Subgroups — #14581188

Order 2624.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050221	C₂	1	1	yes	maximal, characteristic, complemented
2	(C₂×#4223)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#13050139	C₂	1	1	yes	maximal, characteristic
4	C₂²×#4223	—	2	1	no	—
5	C₂×D₁₆₄	—	2	1	no	—
6	C₂×D₁₆₄	C₄	1	1	yes	characteristic, complemented
7	C₂×C₄×D₄₁	C₂²	1	1	yes	characteristic
8	C₂×#66493	—	2	1	no	—
9	C₄⋊C₄×C₄₁	C₄	1	1	yes	characteristic, complemented
10	#10620281	—	2	1	no	—
11	C₂²×D₄₁	—	2	1	no	—
12	C₂²×D₄₁	D₄	1	1	yes	characteristic
13	C₂×#4223	—	2	2	no	—
14	C₂×#4223	—	4	1	no	—
15	C₂×#4223	—	4	1	no	—
16	C₂×C₄×C₄₁	—	2	1	no	—
17	C₂×C₄×C₄₁	C₂×C₄	1	1	yes	characteristic
18	C₂×Dic₄₁	D₄	1	1	yes	characteristic
19	D₁₆₄	—	4	1	no	—
20	D₁₆₄	—	4	1	no	—
21	D₁₆₄	—	4	1	no	—
22	C₄×D₄₁	—	4	1	no	—
23	#66493	—	4	1	no	—
24	C₂²×C₄₁	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×D₄₁	—	2	1	no	—
26	C₂×D₄₁	—	2	1	no	—
27	C₂×D₄₁	—	4	1	no	—
28	C₂×D₄₁	—	4	1	no	—
29	C₂×D₄₁	—	4	1	no	—
30	C₂×D₄₁	D₈	1	1	yes	characteristic
31	C₂×D₄₁	SD₁₆	1	1	yes	characteristic
32	#4223	—	4	2	no	—
33	C₄×C₄₁	—	2	1	no	—
34	C₄×C₄₁	—	4	1	no	—
35	Dic₄₁	—	2	1	no	—
36	C₂×C₄₁	#99	1	1	yes	characteristic
37	C₂×C₄₁	#102	1	1	yes	characteristic
38	C₂×C₄₁	#104	1	1	yes	characteristic
39	D₄₁	—	2	1	no	—
40	D₄₁	—	2	1	no	—
41	D₄₁	—	8	1	no	—
42	#327	—	41	1	no	maximal, Sylow
43	C₄₁	#327	1	1	yes	characteristic, complemented, Sylow
44	#121	—	41	1	no	—
45	C₂²⋊C₈	—	41	1	no	—
46	#95	—	41	1	no	—
47	C₂×D₄	—	41	1	no	—
48	C₂×D₄	—	82	1	no	—
49	C₂²×C₄	—	41	1	no	—
50	C₂²×C₄	—	82	1	no	—
51	C₂×C₈	—	82	1	no	—
52	C₄⋊C₄	#4223	1	1	yes	characteristic, complemented
53	C₂²⋊C₄	—	82	1	no	—
54	C₂³	—	41	1	no	—
55	C₂³	—	82	1	no	—
56	D₄	—	164	1	no	—
57	D₄	—	164	1	no	—
58	D₄	—	164	1	no	—
59	C₂×C₄	—	2	1	no	—
60	C₂×C₄	—	41	1	no	—
61	C₂×C₄	—	82	2	no	—
62	C₂×C₄	—	164	1	no	—
63	C₂×C₄	—	164	1	no	—
64	C₂×C₄	—	164	1	no	—
65	C₂×C₄	C₂×#4223	1	1	yes	characteristic
66	C₈	—	164	1	no	—
67	C₂²	—	41	1	no	—
68	C₂²	—	41	1	no	—
69	C₂²	—	82	1	no	—
70	C₂²	—	82	1	no	—
71	C₂²	—	164	1	no	—
72	C₂²	—	164	1	no	—
73	C₂²	—	164	1	no	—
74	C₂²	C₂²⋊#4223	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	82	1	no	—
78	C₄	—	164	2	no	—
79	C₂	—	328	1	no	—
80	C₂	—	82	1	no	—
81	C₂	—	82	1	no	—
82	C₂	#13050223	1	1	yes	characteristic
83	C₂	#13050226	1	1	yes	characteristic
84	C₂	#13050229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²⋊#4223
commutator	C₂×C₄×C₄₁	C₂×C₄
Frattini	C₂×C₄	C₂×#4223
Fitting	C₄⋊C₄×C₄₁	C₄
socle	C₂²×C₄₁	C₂²⋊C₄
prime radical	#13050139	C₂
cosocle	C₂×C₄×D₄₁	C₂²
hypercenter	C₄⋊C₄	#4223
hypocenter	C₄₁	#327
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. #13050221	C₂	semidirect
2. (C₂×#4223)⋊C₄	C₂	semidirect
6. C₂×D₁₆₄	C₄	semidirect
9. C₄⋊C₄×C₄₁	C₄	semidirect
43. C₄₁	#327	semidirect
52. C₄⋊C₄	#4223	semidirect

Maximal subgroups

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

subgroup	index
#13050221	2
(C₂×#4223)⋊C₄	2
#13050139	2
#327	41