Group #11929227 — subgroups

Subgroups — D₃×Q₈×D₁₃ #11929227

Order 1248.

312 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 122 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	Q₈×D₃₉	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₃×D₁₃	C₂	1	3	yes	maximal
5	D₃₉⋊Q₈	C₂	1	3	yes	maximal, complemented
6	D₃×Dic₂₆	C₂	1	3	yes	maximal, complemented
7	Dic₆×D₁₃	C₂	1	3	yes	maximal, complemented
8	C₂×Q₈×D₁₃	—	3	1	no	maximal, Hall
9	C₂×D₃×D₁₃	C₂²	1	1	yes	characteristic
10	C₃×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
11	C₄×D₃₉	C₂²	1	3	yes	—
12	Dic₇₈	C₂²	1	3	yes	complemented
13	C₄×D₃×C₁₃	C₂²	1	3	yes	—
14	C₁₃×Dic₆	C₂²	1	3	yes	complemented
15	C₃×C₄×D₁₃	C₂²	1	3	yes	—
16	C₃×Dic₂₆	C₂²	1	3	yes	complemented
17	#64605	C₂²	1	6	yes	complemented
18	#64601	C₂²	1	3	yes	—
19	D₃×Dic₁₃	C₂²	1	3	yes	—
20	Dic₃×D₁₃	C₂²	1	3	yes	—
21	C₂×Q₈×C₁₃	—	3	1	no	—
22	Q₈×D₁₃	—	3	1	no	—
23	Q₈×D₁₃	—	3	3	no	—
24	Q₈×D₁₃	—	3	3	no	—
25	Q₈×D₁₃	D₃	1	1	yes	characteristic, direct product factor
26	C₂×C₄×D₁₃	—	3	3	no	—
27	C₂×Dic₂₆	—	3	3	no	—
28	C₂×D₃₉	C₂³	1	1	yes	characteristic
29	C₂×D₃×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₃×D₁₃	C₂³	1	1	yes	characteristic
31	D₃×D₁₃	Q₈	1	4	yes	direct product factor
32	C₃×C₄×C₁₃	C₂³	1	3	yes	—
33	Dic₃₉	C₂³	1	3	yes	—
34	C₃×Dic₁₃	C₂³	1	3	yes	—
35	Dic₃×C₁₃	C₂³	1	3	yes	—
36	C₂²×D₁₃	—	3	1	no	—
37	Q₈×C₁₃	—	3	3	no	—
38	Q₈×C₁₃	C₂×D₃	1	1	yes	characteristic, complemented
39	C₂×C₄×C₁₃	—	3	3	no	—
40	C₂×Dic₁₃	—	3	3	no	—
41	C₄×D₁₃	—	3	3	no	—
42	C₄×D₁₃	—	3	3	no	—
43	C₄×D₁₃	—	3	3	no	—
44	C₄×D₁₃	C₂×D₃	1	3	yes	—
45	Dic₂₆	—	3	3	no	—
46	Dic₂₆	—	3	6	no	—
47	Dic₂₆	C₂×D₃	1	3	yes	complemented
48	C₂×D₃×Q₈	—	13	1	no	maximal, Hall
49	C₂×C₃×C₁₃	C₂⁴	1	1	yes	characteristic
50	D₃₉	C₂×Q₈	1	2	yes	complemented
51	C₃×D₁₃	C₂×Q₈	1	2	yes	complemented
52	D₃×C₁₃	C₂×Q₈	1	2	yes	complemented
53	C₂²×C₁₃	—	3	1	no	—
54	C₂×D₁₃	—	3	1	no	—
55	C₂×D₁₃	—	3	4	no	—
56	C₂×D₁₃	C₂²×D₃	1	1	yes	characteristic
57	C₄×C₁₃	—	3	3	no	—
58	C₄×C₁₃	C₂²×D₃	1	3	yes	—
