Group #28384324 — subgroups

Subgroups — #28384324

Order 14272.

220 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20297296	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297292	C₂	1	1	yes	maximal, characteristic, complemented
3	#20297288	C₂	1	2	yes	maximal, complemented
4	#20297286	C₂	1	2	yes	maximal, complemented
5	#20297281	C₂	1	2	yes	maximal, complemented
6	#20297280	C₂	1	2	yes	maximal, complemented
7	#20297277	C₂	1	2	yes	maximal, complemented
8	#20297276	C₂	1	2	yes	maximal, complemented
9	#124×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Dic₄₄₆	C₂²	1	1	yes	characteristic
11	C₂×D₈₉₂	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₂₂₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₂₂₃	C₂²	1	2	yes	—
14	C₂²×Dic₂₂₃	C₂²	1	2	yes	—
15	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
16	C₄⋊Dic₂₂₃	C₂²	1	2	yes	—
17	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
18	C₄⋊Dic₂₂₃	C₂²	1	1	yes	characteristic
19	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
20	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
21	Dic₂₂₃⋊C₄	C₂²	1	2	yes	—
22	C₂²⋊Dic₂₂₃	C₂²	1	2	yes	—
23	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
25	(C₂×D₂₂₃)⋊C₄	C₂²	1	2	yes	—
26	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic
27	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic
28	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
29	C₂²⋊C₄×C₂₂₃	C₂²	1	2	yes	—
30	C₄²×C₂₂₃	C₂²	1	1	yes	characteristic
31	C₂²×D₂₂₃	C₂³	1	2	yes	—
32	Dic₄₄₆	—	4	1	no	—
33	D₈₉₂	—	4	1	no	—
34	C₂²⋊D₂₂₃	—	4	2	no	—
35	C₂²⋊D₂₂₃	—	4	2	no	—
36	C₂×Dic₂₂₃	—	4	2	no	—
37	C₂×Dic₂₂₃	C₂³	1	2	yes	—
38	C₂×Dic₂₂₃	C₂³	1	2	yes	—
39	C₂×Dic₂₂₃	C₂³	1	2	yes	—
40	C₄×D₂₂₃	—	4	2	no	—
41	C₂³×C₂₂₃	C₂³	1	2	yes	—
42	Q₈×C₂₂₃	—	4	1	no	—
43	D₄×C₂₂₃	—	4	1	no	—
44	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
48	C₂²×C₂₂₃	—	2	2	no	—
49	C₂²×C₂₂₃	—	2	2	no	—
50	C₂²×C₂₂₃	—	2	2	no	—
51	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₂₂₃	—	2	2	no	—
53	C₂×D₂₂₃	—	2	2	no	—
54	C₂×D₂₂₃	—	2	2	no	—
55	C₄×C₂₂₃	—	2	1	no	—
56	C₄×C₂₂₃	—	2	2	no	—
57	C₄×C₂₂₃	—	2	1	no	—
58	C₄×C₂₂₃	—	2	1	no	—
59	Dic₂₂₃	—	2	2	no	—
60	Dic₂₂₃	—	2	2	no	—
61	Dic₂₂₃	—	2	2	no	—
62	C₂×C₂₂₃	—	4	2	no	—
63	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
64	C₂×C₂₂₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₂₂₃	2₋¹⁺⁴	1	1	yes	characteristic
66	D₂₂₃	—	4	2	no	—
67	C₂₂₃	#562	1	1	yes	characteristic, complemented, Sylow
68	#562	—	223	1	no	maximal, Sylow
69	#125	—	223	1	no	—
70	#124	—	223	1	no	—
71	#124	D₂₂₃	1	1	yes	characteristic, complemented
72	#123	—	223	2	no	—
73	#123	—	223	2	no	—
74	C₂²⋊Q₈	—	223	2	no	—
75	C₂²⋊Q₈	—	223	2	no	—
76	#121	—	223	2	no	—
77	#121	—	223	2	no	—
78	C₂×Q₈	—	223	1	no	—
79	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic
80	C₂×D₄	—	223	2	no	—
81	C₂×D₄	—	223	2	no	—
82	C₂×D₄	—	223	1	no	—
83	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic
84	C₂²×C₄	—	223	2	no	—
85	C₂²×C₄	—	223	2	no	—
86	C₄⋊C₄	—	223	2	no	—
87	C₄⋊C₄	—	223	2	no	—
88	C₄⋊C₄	—	223	2	no	—
89	C₄⋊C₄	—	223	2	no	—
90	C₄⋊C₄	—	223	1	no	—
91	C₄⋊C₄	—	223	1	no	—
92	C₂²⋊C₄	—	223	2	no	—
93	C₂²⋊C₄	—	223	2	no	—
94	C₂²⋊C₄	—	223	2	no	—
95	C₂²⋊C₄	—	223	2	no	—
96	C₂²⋊C₄	C₂×D₂₂₃	1	2	yes	—
97	C₂²⋊C₄	C₂×D₂₂₃	1	2	yes	—
98	C₄²	C₂×D₂₂₃	1	1	yes	characteristic
99	C₂³	—	223	2	no	—
100	C₂³	C₂²×D₂₂₃	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	892	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	892	1	no	—
105	D₄	—	892	2	no	—
106	D₄	—	892	2	no	—
107	C₂×C₄	—	892	2	no	—
108	C₂×C₄	—	892	2	no	—
109	C₂×C₄	—	223	2	no	—
110	C₂×C₄	—	223	2	no	—
111	C₂×C₄	—	223	2	no	—
112	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
115	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	446	2	no	—
120	C₂²	—	446	2	no	—
121	C₂²	—	446	2	no	—
122	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	446	2	no	—
128	C₄	—	446	2	no	—
129	C₄	—	446	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	892	2	no	—
132	C₂	#20297349	1	1	yes	characteristic
133	C₂	#20297350	1	1	yes	characteristic
134	C₂	#20297351	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20297296

