Group #28383749 — subgroups

Subgroups — #28383749

Order 14272.

168 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 39 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	#20297350	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297332	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297329	C₂	1	1	yes	maximal, characteristic, complemented
4	#20297301	C₂	1	1	yes	maximal, characteristic, complemented
5	#20297232	C₂	1	1	yes	maximal, characteristic, complemented
6	#20297220	C₂	1	1	yes	maximal, characteristic, complemented
7	#104×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#16813171	—	2	1	no	—
9	#16813170	—	2	1	no	—
10	#16813170	C₂²	1	1	yes	characteristic
11	C₂×D₈₉₂	—	2	1	no	—
12	C₂×D₈₉₂	—	2	2	no	—
13	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
14	C₂×D₈₉₂	C₂²	1	1	yes	characteristic, complemented
15	D₄×D₂₂₃	—	2	2	no	—
16	D₄×D₂₂₃	—	2	1	no	—
17	#16813159	—	2	1	no	—
18	#16813157	—	2	1	no	—
19	D₁₇₈₄	—	2	1	no	—
20	#16813155	—	2	1	no	—
21	#16813154	C₂²	1	1	yes	characteristic, complemented
22	#41×C₂₂₃	C₂²	1	1	yes	characteristic
23	M₁₆×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
24	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×D₂₂₃	—	2	2	no	—
26	C₂²×D₂₂₃	—	2	1	no	—
27	C₂²×D₂₂₃	D₄	1	2	yes	—
28	Dic₄₄₆	D₄	1	1	yes	characteristic, complemented
29	D₈₉₂	—	2	2	no	—
30	D₈₉₂	—	2	1	no	—
31	D₈₉₂	—	2	1	no	—
32	D₈₉₂	—	2	1	no	—
33	D₈₉₂	—	2	1	no	—
34	D₈₉₂	—	4	4	no	—
35	D₈₉₂	D₄	1	1	yes	characteristic, complemented
36	C₂²⋊D₂₂₃	—	2	2	no	—
37	C₂²⋊D₂₂₃	—	2	1	no	—
38	C₄×D₂₂₃	—	2	2	no	—
39	C₄×D₂₂₃	—	2	1	no	—
40	C₂₂₃⋊C₈	—	2	1	no	—
41	Q₈×C₂₂₃	D₄	1	1	yes	characteristic, complemented
42	D₄×C₂₂₃	—	2	1	no	—
43	D₄×C₂₂₃	D₄	1	1	yes	characteristic, complemented
44	C₂×C₄×C₂₂₃	—	2	1	no	—
45	C₂×C₄×C₂₂₃	—	2	1	no	—
46	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
47	C₈×C₂₂₃	—	2	1	no	—
48	C₂²×C₂₂₃	—	2	1	no	—
49	C₂²×C₂₂₃	C₂×D₄	1	1	yes	characteristic
50	C₂×D₂₂₃	—	2	2	no	—
51	C₂×D₂₂₃	—	2	1	no	—
52	C₂×D₂₂₃	—	4	2	no	—
53	C₂×D₂₂₃	—	4	2	no	—
54	C₂×D₂₂₃	—	4	1	no	—
55	C₂×D₂₂₃	—	8	2	no	—
56	C₄×C₂₂₃	—	2	2	no	—
57	C₄×C₂₂₃	—	2	1	no	—
58	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
59	C₄×C₂₂₃	C₂×D₄	1	1	yes	characteristic
60	Dic₂₂₃	—	2	1	no	—
61	C₂×C₂₂₃	—	2	1	no	—
62	C₂×C₂₂₃	—	4	1	no	—
63	C₂×C₂₂₃	#120	1	1	yes	characteristic
64	D₂₂₃	—	4	2	no	—
65	D₂₂₃	—	4	1	no	—
66	D₂₂₃	—	8	1	no	—
67	C₂₂₃	#453	1	1	yes	characteristic, complemented, Sylow
68	#453	—	223	1	no	maximal, Sylow
69	H(5,𝔽₂)	—	223	1	no	—
70	AGL(1,ℤ/8)	—	223	1	no	—
71	AGL(1,ℤ/8)	—	223	1	no	—
72	#127	—	223	1	no	—
73	#104	—	223	1	no	—
74	#104	D₂₂₃	1	1	yes	characteristic, complemented
75	#100	—	223	1	no	—
76	#41	—	446	1	no	—
77	#41	—	446	1	no	—
78	#41	—	223	1	no	—
79	#41	C₂×D₂₂₃	1	1	yes	characteristic
80	C₂×D₄	—	446	2	no	—
81	C₂×D₄	—	446	1	no	—
82	C₂×D₄	—	446	2	no	—
83	C₂×D₄	—	446	1	no	—
84	C₂×D₄	—	223	1	no	—
85	C₂×D₄	—	223	1	no	—
86	SD₁₆	—	446	1	no	—
87	SD₁₆	—	446	1	no	—
88	D₈	—	446	1	no	—
89	D₈	—	446	1	no	—
90	M₁₆	—	223	1	no	—
91	M₁₆	C₂×D₂₂₃	1	1	yes	characteristic, complemented
92	C₄²	C₂×D₂₂₃	1	1	yes	characteristic, complemented
93	C₂³	—	446	2	no	—
94	C₂³	—	446	1	no	—
95	C₂³	—	223	2	no	—
96	Q₈	—	223	1	no	—
97	Q₈	D₈₉₂	1	1	yes	characteristic, complemented
98	D₄	—	2	1	no	—
99	D₄	—	892	4	no	—
100	D₄	—	446	2	no	—
101	D₄	—	446	2	no	—
102	D₄	—	446	1	no	—
103	D₄	—	446	1	no	—
104	D₄	—	446	1	no	—
105	D₄	—	446	1	no	—
106	D₄	—	446	1	no	—
107	D₄	—	223	1	no	—
108	D₄	D₈₉₂	1	1	yes	characteristic, complemented
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	446	2	no	—
112	C₂×C₄	—	446	1	no	—
113	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
114	C₈	—	2	1	no	—
115	C₈	—	446	1	no	—
116	C₂²	—	2	1	no	—
117	C₂²	—	892	2	no	—
118	C₂²	—	892	2	no	—
119	C₂²	—	892	1	no	—
120	C₂²	—	446	2	no	—
121	C₂²	—	446	1	no	—
122	C₂²	—	1784	2	no	—
123	C₂²	D₄×D₂₂₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	1	no	—
126	C₄	—	446	1	no	—
127	C₄	D₄×D₂₂₃	1	1	yes	characteristic
128	C₄	C₂×D₈₉₂	1	1	yes	characteristic
129	C₂	—	2	1	no	—
130	C₂	—	4	1	no	—
131	C₂	—	892	2	no	—
132	C₂	—	892	1	no	—
133	C₂	—	1784	1	no	—
134	C₂	#20297274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20297350

