Group #28372052 — subgroups

Subgroups — #28372052

Order 13952.

402 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×#11724784	C₂	1	2	yes	maximal, complemented
2	#20295104	C₂	1	6	yes	maximal, complemented
3	#20295083	C₂	1	3	yes	maximal, complemented
4	C₄×#13182013	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₁₀₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182013	C₂²	1	6	yes	—
7	C₄⋊#11724784	C₂²	1	6	yes	—
8	C₂²⋊#11724784	C₂²	1	6	yes	—
9	#16812610	C₂²	1	1	yes	characteristic
10	#16812610	C₂²	1	3	yes	—
11	C₄×#11724784	C₂²	1	6	yes	—
12	C₂×#41×C₁₀₉	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₁₀₉	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₁₀₉	C₄	1	3	yes	—
16	C₄×D₄×C₁₀₉	C₂²	1	3	yes	—
17	#117×C₁₀₉	C₄	1	3	yes	—
18	C₂×C₄²×C₁₀₉	C₂²	1	3	yes	—
19	#13182013	—	2	12	no	—
20	C₂×#11724784	C₂³	1	6	yes	—
21	C₂×#11724784	C₂³	1	2	yes	—
22	#41×C₁₀₉	—	2	4	no	—
23	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₁₀₉	C₂×C₄	1	3	yes	—
25	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
26	C₂²×C₄×C₁₀₉	C₂×C₄	1	3	yes	—
27	C₂²×C₄×C₁₀₉	C₂³	1	3	yes	—
28	C₄⋊C₄×C₁₀₉	C₂×C₄	1	3	yes	—
29	C₄⋊C₄×C₁₀₉	C₂×C₄	1	3	yes	—
30	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	6	yes	—
31	C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
32	C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
33	C₄²×C₁₀₉	C₂³	1	1	yes	characteristic
34	C₄²×C₁₀₉	C₂³	1	3	yes	—
35	C₂³×C₁₀₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₁₀₉	M₁₆	1	4	yes	complemented
37	D₄×C₁₀₉	—	2	6	no	—
38	C₂×C₄×C₁₀₉	—	2	6	no	—
39	C₂×C₄×C₁₀₉	—	2	6	no	—
40	C₂×C₄×C₁₀₉	—	4	3	no	—
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
47	#11724784	—	2	6	no	—
48	#11724784	#41	1	4	yes	complemented
49	C₂²×C₁₀₉	—	2	3	no	—
50	C₂²×C₁₀₉	—	2	3	no	—
51	C₂²×C₁₀₉	—	2	3	no	—
52	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₁₀₉	—	2	3	no	—
54	C₄×C₁₀₉	—	2	1	no	—
55	C₄×C₁₀₉	—	2	1	no	—
56	C₄×C₁₀₉	—	2	3	no	—
57	C₄×C₁₀₉	C₂×M₁₆	1	6	yes	—
58	C₄×C₁₀₉	C₂×#41	1	2	yes	—
59	C₂×C₁₀₉	—	4	3	no	—
60	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
61	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
62	C₂×C₁₀₉	#568	1	1	yes	characteristic
63	#2971	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2971	1	1	yes	characteristic, complemented, Sylow
65	C₄×#41	D₁₀₉	1	1	yes	characteristic
66	C₈×Q₈	—	109	2	no	—
67	#436	—	109	6	no	—
68	#432	—	109	3	no	—
69	C₄×M₁₆	—	109	3	no	—
70	C₂×#41	Dic₁₀₉	1	1	yes	characteristic
71	C₂×M₁₆	—	109	6	no	—
72	C₄×Q₈	Dic₁₀₉	1	1	yes	characteristic
73	C₄×Q₈	C₂×D₁₀₉	1	1	yes	characteristic
74	C₄×D₄	Dic₁₀₉	1	3	yes	—
75	C₄×D₄	C₂×D₁₀₉	1	3	yes	—
76	#117	Dic₁₀₉	1	3	yes	—
77	C₂×C₄²	C₂×D₁₀₉	1	3	yes	—
78	C₄⋊C₈	—	109	6	no	—
79	C₂²⋊C₈	—	109	6	no	—
80	#97	—	109	1	no	—
81	#97	—	109	3	no	—
82	C₄×C₈	—	109	6	no	—
83	#41	—	2	4	no	—
84	C₂×Q₈	C₂×Dic₁₀₉	1	1	yes	characteristic
85	C₂×D₄	C₂×Dic₁₀₉	1	3	yes	—
86	C₂²×C₄	C₂×Dic₁₀₉	1	3	yes	—
87	C₂²×C₄	C₂×Dic₁₀₉	1	3	yes	—
88	C₂²×C₄	C₂²×D₁₀₉	1	3	yes	—
89	M₁₆	—	218	12	no	—
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	109	6	no	—
92	C₄⋊C₄	C₂×Dic₁₀₉	1	3	yes	—
93	C₄⋊C₄	C₂×Dic₁₀₉	1	3	yes	—
94	C₂²⋊C₄	C₂×Dic₁₀₉	1	6	yes	—
95	C₄²	C₂×Dic₁₀₉	1	3	yes	—
96	C₄²	C₂×Dic₁₀₉	1	3	yes	—
97	C₄²	C₂²×D₁₀₉	1	3	yes	—
98	C₄²	C₂²×D₁₀₉	1	1	yes	characteristic
99	C₂³	C₂²×Dic₁₀₉	1	3	yes	—
100	Q₈	#13182013	1	4	yes	complemented
101	D₄	—	2	6	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	2	6	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
106	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
107	C₂×C₄	C₂²×Dic₁₀₉	1	3	yes	—
108	C₂×C₄	C₂²×Dic₁₀₉	1	3	yes	—
109	C₂×C₄	C₂²×Dic₁₀₉	1	3	yes	—
110	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
111	C₈	—	109	4	no	—
112	C₈	—	218	6	no	—
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	—	2	3	no	—
116	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
117	C₄	—	2	3	no	—
118	C₄	—	2	1	no	—
119	C₄	—	2	1	no	—
120	C₄	—	2	3	no	—
121	C₄	C₂×#13182013	1	6	yes	—
122	C₄	#41×D₁₀₉	1	2	yes	—
123	C₂	—	4	3	no	—
124	C₂	#41×Dic₁₀₉	1	1	yes	characteristic
125	C₂	C₂²×#13182013	1	1	yes	characteristic
126	C₂	#20295793	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11724784

