Group #30692142 — subgroups

Subgroups — C₆₇×#12891 #30692142

Order 17152.

282 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 166 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×C₄×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1848	C₂	1	2	yes	maximal
3	C₆₇×#1381	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₆₇	C₂²	1	2	yes	—
5	C₂×#108×C₆₇	—	2	2	no	—
6	C₂×#108×C₆₇	C₂²	1	4	yes	—
7	#423×C₆₇	—	2	2	no	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
9	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
10	C₄×C₁₆×C₆₇	C₄	1	2	yes	—
11	C₂×M₁₆×C₆₇	—	2	4	no	—
12	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₆₇	—	2	4	no	—
16	C₂×C₁₆×C₆₇	C₂×C₄	1	4	yes	—
17	#108×C₆₇	—	2	8	no	—
18	#108×C₆₇	—	4	4	no	—
19	C₄⋊C₈×C₆₇	—	4	4	no	—
20	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
23	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
25	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
26	M₁₆×C₆₇	—	4	4	no	—
27	M₁₆×C₆₇	—	4	4	no	—
28	C₂×C₈×C₆₇	—	4	4	no	—
29	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₆₇	SD₁₆	1	4	yes	—
32	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₆₇	#41	1	4	yes	—
35	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₆₇	D₈	1	2	yes	—
38	C₁₆×C₆₇	—	2	4	no	—
39	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	#107	1	2	yes	—
42	C₂×C₄×C₆₇	#107	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#124	1	2	yes	—
45	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
47	C₈×C₆₇	—	8	4	no	—
48	C₈×C₆₇	#106	1	2	yes	—
49	C₈×C₆₇	#106	1	2	yes	—
50	C₈×C₆₇	#117	1	2	yes	—
51	C₈×C₆₇	#125	1	2	yes	—
52	C₂²×C₆₇	—	2	2	no	—
53	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
54	C₂²×C₆₇	#487	1	2	yes	—
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	#368	1	4	yes	—
57	C₄×C₆₇	#382	1	1	yes	characteristic
58	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
59	C₄×C₆₇	#486	1	2	yes	—
60	#12891	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂×C₆₇	—	2	2	no	—
62	C₂×C₆₇	#1834	1	1	yes	characteristic
63	C₂×C₆₇	C₂×#368	1	2	yes	—
64	C₂×C₄×C₁₆	C₂×C₆₇	1	1	yes	characteristic
65	#1848	C₂×C₆₇	1	2	yes	—
66	#1381	C₂×C₆₇	1	4	yes	—
67	C₆₇	#12891	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₁₆	C₂²×C₆₇	1	2	yes	—
69	C₂×#108	—	2	2	no	—
70	C₂×#108	C₂²×C₆₇	1	4	yes	—
71	#423	—	2	2	no	—
72	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
73	C₄×C₁₆	C₄×C₆₇	1	2	yes	—
74	C₄×C₁₆	C₄×C₆₇	1	2	yes	—
75	C₂×M₁₆	—	2	4	no	—
76	C₂²×C₈	D₄×C₆₇	1	1	yes	characteristic
77	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
78	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
79	C₂×C₁₆	—	2	4	no	—
80	C₂×C₁₆	C₂×C₄×C₆₇	1	4	yes	—
81	#108	—	2	8	no	—
82	#108	—	4	4	no	—
83	C₄⋊C₈	—	4	4	no	—
84	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
85	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
86	C₄×C₈	Q₈×C₆₇	1	1	yes	characteristic
87	C₄×C₈	Q₈×C₆₇	1	1	yes	characteristic
88	C₂²×C₄	Q₁₆×C₆₇	1	2	yes	—
89	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
90	M₁₆	—	4	4	no	—
91	M₁₆	—	4	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
94	C₂×C₈	C₄⋊C₄×C₆₇	1	1	yes	characteristic
95	C₂×C₈	SD₁₆×C₆₇	1	4	yes	—
96	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
97	C₂×C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
98	C₂×C₈	#41×C₆₇	1	4	yes	—
99	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
100	C₄²	C₄⋊C₄×C₆₇	1	1	yes	characteristic
101	C₄²	D₈×C₆₇	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	#107×C₆₇	1	2	yes	—
106	C₂×C₄	#107×C₆₇	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
108	C₂×C₄	#124×C₆₇	1	2	yes	—
109	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
110	C₂×C₄	C₂×SD₁₆×C₆₇	1	2	yes	—
111	C₈	—	8	4	no	—
112	C₈	#106×C₆₇	1	2	yes	—
113	C₈	#106×C₆₇	1	2	yes	—
114	C₈	#117×C₆₇	1	2	yes	—
115	C₈	#125×C₆₇	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#107×C₆₇	1	1	yes	characteristic
118	C₂²	#487×C₆₇	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	#368×C₆₇	1	4	yes	—
121	C₄	#382×C₆₇	1	1	yes	characteristic
122	C₄	C₂×#106×C₆₇	1	1	yes	characteristic
123	C₄	#486×C₆₇	1	2	yes	—
124	C₂	—	2	2	no	—
125	C₂	C₆₇×#1834	1	1	yes	characteristic
126	C₂	C₂×#368×C₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₆×C₆₇

