Group #19210252 — subgroups

Subgroups — C₂³×C₂₃×SD₃₂ #19210252

Order 5888.

2334 conjugacy classes of subgroups; only the 1054 normal subgroups are shown.

normal subgroups only

This group has 2334 conjugacy classes of subgroups — too many to draw. Only its 1054 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	C₂³×Q₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₂₃×SD₃₂	C₂	1	28	yes	maximal, direct product factor
4	C₂³×C₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×Q₈×C₂₃	—	2	1	no	—
6	C₂³×D₄×C₂₃	—	2	1	no	—
7	C₂²×Q₁₆×C₂₃	—	2	7	no	—
8	C₂²×Q₁₆×C₂₃	C₂²	1	14	yes	complemented
9	C₂²×D₈×C₂₃	—	2	7	no	—
10	C₂²×D₈×C₂₃	C₂²	1	14	yes	—
11	C₂³×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×SD₃₂	C₂²	1	112	yes	direct product factor
13	C₂²×C₁₆×C₂₃	C₂²	1	14	yes	complemented
14	C₂⁵×C₂₃	—	4	1	no	—
15	C₂²×Q₈×C₂₃	—	2	14	no	—
16	C₂²×Q₈×C₂₃	—	4	7	no	—
17	C₂²×D₄×C₂₃	—	2	14	no	—
18	C₂²×D₄×C₂₃	—	4	7	no	—
19	C₂³×C₄×C₂₃	—	4	1	no	—
20	C₂³×C₄×C₂₃	D₄	1	1	yes	characteristic
21	C₂×Q₁₆×C₂₃	—	2	42	no	—
22	C₂×Q₁₆×C₂₃	C₂³	1	28	yes	complemented
23	C₂×D₈×C₂₃	—	2	42	no	—
24	C₂×D₈×C₂₃	C₂³	1	28	yes	—
25	C₂²×C₈×C₂₃	D₄	1	7	yes	—
26	C₂²×C₈×C₂₃	C₂³	1	7	yes	—
27	C₂₃×SD₃₂	C₂³	1	64	yes	direct product factor
28	C₂×C₁₆×C₂₃	C₂³	1	28	yes	complemented
29	C₂⁴×C₂₃	—	4	14	no	—
30	C₂⁴×C₂₃	—	8	8	no	—
31	C₂⁴×C₂₃	D₈	1	1	yes	characteristic
32	C₂×Q₈×C₂₃	—	2	28	no	—
33	C₂×Q₈×C₂₃	—	4	42	no	—
34	C₂×D₄×C₂₃	—	2	28	no	—
35	C₂×D₄×C₂₃	—	4	42	no	—
36	C₂²×C₄×C₂₃	—	4	14	no	—
37	C₂²×C₄×C₂₃	D₈	1	7	yes	—
38	C₂²×C₄×C₂₃	C₂×D₄	1	7	yes	—
