Group #28335604 — subgroups

Subgroups — #28335604

Order 13696.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#20293343	C₂	1	1	yes	maximal, characteristic
2	#20292914	C₂	1	2	yes	maximal, complemented
3	#20292376	C₂	1	1	yes	maximal, characteristic, complemented
4	#20292356	C₂	1	2	yes	maximal, complemented
5	#494×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#16812098	—	2	2	no	—
7	#16812084	—	2	2	no	—
8	C₄×D₄₂₈	C₂²	1	2	yes	—
9	#16812067	—	2	2	no	—
10	#16812040	—	2	2	no	—
11	C₄⋊(C₁₀₇⋊C₈)	C₂²	1	2	yes	—
12	#16812029	—	2	2	no	—
13	C₂×Q₁₆×C₁₀₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₀₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
17	C₂×D₄₂₈	—	2	2	no	—
18	C₂×D₄₂₈	D₄	1	2	yes	—
19	C₂×C₄×D₁₀₇	—	2	2	no	—
20	C₂×C₁₀₇⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₀₇	D₄	1	2	yes	—
22	(C₂×D₁₀₇)⋊C₄	—	2	4	no	—
23	(C₂×D₁₀₇)⋊C₄	—	2	2	no	—
24	C₄×Dic₁₀₇	—	2	2	no	—
25	C₂×Q₈×C₁₀₇	—	2	1	no	—
26	C₂×Q₈×C₁₀₇	D₄	1	2	yes	—
27	Q₁₆×C₁₀₇	—	4	4	no	—
28	C₂×C₈×C₁₀₇	—	2	1	no	—
29	C₄⋊C₄×C₁₀₇	—	2	2	no	—
30	C₄⋊C₄×C₁₀₇	—	2	2	no	—
31	C₄²×C₁₀₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₀₇	—	2	2	no	—
33	Q₈×C₁₀₇	—	4	2	no	—
34	Q₈×C₁₀₇	—	4	2	no	—
35	C₂×C₄×C₁₀₇	—	2	2	no	—
36	C₂×C₄×C₁₀₇	—	4	1	no	—
37	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₁₀₇	—	2	2	no	—
40	D₄₂₈	—	2	2	no	—
41	D₄₂₈	—	2	2	no	—
42	D₄₂₈	—	8	2	no	—
43	C₄×D₁₀₇	—	4	2	no	—
44	C₄×D₁₀₇	—	4	2	no	—
45	C₈×C₁₀₇	—	2	2	no	—
46	C₁₀₇⋊C₈	—	4	2	no	—
47	C₂²×C₁₀₇	#120	1	1	yes	characteristic
48	C₂×D₁₀₇	—	4	2	no	—
49	C₂×D₁₀₇	—	4	2	no	—
50	C₂×D₁₀₇	—	4	2	no	—
51	C₄×C₁₀₇	—	2	1	no	—
52	C₄×C₁₀₇	—	4	2	no	—
53	C₄×C₁₀₇	—	8	1	no	—
54	C₄×C₁₀₇	#135	1	2	yes	—
55	C₄×C₁₀₇	#137	1	2	yes	—
56	Dic₁₀₇	—	4	2	no	—
57	C₂×C₁₀₇	#452	1	2	yes	—
58	C₂×C₁₀₇	#453	1	1	yes	characteristic
59	D₁₀₇	—	8	2	no	—
60	#1634	—	107	1	no	maximal, Sylow
61	C₁₀₇	#1634	1	1	yes	characteristic, complemented, Sylow
62	#555	—	107	1	no	—
63	#494	D₁₀₇	1	1	yes	characteristic, complemented
64	#476	—	107	2	no	—
65	#333	—	107	1	no	—
66	D₄⋊C₈	—	107	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₁₀₇	1	1	yes	characteristic
70	#124	—	214	2	no	—
71	#121	—	214	2	no	—
72	C₄×D₄	—	107	2	no	—
73	#117	—	214	2	no	—
74	#106	—	214	2	no	—
75	C₄⋊C₈	—	107	2	no	—
76	#102	—	214	2	no	—
77	C₄×C₈	C₂×D₁₀₇	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₁₀₇	1	2	yes	—
80	C₂×D₄	—	107	2	no	—
81	C₂×D₄	—	214	2	no	—
82	C₂²×C₄	—	214	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	214	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	107	2	no	—
89	C₂²⋊C₄	—	214	4	no	—
90	C₂²⋊C₄	—	214	2	no	—
91	C₄²	—	214	2	no	—
92	C₄²	C₂²×D₁₀₇	1	1	yes	characteristic
93	C₂³	—	214	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	856	2	no	—
97	D₄	—	214	2	no	—
98	D₄	—	214	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	428	2	no	—
102	C₂×C₄	—	428	2	no	—
103	C₂×C₄	—	214	2	no	—
104	C₂×C₄	D₄×D₁₀₇	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	428	2	no	—
108	C₂²	—	428	2	no	—
109	C₂²	—	428	2	no	—
110	C₂²	—	428	2	no	—
111	C₂²	#16812077	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	428	2	no	—
116	C₄	#16812132	1	2	yes	—
117	C₄	#16812141	1	2	yes	—
118	C₂	—	856	2	no	—
119	C₂	#20292788	1	2	yes	—
120	C₂	#20292794	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20293343

