Group #41013564 — subgroups

Subgroups — #41013564

Order 27008.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20269910	C₂	1	2	yes	maximal
2	C₂×#20269908	C₂	1	2	yes	maximal
3	C₂×#20269888	C₂	1	1	yes	maximal, characteristic
4	#27213012	C₂	1	1	yes	maximal, characteristic
5	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₁₁	C₂²	1	1	yes	characteristic
9	#20269910	—	2	4	no	—
10	#20269908	—	2	4	no	—
11	#20269888	C₄	1	4	yes	complemented
12	C₂₁₁⋊#95	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
14	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
16	C₂²×Dic₂₁₁	—	2	1	no	—
17	C₂²×Dic₂₁₁	—	2	1	no	—
18	C₂×C₂₁₁⋊C₈	C₂×C₄	1	4	yes	—
19	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
20	C₂×C₂₁₁⋊C₈	Q₈	1	4	yes	—
21	C₄⋊Dic₂₁₁	—	2	2	no	—
22	C₄⋊Dic₂₁₁	—	2	2	no	—
23	C₂²×C₄×C₂₁₁	—	2	2	no	—
24	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
25	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₂₁₁	—	2	2	no	—
27	C₄⋊C₄×C₂₁₁	—	2	2	no	—
28	C₄⋊C₄×C₂₁₁	—	4	4	no	—
29	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
30	C₄²×C₂₁₁	Q₈	1	2	yes	—
31	C₂×Dic₂₁₁	—	4	1	no	—
32	C₂×Dic₂₁₁	—	4	1	no	—
33	C₂×Dic₂₁₁	—	4	2	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂₁₁⋊C₈	C₄⋊C₄	1	8	yes	complemented
36	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₁₁	—	2	2	no	—
38	C₂×C₄×C₂₁₁	—	2	2	no	—
39	C₂×C₄×C₂₁₁	—	4	2	no	—
40	C₂×C₄×C₂₁₁	—	4	4	no	—
41	C₂×C₄×C₂₁₁	C₄⋊C₄	1	4	yes	—
42	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₂₁₁	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₁₁	#127	1	1	yes	characteristic
48	C₂²×C₂₁₁	C₄⋊Q₈	1	1	yes	characteristic
49	C₂²×C₂₁₁	AGL(1,ℤ/8)	1	2	yes	—
50	C₂²×C₂₁₁	#137	1	2	yes	—
51	C₄×C₂₁₁	—	2	4	no	—
52	C₄×C₂₁₁	—	8	2	no	—
53	C₄×C₂₁₁	C₂×C₄⋊C₄	1	2	yes	—
54	C₄×C₂₁₁	C₄⋊Q₈	1	2	yes	—
55	Dic₂₁₁	—	8	1	no	—
56	Dic₂₁₁	—	8	1	no	—
57	C₂×C₂₁₁	#384	1	1	yes	characteristic
58	C₂×C₂₁₁	#428	1	2	yes	—
59	C₂×C₂₁₁	#496	1	1	yes	characteristic
60	C₂×C₂₁₁	#497	1	1	yes	characteristic
61	C₂×C₂₁₁	#501	1	2	yes	—
62	C₂₁₁	#1847	1	1	yes	characteristic, complemented, Sylow
63	#1847	—	211	1	no	maximal, Sylow
64	C₂×#107	—	211	2	no	—
65	C₂×#106	—	211	2	no	—
66	C₂×#97	—	211	1	no	—
67	#384	D₂₁₁	1	1	yes	characteristic
68	#382	—	211	1	no	—
69	C₂²×C₈	—	211	2	no	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	211	1	no	—
72	C₂×C₄⋊C₄	—	211	1	no	—
73	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
74	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
75	#107	—	422	4	no	—
76	#106	—	422	4	no	—
77	#97	—	211	4	no	—
78	#95	—	422	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	422	1	no	—
81	C₂²×C₄	—	422	1	no	—
82	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
83	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
84	C₂×C₈	—	211	4	no	—
85	C₂×C₈	—	211	4	no	—
86	C₂×C₈	—	211	4	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₄⋊C₄	—	422	2	no	—
91	C₄⋊C₄	—	422	2	no	—
92	C₄²	C₄×D₂₁₁	1	2	yes	—
93	C₄²	Dic₄₂₂	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	844	1	no	—
100	C₂×C₄	—	844	1	no	—
101	C₂×C₄	—	844	2	no	—
102	C₂×C₄	—	844	2	no	—
103	C₂×C₄	Dic₂₁₁⋊C₄	1	4	yes	—
104	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
105	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
106	C₂×C₄	Q₈×D₂₁₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₄₂₂	1	1	yes	characteristic
108	C₈	—	211	8	no	—
109	C₂²	C₂×Dic₂₁₁⋊C₄	1	1	yes	characteristic
110	C₂²	#20269973	1	1	yes	characteristic
111	C₂²	#20269975	1	1	yes	characteristic
112	C₂²	#20270003	1	2	yes	—
113	C₂²	#20270008	1	2	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	8	2	no	—
116	C₄	—	1688	1	no	—
117	C₄	—	1688	1	no	—
118	C₄	C₄⋊C₄×D₂₁₁	1	2	yes	—
119	C₄	#20269976	1	2	yes	—
120	C₂	#27213015	1	1	yes	characteristic
121	C₂	#27213186	1	2	yes	—
122	C₂	#27213533	1	1	yes	characteristic
123	C₂	#27213538	1	1	yes	characteristic
124	C₂	#27213552	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269910

