Group #28337305 — subgroups

Subgroups — #28337305

Order 13696.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#16812109	C₂	1	1	yes	maximal, characteristic
2	#20292651	C₂	1	1	yes	maximal, characteristic
3	C₂×#16812027	C₂	1	4	yes	maximal
4	C₄×M₁₆×C₁₀₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₂₁₄	C₂²	1	2	yes	—
6	C₂×#13133802	C₂²	1	2	yes	—
7	#16812109	—	2	2	no	—
8	#16812109	—	2	1	no	—
9	#16812109	—	2	1	no	—
10	C₂×C₄⋊Dic₁₀₇	C₄	1	4	yes	—
11	C₄⋊(C₁₀₇⋊C₈)	—	2	2	no	—
12	#16812027	—	2	8	no	—
13	C₂×M₁₆×C₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₀₇	C₂²	1	1	yes	characteristic
15	#97×C₁₀₇	—	2	1	no	—
16	C₄×C₈×C₁₀₇	—	2	1	no	—
17	C₂×Dic₂₁₄	—	2	2	no	—
18	C₂×Dic₂₁₄	—	2	2	no	—
19	C₂×Dic₂₁₄	—	2	4	no	—
20	C₂×Dic₂₁₄	—	2	4	no	—
21	C₂²×Dic₁₀₇	C₂×C₄	1	4	yes	—
22	#13133802	—	2	4	no	—
23	C₂×C₁₀₇⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₀₇	—	2	4	no	—
25	C₄⋊Dic₁₀₇	—	2	4	no	—
26	C₂²×C₄×C₁₀₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
28	M₁₆×C₁₀₇	—	2	2	no	—
29	M₁₆×C₁₀₇	D₄	1	4	yes	—
30	C₂×C₈×C₁₀₇	—	2	2	no	—
31	C₄²×C₁₀₇	D₄	1	2	yes	—
32	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
33	C₄²×C₁₀₇	D₄	1	1	yes	characteristic
34	C₂³×C₁₀₇	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₀₇	—	2	4	no	—
36	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₁₀₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₀₇	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₀₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₀₇	#41	1	2	yes	—
42	C₂×Dic₁₀₇	—	2	4	no	—
43	C₂×Dic₁₀₇	—	2	8	no	—
44	Dic₂₁₄	—	4	4	no	—
45	Dic₂₁₄	—	4	2	no	—
46	Dic₂₁₄	—	4	2	no	—
47	C₈×C₁₀₇	—	2	4	no	—
48	C₁₀₇⋊C₈	—	4	2	no	—
49	C₂²×C₁₀₇	—	2	2	no	—
50	C₂²×C₁₀₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₀₇	C₄×D₄	1	2	yes	—
52	C₄×C₁₀₇	—	2	2	no	—
53	C₄×C₁₀₇	#101	1	4	yes	complemented
54	C₄×C₁₀₇	#121	1	2	yes	—
55	C₄×C₁₀₇	#124	1	1	yes	characteristic
56	C₄×C₁₀₇	#127	1	1	yes	characteristic
57	Dic₁₀₇	—	4	4	no	—
58	C₂×C₁₀₇	—	2	2	no	—
