Group #15689698 — subgroups

Subgroups — #15689698

Order 3008.

270 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 103 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×#10625354	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#13094022	C₂	1	1	yes	maximal, characteristic, complemented
4	#13094013	C₂	1	1	yes	maximal, characteristic, complemented
5	#13094012	C₂	1	1	yes	maximal, characteristic, complemented
6	#13094009	C₂	1	4	yes	maximal, complemented
7	#13094002	C₂	1	4	yes	maximal, complemented
8	#13093981	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄⋊Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	#10625354	—	2	4	no	—
13	Q₈×D₄₇	—	2	4	no	—
14	C₂×D₁₈₈	C₂²	1	2	yes	—
15	C₂×D₁₈₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄₇	C₂²	1	2	yes	—
17	C₂×C₄×D₄₇	C₂²	1	2	yes	—
18	C₂×C₄×D₄₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₉₄	C₂²	1	2	yes	—
20	C₂×Dic₉₄	C₂²	1	1	yes	characteristic
21	C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
22	C₄²×C₄₇	C₂²	1	1	yes	characteristic
23	(C₂×D₄₇)⋊C₄	C₂²	1	2	yes	—
24	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
25	(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
26	Dic₄₇⋊C₄	C₂²	1	2	yes	—
27	Dic₄₇⋊C₄	C₂²	1	4	yes	—
28	C₄×Dic₄₇	C₂²	1	1	yes	characteristic
29	C₂²×D₄₇	C₂³	1	2	yes	—
30	C₂²×D₄₇	C₂³	1	1	yes	characteristic
31	Q₈×C₄₇	—	2	2	no	—
32	Q₈×C₄₇	—	2	2	no	—
33	C₂×C₄×C₄₇	C₂³	1	2	yes	—
34	C₂×C₄×C₄₇	C₂³	1	2	yes	—
35	C₂×C₄×C₄₇	C₂³	1	2	yes	—
36	C₂×C₄×C₄₇	C₂³	1	1	yes	characteristic
37	C₂×Dic₄₇	C₂³	1	2	yes	—
38	C₂×Dic₄₇	C₂³	1	2	yes	—
39	C₂×Dic₄₇	C₂³	1	1	yes	characteristic
40	D₁₈₈	—	2	4	no	—
41	D₁₈₈	—	2	2	no	—
42	C₄×D₄₇	—	2	4	no	—
43	C₄×D₄₇	—	2	4	no	—
44	C₄×D₄₇	D₄	1	4	yes	—
45	Dic₉₄	—	2	2	no	—
46	Dic₉₄	—	2	4	no	—
47	C₂²×C₄₇	C₂⁴	1	1	yes	characteristic
48	C₂×D₄₇	—	2	2	no	—
49	C₂×D₄₇	—	2	2	no	—
50	C₂×D₄₇	—	2	4	no	—
51	C₂×D₄₇	C₂×D₄	1	2	yes	—
52	C₄×C₄₇	—	2	2	no	—
53	C₄×C₄₇	—	2	2	no	—
54	C₄×C₄₇	—	2	2	no	—
55	C₄×C₄₇	C₂×D₄	1	2	yes	—
56	Dic₄₇	—	2	2	no	—
57	Dic₄₇	—	2	2	no	—
58	Dic₄₇	C₂×D₄	1	2	yes	—
59	C₂×C₄₇	C₂²×D₄	1	1	yes	characteristic
60	C₂×C₄₇	2₋¹⁺⁴	1	2	yes	—
61	D₄₇	—	2	2	no	—
62	D₄₇	—	4	2	no	—
63	#536	—	47	1	no	maximal, Sylow
64	C₄₇	#536	1	1	yes	characteristic, complemented, Sylow
65	C₂×#41	—	47	1	no	—
66	C₂²×Q₈	—	47	1	no	—
67	C₄⋊Q₈	—	47	1	no	—
68	C₄⋊Q₈	D₄₇	1	1	yes	characteristic, complemented
69	#124	—	47	1	no	—
70	#124	—	47	1	no	—
71	#123	—	47	4	no	—
72	C₂²⋊Q₈	—	47	4	no	—
73	#117	—	47	1	no	—
74	#41	—	94	4	no	—
75	C₂×Q₈	—	47	2	no	—
76	C₂×Q₈	—	47	1	no	—
77	C₂×Q₈	—	94	4	no	—
78	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
79	C₂×Q₈	C₂×D₄₇	1	1	yes	characteristic
80	C₂×D₄	—	47	2	no	—
81	C₂×D₄	—	47	1	no	—
82	C₂²×C₄	—	47	1	no	—
83	C₂²×C₄	—	47	2	no	—
84	C₂²×C₄	—	47	2	no	—
85	C₄⋊C₄	—	47	2	no	—
86	C₄⋊C₄	—	47	4	no	—
87	C₄⋊C₄	C₂×D₄₇	1	4	yes	—
88	C₂²⋊C₄	—	47	2	no	—
89	C₂²⋊C₄	—	47	4	no	—
90	C₂²⋊C₄	—	47	4	no	—
91	C₄²	—	47	1	no	—
92	C₄²	C₂×D₄₇	1	1	yes	characteristic
93	C₂³	—	47	1	no	—
94	C₂³	—	47	2	no	—
95	Q₈	—	2	2	no	—
96	Q₈	—	2	2	no	—
97	Q₈	—	94	4	no	—
98	Q₈	—	94	2	no	—
99	D₄	—	94	4	no	—
100	D₄	—	94	2	no	—
101	C₂×C₄	—	47	1	no	—
102	C₂×C₄	—	47	4	no	—
103	C₂×C₄	—	47	2	no	—
104	C₂×C₄	—	47	2	no	—
105	C₂×C₄	—	94	4	no	—
106	C₂×C₄	—	94	4	no	—
107	C₂×C₄	C₂²×D₄₇	1	2	yes	—
108	C₂×C₄	C₂²×D₄₇	1	2	yes	—
109	C₂×C₄	C₂²×D₄₇	1	2	yes	—
110	C₂×C₄	C₂²×D₄₇	1	1	yes	characteristic
111	C₂²	—	47	2	no	—
112	C₂²	—	94	2	no	—
113	C₂²	—	94	2	no	—
114	C₂²	—	94	4	no	—
115	C₂²	C₂³×D₄₇	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	47	2	no	—
120	C₄	—	94	2	no	—
121	C₄	—	94	2	no	—
122	C₄	D₄×D₄₇	1	2	yes	—
123	C₂	—	94	2	no	—
124	C₂	—	188	2	no	—
125	C₂	C₂×D₄×D₄₇	1	1	yes	characteristic
126	C₂	#13094072	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625354

