Group #28379876 — subgroups

Subgroups — #28379876

Order 13952.

402 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 168 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×#11724786	C₂	1	2	yes	maximal, complemented
2	#20296060	C₂	1	6	yes	maximal, complemented
3	#20296048	C₂	1	3	yes	maximal, complemented
4	C₄×#13182039	C₂	1	3	yes	maximal, complemented
5	C₄×#13182030	C₂	1	1	yes	maximal, characteristic
6	C₂×#13182039	C₂²	1	6	yes	—
7	C₄⋊#11724786	C₂²	1	6	yes	—
8	#16812756	C₂²	1	6	yes	—
9	#16812754	C₂²	1	1	yes	characteristic
10	#16812754	C₂²	1	3	yes	—
11	C₄×#11724786	C₂²	1	6	yes	—
12	C₂×#13182030	C₄	1	1	yes	characteristic
13	Q₈×Dic₁₀₉	C₂²	1	1	yes	characteristic
14	C₄×D₄₃₆	C₄	1	3	yes	—
15	#16812661	C₂²	1	3	yes	—
16	#16812657	C₄	1	3	yes	—
17	C₄²×D₁₀₉	C₂²	1	3	yes	—
18	C₄×Q₈×C₁₀₉	C₄	1	1	yes	characteristic
19	#13182039	—	2	12	no	—
20	C₂×#11724786	C₂³	1	6	yes	—
21	C₂×#11724786	C₂³	1	2	yes	—
22	#13182030	—	2	4	no	—
23	C₂×D₄₃₆	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₁₀₉	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₁₀₉	C₂³	1	3	yes	—
27	C₄⋊Dic₁₀₉	C₂×C₄	1	3	yes	—
28	#13182006	C₂×C₄	1	6	yes	—
29	C₄×Dic₁₀₉	C₂×C₄	1	3	yes	—
30	C₄×Dic₁₀₉	C₂³	1	1	yes	characteristic
31	C₄×Dic₁₀₉	C₂³	1	3	yes	—
32	C₂×Q₈×C₁₀₉	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₁₀₉	C₂×C₄	1	3	yes	—
34	C₄²×C₁₀₉	C₂×C₄	1	3	yes	—
35	C₂²×D₁₀₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₁₀₉	M₁₆	1	4	yes	complemented
37	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	3	yes	—
40	C₂×Dic₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×Dic₁₀₉	C₂²×C₄	1	3	yes	—
42	C₂×Dic₁₀₉	C₂⁴	1	1	yes	characteristic
43	D₄₃₆	—	2	6	no	—
44	C₄×D₁₀₉	—	2	6	no	—
45	C₄×D₁₀₉	—	2	6	no	—
46	C₄×D₁₀₉	—	4	3	no	—
47	#11724786	—	2	6	no	—
48	#11724786	#41	1	4	yes	complemented
49	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
50	C₂×D₁₀₉	—	2	3	no	—
51	C₂×D₁₀₉	—	2	3	no	—
52	C₂×D₁₀₉	—	2	3	no	—
53	C₄×C₁₀₉	—	2	3	no	—
54	C₄×C₁₀₉	—	2	1	no	—
55	C₄×C₁₀₉	C₂×M₁₆	1	6	yes	—
56	Dic₁₀₉	—	2	1	no	—
57	Dic₁₀₉	—	2	3	no	—
58	Dic₁₀₉	C₂×#41	1	2	yes	—
59	C₂×C₁₀₉	C₄×#41	1	1	yes	characteristic
60	C₂×C₁₀₉	C₂²×M₁₆	1	1	yes	characteristic
61	C₂×C₁₀₉	#568	1	1	yes	characteristic
62	D₁₀₉	—	4	3	no	—
63	#2971	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2971	1	1	yes	characteristic, complemented, Sylow
65	C₄×#41	—	109	1	no	—
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	—	109	1	no	—
71	C₂×M₁₆	—	109	6	no	—
72	C₄×Q₈	—	109	1	no	—
73	C₄×Q₈	#89137	1	1	yes	characteristic
74	C₄×D₄	—	109	3	no	—
75	C₄×D₄	—	109	3	no	—
76	#117	—	109	3	no	—
77	C₂×C₄²	—	109	3	no	—
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	—	218	4	no	—
84	C₂×Q₈	C₂×#89137	1	1	yes	characteristic
85	C₂×D₄	—	109	3	no	—
86	C₂²×C₄	—	109	3	no	—
87	C₂²×C₄	—	109	3	no	—
88	C₂²×C₄	—	109	3	no	—
89	M₁₆	—	218	12	no	—
90	C₂×C₈	—	109	2	no	—
91	C₂×C₈	—	109	6	no	—
92	C₄⋊C₄	—	109	3	no	—
93	C₄⋊C₄	C₂×#89137	1	3	yes	—
94	C₂²⋊C₄	—	109	6	no	—
95	C₄²	—	109	3	no	—
96	C₄²	—	109	3	no	—
97	C₄²	—	109	1	no	—
98	C₄²	C₂×#89137	1	3	yes	—
99	C₂³	—	109	3	no	—
100	Q₈	#13182039	1	4	yes	complemented
101	D₄	—	218	6	no	—
102	C₂×C₄	—	109	1	no	—
103	C₂×C₄	—	109	1	no	—
104	C₂×C₄	—	109	3	no	—
105	C₂×C₄	—	436	3	no	—
106	C₂×C₄	—	218	6	no	—
107	C₂×C₄	—	218	6	no	—
108	C₂×C₄	C₂²×#89137	1	1	yes	characteristic
109	C₂×C₄	C₂²×#89137	1	3	yes	—
110	C₂×C₄	C₂²×#89137	1	3	yes	—
111	C₈	—	109	4	no	—
112	C₈	—	218	6	no	—
113	C₂²	—	218	3	no	—
114	C₂²	—	218	3	no	—
115	C₂²	—	218	3	no	—
116	C₂²	C₂³×#89137	1	1	yes	characteristic
117	C₄	—	2	3	no	—
118	C₄	—	2	1	no	—
119	C₄	—	109	2	no	—
120	C₄	—	218	1	no	—
121	C₄	—	218	3	no	—
122	C₄	C₂×#13182039	1	6	yes	—
123	C₂	—	436	3	no	—
124	C₂	#41×#89137	1	1	yes	characteristic
125	C₂	C₂²×#13182039	1	1	yes	characteristic
126	C₂	#20296106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#11724786

