Group #28384087 — subgroups

Subgroups — #28384087

Order 14272.

330 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₂²×C₄×D₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#20297285	C₂	1	2	yes	maximal, complemented
3	#20297280	C₂	1	2	yes	maximal, complemented
4	#20297275	C₂	1	2	yes	maximal, complemented
5	#20297272	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊D₂₂₃	C₂	1	4	yes	maximal, complemented
7	#20297263	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#41×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₂₂₃	C₂²	1	1	yes	characteristic
10	C₂×C₂²⋊D₂₂₃	C₂²	1	4	yes	—
11	C₂²×Dic₂₂₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₂₃	—	2	4	no	—
13	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
14	C₂×C₄×D₂₂₃	C₂²	1	2	yes	—
15	C₄⋊Dic₂₂₃	C₂²	1	2	yes	complemented
16	Dic₂₂₃⋊C₄	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₂₂₃	C₂²	1	2	yes	—
18	C₂²⋊Dic₂₂₃	C₂²	1	4	yes	complemented
19	(C₂×D₂₂₃)⋊C₄	C₂²	1	4	yes	—
20	C₄×Dic₂₂₃	C₂²	1	2	yes	complemented
21	#41×C₂₂₃	—	2	4	no	—
22	C₂×Q₈×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
23	C₂×D₄×C₂₂₃	C₂²	1	2	yes	—
24	C₂×D₄×C₂₂₃	C₂²	1	1	yes	characteristic, complemented
25	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₂₂₃	C₂²	1	2	yes	complemented
27	C₂²×D₂₂₃	—	2	2	no	—
28	C₂²×D₂₂₃	—	2	4	no	—
29	C₂²×D₂₂₃	C₂³	1	2	yes	—
30	C₂²⋊D₂₂₃	—	2	8	no	—
31	C₂×Dic₂₂₃	—	2	2	no	—
32	C₂×Dic₂₂₃	C₂³	1	4	yes	—
33	C₂×Dic₂₂₃	C₂³	1	2	yes	—
34	C₄×D₂₂₃	—	2	4	no	—
35	C₄×D₂₂₃	—	2	4	no	—
36	C₂³×C₂₂₃	C₂³	1	2	yes	—
37	C₂³×C₂₂₃	C₂³	1	1	yes	characteristic
38	Q₈×C₂₂₃	—	2	2	no	—
39	D₄×C₂₂₃	—	2	4	no	—
40	D₄×C₂₂₃	—	2	2	no	—
41	C₂×C₄×C₂₂₃	—	2	4	no	—
42	C₂×C₄×C₂₂₃	D₄	1	4	yes	complemented
43	C₂×C₄×C₂₂₃	C₂³	1	2	yes	—
44	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
45	C₂×C₄×C₂₂₃	C₂³	1	1	yes	characteristic
46	C₂²×C₂₂₃	—	2	2	no	—
47	C₂²×C₂₂₃	—	2	2	no	—
48	C₂²×C₂₂₃	—	2	4	no	—
49	C₂²×C₂₂₃	C₂×D₄	1	2	yes	—
50	C₂²×C₂₂₃	C₂⁴	1	1	yes	characteristic
51	C₂×D₂₂₃	—	2	8	no	—
52	C₂×D₂₂₃	—	2	2	no	—
53	C₂×D₂₂₃	—	2	2	no	—
54	C₂×D₂₂₃	#41	1	4	yes	complemented
55	C₄×C₂₂₃	—	2	2	no	—
56	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
