Group #21446967 — subgroups

Subgroups — #21446967

Order 7808.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#14519700	C₂	1	2	yes	maximal
2	#16839668	C₂	1	1	yes	maximal, characteristic
3	#16839524	C₂	1	2	yes	maximal
4	#16839517	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₂²×#92418	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	1	yes	characteristic
9	#14519700	—	2	4	no	—
10	#14519684	—	2	2	no	—
11	#125×C₆₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×C₆₁	—	2	2	no	—
16	C₂²×C₄×C₆₁	D₄	1	2	yes	—
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	1	no	—
19	C₂²×Dic₆₁	—	2	1	no	—
20	C₂×Dic₁₂₂	—	2	2	no	—
21	C₂×Dic₁₂₂	—	2	2	no	—
22	C₂×Dic₁₂₂	—	2	1	no	—
23	C₂×Dic₁₂₂	Q₈	1	2	yes	—
24	C₄⋊C₄×C₆₁	—	2	8	no	—
25	C₄⋊C₄×C₆₁	—	2	2	no	—
26	C₄⋊C₄×C₆₁	D₄	1	4	yes	—
27	C₄²×C₆₁	—	2	2	no	—
28	C₄⋊Dic₆₁	—	2	2	no	—
29	C₂×#92418	—	2	2	no	—
30	C₂×#92418	—	2	2	no	—
31	C₂×#92418	—	2	2	no	—
32	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₆₁	—	2	2	no	—
34	C₂×C₄×C₆₁	—	2	4	no	—
35	C₂×C₄×C₆₁	—	2	2	no	—
36	C₂×C₄×C₆₁	—	2	2	no	—
37	C₂×C₄×C₆₁	—	4	2	no	—
38	C₂×C₄×C₆₁	—	4	2	no	—
39	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₆₁	#41	1	2	yes	—
42	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
43	C₂×Dic₆₁	—	2	2	no	—
44	C₂×Dic₆₁	—	4	1	no	—
45	C₂×Dic₆₁	—	4	1	no	—
46	C₂×Dic₆₁	—	4	1	no	—
47	C₂×Dic₆₁	—	4	2	no	—
48	Dic₁₂₂	—	2	4	no	—
49	Dic₁₂₂	—	4	2	no	—
50	#92418	—	4	2	no	—
51	C₂²×C₆₁	#120	1	1	yes	characteristic
52	C₂²×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₆₁	#124	1	1	yes	characteristic
54	C₂²×C₆₁	#135	1	2	yes	—
55	C₂²×C₆₁	#137	1	2	yes	—
56	C₄×C₆₁	—	4	2	no	—
57	C₄×C₆₁	—	4	4	no	—
58	C₄×C₆₁	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₆₁	#124	1	2	yes	—
60	Dic₆₁	—	4	2	no	—
61	Dic₆₁	—	8	1	no	—
62	#2026	—	61	1	no	maximal, Sylow
63	C₂×C₆₁	#393	1	1	yes	characteristic
64	C₂×C₆₁	#452	1	2	yes	—
65	C₂×C₆₁	#479	1	2	yes	—
66	C₂×C₆₁	#488	1	1	yes	characteristic
67	C₂×C₆₁	#491	1	1	yes	characteristic
68	C₂×#125	D₆₁	1	1	yes	characteristic
69	C₂×#103	—	61	2	no	—
70	#391	—	61	1	no	—
71	#340	—	61	2	no	—
72	#336	—	61	1	no	—
73	C₆₁	#2026	1	1	yes	characteristic, complemented, Sylow
74	C₂²×Q₈	—	61	1	no	—
75	C₂²×C₈	—	61	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	61	1	no	—
79	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
80	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
81	#103	—	122	4	no	—
82	#95	—	122	2	no	—
83	C₂×Q₈	—	61	2	no	—
84	C₂×Q₈	—	122	1	no	—
85	C₂×Q₈	—	122	2	no	—
86	C₂×Q₈	—	122	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	122	1	no	—
89	C₂²×C₄	—	122	1	no	—
90	C₂²×C₄	C₂²⋊D₆₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
92	C₂×C₈	—	122	2	no	—
93	C₂×C₈	—	122	2	no	—
94	C₂×C₈	—	122	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	122	2	no	—
98	C₄⋊C₄	C₂²⋊D₆₁	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
101	Q₈	—	122	4	no	—
102	Q₈	—	244	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	122	2	no	—
110	C₂×C₄	—	244	1	no	—
111	C₂×C₄	—	244	2	no	—
112	C₂×C₄	—	244	1	no	—
113	C₂×C₄	—	244	1	no	—
114	C₂×C₄	Q₈×D₆₁	1	1	yes	characteristic
115	C₂×C₄	#11758226	1	2	yes	—
116	C₂×C₄	#11758226	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
118	C₈	—	244	2	no	—
119	C₂²	#14519746	1	1	yes	characteristic
120	C₂²	#14519753	1	1	yes	characteristic
121	C₂²	#14519766	1	1	yes	characteristic
122	C₂²	#14519797	1	2	yes	—
123	C₂²	#14519807	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	488	1	no	—
127	C₄	—	244	2	no	—
128	C₄	#14519753	1	2	yes	—
129	C₄	#14519766	1	2	yes	—
130	C₂	#16839673	1	1	yes	characteristic
131	C₂	#16839924	1	2	yes	—
132	C₂	#16840066	1	2	yes	—
133	C₂	#16840119	1	1	yes	characteristic
134	C₂	#16840132	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519700

