Group #28373581 — subgroups

Subgroups — #28373581

Order 13952.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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	#20295583	C₂	1	1	yes	maximal, characteristic
2	#20295436	C₂	1	4	yes	maximal, complemented
3	#20295425	C₂	1	4	yes	maximal, complemented
4	#20295066	C₂	1	4	yes	maximal
5	C₄×#13182013	C₂	1	1	yes	maximal, characteristic
6	C₂×#125×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂×#13182013	C₂²	1	2	yes	—
8	#16812699	C₂²	1	2	yes	—
9	#16812691	C₂²	1	2	yes	—
10	#16812678	—	2	2	no	—
11	C₄×Dic₂₁₈	—	2	2	no	—
12	#16812658	C₂²	1	2	yes	—
13	#16812632	C₂²	1	8	yes	—
14	#16812630	C₂²	1	8	yes	—
15	#16812610	C₂²	1	2	yes	—
16	C₄×#11724784	C₂²	1	2	yes	—
17	#125×C₁₀₉	—	2	2	no	—
18	#125×C₁₀₉	C₂²	1	2	yes	—
19	#125×C₁₀₉	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₁₀₉	—	2	2	no	—
21	C₂×C₄⋊C₄×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	2	no	—
24	#13182013	Q₈	1	8	yes	complemented
25	C₂×#11724784	C₂³	1	4	yes	—
26	C₄⋊Dic₁₀₉	—	2	2	no	—
27	C₄⋊Dic₁₀₉	C₂³	1	4	yes	—
28	#13182008	—	2	4	no	—
29	C₂²⋊Dic₁₀₉	—	2	2	no	—
30	C₄×Dic₁₀₉	—	2	2	no	—
31	C₂²×C₄×C₁₀₉	—	2	2	no	—
32	C₂²×C₄×C₁₀₉	D₄	1	2	yes	—
33	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₀₉	—	2	4	no	—
35	C₄⋊C₄×C₁₀₉	—	2	4	no	—
36	C₄⋊C₄×C₁₀₉	—	2	2	no	—
37	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
38	C₄²×C₁₀₉	D₄	1	2	yes	—
39	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	—	2	4	no	—
43	C₂×C₄×C₁₀₉	—	4	2	no	—
44	C₂×C₄×C₁₀₉	—	4	4	no	—
45	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
49	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₁₀₉	—	2	4	no	—
51	Dic₂₁₈	—	4	4	no	—
52	#11724784	C₂×Q₈	1	8	yes	—
53	C₂²×C₁₀₉	—	2	2	no	—
54	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
56	C₄×C₁₀₉	—	2	2	no	—
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	—	4	4	no	—
59	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
61	Dic₁₀₉	—	4	4	no	—
62	C₂×C₁₀₉	—	2	2	no	—
63	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₁₀₉	#575	1	2	yes	—
65	#3166	—	109	1	no	maximal, Sylow
66	C₁₀₉	#3166	1	1	yes	characteristic, complemented, Sylow
67	#533	—	109	1	no	—
68	C₂×#125	D₁₀₉	1	1	yes	characteristic
69	#501	—	109	4	no	—
70	#499	—	109	4	no	—
71	#428	—	109	4	no	—
72	C₄×M₁₆	—	109	1	no	—
73	C₂×M₁₆	—	109	2	no	—
74	C₄⋊Q₈	—	109	2	no	—
75	#125	—	2	2	no	—
76	#125	—	109	2	no	—
77	#125	C₂×D₁₀₉	1	2	yes	—
78	#125	C₂×D₁₀₉	1	2	yes	—
79	C₂²⋊Q₈	—	218	2	no	—
80	C₄×Q₈	—	218	2	no	—
81	#117	—	109	2	no	—
82	C₂×C₄⋊C₄	—	2	2	no	—
83	C₂×C₄⋊C₄	C₂×D₁₀₉	1	2	yes	—
84	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
85	#107	—	109	8	no	—
86	#106	—	109	8	no	—
87	#97	—	109	2	no	—
88	C₄×C₈	—	109	2	no	—
89	C₂×Q₈	—	218	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
92	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
93	M₁₆	—	109	8	no	—
94	C₂×C₈	—	109	4	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	2	no	—
98	C₄⋊C₄	—	109	4	no	—
99	C₄⋊C₄	—	218	4	no	—
100	C₄⋊C₄	—	218	2	no	—
101	C₄⋊C₄	C₂²×D₁₀₉	1	4	yes	—
102	C₂²⋊C₄	—	218	2	no	—
103	C₄²	—	218	2	no	—
104	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
105	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
106	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
107	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
108	Q₈	—	436	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	218	4	no	—
113	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
114	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
115	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
116	C₂×C₄	Q₈×D₁₀₉	1	4	yes	—
117	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
118	C₈	—	109	8	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#16812697	1	2	yes	—
121	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	2	no	—
124	C₄	—	4	4	no	—
125	C₄	—	436	4	no	—
126	C₄	#16812697	1	2	yes	—
127	C₄	C₂×Q₈×D₁₀₉	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	C₂×#16812697	1	1	yes	characteristic
130	C₂	#20295824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295583

