Group #20293138 — subgroups

Subgroups — #20293138

Order 6848.

330 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 119 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×#13133818	C₂	1	1	yes	maximal, characteristic, complemented
2	#16812092	C₂	1	2	yes	maximal, complemented
3	#16812088	C₂	1	2	yes	maximal, complemented
4	#16812083	C₂	1	2	yes	maximal, complemented
5	#16812078	C₂	1	2	yes	maximal, complemented
6	C₄×C₂²⋊D₁₀₇	C₂	1	4	yes	maximal, complemented
7	#16812068	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₄×C₁₀₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#13133818	—	2	4	no	—
10	C₂×Dic₂₁₄	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄₂₈	C₂²	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊D₁₀₇	C₂²	1	4	yes	—
13	C₂×C₂²⋊D₁₀₇	C₂²	1	2	yes	—
14	C₂×C₄×D₁₀₇	C₂²	1	2	yes	complemented
15	C₄⋊Dic₁₀₇	C₂²	1	2	yes	complemented
16	Dic₁₀₇⋊C₄	C₂²	1	4	yes	complemented
17	C₂²⋊Dic₁₀₇	C₂²	1	2	yes	—
18	C₂²⋊Dic₁₀₇	C₂²	1	4	yes	—
19	(C₂×D₁₀₇)⋊C₄	C₂²	1	4	yes	complemented
20	C₄×Dic₁₀₇	C₂²	1	2	yes	complemented
21	C₂⁴×C₁₀₇	C₂²	1	1	yes	characteristic
22	C₂²×C₄×C₁₀₇	—	2	4	no	—
23	C₂²×C₄×C₁₀₇	C₂²	1	2	yes	—
24	C₂²×C₄×C₁₀₇	C₂²	1	2	yes	—
25	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
26	C₂²×C₄×C₁₀₇	C₂²	1	1	yes	characteristic
27	C₂³×C₁₀₇	—	2	2	no	—
28	C₂³×C₁₀₇	—	2	4	no	—
29	C₂³×C₁₀₇	C₂³	1	1	yes	characteristic
30	C₂³×C₁₀₇	C₂³	1	2	yes	—
31	C₂²×D₁₀₇	C₂³	1	2	yes	—
32	C₂×C₄×C₁₀₇	—	2	2	no	—
33	C₂×C₄×C₁₀₇	—	2	4	no	—
34	C₂×C₄×C₁₀₇	—	2	4	no	—
35	C₂×C₄×C₁₀₇	D₄	1	4	yes	complemented
36	C₂×C₄×C₁₀₇	C₂³	1	2	yes	—
37	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
39	C₂²⋊D₁₀₇	—	2	4	no	—
40	C₂²⋊D₁₀₇	—	2	8	no	—
41	C₂×Dic₁₀₇	C₂³	1	4	yes	—
42	C₂×Dic₁₀₇	C₂³	1	2	yes	—
43	D₄₂₈	—	2	2	no	—
44	C₄×D₁₀₇	—	2	4	no	—
45	Dic₂₁₄	—	2	2	no	—
46	C₂²×C₁₀₇	—	2	8	no	—
47	C₂²×C₁₀₇	—	2	2	no	—
48	C₂²×C₁₀₇	—	2	2	no	—
49	C₂²×C₁₀₇	—	2	2	no	—
50	C₂²×C₁₀₇	C₂×D₄	1	2	yes	—
51	C₂²×C₁₀₇	#41	1	4	yes	complemented
52	C₂²×C₁₀₇	C₂⁴	1	1	yes	characteristic
53	C₂×D₁₀₇	—	2	2	no	—
54	C₂×D₁₀₇	—	2	4	no	—
55	C₄×C₁₀₇	—	2	2	no	—
56	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
57	C₄×C₁₀₇	C₂×D₄	1	2	yes	—
58	Dic₁₀₇	—	2	2	no	—
59	Dic₁₀₇	—	2	4	no	—
60	C₂×C₁₀₇	—	2	2	no	—
61	C₂×C₁₀₇	—	2	4	no	—
62	C₂×C₁₀₇	C₂²×D₄	1	1	yes	characteristic
63	C₂×C₁₀₇	C₂×#41	1	2	yes	—
64	D₁₀₇	—	4	2	no	—
65	C₁₀₇	#525	1	1	yes	characteristic, complemented, Sylow
66	#525	—	107	1	no	maximal, Sylow
67	C₂×#41	—	107	1	no	—
68	C₂³×C₄	D₁₀₇	1	1	yes	characteristic, complemented
69	#123	—	107	2	no	—
70	C₂²⋊Q₈	—	107	2	no	—
71	#121	—	107	2	no	—
72	#120	—	107	2	no	—
73	C₄×D₄	—	107	4	no	—
74	#117	—	107	1	no	—
75	C₂⁴	C₂×D₁₀₇	1	1	yes	characteristic
76	#41	—	214	4	no	—
77	C₂×Q₈	—	107	1	no	—
78	C₂×D₄	—	107	4	no	—
79	C₂×D₄	—	107	2	no	—
80	C₂×D₄	—	107	1	no	—
81	C₂²×C₄	—	2	4	no	—
82	C₂²×C₄	—	107	2	no	—
83	C₂²×C₄	C₂×D₁₀₇	1	2	yes	—
84	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₁₀₇	1	2	yes	—
86	C₂²×C₄	C₂×D₁₀₇	1	1	yes	characteristic
87	C₄⋊C₄	—	107	2	no	—
88	C₄⋊C₄	—	107	4	no	—
89	C₂²⋊C₄	—	107	2	no	—
90	C₂²⋊C₄	—	107	4	no	—
91	C₂²⋊C₄	—	107	4	no	—
92	C₄²	—	107	2	no	—
93	C₂³	—	2	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	107	2	no	—
96	C₂³	C₂²×D₁₀₇	1	1	yes	characteristic
97	C₂³	C₂²×D₁₀₇	1	2	yes	—
98	Q₈	—	214	2	no	—
99	D₄	—	214	4	no	—
100	D₄	—	214	8	no	—
101	D₄	—	214	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	107	4	no	—
106	C₂×C₄	—	107	2	no	—
107	C₂×C₄	—	214	4	no	—
108	C₂×C₄	C₂²⋊D₁₀₇	1	4	yes	complemented
109	C₂×C₄	C₂²×D₁₀₇	1	2	yes	—
110	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
111	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
112	C₂²	—	2	8	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	—	214	2	no	—
117	C₂²	—	214	4	no	—
118	C₂²	C₂×C₂²⋊D₁₀₇	1	2	yes	—
119	C₂²	#13133818	1	4	yes	complemented
120	C₂²	C₂³×D₁₀₇	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	214	2	no	—
123	C₄	—	214	4	no	—
124	C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
125	C₄	C₂×C₂²⋊D₁₀₇	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	—	2	4	no	—
128	C₂	—	428	2	no	—
129	C₂	C₂²×C₂²⋊D₁₀₇	1	1	yes	characteristic
130	C₂	C₂×#13133818	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133818

