Group #43251876 — subgroups

Subgroups — #2861×C₂₂₃ #43251876

Order 28544.

524 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 216 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×#125×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#124×C₂₂₃	C₂	1	1	yes	maximal, characteristic
3	#391×C₂₂₃	C₂	1	2	yes	maximal, complemented
4	#390×C₂₂₃	C₂	1	2	yes	maximal
5	#388×C₂₂₃	C₂	1	8	yes	maximal, complemented
6	C₄³×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
8	C₂²×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
9	#125×C₂₂₃	—	2	4	no	—
10	#124×C₂₂₃	—	2	4	no	—
11	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₂₂₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	4	yes	—
14	C₂×C₂²⋊C₄×C₂₂₃	C₂²	1	8	yes	—
15	C₂×C₄²×C₂₂₃	C₂²	1	4	yes	—
16	C₂×C₄²×C₂₂₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
18	#95×C₂₂₃	C₂²	1	8	yes	—
19	C₂⁴×C₂₂₃	C₂³	1	2	yes	—
20	C₂×Q₈×C₂₂₃	—	2	2	no	—
21	C₂×Q₈×C₂₂₃	—	2	4	no	—
22	C₂×D₄×C₂₂₃	—	2	2	no	—
23	C₂×D₄×C₂₂₃	—	2	4	no	—
24	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₂₂₃	C₂³	1	2	yes	—
27	C₂²×C₄×C₂₂₃	C₂³	1	4	yes	—
28	C₂²×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₂₂₃	—	2	8	no	—
30	C₄⋊C₄×C₂₂₃	—	2	4	no	—
31	C₂²⋊C₄×C₂₂₃	—	2	8	no	—
32	C₂²⋊C₄×C₂₂₃	—	2	16	no	—
33	C₄²×C₂₂₃	—	2	8	no	—
34	C₄²×C₂₂₃	—	2	4	no	—
35	C₄²×C₂₂₃	D₄	1	4	yes	—
36	C₂³×C₂₂₃	—	2	2	no	—
37	C₂³×C₂₂₃	—	2	8	no	—
38	C₂³×C₂₂₃	—	2	4	no	—
39	C₂³×C₂₂₃	C₂⁴	1	1	yes	characteristic
40	Q₈×C₂₂₃	—	4	4	no	—
41	D₄×C₂₂₃	—	4	4	no	—
42	C₂×C₄×C₂₂₃	—	2	4	no	—
43	C₂×C₄×C₂₂₃	—	2	4	no	—
44	C₂×C₄×C₂₂₃	—	2	4	no	—
45	C₂×C₄×C₂₂₃	—	2	2	no	—
46	C₂×C₄×C₂₂₃	—	2	4	no	—
47	C₂×C₄×C₂₂₃	—	2	8	no	—
48	C₂×C₄×C₂₂₃	—	2	4	no	—
49	C₂×C₄×C₂₂₃	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₂₂₃	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₂₂₃	#41	1	8	yes	—
52	C₂×C₄×C₂₂₃	#41	1	4	yes	—
53	C₂²×C₂₂₃	—	4	2	no	—
54	C₂²×C₂₂₃	—	4	8	no	—
55	C₂²×C₂₂₃	—	4	4	no	—
56	C₂²×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
57	C₂²×C₂₂₃	C₂×#41	1	2	yes	—
58	C₂²×C₂₂₃	C₂×#41	1	4	yes	—
59	C₄×C₂₂₃	—	2	8	no	—
60	C₄×C₂₂₃	—	2	4	no	—
61	C₄×C₂₂₃	—	4	2	no	—
62	C₄×C₂₂₃	—	4	4	no	—
63	C₄×C₂₂₃	#124	1	4	yes	—
64	C₂×C₂₂₃	—	8	2	no	—
65	C₂×C₂₂₃	C₂×#124	1	1	yes	characteristic