59	Dic₁₃	—	3	3	no	—
60	Dic₁₃	C₂²×D₃	1	3	yes	—
61	C₂×C₃×Q₈	—	13	1	no	—
62	D₃×Q₈	—	13	1	no	—
63	D₃×Q₈	—	13	3	no	—
64	D₃×Q₈	—	13	3	no	—
65	D₃×Q₈	D₁₃	1	1	yes	characteristic, direct product factor
66	C₂×C₄×D₃	—	13	3	no	—
67	C₂×Dic₆	—	13	3	no	—
68	C₃×C₁₃	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
69	C₂²×Q₈	—	39	1	no	Sylow
70	C₂×C₁₃	—	3	2	no	—
71	C₂×C₁₃	C₂³×D₃	1	1	yes	characteristic
72	D₁₃	—	3	2	no	—
73	D₁₃	D₃×Q₈	1	2	yes	direct product factor
74	C₂²×D₃	—	13	1	no	—
75	C₃×Q₈	—	13	3	no	—
76	C₃×Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
77	C₂×C₃×C₄	—	13	3	no	—
78	C₂×Dic₃	—	13	3	no	—
79	C₄×D₃	—	13	3	no	—
80	C₄×D₃	—	13	3	no	—
81	C₄×D₃	—	13	3	no	—
82	C₄×D₃	C₂×D₁₃	1	3	yes	—
83	Dic₆	—	13	3	no	—
84	Dic₆	—	13	6	no	—
85	Dic₆	C₂×D₁₃	1	3	yes	complemented
86	C₂×Q₈	—	3	1	no	—
87	C₂×Q₈	—	13	1	no	—
88	C₂×Q₈	—	39	3	no	—
89	C₂×Q₈	—	39	3	no	—
90	C₂×Q₈	—	39	3	no	—
91	C₂×Q₈	—	39	1	no	—
92	C₂²×C₄	—	39	3	no	—
93	C₁₃	C₂×D₃×Q₈	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₃	—	13	1	no	—
95	C₂×D₃	—	13	1	no	—
96	C₂×D₃	—	13	4	no	—
97	C₂×D₃	C₂²×D₁₃	1	1	yes	characteristic
98	C₃×C₄	—	13	3	no	—
99	C₃×C₄	C₂²×D₁₃	1	3	yes	—
100	Dic₃	—	13	3	no	—
101	Dic₃	C₂²×D₁₃	1	3	yes	—
102	C₂³	—	39	1	no	—
103	Q₈	—	3	3	no	—
104	Q₈	—	13	3	no	—
105	Q₈	—	39	6	no	—
106	Q₈	—	39	3	no	—
107	Q₈	D₃×D₁₃	1	1	yes	characteristic, direct product factor
108	C₂×C₄	—	3	3	no	—
109	C₂×C₄	—	13	3	no	—
110	C₂×C₄	—	39	3	no	—
111	C₂×C₄	—	39	3	no	—
112	C₂×C₄	—	39	3	no	—
113	C₂×C₄	—	39	3	no	—
114	C₂×C₃	—	13	2	no	—
115	C₂×C₃	C₂³×D₁₃	1	1	yes	characteristic
116	D₃	—	13	2	no	—
117	D₃	Q₈×D₁₃	1	2	yes	direct product factor
118	C₂²	—	3	1	no	—
119	C₂²	—	13	1	no	—
120	C₂²	—	39	4	no	—
121	C₂²	—	39	1	no	—
122	C₄	—	3	3	no	—
123	C₄	—	13	3	no	—
124	C₄	—	39	3	no	—
125	C₄	C₂×D₃×D₁₃	1	3	yes	—
126	C₃	C₂×Q₈×D₁₃	1	1	yes	characteristic, complemented, Sylow
127	C₂	—	3	2	no	—
128	C₂	—	13	2	no	—
129	C₂	—	39	2	no	—
130	C₂	C₂²×D₃×D₁₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₃₉

