Group #30688053 — subgroups

Subgroups — C₆₇×#8802 #30688053

Order 17152.

258 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 110 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×#1745	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#1281	C₂	1	2	yes	maximal
3	C₆₇×#1276	C₂	1	2	yes	maximal
4	C₆₇×#1275	C₂	1	2	yes	maximal
5	#431×C₆₇	C₂²	1	4	yes	—
6	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
8	C₄³×C₆₇	C₂²	1	1	yes	characteristic
9	#336×C₆₇	C₄	1	4	yes	—
10	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
11	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
12	C₂×C₄²×C₆₇	—	2	2	no	—
13	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
15	C₄⋊C₈×C₆₇	—	2	4	no	—
16	C₂²⋊C₈×C₆₇	—	2	4	no	—
17	#97×C₆₇	—	2	2	no	—
18	#97×C₆₇	—	2	2	no	—
19	C₄×C₈×C₆₇	—	2	4	no	—
20	C₂²×C₄×C₆₇	—	2	2	no	—
21	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₆₇	#41	1	2	yes	—
23	C₂×C₈×C₆₇	—	2	2	no	—
24	C₂×C₈×C₆₇	—	2	2	no	—
25	C₂×C₈×C₆₇	—	2	4	no	—
26	C₂×C₈×C₆₇	—	2	4	no	—
27	C₂×C₈×C₆₇	—	4	4	no	—
28	C₄²×C₆₇	—	2	4	no	—
29	C₄²×C₆₇	—	4	4	no	—
30	C₄²×C₆₇	#41	1	2	yes	—
31	C₄²×C₆₇	#41	1	2	yes	—
32	C₂³×C₆₇	#117	1	1	yes	characteristic
33	C₂×C₄×C₆₇	—	2	4	no	—
34	C₂×C₄×C₆₇	—	4	2	no	—
35	C₂×C₄×C₆₇	—	4	2	no	—
36	C₂×C₄×C₆₇	—	4	2	no	—
37	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
38	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
39	C₂×C₄×C₆₇	#126	1	4	yes	—
40	C₂×C₄×C₆₇	#131	1	4	yes	—
41	C₈×C₆₇	—	4	2	no	—
42	C₈×C₆₇	—	4	2	no	—
43	C₈×C₆₇	—	8	4	no	—
44	C₂²×C₆₇	—	2	2	no	—
45	C₂²×C₆₇	#383	1	1	yes	characteristic
46	C₂²×C₆₇	#433	1	2	yes	—
47	C₄×C₆₇	—	2	4	no	—
48	C₄×C₆₇	—	4	4	no	—
49	C₄×C₆₇	#433	1	2	yes	—
50	C₄×C₆₇	#433	1	2	yes	—
51	#8802	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×C₆₇	—	2	2	no	—
53	C₂×C₆₇	#1840	1	1	yes	characteristic
54	C₂×C₆₇	#1841	1	2	yes	—
55	#1745	C₂×C₆₇	1	1	yes	characteristic
56	#1281	C₂×C₆₇	1	2	yes	—
57	#1276	C₂×C₆₇	1	2	yes	—
58	#1275	C₂×C₆₇	1	2	yes	—
59	C₆₇	#8802	1	1	yes	characteristic, direct product factor, Sylow
60	#431	C₂²×C₆₇	1	4	yes	—
61	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
62	C₂×#97	C₂²×C₆₇	1	1	yes	characteristic
63	C₄³	C₂²×C₆₇	1	1	yes	characteristic
64	#336	C₄×C₆₇	1	4	yes	—
65	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
66	C₂²×C₈	C₂×C₄×C₆₇	1	2	yes	—
67	C₂×C₄²	—	2	2	no	—
68	C₂×C₄²	C₂×C₄×C₆₇	1	2	yes	—
69	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
70	C₄⋊C₈	—	2	4	no	—
71	C₂²⋊C₈	—	2	4	no	—
72	#97	—	2	2	no	—
73	#97	—	2	2	no	—
74	C₄×C₈	—	2	4	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
77	C₂²×C₄	#41×C₆₇	1	2	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	4	4	no	—
83	C₄²	—	2	4	no	—
84	C₄²	—	4	4	no	—
85	C₄²	#41×C₆₇	1	2	yes	—
86	C₄²	#41×C₆₇	1	2	yes	—
87	C₂³	#117×C₆₇	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
93	C₂×C₄	#117×C₆₇	1	1	yes	characteristic
94	C₂×C₄	#126×C₆₇	1	4	yes	—
95	C₂×C₄	#131×C₆₇	1	4	yes	—
96	C₈	—	4	2	no	—
97	C₈	—	4	2	no	—
98	C₈	—	8	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	#383×C₆₇	1	1	yes	characteristic
101	C₂²	#433×C₆₇	1	2	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	4	4	no	—
104	C₄	#433×C₆₇	1	2	yes	—
105	C₄	#433×C₆₇	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	C₆₇×#1840	1	1	yes	characteristic
108	C₂	C₆₇×#1841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#1745