66	C₂×C₂₂₃	#529	1	4	yes	—
67	C₂×C₂₂₃	#532	1	2	yes	—
68	C₂₂₃	#2861	1	1	yes	characteristic, direct product factor, Sylow
69	#2861	C₂₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
70	C₂×#125	C₂×C₂₂₃	1	1	yes	characteristic, complemented
71	C₂×#124	C₂×C₂₂₃	1	1	yes	characteristic
72	#391	C₂×C₂₂₃	1	2	yes	complemented
73	#390	C₂×C₂₂₃	1	2	yes	—
74	#388	C₂×C₂₂₃	1	8	yes	complemented
75	C₄³	C₂×C₂₂₃	1	1	yes	characteristic, complemented
76	C₂²×Q₈	C₂²×C₂₂₃	1	1	yes	characteristic
77	C₂²×D₄	C₂²×C₂₂₃	1	1	yes	characteristic
78	#125	—	2	4	no	—
79	#124	—	2	4	no	—
80	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	4	yes	—
81	C₂×C₄⋊C₄	C₂²×C₂₂₃	1	2	yes	—
82	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	4	yes	—
83	C₂×C₂²⋊C₄	C₂²×C₂₂₃	1	8	yes	—
84	C₂×C₄²	C₂²×C₂₂₃	1	4	yes	—
85	C₂×C₄²	C₂²×C₂₂₃	1	2	yes	—
86	C₂×C₄²	C₂²×C₂₂₃	1	1	yes	characteristic
87	#95	C₂²×C₂₂₃	1	8	yes	—
88	C₂⁴	C₂³×C₂₂₃	1	2	yes	—
89	C₂×Q₈	—	2	2	no	—
90	C₂×Q₈	—	2	4	no	—
91	C₂×D₄	—	2	2	no	—
92	C₂×D₄	—	2	4	no	—
93	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
94	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
95	C₂²×C₄	C₂³×C₂₂₃	1	2	yes	—
96	C₂²×C₄	C₂³×C₂₂₃	1	4	yes	—
97	C₂²×C₄	C₂³×C₂₂₃	1	1	yes	characteristic
98	C₄⋊C₄	—	2	8	no	—
99	C₄⋊C₄	—	2	4	no	—
100	C₂²⋊C₄	—	2	8	no	—
101	C₂²⋊C₄	—	2	16	no	—
102	C₄²	—	2	8	no	—
103	C₄²	—	2	4	no	—
104	C₄²	D₄×C₂₂₃	1	4	yes	—
105	C₂³	—	2	2	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	4	no	—
108	C₂³	C₂⁴×C₂₂₃	1	1	yes	characteristic
109	Q₈	—	4	4	no	—
110	D₄	—	4	4	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	8	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	C₂×D₄×C₂₂₃	1	4	yes	—
119	C₂×C₄	C₂×D₄×C₂₂₃	1	2	yes	—
120	C₂×C₄	#41×C₂₂₃	1	8	yes	—
121	C₂×C₄	#41×C₂₂₃	1	4	yes	—
122	C₂²	—	4	2	no	—
123	C₂²	—	4	8	no	—
124	C₂²	—	4	4	no	—
125	C₂²	C₂²×D₄×C₂₂₃	1	1	yes	characteristic
126	C₂²	C₂×#41×C₂₂₃	1	2	yes	—
127	C₂²	C₂×#41×C₂₂₃	1	4	yes	—
128	C₄	—	2	8	no	—
129	C₄	—	2	4	no	—
130	C₄	—	4	2	no	—
131	C₄	—	4	4	no	—
132	C₄	#124×C₂₂₃	1	4	yes	—
133	C₂	—	8	2	no	—
134	C₂	C₂×#124×C₂₂₃	1	1	yes	characteristic
135	C₂	#529×C₂₂₃	1	4	yes	—
136	C₂	#532×C₂₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#125×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₂₂₃

