Group #72226117 — subgroups

Subgroups — #72226117

Order 1536.

644 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 149 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	#10709899	C₂	1	1	yes	maximal, characteristic, complemented
2	#10709898	C₂	1	4	yes	maximal, complemented
3	C₁₆×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#12515	C₂	1	1	yes	maximal, characteristic, complemented
5	#123403	—	3	1	no	maximal, Sylow
6	C₃×#2246	—	2	2	no	—
7	C₃×#1712	C₂²	1	2	yes	—
8	#71290	—	2	8	no	—
9	#69060	—	2	2	no	—
10	#68062	C₂²	1	2	yes	—
11	C₃×C₈×C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₃⋊C₁₆	C₂²	1	2	yes	—
13	#12515	—	3	4	no	—
14	#12515	—	3	1	no	—
15	#12515	D₃	1	1	yes	characteristic, complemented
16	C₁₆²	—	3	1	no	—
17	C₂×C₃×D₁₆	—	4	4	no	—
18	C₃×#493	—	2	2	no	—
19	C₃×#493	—	2	4	no	—
20	C₂×#826	—	4	16	no	—
21	C₂×D₄₈	—	4	4	no	—
22	#4695	—	2	4	no	—
23	#4695	—	2	2	no	—
24	C₃×C₄×C₁₆	D₄	1	2	yes	—
25	C₃×C₈²	C₂³	1	1	yes	characteristic
26	C₄×C₃⋊C₁₆	D₄	1	4	yes	—
27	#2246	—	2	2	no	—
28	#2246	—	6	8	no	—
29	#2246	—	6	2	no	—
30	#1712	—	3	2	no	—
31	#1712	C₂×D₃	1	2	yes	—
32	C₈×C₁₆	—	3	2	no	—
33	C₈×C₁₆	C₂×D₃	1	1	yes	characteristic
34	C₂×C₃×D₈	—	4	4	no	—
35	C₂×C₃×D₈	—	4	8	no	—
36	C₃×#127	—	4	2	no	—
37	C₂×D₂₄	—	4	8	no	—
38	C₂×D₂₄	—	4	4	no	—
39	#874	—	4	2	no	—
40	C₃×D₁₆	—	8	8	no	—
41	C₂×C₃×C₁₆	D₈	1	4	yes	—
42	C₃×C₄×C₈	C₂×D₄	1	2	yes	—
43	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
44	#826	—	8	32	no	—
45	C₂×C₃⋊C₁₆	D₈	1	8	yes	—
46	D₄₈	—	8	8	no	—
47	C₂×D₁₆	—	4	4	no	—
48	C₂×D₁₆	—	12	16	no	—
49	C₂×D₁₆	—	12	4	no	—
50	#493	—	2	4	no	—
51	#493	—	2	2	no	—
52	#493	—	6	4	no	—
53	#493	—	6	2	no	—
54	C₄×C₁₆	—	3	4	no	—
55	C₄×C₁₆	#67	1	2	yes	—
56	C₈²	C₂²×D₃	1	1	yes	characteristic
57	C₂×C₃×D₄	—	8	4	no	—
58	C₂×C₃×D₄	—	8	2	no	—
59	C₂×D₁₂	—	8	4	no	—
60	C₂×D₁₂	—	8	2	no	—
61	C₃×D₈	—	8	16	no	—
62	C₃×D₈	—	8	8	no	—
63	C₂×C₃×C₈	C₂×D₈	1	4	yes	—
64	C₂×C₃×C₈	C₂×D₈	1	2	yes	—
