Subgroups — C₄×#19129194 #36342266

Order 21568.

142 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19129194	C₂	1	2	yes	maximal
2	#24925014	C₂	1	2	yes	maximal, complemented
3	C₄×#14582113	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×Dic₃₃₇	C₂	1	1	yes	maximal, characteristic
5	#19129194	C₄	1	8	yes	direct product factor
6	C₂×#14582113	C₂²	1	4	yes	—
7	C₂²×Dic₃₃₇	C₄	1	1	yes	characteristic
8	C₂²×Dic₃₃₇	C₂²	1	1	yes	characteristic
9	C₄×Dic₃₃₇	C₄	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	C₂²×C₄×C₃₃₇	C₄	1	1	yes	characteristic
14	#14582113	C₂×C₄	1	8	yes	complemented
15	C₂×Dic₃₃₇	—	2	2	no	—
16	C₂×Dic₃₃₇	C₂×C₄	1	1	yes	characteristic
17	C₂×Dic₃₃₇	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₃₃₇	C₂×C₄	1	4	yes	—
19	C₂×Dic₃₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Dic₃₃₇	C₂³	1	1	yes	characteristic
21	C₂³×C₃₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₃₃₇	—	2	2	no	—
23	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₃₃₇	—	2	1	no	—
26	C₂²×C₃₃₇	—	2	1	no	—
27	C₂²×C₃₃₇	C₄²	1	2	yes	—
28	C₂²×C₃₃₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₃₇	—	2	1	no	—
30	C₄×C₃₃₇	M₁₆	1	2	yes	complemented
31	Dic₃₃₇	—	2	1	no	—
32	Dic₃₃₇	C₄²	1	2	yes	—
33	Dic₃₃₇	M₁₆	1	2	yes	complemented
34	Dic₃₃₇	C₂²×C₄	1	2	yes	—
35	C₂×C₃₃₇	—	2	2	no	—
36	C₂×C₃₃₇	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₃₃₇	C₂×M₁₆	1	1	yes	characteristic
38	C₂×C₃₃₇	C₂×M₁₆	1	1	yes	characteristic
39	C₃₃₇	C₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
40	C₄×M₁₆	—	337	1	no	maximal, Sylow
41	C₂×M₁₆	—	337	2	no	—
42	C₂×C₄²	—	337	1	no	—
43	#97	—	337	2	no	—
44	C₄×C₈	—	337	2	no	—
45	C₂²×C₄	—	337	1	no	—
46	C₂²×C₄	—	337	1	no	—
47	C₂²×C₄	#13062438	1	1	yes	characteristic
48	M₁₆	—	337	8	no	—
49	C₂×C₈	—	337	4	no	—
50	C₄²	—	337	1	no	—
51	C₄²	—	337	1	no	—
52	C₄²	—	337	1	no	—
53	C₄²	—	337	1	no	—
54	C₂³	C₂×#13062438	1	1	yes	characteristic
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	337	4	no	—
57	C₂×C₄	—	337	1	no	—
58	C₂×C₄	—	337	1	no	—
59	C₂×C₄	—	337	1	no	—
60	C₂×C₄	—	337	1	no	—
61	C₂×C₄	—	674	2	no	—
62	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
63	C₂×C₄	C₂×#13062438	1	1	yes	characteristic
64	C₈	—	337	8	no	—
65	C₂²	—	2	1	no	—
66	C₂²	—	2	1	no	—
67	C₂²	C₄×#13062438	1	2	yes	—
68	C₂²	C₂²×#13062438	1	1	yes	characteristic
69	C₄	—	2	1	no	—
70	C₄	—	337	2	no	—
71	C₄	—	337	2	no	—
72	C₄	—	337	2	no	—
73	C₄	—	674	1	no	—
74	C₄	#19129194	1	2	yes	direct product factor
75	C₂	—	2	2	no	—
76	C₂	C₂×C₄×#13062438	1	1	yes	characteristic
77	C₂	C₂×#19129194	1	1	yes	characteristic
78	C₂	D₃₃₇⋊M₁₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#13062438
commutator	C₂×C₃₃₇	C₂×C₄²
Frattini	C₂²	C₂²×#13062438
Fitting	C₂²×C₄×C₃₃₇	C₄
socle	C₂²×C₃₃₇	C₂²×C₄
prime radical	C₂³×C₃₃₇	C₂×C₄
cosocle	C₂×Dic₃₃₇	C₂³
hypercenter	C₂²×C₄	#13062438
hypocenter	C₃₃₇	C₄×M₁₆
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. #24925014	C₂	semidirect
3. C₄×#14582113	C₂	semidirect
5. #19129194	C₄	direct (N × H)
14. #14582113	C₂×C₄	semidirect
30. C₄×C₃₃₇	M₁₆	semidirect
33. Dic₃₃₇	M₁₆	semidirect
39. C₃₃₇	C₄×M₁₆	semidirect
74. C₄	#19129194	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#19129194	2
#24925014	2
C₄×#14582113	2
C₂×C₄×Dic₃₃₇	2
C₄×M₁₆	337

Subgroups — C4×#19129194 #36342266

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#19129194 #36342266