Group #22570622 — subgroups

Subgroups — #494×D₆₇ #22570622

Order 8576.

552 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 165 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	4	yes	maximal
2	C₄⋊Q₈×D₆₇	C₂	1	2	yes	maximal
3	#16863776	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863775	C₂	1	2	yes	maximal, complemented
5	#16863774	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×D₆₇	C₂	1	1	yes	maximal, characteristic
7	#494×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₇	C₂²	1	4	yes	—
9	C₂×Dic₂₆₈	C₂²	1	4	yes	—
10	C₂×#11761618	C₂²	1	8	yes	—
11	Q₁₆×D₆₇	—	2	16	no	—
12	C₂×C₈×D₆₇	C₂²	1	2	yes	—
13	#14524043	C₂²	1	2	yes	—
14	#14524042	—	2	4	no	—
15	#14524041	C₂²	1	4	yes	—
16	C₄⋊C₄×D₆₇	—	2	4	no	—
17	C₄²×D₆₇	C₂²	1	1	yes	characteristic
18	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₈×Dic₆₇	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₆₇	C₂²	1	4	yes	—
21	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
22	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×Dic₁₃₄	—	2	4	no	—
24	C₂×Dic₁₃₄	C₂³	1	4	yes	—
25	Q₈×D₆₇	—	2	16	no	—
26	C₂×C₄×D₆₇	—	2	4	no	—
27	C₂×C₄×D₆₇	D₄	1	2	yes	—
28	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
29	Dic₂₆₈	—	2	8	no	—
30	#11761618	—	2	16	no	—
31	C₂×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₈×D₆₇	D₄	1	8	yes	—
33	C₄⋊Dic₆₇	—	2	4	no	—
34	Dic₆₇⋊C₄	—	2	8	no	—
35	C₄×Dic₆₇	D₄	1	2	yes	—
36	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
38	Q₁₆×C₆₇	—	2	8	no	—
39	C₂×C₈×C₆₇	C₂³	1	2	yes	—
40	C₄⋊C₄×C₆₇	—	2	4	no	—
41	C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₂²×D₆₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₆₇	—	2	8	no	—
44	C₂×C₄×C₆₇	—	2	4	no	—
45	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₆₇	—	2	4	no	—
48	C₂×Dic₆₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₆₇	C₂×D₄	1	2	yes	—
50	C₄×D₆₇	—	4	8	no	—
51	C₄×D₆₇	Q₁₆	1	8	yes	complemented
52	C₄×D₆₇	C₂×D₄	1	4	yes	—
53	Dic₁₃₄	—	2	8	no	—
54	Dic₁₃₄	—	4	8	no	—
55	C₈×C₆₇	C₂×D₄	1	4	yes	—
56	C₆₇⋊C₈	C₂×D₄	1	4	yes	—
57	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
58	C₂×D₆₇	#127	1	2	yes	—
59	C₂×D₆₇	C₂×Q₁₆	1	4	yes	—
60	C₄×C₆₇	—	4	4	no	—