65	C₃×C₄²	#127	1	1	yes	characteristic
66	D₂₄	—	8	8	no	—
67	D₂₄	—	8	16	no	—
68	C₃×C₁₆	D₁₆	1	8	yes	complemented
69	C₃⋊C₁₆	D₁₆	1	16	yes	complemented
70	C₂×D₈	—	4	8	no	—
71	C₂×D₈	—	4	4	no	—
72	C₂×D₈	—	12	8	no	—
73	C₂×D₈	—	12	4	no	—
74	#127	—	4	2	no	—
75	#127	—	12	2	no	—
76	D₁₆	—	8	8	no	—
77	D₁₆	—	24	32	no	—
78	D₁₆	—	24	8	no	—
79	C₂×C₁₆	—	3	8	no	—
80	C₂×C₁₆	#213	1	4	yes	—
81	C₄×C₈	D₃×D₄	1	2	yes	—
82	C₄×C₈	C₂×#67	1	1	yes	characteristic
83	C₂³×C₃	—	16	2	no	—
84	C₂²×D₃	—	16	2	no	—
85	C₃×D₄	—	16	8	no	—
86	C₃×D₄	—	16	4	no	—
87	C₂×C₃×C₄	#493	1	2	yes	—
88	C₂×C₃×C₄	#493	1	1	yes	characteristic
89	D₁₂	—	16	8	no	—
90	D₁₂	—	16	4	no	—
91	C₃×C₈	C₂×D₁₆	1	4	yes	—
92	C₃×C₈	C₂×D₁₆	1	8	yes	—
93	C₂×D₄	—	8	4	no	—
94	C₂×D₄	—	8	2	no	—
95	C₂×D₄	—	24	4	no	—
96	C₂×D₄	—	24	2	no	—
97	D₈	—	8	8	no	—
98	D₈	—	8	16	no	—
99	D₈	—	24	8	no	—
100	D₈	—	24	16	no	—
101	C₂×C₈	D₃×D₈	1	4	yes	—
102	C₂×C₈	C₂×#213	1	2	yes	—
103	C₄²	#940	1	1	yes	characteristic
104	C₁₆	—	3	16	no	—
105	C₁₆	#826	1	8	yes	complemented
106	C₂²×C₃	—	32	4	no	—
107	C₂²×C₃	—	32	2	no	—
108	C₂²×C₃	#1712	1	1	yes	characteristic
109	C₂×D₃	—	32	4	no	—
110	C₂×D₃	—	32	2	no	—
111	C₃×C₄	#2246	1	4	yes	—
112	C₃×C₄	#2246	1	2	yes	—
113	C₂³	—	16	2	no	—
114	C₂³	—	48	2	no	—
115	D₄	—	16	8	no	—
116	D₄	—	16	4	no	—
117	D₄	—	48	8	no	—
118	D₄	—	48	4	no	—
119	C₂×C₄	#5073	1	1	yes	characteristic
120	C₂×C₄	#5151	1	2	yes	—
121	C₈	D₃×D₁₆	1	8	yes	—
122	C₈	C₂×#826	1	4	yes	—
123	C₂×C₃	—	64	2	no	—
124	C₂×C₃	#12515	1	2	yes	—
125	C₂×C₃	#12515	1	1	yes	characteristic
126	D₃	—	64	2	no	—
127	C₂²	—	32	4	no	—
128	C₂²	—	32	2	no	—
129	C₂²	—	96	4	no	—
130	C₂²	—	96	2	no	—
131	C₂²	#71276	1	1	yes	characteristic
132	C₄	#71290	1	2	yes	—
133	C₄	#71464	1	4	yes	—
134	C₃	#123403	1	1	yes	characteristic, complemented, Sylow
135	C₂	—	64	2	no	—
136	C₂	—	192	2	no	—
137	C₂	#10709897	1	2	yes	—
138	C₂	#10709898	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#10709899

