Subgroups — #37527055

Order 22976.

192 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26040015	C₂	1	1	yes	maximal, characteristic, complemented
2	#26040008	C₂	1	4	yes	maximal
3	#26040007	C₂	1	1	yes	maximal, characteristic, complemented
4	#26040005	C₂	1	4	yes	maximal, complemented
5	#26039994	C₂	1	2	yes	maximal
6	C₄×Dic₇₁₈	C₂	1	2	yes	maximal, complemented
7	#117×C₃₅₉	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₇₁₈	C₂²	1	2	yes	—
9	C₄⋊Dic₃₅₉	C₂²	1	2	yes	—
10	C₄⋊Dic₃₅₉	C₂²	1	4	yes	—
11	Dic₃₅₉⋊C₄	C₂²	1	4	yes	—
12	Dic₃₅₉⋊C₄	C₂²	1	4	yes	—
13	Dic₃₅₉⋊C₄	C₂²	1	4	yes	—
14	C₂²⋊Dic₃₅₉	C₂²	1	4	yes	—
15	C₄×Dic₃₅₉	C₂²	1	4	yes	—
16	C₂²×C₄×C₃₅₉	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₅₉	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₃₅₉	C₂²	1	2	yes	—
19	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
20	C₄²×C₃₅₉	C₂²	1	1	yes	characteristic
21	Dic₇₁₈	—	2	4	no	—
22	C₂×Dic₃₅₉	C₂³	1	4	yes	—
23	C₂×Dic₃₅₉	C₂³	1	4	yes	—
24	C₂³×C₃₅₉	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₃₅₉	—	2	2	no	—
26	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
27	C₂×C₄×C₃₅₉	C₂³	1	2	yes	—
28	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
30	C₂²×C₃₅₉	—	2	1	no	—
31	C₂²×C₃₅₉	—	2	2	no	—
32	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
33	C₄×C₃₅₉	—	2	1	no	—
34	C₄×C₃₅₉	—	2	2	no	—
35	C₄×C₃₅₉	—	2	2	no	—
36	C₄×C₃₅₉	#41	1	2	yes	—
37	Dic₃₅₉	—	2	4	no	—
38	Dic₃₅₉	—	2	4	no	—
39	C₂×C₃₅₉	—	4	1	no	—
40	C₂×C₃₅₉	C₂×#41	1	1	yes	characteristic
41	C₂×C₃₅₉	2₋¹⁺⁴	1	2	yes	—
42	C₃₅₉	#541	1	1	yes	characteristic, complemented, Sylow
43	#541	—	359	1	no	maximal, Sylow
44	C₄⋊Q₈	—	359	1	no	—
45	#126	—	359	4	no	—
46	#125	—	359	4	no	—
47	#125	—	359	1	no	—
48	C₂²⋊Q₈	—	359	2	no	—
49	C₄×Q₈	—	359	2	no	—
50	#117	D₃₅₉	1	1	yes	characteristic
51	C₂×Q₈	—	359	2	no	—
52	C₂²×C₄	C₂×D₃₅₉	1	1	yes	characteristic
53	C₄⋊C₄	—	359	2	no	—
54	C₄⋊C₄	—	359	4	no	—
55	C₄⋊C₄	—	359	4	no	—
56	C₄⋊C₄	—	359	4	no	—
57	C₄⋊C₄	—	359	4	no	—
58	C₄⋊C₄	C₂×D₃₅₉	1	2	yes	—
59	C₂²⋊C₄	—	359	4	no	—
60	C₂²⋊C₄	C₂×D₃₅₉	1	2	yes	—
61	C₄²	—	359	4	no	—
62	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
63	C₄²	C₂×D₃₅₉	1	1	yes	characteristic
64	C₂³	C₂²×D₃₅₉	1	1	yes	characteristic
65	Q₈	—	718	4	no	—
66	C₂×C₄	—	2	2	no	—
67	C₂×C₄	—	359	4	no	—
68	C₂×C₄	—	359	4	no	—
69	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
70	C₂×C₄	C₂²×D₃₅₉	1	2	yes	—
71	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
72	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
73	C₂²	—	2	1	no	—
74	C₂²	—	2	2	no	—
75	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
76	C₄	—	2	1	no	—
77	C₄	—	2	2	no	—
78	C₄	—	2	2	no	—
79	C₄	—	718	4	no	—
80	C₄	—	718	4	no	—
81	C₄	#19153341	1	2	yes	—
82	C₂	—	4	1	no	—
83	C₂	C₂×#19153341	1	1	yes	characteristic
84	C₂	#26040062	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting	#117×C₃₅₉	C₂
prime radical	C₂³×C₃₅₉	C₂³
hypercenter	#117	D₃₅₉
hypocenter	C₃₅₉	#541
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #26040015	C₂	semidirect
3. #26040007	C₂	semidirect
4. #26040005	C₂	semidirect
6. C₄×Dic₇₁₈	C₂	semidirect
42. C₃₅₉	#541	semidirect

Maximal subgroups

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

subgroup	index
#26040015	2
#26040008	2
#26040007	2
#26040005	2
#26039994	2
C₄×Dic₇₁₈	2
#117×C₃₅₉	2
#541	359