Subgroups — #27213623

Order 13504.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Dic₄₂₂	C₂	1	1	yes	maximal, characteristic
2	C₄×Dic₄₂₂	C₂	1	4	yes	maximal, complemented
3	#20269942	C₂	1	4	yes	maximal, complemented
4	#20269936	C₂	1	1	yes	maximal, characteristic
5	#20269933	C₂	1	4	yes	maximal
6	#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
7	C₂×Dic₄₂₂	C₄	1	8	yes	complemented
8	C₂×Dic₄₂₂	C₂²	1	2	yes	—
9	C₂×Dic₄₂₂	C₂²	1	2	yes	—
10	C₂²×Dic₂₁₁	C₂²	1	2	yes	—
11	C₄⋊Dic₂₁₁	C₂²	1	2	yes	—
12	Dic₂₁₁⋊C₄	C₂²	1	8	yes	—
13	C₂²⋊Dic₂₁₁	C₂²	1	2	yes	—
14	C₄×Dic₂₁₁	C₂²	1	8	yes	—
15	C₄×Dic₂₁₁	C₂²	1	2	yes	—
16	C₂²×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₂₁₁	C₂²	1	2	yes	—
19	C₄²×C₂₁₁	C₂²	1	2	yes	—
20	Dic₄₂₂	C₂×C₄	1	8	yes	—
21	Dic₄₂₂	C₂×C₄	1	8	yes	—
22	C₂×Dic₂₁₁	C₂×C₄	1	8	yes	—
23	C₂×Dic₂₁₁	C₂³	1	4	yes	—
24	C₂×Dic₂₁₁	C₂³	1	4	yes	—
25	C₂³×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	C₂×C₄	1	4	yes	—
27	C₂×C₄×C₂₁₁	C₂³	1	4	yes	—
28	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
29	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
30	C₂²×C₂₁₁	—	2	2	no	—
31	C₂²×C₂₁₁	C₂²×C₄	1	2	yes	—
32	C₂²×C₂₁₁	C₂⁴	1	1	yes	characteristic
33	C₄×C₂₁₁	—	2	4	no	—
34	C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
35	C₄×C₂₁₁	C₂²×C₄	1	2	yes	—
36	Dic₂₁₁	—	2	4	no	—
37	Dic₂₁₁	C₂²×C₄	1	8	yes	—
38	C₂×C₂₁₁	—	2	2	no	—
39	C₂×C₂₁₁	C₂³×C₄	1	1	yes	characteristic
40	C₂×C₂₁₁	2₋¹⁺⁴	1	2	yes	—
41	C₂₁₁	#519	1	1	yes	characteristic, complemented, Sylow
42	#519	—	211	1	no	maximal, Sylow
43	C₂²×Q₈	—	211	1	no	—
44	C₄×Q₈	—	211	4	no	—
45	C₄×Q₈	—	211	4	no	—
46	#117	—	211	1	no	—
47	#117	—	211	4	no	—
48	#117	D₂₁₁	1	1	yes	characteristic
49	C₂×Q₈	—	211	8	no	—
50	C₂×Q₈	—	211	2	no	—
51	C₂×Q₈	—	211	2	no	—
52	C₂²×C₄	—	211	2	no	—
53	C₂²×C₄	C₂×D₂₁₁	1	1	yes	characteristic
54	C₄⋊C₄	—	211	8	no	—
55	C₄⋊C₄	—	211	2	no	—
56	C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
57	C₂²⋊C₄	—	211	2	no	—
58	C₂²⋊C₄	C₂×D₂₁₁	1	2	yes	—
59	C₄²	—	211	2	no	—
60	C₄²	—	211	8	no	—
61	C₄²	C₂×D₂₁₁	1	2	yes	—
62	C₂³	C₂²×D₂₁₁	1	1	yes	characteristic
63	Q₈	—	211	8	no	—
64	Q₈	—	211	8	no	—
65	C₂×C₄	—	211	4	no	—
66	C₂×C₄	—	211	8	no	—
67	C₂×C₄	—	211	4	no	—
68	C₂×C₄	C₄×D₂₁₁	1	4	yes	—
69	C₂×C₄	C₂²×D₂₁₁	1	4	yes	—
70	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
71	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
72	C₂²	—	2	2	no	—
73	C₂²	C₂×C₄×D₂₁₁	1	2	yes	—
74	C₂²	C₂³×D₂₁₁	1	1	yes	characteristic
75	C₄	—	2	4	no	—
76	C₄	—	422	4	no	—
77	C₄	—	211	8	no	—
78	C₄	C₂×C₄×D₂₁₁	1	2	yes	—
79	C₄	C₂×C₄×D₂₁₁	1	2	yes	—
80	C₂	—	2	2	no	—
81	C₂	C₂²×C₄×D₂₁₁	1	1	yes	characteristic
82	C₂	#20270024	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₁₁
commutator	C₂×C₂₁₁	C₂³×C₄
Fitting	#117×C₂₁₁	C₂
socle, cosocle	C₂²×C₂₁₁	C₂⁴
prime radical	C₂³×C₂₁₁	C₂³
hypercenter	#117	D₂₁₁
hypocenter	C₂₁₁	#519
LCS	G ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×Dic₄₂₂	C₂	semidirect
3. #20269942	C₂	semidirect
7. C₂×Dic₄₂₂	C₄	semidirect
41. C₂₁₁	#519	semidirect

Maximal subgroups

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

subgroup	index
C₂²×Dic₄₂₂	2
C₄×Dic₄₂₂	2
#20269942	2
#20269936	2
#20269933	2
#117×C₂₁₁	2
#519	211