Group #21461349 — subgroups

Subgroups — #21461349

Order 7808.

228 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 60 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	#16840785	C₂	1	1	yes	maximal, characteristic
2	#16840785	C₂	1	1	yes	maximal, characteristic
3	#16840621	C₂	1	4	yes	maximal, complemented
4	C₄⋊Q₈×D₆₁	C₂	1	1	yes	maximal, characteristic
5	#14519844	C₂²	1	2	yes	—
6	#14519844	C₂²	1	2	yes	—
7	#14519834	—	2	1	no	—
8	#14519834	—	2	1	no	—
9	#14519832	—	2	1	no	—
10	#14519832	—	2	1	no	—
11	C₂×Q₈×D₆₁	—	2	1	no	—
12	#14519777	C₄	1	1	yes	characteristic
13	#14519776	—	2	1	no	—
14	#14519776	C₂²	1	2	yes	—
15	#14519775	—	2	1	no	—
16	C₄⋊C₄×D₆₁	—	2	1	no	—
17	C₄⋊C₄×D₆₁	C₄	1	2	yes	—
18	C₄²×D₆₁	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₆₁	C₄	1	1	yes	characteristic
20	C₂×Q₈×C₆₁	—	2	1	no	—
21	Q₈×D₆₁	—	4	4	no	—
22	C₂×C₄×D₆₁	—	2	2	no	—
23	C₂×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
25	C₂×C₄×D₆₁	D₄	1	1	yes	characteristic
26	C₂×Dic₁₂₂	—	2	1	no	—
27	C₂×Dic₁₂₂	—	2	2	no	—
28	C₂×#92420	—	2	2	no	—
29	C₂×#92420	—	2	2	no	—
30	C₄⋊C₄×C₆₁	—	2	1	no	—
31	C₄⋊C₄×C₆₁	C₂×C₄	1	2	yes	—
32	C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Dic₆₁	—	2	1	no	—
34	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
35	#11758196	—	2	2	no	—
36	#11758196	—	2	2	no	—
37	C₄×Dic₆₁	D₄	1	1	yes	characteristic
38	C₄×Dic₆₁	D₄	1	1	yes	characteristic
39	C₄×Dic₆₁	C₂³	1	1	yes	characteristic
40	C₂²×D₆₁	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₆₁	—	4	2	no	—
42	C₂×C₄×C₆₁	—	2	2	no	—
43	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₆₁	—	2	2	no	—
47	C₂×Dic₆₁	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₆₁	C₂×D₄	1	1	yes	characteristic
50	C₄×D₆₁	—	2	2	no	—
51	C₄×D₆₁	—	2	2	no	—
52	C₄×D₆₁	—	2	2	no	—
53	C₄×D₆₁	—	4	4	no	—
54	Dic₁₂₂	—	4	2	no	—
55	Dic₁₂₂	—	4	4	no	—
56	#92420	—	4	2	no	—
57	#92420	—	4	2	no	—
58	C₂²×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×D₆₁	—	2	1	no	—
60	C₂×D₆₁	—	2	1	no	—
61	C₂×D₆₁	#101	1	2	yes	—
62	C₄×C₆₁	—	2	1	no	—
63	C₄×C₆₁	—	2	1	no	—
64	C₄×C₆₁	—	2	1	no	—
65	C₄×C₆₁	—	4	2	no	—
66	Dic₆₁	—	2	1	no	—
67	Dic₆₁	—	4	2	no	—
68	Dic₆₁	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₆₁	#137	1	2	yes	—
70	#1556	—	61	1	no	maximal, Sylow
71	C₂×C₆₁	C₂×#101	1	1	yes	characteristic
72	C₂×C₆₁	#418	1	1	yes	characteristic
73	C₂×C₆₁	#419	1	1	yes	characteristic
74	D₆₁	—	2	2	no	—
75	C₂×C₄⋊Q₈	—	61	1	no	—
76	#432	—	61	1	no	—
77	#432	—	61	1	no	—
78	#332	—	61	4	no	—
79	C₆₁	#1556	1	1	yes	characteristic, complemented, Sylow
80	C₂²×Q₈	—	122	1	no	—
81	C₄⋊Q₈	—	61	2	no	—
82	C₄⋊Q₈	—	61	1	no	—
83	C₄⋊Q₈	—	122	1	no	—
84	C₄⋊Q₈	—	122	1	no	—
85	C₄⋊Q₈	#6923	1	1	yes	characteristic
86	C₂×C₄⋊C₄	—	61	2	no	—
87	C₂×C₄⋊C₄	—	122	1	no	—
88	C₂×C₄²	—	61	1	no	—
89	C₄⋊C₈	—	61	2	no	—
90	C₄⋊C₈	—	61	2	no	—
91	C₂²⋊C₈	—	122	1	no	—
92	C₂²⋊C₈	—	122	1	no	—
93	#97	—	122	1	no	—
94	#97	—	122	1	no	—
95	C₂×Q₈	—	2	1	no	—
96	C₂×Q₈	—	122	2	no	—
97	C₂×Q₈	—	122	1	no	—
98	C₂×Q₈	—	244	4	no	—
99	C₂²×C₄	—	61	1	no	—
100	C₂²×C₄	—	61	1	no	—
101	C₂²×C₄	—	61	1	no	—
102	C₂²×C₄	—	122	2	no	—
103	C₂×C₈	—	122	2	no	—
104	C₂×C₈	—	122	2	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	61	2	no	—
107	C₄⋊C₄	—	122	2	no	—
108	C₄⋊C₄	—	122	2	no	—
109	C₄⋊C₄	—	122	1	no	—
110	C₄⋊C₄	C₂×#6923	1	2	yes	—
111	C₄²	—	61	1	no	—
112	C₄²	—	61	1	no	—
113	C₄²	—	61	1	no	—
114	C₄²	C₂×#6923	1	1	yes	characteristic
115	C₂³	—	61	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	244	2	no	—
118	Q₈	—	244	4	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	61	1	no	—
121	C₂×C₄	—	61	1	no	—
122	C₂×C₄	—	61	1	no	—
123	C₂×C₄	—	122	2	no	—
124	C₂×C₄	—	122	2	no	—
125	C₂×C₄	—	122	2	no	—
126	C₂×C₄	—	122	2	no	—
127	C₂×C₄	—	244	4	no	—
128	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
129	C₂×C₄	C₂²⋊#6923	1	1	yes	characteristic
130	C₂×C₄	C₂²×#6923	1	1	yes	characteristic
131	C₈	—	244	2	no	—
132	C₈	—	244	2	no	—
133	C₂²	—	61	2	no	—
134	C₂²	—	122	1	no	—
135	C₂²	—	122	1	no	—
136	C₂²	C₂×C₂²⋊#6923	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	4	2	no	—
141	C₄	—	61	2	no	—
142	C₄	—	61	2	no	—
143	C₄	—	122	1	no	—
144	C₄	—	244	2	no	—
145	C₂	—	122	2	no	—
146	C₂	D₆₁⋊#101	1	1	yes	characteristic
147	C₂	#16840754	1	1	yes	characteristic
148	C₂	#16840755	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840785

