Group #41014199 — subgroups

Subgroups — #41014199

Order 27008.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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×#20269977	C₂	1	1	yes	maximal, characteristic
2	C₂×#20269900	C₂	1	4	yes	maximal
3	C₂×C₄×C₂₁₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₂₂	C₂²	1	2	yes	—
6	C₂²×C₂₁₁⋊C₈	C₂²	1	2	yes	—
7	#20269977	—	2	2	no	—
8	#20269977	C₄	1	4	yes	complemented
9	C₂×C₄⋊Dic₂₁₁	—	2	2	no	—
10	#20269900	—	2	8	no	—
11	C₄×C₂₁₁⋊C₈	—	2	2	no	—
12	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₄₂₂	—	2	2	no	—
16	C₂×Dic₄₂₂	—	2	4	no	—
17	C₂×Dic₄₂₂	—	2	4	no	—
18	C₂×Dic₄₂₂	C₂×C₄	1	4	yes	—
19	C₂²×Dic₂₁₁	—	2	2	no	—
20	C₂×C₂₁₁⋊C₈	—	2	2	no	—
21	C₂×C₂₁₁⋊C₈	—	2	2	no	—
22	C₂×C₂₁₁⋊C₈	D₄	1	4	yes	—
23	C₄⋊Dic₂₁₁	—	2	8	no	—
24	C₂²×C₄×C₂₁₁	—	2	2	no	—
25	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
26	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₂₁₁	—	2	4	no	—
28	C₄⋊C₄×C₂₁₁	—	4	4	no	—
29	C₄²×C₂₁₁	C₂×C₄	1	2	yes	—
30	C₄²×C₂₁₁	D₄	1	2	yes	—
31	Dic₄₂₂	—	2	8	no	—
32	Dic₄₂₂	—	4	4	no	—
33	C₂×Dic₂₁₁	—	2	4	no	—
34	C₂×Dic₂₁₁	—	4	2	no	—
35	C₂×Dic₂₁₁	—	4	2	no	—
36	C₂₁₁⋊C₈	—	2	4	no	—
37	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₁₁	—	4	2	no	—
39	C₂×C₄×C₂₁₁	—	4	4	no	—
40	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
41	C₂×C₄×C₂₁₁	SD₁₆	1	4	yes	—
42	C₂×C₄×C₂₁₁	Q₁₆	1	4	yes	—
43	C₂×C₄×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₁₁	#41	1	2	yes	—
47	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₂₁₁	#124	1	1	yes	characteristic
49	C₂²×C₂₁₁	#127	1	1	yes	characteristic
50	C₂²×C₂₁₁	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₂₁₁	C₂×Q₁₆	1	2	yes	—
52	C₄×C₂₁₁	—	8	2	no	—
53	C₄×C₂₁₁	#103	1	8	yes	complemented
54	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
55	C₄×C₂₁₁	#121	1	2	yes	—
56	Dic₂₁₁	—	4	4	no	—
57	C₂×C₂₁₁	#390	1	1	yes	characteristic
58	C₂×C₂₁₁	C₂×#103	1	2	yes	—
59	C₂×C₂₁₁	#487	1	2	yes	—
60	C₂×C₂₁₁	#492	1	1	yes	characteristic
61	C₂×C₂₁₁	#494	1	1	yes	characteristic
62	C₂₁₁	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	211	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	211	1	no	—
65	C₂×#103	—	211	4	no	—
66	C₂×C₄×C₈	—	211	1	no	—
67	#384	D₂₁₁	1	1	yes	characteristic
68	C₂²×Q₈	—	211	2	no	—
69	C₂²×C₈	—	211	2	no	—
70	C₄⋊Q₈	—	422	2	no	—
71	C₄⋊Q₈	—	211	4	no	—
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	422	2	no	—
74	C₂×C₄⋊C₄	C₂×D₂₁₁	1	2	yes	—
75	C₂×C₄²	C₂×D₂₁₁	1	1	yes	characteristic
76	#103	—	422	8	no	—
77	C₄×C₈	—	422	2	no	—
78	C₂×Q₈	—	422	4	no	—
79	C₂×Q₈	—	422	4	no	—
80	C₂×Q₈	—	422	2	no	—
81	C₂×Q₈	—	211	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	422	2	no	—
84	C₂²×C₄	C₂²⋊D₂₁₁	1	2	yes	—
85	C₂²×C₄	C₂²×D₂₁₁	1	1	yes	characteristic
86	C₂×C₈	—	422	2	no	—
87	C₂×C₈	—	422	2	no	—
88	C₂×C₈	—	211	4	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	4	4	no	—
91	C₄⋊C₄	—	422	8	no	—
92	C₄²	C₄×D₂₁₁	1	2	yes	—
93	C₄²	D₈₄₄	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₂₁₁	1	1	yes	characteristic
95	Q₈	—	844	4	no	—
96	Q₈	—	422	8	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	—	844	2	no	—
100	C₂×C₄	—	844	2	no	—
101	C₂×C₄	—	422	4	no	—
102	C₂×C₄	(C₂×D₂₁₁)⋊C₄	1	4	yes	—
103	C₂×C₄	#16809950	1	4	yes	—
104	C₂×C₄	#16809952	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₂₁₁	1	1	yes	characteristic
106	C₂×C₄	D₄×D₂₁₁	1	2	yes	—
107	C₂×C₄	C₂×D₈₄₄	1	1	yes	characteristic
108	C₂×C₄	#16809963	1	2	yes	—
109	C₈	—	422	4	no	—
110	C₂²	C₂×(C₂×D₂₁₁)⋊C₄	1	1	yes	characteristic
111	C₂²	#20269966	1	1	yes	characteristic
112	C₂²	#20269973	1	1	yes	characteristic
113	C₂²	C₂×#16809950	1	2	yes	—
114	C₂²	C₂×#16809952	1	2	yes	—
115	C₄	—	8	2	no	—
116	C₄	—	844	4	no	—
117	C₄	#20269900	1	8	yes	complemented
118	C₄	#20269939	1	2	yes	—
119	C₄	#20269952	1	2	yes	—
120	C₂	#27213047	1	1	yes	characteristic
121	C₂	C₂×#20269900	1	2	yes	—
122	C₂	#27213498	1	2	yes	—
123	C₂	#27213518	1	1	yes	characteristic
124	C₂	#27213525	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20269977

