Group #10621578 — subgroups

Subgroups — D₃×Q₈×D₇ #10621578

Order 672.

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×D₂₁	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×C₇×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×Q₈×D₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×D₃×D₇	C₂	1	3	yes	maximal
5	D₂₁⋊Q₈	C₂	1	3	yes	maximal, complemented
6	D₃×Dic₁₄	C₂	1	3	yes	maximal, complemented
7	D₇×Dic₆	C₂	1	3	yes	maximal, complemented
8	C₂×Q₈×D₇	—	3	1	no	maximal, Hall
9	C₂×D₃×D₇	C₂²	1	1	yes	characteristic
10	C₃×C₇×Q₈	C₂²	1	1	yes	characteristic, complemented
11	C₄×D₂₁	C₂²	1	3	yes	—
12	Dic₄₂	C₂²	1	3	yes	complemented
13	C₄×D₃×C₇	C₂²	1	3	yes	—
14	C₇×Dic₆	C₂²	1	3	yes	complemented
15	C₃×C₄×D₇	C₂²	1	3	yes	—
16	C₃×Dic₁₄	C₂²	1	3	yes	complemented
17	#4248	C₂²	1	6	yes	complemented
18	D₂₁⋊C₄	C₂²	1	3	yes	—
19	D₃×Dic₇	C₂²	1	3	yes	—
20	Dic₃×D₇	C₂²	1	3	yes	—
21	C₂×C₇×Q₈	—	3	1	no	—
22	Q₈×D₇	—	3	1	no	—
23	Q₈×D₇	—	3	3	no	—
24	Q₈×D₇	—	3	3	no	—
25	Q₈×D₇	D₃	1	1	yes	characteristic, direct product factor
26	C₂×C₄×D₇	—	3	3	no	—
27	C₂×Dic₁₄	—	3	3	no	—
28	C₂×D₃×Q₈	—	7	1	no	maximal, Hall
29	C₂×D₂₁	C₂³	1	1	yes	characteristic
30	C₂×D₃×C₇	C₂³	1	1	yes	characteristic
31	C₂×C₃×D₇	C₂³	1	1	yes	characteristic
32	D₃×D₇	Q₈	1	4	yes	direct product factor
33	C₃×C₄×C₇	C₂³	1	3	yes	—
34	Dic₂₁	C₂³	1	3	yes	—
35	C₃×Dic₇	C₂³	1	3	yes	—
36	C₇×Dic₃	C₂³	1	3	yes	—
37	C₂²×D₇	—	3	1	no	—
38	C₇×Q₈	—	3	3	no	—
39	C₇×Q₈	C₂×D₃	1	1	yes	characteristic, complemented
40	C₂×C₄×C₇	—	3	3	no	—
41	C₂×Dic₇	—	3	3	no	—
42	C₄×D₇	—	3	3	no	—
43	C₄×D₇	—	3	3	no	—
44	C₄×D₇	—	3	3	no	—
45	C₄×D₇	C₂×D₃	1	3	yes	—
46	Dic₁₄	—	3	3	no	—
47	Dic₁₄	—	3	6	no	—
48	Dic₁₄	C₂×D₃	1	3	yes	complemented
49	C₂×C₃×Q₈	—	7	1	no	—
50	D₃×Q₈	—	7	1	no	—
51	D₃×Q₈	—	7	3	no	—
52	D₃×Q₈	—	7	3	no	—
53	D₃×Q₈	D₇	1	1	yes	characteristic, direct product factor
54	C₂×C₄×D₃	—	7	3	no	—
55	C₂×Dic₆	—	7	3	no	—
56	C₂×C₃×C₇	C₂⁴	1	1	yes	characteristic
57	D₂₁	C₂×Q₈	1	2	yes	complemented