C₂

yes

maximal, characteristic, complemented

#16812092

C₂

yes

maximal, complemented

#16812088

C₂

yes

maximal, complemented

#16812083

C₂

yes

maximal, complemented

#16812078

C₂

yes

maximal, complemented

C₄×C₂²⋊D₁₀₇

C₂

yes

maximal, complemented

#16812068

C₂

yes

maximal, characteristic, complemented

C₂³×C₄×C₁₀₇

C₂

yes

maximal, characteristic, complemented

#13133818

—

C₂×Dic₂₁₄

C₂²

yes

characteristic, complemented

C₂×D₄₂₈

C₂²

yes

characteristic, complemented

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂×C₂²⋊D₁₀₇

C₂²

yes

—

C₂×C₄×D₁₀₇

C₂²

yes

complemented

C₄⋊Dic₁₀₇

C₂²

yes

complemented

Dic₁₀₇⋊C₄

C₂²

yes

complemented

C₂²⋊Dic₁₀₇

C₂²

yes

—

C₂²⋊Dic₁₀₇

C₂²

yes

—

(C₂×D₁₀₇)⋊C₄

C₂²

yes

complemented

C₄×Dic₁₀₇

C₂²

yes

complemented

C₂⁴×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

—

C₂²×C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

—

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂²×C₄×C₁₀₇

C₂²

yes

characteristic

C₂³×C₁₀₇

—

C₂³×C₁₀₇

—

C₂³×C₁₀₇

C₂³

yes

characteristic

C₂³×C₁₀₇

C₂³

yes

—

C₂²×D₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

—

C₂×C₄×C₁₀₇

D₄

yes

complemented

C₂×C₄×C₁₀₇

C₂³

yes

—

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂×C₄×C₁₀₇

C₂³

yes

characteristic

C₂²⋊D₁₀₇

—

C₂²⋊D₁₀₇

—

C₂×Dic₁₀₇

C₂³

yes

—

C₂×Dic₁₀₇

C₂³

yes

—

D₄₂₈

—

C₄×D₁₀₇

—

Dic₂₁₄

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

—

C₂²×C₁₀₇

C₂×D₄

yes

—

C₂²×C₁₀₇

#41

yes

complemented

C₂²×C₁₀₇

C₂⁴

yes

characteristic

C₂×D₁₀₇

—

C₂×D₁₀₇

—

C₄×C₁₀₇

—

C₄×C₁₀₇

C₂×D₄

yes

—

C₄×C₁₀₇

C₂×D₄

yes

—

Dic₁₀₇

—

Dic₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

—

C₂×C₁₀₇

C₂²×D₄

yes

characteristic

C₂×C₁₀₇

C₂×#41

yes

—

D₁₀₇

—

C₁₀₇

#525

yes

characteristic, complemented, Sylow

#525

—

107

maximal, Sylow

C₂×#41

—

107

—

C₂³×C₄

D₁₀₇

yes

characteristic, complemented

#123

—

107

—

C₂²⋊Q₈

—

107

—

#121

—

107

—

#120

—

107

—

C₄×D₄

—

107

—

#117

—

107

—

C₂⁴

C₂×D₁₀₇

yes

characteristic

#41

—

214

—

C₂×Q₈

—

107

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂×D₄

—

107

—

C₂²×C₄

—

C₂²×C₄

—

107

—

C₂²×C₄

C₂×D₁₀₇

yes

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₂²×C₄

C₂×D₁₀₇

yes

—

C₂²×C₄

C₂×D₁₀₇

yes

characteristic

C₄⋊C₄

—

107

—

C₄⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₂²⋊C₄

—

107

—

C₄²

—

107

—

C₂³

—

C₂³

—

C₂³

—

107

—

C₂³

C₂²×D₁₀₇

yes

characteristic

C₂³

C₂²×D₁₀₇

yes

—

Q₈

—

214

—

