Subgroups — #14581052

Order 2624.

236 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13050153	C₂	1	4	yes	maximal, complemented
2	#13050139	C₂	1	4	yes	maximal, complemented
3	C₄×D₁₆₄	C₂	1	2	yes	maximal, complemented
4	#13050122	C₂	1	4	yes	maximal, complemented
5	C₄⋊Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₁	C₂²	1	2	yes	—
7	C₂×D₁₆₄	C₂²	1	2	yes	—
8	C₂×D₁₆₄	C₂²	1	4	yes	—
9	C₂×C₄×D₄₁	C₂²	1	4	yes	—
10	C₄⋊C₄×C₄₁	C₂²	1	4	yes	—
11	C₄²×C₄₁	C₂²	1	1	yes	characteristic
12	#10620281	C₂²	1	8	yes	—
13	#10620281	C₂²	1	4	yes	—
14	C₄⋊Dic₄₁	C₂²	1	2	yes	—
15	C₄×Dic₄₁	C₂²	1	4	yes	—
16	C₂²×D₄₁	C₂³	1	4	yes	—
17	Q₈×C₄₁	—	4	2	no	—
18	C₂×C₄×C₄₁	C₂³	1	4	yes	—
19	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
20	C₂×C₄×C₄₁	C₂³	1	2	yes	—
21	C₂×Dic₄₁	C₂³	1	4	yes	—
22	D₁₆₄	—	2	4	no	—
23	D₁₆₄	—	4	4	no	—
24	C₄×D₄₁	—	2	8	no	—
25	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
26	C₂×D₄₁	—	2	4	no	—
27	C₂×D₄₁	—	2	4	no	—
28	C₂×D₄₁	—	2	4	no	—
29	C₄×C₄₁	—	2	4	no	—
30	C₄×C₄₁	—	2	1	no	—
31	C₄×C₄₁	#41	1	4	yes	—
32	Dic₄₁	—	2	4	no	—
33	C₂×C₄₁	C₂×#41	1	2	yes	—
34	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
35	D₄₁	—	4	4	no	—
36	#555	—	41	1	no	maximal, Sylow
37	C₄₁	#555	1	1	yes	characteristic, complemented, Sylow
38	C₄⋊Q₈	D₄₁	1	1	yes	characteristic, complemented
39	#124	—	41	4	no	—
40	#121	—	41	4	no	—
41	C₄×D₄	—	41	2	no	—
42	#117	—	41	4	no	—
43	C₂×Q₈	C₂×D₄₁	1	2	yes	—
44	C₂×D₄	—	41	2	no	—
45	C₂×D₄	—	41	4	no	—
46	C₂²×C₄	—	41	4	no	—
47	C₄⋊C₄	—	41	2	no	—
48	C₄⋊C₄	C₂×D₄₁	1	4	yes	—
49	C₂²⋊C₄	—	41	8	no	—
50	C₂²⋊C₄	—	41	4	no	—
51	C₄²	—	41	4	no	—
52	C₄²	C₂×D₄₁	1	1	yes	characteristic
53	C₂³	—	41	4	no	—
54	Q₈	—	4	2	no	—
55	D₄	—	82	4	no	—
56	D₄	—	164	4	no	—
57	C₂×C₄	—	41	4	no	—
58	C₂×C₄	—	82	8	no	—
59	C₂×C₄	C₂²×D₄₁	1	4	yes	—
60	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
61	C₂×C₄	C₂²×D₄₁	1	2	yes	—
62	C₂²	—	82	4	no	—
63	C₂²	—	82	4	no	—
64	C₂²	—	82	4	no	—
65	C₂²	C₂³×D₄₁	1	1	yes	characteristic
66	C₄	—	2	4	no	—
67	C₄	—	2	1	no	—
68	C₄	—	82	4	no	—
69	C₄	#10620310	1	4	yes	—
70	C₂	—	164	4	no	—
71	C₂	C₂×#10620310	1	2	yes	—
72	C₂	#13050209	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₁
commutator, socle, cosocle	C₂²×C₄₁	C₂⁴
Fitting	C₄⋊Q₈×C₄₁	C₂
hypercenter	C₄⋊Q₈	D₄₁
hypocenter	C₄₁	#555
LCS	G ▷ C₂²×C₄₁ ▷ C₄₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. #13050153	C₂	semidirect
2. #13050139	C₂	semidirect
3. C₄×D₁₆₄	C₂	semidirect
4. #13050122	C₂	semidirect
5. C₄⋊Q₈×C₄₁	C₂	semidirect
37. C₄₁	#555	semidirect
38. 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
#13050153	2
#13050139	2
C₄×D₁₆₄	2
#13050122	2
C₄⋊Q₈×C₄₁	2
#555	41