Group #30692974 — subgroups

Subgroups — C₆₇×#13723 #30692974

Order 17152.

194 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 78 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	C₆₇×#2253	C₂	1	1	yes	maximal, characteristic, complemented
2	C₆₇×#2246	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1415	C₂	1	4	yes	maximal, complemented
4	C₆₇×#1397	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₆×C₆₇	—	2	1	no	—
6	C₂×D₁₆×C₆₇	C₂²	1	2	yes	—
7	#499×C₆₇	—	2	1	no	—
8	#493×C₆₇	—	2	1	no	—
9	C₂×C₃₂×C₆₇	C₂²	1	2	yes	—
10	#367×C₆₇	—	2	1	no	—
11	#366×C₆₇	C₂²	1	2	yes	—
12	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×D₈×C₆₇	—	2	2	no	—
14	C₂×D₈×C₆₇	—	4	1	no	—
15	#127×C₆₇	—	4	1	no	—
16	#125×C₆₇	—	4	1	no	—
17	D₁₆×C₆₇	—	2	4	no	—
18	D₁₆×C₆₇	—	4	2	no	—
19	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
20	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
21	#107×C₆₇	—	2	2	no	—
22	#106×C₆₇	—	4	1	no	—
23	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
24	C₃₂×C₆₇	—	2	2	no	—
25	C₂×D₄×C₆₇	—	4	2	no	—
26	C₂×D₄×C₆₇	—	8	2	no	—
27	D₈×C₆₇	—	4	2	no	—
28	D₈×C₆₇	—	4	2	no	—
29	D₈×C₆₇	—	8	2	no	—
30	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
31	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₆₇	—	4	2	no	—
33	C₄⋊C₄×C₆₇	—	8	2	no	—
34	C₄²×C₆₇	D₈	1	1	yes	characteristic
35	C₁₆×C₆₇	SD₁₆	1	2	yes	—
36	C₁₆×C₆₇	#41	1	2	yes	—
37	C₂³×C₆₇	—	8	2	no	—
38	D₄×C₆₇	—	8	2	no	—
39	D₄×C₆₇	—	8	2	no	—
40	D₄×C₆₇	—	16	4	no	—
41	C₂×C₄×C₆₇	—	8	2	no	—
42	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
44	C₈×C₆₇	SD₃₂	1	2	yes	—
45	C₈×C₆₇	#124	1	1	yes	characteristic
46	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
47	C₂²×C₆₇	—	16	2	no	—
48	C₂²×C₆₇	—	16	4	no	—
49	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
50	C₄×C₆₇	—	2	2	no	—
51	C₄×C₆₇	—	16	2	no	—
52	C₄×C₆₇	#486	1	1	yes	characteristic
53	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
54	#13723	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂×C₆₇	—	32	2	no	—
56	C₂×C₆₇	#2240	1	1	yes	characteristic
57	C₂×C₆₇	#2263	1	2	yes	—
58	#2253	C₂×C₆₇	1	1	yes	characteristic, complemented
59	#2246	C₂×C₆₇	1	1	yes	characteristic
60	#1415	C₂×C₆₇	1	4	yes	complemented
61	#1397	C₂×C₆₇	1	1	yes	characteristic, complemented
62	C₆₇	#13723	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×D₁₆	—	2	1	no	—
64	C₂×D₁₆	C₂²×C₆₇	1	2	yes	—
65	#499	—	2	1	no	—
66	#493	—	2	1	no	—
67	C₂×C₃₂	C₂²×C₆₇	1	2	yes	—
68	#367	—	2	1	no	—
69	#366	C₂²×C₆₇	1	2	yes	—
70	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
71	C₂×D₈	—	2	2	no	—
72	C₂×D₈	—	4	1	no	—
73	#127	—	4	1	no	—
74	#125	—	4	1	no	—
75	D₁₆	—	2	4	no	—
76	D₁₆	—	4	2	no	—
77	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
78	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
79	#107	—	2	2	no	—
80	#106	—	4	1	no	—
81	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
82	C₃₂	—	2	2	no	—
83	C₂×D₄	—	4	2	no	—
84	C₂×D₄	—	8	2	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	D₈	—	8	2	no	—
88	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
89	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	8	2	no	—
92	C₄²	D₈×C₆₇	1	1	yes	characteristic
93	C₁₆	SD₁₆×C₆₇	1	2	yes	—
94	C₁₆	#41×C₆₇	1	2	yes	—
95	C₂³	—	8	2	no	—
96	D₄	—	8	2	no	—
97	D₄	—	8	2	no	—
98	D₄	—	16	4	no	—
99	C₂×C₄	—	8	2	no	—
100	C₂×C₄	D₁₆×C₆₇	1	2	yes	—
101	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
102	C₈	SD₃₂×C₆₇	1	2	yes	—
103	C₈	#124×C₆₇	1	1	yes	characteristic
104	C₈	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
105	C₂²	—	16	4	no	—
106	C₂²	—	16	2	no	—
107	C₂²	C₂×D₁₆×C₆₇	1	1	yes	characteristic
108	C₄	—	2	2	no	—
109	C₄	—	16	2	no	—
110	C₄	#486×C₆₇	1	1	yes	characteristic
111	C₄	C₂×SD₃₂×C₆₇	1	1	yes	characteristic
112	C₂	—	32	2	no	—
113	C₂	C₆₇×#2240	1	1	yes	characteristic
114	C₂	C₆₇×#2263	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2253

