Subgroups — #14581008

Order 2624.

312 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#10620310	C₂	1	2	yes	maximal, complemented
2	#13050161	C₂	1	3	yes	maximal, complemented
3	#13050139	C₂	1	6	yes	maximal, complemented
4	C₄×D₁₆₄	C₂	1	3	yes	maximal, complemented
5	C₄×Q₈×C₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×C₄₁	C₂²	1	1	yes	characteristic
7	#10620310	—	2	8	no	—
8	C₂×D₁₆₄	C₂²	1	3	yes	—
9	C₂×D₁₆₄	C₂²	1	6	yes	—
10	C₂×D₁₆₄	C₂²	1	6	yes	—
11	C₂×C₄×D₄₁	C₂²	1	6	yes	—
12	C₄⋊C₄×C₄₁	C₂²	1	3	yes	—
13	C₄²×C₄₁	C₂²	1	3	yes	—
14	#10620281	C₂²	1	6	yes	—
15	C₄⋊Dic₄₁	C₂²	1	1	yes	characteristic
16	C₂²×D₄₁	C₂³	1	6	yes	—
17	Q₈×C₄₁	D₄	1	4	yes	complemented
18	C₂×C₄×C₄₁	C₂³	1	3	yes	—
19	C₂×C₄×C₄₁	C₂³	1	3	yes	—
20	C₂×C₄×C₄₁	C₂³	1	1	yes	characteristic
21	C₂×Dic₄₁	C₂³	1	2	yes	—
22	D₁₆₄	—	2	12	no	—
23	D₁₆₄	—	2	6	no	—
24	D₁₆₄	—	4	6	no	—
25	C₄×D₄₁	—	2	12	no	—
26	C₂²×C₄₁	C₂⁴	1	1	yes	characteristic
27	C₂×D₄₁	—	2	6	no	—
28	C₂×D₄₁	—	2	6	no	—
29	C₂×D₄₁	—	2	6	no	—
30	C₄×C₄₁	—	2	3	no	—
31	C₄×C₄₁	C₂×D₄	1	6	yes	—
32	C₄×C₄₁	#41	1	2	yes	complemented
33	Dic₄₁	—	2	2	no	—
34	C₂×C₄₁	C₂²×D₄	1	1	yes	characteristic
35	C₂×C₄₁	C₂×#41	1	1	yes	characteristic
36	C₂×C₄₁	H(5,𝔽₂)	1	1	yes	characteristic
37	D₄₁	—	4	6	no	—
38	#550	—	41	1	no	maximal, Sylow
39	C₄₁	#550	1	1	yes	characteristic, complemented, Sylow
40	C₂×#41	—	41	2	no	—
41	#127	—	41	3	no	—
42	#121	—	41	6	no	—
43	C₄×Q₈	D₄₁	1	1	yes	characteristic, complemented
44	C₄×D₄	—	41	3	no	—
45	#41	—	82	8	no	—
46	C₂×Q₈	C₂×D₄₁	1	1	yes	characteristic
47	C₂×D₄	—	41	3	no	—
48	C₂×D₄	—	41	6	no	—
49	C₂×D₄	—	41	6	no	—
50	C₂²×C₄	—	41	6	no	—
51	C₄⋊C₄	—	41	1	no	—
52	C₄⋊C₄	C₂×D₄₁	1	3	yes	—
53	C₂²⋊C₄	—	41	6	no	—
54	C₄²	C₂×D₄₁	1	3	yes	—
55	C₂³	—	41	6	no	—
56	Q₈	D₁₆₄	1	4	yes	complemented
57	D₄	—	82	12	no	—
58	D₄	—	82	6	no	—
59	D₄	—	164	6	no	—
60	C₂×C₄	—	41	2	no	—
61	C₂×C₄	—	82	12	no	—
62	C₂×C₄	C₂²×D₄₁	1	3	yes	—
63	C₂×C₄	C₂²×D₄₁	1	3	yes	—
64	C₂×C₄	C₂²×D₄₁	1	1	yes	characteristic
65	C₂²	—	82	6	no	—
66	C₂²	—	82	6	no	—
67	C₂²	—	82	6	no	—
68	C₂²	C₂³×D₄₁	1	1	yes	characteristic
69	C₄	—	2	3	no	—
70	C₄	—	82	2	no	—
71	C₄	C₂×D₁₆₄	1	6	yes	—
72	C₄	#10620310	1	2	yes	complemented
73	C₂	—	164	6	no	—
74	C₂	C₂²×D₁₆₄	1	1	yes	characteristic
75	C₂	C₂×#10620310	1	1	yes	characteristic
76	C₂	#13050210	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	#550
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. C₂×#10620310	C₂	semidirect
2. #13050161	C₂	semidirect
3. #13050139	C₂	semidirect
4. C₄×D₁₆₄	C₂	semidirect
5. C₄×Q₈×C₄₁	C₂	semidirect
17. Q₈×C₄₁	D₄	semidirect
32. C₄×C₄₁	#41	semidirect
39. C₄₁	#550	semidirect
43. C₄×Q₈	D₄₁	semidirect
56. Q₈	D₁₆₄	semidirect
72. C₄	#10620310	semidirect

Maximal subgroups

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

subgroup	index
C₂×#10620310	2
#13050161	2
#13050139	2
C₄×D₁₆₄	2
C₄×Q₈×C₄₁	2
#550	41