Group #10709973 — subgroups

Subgroups — #10709973

Order 768.

272 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 91 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	#69058	C₂	1	2	yes	maximal
2	#69044	C₂	1	2	yes	maximal, complemented
3	#68065	C₂	1	1	yes	maximal, characteristic, complemented
4	#68058	C₂	1	1	yes	maximal, characteristic
5	C₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	#12531	—	3	1	no	maximal, Sylow
7	#4699	—	2	1	no	—
8	#4695	C₂²	1	1	yes	characteristic
9	#4694	—	2	1	no	—
10	#4693	—	2	1	no	—
11	#4682	—	2	1	no	—
12	#4682	C₂²	1	1	yes	characteristic
13	#4682	C₂²	1	2	yes	—
14	C₃×C₄×C₁₆	C₂²	1	2	yes	—
15	C₃×C₈²	C₂²	1	1	yes	characteristic
16	#4509	—	2	4	no	—
17	#4504	—	2	4	no	—
18	#2252	—	3	2	no	—
19	#2240	—	3	2	no	—
20	#1713	—	3	1	no	—
21	#1706	—	3	1	no	—
22	#1312	D₃	1	1	yes	characteristic, complemented
23	C₂×Dic₁₂	—	4	2	no	—
24	C₂×D₂₄	—	2	2	no	—
25	C₂×#204	—	4	2	no	—
26	#874	—	2	1	no	—
27	#869	—	2	2	no	—
28	#869	—	2	1	no	—
29	C₂×C₃×C₁₆	D₄	1	4	yes	—
30	C₃×C₄×C₈	D₄	1	1	yes	characteristic
31	C₃×C₄×C₈	D₄	1	1	yes	characteristic
32	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
33	#821	—	4	2	no	—
34	#818	—	2	2	no	—
35	#818	—	2	2	no	—
36	#818	—	2	2	no	—
37	#818	—	4	2	no	—
38	#500	—	3	1	no	—
39	#500	—	3	2	no	—
40	#500	—	6	1	no	—
41	#494	—	6	1	no	—
42	#493	—	3	1	no	—
43	#492	—	6	1	no	—
44	#486	—	6	1	no	—
45	#366	—	6	4	no	—
46	#357	—	6	4	no	—
47	C₄×C₁₆	C₂×D₃	1	2	yes	—
48	C₈²	C₂×D₃	1	1	yes	characteristic
49	C₂×D₁₂	—	4	1	no	—
50	C₂×D₁₂	—	4	1	no	—
51	C₂×D₁₂	—	4	1	no	—
52	C₂×Dic₆	—	4	1	no	—
53	C₂×Dic₆	—	4	2	no	—
54	C₂×C₃×C₈	D₈	1	2	yes	—
55	C₂×C₃×C₈	D₈	1	2	yes	—
56	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
57	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
58	C₄⋊Dic₃	—	4	1	no	—
59	C₄⋊Dic₃	—	4	2	no	—
60	C₄⋊Dic₃	—	4	2	no	—
61	C₄⋊Dic₃	—	4	1	no	—
62	Dic₁₂	—	8	4	no	—
63	D₂₄	—	4	4	no	—
64	#204	—	8	4	no	—
65	C₃×C₁₆	—	2	4	no	—
66	C₂×Q₁₆	—	12	2	no	—
67	C₂×SD₁₆	—	12	2	no	—
68	C₂×D₈	—	6	2	no	—
69	C₄⋊Q₈	—	6	1	no	—
70	C₄⋊Q₈	—	6	2	no	—
71	#127	—	6	1	no	—
72	C₂×C₁₆	D₁₂	1	4	yes	—
73	#107	—	6	2	no	—
74	#107	—	6	2	no	—
75	#107	—	6	2	no	—
76	#107	—	12	2	no	—
77	#102	—	12	2	no	—
78	C₄×C₈	D₁₂	1	1	yes	characteristic
79	C₄×C₈	D₁₂	1	1	yes	characteristic
80	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
81	C₂²×D₃	—	8	1	no	—
82	C₂×C₃×C₄	#127	1	1	yes	characteristic
83	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
84	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
85	C₂×Dic₃	—	8	2	no	—
86	C₂×Dic₃	—	8	1	no	—
87	D₁₂	—	8	2	no	—
88	D₁₂	—	8	2	no	—
89	D₁₂	—	8	2	no	—
90	Dic₆	—	8	2	no	—
91	Dic₆	—	8	4	no	—
92	C₃×C₈	—	2	2	no	—
93	C₃×C₈	D₁₆	1	4	yes	—
94	C₃×C₈	#137	1	4	yes	—
95	C₂×Q₈	—	12	1	no	—
96	C₂×Q₈	—	12	2	no	—
97	C₂×D₄	—	12	1	no	—
98	C₂×D₄	—	12	1	no	—
99	C₂×D₄	—	12	1	no	—
100	Q₁₆	—	24	4	no	—
101	SD₁₆	—	24	4	no	—
102	D₈	—	12	4	no	—
103	C₂×C₈	D₂₄	1	2	yes	—
104	C₂×C₈	D₂₄	1	2	yes	—
105	C₂×C₈	C₂×D₁₂	1	2	yes	—
106	C₄⋊C₄	—	12	2	no	—
107	C₄⋊C₄	—	12	1	no	—
108	C₄⋊C₄	—	12	1	no	—
109	C₄⋊C₄	—	12	2	no	—
110	C₄²	C₂×D₁₂	1	1	yes	characteristic
111	C₁₆	—	2	4	no	—
112	C₂²×C₃	#493	1	1	yes	characteristic
113	C₂×D₃	—	16	1	no	—
114	C₂×D₃	—	16	1	no	—
115	C₂×D₃	—	16	1	no	—
116	C₃×C₄	#497	1	2	yes	—
117	C₃×C₄	C₂×D₁₆	1	2	yes	—
118	C₃×C₄	#510	1	2	yes	—
119	Dic₃	—	16	1	no	—
120	Dic₃	—	16	2	no	—
121	C₂³	—	24	1	no	—
122	Q₈	—	24	2	no	—
123	Q₈	—	24	4	no	—
124	D₄	—	24	2	no	—
125	D₄	—	24	2	no	—
126	D₄	—	24	2	no	—
127	C₂×C₄	—	24	1	no	—
128	C₂×C₄	—	24	2	no	—
129	C₂×C₄	#874	1	1	yes	characteristic
130	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
131	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
132	C₈	—	2	2	no	—
133	C₈	D₄₈	1	4	yes	—
134	C₈	#909	1	4	yes	—
135	C₂×C₃	#1722	1	1	yes	characteristic
136	C₂×C₃	#2246	1	1	yes	characteristic
137	C₂×C₃	#2251	1	1	yes	characteristic
138	D₃	—	32	1	no	—
139	C₂²	—	48	1	no	—
140	C₂²	—	48	1	no	—
141	C₂²	—	48	1	no	—
142	C₂²	#4695	1	1	yes	characteristic
143	C₄	—	48	1	no	—
144	C₄	—	48	2	no	—
145	C₄	#4715	1	2	yes	—
146	C₄	C₂×D₄₈	1	2	yes	—
147	C₄	#4909	1	2	yes	—
148	C₃	#12531	1	1	yes	characteristic, complemented, Sylow
149	C₂	—	96	1	no	—
150	C₂	#68098	1	1	yes	characteristic
151	C₂	#69060	1	1	yes	characteristic
152	C₂	#69084	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#69058