C₂

yes

maximal, characteristic, complemented

C₆₇×#2246

C₂

yes

maximal, characteristic

C₆₇×#1415

C₂

yes

maximal, complemented

C₆₇×#1397

C₂

yes

maximal, characteristic, complemented

C₂×D₁₆×C₆₇

—

C₂×D₁₆×C₆₇

C₂²

yes

—

#499×C₆₇

—

#493×C₆₇

—

C₂×C₃₂×C₆₇

C₂²

yes

—

#367×C₆₇

—

#366×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×D₈×C₆₇

—

C₂×D₈×C₆₇

—

#127×C₆₇

—

#125×C₆₇

—

D₁₆×C₆₇

—

D₁₆×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

#107×C₆₇

—

#106×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₃₂×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₈

yes

characteristic

C₁₆×C₆₇

SD₁₆

yes

—

C₁₆×C₆₇

#41

yes

—

C₂³×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

D₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₈×C₆₇

SD₃₂

yes

—

C₈×C₆₇

#124

yes

characteristic

C₈×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×D₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#486

yes

characteristic

C₄×C₆₇

C₂×SD₃₂

yes

characteristic

#13723

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#2240

yes

characteristic

C₂×C₆₇

#2263

yes

—

#2253

C₂×C₆₇

yes

characteristic, complemented

#2246

C₂×C₆₇

yes

characteristic

#1415

C₂×C₆₇

yes

complemented

#1397

C₂×C₆₇

yes

characteristic, complemented

C₆₇

#13723

yes

characteristic, direct product factor, Sylow

C₂×D₁₆

—

C₂×D₁₆

C₂²×C₆₇

yes

—

#499

—

#493

—

C₂×C₃₂

C₂²×C₆₇

yes

—

#367

—

#366

C₂²×C₆₇

yes

—

C₄×C₁₆

C₂²×C₆₇

yes

characteristic

C₂×D₈

—

C₂×D₈

—

#127

—

#125

—

D₁₆

—

D₁₆

—

C₂×C₁₆

D₄×C₆₇

yes

characteristic

C₂×C₁₆

C₂³×C₆₇

yes

characteristic

#107

—

#106

—

C₄×C₈

D₄×C₆₇

yes

characteristic

C₃₂

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

D₈×C₆₇

yes

characteristic

C₂×C₈

C₂×D₄×C₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₈×C₆₇

yes

characteristic

C₁₆

SD₁₆×C₆₇

yes

—

C₁₆

#41×C₆₇

yes

—

C₂³

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

D₁₆×C₆₇

yes

—

101

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

102

C₈

SD₃₂×C₆₇

yes

—

103

C₈

#124×C₆₇

yes

characteristic

104

C₈

C₂×SD₁₆×C₆₇

yes

characteristic

105

C₂²

—

106

C₂²

—

107

C₂²

C₂×D₁₆×C₆₇

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

#486×C₆₇

yes

characteristic

111

C₄

C₂×SD₃₂×C₆₇

yes

characteristic

112

C₂

—

113

C₂

C₆₇×#2240

yes

characteristic

114

C₂

C₆₇×#2263

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₇
prime radical	C₆₇×#2246	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₆₇

prime radical

C₆₇×#2246

C₂

LCS

G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ C₄×C₈×C₆₇ ◁ C₄×C₁₆×C₆₇ ◁ G

N	H ≅ G/N	type
1. C₆₇×#2253	C₂	semidirect
3. C₆₇×#1415	C₂	semidirect
4. C₆₇×#1397	C₂	semidirect
54. #13723	C₆₇	direct (N × H)
58. #2253	C₂×C₆₇	semidirect
60. #1415	C₂×C₆₇	semidirect
61. #1397	C₂×C₆₇	semidirect
62. C₆₇	#13723	direct (N × H)

H ≅ G/N

type

1. C₆₇×#2253

C₂

semidirect

3. C₆₇×#1415

C₂

semidirect

4. C₆₇×#1397

C₂

semidirect

54. #13723

C₆₇

direct (N × H)

58. #2253

C₂×C₆₇

semidirect

60. #1415

C₂×C₆₇

semidirect

61. #1397

C₂×C₆₇

semidirect

62. C₆₇

#13723

direct (N × H)

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

subgroup	index
C₆₇×#2253	2
C₆₇×#2246	2
C₆₇×#1415	2
C₆₇×#1397	2
#13723	67

subgroup

index

C₆₇×#2253

C₆₇×#2246

C₆₇×#1415

C₆₇×#1397

#13723

Subgroups — C₆₇×#13723 #30692974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#13723 #30692974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13723 #30692974