Subgroups — #27213351

Order 13504.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16809950	C₂	1	2	yes	maximal
2	#20269977	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊(C₂₁₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
4	#20269900	C₂	1	2	yes	maximal, complemented
5	C₄×D₄×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₂	C₂²	1	2	yes	—
7	#16809950	—	2	4	no	—
8	C₂×C₂₁₁⋊C₈	C₂²	1	2	yes	—
9	C₄⋊Dic₂₁₁	—	2	1	no	—
10	C₄⋊Dic₂₁₁	—	2	1	no	—
11	C₂×D₄×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₂₁₁	—	2	1	no	—
13	C₄⋊C₄×C₂₁₁	C₂²	1	1	yes	characteristic
14	C₂²⋊C₄×C₂₁₁	—	2	1	no	—
15	C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	Dic₄₂₂	—	2	2	no	—
17	Dic₄₂₂	—	2	2	no	—
18	C₂×Dic₂₁₁	—	2	2	no	—
19	C₂₁₁⋊C₈	—	2	2	no	—
20	C₂³×C₂₁₁	—	2	1	no	—
21	D₄×C₂₁₁	—	2	1	no	—
22	D₄×C₂₁₁	D₄	1	2	yes	—
23	C₂×C₄×C₂₁₁	—	2	1	no	—
24	C₂×C₄×C₂₁₁	—	4	2	no	—
25	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
28	C₂²×C₂₁₁	—	2	2	no	—
29	C₂²×C₂₁₁	—	4	1	no	—
30	C₂²×C₂₁₁	—	4	1	no	—
31	C₂²×C₂₁₁	C₂×D₄	1	1	yes	characteristic
32	C₄×C₂₁₁	—	2	1	no	—
33	C₄×C₂₁₁	—	4	1	no	—
34	C₄×C₂₁₁	SD₁₆	1	2	yes	complemented
35	C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
36	C₄×C₂₁₁	#41	1	1	yes	characteristic
37	Dic₂₁₁	—	4	2	no	—
38	C₂×C₂₁₁	—	4	2	no	—
39	C₂×C₂₁₁	#121	1	1	yes	characteristic
40	C₂×C₂₁₁	C₂×SD₁₆	1	1	yes	characteristic
41	C₂×C₂₁₁	#137	1	1	yes	characteristic
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	D₂₁₁	1	1	yes	characteristic
47	C₄⋊C₈	—	211	1	no	—
48	#103	—	211	2	no	—
49	C₂×Q₈	—	211	2	no	—
50	C₂×D₄	C₂×D₂₁₁	1	1	yes	characteristic
51	C₂²×C₄	—	2	1	no	—
52	SD₁₆	—	422	4	no	—
53	C₂×C₈	—	211	2	no	—
54	C₄⋊C₄	—	422	1	no	—
55	C₄⋊C₄	—	422	1	no	—
56	C₄⋊C₄	C₂×D₂₁₁	1	1	yes	characteristic
57	C₂²⋊C₄	—	2	1	no	—
58	C₄²	C₂×D₂₁₁	1	1	yes	characteristic
59	C₂³	—	2	1	no	—
60	Q₈	—	422	2	no	—
61	Q₈	—	422	2	no	—
62	D₄	—	2	1	no	—
63	D₄	D₈₄₄	1	2	yes	—
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	—	422	2	no	—
67	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
68	C₂×C₄	C₂²⋊D₂₁₁	1	1	yes	characteristic
69	C₂×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
70	C₈	—	422	2	no	—
71	C₂²	—	2	2	no	—
72	C₂²	—	4	1	no	—
73	C₂²	—	4	1	no	—
74	C₂²	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	4	1	no	—
77	C₄	—	844	2	no	—
78	C₄	#16809950	1	2	yes	complemented
79	C₄	C₂×D₈₄₄	1	1	yes	characteristic
80	C₄	#16809962	1	1	yes	characteristic
81	C₂	—	4	2	no	—
82	C₂	#20269951	1	1	yes	characteristic
83	C₂	C₂×#16809950	1	1	yes	characteristic
84	C₂	#20270007	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #20269977	C₂	semidirect
3. C₄⋊(C₂₁₁⋊C₈)	C₂	semidirect
4. #20269900	C₂	semidirect
34. C₄×C₂₁₁	SD₁₆	semidirect
42. C₂₁₁	#461	semidirect
78. C₄	#16809950	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16809950	2
#20269977	2
C₄⋊(C₂₁₁⋊C₈)	2
#20269900	2
C₄×D₄×C₂₁₁	2
#461	211