Group #30713200 — subgroups

Subgroups — Q₈×Q₃₂×C₆₇ #30713200

Order 17152.

460 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 234 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×Q₁₆×C₆₇	C₂	1	2	yes	maximal
2	C₆₇×#2252	C₂	1	3	yes	maximal
3	C₆₇×#2227	C₂	1	6	yes	maximal
4	Q₈×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₄×Q₃₂×C₆₇	C₂	1	3	yes	maximal
6	Q₈²×C₆₇	—	2	2	no	—
7	C₂×Q₃₂×C₆₇	C₂²	1	1	yes	characteristic
8	#500×C₆₇	C₂²	1	6	yes	—
9	#477×C₆₇	—	2	6	no	—
10	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₄×Q₁₆×C₆₇	C₂²	1	6	yes	—
12	#366×C₆₇	C₂²	1	3	yes	—
13	#366×C₆₇	C₂²	1	6	yes	—
14	C₄⋊C₁₆×C₆₇	C₂²	1	3	yes	—
15	#358×C₆₇	C₂²	1	6	yes	—
16	C₄×C₁₆×C₆₇	C₂²	1	3	yes	—
17	C₂×Q₁₆×C₆₇	C₂³	1	2	yes	—
18	C₄⋊Q₈×C₆₇	—	2	6	no	—
19	C₄⋊Q₈×C₆₇	—	4	6	no	—
20	C₄×Q₈×C₆₇	—	2	6	no	—
21	C₄×Q₈×C₆₇	—	4	2	no	—
22	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
23	Q₃₂×C₆₇	Q₈	1	4	yes	direct product factor
24	C₂×C₁₆×C₆₇	C₂³	1	3	yes	—
25	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
26	#107×C₆₇	—	2	6	no	—
27	#107×C₆₇	C₂³	1	6	yes	—
28	C₄⋊C₈×C₆₇	D₄	1	3	yes	—
29	#103×C₆₇	—	2	6	no	—
30	C₄×C₈×C₆₇	C₂³	1	3	yes	—
31	C₂×Q₈×C₆₇	—	2	2	no	—
32	C₂×Q₈×C₆₇	—	4	6	no	—
33	C₂×Q₈×C₆₇	D₈	1	1	yes	characteristic
34	Q₁₆×C₆₇	—	2	2	no	—
35	Q₁₆×C₆₇	C₂×Q₈	1	4	yes	—
36	C₂×C₈×C₆₇	C₂×D₄	1	3	yes	—
37	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
38	C₄⋊C₄×C₆₇	—	2	6	no	—
39	C₄⋊C₄×C₆₇	—	4	6	no	—
40	C₄⋊C₄×C₆₇	—	4	6	no	—
41	C₄⋊C₄×C₆₇	D₈	1	3	yes	—
42	C₄²×C₆₇	—	4	6	no	—
43	C₄²×C₆₇	C₂×D₄	1	3	yes	—
44	C₁₆×C₆₇	—	2	3	no	—
45	C₁₆×C₆₇	C₂×Q₈	1	2	yes	—
46	Q₈×C₆₇	—	2	4	no	—
47	Q₈×C₆₇	—	4	2	no	—
48	Q₈×C₆₇	—	16	6	no	—
49	Q₈×C₆₇	Q₃₂	1	4	yes	direct product factor
50	C₂×C₄×C₆₇	—	4	2	no	—
51	C₂×C₄×C₆₇	—	4	6	no	—
52	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
53	C₂×C₄×C₆₇	C₂×D₈	1	3	yes	—
54	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
55	C₈×C₆₇	—	2	3	no	—
56	C₈×C₆₇	C₂²×Q₈	1	1	yes	characteristic
57	C₈×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂²×C₆₇	C₂²×D₈	1	1	yes	characteristic
59	C₄×C₆₇	—	2	3	no	—
60	C₄×C₆₇	—	4	4	no	—
61	C₄×C₆₇	—	8	6	no	—
62	C₄×C₆₇	C₂×Q₃₂	1	6	yes	—
63	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
64	C₄×C₆₇	#578	1	1	yes	characteristic
65	Q₈×Q₃₂	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₆₇	Q₈×D₈	1	1	yes	characteristic
