Group #21454179 — subgroups

Subgroups — #21454179

Order 7808.

472 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14519769	C₂	1	1	yes	maximal, characteristic
2	#16839657	C₂	1	8	yes	maximal
3	#16839631	C₂	1	2	yes	maximal
4	#16839627	C₂	1	2	yes	maximal
5	C₄²×Dic₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₆₁	C₂	1	1	yes	maximal, characteristic
7	#14519769	—	2	4	no	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
9	C₂×#11758196	C₂²	1	4	yes	—
10	C₂×#11758196	C₂²	1	8	yes	—
11	C₂×C₄×Dic₆₁	C₂²	1	4	yes	—
12	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₆₁	C₂²	1	1	yes	characteristic
14	#14519684	C₂²	1	8	yes	—
15	#125×C₆₁	—	2	4	no	—
16	C₂×C₄⋊C₄×C₆₁	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
18	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
21	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
22	C₂²×Dic₆₁	C₂³	1	4	yes	—
23	C₂²×Dic₆₁	C₂³	1	2	yes	—
24	C₂²×Dic₆₁	C₂³	1	2	yes	—
25	C₄⋊C₄×C₆₁	—	2	8	no	—
26	C₄⋊C₄×C₆₁	—	2	4	no	—
27	C₄²×C₆₁	D₄	1	4	yes	—
28	C₄⋊Dic₆₁	—	2	4	no	—
29	#11758196	—	2	8	no	—
30	#11758196	—	2	16	no	—
31	C₄×Dic₆₁	—	2	8	no	—
32	C₄×Dic₆₁	Q₈	1	4	yes	—
33	C₄×Dic₆₁	Q₈	1	4	yes	—
34	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₆₁	—	2	4	no	—
36	C₂×C₄×C₆₁	—	2	4	no	—
37	C₂×C₄×C₆₁	—	2	8	no	—
38	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
41	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
42	C₂×Dic₆₁	—	2	4	no	—
43	C₂×Dic₆₁	—	2	4	no	—
44	C₂×Dic₆₁	—	2	8	no	—
45	C₂×Dic₆₁	C₂×Q₈	1	4	yes	—
46	C₂×Dic₆₁	C₂×Q₈	1	4	yes	—
47	C₂×Dic₆₁	#41	1	8	yes	—
48	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₆₁	C₂×#41	1	4	yes	—
52	C₄×C₆₁	—	2	4	no	—
53	C₄×C₆₁	—	4	4	no	—
54	C₄×C₆₁	C₄⋊Q₈	1	4	yes	—
55	Dic₆₁	—	2	4	no	—
56	Dic₆₁	—	4	4	no	—
57	Dic₆₁	#125	1	8	yes	complemented
58	#2859	—	61	1	no	maximal, Sylow
59	C₂×C₆₁	C₂×#125	1	2	yes	—
60	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₆₁	#532	1	2	yes	—
62	C₂×C₆₁	#533	1	2	yes	—
63	C₂×#125	—	61	1	no	—
64	C₂×#125	D₆₁	1	1	yes	characteristic
65	#389	—	61	8	no	—
66	#384	—	61	2	no	—
67	#382	—	61	2	no	—
68	C₄³	—	61	1	no	—
69	C₆₁	#2859	1	1	yes	characteristic, complemented, Sylow
70	#125	—	2	4	no	—
71	#125	—	122	4	no	—
72	C₂×C₄⋊C₄	—	61	2	no	—
73	C₂×C₄⋊C₄	—	61	4	no	—
74	C₂×C₄⋊C₄	—	61	8	no	—
75	C₂×C₄⋊C₄	C₂×D₆₁	1	2	yes	—
76	C₂×C₄⋊C₄	C₂×D₆₁	1	4	yes	—
77	C₂×C₄²	—	61	1	no	—
78	C₂×C₄²	—	61	1	no	—
79	C₂×C₄²	—	61	4	no	—
80	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
81	#95	—	61	8	no	—
82	C₂²×C₄	—	61	2	no	—
83	C₂²×C₄	—	61	2	no	—
84	C₂²×C₄	—	61	4	no	—
85	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
86	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
87	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
88	C₄⋊C₄	—	2	8	no	—
89	C₄⋊C₄	—	2	4	no	—
90	C₄⋊C₄	—	122	16	no	—
91	C₄⋊C₄	—	122	8	no	—
92	C₄⋊C₄	—	122	4	no	—
93	C₄²	—	61	4	no	—
94	C₄²	—	61	4	no	—
95	C₄²	—	122	8	no	—
96	C₄²	C₂²⋊D₆₁	1	4	yes	—
97	C₂³	C₂³×D₆₁	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	61	4	no	—
102	C₂×C₄	—	61	8	no	—
103	C₂×C₄	—	61	4	no	—
104	C₂×C₄	—	122	4	no	—
105	C₂×C₄	—	122	4	no	—
106	C₂×C₄	—	122	8	no	—
107	C₂×C₄	Q₈×D₆₁	1	2	yes	—
108	C₂×C₄	Q₈×D₆₁	1	2	yes	—
109	C₂×C₄	C₂×C₂²⋊D₆₁	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
111	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
112	C₂²	C₂×Q₈×D₆₁	1	1	yes	characteristic
113	C₂²	C₂×Q₈×D₆₁	1	1	yes	characteristic
114	C₂²	#41×D₆₁	1	4	yes	—
115	C₄	—	2	4	no	—
116	C₄	—	4	4	no	—
117	C₄	—	61	8	no	—
118	C₄	—	122	4	no	—
119	C₄	—	244	4	no	—
120	C₄	#14519775	1	4	yes	—
121	C₂	#125×D₆₁	1	2	yes	—
122	C₂	C₂×#14519775	1	1	yes	characteristic
123	C₂	#16840308	1	2	yes	—
124	C₂	#16840318	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519769

