Group #22626500 — subgroups

Subgroups — C₄×#14556679 #22626500

Order 9344.

572 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 288 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×#14556679	C₂	1	2	yes	maximal
2	#17989552	C₂	1	6	yes	maximal, complemented
3	C₄×#11924038	C₂	1	3	yes	maximal, complemented
4	C₄×#11924036	C₂	1	3	yes	maximal, complemented
5	C₄×#11924029	C₂	1	1	yes	maximal, characteristic
6	#14556679	C₄	1	16	yes	direct product factor
7	C₂×#11924038	C₂²	1	6	yes	—
8	C₂×#11924036	C₂²	1	6	yes	—
9	#14556646	C₂²	1	6	yes	—
10	C₄×#10596833	C₂²	1	1	yes	characteristic
11	C₄×#10596833	C₂²	1	3	yes	—
12	C₄×#10596833	C₂²	1	6	yes	—
13	C₂×#11924029	C₂²	1	1	yes	characteristic
14	Q₈×Dic₇₃	C₄	1	1	yes	characteristic
15	C₄×D₂₉₂	C₄	1	3	yes	—
16	#14556553	C₄	1	3	yes	—
17	#14556549	C₂²	1	3	yes	—
18	C₄²×D₇₃	C₂²	1	3	yes	—
19	C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
20	#11924038	C₂×C₄	1	24	yes	complemented
21	#11924036	C₂×C₄	1	24	yes	complemented
22	C₂×#10596833	C₂³	1	2	yes	—
23	C₂×#10596833	C₂³	1	6	yes	—
24	#11924029	C₂×C₄	1	8	yes	—
25	C₂×D₂₉₂	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₇₃	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₇₃	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₇₃	C₂³	1	3	yes	—
29	C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
30	#11924005	C₂×C₄	1	6	yes	—
31	C₄×Dic₇₃	C₂×C₄	1	3	yes	—
32	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
33	C₄×Dic₇₃	C₂³	1	3	yes	—
34	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₇₃	C₂×C₄	1	3	yes	—
36	C₄²×C₇₃	C₂×C₄	1	3	yes	—
37	C₂²×D₇₃	C₂²×C₄	1	3	yes	—
38	Q₈×C₇₃	C₄²	1	4	yes	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
40	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
41	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
43	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
45	D₂₉₂	C₄²	1	12	yes	—
46	C₄×D₇₃	—	2	6	no	—
47	C₄×D₇₃	—	2	6	no	—
48	C₄×D₇₃	C₂²×C₄	1	12	yes	—
49	#10596833	C₂²×C₄	1	4	yes	—
50	#10596833	C₂²×C₄	1	12	yes	—
51	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
52	C₂×D₇₃	—	2	3	no	—
53	C₂×D₇₃	—	2	3	no	—
54	C₂×D₇₃	C₂×C₄²	1	6	yes	—
55	C₄×C₇₃	—	2	3	no	—
56	C₄×C₇₃	C₂×C₄²	1	6	yes	—
57	C₄×C₇₃	#131	1	2	yes	complemented
58	Dic₇₃	—	2	3	no	—
59	Dic₇₃	#131	1	2	yes	complemented
60	Dic₇₃	C₂³×C₄	1	2	yes	—
61	C₂×C₇₃	C₂²×C₄²	1	1	yes	characteristic
62	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
63	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
64	D₇₃	—	2	6	no	—
65	C₄×#131	—	73	1	no	maximal, Sylow
66	C₇₃	C₄×#131	1	1	yes	characteristic, complemented, Sylow
67	C₂×#131	—	73	2	no	—
68	C₄×#41	—	73	1	no	—
69	#405	—	73	6	no	—
70	C₄×M₁₆	—	73	3	no	—
71	C₂×C₄×C₈	—	73	3	no	—
72	C₂×#41	—	73	1	no	—
73	#131	—	73	16	no	—
74	C₂×M₁₆	—	73	6	no	—
75	C₂²×C₈	—	73	6	no	—
76	C₄×Q₈	—	73	1	no	—
77	C₄×Q₈	#64408	1	1	yes	characteristic
78	C₄×D₄	—	73	3	no	—
79	C₄×D₄	—	73	3	no	—
80	#117	—	73	3	no	—
81	C₂×C₄²	—	73	3	no	—
82	#97	—	73	6	no	—
83	C₄×C₈	—	73	1	no	—
84	C₄×C₈	—	73	3	no	—
85	C₄×C₈	—	73	6	no	—
86	#41	—	73	8	no	—
87	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
88	C₂×D₄	—	73	3	no	—
89	C₂²×C₄	—	73	3	no	—
90	C₂²×C₄	—	73	3	no	—
91	C₂²×C₄	—	73	3	no	—
92	M₁₆	—	73	24	no	—
93	C₂×C₈	—	73	2	no	—
94	C₂×C₈	—	73	6	no	—
95	C₂×C₈	—	73	24	no	—
96	C₄⋊C₄	—	73	3	no	—
97	C₄⋊C₄	C₂×#64408	1	3	yes	—
98	C₂²⋊C₄	—	73	6	no	—
99	C₄²	—	73	3	no	—
100	C₄²	—	73	3	no	—
101	C₄²	—	73	1	no	—
102	C₄²	C₂×#64408	1	3	yes	—
103	C₂³	—	73	3	no	—
104	Q₈	C₄×#64408	1	4	yes	—
105	D₄	—	73	12	no	—
106	C₂×C₄	—	73	1	no	—
107	C₂×C₄	—	73	1	no	—
108	C₂×C₄	—	73	12	no	—
109	C₂×C₄	—	73	3	no	—
110	C₂×C₄	—	146	6	no	—
111	C₂×C₄	—	146	6	no	—
112	C₂×C₄	C₂²×#64408	1	3	yes	—
113	C₂×C₄	C₂²×#64408	1	3	yes	—
114	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
115	C₈	—	73	4	no	—
116	C₈	—	73	12	no	—
117	C₂²	—	73	6	no	—
118	C₂²	—	146	3	no	—
119	C₂²	—	146	3	no	—
120	C₂²	C₂³×#64408	1	1	yes	characteristic
121	C₄	—	2	3	no	—
122	C₄	—	73	2	no	—
123	C₄	—	73	2	no	—
124	C₄	—	146	3	no	—
125	C₄	C₂×C₄×#64408	1	6	yes	—
126	C₄	#14556679	1	2	yes	direct product factor
127	C₂	—	146	6	no	—
128	C₂	C₂²×C₄×#64408	1	1	yes	characteristic
129	C₂	D₇₃⋊#131	1	1	yes	characteristic
130	C₂	C₂×#14556679	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556679