C₂

yes

maximal

C₂×#20269908

C₂

yes

maximal

C₂×#20269888

C₂

yes

maximal, characteristic

#27213012

C₂

yes

maximal, characteristic

#384×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₂₁₁

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₁₁

C₂²

yes

—

—

C₄

yes

complemented

C₂₁₁⋊#95

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂²×Dic₂₁₁

—

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

C₂×C₄

yes

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₂×C₂₁₁⋊C₈

Q₈

yes

—

C₄⋊Dic₂₁₁

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

Q₈

yes

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

C₄⋊C₄

yes

complemented

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₁₁

#127

yes

characteristic

C₂²×C₂₁₁

C₄⋊Q₈

yes

characteristic

C₂²×C₂₁₁

AGL(1,ℤ/8)

yes

—

C₂²×C₂₁₁

#137

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

C₂×C₄⋊C₄

yes

—

C₄×C₂₁₁

C₄⋊Q₈

yes

—

Dic₂₁₁

—

Dic₂₁₁

—

C₂×C₂₁₁

#384

yes

characteristic

C₂×C₂₁₁

#428

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

—

C₂₁₁

#1847

yes

characteristic, complemented, Sylow

#1847

—

211

maximal, Sylow

C₂×#107

—

211

—

C₂×#106

—

211

—

C₂×#97

—

211

—

#384

D₂₁₁

yes

characteristic

#382

—

211

—

C₂²×C₈

—

211

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

—

211

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#107

—

422

—

#106

—

422

—

#97

—

211

—

#95

—

422

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₂×C₈

—

211

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄⋊C₄

—

422

—

C₄²

C₄×D₂₁₁

yes

—

C₄²

Dic₄₂₂

yes

—

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

844

—

100

C₂×C₄

—

844

—

101

C₂×C₄

—

844

—

102

C₂×C₄

—

844

—

103

C₂×C₄

Dic₂₁₁⋊C₄

yes

—

104

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

105

C₂×C₄

D₄×D₂₁₁

yes

—

106

C₂×C₄

Q₈×D₂₁₁

yes

—

107

C₂×C₄

C₂×Dic₄₂₂

yes

characteristic

108

C₈

—

211

—

109

C₂²

C₂×Dic₂₁₁⋊C₄

yes

characteristic

110

C₂²

#20269973

yes

characteristic

111

C₂²

#20269975

yes

characteristic

112

C₂²

#20270003

yes

—

113

C₂²

#20270008

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

1688

—

117

C₄

—

1688

—

118

C₄

C₄⋊C₄×D₂₁₁

yes

—

119

C₄

#20269976

yes

—

120

C₂

#27213015

yes

characteristic

121

C₂

#27213186

yes

—

122

C₂

#27213533

yes

characteristic

123

C₂

#27213538

yes

characteristic

124

C₂

#27213552

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₁₁
commutator	C₂×C₄×C₂₁₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₁₁
Fitting	#384×C₂₁₁	C₂
socle, prime radical	C₂³×C₂₁₁	C₂×D₄
cosocle	C₂²×C₄×C₂₁₁	C₂³
hypercenter	#384	D₂₁₁
hypocenter	C₂₁₁	#1847
LCS	G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₁₁

commutator

C₂×C₄×C₂₁₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₁₁

Fitting

#384×C₂₁₁

C₂

socle, prime radical

C₂³×C₂₁₁

C₂×D₄

cosocle

C₂²×C₄×C₂₁₁

C₂³

D₂₁₁

C₂₁₁

G ▷ C₂×C₄×C₂₁₁ ▷ C₂×C₂₁₁ ▷ C₂₁₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
11. #20269888	C₄	semidirect
35. C₂₁₁⋊C₈	C₄⋊C₄	semidirect
62. C₂₁₁	#1847	semidirect

H ≅ G/N

type

11. #20269888

C₄

semidirect

35. C₂₁₁⋊C₈

C₄⋊C₄

semidirect

62. C₂₁₁

#1847

semidirect

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

subgroup	index
C₂×#20269910	2
C₂×#20269908	2
C₂×#20269888	2
#27213012	2
#384×C₂₁₁	2
#1847	211

subgroup

index

#384×C₂₁₁

211

Subgroups — #41013564

Conjugacy classes of subgroups

Complements

Maximal subgroups