67	C₂×C₆₇	C₂²×Q₃₂	1	1	yes	characteristic
68	C₂×C₆₇	#3415	1	1	yes	characteristic
69	Q₈×Q₁₆	C₂×C₆₇	1	2	yes	—
70	#2252	C₂×C₆₇	1	3	yes	—
71	#2227	C₂×C₆₇	1	6	yes	—
72	Q₈×C₁₆	C₂×C₆₇	1	1	yes	characteristic
73	C₄×Q₃₂	C₂×C₆₇	1	3	yes	—
74	C₆₇	Q₈×Q₃₂	1	1	yes	characteristic, direct product factor, Sylow
75	Q₈²	—	2	2	no	—
76	C₂×Q₃₂	C₂²×C₆₇	1	1	yes	characteristic
77	#500	C₂²×C₆₇	1	6	yes	—
78	#477	—	2	6	no	—
79	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
80	C₄×Q₁₆	C₂²×C₆₇	1	6	yes	—
81	#366	C₂²×C₆₇	1	3	yes	—
82	#366	C₂²×C₆₇	1	6	yes	—
83	C₄⋊C₁₆	C₂²×C₆₇	1	3	yes	—
84	#358	C₂²×C₆₇	1	6	yes	—
85	C₄×C₁₆	C₂²×C₆₇	1	3	yes	—
86	C₂×Q₁₆	C₂³×C₆₇	1	2	yes	—
87	C₄⋊Q₈	—	2	6	no	—
88	C₄⋊Q₈	—	4	6	no	—
89	C₄×Q₈	—	2	6	no	—
90	C₄×Q₈	—	4	2	no	—
91	C₄×Q₈	D₄×C₆₇	1	1	yes	characteristic
92	Q₃₂	Q₈×C₆₇	1	4	yes	direct product factor
93	C₂×C₁₆	C₂³×C₆₇	1	3	yes	—
94	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
95	#107	—	2	6	no	—
96	#107	C₂³×C₆₇	1	6	yes	—
97	C₄⋊C₈	D₄×C₆₇	1	3	yes	—
98	#103	—	2	6	no	—
99	C₄×C₈	C₂³×C₆₇	1	3	yes	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	—	4	6	no	—
102	C₂×Q₈	D₈×C₆₇	1	1	yes	characteristic
103	Q₁₆	—	2	2	no	—
104	Q₁₆	C₂×Q₈×C₆₇	1	4	yes	—
105	C₂×C₈	C₂×D₄×C₆₇	1	3	yes	—
106	C₂×C₈	C₂⁴×C₆₇	1	1	yes	characteristic
107	C₄⋊C₄	—	2	6	no	—
108	C₄⋊C₄	—	4	6	no	—
109	C₄⋊C₄	—	4	6	no	—
110	C₄⋊C₄	D₈×C₆₇	1	3	yes	—
111	C₄²	—	4	6	no	—
112	C₄²	C₂×D₄×C₆₇	1	3	yes	—
113	C₁₆	—	2	3	no	—
114	C₁₆	C₂×Q₈×C₆₇	1	2	yes	—
115	Q₈	—	2	4	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	16	6	no	—
118	Q₈	Q₃₂×C₆₇	1	4	yes	direct product factor
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	4	6	no	—
121	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
122	C₂×C₄	C₂×D₈×C₆₇	1	3	yes	—
123	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
124	C₈	—	2	3	no	—
125	C₈	C₂²×Q₈×C₆₇	1	1	yes	characteristic
126	C₈	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
127	C₂²	C₂²×D₈×C₆₇	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	4	4	no	—
130	C₄	—	8	6	no	—
131	C₄	C₂×Q₃₂×C₆₇	1	6	yes	—
132	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
133	C₄	#578×C₆₇	1	1	yes	characteristic
134	C₂	Q₈×D₈×C₆₇	1	1	yes	characteristic
135	C₂	C₂²×Q₃₂×C₆₇	1	1	yes	characteristic
136	C₂	C₆₇×#3415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Q₁₆×C₆₇