C₂

yes

maximal, characteristic

#20292914

C₂

yes

maximal, complemented

#20292376

C₂

yes

maximal, characteristic, complemented

#20292356

C₂

yes

maximal, complemented

#494×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#16812098

—

#16812084

—

C₄×D₄₂₈

C₂²

yes

—

#16812067

—

#16812040

—

C₄⋊(C₁₀₇⋊C₈)

C₂²

yes

—

#16812029

—

C₂×Q₁₆×C₁₀₇

—

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₀₇

C₂²

yes

characteristic

C₄×C₈×C₁₀₇

C₂²

yes

characteristic

C₂×D₄₂₈

—

C₂×D₄₂₈

D₄

yes

—

C₂×C₄×D₁₀₇

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

D₄

yes

—

(C₂×D₁₀₇)⋊C₄

—

(C₂×D₁₀₇)⋊C₄

—

C₄×Dic₁₀₇

—

C₂×Q₈×C₁₀₇

—

C₂×Q₈×C₁₀₇

D₄

yes

—

Q₁₆×C₁₀₇

—

C₂×C₈×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄⋊C₄×C₁₀₇

—

C₄²×C₁₀₇

C₂³

yes

characteristic

C₂²×D₁₀₇

—

Q₈×C₁₀₇

—

Q₈×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×Dic₁₀₇

—

D₄₂₈

—

D₄₂₈

—

D₄₂₈

—

C₄×D₁₀₇

—

C₄×D₁₀₇

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

#120

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#135

yes

—

C₄×C₁₀₇

#137

yes

—

Dic₁₀₇

—

C₂×C₁₀₇

#452

yes

—

C₂×C₁₀₇

#453

yes

characteristic

D₁₀₇

—

#1634

—

107

maximal, Sylow

C₁₀₇

#1634

yes

characteristic, complemented, Sylow

#555

—

107

—

#494

D₁₀₇

yes

characteristic, complemented

#476

—

107

—

#333

—

107

—

D₄⋊C₈

—

107

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₀₇

yes

characteristic

#124

—

214

—

#121

—

214

—

C₄×D₄

—

107

—

#117

—

214

—

#106

—

214

—

C₄⋊C₈

—

107

—

#102

—

214

—

C₄×C₈

C₂×D₁₀₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₀₇

yes

—

C₂×D₄

—

107

—

C₂×D₄

—

214

—

C₂²×C₄

—

214

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

214

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

107

—

C₂²⋊C₄

—

214

—

C₂²⋊C₄

—

214

—

C₄²

—

214

—

C₄²

C₂²×D₁₀₇

yes

characteristic

C₂³

—

214

—

Q₈

—

Q₈

—

D₄

—

856

—

D₄

—

214

—

D₄

—

214

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

428

—

102

C₂×C₄

—

428

—

103

C₂×C₄

—

214

—

104

C₂×C₄

D₄×D₁₀₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

106

C₈

—

107

C₈

—

428

—

108

C₂²

—

428

—

109

C₂²

—

428

—

110

C₂²

—

428

—

111

C₂²

#16812077

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

428

—

116

C₄

#16812132

yes

—

117

C₄

#16812141

yes

—

118

C₂

—

856

—

119

C₂

#20292788

yes

—

120

C₂

#20292794

yes

characteristic

	subgroup	G/H
center	C₂²	#16812077
commutator, cosocle	C₄²×C₁₀₇	C₂³
Frattini	C₄²	C₂²×D₁₀₇
Fitting	#494×C₁₀₇	C₂
socle	C₂²×C₁₀₇	#120
prime radical	#20293343	C₂
hypercenter	#494	D₁₀₇
hypocenter	C₁₀₇	#1634
LCS	G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#16812077

commutator, cosocle

C₄²×C₁₀₇

C₂³

Frattini

C₄²

C₂²×D₁₀₇

Fitting

#494×C₁₀₇

C₂

C₂²×C₁₀₇

C₂

D₁₀₇

C₁₀₇

G ▷ C₄²×C₁₀₇ ▷ C₂²×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #20292914	C₂	semidirect
3. #20292376	C₂	semidirect
4. #20292356	C₂	semidirect
5. #494×C₁₀₇	C₂	semidirect
61. C₁₀₇	#1634	semidirect
63. #494	D₁₀₇	semidirect

H ≅ G/N

type

2. #20292914

C₂

semidirect

3. #20292376

C₂

semidirect

4. #20292356

C₂

semidirect

5. #494×C₁₀₇

C₂

semidirect

61. C₁₀₇

#1634

semidirect

63. #494

D₁₀₇

semidirect

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

subgroup	index
#20293343	2
#20292914	2
#20292376	2
#20292356	2
#494×C₁₀₇	2
#1634	107

subgroup

index

#494×C₁₀₇

107

Subgroups — #28335604

Conjugacy classes of subgroups

Complements

Maximal subgroups