Subgroups — #37527115

Order 22976.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19153342	C₂	1	2	yes	maximal, complemented
2	#26040012	C₂	1	3	yes	maximal, complemented
3	#26039990	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₄₃₆	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₃₅₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19153342	—	2	8	no	—
7	C₂×D₁₄₃₆	C₂²	1	3	yes	—
8	C₂×D₁₄₃₆	C₂²	1	6	yes	—
9	C₂×D₁₄₃₆	C₂²	1	6	yes	—
10	C₂×C₄×D₃₅₉	C₂²	1	6	yes	—
11	C₄⋊Dic₃₅₉	C₂²	1	1	yes	characteristic
12	(C₂×D₃₅₉)⋊C₄	C₂²	1	6	yes	—
13	C₂×Q₈×C₃₅₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₃₅₉	C₂²	1	3	yes	—
15	C₄²×C₃₅₉	C₂²	1	3	yes	—
16	C₂²×D₃₅₉	C₂³	1	6	yes	—
17	D₁₄₃₆	—	2	12	no	—
18	D₁₄₃₆	—	2	6	no	—
19	D₁₄₃₆	—	4	6	no	—
20	C₂×Dic₃₅₉	C₂³	1	2	yes	—
21	C₄×D₃₅₉	—	2	12	no	—
22	Q₈×C₃₅₉	D₄	1	4	yes	complemented
23	C₂×C₄×C₃₅₉	C₂³	1	3	yes	—
24	C₂×C₄×C₃₅₉	C₂³	1	3	yes	—
25	C₂×C₄×C₃₅₉	C₂³	1	1	yes	characteristic
26	C₂²×C₃₅₉	C₂⁴	1	1	yes	characteristic
27	C₂×D₃₅₉	—	2	6	no	—
28	C₂×D₃₅₉	—	2	6	no	—
29	C₂×D₃₅₉	—	2	6	no	—
30	C₄×C₃₅₉	—	2	3	no	—
31	C₄×C₃₅₉	C₂×D₄	1	6	yes	—
32	C₄×C₃₅₉	#41	1	2	yes	complemented
33	Dic₃₅₉	—	2	2	no	—
34	C₂×C₃₅₉	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₃₅₉	C₂×#41	1	1	yes	characteristic
36	C₂×C₃₅₉	H(5,𝔽₂)	1	1	yes	characteristic
37	D₃₅₉	—	4	6	no	—
38	C₃₅₉	#550	1	1	yes	characteristic, complemented, Sylow
39	#550	—	359	1	no	maximal, Sylow
40	C₂×#41	—	359	2	no	—
41	#127	—	359	3	no	—
42	#121	—	359	6	no	—
43	C₄×Q₈	D₃₅₉	1	1	yes	characteristic, complemented
44	C₄×D₄	—	359	3	no	—
45	#41	—	718	8	no	—
46	C₂×Q₈	C₂×D₃₅₉	1	1	yes	characteristic
47	C₂×D₄	—	359	6	no	—
48	C₂×D₄	—	359	3	no	—
49	C₂×D₄	—	359	6	no	—
50	C₂²×C₄	—	359	6	no	—
51	C₄⋊C₄	—	359	1	no	—
52	C₄⋊C₄	C₂×D₃₅₉	1	3	yes	—
53	C₂²⋊C₄	—	359	6	no	—
54	C₄²	C₂×D₃₅₉	1	3	yes	—
55	C₂³	—	359	6	no	—
56	Q₈	D₁₄₃₆	1	4	yes	complemented
57	D₄	—	1436	6	no	—
58	D₄	—	718	12	no	—
59	D₄	—	718	6	no	—
60	C₂×C₄	—	359	2	no	—
61	C₂×C₄	—	718	12	no	—
62	C₂×C₄	C₂²×D₃₅₉	1	3	yes	—
63	C₂×C₄	C₂²×D₃₅₉	1	3	yes	—
64	C₂×C₄	C₂²×D₃₅₉	1	1	yes	characteristic
65	C₂²	—	718	6	no	—
66	C₂²	—	718	6	no	—
67	C₂²	—	718	6	no	—
68	C₂²	C₂³×D₃₅₉	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	718	2	no	—
71	C₄	C₂×D₁₄₃₆	1	6	yes	—
72	C₄	#19153342	1	2	yes	complemented
73	C₂	—	1436	6	no	—
74	C₂	C₂²×D₁₄₃₆	1	1	yes	characteristic
75	C₂	C₂×#19153342	1	1	yes	characteristic
76	C₂	#26040061	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₅₉
commutator, socle, cosocle	C₂²×C₃₅₉	C₂⁴
Fitting	C₄×Q₈×C₃₅₉	C₂
hypercenter	C₄×Q₈	D₃₅₉
hypocenter	C₃₅₉	#550
LCS	G ▷ C₂²×C₃₅₉ ▷ C₃₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×#19153342	C₂	semidirect
2. #26040012	C₂	semidirect
3. #26039990	C₂	semidirect
4. C₄×D₁₄₃₆	C₂	semidirect
5. C₄×Q₈×C₃₅₉	C₂	semidirect
22. Q₈×C₃₅₉	D₄	semidirect
32. C₄×C₃₅₉	#41	semidirect
38. C₃₅₉	#550	semidirect
43. C₄×Q₈	D₃₅₉	semidirect
56. Q₈	D₁₄₃₆	semidirect
72. C₄	#19153342	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19153342	2
#26040012	2
#26039990	2
C₄×D₁₄₃₆	2
C₄×Q₈×C₃₅₉	2
#550	359