C₂

yes

maximal

#69044

C₂

yes

maximal, complemented

#68065

C₂

yes

maximal, characteristic, complemented

#68058

C₂

yes

maximal, characteristic

C₃×#1312

C₂

yes

maximal, characteristic, complemented

—

—

C₂²

yes

—

—

—

C₂²

yes

characteristic

#4682

C₂²

yes

—

C₃×C₄×C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic, complemented

C₂×Dic₁₂

—

C₂×D₂₄

—

—

—

—

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

C₂³

yes

—

—

—

—

—

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₃

yes

—

C₈²

C₂×D₃

yes

characteristic

C₂×D₁₂

—

C₂×D₁₂

—

C₂×D₁₂

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₁₂

—

D₂₄

—

#204

—

C₃×C₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

D₁₂

yes

—

—

—

—

—

—

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

D₁₂

yes

characteristic

C₄×C₈

C₂²×D₃

yes

characteristic

C₂²×D₃

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

D₁₂

—

D₁₂

—

D₁₂

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

D₁₆

yes

—

C₃×C₈

#137

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

100

Q₁₆

—

101

SD₁₆

—

102

D₈

—

103

C₂×C₈

D₂₄

yes

—

104

C₂×C₈

D₂₄

yes

—

105

C₂×C₈

C₂×D₁₂

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄²

C₂×D₁₂

yes

characteristic

111

C₁₆

—

112

C₂²×C₃

#493

yes

characteristic

113

C₂×D₃

—

114

C₂×D₃

—

115

C₂×D₃

—

116

C₃×C₄

#497

yes

—

117

C₃×C₄

C₂×D₁₆

yes

—

118

C₃×C₄

#510

yes

—

119

Dic₃

—

120

Dic₃

—

121

C₂³

—

122

Q₈

—

123

Q₈

—

124

D₄

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

#874

yes

characteristic

130

C₂×C₄

C₂×D₂₄

yes

characteristic

131

C₂×C₄

C₂×D₂₄

yes

characteristic

132

C₈

—

133

C₈

D₄₈

yes

—

134

C₈

#909

yes

—

135

C₂×C₃

#1722

yes

characteristic

136

C₂×C₃

#2246

yes

characteristic

137

C₂×C₃

#2251

yes

characteristic

138

D₃

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

#4695

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

#4715

yes

—

146

C₄

C₂×D₄₈

yes

—

147

C₄

#4909

yes

—

148

C₃

#12531

yes

characteristic, complemented, Sylow

149

C₂

—

150

C₂

#68098

yes

characteristic

151

C₂

#69060

yes

characteristic

152

C₂

#69084

yes

characteristic

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×#1312	C₂
socle	C₂²×C₃	#493
prime radical	#4695	C₂²
hypercenter	#1312	D₃
hypocenter	C₃	#12531
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#4695

commutator, cosocle

C₃×C₄×C₈

C₂³

Frattini

C₄×C₈

C₂²×D₃

C₂

C₂²×C₃

C₂²

D₃

C₃

G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
2. #69044	C₂	semidirect
3. #68065	C₂	semidirect
5. C₃×#1312	C₂	semidirect
22. #1312	D₃	semidirect
148. C₃	#12531	semidirect

H ≅ G/N

type

2. #69044

C₂

semidirect

3. #68065

C₂

semidirect

5. C₃×#1312

C₂

semidirect

22. #1312

D₃

semidirect

148. C₃

#12531

semidirect

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

subgroup	index
#69058	2
#69044	2
#68065	2
#68058	2
C₃×#1312	2
#12531	3

subgroup

index

Subgroups — #10709973

Conjugacy classes of subgroups

Complements

Maximal subgroups