Group #28371596 — subgroups

Subgroups — #28371596

Order 13952.

484 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 293 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×#13182013	C₂	1	2	yes	maximal
2	#20295053	C₂	1	8	yes	maximal, complemented
3	#20295050	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₁₀₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13182013	—	2	8	no	—
6	C₂×#13182013	C₂²	1	4	yes	—
7	C₂×#13182013	C₂²	1	4	yes	—
8	C₂²×#11724784	C₂²	1	4	yes	—
9	C₄⋊#11724784	C₂²	1	16	yes	complemented
10	C₂³×C₄×C₁₀₉	C₂²	1	1	yes	characteristic
11	#117×C₁₀₉	C₄	1	4	yes	—
12	#117×C₁₀₉	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₁₀₉	C₄	1	2	yes	—
14	C₂×C₂²⋊C₄×C₁₀₉	C₄	1	2	yes	—
15	C₂×C₄²×C₁₀₉	C₂²	1	2	yes	—
16	#13182013	—	2	8	no	—
17	#13182013	—	2	8	no	—
18	C₂×#11724784	—	2	8	no	—
19	C₂×#11724784	C₂³	1	8	yes	—
20	C₂⁴×C₁₀₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₀₉	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₁₀₉	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₀₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₀₉	D₄	1	4	yes	—
25	C₂²×C₄×C₁₀₉	Q₈	1	4	yes	—
26	C₂²×C₄×C₁₀₉	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₁₀₉	C₂³	1	2	yes	—
28	C₄⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
29	C₂²⋊C₄×C₁₀₉	C₂×C₄	1	8	yes	—
30	C₄²×C₁₀₉	C₂×C₄	1	4	yes	—
31	C₄²×C₁₀₉	C₂³	1	4	yes	—
32	C₂³×C₁₀₉	—	4	2	no	—
33	C₂³×C₁₀₉	C₄⋊C₄	1	4	yes	—
34	C₂³×C₁₀₉	C₂²×C₄	1	2	yes	—
35	C₂³×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₀₉	—	4	4	no	—
37	C₂×C₄×C₁₀₉	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₁₀₉	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₁₀₉	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₁₀₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₉	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₁₀₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₁₀₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₀₉	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₁₀₉	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₁₀₉	C₂⁴	1	1	yes	characteristic
47	#11724784	—	2	8	no	—
48	C₂²×C₁₀₉	—	2	2	no	—
49	C₂²×C₁₀₉	—	2	4	no	—
50	C₂²×C₁₀₉	—	4	4	no	—
51	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
52	C₂²×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₁₀₉	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₁₀₉	—	2	8	no	—
55	C₄×C₁₀₉	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₁₀₉	C₂×C₄⋊C₄	1	4	yes	—
57	C₄×C₁₀₉	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₀₉	C₂²×Q₈	1	1	yes	characteristic
59	C₂×C₁₀₉	—	2	2	no	—
60	C₂×C₁₀₉	—	2	4	no	—
61	C₂×C₁₀₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₁₀₉	#568	1	2	yes	—
63	#2905	—	109	1	no	maximal, Sylow
64	C₁₀₉	#2905	1	1	yes	characteristic, complemented, Sylow
65	C₂²×M₁₆	—	109	2	no	—
66	C₂×#117	D₁₀₉	1	1	yes	characteristic
67	#424	—	109	8	no	—
68	#423	—	109	4	no	—
69	C₂³×C₄	C₂×D₁₀₉	1	1	yes	characteristic
70	C₂×M₁₆	—	109	4	no	—
71	C₂×M₁₆	—	109	4	no	—
72	C₂×M₁₆	—	218	8	no	—
73	C₂²×C₈	—	109	4	no	—
74	#117	Dic₁₀₉	1	4	yes	—
75	#117	C₂×D₁₀₉	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₁₀₉	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₁₀₉	1	2	yes	—
78	C₂×C₄²	C₂×D₁₀₉	1	2	yes	—
79	C₄⋊C₈	—	109	16	no	—
80	C₂⁴	C₂×Dic₁₀₉	1	1	yes	characteristic
81	C₂²×C₄	Dic₂₁₈	1	4	yes	—
82	C₂²×C₄	D₄₃₆	1	4	yes	—
83	C₂²×C₄	C₂×Dic₁₀₉	1	4	yes	—
84	C₂²×C₄	C₂×Dic₁₀₉	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₁₀₉	1	2	yes	—
86	C₂²×C₄	C₂²×D₁₀₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₀₉	1	2	yes	—
88	M₁₆	—	218	8	no	—
89	M₁₆	—	218	8	no	—
90	C₂×C₈	—	109	8	no	—
91	C₂×C₈	—	218	8	no	—
92	C₄⋊C₄	C₂×Dic₁₀₉	1	8	yes	—
93	C₂²⋊C₄	C₂×Dic₁₀₉	1	8	yes	—
94	C₄²	C₂×Dic₁₀₉	1	4	yes	—
95	C₄²	C₂²×D₁₀₉	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊Dic₁₀₉	1	4	yes	—
98	C₂³	C₂²×Dic₁₀₉	1	2	yes	—
99	C₂³	C₂²×Dic₁₀₉	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊Dic₁₀₉	1	4	yes	—
102	C₂×C₄	C₄⋊Dic₁₀₉	1	8	yes	—
103	C₂×C₄	C₂²×Dic₁₀₉	1	8	yes	—
104	C₂×C₄	C₂²×Dic₁₀₉	1	1	yes	characteristic
105	C₂×C₄	C₂²×Dic₁₀₉	1	2	yes	—
106	C₂×C₄	C₂×D₄₃₆	1	4	yes	—
107	C₂×C₄	C₂×D₄₃₆	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₁₈	1	2	yes	—
109	C₂×C₄	C₂×Dic₂₁₈	1	4	yes	—
110	C₂×C₄	C₂³×D₁₀₉	1	1	yes	characteristic
111	C₈	—	218	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊Dic₁₀₉	1	4	yes	—
116	C₂²	C₂×C₄⋊Dic₁₀₉	1	2	yes	—
117	C₂²	C₂³×Dic₁₀₉	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊Dic₁₀₉	1	2	yes	—
120	C₄	C₂×C₄⋊Dic₁₀₉	1	4	yes	—
121	C₄	C₂²×D₄₃₆	1	1	yes	characteristic
122	C₄	C₂²×Dic₂₁₈	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊Dic₁₀₉	1	1	yes	characteristic
126	C₂	#20295793	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#13182013

