Group #41009576 — subgroups

Subgroups — #1956×C₂₁₁ #41009576

Order 27008.

332 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 160 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×C₄⋊Q₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₂₁₁	C₂	1	4	yes	maximal
3	C₂×C₄×C₈×C₂₁₁	C₂	1	1	yes	maximal, characteristic
4	#384×C₂₁₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₈×C₂₁₁	C₂²	1	2	yes	—
6	C₂²×C₈×C₂₁₁	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₂₁₁	—	2	2	no	—
8	C₄⋊Q₈×C₂₁₁	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₁₁	—	2	2	no	—
11	C₂×C₄⋊C₄×C₂₁₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
13	#103×C₂₁₁	—	2	8	no	—
14	C₄×C₈×C₂₁₁	—	2	2	no	—
15	C₂×Q₈×C₂₁₁	—	2	2	no	—
16	C₂×Q₈×C₂₁₁	—	2	4	no	—
17	C₂×Q₈×C₂₁₁	—	2	4	no	—
18	C₂×Q₈×C₂₁₁	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₂₁₁	—	2	2	no	—
20	C₂²×C₄×C₂₁₁	—	2	2	no	—
21	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
22	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₁₁	—	2	2	no	—
24	C₂×C₈×C₂₁₁	—	2	2	no	—
25	C₂×C₈×C₂₁₁	D₄	1	4	yes	—
26	C₄⋊C₄×C₂₁₁	—	2	8	no	—
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	C₂³×C₂₁₁	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₂₁₁	—	2	8	no	—
33	Q₈×C₂₁₁	—	4	4	no	—
34	C₂×C₄×C₂₁₁	—	2	4	no	—
35	C₂×C₄×C₂₁₁	—	4	2	no	—
36	C₂×C₄×C₂₁₁	—	4	2	no	—
37	C₂×C₄×C₂₁₁	—	4	2	no	—
38	C₂×C₄×C₂₁₁	—	4	4	no	—
39	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₁₁	SD₁₆	1	4	yes	—
41	C₂×C₄×C₂₁₁	Q₁₆	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×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	#41	1	2	yes	—
46	C₈×C₂₁₁	—	2	4	no	—
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	—	4	4	no	—
53	C₄×C₂₁₁	—	8	2	no	—
54	C₄×C₂₁₁	#103	1	8	yes	complemented
55	C₄×C₂₁₁	C₄×D₄	1	2	yes	—
56	C₄×C₂₁₁	#121	1	2	yes	—
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, direct product factor, Sylow
63	#1956	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂×C₄⋊Q₈	C₂×C₂₁₁	1	1	yes	characteristic
65	C₂×#103	C₂×C₂₁₁	1	4	yes	—
66	C₂×C₄×C₈	C₂×C₂₁₁	1	1	yes	characteristic
67	#384	C₂×C₂₁₁	1	1	yes	characteristic
68	C₂²×Q₈	C₂²×C₂₁₁	1	2	yes	—
69	C₂²×C₈	C₂²×C₂₁₁	1	2	yes	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	C₄×C₂₁₁	1	4	yes	complemented
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	2	2	no	—
74	C₂×C₄⋊C₄	C₂²×C₂₁₁	1	2	yes	—
75	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
76	#103	—	2	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	C₂×C₄×C₂₁₁	1	4	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
85	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₄×C₂₁₁	1	4	yes	—
89	C₄⋊C₄	—	2	8	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	4	4	no	—
92	C₄²	C₂×C₄×C₂₁₁	1	2	yes	—
93	C₄²	D₄×C₂₁₁	1	2	yes	—
94	C₂³	C₂×D₄×C₂₁₁	1	1	yes	characteristic
95	Q₈	—	2	8	no	—
96	Q₈	—	4	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	4	no	—
102	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	4	yes	—
103	C₂×C₄	SD₁₆×C₂₁₁	1	4	yes	—
104	C₂×C₄	Q₁₆×C₂₁₁	1	4	yes	—
105	C₂×C₄	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
107	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
108	C₂×C₄	#41×C₂₁₁	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
111	C₂²	#124×C₂₁₁	1	1	yes	characteristic
112	C₂²	#127×C₂₁₁	1	1	yes	characteristic
113	C₂²	C₂×SD₁₆×C₂₁₁	1	2	yes	—
114	C₂²	C₂×Q₁₆×C₂₁₁	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	—	8	2	no	—
117	C₄	#103×C₂₁₁	1	8	yes	complemented
118	C₄	C₄×D₄×C₂₁₁	1	2	yes	—
119	C₄	#121×C₂₁₁	1	2	yes	—
120	C₂	#390×C₂₁₁	1	1	yes	characteristic
121	C₂	C₂×#103×C₂₁₁	1	2	yes	—
122	C₂	#487×C₂₁₁	1	2	yes	—
123	C₂	#492×C₂₁₁	1	1	yes	characteristic
124	C₂	#494×C₂₁₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₂₁₁