57	C₄×C₂₂₃	C₂×D₄	1	2	yes	—
58	Dic₂₂₃	—	2	2	no	—
59	Dic₂₂₃	—	2	4	no	—
60	C₂×C₂₂₃	—	2	2	no	—
61	C₂×C₂₂₃	—	4	2	no	—
62	C₂×C₂₂₃	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₂₂₃	C₂×#41	1	2	yes	—
64	D₂₂₃	—	2	4	no	—
65	C₂₂₃	#525	1	1	yes	characteristic, complemented, Sylow
66	#525	—	223	1	no	maximal, Sylow
67	C₂×#41	D₂₂₃	1	1	yes	characteristic, complemented
68	C₂³×C₄	—	223	1	no	—
69	#123	—	223	2	no	—
70	C₂²⋊Q₈	—	223	2	no	—
71	#121	—	223	2	no	—
72	#120	—	223	2	no	—
73	C₄×D₄	—	223	4	no	—
74	#117	—	223	1	no	—
75	C₂⁴	—	223	1	no	—
76	#41	—	2	4	no	—
77	C₂×Q₈	C₂×D₂₂₃	1	1	yes	characteristic, complemented
78	C₂×D₄	—	223	4	no	—
79	C₂×D₄	C₂×D₂₂₃	1	2	yes	—
80	C₂×D₄	C₂×D₂₂₃	1	1	yes	characteristic, complemented
81	C₂²×C₄	—	446	4	no	—
82	C₂²×C₄	—	223	1	no	—
83	C₂²×C₄	—	223	2	no	—
84	C₂²×C₄	—	223	2	no	—
85	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
86	C₂²×C₄	C₂×D₂₂₃	1	2	yes	complemented
87	C₄⋊C₄	—	223	4	no	—
88	C₄⋊C₄	—	223	2	no	—
89	C₂²⋊C₄	—	223	2	no	—
90	C₂²⋊C₄	—	223	4	no	—
91	C₂²⋊C₄	—	223	4	no	—
92	C₄²	—	223	2	no	—
93	C₂³	—	446	4	no	—
94	C₂³	—	446	2	no	—
95	C₂³	—	223	2	no	—
96	C₂³	C₂²×D₂₂₃	1	2	yes	—
97	C₂³	C₂²×D₂₂₃	1	1	yes	characteristic
98	Q₈	—	2	2	no	—
99	D₄	—	2	4	no	—
100	D₄	—	2	2	no	—
101	D₄	—	446	8	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	446	2	no	—
104	C₂×C₄	—	446	4	no	—
105	C₂×C₄	—	446	4	no	—
106	C₂×C₄	—	223	4	no	—
107	C₂×C₄	—	223	2	no	—
108	C₂×C₄	C₂²⋊D₂₂₃	1	4	yes	complemented
109	C₂×C₄	C₂²×D₂₂₃	1	2	yes	—
110	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₂₂₃	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	4	no	—
115	C₂²	—	446	8	no	—
116	C₂²	—	446	2	no	—
117	C₂²	—	446	2	no	—
118	C₂²	—	223	4	no	—
119	C₂²	C₂×C₂²⋊D₂₂₃	1	2	yes	—
120	C₂²	C₂³×D₂₂₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	446	2	no	—
123	C₄	—	446	4	no	—
124	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
125	C₄	C₂×C₂²⋊D₂₂₃	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	4	2	no	—
128	C₂	—	446	4	no	—
129	C₂	C₂²×C₂²⋊D₂₂₃	1	1	yes	characteristic
130	C₂	#41×D₂₂₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₂₂₃

