Group #41009418 — subgroups

Subgroups — #1798×C₂₁₁ #41009418

Order 27008.

312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 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×#117×C₂₁₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊C₈×C₂₁₁	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊C₈×C₂₁₁	C₂	1	2	yes	maximal
4	#336×C₂₁₁	C₂	1	2	yes	maximal, complemented
5	C₂³×C₄×C₂₁₁	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₂₁₁	C₂²	1	4	yes	—
7	#117×C₂₁₁	—	2	4	no	—
8	C₂×C₄⋊C₄×C₂₁₁	C₄	1	2	yes	—
9	C₂×C₂²⋊C₄×C₂₁₁	C₄	1	2	yes	—
10	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₂₁₁	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₂₁₁	—	2	4	no	—
13	C₂²⋊C₈×C₂₁₁	—	2	4	no	—
14	C₂⁴×C₂₁₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₁₁	—	2	2	no	—
16	C₂²×C₄×C₂₁₁	—	2	2	no	—
17	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₂₁₁	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₂₁₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₁₁	D₄	1	2	yes	—
21	C₂²×C₄×C₂₁₁	Q₈	1	2	yes	—
22	C₂²×C₄×C₂₁₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₂₁₁	—	2	4	no	—
24	C₂×C₈×C₂₁₁	—	2	4	no	—
25	C₂×C₈×C₂₁₁	—	2	4	no	—
26	C₄⋊C₄×C₂₁₁	—	2	4	no	—
27	C₂²⋊C₄×C₂₁₁	—	2	4	no	—
28	C₄²×C₂₁₁	—	2	2	no	—
29	C₄²×C₂₁₁	D₄	1	4	yes	—
30	C₂³×C₂₁₁	—	2	1	no	—
31	C₂³×C₂₁₁	—	2	1	no	—
32	C₂³×C₂₁₁	—	2	4	no	—
33	C₂³×C₂₁₁	C₄⋊C₄	1	2	yes	—
34	C₂³×C₂₁₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₁₁	—	2	1	no	—
36	C₂×C₄×C₂₁₁	—	2	1	no	—
37	C₂×C₄×C₂₁₁	—	2	4	no	—
38	C₂×C₄×C₂₁₁	—	2	4	no	—
39	C₂×C₄×C₂₁₁	—	2	4	no	—
40	C₂×C₄×C₂₁₁	—	2	2	no	—
41	C₂×C₄×C₂₁₁	—	2	4	no	—
42	C₂×C₄×C₂₁₁	C₂²⋊C₄	1	4	yes	—
43	C₂×C₄×C₂₁₁	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₁₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₂₁₁	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₂₁₁	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₁₁	#41	1	2	yes	—
48	C₈×C₂₁₁	—	4	4	no	—
49	C₂²×C₂₁₁	—	2	2	no	—
50	C₂²×C₂₁₁	—	2	4	no	—
51	C₂²×C₂₁₁	—	4	2	no	—
52	C₂²×C₂₁₁	—	4	2	no	—
53	C₂²×C₂₁₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₂₁₁	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂²×C₂₁₁	#117	1	1	yes	characteristic
56	C₂²×C₂₁₁	#131	1	4	yes	—
57	C₄×C₂₁₁	—	2	2	no	—
