Group #21458491 — subgroups

Subgroups — #21458491

Order 7808.

392 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 139 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	Q₈×Dic₁₂₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#16840434	C₂	1	1	yes	maximal, characteristic
3	#16840109	C₂	1	3	yes	maximal, complemented
4	#16840053	C₂	1	3	yes	maximal, complemented
5	#16839963	C₂	1	3	yes	maximal, complemented
6	C₄×#11758190	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11758226	—	2	1	no	—
9	C₂×#11758190	C₂²	1	3	yes	—
10	#14519777	C₂²	1	3	yes	—
11	#14519776	—	2	3	no	—
12	#14519774	C₂²	1	3	yes	—
13	#14519752	—	2	3	no	—
14	Q₈×Dic₆₁	—	2	1	no	—
15	C₄×Dic₁₂₂	C₂²	1	3	yes	—
16	C₄×D₂₄₄	C₂²	1	3	yes	—
17	#14519709	C₂²	1	3	yes	—
18	#14519702	C₂²	1	1	yes	characteristic
19	#14519699	C₂²	1	6	yes	—
20	#14519699	C₂²	1	3	yes	—
21	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₆₁	C₂²	1	3	yes	—
23	C₄×C₈×C₆₁	C₂²	1	3	yes	—
24	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
25	#11758226	—	4	4	no	—
26	C₂×D₂₄₄	—	2	3	no	—
27	C₂×D₂₄₄	—	2	3	no	—
28	C₂×D₂₄₄	C₂³	1	3	yes	—
29	C₂×C₄×D₆₁	—	2	3	no	—
30	C₂×Dic₁₂₂	—	2	3	no	—
31	C₂×Dic₁₂₂	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₂³	1	3	yes	—
34	C₄⋊C₄×C₆₁	D₄	1	3	yes	—
35	C₄²×C₆₁	C₂³	1	3	yes	—
36	#11758198	—	2	3	no	—
37	C₄⋊Dic₆₁	—	2	3	no	—
38	C₄⋊Dic₆₁	C₂³	1	3	yes	—
39	C₄⋊Dic₆₁	C₂³	1	1	yes	characteristic
40	#11758196	—	2	3	no	—
41	C₄×Dic₆₁	—	2	3	no	—
42	#11758190	—	2	6	no	—
43	C₂²×D₆₁	—	2	3	no	—
44	Q₈×C₆₁	SD₁₆	1	4	yes	complemented
45	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
46	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
47	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
48	C₂×Dic₆₁	—	2	1	no	—
49	C₂×Dic₆₁	—	2	3	no	—
50	C₂×Dic₆₁	—	2	1	no	—
51	D₂₄₄	—	2	3	no	—
52	D₂₄₄	—	2	3	no	—
53	D₂₄₄	—	4	6	no	—
54	D₂₄₄	—	8	3	no	—
55	C₄×D₆₁	—	4	6	no	—
56	Dic₁₂₂	—	2	1	no	—
57	Dic₁₂₂	—	8	3	no	—
58	Dic₁₂₂	#41	1	2	yes	complemented
59	C₈×C₆₁	—	2	3	no	—
60	C₈×C₆₁	—	2	1	no	—
61	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
62	C₂×D₆₁	—	4	3	no	—
63	C₂×D₆₁	—	4	3	no	—
64	C₂×D₆₁	—	4	3	no	—
65	C₄×C₆₁	—	2	3	no	—
66	C₄×C₆₁	C₂×SD₁₆	1	6	yes	—
67	C₄×C₆₁	C₂×#41	1	1	yes	characteristic
68	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₆₁	—	2	2	no	—
70	Dic₆₁	—	4	1	no	—