C₂

yes

maximal, characteristic, complemented

#10709898

C₂

yes

maximal, complemented

C₁₆×C₃⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₃×#12515

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

#71290

—

#69060

—

#68062

C₂²

yes

—

C₃×C₈×C₁₆

C₂²

yes

characteristic

C₈×C₃⋊C₁₆

C₂²

yes

—

#12515

—

#12515

—

#12515

D₃

yes

characteristic, complemented

C₁₆²

—

C₂×C₃×D₁₆

—

C₃×#493

—

C₃×#493

—

C₂×#826

—

C₂×D₄₈

—

#4695

—

#4695

—

C₃×C₄×C₁₆

D₄

yes

—

C₃×C₈²

C₂³

yes

characteristic

C₄×C₃⋊C₁₆

D₄

yes

—

—

—

—

—

C₂×D₃

yes

—

C₈×C₁₆

—

C₈×C₁₆

C₂×D₃

yes

characteristic

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₃×#127

—

C₂×D₂₄

—

C₂×D₂₄

—

#874

—

C₃×D₁₆

—

C₂×C₃×C₁₆

D₈

yes

—

C₃×C₄×C₈

C₂×D₄

yes

—

C₃×C₄×C₈

C₂×D₄

yes

characteristic

#826

—

C₂×C₃⋊C₁₆

D₈

yes

—

D₄₈

—

C₂×D₁₆

—

C₂×D₁₆

—

C₂×D₁₆

—

—

—

—

—

C₄×C₁₆

—

C₄×C₁₆

#67

yes

—

C₈²

C₂²×D₃

yes

characteristic

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×D₁₂

—

C₂×D₁₂

—

C₃×D₈

—

C₃×D₈

—

C₂×C₃×C₈

C₂×D₈

yes

—

C₂×C₃×C₈

C₂×D₈

yes

—

C₃×C₄²

#127

yes

characteristic

D₂₄

—

D₂₄

—

C₃×C₁₆

D₁₆

yes

complemented

C₃⋊C₁₆

D₁₆

yes

complemented

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

#127

—

#127

—

D₁₆

—

D₁₆

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

#213

yes

—

C₄×C₈

D₃×D₄

yes

—

C₄×C₈

C₂×#67

yes

characteristic

C₂³×C₃

—

C₂²×D₃

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

#493

yes

—

C₂×C₃×C₄

#493

yes

characteristic

D₁₂

—

D₁₂

—

C₃×C₈

C₂×D₁₆

yes

—

C₃×C₈

C₂×D₁₆

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

100

D₈

—

101

C₂×C₈

D₃×D₈

yes

—

102

C₂×C₈

C₂×#213

yes

—

103

C₄²

#940

yes

characteristic

104

C₁₆

—

105

C₁₆

#826

yes

complemented

106

C₂²×C₃

—

107

C₂²×C₃

—

108

C₂²×C₃

#1712

yes

characteristic

109

C₂×D₃

—

110

C₂×D₃

—

111

C₃×C₄

#2246

yes

—

112

C₃×C₄

#2246

yes

—

113

C₂³

—

114

C₂³

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

C₂×C₄

#5073

yes

characteristic

120

C₂×C₄

#5151

yes

—

121

C₈

D₃×D₁₆

yes

—

122

C₈

C₂×#826

yes

—

123

C₂×C₃

—

124

C₂×C₃

#12515

yes

—

125

C₂×C₃

#12515

yes

characteristic

126

D₃

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

#71276

yes

characteristic

132

C₄

#71290

yes

—

133

C₄

#71464

yes

—

134

C₃

#123403

yes

characteristic, complemented, Sylow

135

C₂

—

136

C₂

—

192

—

137

C₂

#10709897

yes

—

138

C₂

#10709898

yes

characteristic

	subgroup	G/H
center	C₂²	#71276
commutator, cosocle	C₃×C₈²	C₂³
Frattini	C₈²	C₂²×D₃
Fitting	C₃×#12515	C₂
socle	C₂²×C₃	#1712
hypercenter	#12515	D₃
hypocenter	C₃	#123403
LCS	G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #12515

subgroup

G/H

center

C₂²

#71276

commutator, cosocle

C₃×C₈²

C₂³

Frattini

C₈²

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #12515

N	H ≅ G/N	type
1. #10709899	C₂	semidirect
2. #10709898	C₂	semidirect
3. C₁₆×C₃⋊C₁₆	C₂	semidirect
4. C₃×#12515	C₂	semidirect
15. #12515	D₃	semidirect
68. C₃×C₁₆	D₁₆	semidirect
69. C₃⋊C₁₆	D₁₆	semidirect
105. C₁₆	#826	semidirect
134. C₃	#123403	semidirect

H ≅ G/N

type

1. #10709899

C₂

semidirect

2. #10709898

C₂

semidirect

3. C₁₆×C₃⋊C₁₆

C₂

semidirect

4. C₃×#12515

C₂

semidirect

15. #12515

D₃

semidirect

68. C₃×C₁₆

D₁₆

semidirect

69. C₃⋊C₁₆

D₁₆

semidirect

105. C₁₆

#826

semidirect

134. C₃

#123403

semidirect

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

subgroup	index
#10709899	2
#10709898	2
C₁₆×C₃⋊C₁₆	2
C₃×#12515	2
#123403	3

subgroup

index

C₁₆×C₃⋊C₁₆

Subgroups — #72226117

Conjugacy classes of subgroups

Complements

Maximal subgroups