C₂

yes

maximal, characteristic, complemented

#20297292

C₂

yes

maximal, characteristic, complemented

#20297288

C₂

yes

maximal, complemented

#20297286

C₂

yes

maximal, complemented

#20297281

C₂

yes

maximal, complemented

#20297280

C₂

yes

maximal, complemented

#20297277

C₂

yes

maximal, complemented

#20297276

C₂

yes

maximal, complemented

#124×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂×Dic₄₄₆

C₂²

yes

characteristic

C₂×D₈₉₂

C₂²

yes

characteristic

C₂×C₂²⋊D₂₂₃

C₂²

yes

—

C₂×C₂²⋊D₂₂₃

C₂²

yes

—

C₂²×Dic₂₂₃

C₂²

yes

—

C₂×C₄×D₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

characteristic

C₄⋊Dic₂₂₃

C₂²

yes

characteristic

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

Dic₂₂₃⋊C₄

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

C₂×Q₈×C₂₂₃

C₂²

yes

characteristic

C₂×D₄×C₂₂₃

C₂²

yes

characteristic

C₂²⋊C₄×C₂₂₃

C₂²

yes

—

C₂²⋊C₄×C₂₂₃

C₂²

yes

—

C₄²×C₂₂₃

C₂²

yes

characteristic

C₂²×D₂₂₃

C₂³

yes

—

Dic₄₄₆

—

D₈₉₂

—

C₂²⋊D₂₂₃

—

C₂²⋊D₂₂₃

—

C₂×Dic₂₂₃

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₄×D₂₂₃

—

C₂³×C₂₂₃

C₂³

yes

—

Q₈×C₂₂₃

—

D₄×C₂₂₃

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

C₄×C₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

yes

C₂×C₂₂₃

yes

C₂×C₂₂₃

2₋¹⁺⁴

yes

characteristic

D₂₂₃

—

C₂₂₃

#562

yes

characteristic, complemented, Sylow

#562

—

223

maximal, Sylow

#125

—

223

—

#124

—

223

—

#124

D₂₂₃

yes

characteristic, complemented

#123

—

223

—

#123

—

223

—

C₂²⋊Q₈

—

223

—

C₂²⋊Q₈

—

223

—

#121

—

223

—

#121

—

223

—

C₂×Q₈

—

223

—

C₂×Q₈

C₂×D₂₂₃

yes

characteristic

C₂×D₄

—

223

—

C₂×D₄

—

223

—

C₂×D₄

—

223

—

C₂×D₄

C₂×D₂₂₃

yes

characteristic

C₂²×C₄

—

223

—

C₂²×C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

C₂×D₂₂₃

yes

—

C₂²⋊C₄

C₂×D₂₂₃

yes

—

C₄²

C₂×D₂₂₃

yes

characteristic

C₂³

—

223

—

100

C₂³

C₂²×D₂₂₃

yes

—

101

Q₈

—

102

Q₈

—

892

—

103

D₄

—

104

D₄

—

892

—

105

D₄

—

892

—

106

D₄

—

892

—

107

C₂×C₄

—

892

—

108

C₂×C₄

—

892

—

109

C₂×C₄

—

223

—

110

C₂×C₄

—

223

—

111

C₂×C₄

—

223

—

112

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

113

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

114

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

115

C₂×C₄

C₂²×D₂₂₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

446

—

120

C₂²

—

446

—

121

C₂²

—

446

—

122

C₂²

C₂³×D₂₂₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

446

—

128

C₄

—

446

—

129

C₄

—

446

—

130

C₂

—

131

C₂

—

892

—

132

C₂

#20297349

yes

characteristic

133

C₂

#20297350

yes

characteristic

134

C₂

#20297351

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Fitting	#124×C₂₂₃	C₂
hypercenter	#124	D₂₂₃
hypocenter	C₂₂₃	#562
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₂₂₃

commutator, socle, cosocle

C₂²×C₂₂₃

C₂⁴

Fitting

#124×C₂₂₃

C₂

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #20297296	C₂	semidirect
2. #20297292	C₂	semidirect
3. #20297288	C₂	semidirect
4. #20297286	C₂	semidirect
5. #20297281	C₂	semidirect
6. #20297280	C₂	semidirect
7. #20297277	C₂	semidirect
8. #20297276	C₂	semidirect
9. #124×C₂₂₃	C₂	semidirect
67. C₂₂₃	#562	semidirect
71. #124	D₂₂₃	semidirect

H ≅ G/N

type

1. #20297296

C₂

semidirect

2. #20297292

C₂

semidirect

3. #20297288

C₂

semidirect

4. #20297286

C₂

semidirect

5. #20297281

C₂

semidirect

6. #20297280

C₂

semidirect

7. #20297277

C₂

semidirect

8. #20297276

C₂

semidirect

9. #124×C₂₂₃

C₂

semidirect

67. C₂₂₃

#562

semidirect

71. #124

D₂₂₃

semidirect

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

subgroup	index
#20297296	2
#20297292	2
#20297288	2
#20297286	2
#20297281	2
#20297280	2
#20297277	2
#20297276	2
#124×C₂₂₃	2
#562	223

subgroup

index

#124×C₂₂₃

223

Subgroups — #28384324

Conjugacy classes of subgroups

Complements

Maximal subgroups