C₂

yes

maximal

#17989552

C₂

yes

maximal, complemented

C₄×#11924038

C₂

yes

maximal, complemented

C₄×#11924036

C₂

yes

maximal, complemented

C₄×#11924029

C₂

yes

maximal, characteristic

#14556679

C₄

yes

direct product factor

C₂×#11924038

C₂²

yes

—

C₂×#11924036

C₂²

yes

—

#14556646

C₂²

yes

—

C₄×#10596833

C₂²

yes

characteristic

C₄×#10596833

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

C₂×#11924029

C₂²

yes

characteristic

Q₈×Dic₇₃

C₄

yes

characteristic

C₄×D₂₉₂

C₄

yes

—

#14556553

C₄

yes

—

#14556549

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

—

C₄×Q₈×C₇₃

C₄

yes

characteristic

#11924038

C₂×C₄

yes

complemented

#11924036

C₂×C₄

yes

complemented

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

#11924029

C₂×C₄

yes

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924005

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

—

C₂²×D₇₃

C₂²×C₄

yes

—

Q₈×C₇₃

C₄²

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂⁴

yes

characteristic

D₂₉₂

C₄²

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂²×C₄

yes

—

#10596833

C₂²×C₄

yes

—

#10596833

C₂²×C₄

yes

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂×C₄²

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄²

yes

—

C₄×C₇₃

#131

yes

complemented

Dic₇₃

—

Dic₇₃

#131

yes

complemented

Dic₇₃

C₂³×C₄

yes

—

C₂×C₇₃

C₂²×C₄²

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

C₄×#131

—

maximal, Sylow

C₇₃

C₄×#131

yes

characteristic, complemented, Sylow

C₂×#131

—

C₄×#41

—

#405

—

C₄×M₁₆

—

C₂×C₄×C₈

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

#64408

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×#64408

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#64408

yes

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₂×#64408

yes

—

103

C₂³

—

104

Q₈

C₄×#64408

yes

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

146

—

111

C₂×C₄

—

146

—

112

C₂×C₄

C₂²×#64408

yes

—

113

C₂×C₄

C₂²×#64408

yes

—

114

C₂×C₄

C₂²×#64408

yes

characteristic

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

146

—

119

C₂²

—

146

—

120

C₂²

C₂³×#64408

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

146

—

125

C₄

C₂×C₄×#64408

yes

—

126

C₄

#14556679

yes

direct product factor

127

C₂

—

146

—

128

C₂

C₂²×C₄×#64408

yes

characteristic

129

C₂

D₇₃⋊#131

yes

characteristic

130

C₂

C₂×#14556679

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×#64408
commutator	C₂×C₇₃	C₂²×C₄²
Frattini	C₂²	C₂³×#64408
Fitting	C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₄
hypercenter	C₄×Q₈	#64408
hypocenter	C₇₃	C₄×#131
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×Q₈

subgroup

G/H

center

C₂×C₄

C₂²×#64408

commutator

C₂×C₇₃

C₂²×C₄²

Frattini

C₂²

C₂³×#64408

Fitting

C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂³×C₄

C₄×Q₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ C₄×Q₈

N	H ≅ G/N	type
2. #17989552	C₂	semidirect
3. C₄×#11924038	C₂	semidirect
4. C₄×#11924036	C₂	semidirect
6. #14556679	C₄	direct (N × H)
20. #11924038	C₂×C₄	semidirect
21. #11924036	C₂×C₄	semidirect
57. C₄×C₇₃	#131	semidirect
59. Dic₇₃	#131	semidirect
66. C₇₃	C₄×#131	semidirect
126. C₄	#14556679	direct (N × H)

H ≅ G/N

type

2. #17989552

C₂

semidirect

3. C₄×#11924038

C₂

semidirect

4. C₄×#11924036

C₂

semidirect

6. #14556679

C₄

direct (N × H)

20. #11924038

C₂×C₄

semidirect

21. #11924036

C₂×C₄

semidirect

57. C₄×C₇₃

#131

semidirect

59. Dic₇₃

#131

semidirect

66. C₇₃

C₄×#131

semidirect

126. C₄

#14556679

direct (N × H)

Subgroups — C₄×#14556679 #22626500

Conjugacy classes of subgroups

Complements

Subgroups — C4×#14556679 #22626500

Conjugacy classes of subgroups

Complements

Subgroups — C₄×#14556679 #22626500