Subgroups — #27213703

Order 13504.

222 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20269977	C₂	1	2	yes	maximal, complemented
2	#20269969	C₂	1	2	yes	maximal, complemented
3	#20269956	C₂	1	4	yes	maximal
4	C₄×Dic₄₂₂	C₂	1	4	yes	maximal, complemented
5	#20269936	C₂	1	2	yes	maximal
6	C₂×C₄²×C₂₁₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₄₂₂	C₂²	1	4	yes	—
8	C₄⋊Dic₂₁₁	C₂²	1	4	yes	—
9	C₄⋊Dic₂₁₁	C₂²	1	4	yes	—
10	Dic₂₁₁⋊C₄	C₂²	1	8	yes	—
11	C₂²⋊Dic₂₁₁	C₂²	1	4	yes	—
12	C₄×Dic₂₁₁	C₂²	1	4	yes	—
13	C₂²×C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₄²×C₂₁₁	C₂²	1	2	yes	—
16	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
18	Dic₄₂₂	—	2	8	no	—
19	C₂×Dic₂₁₁	C₂³	1	8	yes	—
20	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
21	C₂×C₄×C₂₁₁	—	2	4	no	—
22	C₂×C₄×C₂₁₁	Q₈	1	4	yes	complemented
23	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
24	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	C₂³	1	2	yes	—
26	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₂²×C₂₁₁	—	2	2	no	—
28	C₂²×C₂₁₁	C₂×Q₈	1	2	yes	—
29	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
30	C₄×C₂₁₁	—	2	2	no	—
31	C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
32	C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
33	C₄×C₂₁₁	#41	1	4	yes	—
34	Dic₂₁₁	—	2	8	no	—
35	C₂×C₂₁₁	—	2	2	no	—
36	C₂×C₂₁₁	C₂²×Q₈	1	1	yes	characteristic
37	C₂×C₂₁₁	C₂×#41	1	2	yes	—
38	C₂₁₁	#533	1	1	yes	characteristic, complemented, Sylow
39	#533	—	211	1	no	maximal, Sylow
40	C₄⋊Q₈	—	211	2	no	—
41	#125	—	211	2	no	—
42	C₂²⋊Q₈	—	211	4	no	—
43	C₄×Q₈	—	211	4	no	—
44	#117	—	211	2	no	—
45	C₂×C₄²	D₂₁₁	1	1	yes	characteristic
46	C₂×Q₈	—	211	4	no	—
47	C₂²×C₄	C₂×D₂₁₁	1	2	yes	—
48	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
49	C₄⋊C₄	—	211	4	no	—
50	C₄⋊C₄	—	211	8	no	—
51	C₄⋊C₄	—	211	4	no	—
52	C₂²⋊C₄	—	211	4	no	—
53	C₄²	—	211	4	no	—
54	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
55	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
56	C₄²	C₂×D₂₁₁	1	2	yes	—
57	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
58	Q₈	—	422	8	no	—
59	C₂×C₄	—	2	4	no	—
60	C₂×C₄	—	211	8	no	—
61	C₂×C₄	Dic₄₂₂	1	4	yes	complemented
62	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
63	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
64	C₂×C₄	C₂²×D₂₁₁	1	2	yes	—
65	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
66	C₂²	—	2	2	no	—
67	C₂²	C₂×Dic₄₂₂	1	2	yes	—
68	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
69	C₄	—	2	2	no	—
70	C₄	—	422	8	no	—
71	C₄	C₂×Dic₄₂₂	1	2	yes	—
72	C₄	C₂×Dic₄₂₂	1	2	yes	—
73	C₄	#16809962	1	4	yes	—
74	C₂	—	2	2	no	—
75	C₂	C₂²×Dic₄₂₂	1	1	yes	characteristic
76	C₂	C₂×#16809962	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₁₁
commutator, socle, cosocle	C₂²×C₂₁₁	C₂⁴
Frattini	C₂²	C₂³×D₂₁₁
Fitting	C₂×C₄²×C₂₁₁	C₂
prime radical	C₂³×C₂₁₁	C₂³
hypercenter	C₂×C₄²	D₂₁₁
hypocenter	C₂₁₁	#533
LCS	G ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #20269977	C₂	semidirect
2. #20269969	C₂	semidirect
4. C₄×Dic₄₂₂	C₂	semidirect
22. C₂×C₄×C₂₁₁	Q₈	semidirect
38. C₂₁₁	#533	semidirect
61. C₂×C₄	Dic₄₂₂	semidirect

Maximal subgroups

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

subgroup	index
#20269977	2
#20269969	2
#20269956	2
C₄×Dic₄₂₂	2
#20269936	2
C₂×C₄²×C₂₁₁	2
#533	211