C₂

yes

maximal, characteristic

C₂×Q₈×D₄₇

C₂

yes

maximal, characteristic, complemented

#13094022

C₂

yes

maximal, characteristic, complemented

#13094013

C₂

yes

maximal, characteristic, complemented

#13094012

C₂

yes

maximal, characteristic, complemented

#13094009

C₂

yes

maximal, complemented

#13094002

C₂

yes

maximal, complemented

#13093981

C₂

yes

maximal, characteristic, complemented

C₄⋊Q₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂²

yes

characteristic

#10625354

—

Q₈×D₄₇

—

C₂×D₁₈₈

C₂²

yes

—

C₂×D₁₈₈

C₂²

yes

characteristic

C₂×C₄×D₄₇

C₂²

yes

—

C₂×C₄×D₄₇

C₂²

yes

—

C₂×C₄×D₄₇

C₂²

yes

characteristic

C₂×Dic₉₄

C₂²

yes

—

C₂×Dic₉₄

C₂²

yes

characteristic

C₄⋊C₄×C₄₇

C₂²

yes

—

C₄²×C₄₇

C₂²

yes

characteristic

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

(C₂×D₄₇)⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

Dic₄₇⋊C₄

C₂²

yes

—

C₄×Dic₄₇

C₂²

yes

characteristic

C₂²×D₄₇

C₂³

yes

—

C₂²×D₄₇

C₂³

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

—

C₂×C₄×C₄₇

C₂³

yes

characteristic

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

—

C₂×Dic₄₇

C₂³

yes

characteristic

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

D₄

yes

—

Dic₉₄

—

Dic₉₄

—

C₂²×C₄₇

C₂⁴

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×D₄

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₄

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×D₄

yes

—

C₂×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄₇

2₋¹⁺⁴

yes

—

D₄₇

—

D₄₇

—

#536

—

maximal, Sylow

C₄₇

#536

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄₇

yes

characteristic, complemented

#124

—

#124

—

#123

—

C₂²⋊Q₈

—

#117

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₄₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄₇

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂²×D₄₇

yes

—

108

C₂×C₄

C₂²×D₄₇

yes

—

109

C₂×C₄

C₂²×D₄₇

yes

—

110

C₂×C₄

C₂²×D₄₇

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂³×D₄₇

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

D₄×D₄₇

yes

—

123

C₂

—

124

C₂

—

188

—

125

C₂

C₂×D₄×D₄₇

yes

characteristic

126

C₂

#13094072

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₄₇
commutator, socle, cosocle	C₂²×C₄₇	C₂⁴
Fitting	C₄⋊Q₈×C₄₇	C₂
prime radical	C₂×#10625354	C₂
hypercenter	C₄⋊Q₈	D₄₇
hypocenter	C₄₇	#536
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₄₇

commutator, socle, cosocle

C₂²×C₄₇

C₂⁴

Fitting

C₄⋊Q₈×C₄₇

C₂

prime radical

C₂×#10625354

C₂

hypercenter

C₄⋊Q₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
2. C₂×Q₈×D₄₇	C₂	semidirect
3. #13094022	C₂	semidirect
4. #13094013	C₂	semidirect
5. #13094012	C₂	semidirect
6. #13094009	C₂	semidirect
7. #13094002	C₂	semidirect
8. #13093981	C₂	semidirect
9. C₄⋊Q₈×C₄₇	C₂	semidirect
64. C₄₇	#536	semidirect
68. C₄⋊Q₈	D₄₇	semidirect

H ≅ G/N

type

2. C₂×Q₈×D₄₇

C₂

semidirect

3. #13094022

C₂

semidirect

4. #13094013

C₂

semidirect

5. #13094012

C₂

semidirect

6. #13094009

C₂

semidirect

7. #13094002

C₂

semidirect

8. #13093981

C₂

semidirect

9. C₄⋊Q₈×C₄₇

C₂

semidirect

64. C₄₇

#536

semidirect

68. C₄⋊Q₈

D₄₇

semidirect

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

subgroup	index
C₂×#10625354	2
C₂×Q₈×D₄₇	2
#13094022	2
#13094013	2
#13094012	2
#13094009	2
#13094002	2
#13093981	2
C₄⋊Q₈×C₄₇	2
#536	47

subgroup

index

C₂×Q₈×D₄₇

C₄⋊Q₈×C₄₇

Subgroups — #15689698

Conjugacy classes of subgroups

Complements

Maximal subgroups