C₂

yes

maximal, characteristic

#16839657

C₂

yes

maximal

#16839631

C₂

yes

maximal

#16839627

C₂

yes

maximal

C₄²×Dic₆₁

C₂

yes

maximal, characteristic

C₂×#125×C₆₁

C₂

yes

maximal, characteristic

#14519769

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

C₂×C₄×Dic₆₁

C₂²

yes

characteristic

#14519684

C₂²

yes

—

#125×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

#11758196

—

#11758196

—

C₄×Dic₆₁

—

C₄×Dic₆₁

Q₈

yes

—

C₄×Dic₆₁

Q₈

yes

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

#41

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

C₂×#41

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

Dic₆₁

—

Dic₆₁

—

Dic₆₁

yes

—

C₂×C₆₁

yes

—

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

#532

yes

—

C₂×C₆₁

#533

yes

—

C₂×#125

—

C₂×#125

D₆₁

yes

—

—

—

C₄³

—

C₆₁

#2859

yes

characteristic, complemented, Sylow

#125

—

#125

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

C₄²

—

C₄²

—

122

—

C₄²

C₂²⋊D₆₁

yes

—

C₂³

C₂³×D₆₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

122

—

105

C₂×C₄

—

122

—

106

C₂×C₄

—

122

—

107

C₂×C₄

Q₈×D₆₁

yes

—

108

C₂×C₄

Q₈×D₆₁

yes

—

109

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

110

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

111

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

112

C₂²

C₂×Q₈×D₆₁

yes

characteristic

113

C₂²

C₂×Q₈×D₆₁

yes

characteristic

114

C₂²

#41×D₆₁

yes

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

122

—

119

C₄

—

244

—

120

C₄

#14519775

yes

—

121

C₂

#125×D₆₁

yes

—

122

C₂

C₂×#14519775

yes

characteristic

123

C₂

#16840308

yes

—

124

C₂

#16840318

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₂×#125×C₆₁	C₂
hypercenter	C₂×#125	D₆₁
hypocenter	C₆₁	#2859
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₂×#125×C₆₁

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
57. Dic₆₁	#125	semidirect
69. C₆₁	#2859	semidirect

H ≅ G/N

type

57. Dic₆₁

#125

semidirect

69. C₆₁

#2859

semidirect

Subgroups — #21454179

Conjugacy classes of subgroups

Complements