58	C₃×D₇	C₂×Q₈	1	2	yes	complemented
59	D₃×C₇	C₂×Q₈	1	2	yes	complemented
60	C₂²×Q₈	—	21	1	no	Sylow
61	C₂²×C₇	—	3	1	no	—
62	C₂×D₇	—	3	1	no	—
63	C₂×D₇	—	3	4	no	—
64	C₂×D₇	C₂²×D₃	1	1	yes	characteristic
65	C₄×C₇	—	3	3	no	—
66	C₄×C₇	C₂²×D₃	1	3	yes	—
67	Dic₇	—	3	3	no	—
68	Dic₇	C₂²×D₃	1	3	yes	—
69	C₂²×D₃	—	7	1	no	—
70	C₃×Q₈	—	7	3	no	—
71	C₃×Q₈	C₂×D₇	1	1	yes	characteristic, complemented
72	C₂×C₃×C₄	—	7	3	no	—
73	C₂×Dic₃	—	7	3	no	—
74	C₄×D₃	—	7	3	no	—
75	C₄×D₃	—	7	3	no	—
76	C₄×D₃	—	7	3	no	—
77	C₄×D₃	C₂×D₇	1	3	yes	—
78	Dic₆	—	7	3	no	—
79	Dic₆	—	7	6	no	—
80	Dic₆	C₂×D₇	1	3	yes	complemented
81	C₃×C₇	C₂²×Q₈	1	1	yes	characteristic, complemented, Hall
82	C₂×Q₈	—	3	1	no	—
83	C₂×Q₈	—	7	1	no	—
84	C₂×Q₈	—	21	3	no	—
85	C₂×Q₈	—	21	3	no	—
86	C₂×Q₈	—	21	3	no	—
87	C₂×Q₈	—	21	1	no	—
88	C₂²×C₄	—	21	3	no	—
89	C₂×C₇	—	3	2	no	—
90	C₂×C₇	C₂³×D₃	1	1	yes	characteristic
91	D₇	—	3	2	no	—
92	D₇	D₃×Q₈	1	2	yes	direct product factor
93	C₂²×C₃	—	7	1	no	—
94	C₂×D₃	—	7	1	no	—
95	C₂×D₃	—	7	4	no	—
96	C₂×D₃	C₂²×D₇	1	1	yes	characteristic
97	C₃×C₄	—	7	3	no	—
98	C₃×C₄	C₂²×D₇	1	3	yes	—
99	Dic₃	—	7	3	no	—
100	Dic₃	C₂²×D₇	1	3	yes	—
101	C₂³	—	21	1	no	—
102	Q₈	—	3	3	no	—
103	Q₈	—	7	3	no	—
104	Q₈	—	21	6	no	—
105	Q₈	—	21	3	no	—
106	Q₈	D₃×D₇	1	1	yes	characteristic, direct product factor
107	C₂×C₄	—	3	3	no	—
108	C₂×C₄	—	7	3	no	—
109	C₂×C₄	—	21	3	no	—
110	C₂×C₄	—	21	3	no	—
111	C₂×C₄	—	21	3	no	—
112	C₂×C₄	—	21	3	no	—
113	C₇	C₂×D₃×Q₈	1	1	yes	characteristic, complemented, Sylow
114	C₂×C₃	—	7	2	no	—
115	C₂×C₃	C₂³×D₇	1	1	yes	characteristic
116	D₃	—	7	2	no	—
117	D₃	Q₈×D₇	1	2	yes	direct product factor
118	C₂²	—	3	1	no	—
119	C₂²	—	7	1	no	—
120	C₂²	—	21	4	no	—
121	C₂²	—	21	1	no	—
122	C₄	—	3	3	no	—
123	C₄	—	7	3	no	—
124	C₄	—	21	3	no	—
125	C₄	C₂×D₃×D₇	1	3	yes	—
126	C₃	C₂×Q₈×D₇	1	1	yes	characteristic, complemented, Sylow
127	C₂	—	3	2	no	—
128	C₂	—	7	2	no	—
129	C₂	—	21	2	no	—
130	C₂	C₂²×D₃×D₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₁