C₂

yes

maximal, complemented

#20296060

C₂

yes

maximal, complemented

#20296048

C₂

yes

maximal, complemented

C₄×#13182039

C₂

yes

maximal, complemented

C₄×#13182030

C₂

yes

maximal, characteristic

C₂×#13182039

C₂²

yes

—

C₄⋊#11724786

C₂²

yes

—

#16812756

C₂²

yes

—

#16812754

C₂²

yes

characteristic

#16812754

C₂²

yes

—

C₄×#11724786

C₂²

yes

—

C₂×#13182030

C₄

yes

characteristic

Q₈×Dic₁₀₉

C₂²

yes

characteristic

C₄×D₄₃₆

C₄

yes

—

#16812661

C₂²

yes

—

#16812657

C₄

yes

—

C₄²×D₁₀₉

C₂²

yes

—

C₄×Q₈×C₁₀₉

C₄

yes

characteristic

#13182039

—

C₂×#11724786

C₂³

yes

—

C₂×#11724786

C₂³

yes

—

#13182030

—

C₂×D₄₃₆

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂×C₄

yes

—

C₂×C₄×D₁₀₉

C₂³

yes

—

C₄⋊Dic₁₀₉

C₂×C₄

yes

—

#13182006

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂×C₄

yes

—

C₄×Dic₁₀₉

C₂³

yes

characteristic

C₄×Dic₁₀₉

C₂³

yes

—

C₂×Q₈×C₁₀₉

C₂×C₄

yes

characteristic

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₂²×D₁₀₉

C₂²×C₄

yes

—

Q₈×C₁₀₉

M₁₆

yes

complemented

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂²×C₄

yes

characteristic

C₂×Dic₁₀₉

C₂²×C₄

yes

—

C₂×Dic₁₀₉

C₂⁴

yes

characteristic

D₄₃₆

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

C₄×D₁₀₉

—

—

yes

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₂×D₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×M₁₆

