Group #72227045 — subgroups

Subgroups — #72227045

Order 2048.

376 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 128 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	#72225400	C₂	1	2	yes	maximal
2	#72216197	C₂	1	1	yes	maximal, characteristic
3	#151042	—	2	2	no	—
4	#135778	—	2	2	no	—
5	#135778	C₄	1	2	yes	—
6	C₂×C₁₆²	C₂²	1	1	yes	characteristic
7	#93883	—	2	2	no	—
8	#93879	C₄	1	4	yes	complemented
9	#13640	—	4	2	no	—
10	#12906	—	4	2	no	—
11	#12906	—	4	2	no	—
12	#12906	—	4	2	no	—
13	C₂×C₈×C₁₆	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆	D₄	1	2	yes	—
15	#8818	—	2	2	no	—
16	#7458	—	2	4	no	—
17	#7455	—	4	2	no	—
18	C₁₆²	C₂×C₄	1	2	yes	—
19	C₁₆²	D₄	1	1	yes	characteristic
20	C₁₆²	Q₈	1	1	yes	characteristic
21	C₂×#367	—	8	4	no	—
22	C₂×#367	—	8	4	no	—
23	C₂×#367	—	8	2	no	—
24	C₂×#367	—	8	2	no	—
25	C₂×C₄×C₁₆	—	2	1	no	—
26	C₂×C₄×C₁₆	D₈	1	2	yes	—
27	C₂×C₄×C₁₆	SD₁₆	1	2	yes	—
28	#1845	—	4	2	no	—
29	#1845	—	4	4	no	—
30	C₂×C₈²	C₂²⋊C₄	1	1	yes	characteristic
31	C₄⋊C₃₂	—	4	4	no	—
32	C₂²⋊C₃₂	—	8	2	no	—
33	#1397	—	8	2	no	—
34	C₈×C₁₆	—	2	1	no	—
35	C₈×C₁₆	C₄²	1	1	yes	characteristic
36	C₈×C₁₆	C₂²⋊C₄	1	2	yes	—
37	C₈×C₁₆	C₄⋊C₄	1	1	yes	characteristic
38	C₈×C₁₆	C₄⋊C₄	1	2	yes	—
39	C₈×C₁₆	SD₁₆	1	2	yes	—
40	C₈×C₁₆	Q₁₆	1	2	yes	—
41	C₂²×C₁₆	—	2	2	no	—
42	C₂²×C₁₆	—	2	2	no	—
43	C₂²×C₁₆	D₁₆	1	2	yes	—
44	C₂²×C₁₆	SD₃₂	1	2	yes	—
45	C₂×#107	—	8	4	no	—
46	C₂×#107	—	8	4	no	—
47	C₂×#107	—	8	4	no	—
48	C₂×C₄×C₈	#100	1	1	yes	characteristic
49	C₂×C₄×C₈	#102	1	2	yes	—
50	#384	—	8	2	no	—
51	C₂×C₃₂	—	8	4	no	—
52	#367	—	16	8	no	—
53	#367	—	16	8	no	—
54	#367	—	16	4	no	—
55	#367	—	16	4	no	—
56	C₄×C₁₆	—	2	2	no	—
57	C₄×C₁₆	—	2	1	no	—
58	C₄×C₁₆	—	2	1	no	—
59	C₄×C₁₆	—	2	2	no	—
60	C₄×C₁₆	#102	1	2	yes	—
61	C₄×C₁₆	#103	1	2	yes	—
62	C₄×C₁₆	#106	1	2	yes	—
63	C₄×C₁₆	#107	1	2	yes	—
64	C₄×C₁₆	SD₃₂	1	2	yes	—
65	C₄×C₁₆	Q₃₂	1	2	yes	—
66	C₈²	#95	1	1	yes	characteristic
67	C₈²	#101	1	1	yes	characteristic
68	C₈²	#103	1	2	yes	—
69	C₂²×C₈	—	2	1	no	—
70	C₂²×C₈	#357	1	2	yes	—
71	C₂²×C₈	#361	1	2	yes	—
72	C₂×C₄⋊C₄	—	16	2	no	—
73	C₂×C₄⋊C₄	—	16	4	no	—