C₂

yes

maximal, characteristic, complemented

D₃×Q₈×C₁₃

C₂

yes

maximal, characteristic, complemented

C₃×Q₈×D₁₃

C₂

yes

maximal, characteristic, complemented

C₄×D₃×D₁₃

C₂

yes

maximal

D₃₉⋊Q₈

C₂

yes

maximal, complemented

D₃×Dic₂₆

C₂

yes

maximal, complemented

Dic₆×D₁₃

C₂

yes

maximal, complemented

C₂×Q₈×D₁₃

—

maximal, Hall

C₂×D₃×D₁₃

C₂²

yes

characteristic

C₃×Q₈×C₁₃

C₂²

yes

characteristic, complemented

C₄×D₃₉

C₂²

yes

—

Dic₇₈

C₂²

yes

complemented

C₄×D₃×C₁₃

C₂²

yes

—

C₁₃×Dic₆

C₂²

yes

complemented

C₃×C₄×D₁₃

C₂²

yes

—

C₃×Dic₂₆

C₂²

yes

complemented

#64605

C₂²

yes

complemented

#64601

C₂²

yes

—

D₃×Dic₁₃

C₂²

yes

—

Dic₃×D₁₃

C₂²

yes

—

C₂×Q₈×C₁₃

—

Q₈×D₁₃

—

Q₈×D₁₃

—

Q₈×D₁₃

—

Q₈×D₁₃

D₃

yes

characteristic, direct product factor

C₂×C₄×D₁₃

—

C₂×Dic₂₆

—

C₂×D₃₉

C₂³

yes

characteristic

C₂×D₃×C₁₃

C₂³

yes

characteristic

C₂×C₃×D₁₃

C₂³

yes

characteristic

D₃×D₁₃

Q₈

yes

direct product factor

C₃×C₄×C₁₃

C₂³

yes

—

Dic₃₉

C₂³

yes

—

C₃×Dic₁₃

C₂³

yes

—

Dic₃×C₁₃

C₂³

yes

—

C₂²×D₁₃

—

Q₈×C₁₃

—

Q₈×C₁₃

C₂×D₃

yes

characteristic, complemented

C₂×C₄×C₁₃

—

C₂×Dic₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

—

C₄×D₁₃

C₂×D₃

yes

—

Dic₂₆

—

Dic₂₆

—

Dic₂₆

C₂×D₃

yes

complemented

C₂×D₃×Q₈

—

maximal, Hall

C₂×C₃×C₁₃

C₂⁴

yes

characteristic

D₃₉

C₂×Q₈

yes

complemented

C₃×D₁₃

C₂×Q₈

yes

complemented

D₃×C₁₃

C₂×Q₈

yes

complemented

C₂²×C₁₃

—

C₂×D₁₃

—

C₂×D₁₃

—

C₂×D₁₃

C₂²×D₃

yes

characteristic

C₄×C₁₃

—

C₄×C₁₃

C₂²×D₃

yes

—

Dic₁₃

—

Dic₁₃

C₂²×D₃

yes

—

C₂×C₃×Q₈

—

D₃×Q₈

—

D₃×Q₈

—

D₃×Q₈

—

D₃×Q₈

D₁₃

yes

characteristic, direct product factor

C₂×C₄×D₃

—

C₂×Dic₆

—

C₃×C₁₃

C₂²×Q₈

yes

characteristic, complemented, Hall

C₂²×Q₈

—

Sylow

C₂×C₁₃

—

C₂×C₁₃

C₂³×D₃

yes

characteristic

D₁₃

—

D₁₃

D₃×Q₈

yes

direct product factor

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

C₂×D₁₃

yes

characteristic, complemented

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

C₂×D₁₃

yes

—

Dic₆

—

Dic₆

—

Dic₆

C₂×D₁₃

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₁₃

C₂×D₃×Q₈

yes

characteristic, complemented, Sylow

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

C₂²×D₁₃

yes

characteristic

C₃×C₄

—

C₃×C₄

C₂²×D₁₃

yes

—

100

Dic₃

—

101

Dic₃

C₂²×D₁₃

yes

—

102

C₂³

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

Q₈

D₃×D₁₃

yes

characteristic, direct product factor

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₃

—

115

C₂×C₃

C₂³×D₁₃

yes

characteristic

116

D₃

—

117

D₃

Q₈×D₁₃

yes

direct product factor

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂×D₃×D₁₃

yes

—

126

C₃

C₂×Q₈×D₁₃

yes

characteristic, complemented, Sylow

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

C₂²×D₃×D₁₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃×D₁₃
commutator, socle, cosocle	C₂×C₃×C₁₃	C₂⁴
Fitting	C₃×Q₈×C₁₃	C₂²
prime radical	C₂×D₃×D₁₃	C₂²
hypercenter	Q₈	D₃×D₁₃
hypocenter	C₃×C₁₃	C₂²×Q₈
LCS	G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃
UCS	C₁ ◁ C₂ ◁ Q₈

