Subgroups — D₄×#14582115 #36342419

Order 21568.

134 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14582115	C₂	1	2	yes	maximal, complemented
2	C₄⋊#14582115	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²⋊#14582115	C₂	1	2	yes	maximal, complemented
4	C₄×#14582115	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×#13062438	C₂	1	1	yes	maximal, characteristic
6	C₂×#14582115	—	2	4	no	—
7	C₂×#14582115	C₂²	1	1	yes	characteristic
8	C₂×#14582115	C₂²	1	2	yes	—
9	C₂×#14582115	C₂²	1	1	yes	characteristic
10	C₂²×#13062438	C₂²	1	2	yes	—
11	C₄⋊#13062438	C₄	1	1	yes	characteristic
12	C₂²⋊#13062438	C₄	1	2	yes	—
13	C₄×#13062438	C₂²	1	1	yes	characteristic
14	D₄×D₃₃₇	C₄	1	1	yes	characteristic
15	#14582115	—	2	2	no	—
16	#14582115	—	2	1	no	—
17	#14582115	D₄	1	2	yes	direct product factor
18	C₂×#13062438	—	2	2	no	—
19	C₂×#13062438	—	2	2	no	—
20	C₂×#13062438	C₂×C₄	1	1	yes	characteristic
21	C₂×#13062438	C₂×C₄	1	2	yes	—
22	C₂×#13062438	C₂³	1	1	yes	characteristic
23	C₂²×D₃₃₇	C₂×C₄	1	2	yes	—
24	D₁₃₄₈	C₈	1	1	yes	characteristic, complemented
25	C₂²⋊D₃₃₇	C₈	1	2	yes	complemented
26	C₄×D₃₃₇	C₂×C₄	1	1	yes	characteristic
27	D₄×C₃₃₇	C₈	1	1	yes	characteristic, complemented
28	C₂²×C₃₃₇	C₂×C₈	1	2	yes	complemented
29	C₂×D₃₃₇	—	2	2	no	—
30	C₂×D₃₃₇	—	2	2	no	—
31	C₂×D₃₃₇	C₂×C₈	1	2	yes	complemented
32	C₂×D₃₃₇	C₂²×C₄	1	1	yes	characteristic
33	#13062438	—	2	2	no	—
34	#13062438	—	2	1	no	—
35	#13062438	C₂×D₄	1	1	yes	characteristic
36	#13062438	#41	1	1	yes	characteristic
37	C₄×C₃₃₇	C₂×C₈	1	1	yes	characteristic, complemented
38	Dic₃₃₇	C₂×C₈	1	1	yes	characteristic, complemented
39	C₂×C₃₃₇	—	2	2	no	—
40	C₂×C₃₃₇	C₂²×C₈	1	1	yes	characteristic
41	D₃₃₇	—	2	2	no	—
42	D₃₃₇	C₄×D₄	1	1	yes	characteristic
43	D₃₃₇	#131	1	1	yes	characteristic
44	C₃₃₇	C₈×D₄	1	1	yes	characteristic, complemented, Sylow
45	C₈×D₄	—	337	1	no	maximal, Sylow
46	C₂²×C₈	—	337	2	no	—
47	C₄×D₄	—	337	1	no	—
48	C₄⋊C₈	—	337	1	no	—
49	C₂²⋊C₈	—	337	2	no	—
50	C₄×C₈	—	337	1	no	—
51	C₂×D₄	—	337	1	no	—
52	C₂²×C₄	—	337	2	no	—
53	C₂×C₈	—	337	1	no	—
54	C₂×C₈	—	337	1	no	—
55	C₂×C₈	—	337	2	no	—
56	C₂×C₈	—	674	4	no	—
57	C₄⋊C₄	—	337	1	no	—
58	C₂²⋊C₄	—	337	2	no	—
59	C₄²	—	337	1	no	—
60	C₂³	—	337	2	no	—
61	D₄	—	337	2	no	—
62	D₄	—	337	1	no	—
63	D₄	#14582115	1	1	yes	characteristic, direct product factor
64	C₂×C₄	—	337	1	no	—
65	C₂×C₄	—	337	1	no	—
66	C₂×C₄	—	337	1	no	—
67	C₂×C₄	—	337	2	no	—
68	C₂×C₄	—	674	2	no	—
69	C₂×C₄	—	674	2	no	—
70	C₈	—	337	2	no	—
71	C₈	—	674	2	no	—
72	C₈	—	674	1	no	—
73	C₂²	—	337	1	no	—
74	C₂²	—	337	2	no	—
75	C₂²	—	674	2	no	—
76	C₂²	—	674	2	no	—
77	C₂²	C₂×#14582115	1	2	yes	complemented
78	C₄	—	337	1	no	—
79	C₄	—	337	1	no	—
80	C₄	—	337	1	no	—
81	C₄	—	674	2	no	—
82	C₄	—	674	1	no	—
83	C₄	C₂×#14582115	1	1	yes	characteristic, complemented
84	C₂	—	2	2	no	—
85	C₂	—	337	1	no	—
86	C₂	—	337	1	no	—
87	C₂	—	674	2	no	—
88	C₂	C₂²×#14582115	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×#14582115
commutator, socle	C₂×C₃₃₇	C₂²×C₈
Fitting	D₄×C₃₃₇	C₈
prime radical	D₄×D₃₃₇	C₄
cosocle	C₂×#13062438	C₂³
hypercenter	D₄	#14582115
hypocenter	C₃₃₇	C₈×D₄
LCS	G ▷ C₂×C₃₃₇ ▷ C₃₃₇
UCS	C₁ ◁ C₂ ◁ D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14582115	C₂	semidirect
2. C₄⋊#14582115	C₂	semidirect
3. C₂²⋊#14582115	C₂	semidirect
4. C₄×#14582115	C₂	semidirect
17. #14582115	D₄	direct (N × H)
24. D₁₃₄₈	C₈	semidirect
25. C₂²⋊D₃₃₇	C₈	semidirect
27. D₄×C₃₃₇	C₈	semidirect
28. C₂²×C₃₃₇	C₂×C₈	semidirect
31. C₂×D₃₃₇	C₂×C₈	semidirect
37. C₄×C₃₃₇	C₂×C₈	semidirect
38. Dic₃₃₇	C₂×C₈	semidirect
44. C₃₃₇	C₈×D₄	semidirect
63. D₄	#14582115	direct (N × H)
77. C₂²	C₂×#14582115	semidirect
83. C₄	C₂×#14582115	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14582115	2
C₄⋊#14582115	2
C₂²⋊#14582115	2
C₄×#14582115	2
D₄×#13062438	2
C₈×D₄	337

Subgroups — D4×#14582115 #36342419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#14582115 #36342419