Subgroups — #36342306

Order 21568.

138 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19129193	C₂	1	2	yes	maximal
2	#24925026	C₂	1	2	yes	maximal, complemented
3	C₄⋊#14582113	C₂	1	2	yes	maximal, complemented
4	C₄²×D₃₃₇	C₂	1	1	yes	maximal, characteristic
5	#19129193	—	2	4	no	—
6	C₂×#14582113	C₂²	1	4	yes	—
7	C₂×C₄×D₃₃₇	C₄	1	1	yes	characteristic
8	C₂×C₄×D₃₃₇	C₄	1	1	yes	characteristic
9	C₂×C₄×D₃₃₇	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₃₇	C₄	1	1	yes	characteristic
11	C₄×Dic₃₃₇	C₂²	1	1	yes	characteristic
12	C₄×Dic₃₃₇	C₂²	1	1	yes	characteristic
13	C₄²×C₃₃₇	C₄	1	1	yes	characteristic
14	#14582113	—	2	4	no	—
15	C₂²×D₃₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂×Dic₃₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₃₇	C₂³	1	1	yes	characteristic
19	C₄×D₃₃₇	—	2	2	no	—
20	C₄×D₃₃₇	C₈	1	4	yes	complemented
21	C₄×D₃₃₇	D₄	1	2	yes	—
22	C₄×D₃₃₇	Q₈	1	2	yes	—
23	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₃₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×D₃₃₇	C₄⋊C₄	1	2	yes	—
28	C₂×D₃₃₇	C₂×C₈	1	2	yes	—
29	C₂×D₃₃₇	M₁₆	1	2	yes	—
30	C₄×C₃₃₇	—	2	1	no	—
31	C₄×C₃₃₇	C₄⋊C₄	1	2	yes	—
32	C₄×C₃₃₇	C₂×C₈	1	2	yes	complemented
33	Dic₃₃₇	—	2	1	no	—
34	Dic₃₃₇	C₂×C₈	1	2	yes	complemented
35	Dic₃₃₇	C₂×D₄	1	1	yes	characteristic
36	Dic₃₃₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₃₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₃₃₇	C₂²×C₈	1	1	yes	characteristic
39	C₂×C₃₃₇	C₂×M₁₆	1	1	yes	characteristic
40	D₃₃₇	C₄⋊C₈	1	4	yes	complemented
41	C₃₃₇	C₂×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₄⋊C₈	—	337	1	no	maximal, Sylow
43	C₂²×C₈	—	337	2	no	—
44	C₂×C₄²	—	337	1	no	—
45	C₄⋊C₈	—	337	2	no	—
46	C₄⋊C₈	—	337	2	no	—
47	C₂²×C₄	—	337	1	no	—
48	C₂²×C₄	—	337	1	no	—
49	C₂²×C₄	—	337	1	no	—
50	C₂×C₈	—	337	4	no	—
51	C₂×C₈	—	674	4	no	—
52	C₄²	—	337	1	no	—
53	C₄²	—	337	1	no	—
54	C₄²	—	337	1	no	—
55	C₄²	#13062438	1	1	yes	characteristic
56	C₂³	—	337	1	no	—
57	C₂×C₄	—	337	1	no	—
58	C₂×C₄	—	337	1	no	—
59	C₂×C₄	—	337	2	no	—
60	C₂×C₄	—	337	2	no	—
61	C₂×C₄	—	337	4	no	—
62	C₂×C₄	—	337	1	no	—
63	C₂×C₄	—	674	2	no	—
64	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
65	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
66	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
67	C₈	—	674	4	no	—
68	C₂²	—	337	2	no	—
69	C₂²	—	337	2	no	—
70	C₂²	—	337	2	no	—
71	C₂²	C₂²×#13062438	1	1	yes	characteristic
72	C₄	—	2	1	no	—
73	C₄	—	337	1	no	—
74	C₄	—	337	1	no	—
75	C₄	—	337	2	no	—
76	C₄	—	674	1	no	—
77	C₄	C₄⋊#13062438	1	2	yes	—
78	C₄	#19129193	1	2	yes	complemented
79	C₂	—	337	4	no	—
80	C₂	C₂×C₄⋊#13062438	1	1	yes	characteristic
81	C₂	C₂×#19129193	1	1	yes	characteristic
82	C₂	D₃₃₇⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13062438
commutator	C₂×C₃₃₇	C₂²×C₈
Frattini	C₂²	C₂²×#13062438
Fitting	C₄²×C₃₃₇	C₄
socle	C₂²×C₃₃₇	C₂²×C₄
prime radical	C₂²×D₃₃₇	C₂×C₄
cosocle	C₂×Dic₃₃₇	C₂³
hypercenter	C₄²	#13062438
hypocenter	C₃₃₇	C₂×C₄⋊C₈
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄²

Complements

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

N	H ≅ G/N	type
2. #24925026	C₂	semidirect
3. C₄⋊#14582113	C₂	semidirect
20. C₄×D₃₃₇	C₈	semidirect
32. C₄×C₃₃₇	C₂×C₈	semidirect
34. Dic₃₃₇	C₂×C₈	semidirect
40. D₃₃₇	C₄⋊C₈	semidirect
41. C₃₃₇	C₂×C₄⋊C₈	semidirect
78. C₄	#19129193	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19129193	2
#24925026	2
C₄⋊#14582113	2
C₄²×D₃₃₇	2
C₂×C₄⋊C₈	337