71	Dic₆₁	—	4	3	no	—
72	#3362	—	61	1	no	maximal, Sylow
73	C₂×C₆₁	#548	1	1	yes	characteristic
74	C₂×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
75	C₂×C₆₁	#576	1	1	yes	characteristic
76	D₆₁	—	8	3	no	—
77	Q₈²	—	61	1	no	—
78	#550	—	61	1	no	—
79	#486	—	61	3	no	—
80	#476	—	61	3	no	—
81	#460	—	61	3	no	—
82	C₈×Q₈	D₆₁	1	1	yes	characteristic, complemented
83	C₄×SD₁₆	—	61	3	no	—
84	C₆₁	#3362	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	122	1	no	—
86	C₂×SD₁₆	—	61	3	no	—
87	C₄⋊Q₈	—	61	3	no	—
88	C₄⋊Q₈	—	122	3	no	—
89	#127	—	61	3	no	—
90	#121	—	122	3	no	—
91	C₄×Q₈	—	61	3	no	—
92	C₄×Q₈	—	122	1	no	—
93	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
94	C₄×D₄	—	61	3	no	—
95	#106	—	61	3	no	—
96	C₄⋊C₈	C₂×D₆₁	1	3	yes	—
97	#103	—	61	1	no	—
98	#102	—	61	3	no	—
99	#102	—	61	6	no	—
100	C₄×C₈	C₂×D₆₁	1	3	yes	—
101	#41	—	244	4	no	—
102	C₂×Q₈	—	61	1	no	—
103	C₂×Q₈	—	122	3	no	—
104	C₂×Q₈	D₂₄₄	1	1	yes	characteristic
105	C₂×D₄	—	61	3	no	—
106	C₂×D₄	—	122	3	no	—
107	C₂×D₄	—	122	3	no	—
108	C₂²×C₄	—	122	3	no	—
109	SD₁₆	—	122	6	no	—
110	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
111	C₂×C₈	C₂²×D₆₁	1	3	yes	—
112	C₄⋊C₄	—	61	3	no	—
113	C₄⋊C₄	—	61	1	no	—
114	C₄⋊C₄	—	122	3	no	—
115	C₄⋊C₄	—	122	3	no	—
116	C₄⋊C₄	D₂₄₄	1	3	yes	—
117	C₂²⋊C₄	—	122	3	no	—
118	C₄²	—	122	3	no	—
119	C₄²	C₂²×D₆₁	1	3	yes	—
120	C₂³	—	122	3	no	—
121	Q₈	—	61	2	no	—
122	Q₈	—	122	1	no	—
123	Q₈	—	488	3	no	—
124	Q₈	#11758190	1	4	yes	complemented
125	D₄	—	122	3	no	—
126	D₄	—	122	3	no	—
127	D₄	—	488	3	no	—
128	D₄	—	244	6	no	—
129	C₂×C₄	—	122	1	no	—
130	C₂×C₄	—	122	3	no	—
131	C₂×C₄	—	122	1	no	—
132	C₂×C₄	—	244	6	no	—
133	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
134	C₂×C₄	C₂×D₂₄₄	1	3	yes	—
135	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
136	C₈	—	2	3	no	—
137	C₈	—	2	1	no	—
138	C₂²	—	244	3	no	—
139	C₂²	—	244	3	no	—
140	C₂²	—	244	3	no	—
141	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
142	C₄	—	2	3	no	—
143	C₄	—	122	2	no	—
144	C₄	—	244	1	no	—
145	C₄	—	244	3	no	—
146	C₄	C₂×#11758190	1	6	yes	—
147	C₄	#41×D₆₁	1	1	yes	characteristic
148	C₄	#14519825	1	1	yes	characteristic
149	C₂	—	488	3	no	—
150	C₂	#16840424	1	1	yes	characteristic
151	C₂	C₂²×#11758190	1	1	yes	characteristic
152	C₂	#16840567	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₂₂