C₂

yes

maximal, characteristic, complemented

#20297285

C₂

yes

maximal, complemented

#20297280

C₂

yes

maximal, complemented

#20297275

C₂

yes

maximal, complemented

#20297272

C₂

yes

maximal, complemented

C₄×C₂²⋊D₂₂₃

C₂

yes

maximal, complemented

#20297263

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₂₂₃

C₂

yes

maximal, characteristic, complemented

C₂³×D₂₂₃

C₂²

yes

characteristic

C₂×C₂²⋊D₂₂₃

C₂²

yes

—

C₂²×Dic₂₂₃

C₂²

yes

characteristic

C₂×C₄×D₂₂₃

—

C₂×C₄×D₂₂₃

C₂²

yes

—

C₂×C₄×D₂₂₃

C₂²

yes

—

C₄⋊Dic₂₂₃

C₂²

yes

complemented

Dic₂₂₃⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₂₂₃

C₂²

yes

—

C₂²⋊Dic₂₂₃

C₂²

yes

complemented

(C₂×D₂₂₃)⋊C₄

C₂²

yes

—

C₄×Dic₂₂₃

C₂²

yes

complemented

#41×C₂₂₃

—

C₂×Q₈×C₂₂₃

C₂²

yes

characteristic, complemented

C₂×D₄×C₂₂₃

C₂²

yes

—

C₂×D₄×C₂₂₃

C₂²

yes

characteristic, complemented

C₂²×C₄×C₂₂₃

C₂²

yes

characteristic

C₂²×C₄×C₂₂₃

C₂²

yes

complemented

C₂²×D₂₂₃

—

C₂²×D₂₂₃

—

C₂²×D₂₂₃

C₂³

yes

—

C₂²⋊D₂₂₃

—

C₂×Dic₂₂₃

—

C₂×Dic₂₂₃

C₂³

yes

—

C₂×Dic₂₂₃

C₂³

yes

—

C₄×D₂₂₃

—

C₄×D₂₂₃

—

C₂³×C₂₂₃

C₂³

yes

—

C₂³×C₂₂₃

C₂³

yes

characteristic

Q₈×C₂₂₃

—

D₄×C₂₂₃

—

D₄×C₂₂₃

—

C₂×C₄×C₂₂₃

—

C₂×C₄×C₂₂₃

D₄

yes

complemented

C₂×C₄×C₂₂₃

C₂³

yes

—

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂×C₄×C₂₂₃

C₂³

yes

characteristic

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

—

C₂²×C₂₂₃

C₂×D₄

yes

—

C₂²×C₂₂₃

C₂⁴

yes

characteristic

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

—

C₂×D₂₂₃

#41

yes

complemented

C₄×C₂₂₃

—

C₄×C₂₂₃

C₂×D₄

yes

—

C₄×C₂₂₃

C₂×D₄

yes

—

Dic₂₂₃

—

Dic₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

—

C₂×C₂₂₃

C₂²×D₄

yes

characteristic

C₂×C₂₂₃

C₂×#41

yes

—

D₂₂₃

—

C₂₂₃

#525

yes

characteristic, complemented, Sylow

#525

—

223

maximal, Sylow

C₂×#41

D₂₂₃

yes

characteristic, complemented

C₂³×C₄

—

223

—

#123

—

223

—

C₂²⋊Q₈

—

223

—

#121

—

223

—

#120

—

223

—

C₄×D₄

—

223

—

#117

—

223

—

C₂⁴

—

223

—

#41

—

C₂×Q₈

C₂×D₂₂₃

yes

characteristic, complemented

C₂×D₄

—

223

—

C₂×D₄

C₂×D₂₂₃

yes

—

C₂×D₄

C₂×D₂₂₃

yes

characteristic, complemented

C₂²×C₄

—

446

—

C₂²×C₄

—

223

—

C₂²×C₄

—

223

—

C₂²×C₄

—

223

—

C₂²×C₄

C₂×D₂₂₃

yes

characteristic

C₂²×C₄

C₂×D₂₂₃

yes

complemented

C₄⋊C₄

—

223

—

C₄⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₂²⋊C₄

—

223

—

C₄²

—

223

—

C₂³

—

446

—

C₂³

—

446

—

C₂³

—

223

—

C₂³

C₂²×D₂₂₃

yes

—

C₂³

C₂²×D₂₂₃

yes

characteristic

Q₈

—

D₄

—

100

D₄

—

101

D₄

—

446

—

102

C₂×C₄

—

103

C₂×C₄

—

446

—

104

C₂×C₄

—

446

—

105

C₂×C₄

—

446

—

106

C₂×C₄

—

223

—

107

C₂×C₄

—

223

—

108

C₂×C₄

C₂²⋊D₂₂₃

yes

complemented

109

C₂×C₄

C₂²×D₂₂₃

yes

—

110

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

111

C₂×C₄

C₂²×D₂₂₃

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

446

—

116

C₂²

—

446

—

117

C₂²

—

446

—

118

C₂²

—

223

—

119

C₂²

C₂×C₂²⋊D₂₂₃

yes

—

120

C₂²

C₂³×D₂₂₃

yes

characteristic

121

C₄

—

122

C₄

—

446

—

123

C₄

—

446

—

124

C₄

C₂×C₂²⋊D₂₂₃

yes

—

125

C₄

C₂×C₂²⋊D₂₂₃

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

446

—

129

C₂

C₂²×C₂²⋊D₂₂₃

yes

characteristic

130

C₂

#41×D₂₂₃

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₂₃
commutator, socle, cosocle	C₂²×C₂₂₃	C₂⁴
Frattini	C₂²	C₂³×D₂₂₃
Fitting	C₂×#41×C₂₂₃	C₂
hypercenter	C₂×#41	D₂₂₃
hypocenter	C₂₂₃	#525
LCS	G ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