C₂

yes

maximal, characteristic

C₂×#20269900

C₂

yes

maximal

C₂×C₄×C₂₁₁⋊C₈

C₂

yes

maximal, characteristic

#384×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Dic₄₂₂

C₂²

yes

—

C₂²×C₂₁₁⋊C₈

C₂²

yes

—

#20269977

—

#20269977

C₄

yes

complemented

C₂×C₄⋊Dic₂₁₁

—

#20269900

—

C₄×C₂₁₁⋊C₈

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

—

C₂×Dic₄₂₂

C₂×C₄

yes

—

C₂²×Dic₂₁₁

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

—

C₂×C₂₁₁⋊C₈

D₄

yes

—

C₄⋊Dic₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

Dic₄₂₂

—

Dic₄₂₂

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂×Dic₂₁₁

—

C₂₁₁⋊C₈

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₁₁

SD₁₆

yes

—

C₂×C₄×C₂₁₁

Q₁₆

yes

—

C₂×C₄×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

C₂×SD₁₆

yes

—

C₂²×C₂₁₁

C₂×Q₁₆

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#103

yes

complemented

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

#121

yes

—

Dic₂₁₁

—

C₂×C₂₁₁

#390

yes

characteristic

C₂×C₂₁₁

C₂×#103

yes

—

C₂×C₂₁₁

#487

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, complemented, Sylow

#1956

—

211

maximal, Sylow

C₂×C₄⋊Q₈

—

211

—

C₂×#103

—

211

—

C₂×C₄×C₈

—

211

—

#384

D₂₁₁

yes

characteristic

C₂²×Q₈

—

211

—

C₂²×C₈

—

211

—

C₄⋊Q₈

—

422

—

C₄⋊Q₈

—

211

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

422

—

C₂×C₄⋊C₄

C₂×D₂₁₁

yes

—

C₂×C₄²

C₂×D₂₁₁

yes

characteristic

#103

—

422

—

C₄×C₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

422

—

C₂×Q₈

—

211

—

C₂²×C₄

—

C₂²×C₄

—

422

—

C₂²×C₄

C₂²⋊D₂₁₁

yes

—

C₂²×C₄

C₂²×D₂₁₁

yes

characteristic

C₂×C₈

—

422

—

C₂×C₈

—

422

—

C₂×C₈

—

211

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

422

—

C₄²

C₄×D₂₁₁

yes

—

C₄²

D₈₄₄

yes

—

C₂³

C₂×C₂²⋊D₂₁₁

yes

characteristic

Q₈

—

844

—

Q₈

—

422

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

844

—

100

C₂×C₄

—

844

—

101

C₂×C₄

—

422

—

102

C₂×C₄

(C₂×D₂₁₁)⋊C₄

yes

—

103

C₂×C₄

#16809950

yes

—

104

C₂×C₄

#16809952

yes

—

105

C₂×C₄

C₂×C₄×D₂₁₁

yes

characteristic

106

C₂×C₄

D₄×D₂₁₁

yes

—

107

C₂×C₄

C₂×D₈₄₄

yes

characteristic

108

C₂×C₄

#16809963

yes

—

109

C₈

—

422

—

110

C₂²

C₂×(C₂×D₂₁₁)⋊C₄

yes

characteristic

111

C₂²

#20269966

yes

characteristic

112

C₂²

#20269973

yes

characteristic

113

C₂²

C₂×#16809950

yes

—

114

C₂²

C₂×#16809952

yes

—

115

C₄

—

116

C₄

—

844

—

117

C₄

#20269900

yes

complemented

118

C₄

#20269939

yes

—

119

C₄

#20269952

yes

—

120

C₂

#27213047

yes

characteristic

121

C₂

C₂×#20269900

yes

—

122

C₂

#27213498

yes

—

123

C₂

#27213518

yes

characteristic

124

C₂

#27213525

yes

characteristic

	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	#1956
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
8. #20269977	C₄	semidirect
53. C₄×C₂₁₁	#103	semidirect
62. C₂₁₁	#1956	semidirect
117. C₄	#20269900	semidirect

H ≅ G/N

type

8. #20269977

C₄

semidirect

53. C₄×C₂₁₁

#103

semidirect

62. C₂₁₁

#1956

semidirect

117. C₄

#20269900

semidirect

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

subgroup	index
C₂×#20269977	2
C₂×#20269900	2
C₂×C₄×C₂₁₁⋊C₈	2
#384×C₂₁₁	2
#1956	211

subgroup

index

C₂×#20269977

C₂×#20269900

C₂×C₄×C₂₁₁⋊C₈

#384×C₂₁₁

#1956

211

Subgroups — #41014199

Conjugacy classes of subgroups

Complements

Maximal subgroups