39	Q₁₆×C₂₃	—	2	28	no	—
40	Q₁₆×C₂₃	C₂⁴	1	8	yes	complemented
41	D₈×C₂₃	—	2	28	no	—
42	D₈×C₂₃	C₂⁴	1	8	yes	—
43	C₂×C₈×C₂₃	C₂×D₄	1	21	yes	—
44	C₂×C₈×C₂₃	C₂⁴	1	7	yes	—
45	C₁₆×C₂₃	C₂⁴	1	8	yes	complemented
46	C₂³×SD₃₂	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂³×C₂₃	—	4	28	no	—
48	C₂³×C₂₃	—	8	56	no	—
49	C₂³×C₂₃	SD₃₂	1	8	yes	direct product factor
50	C₂³×C₂₃	C₂×D₈	1	7	yes	—
51	Q₈×C₂₃	—	2	8	no	—
52	Q₈×C₂₃	—	4	28	no	—
53	D₄×C₂₃	—	2	8	no	—
54	D₄×C₂₃	—	4	28	no	—
55	C₂×C₄×C₂₃	—	4	28	no	—
56	C₂×C₄×C₂₃	C₂×D₈	1	21	yes	—
57	C₂×C₄×C₂₃	C₂²×D₄	1	7	yes	—
58	C₈×C₂₃	C₂²×D₄	1	7	yes	—
59	C₈×C₂₃	C₂⁵	1	1	yes	characteristic
60	C₂³×Q₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
61	C₂³×D₈	C₂×C₂₃	1	1	yes	characteristic
62	C₂²×SD₃₂	C₂×C₂₃	1	28	yes	direct product factor
63	C₂³×C₁₆	C₂×C₂₃	1	1	yes	characteristic, complemented
64	C₂²×C₂₃	—	4	8	no	—
65	C₂²×C₂₃	—	8	56	no	—
66	C₂²×C₂₃	C₂×SD₃₂	1	28	yes	direct product factor
67	C₂²×C₂₃	C₂²×D₈	1	7	yes	—
68	C₄×C₂₃	—	4	8	no	—
69	C₄×C₂₃	C₂²×D₈	1	7	yes	—
70	C₄×C₂₃	C₂³×D₄	1	1	yes	characteristic
71	C₂³×Q₈	—	2	1	no	—
72	C₂³×D₄	—	2	1	no	—
73	C₂²×Q₁₆	—	2	7	no	—
74	C₂²×Q₁₆	C₂²×C₂₃	1	14	yes	complemented
75	C₂²×D₈	—	2	7	no	—
76	C₂²×D₈	C₂²×C₂₃	1	14	yes	—
77	C₂³×C₈	C₂²×C₂₃	1	1	yes	characteristic
78	C₂×SD₃₂	C₂²×C₂₃	1	112	yes	direct product factor
79	C₂²×C₁₆	C₂²×C₂₃	1	14	yes	complemented
80	C₂×C₂₃	—	8	8	no	—
81	C₂×C₂₃	C₂²×SD₃₂	1	14	yes	direct product factor
82	C₂×C₂₃	C₂³×D₈	1	1	yes	characteristic
83	C₂⁵	—	4	1	no	—
84	C₂²×Q₈	—	2	14	no	—
85	C₂²×Q₈	—	4	7	no	—
86	C₂²×D₄	—	2	14	no	—
87	C₂²×D₄	—	4	7	no	—