C₂

yes

maximal, characteristic

C₆₇×#1848

C₂

yes

maximal

C₆₇×#1381

C₂

yes

maximal

C₂²×C₁₆×C₆₇

C₂²

yes

—

C₂×#108×C₆₇

—

C₂×#108×C₆₇

C₂²

yes

—

#423×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄×C₁₆×C₆₇

C₄

yes

—

C₄×C₁₆×C₆₇

C₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

Q₈

yes

characteristic

C₄×C₈×C₆₇

Q₈

yes

characteristic

C₂²×C₄×C₆₇

Q₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₄⋊C₄

yes

characteristic

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄²×C₆₇

C₄⋊C₄

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

characteristic

C₄²×C₆₇

D₈

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

#107

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

#106

yes

—

C₈×C₆₇

#106

yes

—

C₈×C₆₇

#117

yes

—

C₈×C₆₇

#125

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#487

yes

—

C₄×C₆₇

—

C₄×C₆₇

#368

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

#12891

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1834

yes

characteristic

C₂×C₆₇

C₂×#368

yes

—

C₂×C₄×C₁₆

C₂×C₆₇

yes

characteristic

#1848

C₂×C₆₇

yes

—

#1381

C₂×C₆₇

yes

—

C₆₇

#12891

yes

characteristic, direct product factor, Sylow

C₂²×C₁₆

C₂²×C₆₇

yes

—

C₂×#108

—

C₂×#108

C₂²×C₆₇

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₆₇

yes

characteristic

C₄×C₁₆

C₄×C₆₇

yes

—

C₄×C₁₆

C₄×C₆₇

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₆₇

yes

characteristic

C₂²×C₈

C₂³×C₆₇

yes

characteristic

C₂×C₄²

D₄×C₆₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₆₇

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

C₄×C₈

C₂×C₄×C₆₇

yes

characteristic

C₄×C₈

Q₈×C₆₇

yes

characteristic

C₄×C₈

Q₈×C₆₇

yes

characteristic

C₂²×C₄

Q₁₆×C₆₇

yes

—

C₂²×C₄

C₂×D₄×C₆₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₆₇

yes

characteristic

C₂×C₈

C₄⋊C₄×C₆₇

yes

characteristic

C₂×C₈

SD₁₆×C₆₇

yes

—

C₂×C₈

C₂²×C₄×C₆₇

yes

characteristic

C₂×C₈

C₂×Q₈×C₆₇

yes

characteristic

C₂×C₈

#41×C₆₇

yes

—

C₄²

C₄⋊C₄×C₆₇

yes

characteristic

100

C₄²

C₄⋊C₄×C₆₇

yes

characteristic

101

C₄²

D₈×C₆₇

yes

—

102

C₁₆

—

103

C₂³

C₂×Q₁₆×C₆₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#107×C₆₇

yes

—

106

C₂×C₄

#107×C₆₇

yes

—

107

C₂×C₄

C₂×C₄⋊C₄×C₆₇

yes

characteristic

108

C₂×C₄

#124×C₆₇

yes

—

109

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

110

C₂×C₄

C₂×SD₁₆×C₆₇

yes

—

111

C₈

—

112

C₈

#106×C₆₇

yes

—

113

C₈

#106×C₆₇

yes

—

114

C₈

#117×C₆₇

yes

—

115

C₈

#125×C₆₇

yes

—

116

C₂²

—

117

C₂²

C₂×#107×C₆₇

yes

characteristic

118

C₂²

#487×C₆₇

yes

—

119

C₄

—

120

C₄

#368×C₆₇

yes

—

121

C₄

#382×C₆₇

yes

characteristic

122

C₄

C₂×#106×C₆₇

yes

characteristic

123

C₄

#486×C₆₇

yes

—

124

C₂

—

125

C₂

C₆₇×#1834

yes

characteristic

126

C₂

C₂×#368×C₆₇

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂²×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×C₆₇	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₇

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₆₇

Frattini, cosocle

C₂²×C₈

C₂³×C₆₇

socle

C₂²×C₆₇

C₂×#107

prime radical

C₂³×C₆₇

C₂×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ C₂×C₄×C₈×C₆₇ ◁ G

N	H ≅ G/N	type
60. #12891	C₆₇	direct (N × H)
67. C₆₇	#12891	direct (N × H)

H ≅ G/N

type

60. #12891

C₆₇

direct (N × H)

67. C₆₇

#12891

direct (N × H)

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

subgroup	index
C₂×C₄×C₁₆×C₆₇	2
C₆₇×#1848	2
C₆₇×#1381	2
#12891	67

subgroup

index

C₂×C₄×C₁₆×C₆₇

C₆₇×#1848

C₆₇×#1381

#12891

Subgroups — C₆₇×#12891 #30692142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#12891 #30692142

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#12891 #30692142