C₂

yes

maximal, characteristic

C₂×#103×C₂₁₁

C₂

yes

maximal

C₂×C₄×C₈×C₂₁₁

C₂

yes

maximal, characteristic

#384×C₂₁₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₂₁₁

C₂²

yes

—

C₂²×C₈×C₂₁₁

C₂²

yes

—

C₄⋊Q₈×C₂₁₁

—

C₄⋊Q₈×C₂₁₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₂²

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

#103×C₂₁₁

—

C₄×C₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

—

C₂×Q₈×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

D₄

yes

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

C₂×C₄

yes

—

C₄²×C₂₁₁

D₄

yes

—

C₂³×C₂₁₁

C₂×D₄

yes

characteristic

Q₈×C₂₁₁

—

Q₈×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

C₂×C₄×C₂₁₁

—

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

—

C₈×C₂₁₁

—

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

—

C₄×C₂₁₁

#103

yes

complemented

C₄×C₂₁₁

C₄×D₄

yes

—

C₄×C₂₁₁

#121

yes

—

C₂×C₂₁₁

#390

yes

characteristic

C₂×C₂₁₁

C₂×#103

yes

—

C₂×C₂₁₁

#487

yes

—

C₂×C₂₁₁

yes

C₂×C₂₁₁

yes

C₂₁₁

yes

characteristic, direct product factor, Sylow

#1956

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₂₁₁

yes

characteristic

C₂×#103

C₂×C₂₁₁

yes

—

C₂×C₄×C₈

C₂×C₂₁₁

yes

characteristic

#384

C₂×C₂₁₁

yes

characteristic

C₂²×Q₈

C₂²×C₂₁₁

yes

—

C₂²×C₈

C₂²×C₂₁₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₂₁₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₁₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×C₄×C₂₁₁

yes

—

C₄²

D₄×C₂₁₁

yes

—

C₂³

C₂×D₄×C₂₁₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

103

C₂×C₄

SD₁₆×C₂₁₁

yes

—

104

C₂×C₄

Q₁₆×C₂₁₁

yes

—

105

C₂×C₄

C₂²×C₄×C₂₁₁

yes

characteristic

106

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

107

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

108

C₂×C₄

#41×C₂₁₁

yes

—

109

C₈

—

110

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

111

C₂²

#124×C₂₁₁

yes

characteristic

112

C₂²

#127×C₂₁₁

yes

characteristic

113

C₂²

C₂×SD₁₆×C₂₁₁

yes

—

114

C₂²

C₂×Q₁₆×C₂₁₁

yes

—

115

C₄

—

116

C₄

—

117

C₄

#103×C₂₁₁

yes

complemented

118

C₄

C₄×D₄×C₂₁₁

yes

—

119

C₄

#121×C₂₁₁

yes

—

120

C₂

#390×C₂₁₁

yes

characteristic

121

C₂

C₂×#103×C₂₁₁

yes

—

122

C₂

#487×C₂₁₁

yes

—

123

C₂

#492×C₂₁₁

yes

characteristic

124

C₂

#494×C₂₁₁

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₁₁	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₁₁

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₁₁ ◁ C₂×C₄²×C₂₁₁ ◁ G

N	H ≅ G/N	type
8. C₄⋊Q₈×C₂₁₁	C₄	semidirect
54. C₄×C₂₁₁	#103	semidirect
62. C₂₁₁	#1956	direct (N × H)
63. #1956	C₂₁₁	direct (N × H)
71. C₄⋊Q₈	C₄×C₂₁₁	semidirect
117. C₄	#103×C₂₁₁	semidirect

H ≅ G/N

type

8. C₄⋊Q₈×C₂₁₁

C₄

semidirect

54. C₄×C₂₁₁

#103

semidirect

62. C₂₁₁

#1956

direct (N × H)

63. #1956

C₂₁₁

direct (N × H)

71. C₄⋊Q₈

C₄×C₂₁₁

semidirect

117. C₄

#103×C₂₁₁

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₂₁₁	2
C₂×#103×C₂₁₁	2
C₂×C₄×C₈×C₂₁₁	2
#384×C₂₁₁	2
#1956	211

subgroup

index

C₂×C₄⋊Q₈×C₂₁₁

C₂×#103×C₂₁₁

C₂×C₄×C₈×C₂₁₁

#384×C₂₁₁

#1956

211

Subgroups — #1956×C₂₁₁ #41009576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1956×C211 #41009576

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1956×C₂₁₁ #41009576