C₂

yes

maximal

#20295053

C₂

yes

maximal, complemented

#20295050

C₂

yes

maximal, complemented

C₂×#117×C₁₀₉

C₂

yes

maximal, characteristic

C₂×#13182013

—

C₂×#13182013

C₂²

yes

—

C₂×#13182013

C₂²

yes

—

C₂²×#11724784

C₂²

yes

—

C₄⋊#11724784

C₂²

yes

complemented

C₂³×C₄×C₁₀₉

C₂²

yes

characteristic

#117×C₁₀₉

C₄

yes

—

#117×C₁₀₉

C₂²

yes

—

C₂×C₄⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₂²⋊C₄×C₁₀₉

C₄

yes

—

C₂×C₄²×C₁₀₉

C₂²

yes

—

—

—

—

C₂³

yes

—

C₂⁴×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₀₉

C₂×C₄

yes

—

C₂²×C₄×C₁₀₉

D₄

yes

—

C₂²×C₄×C₁₀₉

Q₈

yes

—

C₂²×C₄×C₁₀₉

C₂³

yes

characteristic

C₂²×C₄×C₁₀₉

C₂³

yes

—

C₄⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₂²⋊C₄×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂×C₄

yes

—

C₄²×C₁₀₉

C₂³

yes

—

C₂³×C₁₀₉

—

C₂³×C₁₀₉

C₄⋊C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

—

C₂³×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₄⋊C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₉