88	C₂³×C₄	—	4	1	no	—
89	C₂³×C₄	D₄×C₂₃	1	1	yes	characteristic
90	C₂×Q₁₆	—	2	42	no	—
91	C₂×Q₁₆	C₂³×C₂₃	1	28	yes	complemented
92	C₂×D₈	—	2	42	no	—
93	C₂×D₈	C₂³×C₂₃	1	28	yes	—
94	C₂²×C₈	D₄×C₂₃	1	7	yes	—
95	C₂²×C₈	C₂³×C₂₃	1	7	yes	—
96	SD₃₂	C₂³×C₂₃	1	64	yes	direct product factor
97	C₂×C₁₆	C₂³×C₂₃	1	28	yes	complemented
98	C₂₃	C₂³×SD₃₂	1	1	yes	characteristic, direct product factor, Sylow
99	C₂⁴	—	4	14	no	—
100	C₂⁴	—	8	8	no	—
101	C₂⁴	D₈×C₂₃	1	1	yes	characteristic
102	C₂×Q₈	—	2	28	no	—
103	C₂×Q₈	—	4	42	no	—
104	C₂×D₄	—	2	28	no	—
105	C₂×D₄	—	4	42	no	—
106	C₂²×C₄	—	4	14	no	—
107	C₂²×C₄	D₈×C₂₃	1	7	yes	—
108	C₂²×C₄	C₂×D₄×C₂₃	1	7	yes	—
109	Q₁₆	—	2	28	no	—
110	Q₁₆	C₂⁴×C₂₃	1	8	yes	complemented
111	D₈	—	2	28	no	—
112	D₈	C₂⁴×C₂₃	1	8	yes	—
113	C₂×C₈	C₂×D₄×C₂₃	1	21	yes	—
114	C₂×C₈	C₂⁴×C₂₃	1	7	yes	—
115	C₁₆	C₂⁴×C₂₃	1	8	yes	complemented
116	C₂³	—	4	28	no	—
117	C₂³	—	8	56	no	—
118	C₂³	C₂₃×SD₃₂	1	8	yes	direct product factor
119	C₂³	C₂×D₈×C₂₃	1	7	yes	—
120	Q₈	—	2	8	no	—
121	Q₈	—	4	28	no	—
122	D₄	—	2	8	no	—
123	D₄	—	4	28	no	—
124	C₂×C₄	—	4	28	no	—
125	C₂×C₄	C₂×D₈×C₂₃	1	21	yes	—
126	C₂×C₄	C₂²×D₄×C₂₃	1	7	yes	—
127	C₈	C₂²×D₄×C₂₃	1	7	yes	—
128	C₈	C₂⁵×C₂₃	1	1	yes	characteristic
129	C₂²	—	4	8	no	—
130	C₂²	—	8	56	no	—
131	C₂²	C₂×C₂₃×SD₃₂	1	28	yes	direct product factor
132	C₂²	C₂²×D₈×C₂₃	1	7	yes	—
133	C₄	—	4	8	no	—
134	C₄	C₂²×D₈×C₂₃	1	7	yes	—
135	C₄	C₂³×D₄×C₂₃	1	1	yes	characteristic
136	C₂	—	8	8	no	—
137	C₂	C₂²×C₂₃×SD₃₂	1	14	yes	direct product factor
138	C₂	C₂³×D₈×C₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₂₃×SD₃₂

