Subgroups — #37565875

Order 24256.

206 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27131330	C₂	1	4	yes	maximal, complemented
2	#27131322	C₂	1	4	yes	maximal, complemented
3	#27131310	C₂	1	4	yes	maximal
4	#27131290	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₃₇₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₇₅₈	C₂²	1	4	yes	—
7	C₄⋊Dic₃₇₉	C₂²	1	2	yes	—
8	C₄⋊Dic₃₇₉	C₂²	1	2	yes	—
9	C₄⋊Dic₃₇₉	C₂²	1	4	yes	—
10	Dic₃₇₉⋊C₄	C₂²	1	8	yes	—
11	C₂²⋊Dic₃₇₉	C₂²	1	4	yes	—
12	C₄×Dic₃₇₉	C₂²	1	4	yes	—
13	C₂²×C₄×C₃₇₉	C₂²	1	2	yes	—
14	C₂²×C₄×C₃₇₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₃₇₉	C₂²	1	4	yes	—
16	Dic₇₅₈	—	4	4	no	—
17	C₂×Dic₃₇₉	C₂³	1	4	yes	—
18	C₂×Dic₃₇₉	C₂³	1	4	yes	—
19	C₂³×C₃₇₉	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₃₇₉	—	4	2	no	—
21	C₂×C₄×C₃₇₉	Q₈	1	4	yes	complemented
22	C₂×C₄×C₃₇₉	C₂³	1	4	yes	—
23	C₂×C₄×C₃₇₉	C₂³	1	2	yes	—
24	C₂²×C₃₇₉	—	2	1	no	—
25	C₂²×C₃₇₉	—	2	1	no	—
26	C₂²×C₃₇₉	C₂×Q₈	1	2	yes	—
27	C₂²×C₃₇₉	C₂⁴	1	1	yes	characteristic
28	C₄×C₃₇₉	—	2	4	no	—
29	C₄×C₃₇₉	C₂×Q₈	1	4	yes	—
30	Dic₃₇₉	—	2	4	no	—
31	Dic₃₇₉	—	2	4	no	—
32	C₂×C₃₇₉	—	2	2	no	—
33	C₂×C₃₇₉	C₂²×Q₈	1	1	yes	characteristic
34	C₂×C₃₇₉	H(5,𝔽₂)	1	1	yes	characteristic
35	C₂×C₃₇₉	2₋¹⁺⁴	1	1	yes	characteristic
36	C₃₇₉	#544	1	1	yes	characteristic, complemented, Sylow
37	#544	—	379	1	no	maximal, Sylow
38	C₄⋊Q₈	—	379	4	no	—
39	#125	—	379	4	no	—
40	C₂²⋊Q₈	—	379	4	no	—
41	#117	—	379	2	no	—
42	C₂×C₄⋊C₄	D₃₇₉	1	1	yes	characteristic
43	C₂×Q₈	—	379	4	no	—
44	C₂²×C₄	C₂×D₃₇₉	1	2	yes	—
45	C₂²×C₄	C₂×D₃₇₉	1	1	yes	characteristic
46	C₄⋊C₄	—	379	8	no	—
47	C₄⋊C₄	—	379	2	no	—
48	C₄⋊C₄	—	379	2	no	—
49	C₄⋊C₄	—	379	4	no	—
50	C₄⋊C₄	C₂×D₃₇₉	1	4	yes	—
51	C₂²⋊C₄	—	379	4	no	—
52	C₄²	—	379	4	no	—
53	C₂³	C₂²×D₃₇₉	1	1	yes	characteristic
54	Q₈	—	1516	4	no	—
55	C₂×C₄	—	4	2	no	—
56	C₂×C₄	—	379	4	no	—
57	C₂×C₄	—	379	4	no	—
58	C₂×C₄	Dic₇₅₈	1	4	yes	complemented
59	C₂×C₄	C₂²×D₃₇₉	1	4	yes	—
60	C₂×C₄	C₂²×D₃₇₉	1	2	yes	—
61	C₂²	—	2	1	no	—
62	C₂²	—	2	1	no	—
63	C₂²	C₂×Dic₇₅₈	1	2	yes	—
64	C₂²	C₂³×D₃₇₉	1	1	yes	characteristic
65	C₄	—	2	4	no	—
66	C₄	—	758	4	no	—
67	C₄	—	758	4	no	—
68	C₄	C₂×Dic₇₅₈	1	4	yes	—
69	C₂	—	2	2	no	—
70	C₂	C₂²×Dic₇₅₈	1	1	yes	characteristic
71	C₂	#27131376	1	1	yes	characteristic
72	C₂	#27131379	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₇₉
commutator, socle, cosocle	C₂²×C₃₇₉	C₂⁴
Fitting	C₂×C₄⋊C₄×C₃₇₉	C₂
prime radical	C₂³×C₃₇₉	C₂³
hypercenter	C₂×C₄⋊C₄	D₃₇₉
hypocenter	C₃₇₉	#544
LCS	G ▷ C₂²×C₃₇₉ ▷ C₃₇₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #27131330	C₂	semidirect
2. #27131322	C₂	semidirect
21. C₂×C₄×C₃₇₉	Q₈	semidirect
36. C₃₇₉	#544	semidirect
58. C₂×C₄	Dic₇₅₈	semidirect

Maximal subgroups

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

subgroup	index
#27131330	2
#27131322	2
#27131310	2
#27131290	2
C₂×C₄⋊C₄×C₃₇₉	2
#544	379