C₂

yes

maximal

#16839668

C₂

yes

maximal, characteristic

#16839524

C₂

yes

maximal

#16839517

C₂

yes

maximal, characteristic

C₂×#125×C₆₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂²×#92418

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

characteristic

#14519700

—

#14519684

—

#125×C₆₁

—

C₂×C₄⋊C₄×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂²×Dic₆₁

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

Q₈

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄²×C₆₁

—

C₄⋊Dic₆₁

—

C₂×#92418

—

C₂×#92418

—

C₂×#92418

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₁

#41

yes

—

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

#92418

—

C₂²×C₆₁

#120

yes

characteristic

C₂²×C₆₁

C₂²⋊Q₈

yes

C₂²×C₆₁

yes

C₂²×C₆₁

yes

—

C₂²×C₆₁

#137

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂²⋊Q₈

yes

—

C₄×C₆₁

#124

yes

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#479

yes

—

C₂×C₆₁

yes

C₂×C₆₁

yes

D₆₁

yes

—

—

—

—

C₆₁

#2026

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#103

—

122

—

#95

—

122

—

C₂×Q₈

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂×Q₈

—

122

—

C₂²×C₄

—

C₂²×C₄

—

122

—

C₂²×C₄

—

122

—

C₂²×C₄

C₂²⋊D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

C₂²⋊D₆₁

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₆₁

yes

characteristic

101

Q₈

—

122

—

102

Q₈

—

244

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

122

—

110

C₂×C₄

—

244

—

111

C₂×C₄

—

244

—

112

C₂×C₄

—

244

—

113

C₂×C₄

—

244

—

114

C₂×C₄

Q₈×D₆₁

yes

characteristic

115

C₂×C₄

#11758226

yes

—

116

C₂×C₄

#11758226

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

118

C₈

—

244

—

119

C₂²

#14519746

yes

characteristic

120

C₂²

#14519753

yes

characteristic

121

C₂²

#14519766

yes

characteristic

122

C₂²

#14519797

yes

—

123

C₂²

#14519807

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

488

—

127

C₄

—

244

—

128

C₄

#14519753

yes

—

129

C₄

#14519766

yes

—

130

C₂

#16839673

yes

characteristic

131

C₂

#16839924

yes

—

132

C₂

#16840066

yes

—

133

C₂

#16840119

yes

characteristic

134

C₂

#16840132

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₆₁
commutator	C₂²×C₄×C₆₁	C₂³
Frattini	C₂²×C₄	C₂²×D₆₁
Fitting	C₂×#125×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×#125	D₆₁
hypocenter	C₆₁	#2026
LCS	G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₆₁

commutator

C₂²×C₄×C₆₁

C₂³

Frattini

C₂²×C₄

C₂²×D₆₁

Fitting

C₂×#125×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

D₆₁

C₆₁

G ▷ C₂²×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
73. C₆₁	#2026	semidirect

H ≅ G/N

type

73. C₆₁

#2026

semidirect

Subgroups — #21446967

Conjugacy classes of subgroups

Complements