C₂

yes

maximal, direct product factor

C₂³×C₁₆×C₂₃

C₂

yes

maximal, characteristic, complemented

C₂³×Q₈×C₂₃

—

C₂³×D₄×C₂₃

—

C₂²×Q₁₆×C₂₃

—

C₂²×Q₁₆×C₂₃

C₂²

yes

complemented

C₂²×D₈×C₂₃

—

C₂²×D₈×C₂₃

C₂²

yes

—

C₂³×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×SD₃₂

C₂²

112

yes

direct product factor

C₂²×C₁₆×C₂₃

C₂²

yes

complemented

C₂⁵×C₂₃

—

C₂²×Q₈×C₂₃

—

C₂²×Q₈×C₂₃

—

C₂²×D₄×C₂₃

—

C₂²×D₄×C₂₃

—

C₂³×C₄×C₂₃

—

C₂³×C₄×C₂₃

D₄

yes

characteristic

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

C₂³

yes

complemented

C₂×D₈×C₂₃

—

C₂×D₈×C₂₃

C₂³

yes

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

C₂³

yes

—

C₂₃×SD₃₂

C₂³

yes

direct product factor

C₂×C₁₆×C₂₃

C₂³

yes

complemented

C₂⁴×C₂₃

—

C₂⁴×C₂₃

—

C₂⁴×C₂₃

D₈

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₈

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

C₂⁴

yes

complemented

D₈×C₂₃

—

D₈×C₂₃

C₂⁴

yes

—

C₂×C₈×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

C₂⁴

yes

—

C₁₆×C₂₃

C₂⁴

yes

complemented

C₂³×SD₃₂

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

SD₃₂

yes

direct product factor

C₂³×C₂₃

C₂×D₈

yes

—

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₈

yes

—

C₂×C₄×C₂₃

C₂²×D₄

yes

—

C₈×C₂₃

C₂²×D₄

yes

—

C₈×C₂₃

C₂⁵

yes

characteristic

C₂³×Q₁₆

C₂×C₂₃

yes

characteristic, complemented

C₂³×D₈

C₂×C₂₃

yes

characteristic

C₂²×SD₃₂

C₂×C₂₃

yes

direct product factor

C₂³×C₁₆

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×SD₃₂

yes

direct product factor

C₂²×C₂₃

C₂²×D₈

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂²×D₈

yes

—

C₄×C₂₃

C₂³×D₄

yes

characteristic

C₂³×Q₈

—

C₂³×D₄

—

C₂²×Q₁₆

—

C₂²×Q₁₆

C₂²×C₂₃

yes

complemented

C₂²×D₈

—

C₂²×D₈

C₂²×C₂₃

yes

—

C₂³×C₈

C₂²×C₂₃

yes

characteristic

C₂×SD₃₂

C₂²×C₂₃

112

yes

direct product factor

C₂²×C₁₆

C₂²×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

C₂²×SD₃₂

yes

direct product factor

C₂×C₂₃

C₂³×D₈

yes

characteristic

C₂⁵

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×D₄

—

C₂²×D₄

—

C₂³×C₄

—

C₂³×C₄

D₄×C₂₃

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

C₂³×C₂₃

yes

complemented

C₂×D₈

—

C₂×D₈

C₂³×C₂₃

yes

—

C₂²×C₈

D₄×C₂₃

yes

—

C₂²×C₈

C₂³×C₂₃

yes

—

SD₃₂

C₂³×C₂₃

yes

direct product factor

C₂×C₁₆

C₂³×C₂₃

yes

complemented

C₂₃

C₂³×SD₃₂

yes

characteristic, direct product factor, Sylow

C₂⁴

—

100

C₂⁴

—

101

C₂⁴

D₈×C₂₃

yes

characteristic

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

D₈×C₂₃

yes

—

108

C₂²×C₄

C₂×D₄×C₂₃

yes

—

109

Q₁₆

—

110

Q₁₆

C₂⁴×C₂₃

yes

complemented

111

D₈

—

112

D₈

C₂⁴×C₂₃

yes

—

113

C₂×C₈

C₂×D₄×C₂₃

yes

—

114

C₂×C₈

C₂⁴×C₂₃

yes

—

115

C₁₆

C₂⁴×C₂₃

yes

complemented

116

C₂³

—

117

C₂³

—

118

C₂³

C₂₃×SD₃₂

yes

direct product factor

119

C₂³

C₂×D₈×C₂₃

yes

—

120

Q₈

—

121

Q₈

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

C₂×D₈×C₂₃

yes

—

126

C₂×C₄

C₂²×D₄×C₂₃

yes

—

127

C₈

C₂²×D₄×C₂₃

yes

—

128

C₈

C₂⁵×C₂₃

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

C₂×C₂₃×SD₃₂

yes

direct product factor

132

C₂²

C₂²×D₈×C₂₃

yes

—

133

C₄

—

134

C₄

C₂²×D₈×C₂₃

yes

—

135

C₄

C₂³×D₄×C₂₃

yes

characteristic

136

C₂

—

137

C₂

C₂²×C₂₃×SD₃₂

yes

direct product factor

138

C₂

C₂³×D₈×C₂₃

yes

characteristic

	subgroup	G/H
center, socle	C₂⁴×C₂₃	D₈
commutator, Frattini, cosocle	C₈	C₂⁵×C₂₃
prime radical	C₂³×D₈×C₂₃	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂⁴×C₂₃