61	C₄×C₆₇	C₂×Q₁₆	1	4	yes	complemented
62	C₄×C₆₇	C₂²×D₄	1	2	yes	—
63	Dic₆₇	—	4	4	no	—
64	Dic₆₇	#127	1	2	yes	—
65	Dic₆₇	C₂×Q₁₆	1	4	yes	complemented
66	C₂×C₆₇	C₂×#127	1	1	yes	characteristic
67	C₂×C₆₇	C₂²×Q₁₆	1	2	yes	—
68	D₆₇	#494	1	4	yes	direct product factor
69	C₂×#494	—	67	1	no	maximal, Sylow
70	C₆₇	C₂×#494	1	1	yes	characteristic, complemented, Sylow
71	C₂²×Q₁₆	—	67	4	no	—
72	C₂×C₄⋊Q₈	—	67	2	no	—
73	#494	—	67	4	no	—
74	#494	—	67	2	no	—
75	#494	—	67	1	no	—
76	#494	D₆₇	1	1	yes	characteristic, direct product factor
77	C₂×C₄×C₈	—	67	1	no	—
78	C₂²×Q₈	—	67	4	no	—
79	C₂×Q₁₆	—	67	8	no	—
80	C₂×Q₁₆	—	67	4	no	—
81	C₂×Q₁₆	—	134	16	no	—
82	C₂×Q₁₆	C₂×D₆₇	1	4	yes	—
83	C₂²×C₈	—	67	2	no	—
84	C₄⋊Q₈	—	67	4	no	—
85	C₄⋊Q₈	—	67	2	no	—
86	C₄⋊Q₈	—	134	4	no	—
87	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
88	C₂×C₄⋊C₄	—	134	4	no	—
89	C₂×C₄²	—	67	1	no	—
90	C₄×C₈	—	67	2	no	—
91	C₄×C₈	—	67	1	no	—
92	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
93	C₂×Q₈	—	67	4	no	—
94	C₂×Q₈	—	134	4	no	—
95	C₂×Q₈	—	134	16	no	—
96	C₂×Q₈	C₂²×D₆₇	1	4	yes	—
97	C₂²×C₄	—	67	2	no	—
98	C₂²×C₄	—	67	1	no	—
99	C₂²×C₄	—	134	4	no	—
100	Q₁₆	—	2	8	no	—
101	Q₁₆	—	134	16	no	—
102	Q₁₆	—	134	8	no	—
103	C₂×C₈	—	67	2	no	—
104	C₂×C₈	—	67	8	no	—
105	C₂×C₈	C₂²×D₆₇	1	2	yes	—
106	C₄⋊C₄	—	2	4	no	—
107	C₄⋊C₄	—	134	8	no	—
108	C₄⋊C₄	—	134	4	no	—
109	C₄²	—	67	2	no	—
110	C₄²	—	67	1	no	—
111	C₄²	C₂²×D₆₇	1	1	yes	characteristic
112	C₂³	—	67	1	no	—
113	Q₈	—	2	8	no	—
114	Q₈	—	268	8	no	—
115	Q₈	—	134	8	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	268	8	no	—
118	C₂×C₄	—	67	2	no	—
119	C₂×C₄	—	67	1	no	—
120	C₂×C₄	—	67	8	no	—
121	C₂×C₄	—	67	4	no	—
122	C₂×C₄	—	134	4	no	—
123	C₂×C₄	D₄×D₆₇	1	2	yes	—
124	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
125	C₈	—	67	4	no	—
126	C₈	D₄×D₆₇	1	4	yes	—
127	C₂²	—	67	4	no	—
128	C₂²	—	67	2	no	—
129	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
130	C₄	—	4	4	no	—
131	C₄	—	268	4	no	—
132	C₄	—	67	4	no	—
133	C₄	—	67	2	no	—
134	C₄	Q₁₆×D₆₇	1	4	yes	complemented
135	C₄	C₂×D₄×D₆₇	1	2	yes	—
136	C₂	—	67	4	no	—
137	C₂	#127×D₆₇	1	1	yes	characteristic
138	C₂	C₂×Q₁₆×D₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₁₆×D₆₇