C₂

yes

maximal, characteristic, complemented

#20297332

C₂

yes

maximal, characteristic, complemented

#20297329

C₂

yes

maximal, characteristic, complemented

#20297301

C₂

yes

maximal, characteristic, complemented

#20297232

C₂

yes

maximal, characteristic, complemented

#20297220

C₂

yes

maximal, characteristic, complemented

#104×C₂₂₃

C₂

yes

maximal, characteristic, complemented

#16813171

—

#16813170

—

#16813170

C₂²

yes

characteristic

C₂×D₈₉₂

—

C₂×D₈₉₂

—

C₂×D₈₉₂

C₂²

yes

characteristic

C₂×D₈₉₂

C₂²

yes

characteristic, complemented

D₄×D₂₂₃

—

D₄×D₂₂₃

—

#16813159

—

#16813157

—

D₁₇₈₄

—

#16813155

—

#16813154

C₂²

yes

characteristic, complemented

#41×C₂₂₃

C₂²

yes

characteristic

M₁₆×C₂₂₃

C₂²

yes

characteristic, complemented

C₄²×C₂₂₃

C₂²

yes

characteristic, complemented

C₂²×D₂₂₃

—

C₂²×D₂₂₃

—

C₂²×D₂₂₃

D₄

yes

—

Dic₄₄₆

D₄

yes

characteristic, complemented

D₈₉₂

—

D₈₉₂

—

D₈₉₂

—

D₈₉₂

—

D₈₉₂

—

D₈₉₂

—

D₈₉₂

D₄

yes

characteristic, complemented

C₂²⋊D₂₂₃

—

C₂²⋊D₂₂₃

—

C₄×D₂₂₃

—

C₄×D₂₂₃

—

C₂₂₃⋊C₈

—

Q₈×C₂₂₃

D₄

yes

characteristic, complemented

D₄×C₂₂₃

—

D₄×C₂₂₃

D₄

yes

characteristic, complemented

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₈×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂×D₄

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×D₄

yes

characteristic

C₄×C₂₂₃

C₂×D₄

yes

characteristic

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

#120

yes

characteristic

D₂₂₃

—

D₂₂₃

—

D₂₂₃

—

C₂₂₃

#453

yes

characteristic, complemented, Sylow

#453

—

223

maximal, Sylow

H(5,𝔽₂)

—

223

—

AGL(1,ℤ/8)

—

223

—

AGL(1,ℤ/8)