C₂

yes

maximal, characteristic

#20295436

C₂

yes

maximal, complemented

#20295425

C₂

yes

maximal, complemented

#20295066

C₂

yes

maximal

C₄×#13182013

C₂

yes

maximal, characteristic

C₂×#125×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

#16812699

C₂²

yes

—

#16812691

C₂²

yes

—

#16812678

—

C₄×Dic₂₁₈

—

#16812658

C₂²

yes

—

#16812632

C₂²

yes

—

#16812630

C₂²

yes

—

#16812610

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

#125×C₁₀₉

—

#125×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

—

C₂×C₄⋊C₄×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

#13182013

Q₈

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂²×C₄×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

C₂×Dic₁₀₉

—

Dic₂₁₈

—

#11724784

C₂×Q₈

yes

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₄⋊Q₈

yes

—

C₂²×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₄⋊Q₈

yes

—

C₄×C₁₀₉

C₂²×Q₈

yes

—

Dic₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₀₉

#575

yes

—

#3166

—

109

maximal, Sylow

C₁₀₉

#3166

yes

characteristic, complemented, Sylow

#533

—

109

—

C₂×#125

D₁₀₉

yes

characteristic

#501

—

109

—

#499

—

109

—

#428

—

109

—

C₄×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₄⋊Q₈

—

109

—

#125

—

#125

—

109

—

#125

C₂×D₁₀₉

yes

—

#125

C₂×D₁₀₉

yes

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

218

—

#117

—

109

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#106

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

C₂×Q₈

—

218

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

M₁₆

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

100

C₄⋊C₄

—

218

—

101

C₄⋊C₄

C₂²×D₁₀₉

yes

—

102

C₂²⋊C₄

—

218

—

103

C₄²

—

218

—

104

C₄²

C₂²⋊D₁₀₉

yes

—

105

C₄²

C₂²×D₁₀₉

yes

characteristic

106

C₄²

C₂²×D₁₀₉

yes

characteristic

107

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

108

Q₈

—

436

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

218

—

113

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

114

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

115

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

116

C₂×C₄

Q₈×D₁₀₉

yes

—

117

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

118

C₈

—

109

—

119

C₂²

—

120

C₂²

#16812697

yes

—

121

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

436

—

126

C₄

#16812697

yes

—

127

C₄

C₂×Q₈×D₁₀₉

yes

—

128

C₂

—

129

C₂

C₂×#16812697

yes

characteristic

130

C₂

#20295824

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	C₂×#125×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	C₂×#125	D₁₀₉
hypocenter	C₁₀₉	#3166
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

C₂×#125×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #20295436	C₂	semidirect
3. #20295425	C₂	semidirect
24. #13182013	Q₈	semidirect
66. C₁₀₉	#3166	semidirect

H ≅ G/N

type

2. #20295436

C₂

semidirect

3. #20295425

C₂

semidirect

24. #13182013

Q₈

semidirect

66. C₁₀₉

#3166

semidirect

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

subgroup	index
#20295583	2
#20295436	2
#20295425	2
#20295066	2
C₄×#13182013	2
C₂×#125×C₁₀₉	2
#3166	109

subgroup

index

C₂×#125×C₁₀₉

109

Subgroups — #28373581

Conjugacy classes of subgroups

Complements

Maximal subgroups