Group #22573320 — subgroups

Subgroups — C₂×Q₃₂×D₆₇ #22573320

Order 8576.

360 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 125 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×Q₁₆×D₆₇	C₂	1	2	yes	maximal
2	C₂×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14523989	C₂	1	2	yes	maximal, complemented
4	Q₃₂×D₆₇	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₃₂×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Q₈×D₆₇	—	2	2	no	—
8	C₂×Dic₂₆₈	C₂²	1	2	yes	—
9	C₂×#11761618	—	2	2	no	—
10	Q₁₆×D₆₇	—	2	4	no	—
11	Q₁₆×D₆₇	C₂²	1	8	yes	—
12	C₂×C₈×D₆₇	C₂²	1	1	yes	characteristic
13	Dic₅₃₆	C₂²	1	4	yes	complemented
14	#14523989	C₂²	1	8	yes	complemented
15	C₂×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₁₆×D₆₇	C₂²	1	4	yes	—
17	C₂×Q₁₆×C₆₇	C₂²	1	2	yes	—
18	Q₃₂×C₆₇	C₂²	1	4	yes	complemented
19	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₁₃₄	—	4	2	no	—
22	Q₈×D₆₇	—	2	8	no	—
23	Q₈×D₆₇	—	4	4	no	—
24	C₂×C₄×D₆₇	—	4	2	no	—
25	C₂×C₄×D₆₇	D₄	1	1	yes	characteristic
26	Dic₂₆₈	—	2	2	no	—
27	Dic₂₆₈	C₂³	1	4	yes	—
28	#11761618	—	2	8	no	—
29	C₂×C₆₇⋊C₈	D₄	1	1	yes	characteristic
30	C₈×D₆₇	D₄	1	2	yes	—
31	C₈×D₆₇	C₂³	1	2	yes	—
32	C₆₇⋊C₁₆	C₂³	1	2	yes	—
33	C₂×Q₈×C₆₇	—	2	2	no	—
34	Q₁₆×C₆₇	—	2	2	no	—
35	Q₁₆×C₆₇	C₂³	1	4	yes	—
36	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₁₆×C₆₇	C₂³	1	2	yes	—
38	C₂²×D₆₇	D₈	1	1	yes	characteristic
39	Q₈×C₆₇	—	2	4	no	—
40	Q₈×C₆₇	—	4	2	no	—
41	C₂×C₄×C₆₇	—	4	2	no	—
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×Dic₆₇	—	4	2	no	—
44	C₂×Dic₆₇	D₈	1	1	yes	characteristic
45	C₄×D₆₇	—	4	8	no	—
46	C₄×D₆₇	D₈	1	2	yes	—
47	C₄×D₆₇	C₂×D₄	1	2	yes	—
48	Dic₁₃₄	—	2	4	no	—
49	Dic₁₃₄	—	4	2	no	—
50	Dic₁₃₄	—	4	8	no	—
51	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
52	C₈×C₆₇	C₂⁴	1	1	yes	characteristic
53	C₆₇⋊C₈	C₂×D₄	1	2	yes	—
54	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
55	C₂×D₆₇	Q₃₂	1	4	yes	direct product factor
56	C₂×D₆₇	C₂×D₈	1	2	yes	—
57	C₄×C₆₇	—	4	4	no	—
58	C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
59	C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
