	2	1	no	—
25	C₄×C₃₇₉	—	2	1	no	—
26	Dic₃₇₉	—	2	1	no	—
27	Dic₃₇₉	—	2	1	no	—
28	Dic₃₇₉	—	2	1	no	—
29	C₂×C₃₇₉	—	4	1	no	—
30	C₂×C₃₇₉	C₂×D₄	1	1	yes	characteristic
31	C₂×C₃₇₉	#41	1	1	yes	characteristic
32	C₂×C₃₇₉	#41	1	1	yes	characteristic
33	D₃₇₉	—	2	2	no	—
34	C₃₇₉	#123	1	1	yes	characteristic, complemented, Sylow
35	#123	—	379	1	no	maximal, Sylow
36	C₂×D₄	—	379	1	no	—
37	C₂²×C₄	—	379	1	no	—
38	C₄⋊C₄	—	379	1	no	—
39	C₄⋊C₄	—	379	1	no	—
40	C₂²⋊C₄	—	379	1	no	—
41	C₂²⋊C₄	—	379	1	no	—
42	C₂²⋊C₄	D₃₇₉	1	1	yes	characteristic, complemented
43	C₂³	—	379	1	no	—
44	C₂³	C₂×D₃₇₉	1	1	yes	characteristic
45	D₄	—	758	2	no	—
46	C₂×C₄	—	379	1	no	—
47	C₂×C₄	—	379	1	no	—
48	C₂×C₄	—	379	1	no	—
49	C₂×C₄	—	758	2	no	—
50	C₂×C₄	C₂×D₃₇₉	1	1	yes	characteristic
51	C₂×C₄	C₂×D₃₇₉	1	1	yes	characteristic
52	C₂²	—	2	1	no	—
53	C₂²	—	2	1	no	—
54	C₂²	—	2	1	no	—
55	C₂²	—	379	2	no	—
56	C₂²	—	758	1	no	—
57	C₂²	—	758	1	no	—
58	C₂²	C₂²×D₃₇₉	1	1	yes	characteristic
59	C₄	—	2	1	no	—
60	C₄	—	2	1	no	—
61	C₄	—	758	1	no	—
62	C₄	—	758	1	no	—
63	C₄	—	758	1	no	—
64	C₂	—	4	1	no	—
65	C₂	—	758	2	no	—
66	C₂	D₄×D₃₇₉	1	1	yes	characteristic
67	C₂	#20258605	1	1	yes	characteristic
68	C₂	#20258606	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₃₇₉
commutator, socle, cosocle	C₂²×C₃₇₉	C₂³
Fitting	C₂²⋊C₄×C₃₇₉	C₂
prime radical	C₂×C₂²⋊D₃₇₉	C₂
hypercenter	C₂²⋊C₄	D₃₇₉
hypocenter	C₃₇₉	#123
LCS	G ▷ C₂²×C₃₇₉ ▷ C₃₇₉
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×D₃₇₉	C₂	semidirect
3. C₄⋊Dic₃₇₉	C₂	semidirect
4. Dic₃₇₉⋊C₄	C₂	semidirect
5. C₂²⋊Dic₃₇₉	C₂	semidirect
6. (C₂×D₃₇₉)⋊C₄	C₂	semidirect
7. C₂²⋊C₄×C₃₇₉	C₂	semidirect
34. C₃₇₉	#123	semidirect
42. C₂²⋊C₄	D₃₇₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊D₃₇₉	2
C₂×C₄×D₃₇₉	2
C₄⋊Dic₃₇₉	2
Dic₃₇₉⋊C₄	2
C₂²⋊Dic₃₇₉	2
(C₂×D₃₇₉)⋊C₄	2
C₂²⋊C₄×C₃₇₉	2
#123	379