Subgroups — #41012025

Order 27008.

196 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#27213203	C₂	1	2	yes	maximal
2	#27212888	C₂	1	4	yes	maximal, complemented
3	C₂×#125×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₂₁₁⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₂₁₁⋊C₈)	—	2	2	no	—
6	#20269888	C₂²	1	4	yes	—
7	#125×C₂₁₁	—	2	2	no	—
8	#125×C₂₁₁	C₄	1	2	yes	—
9	#125×C₂₁₁	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₁₁	C₄	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	C₂×C₂₁₁⋊C₈	—	2	4	no	—
14	C₂²×C₄×C₂₁₁	—	2	2	no	—
15	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
17	C₄⋊C₄×C₂₁₁	—	2	4	no	—
18	C₄⋊C₄×C₂₁₁	—	2	2	no	—
19	C₄⋊C₄×C₂₁₁	—	2	4	no	—
20	C₄⋊C₄×C₂₁₁	C₂×C₄	1	4	yes	—
21	C₄²×C₂₁₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₂₁₁	D₄	1	2	yes	—
23	C₄²×C₂₁₁	C₂³	1	1	yes	characteristic
24	C₂₁₁⋊C₈	—	4	4	no	—
25	C₂³×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₁₁	—	2	2	no	—
27	C₂×C₄×C₂₁₁	—	2	4	no	—
28	C₂×C₄×C₂₁₁	—	4	2	no	—
29	C₂×C₄×C₂₁₁	—	4	4	no	—
30	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	2	yes	—
31	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
34	C₂²×C₂₁₁	—	2	2	no	—
35	C₂²×C₂₁₁	#101	1	2	yes	—
36	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
37	C₄×C₂₁₁	—	2	1	no	—
38	C₄×C₂₁₁	—	2	1	no	—
39	C₄×C₂₁₁	—	2	2	no	—
40	C₄×C₂₁₁	—	2	2	no	—
41	C₄×C₂₁₁	—	4	4	no	—
42	C₂×C₂₁₁	—	2	2	no	—
43	C₂×C₂₁₁	C₂×#101	1	1	yes	characteristic
44	C₂×C₂₁₁	#421	1	2	yes	—
45	C₂₁₁	#1534	1	1	yes	characteristic, complemented, Sylow
46	#1534	—	211	1	no	maximal, Sylow
47	C₂×#125	D₂₁₁	1	1	yes	characteristic
48	#432	—	211	2	no	—
49	#330	—	211	4	no	—
50	#125	—	2	2	no	—
51	#125	Dic₂₁₁	1	2	yes	—
52	#125	C₂×D₂₁₁	1	2	yes	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄⋊C₄	Dic₂₁₁	1	2	yes	—
55	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
56	C₄⋊C₈	—	422	2	no	—
57	C₂²⋊C₈	—	422	2	no	—
58	#97	—	211	4	no	—
59	C₂²×C₄	—	2	2	no	—
60	C₂²×C₄	C₂×Dic₂₁₁	1	1	yes	characteristic
61	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
62	C₂×C₈	—	422	4	no	—
63	C₄⋊C₄	—	2	4	no	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	4	no	—
66	C₄⋊C₄	C₂×Dic₂₁₁	1	4	yes	—
67	C₄²	C₂×Dic₂₁₁	1	1	yes	characteristic
68	C₄²	C₂²⋊D₂₁₁	1	2	yes	—
69	C₄²	C₂²×D₂₁₁	1	1	yes	characteristic
70	C₂³	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	4	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	—	4	4	no	—
75	C₂×C₄	C₂²⋊Dic₂₁₁	1	2	yes	—
76	C₂×C₄	C₂²⋊Dic₂₁₁	1	1	yes	characteristic
77	C₂×C₄	C₂²×Dic₂₁₁	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₂₁₁	1	2	yes	—
79	C₈	—	844	4	no	—
80	C₂²	—	2	2	no	—
81	C₂²	#20269896	1	2	yes	—
82	C₂²	C₂×C₂²⋊Dic₂₁₁	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	2	1	no	—
85	C₄	—	2	2	no	—
86	C₄	—	2	2	no	—
87	C₄	—	4	4	no	—
88	C₂	—	2	2	no	—
89	C₂	C₂×#20269896	1	1	yes	characteristic
90	C₂	#27213163	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₄²	C₂²×D₂₁₁
Fitting	C₂×#125×C₂₁₁	C₂
socle	C₂²×C₂₁₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₁₁	C₂²⋊C₄
cosocle	C₄²×C₂₁₁	C₂³
hypercenter	C₂×#125	D₂₁₁
hypocenter	C₂₁₁	#1534
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂²×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

Complements

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

N	H ≅ G/N	type
2. #27212888	C₂	semidirect
45. C₂₁₁	#1534	semidirect

Maximal subgroups

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

subgroup	index
#27213203	2
#27212888	2
C₂×#125×C₂₁₁	2
#1534	211