C₂

yes

maximal, complemented

#20295104

C₂

yes

maximal, complemented

#20295083

C₂

yes

maximal, complemented

C₄×#13182013

C₂

yes

maximal, complemented

C₄×#41×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

—

C₂²⋊#11724784

C₂²

yes

—

#16812610

C₂²

yes

characteristic

#16812610

C₂²

yes

—

C₄×#11724784

C₂²

yes

—

C₂×#41×C₁₀₉

C₄

yes

characteristic

C₄×Q₈×C₁₀₉

C₄

yes

characteristic

C₄×Q₈×C₁₀₉

C₂²

yes

characteristic

C₄×D₄×C₁₀₉

C₄

yes

—

C₄×D₄×C₁₀₉

C₂²

yes

—

#117×C₁₀₉

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

#13182013

—

C₂×#11724784

C₂³

yes

—

C₂×#11724784

C₂³

yes

—

#41×C₁₀₉

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₂×D₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂³

yes

characteristic

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

—

Q₈×C₁₀₉

M₁₆

yes

complemented

D₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

—

yes

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

C₄×C₁₀₉

C₂×#41

yes

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₄×#41

yes

characteristic

C₂×C₁₀₉

C₂²×M₁₆

yes

C₂×C₁₀₉

yes

—

109

maximal, Sylow

C₁₀₉

#2971

yes

characteristic, complemented, Sylow

C₄×#41

D₁₀₉

yes

characteristic

C₈×Q₈

—

109

—

#436

—

109

—

#432

—

109

—

C₄×M₁₆

—

109

—

C₂×#41

Dic₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₄×Q₈

Dic₁₀₉

yes

characteristic

C₄×Q₈

C₂×D₁₀₉

yes

characteristic

C₄×D₄

Dic₁₀₉

yes

—

C₄×D₄

C₂×D₁₀₉

yes

—

#117

Dic₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

#41

—

C₂×Q₈

C₂×Dic₁₀₉

yes

characteristic

C₂×D₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₂²⋊C₄

C₂×Dic₁₀₉

yes

—

C₄²

C₂×Dic₁₀₉

yes

—

C₄²

C₂×Dic₁₀₉

yes

—

C₄²

C₂²×D₁₀₉

yes

—

C₄²

C₂²×D₁₀₉

yes

characteristic

C₂³

C₂²×Dic₁₀₉

yes

—

100

Q₈

#13182013

yes

complemented

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

106

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

107

C₂×C₄

C₂²×Dic₁₀₉

yes

—

108

C₂×C₄

C₂²×Dic₁₀₉

yes

—

109

C₂×C₄

C₂²×Dic₁₀₉

yes

—

110

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

111

C₈

—

109

—

112

C₈

—

218

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×Dic₁₀₉

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×#13182013

yes

—

122

C₄

#41×D₁₀₉

yes

—

123

C₂

—

124

C₂

#41×Dic₁₀₉

yes

characteristic

125

C₂

C₂²×#13182013

yes

characteristic

126

C₂

#20295793

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄×#41×C₁₀₉	C₂
prime radical	C₂×#41×C₁₀₉	C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₄×#41	D₁₀₉
hypocenter	C₁₀₉	#2971
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₄×#41×C₁₀₉

C₂

prime radical

C₂×#41×C₁₀₉

C₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

hypercenter

C₄×#41

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×#11724784	C₂	semidirect
2. #20295104	C₂	semidirect
3. #20295083	C₂	semidirect
4. C₄×#13182013	C₂	semidirect
36. Q₈×C₁₀₉	M₁₆	semidirect
48. #11724784	#41	semidirect
64. C₁₀₉	#2971	semidirect
100. Q₈	#13182013	semidirect

H ≅ G/N

type

1. Q₈×#11724784

C₂

semidirect

2. #20295104

C₂

semidirect

3. #20295083

C₂

semidirect

4. C₄×#13182013

C₂

semidirect

36. Q₈×C₁₀₉

M₁₆

semidirect

48. #11724784

#41

semidirect

64. C₁₀₉

#2971

semidirect

100. Q₈

#13182013

semidirect

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

subgroup	index
Q₈×#11724784	2
#20295104	2
#20295083	2
C₄×#13182013	2
C₄×#41×C₁₀₉	2
#2971	109

subgroup

index

C₄×#41×C₁₀₉

109

Subgroups — #28372052

Conjugacy classes of subgroups

Complements

Maximal subgroups