74	C₂×C₄²	#331	1	1	yes	characteristic
75	C₂×C₁₆	—	2	2	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	2	8	no	—
78	C₂×C₁₆	—	2	8	no	—
79	C₂×C₁₆	—	2	2	no	—
80	C₂×C₁₆	—	2	2	no	—
81	C₂×C₁₆	—	2	4	no	—
82	C₂×C₁₆	—	2	4	no	—
83	C₂×C₁₆	#357	1	2	yes	—
84	C₂×C₁₆	#358	1	2	yes	—
85	C₂×C₁₆	#366	1	2	yes	—
86	C₂×C₁₆	#367	1	2	yes	—
87	#107	—	16	8	no	—
88	#107	—	16	8	no	—
89	#107	—	16	4	no	—
90	#107	—	16	4	no	—
91	C₄×C₈	—	2	1	no	—
92	C₄×C₈	#332	1	2	yes	—
93	C₄×C₈	#333	1	1	yes	characteristic
94	C₄×C₈	#340	1	2	yes	—
95	C₄×C₈	#343	1	1	yes	characteristic
96	C₄×C₈	#358	1	2	yes	—
97	C₄×C₈	#362	1	2	yes	—
98	C₃₂	—	16	4	no	—
99	C₂²×C₄	—	32	2	no	—
100	C₂²×C₄	#1357	1	1	yes	characteristic
101	C₂²×C₄	#1361	1	2	yes	—
102	C₂×C₈	—	2	4	no	—
103	C₂×C₈	—	2	1	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	2	4	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	#1364	1	2	yes	—
109	C₂×C₈	#1365	1	2	yes	—
110	C₂×C₈	#1380	1	2	yes	—
111	C₂×C₈	#1387	1	2	yes	—
112	C₄⋊C₄	—	32	4	no	—
113	C₄⋊C₄	—	32	4	no	—
114	C₄⋊C₄	—	32	4	no	—
115	C₄²	#1296	1	1	yes	characteristic
116	C₄²	#1358	1	1	yes	characteristic
117	C₄²	#1362	1	2	yes	—
118	C₁₆	—	2	4	no	—
119	C₁₆	—	2	4	no	—
120	C₁₆	—	2	4	no	—
121	C₁₆	—	2	4	no	—
122	C₁₆	—	2	2	no	—
123	C₁₆	—	2	2	no	—
124	C₁₆	#1418	1	4	yes	—
125	C₁₆	#1426	1	4	yes	—
126	C₂³	#7096	1	1	yes	characteristic
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	64	4	no	—
131	C₂×C₄	—	64	2	no	—
132	C₂×C₄	#7097	1	2	yes	—
133	C₂×C₄	#7098	1	1	yes	characteristic
134	C₂×C₄	#7269	1	1	yes	characteristic
135	C₂×C₄	#7274	1	2	yes	—
136	C₈	—	2	2	no	—
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	#7453	1	4	yes	—
141	C₈	#7474	1	2	yes	—
142	C₈	#7477	1	2	yes	—
143	C₂²	—	2	2	no	—
144	C₂²	#93064	1	1	yes	characteristic
145	C₂²	#93868	1	2	yes	—
146	C₄	—	2	2	no	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	—	128	2	no	—
150	C₄	#94355	1	2	yes	—
151	C₄	#94364	1	2	yes	—
152	C₂	—	2	2	no	—
153	C₂	#72216560	1	1	yes	characteristic
154	C₂	#72227044	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#72225400