60	Dic₆₇	—	4	4	no	—
61	Dic₆₇	C₂×D₈	1	2	yes	—
62	C₂×C₆₇	C₂×Q₃₂	1	2	yes	complemented
63	C₂×C₆₇	C₂²×D₈	1	1	yes	characteristic
64	D₆₇	C₂×Q₃₂	1	4	yes	direct product factor
65	C₂²×Q₃₂	—	67	1	no	maximal, Sylow
66	C₆₇	C₂²×Q₃₂	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₁₆	—	67	2	no	—
68	C₂×Q₃₂	—	67	2	no	—
69	C₂×Q₃₂	—	67	8	no	—
70	C₂×Q₃₂	—	67	1	no	—
71	C₂×Q₃₂	D₆₇	1	1	yes	characteristic, direct product factor
72	C₂²×C₁₆	—	67	1	no	—
73	C₂²×Q₈	—	134	2	no	—
74	C₂×Q₁₆	—	67	8	no	—
75	C₂×Q₁₆	—	67	2	no	—
76	C₂×Q₁₆	—	134	4	no	—
77	C₂×Q₁₆	—	134	2	no	—
78	C₂×Q₁₆	C₂×D₆₇	1	2	yes	—
79	C₂²×C₈	—	67	1	no	—
80	Q₃₂	—	67	8	no	—
81	Q₃₂	—	67	4	no	—
82	Q₃₂	C₂×D₆₇	1	4	yes	direct product factor
83	C₂×C₁₆	—	67	1	no	—
84	C₂×C₁₆	—	67	4	no	—
85	C₂×C₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	—	268	2	no	—
88	C₂×Q₈	—	268	4	no	—
89	C₂×Q₈	—	134	8	no	—
90	C₂×Q₈	—	134	2	no	—
91	C₂²×C₄	—	268	2	no	—
92	C₂²×C₄	—	67	1	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	—	67	4	no	—
95	Q₁₆	—	134	8	no	—
96	Q₁₆	—	134	2	no	—
97	Q₁₆	C₂²×D₆₇	1	4	yes	—
98	C₂×C₈	—	67	1	no	—
99	C₂×C₈	—	67	2	no	—
100	C₂×C₈	—	67	2	no	—
101	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
102	C₁₆	—	67	2	no	—
103	C₁₆	C₂²×D₆₇	1	2	yes	—
104	C₂³	—	67	1	no	—
105	Q₈	—	2	4	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	268	8	no	—
108	Q₈	—	268	2	no	—
109	Q₈	—	134	4	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	268	8	no	—
112	C₂×C₄	—	268	2	no	—
113	C₂×C₄	—	67	1	no	—
114	C₂×C₄	—	67	2	no	—
115	C₂×C₄	—	67	2	no	—
116	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
117	C₈	—	67	2	no	—
118	C₈	D₄×D₆₇	1	1	yes	characteristic
119	C₈	C₂³×D₆₇	1	1	yes	characteristic
120	C₂²	—	67	4	no	—
121	C₂²	—	67	2	no	—
122	C₂²	D₈×D₆₇	1	1	yes	characteristic
123	C₄	—	4	4	no	—
124	C₄	—	268	4	no	—
125	C₄	—	67	2	no	—
126	C₄	D₈×D₆₇	1	1	yes	characteristic
127	C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
128	C₂	—	67	4	no	—
129	C₂	Q₃₂×D₆₇	1	2	yes	direct product factor
130	C₂	C₂×D₈×D₆₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₆₇

