Group #28373554 — subgroups

Subgroups — #28373554

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	#20295432	C₂	1	4	yes	maximal, complemented
4	#20295062	C₂	1	4	yes	maximal
5	C₂×#16812610	C₂	1	1	yes	maximal, characteristic
6	#533×C₁₀₉	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11724784	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	#16812610	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₁₀₉	C₂²	1	2	yes	—
18	#125×C₁₀₉	C₂²	1	2	yes	—
19	C₂²⋊Q₈×C₁₀₉	—	2	2	no	—
20	C₄×Q₈×C₁₀₉	—	2	2	no	—
21	#117×C₁₀₉	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₀₉	C₂²	1	1	yes	characteristic
23	C₂×Dic₂₁₈	—	2	2	no	—
24	C₂×#11724784	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	C₂×Q₈×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	2	no	—
36	C₄⋊C₄×C₁₀₉	C₂³	1	4	yes	—
37	C₂²⋊C₄×C₁₀₉	—	2	2	no	—
38	C₄²×C₁₀₉	—	2	2	no	—
39	C₄²×C₁₀₉	D₄	1	2	yes	—
40	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
41	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
42	C₂³×C₁₀₉	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₁₀₉	—	4	4	no	—
44	C₂×C₄×C₁₀₉	—	2	4	no	—
45	C₂×C₄×C₁₀₉	—	4	2	no	—
46	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₀₉	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₁₀₉	—	2	4	no	—
52	Dic₂₁₈	—	4	4	no	—
53	#11724784	C₂×Q₈	1	8	yes	—
54	C₂²×C₁₀₉	—	2	2	no	—
55	C₂²×C₁₀₉	C₄⋊Q₈	1	2	yes	—
56	C₂²×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
57	C₄×C₁₀₉	—	2	2	no	—
58	C₄×C₁₀₉	—	2	2	no	—
59	C₄×C₁₀₉	—	4	4	no	—
60	C₄×C₁₀₉	C₄⋊Q₈	1	2	yes	—
61	C₄×C₁₀₉	C₂²×Q₈	1	2	yes	—
62	Dic₁₀₉	—	4	4	no	—
63	C₂×C₁₀₉	—	2	2	no	—
64	C₂×C₁₀₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₀₉	#575	1	2	yes	—
66	#3162	—	109	1	no	maximal, Sylow
67	C₁₀₉	#3162	1	1	yes	characteristic, complemented, Sylow
68	#533	—	109	1	no	—
69	#533	D₁₀₉	1	1	yes	characteristic
70	#501	—	109	4	no	—
71	#501	—	109	4	no	—
72	#427	—	109	4	no	—
73	C₂×#97	—	109	1	no	—
74	C₂²×C₈	—	109	2	no	—
75	C₄⋊Q₈	—	109	2	no	—
76	C₄⋊Q₈	C₂×D₁₀₉	1	2	yes	—
77	#125	—	109	2	no	—
78	#125	C₂×D₁₀₉	1	2	yes	—
79	C₂²⋊Q₈	—	2	2	no	—
80	C₂²⋊Q₈	—	218	2	no	—
81	C₄×Q₈	—	2	2	no	—
82	C₄×Q₈	—	218	2	no	—
83	#117	—	109	2	no	—
84	#117	C₂×D₁₀₉	1	2	yes	—
85	C₂×C₄²	C₂×D₁₀₉	1	1	yes	characteristic
86	#107	—	109	8	no	—
87	#106	—	109	8	no	—
88	#97	—	109	2	no	—
89	#97	—	109	2	no	—
90	C₂×Q₈	—	2	2	no	—
91	C₂×Q₈	—	218	2	no	—
92	C₂²×C₄	C₂²⋊D₁₀₉	1	2	yes	—
93	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
94	C₂×C₈	—	109	8	no	—
95	C₂×C₈	—	109	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	—	2	2	no	—
103	C₂²⋊C₄	—	218	2	no	—
104	C₄²	—	2	2	no	—
105	C₄²	—	218	2	no	—
106	C₄²	C₂²⋊D₁₀₉	1	2	yes	—
107	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
108	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
109	C₂³	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
110	Q₈	—	4	4	no	—
111	Q₈	—	436	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	2	no	—
114	C₂×C₄	—	218	4	no	—
115	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
116	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	2	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₀₉	1	1	yes	characteristic
118	C₂×C₄	Q₈×D₁₀₉	1	4	yes	—
119	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
120	C₈	—	109	8	no	—
121	C₂²	—	2	2	no	—
122	C₂²	#16812697	1	2	yes	—
123	C₂²	C₂²×C₂²⋊D₁₀₉	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	436	4	no	—
128	C₄	#16812697	1	2	yes	—
129	C₄	C₂×Q₈×D₁₀₉	1	2	yes	—
130	C₂	—	2	2	no	—
131	C₂	C₂×#16812697	1	1	yes	characteristic
132	C₂	#20295824	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20295583