C₂

yes

maximal, characteristic, complemented

D₃×C₇×Q₈

C₂

yes

maximal, characteristic, complemented

C₃×Q₈×D₇

C₂

yes

maximal, characteristic, complemented

C₄×D₃×D₇

C₂

yes

maximal

D₂₁⋊Q₈

C₂

yes

maximal, complemented

D₃×Dic₁₄

C₂

yes

maximal, complemented

D₇×Dic₆

C₂

yes

maximal, complemented

C₂×Q₈×D₇

—

maximal, Hall

C₂×D₃×D₇

C₂²

yes

characteristic

C₃×C₇×Q₈

C₂²

yes

characteristic, complemented

C₄×D₂₁

C₂²

yes

—

Dic₄₂

C₂²

yes

complemented

C₄×D₃×C₇

C₂²

yes

—

C₇×Dic₆

C₂²

yes

complemented

C₃×C₄×D₇

C₂²

yes

—

C₃×Dic₁₄

C₂²

yes

complemented

#4248

C₂²

yes

complemented

D₂₁⋊C₄

C₂²

yes

—

D₃×Dic₇

C₂²

yes

—

Dic₃×D₇

C₂²

yes

—

C₂×C₇×Q₈

—

Q₈×D₇

—

Q₈×D₇

—

Q₈×D₇

—

Q₈×D₇

D₃

yes

characteristic, direct product factor

C₂×C₄×D₇

—

C₂×Dic₁₄

—

C₂×D₃×Q₈

—

maximal, Hall

C₂×D₂₁

C₂³

yes

characteristic

C₂×D₃×C₇

C₂³

yes

characteristic

C₂×C₃×D₇

C₂³

yes

characteristic

D₃×D₇

Q₈

yes

direct product factor

C₃×C₄×C₇

C₂³

yes

—

Dic₂₁

C₂³

yes

—

C₃×Dic₇

C₂³

yes

—

C₇×Dic₃

C₂³

yes

—

C₂²×D₇

—

C₇×Q₈

—

C₇×Q₈

C₂×D₃

yes

characteristic, complemented

C₂×C₄×C₇

—

C₂×Dic₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

—

C₄×D₇

C₂×D₃

yes

—

Dic₁₄

—

Dic₁₄

—

Dic₁₄

C₂×D₃

yes

complemented

C₂×C₃×Q₈

—

D₃×Q₈

—

D₃×Q₈

—

D₃×Q₈

—

D₃×Q₈

D₇

yes

characteristic, direct product factor

C₂×C₄×D₃

—

C₂×Dic₆

—

C₂×C₃×C₇

C₂⁴

yes

characteristic

D₂₁

C₂×Q₈

yes

complemented

C₃×D₇

C₂×Q₈

yes

complemented

D₃×C₇

C₂×Q₈

yes

complemented

C₂²×Q₈

—

Sylow

C₂²×C₇

—

C₂×D₇

—

C₂×D₇

—

C₂×D₇

C₂²×D₃

yes

characteristic

C₄×C₇

—

C₄×C₇

C₂²×D₃

yes

—

Dic₇

—

Dic₇

C₂²×D₃

yes

—

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

C₂×D₇

yes

characteristic, complemented

C₂×C₃×C₄

—

C₂×Dic₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

C₂×D₇

yes

—

Dic₆

—

Dic₆

—

Dic₆

C₂×D₇

yes

complemented

C₃×C₇

C₂²×Q₈

yes

characteristic, complemented, Hall

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂×C₇

—

C₂×C₇

C₂³×D₃

yes

characteristic

D₇

—

D₇

D₃×Q₈

yes

direct product factor

C₂²×C₃

—

C₂×D₃

—

C₂×D₃

—

C₂×D₃

C₂²×D₇

yes

characteristic

C₃×C₄

—

C₃×C₄

C₂²×D₇

yes

—

Dic₃

—

100

Dic₃

C₂²×D₇

yes

—

101

C₂³

—

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

Q₈

D₃×D₇

yes

characteristic, direct product factor

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₇

C₂×D₃×Q₈

yes

characteristic, complemented, Sylow

114

C₂×C₃

—

115

C₂×C₃

C₂³×D₇

yes

characteristic

116

D₃

—

117

D₃

Q₈×D₇

yes

direct product factor

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂×D₃×D₇

yes

—

126

C₃

C₂×Q₈×D₇

yes

characteristic, complemented, Sylow

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

C₂²×D₃×D₇

yes

characteristic

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃×D₇
commutator, socle, cosocle	C₂×C₃×C₇	C₂⁴
Fitting	C₃×C₇×Q₈	C₂²
prime radical	C₂×D₃×D₇	C₂²
hypercenter	Q₈	D₃×D₇
hypocenter	C₃×C₇	C₂²×Q₈
LCS	G ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ Q₈