subgroup

G/H

center, Frattini

C₂

C₂²×D₃×D₁₃

commutator, socle, cosocle

C₂×C₃×C₁₃

C₂⁴

Fitting

C₃×Q₈×C₁₃

C₂²

prime radical

C₂×D₃×D₁₃

C₂²

hypercenter

Q₈

D₃×D₁₃

hypocenter

C₃×C₁₃

C₂²×Q₈

LCS

G ▷ C₂×C₃×C₁₃ ▷ C₃×C₁₃

UCS

C₁ ◁ C₂ ◁ Q₈

N	H ≅ G/N	type
1. Q₈×D₃₉	C₂	semidirect
2. D₃×Q₈×C₁₃	C₂	semidirect
3. C₃×Q₈×D₁₃	C₂	semidirect
5. D₃₉⋊Q₈	C₂	semidirect
6. D₃×Dic₂₆	C₂	semidirect
7. Dic₆×D₁₃	C₂	semidirect
10. C₃×Q₈×C₁₃	C₂²	semidirect
12. Dic₇₈	C₂²	semidirect
14. C₁₃×Dic₆	C₂²	semidirect
16. C₃×Dic₂₆	C₂²	semidirect
17. #64605	C₂²	semidirect
25. Q₈×D₁₃	D₃	direct (N × H)
31. D₃×D₁₃	Q₈	direct (N × H)
38. Q₈×C₁₃	C₂×D₃	semidirect
47. Dic₂₆	C₂×D₃	semidirect
50. D₃₉	C₂×Q₈	semidirect
51. C₃×D₁₃	C₂×Q₈	semidirect
52. D₃×C₁₃	C₂×Q₈	semidirect
65. D₃×Q₈	D₁₃	direct (N × H)
68. C₃×C₁₃	C₂²×Q₈	semidirect
73. D₁₃	D₃×Q₈	direct (N × H)
76. C₃×Q₈	C₂×D₁₃	semidirect
85. Dic₆	C₂×D₁₃	semidirect
93. C₁₃	C₂×D₃×Q₈	semidirect
107. Q₈	D₃×D₁₃	direct (N × H)
117. D₃	Q₈×D₁₃	direct (N × H)
126. C₃	C₂×Q₈×D₁₃	semidirect

H ≅ G/N

type

1. Q₈×D₃₉

C₂

semidirect

2. D₃×Q₈×C₁₃

C₂

semidirect

3. C₃×Q₈×D₁₃

C₂

semidirect

5. D₃₉⋊Q₈

C₂

semidirect

6. D₃×Dic₂₆

C₂

semidirect

7. Dic₆×D₁₃

C₂

semidirect

10. C₃×Q₈×C₁₃

C₂²

semidirect

12. Dic₇₈

C₂²

semidirect

14. C₁₃×Dic₆

C₂²

semidirect

16. C₃×Dic₂₆

C₂²

semidirect

17. #64605

C₂²

semidirect

25. Q₈×D₁₃

D₃

direct (N × H)

31. D₃×D₁₃

Q₈

direct (N × H)

38. Q₈×C₁₃

C₂×D₃

semidirect

47. Dic₂₆

C₂×D₃

semidirect

50. D₃₉

C₂×Q₈

semidirect

51. C₃×D₁₃

C₂×Q₈

semidirect

52. D₃×C₁₃

C₂×Q₈

semidirect

65. D₃×Q₈

D₁₃

direct (N × H)

68. C₃×C₁₃

C₂²×Q₈

semidirect

73. D₁₃

D₃×Q₈

direct (N × H)

76. C₃×Q₈

C₂×D₁₃

semidirect

85. Dic₆

C₂×D₁₃

semidirect

93. C₁₃

C₂×D₃×Q₈

semidirect

107. Q₈

D₃×D₁₃

direct (N × H)

117. D₃

Q₈×D₁₃

direct (N × H)

126. C₃

C₂×Q₈×D₁₃

semidirect

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

subgroup	index
Q₈×D₃₉	2
D₃×Q₈×C₁₃	2
C₃×Q₈×D₁₃	2
C₄×D₃×D₁₃	2
D₃₉⋊Q₈	2
D₃×Dic₂₆	2
Dic₆×D₁₃	2
C₂×Q₈×D₁₃	3
C₂×D₃×Q₈	13

subgroup

index

Q₈×D₃₉

D₃×Q₈×C₁₃

C₃×Q₈×D₁₃

C₄×D₃×D₁₃

D₃₉⋊Q₈

D₃×Dic₂₆

Dic₆×D₁₃

C₂×Q₈×D₁₃

C₂×D₃×Q₈

Subgroups — D₃×Q₈×D₁₃ #11929227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×Q8×D13 #11929227

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×Q₈×D₁₃ #11929227