Group #21456325 — subgroups

Subgroups — C₂×#16840165 #21456325

Order 7808.

424 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂×#14519777	C₂	1	1	yes	maximal, characteristic
2	#16840165	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519710	C₂	1	4	yes	maximal
4	C₂×C₄×#92418	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	—	2	1	no	—
7	C₂²×#92418	C₂²	1	2	yes	—
8	#14519777	—	2	2	no	—
9	#14519777	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₆₁	—	2	1	no	—
11	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
12	#14519710	C₂²	1	16	yes	—
13	C₄×#92418	C₂²	1	4	yes	—
14	C₂²×Q₈×C₆₁	—	2	1	no	—
15	C₄⋊Q₈×C₆₁	—	2	2	no	—
16	C₄⋊Q₈×C₆₁	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
18	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
19	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₆₁	—	2	4	no	—
21	C₂×Q₈×C₆₁	—	4	4	no	—
22	C₂²×C₄×C₆₁	—	2	2	no	—
23	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₆₁	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
26	C₂²×Dic₆₁	—	2	2	no	—
27	C₂×Dic₁₂₂	—	2	4	no	—
28	C₂×Dic₁₂₂	—	4	4	no	—
29	C₄⋊C₄×C₆₁	—	2	4	no	—
30	C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₄⋊C₄×C₆₁	C₂³	1	4	yes	—
32	C₄²×C₆₁	D₄	1	2	yes	—
33	C₄²×C₆₁	C₂³	1	2	yes	—
34	C₄⋊Dic₆₁	—	2	4	no	—
35	C₄⋊Dic₆₁	—	2	2	no	—
36	C₄⋊Dic₆₁	C₂³	1	4	yes	—
37	C₂×#92418	Q₈	1	8	yes	complemented
38	C₂×#92418	C₂³	1	4	yes	—
39	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₆₁	—	4	8	no	—
41	C₂×C₄×C₆₁	—	2	4	no	—
42	C₂×C₄×C₆₁	—	4	4	no	—
43	C₂×C₄×C₆₁	D₈	1	4	yes	—
44	C₂×C₄×C₆₁	Q₁₆	1	4	yes	—
45	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₆₁	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₆₁	—	2	4	no	—
51	C₂×Dic₆₁	—	4	4	no	—
52	Dic₁₂₂	—	4	8	no	—
53	#92418	C₂×Q₈	1	8	yes	complemented
54	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₆₁	C₂×D₈	1	2	yes	—
56	C₂²×C₆₁	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₆₁	—	4	4	no	—
59	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
60	C₄×C₆₁	C₂×D₈	1	4	yes	—
61	C₄×C₆₁	C₂×Q₁₆	1	4	yes	—
62	C₄×C₆₁	C₂²×Q₈	1	2	yes	—
63	Dic₆₁	—	4	4	no	—
64	C₂×#500	—	61	1	no	maximal, Sylow
65	C₂×C₆₁	#500	1	4	yes	complemented
66	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₆₁	C₂²×D₈	1	1	yes	characteristic
68	C₂×C₆₁	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×C₄⋊Q₈	—	61	1	no	—
70	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
71	#500	—	61	8	no	—
72	C₂×#107	—	61	4	no	—
73	C₂×C₄×C₈	—	61	1	no	—
74	C₆₁	C₂×#500	1	1	yes	characteristic, complemented, Sylow
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	122	1	no	—
77	C₂²×C₈	—	61	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	61	4	no	—
80	C₄⋊Q₈	—	122	2	no	—
81	C₄⋊Q₈	C₂×D₆₁	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	61	2	no	—
84	C₂×C₄⋊C₄	—	122	1	no	—
85	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
86	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
87	#107	—	61	16	no	—
88	C₄×C₈	—	61	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	122	4	no	—
92	C₂×Q₈	—	244	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	122	2	no	—
95	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₆₁	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
98	C₂×C₈	—	61	8	no	—
99	C₂×C₈	—	61	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	61	4	no	—
103	C₄⋊C₄	—	122	2	no	—
104	C₄⋊C₄	—	122	4	no	—
105	C₄⋊C₄	C₂²×D₆₁	1	4	yes	—
106	C₄²	C₂²⋊D₆₁	1	2	yes	—
107	C₄²	C₂²×D₆₁	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	244	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	122	4	no	—
114	C₂×C₄	—	244	4	no	—
115	C₂×C₄	#11758199	1	4	yes	—
116	C₂×C₄	#11758202	1	4	yes	—
117	C₂×C₄	Q₈×D₆₁	1	4	yes	—
118	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
119	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
120	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
121	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
122	C₈	—	61	8	no	—
123	C₂²	#14519775	1	2	yes	—
124	C₂²	C₂×#11758199	1	2	yes	—
125	C₂²	C₂×#11758202	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	244	4	no	—
129	C₄	#14519775	1	2	yes	—
130	C₄	C₂×#11758199	1	4	yes	—
131	C₄	C₂×#11758202	1	4	yes	—
132	C₄	C₂×Q₈×D₆₁	1	2	yes	—
133	C₂	#16840165	1	4	yes	direct product factor
134	C₂	C₂×#14519775	1	1	yes	characteristic
135	C₂	C₂²×#11758199	1	1	yes	characteristic
136	C₂	C₂²×#11758202	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519777