58	C₄×C₂₁₁	—	2	4	no	—
59	C₄×C₂₁₁	—	4	2	no	—
60	C₄×C₂₁₁	#121	1	2	yes	—
61	C₄×C₂₁₁	C₂²⋊Q₈	1	2	yes	—
62	C₂×C₂₁₁	—	4	2	no	—
63	C₂×C₂₁₁	#380	1	1	yes	characteristic
64	C₂×C₂₁₁	#408	1	2	yes	—
65	C₂×C₂₁₁	#424	1	2	yes	—
66	C₂×C₂₁₁	#433	1	2	yes	—
67	C₂₁₁	#1798	1	1	yes	characteristic, direct product factor, Sylow
68	#1798	C₂₁₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂×#117	C₂×C₂₁₁	1	1	yes	characteristic
70	C₂×C₄⋊C₈	C₂×C₂₁₁	1	2	yes	complemented
71	C₂×C₂²⋊C₈	C₂×C₂₁₁	1	2	yes	—
72	#336	C₂×C₂₁₁	1	2	yes	complemented
73	C₂³×C₄	C₂²×C₂₁₁	1	1	yes	characteristic
74	C₂²×C₈	C₂²×C₂₁₁	1	4	yes	—
75	#117	—	2	4	no	—
76	C₂×C₄⋊C₄	C₄×C₂₁₁	1	2	yes	—
77	C₂×C₂²⋊C₄	C₄×C₂₁₁	1	2	yes	—
78	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
79	C₂×C₄²	C₂²×C₂₁₁	1	1	yes	characteristic
80	C₄⋊C₈	—	2	4	no	—
81	C₂²⋊C₈	—	2	4	no	—
82	C₂⁴	C₂×C₄×C₂₁₁	1	1	yes	characteristic
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
86	C₂²×C₄	C₂×C₄×C₂₁₁	1	2	yes	—
87	C₂²×C₄	C₂×C₄×C₂₁₁	1	1	yes	characteristic
88	C₂²×C₄	D₄×C₂₁₁	1	2	yes	—
89	C₂²×C₄	Q₈×C₂₁₁	1	2	yes	—
90	C₂²×C₄	C₂³×C₂₁₁	1	1	yes	characteristic
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	2	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₄²	—	2	2	no	—
97	C₄²	D₄×C₂₁₁	1	4	yes	—
98	C₂³	—	2	1	no	—
99	C₂³	—	2	1	no	—
100	C₂³	—	2	4	no	—
101	C₂³	C₄⋊C₄×C₂₁₁	1	2	yes	—
102	C₂³	C₂²×C₄×C₂₁₁	1	1	yes	characteristic
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	C₂²⋊C₄×C₂₁₁	1	4	yes	—
111	C₂×C₄	C₄⋊C₄×C₂₁₁	1	2	yes	—
112	C₂×C₄	C₂×D₄×C₂₁₁	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₂₁₁	1	2	yes	—
114	C₂×C₄	C₂×Q₈×C₂₁₁	1	1	yes	characteristic
115	C₂×C₄	#41×C₂₁₁	1	2	yes	—
116	C₈	—	4	4	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	4	2	no	—
120	C₂²	—	4	2	no	—
121	C₂²	C₂×C₂²⋊C₄×C₂₁₁	1	1	yes	characteristic
122	C₂²	C₂×C₄⋊C₄×C₂₁₁	1	1	yes	characteristic
123	C₂²	#117×C₂₁₁	1	1	yes	characteristic
124	C₂²	#131×C₂₁₁	1	4	yes	—
125	C₄	—	2	2	no	—
126	C₄	—	2	4	no	—
127	C₄	—	4	2	no	—
128	C₄	#121×C₂₁₁	1	2	yes	—
129	C₄	C₂²⋊Q₈×C₂₁₁	1	2	yes	—
130	C₂	—	4	2	no	—
131	C₂	#380×C₂₁₁	1	1	yes	characteristic
132	C₂	#408×C₂₁₁	1	2	yes	—
133	C₂	#424×C₂₁₁	1	2	yes	—
134	C₂	#433×C₂₁₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#117×C₂₁₁