yes

—

Dic₁₀₉

—

Dic₁₀₉

—

Dic₁₀₉

C₂×#41

yes

—

C₂×C₁₀₉

C₄×#41

yes

characteristic

C₂×C₁₀₉

C₂²×M₁₆

yes

characteristic

C₂×C₁₀₉

#568

yes

characteristic

D₁₀₉

—

#2971

—

109

maximal, Sylow

C₁₀₉

#2971

yes

characteristic, complemented, Sylow

C₄×#41

—

109

—

C₈×Q₈

—

109

—

#436

—

109

—

#432

—

109

—

C₄×M₁₆

—

109

—

C₂×#41

—

109

—

C₂×M₁₆

—

109

—

C₄×Q₈

—

109

—

C₄×Q₈

#89137

yes

characteristic

C₄×D₄

—

109

—

C₄×D₄

—

109

—

#117

—

109

—

C₂×C₄²

—

109

—

C₄⋊C₈

—

109

—

C₂²⋊C₈

—

109

—

#97

—

109

—

#97

—

109

—

C₄×C₈

—

109

—

#41

—

218

—

C₂×Q₈

C₂×#89137

yes

characteristic

C₂×D₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

C₂²×C₄

—

109

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

109

—

C₄⋊C₄

—

109

—

C₄⋊C₄

C₂×#89137

yes

—

C₂²⋊C₄

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

—

109

—

C₄²

C₂×#89137

yes

—

C₂³

—

109

—

100

Q₈

#13182039

yes

complemented

101

D₄

—

218

—

102

C₂×C₄

—

109

—

103

C₂×C₄

—

109

—

104

C₂×C₄

—

109

—

105

C₂×C₄

—

436

—

106

C₂×C₄

—

218

—

107

C₂×C₄

—

218

—

108

C₂×C₄

C₂²×#89137

yes

characteristic

109

C₂×C₄

C₂²×#89137

yes

—

110

C₂×C₄

C₂²×#89137

yes

—

111

C₈

—

109

—

112

C₈

—

218

—

113

C₂²

—

218

—

114

C₂²

—

218

—

115

C₂²

—

218

—

116

C₂²

C₂³×#89137

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

109

—

120

C₄

—

218

—

121

C₄

—

218

—

122

C₄

C₂×#13182039

yes

—

123

C₂

—

436

—

124

C₂

#41×#89137

yes

characteristic

125

C₂

C₂²×#13182039

yes

characteristic

126

C₂

#20296106

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#89137
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₄×Q₈×C₁₀₉	C₄
prime radical	C₂×#13182030	C₄
cosocle	C₂×Dic₁₀₉	C₂⁴
hypercenter	C₄×Q₈	#89137
hypocenter	C₁₀₉	#2971
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#89137

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₄×Q₈×C₁₀₉

C₄

prime radical

C₂×#13182030

C₄

cosocle

C₂×Dic₁₀₉

C₂⁴

C₄×Q₈

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#11724786	C₂	semidirect
2. #20296060	C₂	semidirect
3. #20296048	C₂	semidirect
4. C₄×#13182039	C₂	semidirect
36. Q₈×C₁₀₉	M₁₆	semidirect
48. #11724786	#41	semidirect
64. C₁₀₉	#2971	semidirect
100. Q₈	#13182039	semidirect

H ≅ G/N

type

1. Q₈×#11724786

C₂

semidirect

2. #20296060

C₂

semidirect

3. #20296048

C₂

semidirect

4. C₄×#13182039

C₂

semidirect

36. Q₈×C₁₀₉

M₁₆

semidirect

48. #11724786

#41

semidirect

64. C₁₀₉

#2971

semidirect

100. Q₈

#13182039

semidirect

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

subgroup	index
Q₈×#11724786	2
#20296060	2
#20296048	2
C₄×#13182039	2
C₄×#13182030	2
#2971	109

subgroup

index

109

Subgroups — #28379876

Conjugacy classes of subgroups

Complements

Maximal subgroups