C₂

yes

maximal, characteristic

#16840165

C₂

yes

maximal, direct product factor

C₂×#14519710

C₂

yes

maximal

C₂×C₄×#92418

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

—

C₂²×#92418

C₂²

yes

—

#14519777

—

#14519777

C₂²

yes

—

C₂×C₄⋊Dic₆₁

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

#14519710

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₂²×Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

D₄

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₂×#92418

Q₈

yes

complemented

C₂×#92418

C₂³

yes

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

D₈

yes

—

C₂×C₄×C₆₁

Q₁₆

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

#92418

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

C₂×D₈

yes

—

C₂²×C₆₁

C₂×Q₁₆

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂×D₈

yes

—

C₄×C₆₁

C₂×Q₁₆

yes

—

C₄×C₆₁

C₂²×Q₈

yes

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂²×D₈

yes

characteristic

C₂×C₆₁

C₂²×Q₁₆

yes

characteristic

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₆₁

yes

characteristic

#500

—

C₂×#107

—

C₂×C₄×C₈

—

C₆₁

C₂×#500

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

122

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

122

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#107

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

C₂²×C₄

C₂²⋊D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

122

—

104

C₄⋊C₄

—

122

—

105

C₄⋊C₄

C₂²×D₆₁

yes

—

106

C₄²

C₂²⋊D₆₁

yes

—

107

C₄²

C₂²×D₆₁

yes

—

108

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

109

Q₈

—

110

Q₈

—

244

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

122

—

114

C₂×C₄

—

244

—

115

C₂×C₄

#11758199

yes

—

116

C₂×C₄

#11758202

yes

—

117

C₂×C₄

Q₈×D₆₁

yes

—

118

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

119

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

120

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

121

C₂×C₄

C₂³×D₆₁

yes

characteristic

122

C₈

—

123

C₂²

#14519775

yes

—

124

C₂²

C₂×#11758199

yes

—

125

C₂²

C₂×#11758202

yes

—

126

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

127

C₄

—

128

C₄

—

244

—

129

C₄

#14519775

yes

—

130

C₄

C₂×#11758199

yes

—

131

C₄

C₂×#11758202

yes

—

132

C₄

C₂×Q₈×D₆₁

yes

—

133

C₂

#16840165

yes

direct product factor

134

C₂

C₂×#14519775

yes

characteristic

135

C₂

C₂²×#11758199

yes

characteristic

136

C₂

C₂²×#11758202

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	C₂×#500
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₆₁

hypocenter

C₆₁

C₂×#500

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #16840165	C₂	direct (N × H)
37. C₂×#92418	Q₈	semidirect
53. #92418	C₂×Q₈	semidirect
65. C₂×C₆₁	#500	semidirect
74. C₆₁	C₂×#500	semidirect
133. C₂	#16840165	direct (N × H)

H ≅ G/N

type

2. #16840165

C₂

direct (N × H)

37. C₂×#92418

Q₈

semidirect

53. #92418

C₂×Q₈

semidirect

65. C₂×C₆₁

#500

semidirect

74. C₆₁

C₂×#500

semidirect

133. C₂

#16840165

direct (N × H)

Subgroups — C₂×#16840165 #21456325

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840165 #21456325

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840165 #21456325