D₈

commutator, Frattini, cosocle

C₈

C₂⁵×C₂₃

prime radical

C₂³×D₈×C₂₃

C₂

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ C₂³×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂³×Q₁₆×C₂₃	C₂	semidirect
3. C₂²×C₂₃×SD₃₂	C₂	direct (N × H)
4. C₂³×C₁₆×C₂₃	C₂	semidirect
8. C₂²×Q₁₆×C₂₃	C₂²	semidirect
12. C₂×C₂₃×SD₃₂	C₂²	direct (N × H)
13. C₂²×C₁₆×C₂₃	C₂²	semidirect
22. C₂×Q₁₆×C₂₃	C₂³	semidirect
27. C₂₃×SD₃₂	C₂³	direct (N × H)
28. C₂×C₁₆×C₂₃	C₂³	semidirect
40. Q₁₆×C₂₃	C₂⁴	semidirect
45. C₁₆×C₂₃	C₂⁴	semidirect
46. C₂³×SD₃₂	C₂₃	direct (N × H)
49. C₂³×C₂₃	SD₃₂	direct (N × H)
60. C₂³×Q₁₆	C₂×C₂₃	semidirect
62. C₂²×SD₃₂	C₂×C₂₃	direct (N × H)
63. C₂³×C₁₆	C₂×C₂₃	semidirect
66. C₂²×C₂₃	C₂×SD₃₂	direct (N × H)
74. C₂²×Q₁₆	C₂²×C₂₃	semidirect
78. C₂×SD₃₂	C₂²×C₂₃	direct (N × H)
79. C₂²×C₁₆	C₂²×C₂₃	semidirect
81. C₂×C₂₃	C₂²×SD₃₂	direct (N × H)
91. C₂×Q₁₆	C₂³×C₂₃	semidirect
96. SD₃₂	C₂³×C₂₃	direct (N × H)
97. C₂×C₁₆	C₂³×C₂₃	semidirect
98. C₂₃	C₂³×SD₃₂	direct (N × H)
110. Q₁₆	C₂⁴×C₂₃	semidirect
115. C₁₆	C₂⁴×C₂₃	semidirect
118. C₂³	C₂₃×SD₃₂	direct (N × H)
131. C₂²	C₂×C₂₃×SD₃₂	direct (N × H)
137. C₂	C₂²×C₂₃×SD₃₂	direct (N × H)

H ≅ G/N

type

1. C₂³×Q₁₆×C₂₃

C₂

semidirect

3. C₂²×C₂₃×SD₃₂

C₂

direct (N × H)

4. C₂³×C₁₆×C₂₃

C₂

semidirect

8. C₂²×Q₁₆×C₂₃

C₂²

semidirect

12. C₂×C₂₃×SD₃₂

C₂²

direct (N × H)

13. C₂²×C₁₆×C₂₃

C₂²

semidirect

22. C₂×Q₁₆×C₂₃

C₂³

semidirect

27. C₂₃×SD₃₂

C₂³

direct (N × H)

28. C₂×C₁₆×C₂₃

C₂³

semidirect

40. Q₁₆×C₂₃

C₂⁴

semidirect

45. C₁₆×C₂₃

C₂⁴

semidirect

46. C₂³×SD₃₂

C₂₃

direct (N × H)

49. C₂³×C₂₃

SD₃₂

direct (N × H)

60. C₂³×Q₁₆

C₂×C₂₃

semidirect

62. C₂²×SD₃₂

C₂×C₂₃

direct (N × H)

63. C₂³×C₁₆

C₂×C₂₃

semidirect

66. C₂²×C₂₃

C₂×SD₃₂

direct (N × H)

74. C₂²×Q₁₆

C₂²×C₂₃

semidirect

78. C₂×SD₃₂

C₂²×C₂₃

direct (N × H)

79. C₂²×C₁₆

C₂²×C₂₃

semidirect

81. C₂×C₂₃

C₂²×SD₃₂

direct (N × H)

91. C₂×Q₁₆

C₂³×C₂₃

semidirect

96. SD₃₂

C₂³×C₂₃

direct (N × H)

97. C₂×C₁₆

C₂³×C₂₃

semidirect

98. C₂₃

C₂³×SD₃₂

direct (N × H)

110. Q₁₆

C₂⁴×C₂₃

semidirect

115. C₁₆

C₂⁴×C₂₃

semidirect

118. C₂³

C₂₃×SD₃₂

direct (N × H)

131. C₂²

C₂×C₂₃×SD₃₂

direct (N × H)

137. C₂

C₂²×C₂₃×SD₃₂

direct (N × H)

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

subgroup	index
C₂³×Q₁₆×C₂₃	2
C₂³×D₈×C₂₃	2
C₂²×C₂₃×SD₃₂	2
C₂³×C₁₆×C₂₃	2
C₂³×SD₃₂	23

subgroup

index

C₂³×Q₁₆×C₂₃

C₂³×D₈×C₂₃

C₂²×C₂₃×SD₃₂

C₂³×C₁₆×C₂₃

C₂³×SD₃₂

Subgroups — C₂³×C₂₃×SD₃₂ #19210252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×C23×SD32 #19210252

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₂₃×SD₃₂ #19210252