subgroup

G/H

center

C₂×C₄

C₂²×D₂₂₃

commutator, socle, cosocle

C₂²×C₂₂₃

C₂⁴

Frattini

C₂²

C₂³×D₂₂₃

Fitting

C₂×#41×C₂₂₃

C₂

hypercenter

C₂×#41

D₂₂₃

C₂₂₃

G ▷ C₂²×C₂₂₃ ▷ C₂₂₃

C₁ ◁ C₂×C₄ ◁ C₂×#41

N	H ≅ G/N	type
1. C₂²×C₄×D₂₂₃	C₂	semidirect
2. #20297285	C₂	semidirect
3. #20297280	C₂	semidirect
4. #20297275	C₂	semidirect
5. #20297272	C₂	semidirect
6. C₄×C₂²⋊D₂₂₃	C₂	semidirect
7. #20297263	C₂	semidirect
8. C₂×#41×C₂₂₃	C₂	semidirect
15. C₄⋊Dic₂₂₃	C₂²	semidirect
16. Dic₂₂₃⋊C₄	C₂²	semidirect
18. C₂²⋊Dic₂₂₃	C₂²	semidirect
20. C₄×Dic₂₂₃	C₂²	semidirect
22. C₂×Q₈×C₂₂₃	C₂²	semidirect
24. C₂×D₄×C₂₂₃	C₂²	semidirect
26. C₂²×C₄×C₂₂₃	C₂²	semidirect
42. C₂×C₄×C₂₂₃	D₄	semidirect
54. C₂×D₂₂₃	#41	semidirect
65. C₂₂₃	#525	semidirect
67. C₂×#41	D₂₂₃	semidirect
77. C₂×Q₈	C₂×D₂₂₃	semidirect
80. C₂×D₄	C₂×D₂₂₃	semidirect
86. C₂²×C₄	C₂×D₂₂₃	semidirect
108. C₂×C₄	C₂²⋊D₂₂₃	semidirect

H ≅ G/N

type

1. C₂²×C₄×D₂₂₃

C₂

semidirect

2. #20297285

C₂

semidirect

3. #20297280

C₂

semidirect

4. #20297275

C₂

semidirect

5. #20297272

C₂

semidirect

6. C₄×C₂²⋊D₂₂₃

C₂

semidirect

7. #20297263

C₂

semidirect

8. C₂×#41×C₂₂₃

C₂

semidirect

15. C₄⋊Dic₂₂₃

C₂²

semidirect

16. Dic₂₂₃⋊C₄

C₂²

semidirect

18. C₂²⋊Dic₂₂₃

C₂²

semidirect

20. C₄×Dic₂₂₃

C₂²

semidirect

22. C₂×Q₈×C₂₂₃

C₂²

semidirect

24. C₂×D₄×C₂₂₃

C₂²

semidirect

26. C₂²×C₄×C₂₂₃

C₂²

semidirect

42. C₂×C₄×C₂₂₃

D₄

semidirect

54. C₂×D₂₂₃

#41

semidirect

65. C₂₂₃

#525

semidirect

67. C₂×#41

D₂₂₃

semidirect

77. C₂×Q₈

C₂×D₂₂₃

semidirect

80. C₂×D₄

C₂×D₂₂₃

semidirect

86. C₂²×C₄

C₂×D₂₂₃

semidirect

108. C₂×C₄

C₂²⋊D₂₂₃

semidirect

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

subgroup	index
C₂²×C₄×D₂₂₃	2
#20297285	2
#20297280	2
#20297275	2
#20297272	2
C₄×C₂²⋊D₂₂₃	2
#20297263	2
C₂×#41×C₂₂₃	2
#525	223

subgroup

index

C₂²×C₄×D₂₂₃

C₄×C₂²⋊D₂₂₃

C₂×#41×C₂₂₃

223

Subgroups — #28384087

Conjugacy classes of subgroups

Complements

Maximal subgroups