C₂

yes

maximal, characteristic

C₆₇×#1281

C₂

yes

maximal

C₆₇×#1276

C₂

yes

maximal

C₆₇×#1275

C₂

yes

maximal

#431×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×#97×C₆₇

C₂²

yes

characteristic

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

#97×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#383

yes

characteristic

C₂²×C₆₇

#433

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

#433

yes

—

#8802

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1840

yes

characteristic

C₂×C₆₇

#1841

yes

—

#1745

C₂×C₆₇

yes

characteristic

#1281

C₂×C₆₇

yes

—

#1276

C₂×C₆₇

yes

—

#1275

C₂×C₆₇

yes

—

C₆₇

#8802

yes

characteristic, direct product factor, Sylow

#431

C₂²×C₆₇

yes

—

C₂×#97

C₂²×C₆₇

yes

characteristic

C₂×#97

C₂²×C₆₇

yes

characteristic

C₄³

C₂²×C₆₇

yes

characteristic

#336

C₄×C₆₇

yes

—

C₂²×C₈

C₂×C₄×C₆₇

yes

—

C₂²×C₈

C₂×C₄×C₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₆₇

yes

—

C₂×C₄²

C₂³×C₆₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₆₇

yes

characteristic

C₂²×C₄

#41×C₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

#41×C₆₇

yes

—

C₄²

#41×C₆₇

yes

—

C₂³

#117×C₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#117×C₆₇

yes

characteristic

C₂×C₄

#117×C₆₇

yes

characteristic

C₂×C₄

#126×C₆₇

yes

—

C₂×C₄

#131×C₆₇

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

#383×C₆₇

yes

characteristic

101

C₂²

#433×C₆₇

yes

—

102

C₄

—

103

C₄

—

104

C₄

#433×C₆₇

yes

—

105

C₄

#433×C₆₇

yes

—

106

C₂

—

107

C₂

C₆₇×#1840

yes

characteristic

108

C₂

C₆₇×#1841

yes

—

	subgroup	G/H
center	C₂×C₄×C₆₇	#117
commutator	C₂²×C₄	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
socle	C₂²×C₆₇	#383
prime radical	C₂³×C₆₇	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₆₇

#117

commutator

C₂²×C₄

C₂²×C₄×C₆₇

Frattini, cosocle

C₂×C₄²

C₂³×C₆₇

C₂²×C₆₇

C₂³×C₆₇

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₆₇ ◁ C₂×#97×C₆₇ ◁ G

N	H ≅ G/N	type
51. #8802	C₆₇	direct (N × H)
59. C₆₇	#8802	direct (N × H)

H ≅ G/N

type

51. #8802

C₆₇

direct (N × H)

59. C₆₇

#8802

direct (N × H)

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

subgroup	index
C₆₇×#1745	2
C₆₇×#1281	2
C₆₇×#1276	2
C₆₇×#1275	2
#8802	67

subgroup

index

C₆₇×#1745

C₆₇×#1281

C₆₇×#1276

C₆₇×#1275

#8802

Subgroups — C₆₇×#8802 #30688053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#8802 #30688053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#8802 #30688053