Group #30692975 — subgroups

Subgroups — C₆₇×#13724 #30692975

Order 17152.

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

normal subgroups only

This group has 174 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
2	C₆₇×#2247	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1416	C₂	1	4	yes	maximal
4	C₆₇×#1397	C₂	1	1	yes	maximal, characteristic
5	C₂×Q₃₂×C₆₇	—	2	1	no	—
6	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
7	#499×C₆₇	—	2	1	no	—
8	#494×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×Q₁₆×C₆₇	—	2	2	no	—
14	C₂×Q₁₆×C₆₇	—	4	1	no	—
15	C₄⋊Q₈×C₆₇	—	4	1	no	—
16	#125×C₆₇	—	4	1	no	—
17	Q₃₂×C₆₇	—	2	4	no	—
18	Q₃₂×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×Q₈×C₆₇	—	4	2	no	—
26	Q₁₆×C₆₇	—	4	2	no	—
27	Q₁₆×C₆₇	—	4	2	no	—
28	Q₁₆×C₆₇	—	8	2	no	—
29	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
30	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
31	C₄⋊C₄×C₆₇	—	4	2	no	—
32	C₄⋊C₄×C₆₇	—	8	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	Q₈×C₆₇	—	8	2	no	—
38	Q₈×C₆₇	—	8	2	no	—
39	C₂×C₄×C₆₇	—	8	2	no	—
40	C₂×C₄×C₆₇	—	8	2	no	—
41	C₂×C₄×C₆₇	D₁₆	1	2	yes	—
42	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
43	C₈×C₆₇	SD₃₂	1	2	yes	—
44	C₈×C₆₇	#124	1	1	yes	characteristic
45	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
46	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
47	C₄×C₆₇	—	2	2	no	—
48	C₄×C₆₇	—	16	2	no	—
49	C₄×C₆₇	—	16	2	no	—
50	C₄×C₆₇	#486	1	1	yes	characteristic
51	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
52	#13724	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂×C₆₇	#2240	1	1	yes	characteristic
54	C₂×C₆₇	#2264	1	2	yes	—
55	#2253	C₂×C₆₇	1	1	yes	characteristic
56	#2247	C₂×C₆₇	1	1	yes	characteristic
57	#1416	C₂×C₆₇	1	4	yes	—
58	#1397	C₂×C₆₇	1	1	yes	characteristic
59	C₆₇	#13724	1	1	yes	characteristic, direct product factor, Sylow
60	C₂×Q₃₂	—	2	1	no	—
61	C₂×Q₃₂	C₂²×C₆₇	1	2	yes	—
62	#499	—	2	1	no	—
63	#494	—	2	1	no	—
64	C₂×C₃₂	C₂²×C₆₇	1	2	yes	—
65	#367	—	2	1	no	—
66	#366	C₂²×C₆₇	1	2	yes	—
67	C₄×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
68	C₂×Q₁₆	—	2	2	no	—
69	C₂×Q₁₆	—	4	1	no	—
70	C₄⋊Q₈	—	4	1	no	—
71	#125	—	4	1	no	—
72	Q₃₂	—	2	4	no	—
73	Q₃₂	—	4	2	no	—
74	C₂×C₁₆	D₄×C₆₇	1	1	yes	characteristic
75	C₂×C₁₆	C₂³×C₆₇	1	1	yes	characteristic
76	#107	—	2	2	no	—
77	#106	—	4	1	no	—
78	C₄×C₈	D₄×C₆₇	1	1	yes	characteristic
79	C₃₂	—	2	2	no	—
80	C₂×Q₈	—	4	2	no	—
81	Q₁₆	—	4	2	no	—
82	Q₁₆	—	4	2	no	—
83	Q₁₆	—	8	2	no	—
84	C₂×C₈	D₈×C₆₇	1	1	yes	characteristic
85	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
86	C₄⋊C₄	—	4	2	no	—
87	C₄⋊C₄	—	8	2	no	—
88	C₄⋊C₄	—	8	2	no	—
89	C₄²	D₈×C₆₇	1	1	yes	characteristic
90	C₁₆	SD₁₆×C₆₇	1	2	yes	—
91	C₁₆	#41×C₆₇	1	2	yes	—
92	Q₈	—	8	2	no	—
93	Q₈	—	8	2	no	—
94	C₂×C₄	—	8	2	no	—
95	C₂×C₄	—	8	2	no	—
96	C₂×C₄	D₁₆×C₆₇	1	2	yes	—
97	C₂×C₄	C₂×D₈×C₆₇	1	1	yes	characteristic
98	C₈	SD₃₂×C₆₇	1	2	yes	—
99	C₈	#124×C₆₇	1	1	yes	characteristic
100	C₈	C₂×SD₁₆×C₆₇	1	1	yes	characteristic
101	C₂²	C₂×D₁₆×C₆₇	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	16	2	no	—
104	C₄	—	16	2	no	—
105	C₄	#486×C₆₇	1	1	yes	characteristic
106	C₄	C₂×SD₃₂×C₆₇	1	1	yes	characteristic
107	C₂	C₆₇×#2240	1	1	yes	characteristic
108	C₂	C₆₇×#2264	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2253