C₂

yes

maximal, characteristic

#391×C₂₂₃

C₂

yes

maximal, complemented

#390×C₂₂₃

C₂

yes

maximal

#388×C₂₂₃

C₂

yes

maximal, complemented

C₄³×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₂₂₃

C₂²

yes

characteristic

C₂²×D₄×C₂₂₃

C₂²

yes

characteristic

#125×C₂₂₃

—

#124×C₂₂₃

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₂₃

C₂²

yes

—

C₂×C₄²×C₂₂₃

C₂²

yes

—

C₂×C₄²×C₂₂₃

C₂²

yes

—

C₂×C₄²×C₂₂₃

C₂²

yes

characteristic

#95×C₂₂₃

C₂²

yes

—

C₂⁴×C₂₂₃

C₂³

yes

—

C₂×Q₈×C₂₂₃

—

C₂×Q₈×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂×D₄×C₂₂₃

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₄×C₂₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₂₃

—

C₄⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₂²⋊C₄×C₂₂₃

—

C₄²×C₂₂₃

—

C₄²×C₂₂₃

—

C₄²×C₂₂₃

D₄

yes

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

—

C₂³×C₂₂₃

C₂⁴

yes

characteristic

Q₈×C₂₂₃

—

D₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₂₃

#41

yes

—

C₂×C₄×C₂₂₃

#41

yes

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂²×D₄

yes

characteristic

C₂²×C₂₂₃

C₂×#41

yes

—

C₂²×C₂₂₃

C₂×#41

yes

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

#124

yes

—

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂×#124

yes

characteristic

C₂×C₂₂₃

#529

yes

—

C₂×C₂₂₃

#532

yes

—

C₂₂₃

#2861

yes

characteristic, direct product factor, Sylow

#2861

C₂₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×#125

C₂×C₂₂₃

yes

characteristic, complemented

C₂×#124

C₂×C₂₂₃

yes

characteristic

#391

C₂×C₂₂₃

yes

complemented

#390

C₂×C₂₂₃

yes

—

#388

C₂×C₂₂₃

yes

complemented

C₄³

C₂×C₂₂₃

yes

characteristic, complemented

C₂²×Q₈

C₂²×C₂₂₃

yes

characteristic

C₂²×D₄

C₂²×C₂₂₃

yes

characteristic

#125

—

#124

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₂₃

yes

—

C₂×C₄²

C₂²×C₂₂₃

yes

—

C₂×C₄²

C₂²×C₂₂₃

yes

—

C₂×C₄²

C₂²×C₂₂₃

yes

characteristic

#95

C₂²×C₂₂₃

yes

—

C₂⁴

C₂³×C₂₂₃

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

—

C₂²×C₄

C₂³×C₂₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₄²

—

103

C₄²

—

104

C₄²

D₄×C₂₂₃

yes

—

105

C₂³

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂⁴×C₂₂₃

yes

characteristic

109

Q₈

—

110

D₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×D₄×C₂₂₃

yes

—

119

C₂×C₄

C₂×D₄×C₂₂₃

yes

—

120

C₂×C₄

#41×C₂₂₃

yes

—

121

C₂×C₄

#41×C₂₂₃

yes

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

C₂²×D₄×C₂₂₃

yes

characteristic

126

C₂²

C₂×#41×C₂₂₃

yes

—

127

C₂²

C₂×#41×C₂₂₃

yes

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#124×C₂₂₃

yes

—

133

C₂

—

134

C₂

C₂×#124×C₂₂₃

yes

characteristic

135

C₂

#529×C₂₂₃

yes

—

136

C₂

#532×C₂₂₃

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₂₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₂₂₃
prime radical	C₂²×D₄×C₂₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₂₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₂₂₃

prime radical

C₂²×D₄×C₂₂₃

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₂₃ ◁ G

N	H ≅ G/N	type
1. C₂×#125×C₂₂₃	C₂	semidirect
3. #391×C₂₂₃	C₂	semidirect
5. #388×C₂₂₃	C₂	semidirect
6. C₄³×C₂₂₃	C₂	semidirect
68. C₂₂₃	#2861	direct (N × H)
69. #2861	C₂₂₃	direct (N × H)
70. C₂×#125	C₂×C₂₂₃	semidirect
72. #391	C₂×C₂₂₃	semidirect
74. #388	C₂×C₂₂₃	semidirect
75. C₄³	C₂×C₂₂₃	semidirect

H ≅ G/N

type

1. C₂×#125×C₂₂₃

C₂

semidirect

3. #391×C₂₂₃

C₂

semidirect

5. #388×C₂₂₃

C₂

semidirect

6. C₄³×C₂₂₃

C₂

semidirect

68. C₂₂₃

#2861

direct (N × H)

69. #2861

C₂₂₃

direct (N × H)

70. C₂×#125

C₂×C₂₂₃

semidirect

72. #391

C₂×C₂₂₃

semidirect

74. #388

C₂×C₂₂₃

semidirect

75. C₄³

C₂×C₂₂₃

semidirect

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

subgroup	index
C₂×#125×C₂₂₃	2
C₂×#124×C₂₂₃	2
#391×C₂₂₃	2
#390×C₂₂₃	2
#388×C₂₂₃	2
C₄³×C₂₂₃	2
#2861	223

subgroup

index

C₂×#125×C₂₂₃

C₂×#124×C₂₂₃

#391×C₂₂₃

#390×C₂₂₃

#388×C₂₂₃

C₄³×C₂₂₃

#2861

223

Subgroups — #2861×C₂₂₃ #43251876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2861×C223 #43251876

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2861×C₂₂₃ #43251876