C₂

yes

maximal, characteristic

#20295436

C₂

yes

maximal, complemented

#20295432

C₂

yes

maximal, complemented

#20295062

C₂

yes

maximal

C₂×#16812610

C₂

yes

maximal, characteristic

#533×C₁₀₉

C₂

yes

maximal, characteristic

C₂²×#11724784

C₂²

yes

—

#16812699

C₂²

yes

—

#16812691

C₂²

yes

—

#16812678

—

C₄×Dic₂₁₈

—

#16812658

C₂²

yes

—

#16812632

C₂²

yes

—

#16812630

C₂²

yes

—

#16812610

C₂²

yes

—

#16812610

C₂²

yes

—

C₄⋊Q₈×C₁₀₉

C₂²

yes

—

#125×C₁₀₉

C₂²

yes

—

C₂²⋊Q₈×C₁₀₉

—

C₄×Q₈×C₁₀₉

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

characteristic

C₂×Dic₂₁₈

—

C₂×#11724784

Q₈

yes

complemented

C₂×#11724784

C₂³

yes

—

C₄⋊Dic₁₀₉

—

C₄⋊Dic₁₀₉

C₂³

yes

—

#13182008

—

C₂²⋊Dic₁₀₉

—

C₄×Dic₁₀₉

—

C₂×Q₈×C₁₀₉

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

—

C₄⋊C₄×C₁₀₉

C₂³

yes

—

C₂²⋊C₄×C₁₀₉

—

C₄²×C₁₀₉

—

C₄²×C₁₀₉

D₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

characteristic

C₂³×C₁₀₉

C₂×D₄

yes

characteristic

Q₈×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

—

#3162

—

109

maximal, Sylow

C₁₀₉

#3162

yes

characteristic, complemented, Sylow

#533

—

109

—

#533

D₁₀₉

yes

characteristic

#501

—

109

—

#501

—

109

—

#427

—

109

—

C₂×#97

—

109

—

C₂²×C₈

—

109

—

C₄⋊Q₈

—

109

—

C₄⋊Q₈

C₂×D₁₀₉

yes

—

#125

—

109

—

#125

C₂×D₁₀₉

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

218

—

C₄×Q₈

—

C₄×Q₈

—

218

—

#117

—

109

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

characteristic

#107

—

109

—

#106

—

109

—

#97

—

109

—

#97

—

109

—

C₂×Q₈

—

C₂×Q₈

—

218

—

C₂²×C₄

C₂²⋊D₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

109

—

C₄⋊C₄

—

218

—

100

C₄⋊C₄

—

218

—

101

C₄⋊C₄

C₂²×D₁₀₉

yes

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

218

—

104

C₄²

—

105

C₄²

—

218

—

106

C₄²

C₂²⋊D₁₀₉

yes

—

107

C₄²

C₂²×D₁₀₉

yes

characteristic

108

C₄²

C₂²×D₁₀₉

yes

characteristic

109

C₂³

C₂×C₂²⋊D₁₀₉

yes

characteristic

110

Q₈

—

111

Q₈

—

436

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

218

—

115

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

116

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₀₉

yes

characteristic

118

C₂×C₄

Q₈×D₁₀₉

yes

—

119

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

120

C₈

—

109

—

121

C₂²

—

122

C₂²

#16812697

yes

—

123

C₂²

C₂²×C₂²⋊D₁₀₉

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

436

—

128

C₄

#16812697

yes

—

129

C₄

C₂×Q₈×D₁₀₉

yes

—

130

C₂

—

131

C₂

C₂×#16812697

yes

characteristic

132

C₂

#20295824

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₀₉
commutator, cosocle	C₂×C₄×C₁₀₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₀₉
Fitting	#533×C₁₀₉	C₂
socle	C₂²×C₁₀₉	C₂²×D₄
prime radical	C₂³×C₁₀₉	C₂×D₄
hypercenter	#533	D₁₀₉
hypocenter	C₁₀₉	#3162
LCS	G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₁₀₉

commutator, cosocle

C₂×C₄×C₁₀₉

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₀₉

Fitting

#533×C₁₀₉

C₂

socle

C₂²×C₁₀₉

C₂²×D₄

prime radical

C₂³×C₁₀₉

C₂×D₄

D₁₀₉

C₁₀₉

G ▷ C₂×C₄×C₁₀₉ ▷ C₂×C₁₀₉ ▷ C₁₀₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #20295436	C₂	semidirect
3. #20295432	C₂	semidirect
24. C₂×#11724784	Q₈	semidirect
67. C₁₀₉	#3162	semidirect

H ≅ G/N

type

2. #20295436

C₂

semidirect

3. #20295432

C₂

semidirect

24. C₂×#11724784

Q₈

semidirect

67. C₁₀₉

#3162

semidirect

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

subgroup	index
#20295583	2
#20295436	2
#20295432	2
#20295062	2
C₂×#16812610	2
#533×C₁₀₉	2
#3162	109

subgroup

index

#533×C₁₀₉

109

Subgroups — #28373554

Conjugacy classes of subgroups

Complements

Maximal subgroups