C₂

yes

maximal

C₆₇×#2252

C₂

yes

maximal

C₆₇×#2227

C₂

yes

maximal

Q₈×C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₄×Q₃₂×C₆₇

C₂

yes

maximal

Q₈²×C₆₇

—

C₂×Q₃₂×C₆₇

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#477×C₆₇

—

C₈×Q₈×C₆₇

C₂²

yes

characteristic

C₄×Q₁₆×C₆₇

C₂²

yes

—

#366×C₆₇

C₂²

yes

—

#366×C₆₇

C₂²

yes

—

C₄⋊C₁₆×C₆₇

C₂²

yes

—

#358×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂³

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

D₄

yes

characteristic

Q₃₂×C₆₇

Q₈

yes

direct product factor

C₂×C₁₆×C₆₇

C₂³

yes

—

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

#107×C₆₇

C₂³

yes

—

C₄⋊C₈×C₆₇

D₄

yes

—

#103×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

D₈

yes

characteristic

Q₁₆×C₆₇

—

Q₁₆×C₆₇

C₂×Q₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂⁴

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₈

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂×Q₈

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

Q₃₂

yes

direct product factor

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

C₂²×Q₈

yes

characteristic

C₈×C₆₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₆₇

C₂²×D₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₃₂

yes

—

C₄×C₆₇

D₄×Q₈

yes

characteristic

C₄×C₆₇

#578

yes

characteristic

Q₈×Q₃₂

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

Q₈×D₈

yes

characteristic

C₂×C₆₇

C₂²×Q₃₂

yes

characteristic

C₂×C₆₇

#3415

yes

characteristic

Q₈×Q₁₆

C₂×C₆₇

yes

—

#2252

C₂×C₆₇

yes

—

#2227

C₂×C₆₇

yes

—

Q₈×C₁₆

C₂×C₆₇

yes

characteristic

C₄×Q₃₂

C₂×C₆₇

yes

—

C₆₇

Q₈×Q₃₂

yes

characteristic, direct product factor, Sylow

Q₈²

—

C₂×Q₃₂

C₂²×C₆₇

yes

characteristic

#500

C₂²×C₆₇

yes

—

#477

—

C₈×Q₈

C₂²×C₆₇

yes

characteristic

C₄×Q₁₆

C₂²×C₆₇

yes

—

#366

C₂²×C₆₇

yes

—

#366

C₂²×C₆₇

yes

—

C₄⋊C₁₆

C₂²×C₆₇

yes

—

#358

C₂²×C₆₇

yes

—

C₄×C₁₆

C₂²×C₆₇

yes

—

C₂×Q₁₆

C₂³×C₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₄×C₆₇

yes

characteristic

Q₃₂

Q₈×C₆₇

yes

direct product factor

C₂×C₁₆

C₂³×C₆₇

yes

—

C₂×C₁₆

C₂³×C₆₇

yes

characteristic

#107

—

#107

C₂³×C₆₇

yes

—

C₄⋊C₈

D₄×C₆₇

yes

—

#103

—

C₄×C₈

C₂³×C₆₇

yes

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

D₈×C₆₇

yes

characteristic

103

Q₁₆

—

104

Q₁₆

C₂×Q₈×C₆₇

yes

—

105

C₂×C₈

C₂×D₄×C₆₇

yes

—

106

C₂×C₈

C₂⁴×C₆₇

yes

characteristic

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

D₈×C₆₇

yes

—

111

C₄²

—

112

C₄²

C₂×D₄×C₆₇

yes

—

113

C₁₆

—

114

C₁₆

C₂×Q₈×C₆₇

yes

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

Q₈

Q₃₂×C₆₇

yes

direct product factor

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

C₂×D₈×C₆₇

yes

—

122

C₂×C₄

C₂×D₈×C₆₇

yes

—

123

C₂×C₄

C₂²×D₄×C₆₇

yes

characteristic

124

C₈

—

125

C₈

C₂²×Q₈×C₆₇

yes

characteristic

126

C₈

2₋¹⁺⁴×C₆₇

yes

characteristic

127

C₂²

C₂²×D₈×C₆₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

C₂×Q₃₂×C₆₇

yes

—

132

C₄

D₄×Q₈×C₆₇

yes

characteristic

133

C₄

#578×C₆₇

yes

characteristic

134

C₂

Q₈×D₈×C₆₇

yes

characteristic

135

C₂

C₂²×Q₃₂×C₆₇

yes

characteristic

136

C₂

C₆₇×#3415

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₆₇
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₆₇

C₂²×D₈

commutator, Frattini, cosocle

C₂×C₈

C₂⁴×C₆₇

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₄×Q₈×C₆₇ ◁ C₈×Q₈×C₆₇ ◁ G

N	H ≅ G/N	type
23. Q₃₂×C₆₇	Q₈	direct (N × H)
49. Q₈×C₆₇	Q₃₂	direct (N × H)
65. Q₈×Q₃₂	C₆₇	direct (N × H)
74. C₆₇	Q₈×Q₃₂	direct (N × H)
92. Q₃₂	Q₈×C₆₇	direct (N × H)
118. Q₈	Q₃₂×C₆₇	direct (N × H)

H ≅ G/N

type

23. Q₃₂×C₆₇

Q₈

direct (N × H)

49. Q₈×C₆₇

Q₃₂

direct (N × H)

65. Q₈×Q₃₂

C₆₇

direct (N × H)

74. C₆₇

Q₈×Q₃₂

direct (N × H)

92. Q₃₂

Q₈×C₆₇

direct (N × H)

118. Q₈

Q₃₂×C₆₇

direct (N × H)

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

subgroup	index
Q₈×Q₁₆×C₆₇	2
C₆₇×#2252	2
C₆₇×#2227	2
Q₈×C₁₆×C₆₇	2
C₄×Q₃₂×C₆₇	2
Q₈×Q₃₂	67

subgroup

index

Q₈×Q₁₆×C₆₇

C₆₇×#2252

C₆₇×#2227

Q₈×C₁₆×C₆₇

C₄×Q₃₂×C₆₇

Q₈×Q₃₂

Subgroups — Q₈×Q₃₂×C₆₇ #30713200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×Q32×C67 #30713200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Q₃₂×C₆₇ #30713200