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₁₁

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₂₁₁

C₂

yes

maximal

#336×C₂₁₁

C₂

yes

maximal, complemented

C₂³×C₄×C₂₁₁

C₂²

yes

characteristic

C₂²×C₈×C₂₁₁

C₂²

yes

—

#117×C₂₁₁

—

C₂×C₄⋊C₄×C₂₁₁

C₄

yes

—

C₂×C₂²⋊C₄×C₂₁₁

C₄

yes

—

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₂×C₄²×C₂₁₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₁₁

—

C₂²⋊C₈×C₂₁₁

—

C₂⁴×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

—

C₂²×C₄×C₂₁₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₁₁

D₄

yes

—

C₂²×C₄×C₂₁₁

Q₈

yes

—

C₂²×C₄×C₂₁₁

C₂³

yes

characteristic

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₂×C₈×C₂₁₁

—

C₄⋊C₄×C₂₁₁

—

C₂²⋊C₄×C₂₁₁

—

C₄²×C₂₁₁

—

C₄²×C₂₁₁

D₄

yes

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

—

C₂³×C₂₁₁

C₄⋊C₄

yes

—

C₂³×C₂₁₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₁₁

—

C₂×C₄×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

C₄⋊C₄

yes

—

C₂×C₄×C₂₁₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₁₁

C₂×D₄

yes

—

C₂×C₄×C₂₁₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₁₁

#41

yes

—

C₈×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

—

C₂²×C₂₁₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₁₁

C₂×C₄⋊C₄

yes

C₂²×C₂₁₁

yes

C₂²×C₂₁₁

yes

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

—

C₄×C₂₁₁

#121

yes

—

C₄×C₂₁₁

C₂²⋊Q₈

yes

—

C₂×C₂₁₁

—

C₂×C₂₁₁

#380

yes

characteristic

C₂×C₂₁₁

#408

yes

—

C₂×C₂₁₁

#424

yes

—

C₂×C₂₁₁

#433

yes

—

C₂₁₁

#1798

yes

characteristic, direct product factor, Sylow

#1798

C₂₁₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×#117

C₂×C₂₁₁

yes

characteristic

C₂×C₄⋊C₈

C₂×C₂₁₁

yes

complemented

C₂×C₂²⋊C₈

C₂×C₂₁₁

yes

—

#336

C₂×C₂₁₁

yes

complemented

C₂³×C₄

C₂²×C₂₁₁

yes

characteristic

C₂²×C₈

C₂²×C₂₁₁

yes

—

#117

—

C₂×C₄⋊C₄

C₄×C₂₁₁

yes

—

C₂×C₂²⋊C₄

C₄×C₂₁₁

yes

—

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₂×C₄²

C₂²×C₂₁₁

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂⁴

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

—

C₂²×C₄

C₂×C₄×C₂₁₁

yes

characteristic

C₂²×C₄

D₄×C₂₁₁

yes

—

C₂²×C₄

Q₈×C₂₁₁

yes

—

C₂²×C₄

C₂³×C₂₁₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₄×C₂₁₁

yes

—

C₂³

—

C₂³

—

100

C₂³

—

101

C₂³

C₄⋊C₄×C₂₁₁

yes

—

102

C₂³

C₂²×C₄×C₂₁₁

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₂²⋊C₄×C₂₁₁

yes

—

111

C₂×C₄

C₄⋊C₄×C₂₁₁

yes

—

112

C₂×C₄

C₂×D₄×C₂₁₁

yes

characteristic

113

C₂×C₄

C₂×D₄×C₂₁₁

yes

—

114

C₂×C₄

C₂×Q₈×C₂₁₁

yes

characteristic

115

C₂×C₄

#41×C₂₁₁

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

C₂×C₂²⋊C₄×C₂₁₁

yes

characteristic

122

C₂²

C₂×C₄⋊C₄×C₂₁₁

yes

characteristic

123

C₂²

#117×C₂₁₁

yes

characteristic

124

C₂²

#131×C₂₁₁

yes

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#121×C₂₁₁

yes

—

129

C₄

C₂²⋊Q₈×C₂₁₁

yes

—

130

C₂

—

131

C₂

#380×C₂₁₁

yes

characteristic

132

C₂

#408×C₂₁₁

yes

—

133

C₂

#424×C₂₁₁

yes

—

134

C₂

#433×C₂₁₁

yes

—

	subgroup	G/H
center	C₂²×C₄×C₂₁₁	C₂³
commutator	C₂³	C₂²×C₄×C₂₁₁
Frattini, cosocle	C₂²×C₄	C₂³×C₂₁₁
socle	C₂³×C₂₁₁	C₂²×C₄
prime radical	C₂⁴×C₂₁₁	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

subgroup

G/H

center

C₂²×C₄×C₂₁₁

C₂³

commutator

C₂³

C₂²×C₄×C₂₁₁

Frattini, cosocle

C₂²×C₄

C₂³×C₂₁₁

socle

C₂³×C₂₁₁

C₂²×C₄

prime radical

C₂⁴×C₂₁₁

C₂×C₄

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₂₁₁ ◁ G

N	H ≅ G/N	type
2. C₂×C₄⋊C₈×C₂₁₁	C₂	semidirect
4. #336×C₂₁₁	C₂	semidirect
67. C₂₁₁	#1798	direct (N × H)
68. #1798	C₂₁₁	direct (N × H)
70. C₂×C₄⋊C₈	C₂×C₂₁₁	semidirect
72. #336	C₂×C₂₁₁	semidirect

H ≅ G/N

type

2. C₂×C₄⋊C₈×C₂₁₁

C₂

semidirect

4. #336×C₂₁₁

C₂

semidirect

67. C₂₁₁

#1798

direct (N × H)

68. #1798

C₂₁₁

direct (N × H)

70. C₂×C₄⋊C₈

C₂×C₂₁₁

semidirect

72. #336

C₂×C₂₁₁

semidirect

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

subgroup	index
C₂×#117×C₂₁₁	2
C₂×C₄⋊C₈×C₂₁₁	2
C₂×C₂²⋊C₈×C₂₁₁	2
#336×C₂₁₁	2
#1798	211

subgroup

index

C₂×#117×C₂₁₁

C₂×C₄⋊C₈×C₂₁₁

C₂×C₂²⋊C₈×C₂₁₁

#336×C₂₁₁

#1798

211

Subgroups — #1798×C₂₁₁ #41009418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1798×C211 #41009418

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1798×C₂₁₁ #41009418