C₂

yes

maximal, characteristic, complemented

#16840434

C₂

yes

maximal, characteristic

#16840109

C₂

yes

maximal, complemented

#16840053

C₂

yes

maximal, complemented

#16839963

C₂

yes

maximal, complemented

C₄×#11758190

C₂

yes

maximal, complemented

C₈×Q₈×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×#11758226

—

C₂×#11758190

C₂²

yes

—

#14519777

C₂²

yes

—

#14519776

—

#14519774

C₂²

yes

—

#14519752

—

Q₈×Dic₆₁

—

C₄×Dic₁₂₂

C₂²

yes

—

C₄×D₂₄₄

C₂²

yes

—

#14519709

C₂²

yes

—

#14519702

C₂²

yes

characteristic

#14519699

C₂²

yes

—

#14519699

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₄⋊C₈×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

#11758226

—

C₂×D₂₄₄

—

C₂×D₂₄₄

—

C₂×D₂₄₄

C₂³

yes

—

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₄⋊Dic₆₁

C₂³

yes

characteristic

#11758196

—

C₄×Dic₆₁

—

#11758190

—

C₂²×D₆₁

—

Q₈×C₆₁

SD₁₆

yes

complemented

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

D₂₄₄

—

C₄×D₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

#41

yes

complemented

C₈×C₆₁

—

C₈×C₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

C₂×#41

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂²×SD₁₆

yes

characteristic

C₂×C₆₁

#576

yes

characteristic

D₆₁

—

Q₈²

—

—

—

—

—

C₈×Q₈

D₆₁

yes

characteristic, complemented

C₄×SD₁₆

—

C₆₁

#3362

yes

characteristic, complemented, Sylow

C₂×#41

—

122

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

#127

—

#121

—

122

—

C₄×Q₈

—

C₄×Q₈

—

122

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

#106

—

C₄⋊C₈

C₂×D₆₁

yes

—

#103

—

#102

—

#102

—

100

C₄×C₈

C₂×D₆₁

yes

—

101

#41

—

244

—

102

C₂×Q₈

—

103

C₂×Q₈

—

122

—

104

C₂×Q₈

D₂₄₄

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

122

—

107

C₂×D₄

—

122

—

108

C₂²×C₄

—

122

—

109

SD₁₆

—

122

—

110

C₂×C₈

C₂²×D₆₁

yes

characteristic

111

C₂×C₈

C₂²×D₆₁

yes

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

122

—

115

C₄⋊C₄

—

122

—

116

C₄⋊C₄

D₂₄₄

yes

—

117

C₂²⋊C₄

—

122

—

118

C₄²

—

122

—

119

C₄²

C₂²×D₆₁

yes

—

120

C₂³

—

122

—

121

Q₈

—

122

Q₈

—

122

—

123

Q₈

—

488

—

124

Q₈

#11758190

yes

complemented

125

D₄

—

122

—

126

D₄

—

122

—

127

D₄

—

488

—

128

D₄

—

244

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

122

—

131

C₂×C₄

—

122

—

132

C₂×C₄

—

244

—

133

C₂×C₄

C₂×D₂₄₄

yes

—

134

C₂×C₄

C₂×D₂₄₄

yes

—

135

C₂×C₄

C₂³×D₆₁

yes

characteristic

136

C₈

—

137

C₈

—

138

C₂²

—

244

—

139

C₂²

—

244

—

140

C₂²

—

244

—

141

C₂²

C₂²×D₂₄₄

yes

characteristic

142

C₄

—

143

C₄

—

122

—

144

C₄

—

244

—

145

C₄

—

244

—

146

C₄

C₂×#11758190

yes

—

147

C₄

#41×D₆₁

yes

characteristic

148

C₄

#14519825

yes

characteristic

149

C₂

—

488

—

150

C₂

#16840424

yes

characteristic

151

C₂

C₂²×#11758190

yes

characteristic

152

C₂

#16840567

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₈×Q₈×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	C₈×Q₈	D₆₁
hypocenter	C₆₁	#3362
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₈×Q₈×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

C₈×Q₈

D₆₁

hypocenter

C₆₁

#3362

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₂₂	C₂	semidirect
3. #16840109	C₂	semidirect
4. #16840053	C₂	semidirect
5. #16839963	C₂	semidirect
6. C₄×#11758190	C₂	semidirect
7. C₈×Q₈×C₆₁	C₂	semidirect
44. Q₈×C₆₁	SD₁₆	semidirect
58. Dic₁₂₂	#41	semidirect
82. C₈×Q₈	D₆₁	semidirect
84. C₆₁	#3362	semidirect
124. Q₈	#11758190	semidirect

H ≅ G/N

type

1. Q₈×Dic₁₂₂

C₂

semidirect

3. #16840109

C₂

semidirect

4. #16840053

C₂

semidirect

5. #16839963

C₂

semidirect

6. C₄×#11758190

C₂

semidirect

7. C₈×Q₈×C₆₁

C₂

semidirect

44. Q₈×C₆₁

SD₁₆

semidirect

58. Dic₁₂₂

#41

semidirect

82. C₈×Q₈

D₆₁

semidirect

84. C₆₁

#3362

semidirect

124. Q₈

#11758190

semidirect

Subgroups — #21458491

Conjugacy classes of subgroups

Complements