Subgroups — #14580684

Order 2624.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050161	C₂	1	1	yes	maximal, characteristic
2	#13050084	C₂	1	4	yes	maximal, complemented
3	C₄×#66491	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×C₄₁	—	2	1	no	—
6	C₂×D₁₆₄	—	2	1	no	—
7	C₂×D₁₆₄	—	2	1	no	—
8	C₂×D₁₆₄	C₂²	1	2	yes	—
9	C₄⋊C₄×C₄₁	—	2	1	no	—
10	C₄⋊C₄×C₄₁	C₂²	1	2	yes	—
11	C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	C₂×#66491	C₂²	1	2	yes	—
13	C₂²×D₄₁	—	2	2	no	—
14	Q₈×C₄₁	—	4	2	no	—
15	C₂×C₄×C₄₁	—	2	2	no	—
16	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
17	C₂×C₄×C₄₁	D₄	1	1	yes	characteristic
18	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
19	D₁₆₄	—	2	4	no	—
20	D₁₆₄	—	4	2	no	—
21	D₁₆₄	—	4	2	no	—
22	#66491	—	2	2	no	—
23	C₂²×C₄₁	C₂×D₄	1	1	yes	characteristic
24	C₂×D₄₁	—	4	2	no	—
25	C₂×D₄₁	—	4	2	no	—
26	C₂×D₄₁	—	4	2	no	—
27	C₄×C₄₁	—	4	2	no	—
28	C₄×C₄₁	D₈	1	2	yes	—
29	C₄×C₄₁	SD₁₆	1	2	yes	—
30	C₄×C₄₁	#41	1	2	yes	—
31	C₂×C₄₁	#124	1	1	yes	characteristic
32	C₂×C₄₁	C₂×D₈	1	1	yes	characteristic
33	C₂×C₄₁	C₂×SD₁₆	1	1	yes	characteristic
34	D₄₁	—	8	2	no	—
35	#486	—	41	1	no	maximal, Sylow
36	C₄₁	#486	1	1	yes	characteristic, complemented, Sylow
37	C₄⋊Q₈	D₄₁	1	1	yes	characteristic, complemented
38	#127	—	41	1	no	—
39	#102	—	41	4	no	—
40	C₄×C₈	—	41	1	no	—
41	C₂×Q₈	—	2	1	no	—
42	C₂×D₄	—	41	2	no	—
43	C₂×D₄	—	82	1	no	—
44	C₂×D₄	—	82	1	no	—
45	C₂×C₈	—	41	2	no	—
46	C₄⋊C₄	—	2	1	no	—
47	C₄⋊C₄	C₂×D₄₁	1	2	yes	—
48	C₄²	C₂×D₄₁	1	1	yes	characteristic
49	C₂³	—	82	2	no	—
50	Q₈	—	4	2	no	—
51	D₄	—	82	4	no	—
52	D₄	—	164	2	no	—
53	D₄	—	164	2	no	—
54	C₂×C₄	—	2	2	no	—
55	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
56	C₂×C₄	C₂²⋊D₄₁	1	1	yes	characteristic
57	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
58	C₈	—	82	2	no	—
59	C₂²	—	164	2	no	—
60	C₂²	—	164	2	no	—
61	C₂²	—	164	2	no	—
62	C₂²	C₂×C₂²⋊D₄₁	1	1	yes	characteristic
63	C₄	—	4	2	no	—
64	C₄	#10620282	1	2	yes	—
65	C₄	#10620284	1	2	yes	—
66	C₄	#10620310	1	2	yes	—
67	C₂	—	328	2	no	—
68	C₂	#13050153	1	1	yes	characteristic
69	C₂	C₂×#10620282	1	1	yes	characteristic
70	C₂	C₂×#10620284	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₄₁
commutator, cosocle	C₂×C₄×C₄₁	C₂³
Frattini	C₂×C₄	C₂²×D₄₁
Fitting	C₄⋊Q₈×C₄₁	C₂
socle	C₂²×C₄₁	C₂×D₄
prime radical	#13050161	C₂
hypercenter	C₄⋊Q₈	D₄₁
hypocenter	C₄₁	#486
LCS	G ▷ C₂×C₄×C₄₁ ▷ C₂×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
2. #13050084	C₂	semidirect
3. C₄×#66491	C₂	semidirect
4. C₄⋊Q₈×C₄₁	C₂	semidirect
36. C₄₁	#486	semidirect
37. C₄⋊Q₈	D₄₁	semidirect

Maximal subgroups

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

subgroup	index
#13050161	2
#13050084	2
C₄×#66491	2
C₄⋊Q₈×C₄₁	2
#486	41