C₂

yes

maximal, characteristic

C₆₇×#2247

C₂

yes

maximal, characteristic

C₆₇×#1416

C₂

yes

maximal

C₆₇×#1397

C₂

yes

maximal, characteristic

C₂×Q₃₂×C₆₇

—

C₂×Q₃₂×C₆₇

C₂²

yes

—

#499×C₆₇

—

#494×C₆₇

—

C₂×C₃₂×C₆₇

C₂²

yes

—

#367×C₆₇

—

#366×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

#125×C₆₇

—

Q₃₂×C₆₇

—

Q₃₂×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×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₈

yes

characteristic

C₁₆×C₆₇

SD₁₆

yes

—

C₁₆×C₆₇

#41

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×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

C₂×D₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#486

yes

characteristic

C₄×C₆₇

C₂×SD₃₂

yes

characteristic

#13724

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

#2240

yes

characteristic

C₂×C₆₇

#2264

yes

—

#2253

C₂×C₆₇

yes

characteristic

#2247

C₂×C₆₇

yes

characteristic

#1416

C₂×C₆₇

yes

—

#1397

C₂×C₆₇

yes

characteristic

C₆₇

#13724

yes

characteristic, direct product factor, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂²×C₆₇

yes

—

#499

—

#494

—

C₂×C₃₂

C₂²×C₆₇

yes

—

#367

—

#366

C₂²×C₆₇

yes

—

C₄×C₁₆

C₂²×C₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

#125

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

D₄×C₆₇

yes

characteristic

C₂×C₁₆

C₂³×C₆₇

yes

characteristic

#107

—

#106

—

C₄×C₈

D₄×C₆₇

yes

characteristic

C₃₂

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

D₈×C₆₇

yes

characteristic

C₂×C₈

C₂×D₄×C₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₈×C₆₇

yes

characteristic

C₁₆

SD₁₆×C₆₇

yes

—

C₁₆

#41×C₆₇

yes

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₆×C₆₇

yes

—

C₂×C₄

C₂×D₈×C₆₇

yes

characteristic

C₈

SD₃₂×C₆₇

yes

—

C₈

#124×C₆₇

yes

characteristic

100

C₈

C₂×SD₁₆×C₆₇

yes

characteristic

101

C₂²

C₂×D₁₆×C₆₇

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#486×C₆₇

yes

characteristic

106

C₄

C₂×SD₃₂×C₆₇

yes

characteristic

107

C₂

C₆₇×#2240

yes

characteristic

108

C₂

C₆₇×#2264

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²×C₆₇	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₆₇
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, prime radical

C₂²×C₆₇

C₂×D₁₆

commutator, Frattini, cosocle

C₂×C₁₆

C₂³×C₆₇

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
52. #13724	C₆₇	direct (N × H)
59. C₆₇	#13724	direct (N × H)

H ≅ G/N

type

52. #13724

C₆₇

direct (N × H)

59. C₆₇

#13724

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×#2247	2
C₆₇×#1416	2
C₆₇×#1397	2
#13724	67

subgroup

index

C₆₇×#2253

C₆₇×#2247

C₆₇×#1416

C₆₇×#1397

#13724

Subgroups — C₆₇×#13724 #30692975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#13724 #30692975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#13724 #30692975