C₂

yes

maximal

C₂×Dic₅₃₆

C₂

yes

maximal, characteristic, complemented

C₂×#14523989

C₂

yes

maximal, complemented

Q₃₂×D₆₇

C₂

yes

maximal, direct product factor

C₂×C₁₆×D₆₇

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₇

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×#11761618

—

Q₁₆×D₆₇

—

Q₁₆×D₆₇

C₂²

yes

—

C₂×C₈×D₆₇

C₂²

yes

characteristic

Dic₅₃₆

C₂²

yes

complemented

#14523989

C₂²

yes

complemented

C₂×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂²

yes

—

Q₃₂×C₆₇

C₂²

yes

complemented

C₂×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Q₈×D₆₇

—

Q₈×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

D₄

yes

characteristic

Dic₂₆₈

—

Dic₂₆₈

C₂³

yes

—

#11761618

—

C₂×C₆₇⋊C₈

D₄

yes

characteristic

C₈×D₆₇

D₄

yes

—

C₈×D₆₇

C₂³

yes

—

C₆₇⋊C₁₆

C₂³

yes

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₁₆×C₆₇

C₂³

yes

—

C₂²×D₆₇

D₈

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

D₈

yes

characteristic

C₄×D₆₇

—

C₄×D₆₇

D₈

yes

—

C₄×D₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

characteristic

C₈×C₆₇

C₂⁴

yes

characteristic

C₆₇⋊C₈

C₂×D₄

yes

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂×D₆₇

Q₃₂

yes

direct product factor

C₂×D₆₇

C₂×D₈

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×D₈

yes

characteristic

C₄×C₆₇

C₂²×D₄

yes

characteristic

Dic₆₇

—

Dic₆₇

C₂×D₈

yes

—

C₂×C₆₇

C₂×Q₃₂

yes

complemented

C₂×C₆₇

C₂²×D₈

yes

characteristic

D₆₇

C₂×Q₃₂

yes

direct product factor

C₂²×Q₃₂

—

maximal, Sylow

C₆₇

C₂²×Q₃₂

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

D₆₇

yes

characteristic, direct product factor

C₂²×C₁₆

—

C₂²×Q₈

—

134

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

Q₃₂

—

Q₃₂

—

Q₃₂

C₂×D₆₇

yes

direct product factor

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×Q₈

—

268

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

268

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

134

—

Q₁₆

—

134

—

Q₁₆

C₂²×D₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₆₇

yes

characteristic

102

C₁₆

—

103

C₁₆

C₂²×D₆₇

yes

—

104

C₂³

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

268

—

108

Q₈

—

268

—

109

Q₈

—

134

—

110

C₂×C₄

—

111

C₂×C₄

—

268

—

112

C₂×C₄

—

268

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

D₄×D₆₇

yes

characteristic

117

C₈

—

118

C₈

D₄×D₆₇

yes

characteristic

119

C₈

C₂³×D₆₇

yes

characteristic

120

C₂²

—

121

C₂²

—

122

C₂²

D₈×D₆₇

yes

characteristic

123

C₄

—

124

C₄

—

268

—

125

C₄

—

126

C₄

D₈×D₆₇

yes

characteristic

127

C₄

C₂×D₄×D₆₇

yes

characteristic

128

C₂

—

129

C₂

Q₃₂×D₆₇

yes

direct product factor

130

C₂

C₂×D₈×D₆₇

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₆₇
commutator	C₈×C₆₇	C₂⁴
Frattini	C₈	C₂³×D₆₇
Fitting	C₂×Q₃₂×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	C₂×Q₃₂	D₆₇
hypocenter	C₆₇	C₂²×Q₃₂
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₆₇

commutator

C₈×C₆₇

C₂⁴

Frattini

C₈

C₂³×D₆₇

Fitting

C₂×Q₃₂×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

hypercenter

C₂×Q₃₂

D₆₇

hypocenter

C₆₇

C₂²×Q₃₂

LCS

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
2. C₂×Dic₅₃₆	C₂	semidirect
3. C₂×#14523989	C₂	semidirect
4. Q₃₂×D₆₇	C₂	direct (N × H)
6. C₂×Q₃₂×C₆₇	C₂	semidirect
13. Dic₅₃₆	C₂²	semidirect
14. #14523989	C₂²	semidirect
18. Q₃₂×C₆₇	C₂²	semidirect
55. C₂×D₆₇	Q₃₂	direct (N × H)
62. C₂×C₆₇	C₂×Q₃₂	semidirect
64. D₆₇	C₂×Q₃₂	direct (N × H)
66. C₆₇	C₂²×Q₃₂	semidirect
71. C₂×Q₃₂	D₆₇	direct (N × H)
82. Q₃₂	C₂×D₆₇	direct (N × H)
129. C₂	Q₃₂×D₆₇	direct (N × H)

H ≅ G/N

type

2. C₂×Dic₅₃₆

C₂

semidirect

3. C₂×#14523989

C₂

semidirect

4. Q₃₂×D₆₇

C₂

direct (N × H)

6. C₂×Q₃₂×C₆₇

C₂

semidirect

13. Dic₅₃₆

C₂²

semidirect

14. #14523989

C₂²

semidirect

18. Q₃₂×C₆₇

C₂²

semidirect

55. C₂×D₆₇

Q₃₂

direct (N × H)

62. C₂×C₆₇

C₂×Q₃₂

semidirect

64. D₆₇

C₂×Q₃₂

direct (N × H)

66. C₆₇

C₂²×Q₃₂

semidirect

71. C₂×Q₃₂

D₆₇

direct (N × H)

82. Q₃₂

C₂×D₆₇

direct (N × H)

129. C₂

Q₃₂×D₆₇

direct (N × H)

Subgroups — C₂×Q₃₂×D₆₇ #22573320

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q32×D67 #22573320

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₃₂×D₆₇ #22573320