59	C₂×C₁₀₇	#390	1	1	yes	characteristic
60	C₂×C₁₀₇	C₂×#101	1	2	yes	—
61	#1966	—	107	1	no	maximal, Sylow
62	C₁₀₇	#1966	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊Q₈	—	107	1	no	—
64	#423	—	107	1	no	—
65	C₂×#101	—	107	4	no	—
66	C₄×M₁₆	D₁₀₇	1	1	yes	characteristic
67	C₂²×Q₈	—	107	2	no	—
68	C₂×M₁₆	—	107	2	no	—
69	C₂×M₁₆	C₂×D₁₀₇	1	2	yes	—
70	C₄⋊Q₈	—	214	2	no	—
71	C₄⋊Q₈	—	214	1	no	—
72	C₄⋊Q₈	—	214	1	no	—
73	C₂×C₄⋊C₄	—	107	4	no	—
74	C₂×C₄²	C₂×D₁₀₇	1	1	yes	characteristic
75	C₄⋊C₈	—	214	2	no	—
76	#101	—	214	8	no	—
77	#97	—	2	1	no	—
78	C₄×C₈	—	2	1	no	—
79	C₂×Q₈	—	214	2	no	—
80	C₂×Q₈	—	214	4	no	—
81	C₂×Q₈	—	214	2	no	—
82	C₂×Q₈	—	214	4	no	—
83	C₂²×C₄	—	107	4	no	—
84	C₂²×C₄	C₄×D₁₀₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
86	M₁₆	—	2	2	no	—
87	M₁₆	—	214	4	no	—
88	M₁₆	D₄₂₈	1	4	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	214	2	no	—
91	C₄⋊C₄	—	214	4	no	—
92	C₄⋊C₄	—	214	4	no	—
93	C₄²	D₄₂₈	1	2	yes	—
94	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
95	C₄²	C₂²⋊D₁₀₇	1	1	yes	characteristic
96	C₂³	C₂×C₄×D₁₀₇	1	1	yes	characteristic
97	Q₈	—	428	4	no	—
98	Q₈	—	428	2	no	—
99	Q₈	—	428	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	214	4	no	—
102	C₂×C₄	—	214	8	no	—
103	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
104	C₂×C₄	(C₂×D₁₀₇)⋊C₄	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₀₇	1	1	yes	characteristic
106	C₂×C₄	C₂×D₄₂₈	1	2	yes	—
107	C₂×C₄	C₂×D₄₂₈	1	1	yes	characteristic
108	C₂×C₄	#13133818	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	428	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×(C₂×D₁₀₇)⋊C₄	1	1	yes	characteristic
113	C₂²	C₄×D₄₂₈	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	428	4	no	—
116	C₄	#16812027	1	4	yes	complemented
117	C₄	#16812083	1	2	yes	—
118	C₄	#16812097	1	1	yes	characteristic
119	C₄	#16812106	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#20292529	1	1	yes	characteristic
122	C₂	C₂×#16812027	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16812109