C₂

yes

maximal

#72216197

C₂

yes

maximal, characteristic

#151042

—

#135778

—

#135778

C₄

yes

—

C₂×C₁₆²

C₂²

yes

characteristic

#93883

—

#93879

C₄

yes

—

—

—

—

C₂×C₈×C₁₆

C₂×C₄

yes

characteristic

C₂×C₈×C₁₆

D₄

yes

—

#8818

—

#7458

—

#7455

—

C₁₆²

C₂×C₄

yes

—

C₁₆²

D₄

yes

characteristic

C₁₆²

Q₈

yes

—

—

—

—

C₂×C₄×C₁₆

—

C₂×C₄×C₁₆

D₈

yes

—

C₂×C₄×C₁₆

SD₁₆

yes

—

#1845

—

#1845

—

C₂×C₈²

C₂²⋊C₄

yes

characteristic

C₄⋊C₃₂

—

C₂²⋊C₃₂

—

#1397

—

C₈×C₁₆

—

C₈×C₁₆

C₄²

yes

characteristic

C₈×C₁₆

C₂²⋊C₄

yes

—

C₈×C₁₆

C₄⋊C₄

yes

characteristic

C₈×C₁₆

C₄⋊C₄

yes

—

C₈×C₁₆

SD₁₆

yes

—

C₈×C₁₆

Q₁₆

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂²×C₁₆

D₁₆

yes

—

C₂²×C₁₆

SD₃₂

yes

—

C₂×#107

—

C₂×#107

—

C₂×#107

—

C₂×C₄×C₈

#100

yes

characteristic

C₂×C₄×C₈

#102

yes

—

#384

—

C₂×C₃₂

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

#102

yes

—

C₄×C₁₆

#103

yes

—

C₄×C₁₆

#106

yes

—

C₄×C₁₆

#107

yes

—

C₄×C₁₆

SD₃₂

yes

—

C₄×C₁₆

Q₃₂

yes

—

C₈²

yes

C₈²

yes

C₈²

yes

—

C₂²×C₈

—

C₂²×C₈

#357

yes

—

C₂²×C₈

#361

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

#331

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#357

yes

—

C₂×C₁₆

#358

yes

—

C₂×C₁₆

#366

yes

—

C₂×C₁₆

#367

yes

—

—

—

—

—

C₄×C₈

—

C₄×C₈

#332

yes

—

C₄×C₈

#333

yes

characteristic

C₄×C₈

#340

yes

—

C₄×C₈

#343

yes

characteristic

C₄×C₈

#358

yes

—

C₄×C₈

#362

yes

—

C₃₂

—

C₂²×C₄

—

100

C₂²×C₄

#1357

yes

characteristic

101

C₂²×C₄

#1361

yes

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

#1364

yes

—

109

C₂×C₈

#1365

yes

—

110

C₂×C₈

#1380

yes

—

111

C₂×C₈

#1387

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

#1296

yes

characteristic

116

C₄²

#1358

yes

characteristic

117

C₄²

#1362

yes

—

118

C₁₆

—

119

C₁₆

—

120

C₁₆

—

121

C₁₆

—

122

C₁₆

—

123

C₁₆

—

124

C₁₆

#1418

yes

—

125

C₁₆

#1426

yes

—

126

C₂³

#7096

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

#7097

yes

—

133

C₂×C₄

#7098

yes

characteristic

134

C₂×C₄

#7269

yes

characteristic

135

C₂×C₄

#7274

yes

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

#7453

yes

—

141

C₈

#7474

yes

—

142

C₈

#7477

yes

—

143

C₂²

—

144

C₂²

#93064

yes

characteristic

145

C₂²

#93868

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

128

—

150

C₄

#94355

yes

—

151

C₄

#94364

yes

—

152

C₂

—

153

C₂

#72216560

yes

characteristic

154

C₂

#72227044

yes

—

	subgroup	G/H
center, socle	C₂²	#93064
commutator	C₈×C₁₆	C₄²
Frattini, cosocle	C₂×C₁₆²	C₂²
prime radical	C₂³	#7096
LCS	G ▷ C₈×C₁₆ ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G
LECS	G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#93064

commutator

C₈×C₁₆

C₄²

Frattini, cosocle

C₂×C₁₆²

C₂²

prime radical

C₂³

#7096

LCS

G ▷ C₈×C₁₆ ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ C₂×C₁₆² ◁ G

LECS

G ▷ C₂×C₁₆² ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
8. #93879	C₄	semidirect

H ≅ G/N

type

8. #93879

C₄

semidirect

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

subgroup	index
#72225400	2
#72216197	2

subgroup

index

#72225400

#72216197

Subgroups — #72227045

Conjugacy classes of subgroups

Complements

Maximal subgroups