D₄

—

214

—

100

D₄

—

214

—

101

D₄

—

214

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

107

—

106

C₂×C₄

—

107

—

107

C₂×C₄

—

214

—

108

C₂×C₄

C₂²⋊D₁₀₇

yes

complemented

109

C₂×C₄

C₂²×D₁₀₇

yes

—

110

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

111

C₂×C₄

C₂²×D₁₀₇

yes

characteristic

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

214

—

117

C₂²

—

214

—

118

C₂²

C₂×C₂²⋊D₁₀₇

yes

—

119

C₂²

#13133818

yes

complemented

120

C₂²

C₂³×D₁₀₇

yes

characteristic

121

C₄

—

122

C₄

—

214

—

123

C₄

—

214

—

124

C₄

C₂×C₂²⋊D₁₀₇

yes

—

125

C₄

C₂×C₂²⋊D₁₀₇

yes

—

126

C₂

—

127

C₂

—

128

C₂

—

428

—

129

C₂

C₂²×C₂²⋊D₁₀₇

yes

characteristic

130

C₂

C₂×#13133818

yes

—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₀₇
commutator, socle	C₂²×C₁₀₇	C₂⁴
Frattini	C₂²	C₂³×D₁₀₇
Fitting	C₂³×C₄×C₁₀₇	C₂
hypercenter	C₂³×C₄	D₁₀₇
hypocenter	C₁₀₇	#525
LCS	G ▷ C₂²×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄

subgroup

G/H

center

C₂×C₄

C₂²×D₁₀₇

commutator, socle

C₂²×C₁₀₇

C₂⁴

Frattini

C₂²

C₂³×D₁₀₇

Fitting

C₂³×C₄×C₁₀₇

C₂

hypercenter

C₂³×C₄

D₁₀₇

C₁₀₇

G ▷ C₂²×C₁₀₇ ▷ C₁₀₇

C₁ ◁ C₂×C₄ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂×#13133818	C₂	semidirect
2. #16812092	C₂	semidirect
3. #16812088	C₂	semidirect
4. #16812083	C₂	semidirect
5. #16812078	C₂	semidirect
6. C₄×C₂²⋊D₁₀₇	C₂	semidirect
7. #16812068	C₂	semidirect
8. C₂³×C₄×C₁₀₇	C₂	semidirect
10. C₂×Dic₂₁₄	C₂²	semidirect
11. C₂×D₄₂₈	C₂²	semidirect
14. C₂×C₄×D₁₀₇	C₂²	semidirect
15. C₄⋊Dic₁₀₇	C₂²	semidirect
16. Dic₁₀₇⋊C₄	C₂²	semidirect
19. (C₂×D₁₀₇)⋊C₄	C₂²	semidirect
20. C₄×Dic₁₀₇	C₂²	semidirect
35. C₂×C₄×C₁₀₇	D₄	semidirect
51. C₂²×C₁₀₇	#41	semidirect
65. C₁₀₇	#525	semidirect
68. C₂³×C₄	D₁₀₇	semidirect
108. C₂×C₄	C₂²⋊D₁₀₇	semidirect
119. C₂²	#13133818	semidirect

H ≅ G/N

type

1. C₂×#13133818

C₂

semidirect

2. #16812092

C₂

semidirect

3. #16812088

C₂

semidirect

4. #16812083

C₂

semidirect

5. #16812078

C₂

semidirect

6. C₄×C₂²⋊D₁₀₇

C₂

semidirect

7. #16812068

C₂

semidirect

8. C₂³×C₄×C₁₀₇

C₂

semidirect

10. C₂×Dic₂₁₄

C₂²

semidirect

11. C₂×D₄₂₈

C₂²

semidirect

14. C₂×C₄×D₁₀₇

C₂²

semidirect

15. C₄⋊Dic₁₀₇

C₂²

semidirect

16. Dic₁₀₇⋊C₄

C₂²

semidirect

19. (C₂×D₁₀₇)⋊C₄

C₂²

semidirect

20. C₄×Dic₁₀₇

C₂²

semidirect

35. C₂×C₄×C₁₀₇

D₄

semidirect

51. C₂²×C₁₀₇

#41

semidirect

65. C₁₀₇

#525

semidirect

68. C₂³×C₄

D₁₀₇

semidirect

108. C₂×C₄

C₂²⋊D₁₀₇

semidirect

119. C₂²

#13133818

semidirect

Subgroups — #20293138

Conjugacy classes of subgroups

Complements