C₂

yes

maximal

C₄⋊Q₈×D₆₇

C₂

yes

maximal

#16863776

C₂

yes

maximal, characteristic, complemented

#16863775

C₂

yes

maximal, complemented

#16863774

C₂

yes

maximal, complemented

C₄×C₈×D₆₇

C₂

yes

maximal, characteristic

#494×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₇

C₂²

yes

—

C₂×Dic₂₆₈

C₂²

yes

—

C₂×#11761618

C₂²

yes

—

Q₁₆×D₆₇

—

C₂×C₈×D₆₇

C₂²

yes

—

#14524043

C₂²

yes

—

#14524042

—

#14524041

C₂²

yes

—

C₄⋊C₄×D₆₇

—

C₄²×D₆₇

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₆₇

C₂²

yes

—

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

Q₈×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

D₄

yes

—

C₂×C₄×D₆₇

C₂³

yes

characteristic

Dic₂₆₈

—

#11761618

—

C₂×C₆₇⋊C₈

C₂³

yes

—

C₈×D₆₇

D₄

yes

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

D₄

yes

—

C₄×Dic₆₇

C₂³

yes

characteristic

C₂×Q₈×C₆₇

C₂³

yes

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

C₂×D₄

yes

—

C₄×D₆₇

—

C₄×D₆₇

Q₁₆

yes

complemented

C₄×D₆₇

C₂×D₄

yes

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×D₄

yes

—

C₆₇⋊C₈

C₂×D₄

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂×D₆₇

#127

yes

—

C₂×D₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×Q₁₆

yes

complemented

C₄×C₆₇

C₂²×D₄

yes

—

Dic₆₇

—

Dic₆₇

#127

yes

—

Dic₆₇

C₂×Q₁₆

yes

complemented

C₂×C₆₇

C₂×#127

yes

characteristic

C₂×C₆₇

C₂²×Q₁₆

yes

—

D₆₇

#494

yes

direct product factor

C₂×#494

—

maximal, Sylow

C₆₇

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

—

—

—

D₆₇

yes

characteristic, direct product factor

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

C₂×D₆₇

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

C₂²×D₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

100

Q₁₆

—

101

Q₁₆

—

134

—

102

Q₁₆

—

134

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

C₂²×D₆₇

yes

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

134

—

108

C₄⋊C₄

—

134

—

109

C₄²

—

110

C₄²

—

111

C₄²

C₂²×D₆₇

yes

characteristic

112

C₂³

—

113

Q₈

—

114

Q₈

—

268

—

115

Q₈

—

134

—

116

C₂×C₄

—

117

C₂×C₄

—

268

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

134

—

123

C₂×C₄

D₄×D₆₇

yes

—

124

C₂×C₄

C₂³×D₆₇

yes

characteristic

125

C₈

—

126

C₈

D₄×D₆₇

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×D₄×D₆₇

yes

characteristic

130

C₄

—

131

C₄

—

268

—

132

C₄

—

133

C₄

—

134

C₄

Q₁₆×D₆₇

yes

complemented

135

C₄

C₂×D₄×D₆₇

yes

—

136

C₂

—

137

C₂

#127×D₆₇

yes

characteristic

138

C₂

C₂×Q₁₆×D₆₇

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	#494×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×D₄
hypercenter	#494	D₆₇
hypocenter	C₆₇	C₂×#494
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

C₂×D₄×D₆₇

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

#494×C₆₇

C₂

socle

C₂²×C₆₇

C₂²×D₄

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
3. #16863776	C₂	semidirect
4. #16863775	C₂	semidirect
5. #16863774	C₂	semidirect
7. #494×C₆₇	C₂	semidirect
51. C₄×D₆₇	Q₁₆	semidirect
61. C₄×C₆₇	C₂×Q₁₆	semidirect
65. Dic₆₇	C₂×Q₁₆	semidirect
68. D₆₇	#494	direct (N × H)
70. C₆₇	C₂×#494	semidirect
76. #494	D₆₇	direct (N × H)
134. C₄	Q₁₆×D₆₇	semidirect

H ≅ G/N

type

3. #16863776

C₂

semidirect

4. #16863775

C₂

semidirect

5. #16863774

C₂

semidirect

7. #494×C₆₇

C₂

semidirect

51. C₄×D₆₇

Q₁₆

semidirect

61. C₄×C₆₇

C₂×Q₁₆

semidirect

65. Dic₆₇

C₂×Q₁₆

semidirect

68. D₆₇

#494

direct (N × H)

70. C₆₇

C₂×#494

semidirect

76. #494

D₆₇

direct (N × H)

134. C₄

Q₁₆×D₆₇

semidirect

Subgroups — #494×D₆₇ #22570622

Conjugacy classes of subgroups

Complements

Subgroups — #494×D67 #22570622

Conjugacy classes of subgroups

Complements

Subgroups — #494×D₆₇ #22570622