—

223

—

#127

—

223

—

#104

—

223

—

#104

D₂₂₃

yes

characteristic, complemented

#100

—

223

—

#41

—

446

—

#41

—

446

—

#41

—

223

—

#41

C₂×D₂₂₃

yes

characteristic

C₂×D₄

—

446

—

C₂×D₄

—

446

—

C₂×D₄

—

446

—

C₂×D₄

—

446

—

C₂×D₄

—

223

—

C₂×D₄

—

223

—

SD₁₆

—

446

—

SD₁₆

—

446

—

D₈

—

446

—

D₈

—

446

—

M₁₆

—

223

—

M₁₆

C₂×D₂₂₃

yes

characteristic, complemented

C₄²

C₂×D₂₂₃

yes

characteristic, complemented

C₂³

—

446

—

C₂³

—

446

—

C₂³

—

223

—

Q₈

—

223

—

Q₈

D₈₉₂

yes

characteristic, complemented

D₄

—

D₄

—

892

—

100

D₄

—

446

—

101

D₄

—

446

—

102

D₄

—

446

—

103

D₄

—

446

—

104

D₄

—

446

—

105

D₄

—

446

—

106

D₄

—

446

—

107

D₄

—

223

—

108

D₄

D₈₉₂

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

446

—

112

C₂×C₄

—

446

—

113

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

114

C₈

—

115

C₈

—

446

—

116

C₂²

—

117

C₂²

—

892

—

118

C₂²

—

892

—

119

C₂²

—

892

—

120

C₂²

—

446

—

121

C₂²

—

446

—

122

C₂²

—

1784

—

123

C₂²

D₄×D₂₂₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

446

—

127

C₄

D₄×D₂₂₃

yes

characteristic

128

C₄

C₂×D₈₉₂

yes

characteristic

129

C₂

—

130

C₂

—

131

C₂

—

892

—

132

C₂

—

892

—

133

C₂

—

1784

—

134

C₂

#20297274

yes

characteristic

	subgroup	G/H
center	C₂	#20297274
commutator, cosocle	C₂×C₄×C₂₂₃	C₂³
Frattini	C₂×C₄	C₂²×D₂₂₃
Fitting	#104×C₂₂₃	C₂
socle	C₂×C₂₂₃	#120
hypercenter	#104	D₂₂₃
hypocenter	C₂₂₃	#453
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

subgroup

G/H

center

C₂

#20297274

commutator, cosocle

C₂×C₄×C₂₂₃

C₂³

Frattini

C₂×C₄

C₂²×D₂₂₃

Fitting

#104×C₂₂₃

C₂

C₂×C₂₂₃

D₂₂₃

C₂₂₃

G ▷ C₂×C₄×C₂₂₃ ▷ C₂×C₂₂₃ ▷ C₂₂₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ #104

N	H ≅ G/N	type
1. #20297350	C₂	semidirect
2. #20297332	C₂	semidirect
3. #20297329	C₂	semidirect
4. #20297301	C₂	semidirect
5. #20297232	C₂	semidirect
6. #20297220	C₂	semidirect
7. #104×C₂₂₃	C₂	semidirect
14. C₂×D₈₉₂	C₂²	semidirect
21. #16813154	C₂²	semidirect
23. M₁₆×C₂₂₃	C₂²	semidirect
24. C₄²×C₂₂₃	C₂²	semidirect
28. Dic₄₄₆	D₄	semidirect
35. D₈₉₂	D₄	semidirect
41. Q₈×C₂₂₃	D₄	semidirect
43. D₄×C₂₂₃	D₄	semidirect
67. C₂₂₃	#453	semidirect
74. #104	D₂₂₃	semidirect
91. M₁₆	C₂×D₂₂₃	semidirect
92. C₄²	C₂×D₂₂₃	semidirect
97. Q₈	D₈₉₂	semidirect
108. D₄	D₈₉₂	semidirect

H ≅ G/N

type

1. #20297350

C₂

semidirect

2. #20297332

C₂

semidirect

3. #20297329

C₂

semidirect

4. #20297301

C₂

semidirect

5. #20297232

C₂

semidirect

6. #20297220

C₂

semidirect

7. #104×C₂₂₃

C₂

semidirect

14. C₂×D₈₉₂

C₂²

semidirect

21. #16813154

C₂²

semidirect

23. M₁₆×C₂₂₃

C₂²

semidirect

24. C₄²×C₂₂₃

C₂²

semidirect

28. Dic₄₄₆

D₄

semidirect

35. D₈₉₂

D₄

semidirect

41. Q₈×C₂₂₃

D₄

semidirect

43. D₄×C₂₂₃

D₄

semidirect

67. C₂₂₃

#453

semidirect

74. #104

D₂₂₃

semidirect

91. M₁₆

C₂×D₂₂₃

semidirect

92. C₄²

C₂×D₂₂₃

semidirect

97. Q₈

D₈₉₂

semidirect

108. D₄

D₈₉₂

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20297350	2
#20297332	2
#20297329	2
#20297301	2
#20297232	2
#20297220	2
#104×C₂₂₃	2
#453	223

subgroup

index

#104×C₂₂₃

223

Subgroups — #28383749

Conjugacy classes of subgroups

Complements

Maximal subgroups