C₂²×C₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×D₄

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂×Q₈

yes

—

C₂×C₄×C₁₀₉

C₂⁴

yes

characteristic

#11724784

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₂²×C₁₀₉

C₂³×C₄

yes

characteristic

C₄×C₁₀₉

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂×C₄⋊C₄

yes

—

C₄×C₁₀₉

C₂²×D₄

yes

characteristic

C₄×C₁₀₉

C₂²×Q₈

yes

characteristic

C₂×C₁₀₉

—

C₂×C₁₀₉

—

C₂×C₁₀₉

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₁₀₉

#568

yes

—

#2905

—

109

maximal, Sylow

C₁₀₉

#2905

yes

characteristic, complemented, Sylow

C₂²×M₁₆

—

109

—

C₂×#117

D₁₀₉

yes

characteristic

#424

—

109

—

#423

—

109

—

C₂³×C₄

C₂×D₁₀₉

yes

characteristic

C₂×M₁₆

—

109

—

C₂×M₁₆

—

109

—

C₂×M₁₆

—

218

—

C₂²×C₈

—

109

—

#117

Dic₁₀₉

yes

—

#117

C₂×D₁₀₉

yes

—

C₂×C₄⋊C₄

Dic₁₀₉

yes

—

C₂×C₂²⋊C₄

Dic₁₀₉

yes

—

C₂×C₄²

C₂×D₁₀₉

yes

—

C₄⋊C₈

—

109

—

C₂⁴

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

Dic₂₁₈

yes

—

C₂²×C₄

D₄₃₆

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂×Dic₁₀₉

yes

characteristic

C₂²×C₄

C₂×Dic₁₀₉

yes

—

C₂²×C₄

C₂²×D₁₀₉

yes

characteristic

C₂²×C₄

C₂²×D₁₀₉

yes

—

M₁₆

—

218

—

M₁₆

—

218

—

C₂×C₈

—

109

—

C₂×C₈

—

218

—

C₄⋊C₄

C₂×Dic₁₀₉

yes

—

C₂²⋊C₄

C₂×Dic₁₀₉

yes

—

C₄²

C₂×Dic₁₀₉

yes

—

C₄²

C₂²×D₁₀₉

yes

—

C₂³

—

C₂³

C₄⋊Dic₁₀₉

yes

—

C₂³

C₂²×Dic₁₀₉

yes

—

C₂³

C₂²×Dic₁₀₉

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₁₀₉

yes

—

102

C₂×C₄

C₄⋊Dic₁₀₉

yes

—

103

C₂×C₄

C₂²×Dic₁₀₉

yes

—

104

C₂×C₄

C₂²×Dic₁₀₉

yes

characteristic

105

C₂×C₄

C₂²×Dic₁₀₉

yes

—

106

C₂×C₄

C₂×D₄₃₆

yes

—

107

C₂×C₄

C₂×D₄₃₆

yes

—

108

C₂×C₄

C₂×Dic₂₁₈

yes

—

109

C₂×C₄

C₂×Dic₂₁₈

yes

—

110

C₂×C₄

C₂³×D₁₀₉

yes

characteristic

111

C₈

—

218

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊Dic₁₀₉

yes

—

116

C₂²

C₂×C₄⋊Dic₁₀₉

yes

—

117

C₂²

C₂³×Dic₁₀₉

yes

characteristic

118

C₄

—

119

C₄

C₂×C₄⋊Dic₁₀₉

yes

—

120

C₄

C₂×C₄⋊Dic₁₀₉

yes

—

121

C₄

C₂²×D₄₃₆

yes

characteristic

122

C₄

C₂²×Dic₂₁₈

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂²×C₄⋊Dic₁₀₉

yes

characteristic

126

C₂

#20295793

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₀₉
commutator, socle	C₂²×C₁₀₉	C₂³×C₄
Fitting	C₂×#117×C₁₀₉	C₂
prime radical	C₂⁴×C₁₀₉	C₂×C₄
cosocle	C₂×C₄×C₁₀₉	C₂⁴
hypercenter	C₂×#117	D₁₀₉
hypocenter	C₁₀₉	#2905
LCS	G ▷ C₂²×C₁₀₉ ▷ C₁₀₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₀₉

commutator, socle

C₂²×C₁₀₉

C₂³×C₄

Fitting

C₂×#117×C₁₀₉

C₂

prime radical

C₂⁴×C₁₀₉

C₂×C₄

cosocle

C₂×C₄×C₁₀₉

C₂⁴

D₁₀₉

C₁₀₉

G ▷ C₂²×C₁₀₉ ▷ C₁₀₉

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. #20295053	C₂	semidirect
3. #20295050	C₂	semidirect
9. C₄⋊#11724784	C₂²	semidirect
64. C₁₀₉	#2905	semidirect

H ≅ G/N

type

2. #20295053

C₂

semidirect

3. #20295050

C₂

semidirect

9. C₄⋊#11724784

C₂²

semidirect

64. C₁₀₉

#2905

semidirect

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

subgroup	index
C₂²×#13182013	2
#20295053	2
#20295050	2
C₂×#117×C₁₀₉	2
#2905	109

subgroup

index

C₂×#117×C₁₀₉

109

Subgroups — #28371596

Conjugacy classes of subgroups

Complements

Maximal subgroups