Subgroups — #27213352

Order 13504.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809951	C₂	1	2	yes	maximal
2	#20269977	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₈₄₄	C₂	1	1	yes	maximal, characteristic
4	#20269902	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₄₂₂	C₂²	1	2	yes	—
7	C₂×D₈₄₄	C₂²	1	1	yes	characteristic
8	C₂×C₄×D₂₁₁	—	2	1	no	—
9	#16809951	—	2	4	no	—
10	C₄⋊Dic₂₁₁	—	2	1	no	—
11	C₄⋊Dic₂₁₁	—	2	1	no	—
12	C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
13	(C₂×D₂₁₁)⋊C₄	—	2	1	no	—
14	C₂×C₈×C₂₁₁	C₂²	1	2	yes	—
15	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂²×D₂₁₁	—	2	1	no	—
17	Dic₄₂₂	—	2	2	no	—
18	Dic₄₂₂	—	2	2	no	—
19	D₈₄₄	—	2	1	no	—
20	D₈₄₄	D₄	1	2	yes	—
21	C₂×Dic₂₁₁	—	2	1	no	—
22	C₂×Dic₂₁₁	—	2	2	no	—
23	C₄×D₂₁₁	—	4	2	no	—
24	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
25	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₈×C₂₁₁	—	2	2	no	—
28	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
29	C₂×D₂₁₁	—	2	2	no	—
30	C₂×D₂₁₁	—	4	1	no	—
31	C₂×D₂₁₁	—	4	1	no	—
32	C₄×C₂₁₁	—	2	1	no	—
33	C₄×C₂₁₁	SD₁₆	1	2	yes	complemented
34	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
35	C₄×C₂₁₁	#41	1	1	yes	characteristic
36	Dic₂₁₁	—	4	2	no	—
37	Dic₂₁₁	—	4	1	no	—
38	C₂×C₂₁₁	#121	1	1	yes	characteristic
39	C₂×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
40	C₂×C₂₁₁	#137	1	1	yes	characteristic
41	D₂₁₁	—	4	2	no	—
42	C₂₁₁	#461	1	1	yes	characteristic, complemented, Sylow
43	#461	—	211	1	no	maximal, Sylow
44	C₂×SD₁₆	—	211	2	no	—
45	C₄⋊Q₈	—	211	1	no	—
46	C₄×D₄	—	211	1	no	—
47	C₄⋊C₈	D₂₁₁	1	1	yes	characteristic, complemented
48	#103	—	211	2	no	—
49	C₂×Q₈	—	211	2	no	—
50	C₂×D₄	—	211	1	no	—
51	C₂²×C₄	—	422	1	no	—
52	SD₁₆	—	422	4	no	—
53	C₂×C₈	C₂×D₂₁₁	1	2	yes	—
54	C₄⋊C₄	—	422	1	no	—
55	C₄⋊C₄	—	422	1	no	—
56	C₄⋊C₄	—	211	1	no	—
57	C₂²⋊C₄	—	422	1	no	—
58	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
59	C₂³	—	422	1	no	—
60	Q₈	—	422	2	no	—
61	Q₈	—	422	2	no	—
62	D₄	—	422	1	no	—
63	D₄	—	211	2	no	—
64	C₂×C₄	—	844	2	no	—
65	C₂×C₄	—	422	1	no	—
66	C₂×C₄	—	422	2	no	—
67	C₂×C₄	D₈₄₄	1	1	yes	characteristic
68	C₂×C₄	D₈₄₄	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
70	C₈	—	2	2	no	—
71	C₂²	—	844	1	no	—
72	C₂²	—	844	1	no	—
73	C₂²	—	422	2	no	—
74	C₂²	C₂×D₈₄₄	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	844	2	no	—
77	C₄	—	844	1	no	—
78	C₄	#16809951	1	2	yes	complemented
79	C₄	D₄×D₂₁₁	1	1	yes	characteristic
80	C₄	#16809963	1	1	yes	characteristic
81	C₂	—	844	2	no	—
82	C₂	#20269952	1	1	yes	characteristic
83	C₂	C₂×#16809951	1	1	yes	characteristic
84	C₂	#20270010	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈₄₄
commutator, cosocle	C₂×C₄×C₂₁₁	C₂³
Frattini	C₂×C₄	C₂²×D₂₁₁
Fitting	C₄⋊C₈×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×D₄
prime radical	C₂×D₈₄₄	C₂²
hypercenter	C₄⋊C₈	D₂₁₁
hypocenter	C₂₁₁	#461
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
2. #20269977	C₂	semidirect
4. #20269902	C₂	semidirect
5. C₄⋊C₈×C₂₁₁	C₂	semidirect
33. C₄×C₂₁₁	SD₁₆	semidirect
42. C₂₁₁	#461	semidirect
47. C₄⋊C₈	D₂₁₁	semidirect
78. C₄	#16809951	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809951	2
#20269977	2
C₄×D₈₄₄	2
#20269902	2
C₄⋊C₈×C₂₁₁	2
#461	211