C₂

yes

maximal, characteristic

#16840785

C₂

yes

maximal, characteristic

#16840621

C₂

yes

maximal, complemented

C₄⋊Q₈×D₆₁

C₂

yes

maximal, characteristic

#14519844

C₂²

yes

—

#14519844

C₂²

yes

—

—

—

—

—

C₂×Q₈×D₆₁

—

#14519777

C₄

yes

characteristic

#14519776

—

#14519776

C₂²

yes

—

#14519775

—

C₄⋊C₄×D₆₁

—

C₄⋊C₄×D₆₁

C₄

yes

—

C₄²×D₆₁

C₂²

yes

characteristic

C₄⋊Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

—

Q₈×D₆₁

—

C₂×C₄×D₆₁

—

C₂×C₄×D₆₁

C₂×C₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×C₄×D₆₁

D₄

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×#92420

—

C₂×#92420

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂×C₄

yes

—

C₄²×C₆₁

C₂×C₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

#11758196

—

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

D₄

yes

characteristic

C₄×Dic₆₁

C₂³

yes

characteristic

C₂²×D₆₁

C₂²⋊C₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

C₂×D₄

yes

characteristic

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

#92420

—

#92420

—

C₂²×C₆₁

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

#101

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

AGL(1,ℤ/8)

yes

—

Dic₆₁

#137

yes

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

—

C₂×C₄⋊Q₈

—

#432

—

#432

—

#332

—

C₆₁

#1556

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

#6923

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

122

—

C₂²⋊C₈

—

122

—

#97

—

122

—

#97

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

122

—

103

C₂×C₈

—

122

—

104

C₂×C₈

—

122

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

122

—

108

C₄⋊C₄

—

122

—

109

C₄⋊C₄

—

122

—

110

C₄⋊C₄

C₂×#6923

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₂×#6923

yes

characteristic

115

C₂³

—

116

Q₈

—

117

Q₈

—

244

—

118

Q₈

—

244

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

122

—

124

C₂×C₄

—

122

—

125

C₂×C₄

—

122

—

126

C₂×C₄

—

122

—

127

C₂×C₄

—

244

—

128

C₂×C₄

C₂²⋊#6923

yes

characteristic

129

C₂×C₄

C₂²⋊#6923

yes

characteristic

130

C₂×C₄

C₂²×#6923

yes

characteristic

131

C₈

—

244

—

132

C₈

—

244

—

133

C₂²

—

134

C₂²

—

122

—

135

C₂²

—

122

—

136

C₂²

C₂×C₂²⋊#6923

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

122

—

144

C₄

—

244

—

145

C₂

—

122

—

146

C₂

D₆₁⋊#101

yes

characteristic

147

C₂

#16840754

yes

characteristic

148

C₂

#16840755

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#6923
commutator	C₂×C₄×C₆₁	C₂²×C₄
Frattini	C₂×C₄	C₂²×#6923
Fitting	C₄⋊Q₈×C₆₁	C₄
socle	C₂²×C₆₁	C₂×C₂²⋊C₄
hypercenter	C₄⋊Q₈	#6923
hypocenter	C₆₁	#1556
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#6923

commutator

C₂×C₄×C₆₁

C₂²×C₄

Frattini

C₂×C₄

C₂²×#6923

Fitting

C₄⋊Q₈×C₆₁

C₄

socle

C₂²×C₆₁

C₂×C₂²⋊C₄

C₄⋊Q₈

C₆₁

G ▷ C₂×C₄×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
3. #16840621	C₂	semidirect
79. C₆₁	#1556	semidirect

H ≅ G/N

type

3. #16840621

C₂

semidirect

79. C₆₁

#1556

semidirect

Subgroups — #21461349

Conjugacy classes of subgroups

Complements