C₂

yes

maximal, characteristic

#20292651

C₂

yes

maximal, characteristic

C₂×#16812027

C₂

yes

maximal

C₄×M₁₆×C₁₀₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₁₄

C₂²

yes

—

C₂×#13133802

C₂²

yes

—

#16812109

—

#16812109

—

#16812109

—

C₂×C₄⋊Dic₁₀₇

C₄

yes

—

C₄⋊(C₁₀₇⋊C₈)

—

#16812027

—

C₂×M₁₆×C₁₀₇

C₂²

yes

—

C₂×C₄²×C₁₀₇

C₂²

yes

characteristic

#97×C₁₀₇

—

C₄×C₈×C₁₀₇

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂×Dic₂₁₄

—

C₂²×Dic₁₀₇

C₂×C₄

yes

—

#13133802

—

C₂×C₁₀₇⋊C₈

—

C₄⋊Dic₁₀₇

—

C₄⋊Dic₁₀₇

—

C₂²×C₄×C₁₀₇

C₂×C₄

yes

—

C₂²×C₄×C₁₀₇

C₂³

yes

characteristic

M₁₆×C₁₀₇

—

M₁₆×C₁₀₇

D₄

yes

—

C₂×C₈×C₁₀₇

—

C₄²×C₁₀₇

D₄

yes

—

C₄²×C₁₀₇

D₄

yes

characteristic

C₄²×C₁₀₇

D₄

yes

characteristic

C₂³×C₁₀₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

C₂×D₄

yes

—

C₂×C₄×C₁₀₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₀₇

#41

yes

—

C₂×Dic₁₀₇

—

C₂×Dic₁₀₇

—

Dic₂₁₄

—

Dic₂₁₄

—

Dic₂₁₄

—

C₈×C₁₀₇

—

C₁₀₇⋊C₈

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₀₇

C₄×D₄

yes

—

C₄×C₁₀₇

—

C₄×C₁₀₇

#101

yes

complemented

C₄×C₁₀₇

#121

yes

—

C₄×C₁₀₇

yes

C₄×C₁₀₇

yes

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

#390

yes

characteristic

C₂×C₁₀₇

C₂×#101

yes

—

#1966

—

107

maximal, Sylow

C₁₀₇

#1966

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

107

—

#423

—

107

—

C₂×#101

—

107

—

C₄×M₁₆

D₁₀₇

yes

characteristic

C₂²×Q₈

—

107

—

C₂×M₁₆

—

107

—

C₂×M₁₆

C₂×D₁₀₇

yes

—

C₄⋊Q₈

—

214

—

C₄⋊Q₈

—

214

—

C₄⋊Q₈

—

214

—

C₂×C₄⋊C₄

—

107

—

C₂×C₄²

C₂×D₁₀₇

yes

characteristic

C₄⋊C₈

—

214

—

#101

—

214

—

#97

—

C₄×C₈

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂×Q₈

—

214

—

C₂²×C₄

—

107

—

C₂²×C₄

C₄×D₁₀₇

yes

—

C₂²×C₄

C₂²×D₁₀₇

yes

characteristic

M₁₆

—

M₁₆

—

214

—

M₁₆

D₄₂₈

yes

—

C₂×C₈

—

C₂×C₈

—

214

—

C₄⋊C₄

—

214

—

C₄⋊C₄

—

214

—

C₄²

D₄₂₈

yes

—

C₄²

C₂²⋊D₁₀₇

yes

characteristic

C₄²

C₂²⋊D₁₀₇

yes

characteristic

C₂³

C₂×C₄×D₁₀₇

yes

characteristic

Q₈

—

428

—

Q₈

—

428

—

Q₈

—

428

—

100

C₂×C₄

—

101

C₂×C₄

—

214

—

102

C₂×C₄

—

214

—

103

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

104

C₂×C₄

(C₂×D₁₀₇)⋊C₄

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₀₇

yes

characteristic

106

C₂×C₄

C₂×D₄₂₈

yes

—

107

C₂×C₄

C₂×D₄₂₈

yes

characteristic

108

C₂×C₄

#13133818

yes

—

109

C₈

—

110

C₈

—

428

—

111

C₂²

—

112

C₂²

C₂×(C₂×D₁₀₇)⋊C₄

yes

characteristic

113

C₂²

C₄×D₄₂₈

yes

—

114

C₄

—

115

C₄

—

428

—

116

C₄

#16812027

yes

complemented

117

C₄

#16812083

yes

—

118

C₄

#16812097

yes

characteristic

119

C₄

#16812106

yes

characteristic

120

C₂

—

121

C₂

#20292529

yes

characteristic

122

C₂

C₂×#16812027

yes

—

	subgroup	G/H
center	C₂²	C₂×(C₂×D₁₀₇)⋊C₄
commutator, prime radical	C₂³×C₁₀₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₀₇
Fitting	C₄×M₁₆×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₀₇	C₂³
hypercenter	C₄×M₁₆	D₁₀₇
hypocenter	C₁₀₇	#1966
LCS	G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×(C₂×D₁₀₇)⋊C₄

commutator, prime radical

C₂³×C₁₀₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₀₇

Fitting

C₄×M₁₆×C₁₀₇

C₂

socle

C₂²×C₁₀₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₀₇

C₂³

hypercenter

C₄×M₁₆

D₁₀₇

hypocenter

C₁₀₇

#1966

LCS

G ▷ C₂³×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
53. C₄×C₁₀₇	#101	semidirect
62. C₁₀₇	#1966	semidirect
116. C₄	#16812027	semidirect

H ≅ G/N

type

53. C₄×C₁₀₇

#101

semidirect

62. C₁₀₇

#1966

semidirect

116. C₄

#16812027

semidirect

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

subgroup	index
C₂×#16812109	2
#20292651	2
C₂×#16812027	2
C₄×M₁₆×C₁₀₇	2
#1966	107

subgroup

index

C₄×M₁₆×C₁₀₇

107

Subgroups — #28337305

Conjugacy classes of subgroups

Complements

Maximal subgroups