subgroup

G/H

center, Frattini

C₂

C₂²×D₃×D₇

commutator, socle, cosocle

C₂×C₃×C₇

C₂⁴

Fitting

C₃×C₇×Q₈

C₂²

prime radical

C₂×D₃×D₇

C₂²

hypercenter

Q₈

D₃×D₇

hypocenter

C₃×C₇

C₂²×Q₈

LCS

G ▷ C₂×C₃×C₇ ▷ C₃×C₇

UCS

C₁ ◁ C₂ ◁ Q₈

N	H ≅ G/N	type
1. Q₈×D₂₁	C₂	semidirect
2. D₃×C₇×Q₈	C₂	semidirect
3. C₃×Q₈×D₇	C₂	semidirect
5. D₂₁⋊Q₈	C₂	semidirect
6. D₃×Dic₁₄	C₂	semidirect
7. D₇×Dic₆	C₂	semidirect
10. C₃×C₇×Q₈	C₂²	semidirect
12. Dic₄₂	C₂²	semidirect
14. C₇×Dic₆	C₂²	semidirect
16. C₃×Dic₁₄	C₂²	semidirect
17. #4248	C₂²	semidirect
25. Q₈×D₇	D₃	direct (N × H)
32. D₃×D₇	Q₈	direct (N × H)
39. C₇×Q₈	C₂×D₃	semidirect
48. Dic₁₄	C₂×D₃	semidirect
53. D₃×Q₈	D₇	direct (N × H)
57. D₂₁	C₂×Q₈	semidirect
58. C₃×D₇	C₂×Q₈	semidirect
59. D₃×C₇	C₂×Q₈	semidirect
71. C₃×Q₈	C₂×D₇	semidirect
80. Dic₆	C₂×D₇	semidirect
81. C₃×C₇	C₂²×Q₈	semidirect
92. D₇	D₃×Q₈	direct (N × H)
106. Q₈	D₃×D₇	direct (N × H)
113. C₇	C₂×D₃×Q₈	semidirect
117. D₃	Q₈×D₇	direct (N × H)
126. C₃	C₂×Q₈×D₇	semidirect

H ≅ G/N

type

1. Q₈×D₂₁

C₂

semidirect

2. D₃×C₇×Q₈

C₂

semidirect

3. C₃×Q₈×D₇

C₂

semidirect

5. D₂₁⋊Q₈

C₂

semidirect

6. D₃×Dic₁₄

C₂

semidirect

7. D₇×Dic₆

C₂

semidirect

10. C₃×C₇×Q₈

C₂²

semidirect

12. Dic₄₂

C₂²

semidirect

14. C₇×Dic₆

C₂²

semidirect

16. C₃×Dic₁₄

C₂²

semidirect

17. #4248

C₂²

semidirect

25. Q₈×D₇

D₃

direct (N × H)

32. D₃×D₇

Q₈

direct (N × H)

39. C₇×Q₈

C₂×D₃

semidirect

48. Dic₁₄

C₂×D₃

semidirect

53. D₃×Q₈

D₇

direct (N × H)

57. D₂₁

C₂×Q₈

semidirect

58. C₃×D₇

C₂×Q₈

semidirect

59. D₃×C₇

C₂×Q₈

semidirect

71. C₃×Q₈

C₂×D₇

semidirect

80. Dic₆

C₂×D₇

semidirect

81. C₃×C₇

C₂²×Q₈

semidirect

92. D₇

D₃×Q₈

direct (N × H)

106. Q₈

D₃×D₇

direct (N × H)

113. C₇

C₂×D₃×Q₈

semidirect

117. D₃

Q₈×D₇

direct (N × H)

126. C₃

C₂×Q₈×D₇

semidirect

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

subgroup	index
Q₈×D₂₁	2
D₃×C₇×Q₈	2
C₃×Q₈×D₇	2
C₄×D₃×D₇	2
D₂₁⋊Q₈	2
D₃×Dic₁₄	2
D₇×Dic₆	2
C₂×Q₈×D₇	3
C₂×D₃×Q₈	7

subgroup

index

Q₈×D₂₁

D₃×C₇×Q₈

C₃×Q₈×D₇

C₄×D₃×D₇

D₂₁⋊Q₈

D₃×Dic₁₄

D₇×Dic₆

C₂×Q₈×D₇

C₂×D₃×Q₈

Subgroups — D₃×Q₈×D₇ #10621578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×Q8×D7 #10621578

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×Q₈×D₇ #10621578