Group #30712566 — subgroups

Subgroups — C₆₇×#33315 #30712566

Order 17152.

806 conjugacy classes of subgroups; only the 382 normal subgroups are shown.

normal subgroups only

This group has 806 conjugacy classes of subgroups — too many to draw. Only its 382 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×#499×C₆₇	C₂	1	4	yes	maximal
2	C₆₇×#2859	C₂	1	2	yes	maximal
3	C₆₇×#1847	C₂	1	8	yes	maximal
4	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₆₇	C₂²	1	4	yes	—
6	#499×C₆₇	—	2	16	no	—
7	C₂×#107×C₆₇	C₂²	1	8	yes	—
8	C₂×#106×C₆₇	C₂²	1	8	yes	—
9	C₂×#97×C₆₇	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
11	#389×C₆₇	—	2	8	no	—
12	#384×C₆₇	C₂²	1	4	yes	—
13	#382×C₆₇	C₂²	1	4	yes	—
14	C₄³×C₆₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
16	#125×C₆₇	—	2	16	no	—
17	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
18	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
19	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
20	C₂×C₄²×C₆₇	D₄	1	4	yes	—
21	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
23	#107×C₆₇	—	2	16	no	—
24	#106×C₆₇	—	2	16	no	—
25	#97×C₆₇	Q₈	1	16	yes	—
26	C₄×C₈×C₆₇	D₄	1	8	yes	—
27	#95×C₆₇	—	2	8	no	—
28	C₂²×C₄×C₆₇	—	2	8	no	—
29	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
30	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
31	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
32	C₂×C₈×C₆₇	C₂×D₄	1	8	yes	—
33	C₂×C₈×C₆₇	C₂×Q₈	1	16	yes	—
34	C₄⋊C₄×C₆₇	—	2	16	no	—
35	C₄⋊C₄×C₆₇	—	4	16	no	—
36	C₄⋊C₄×C₆₇	—	4	16	no	—
37	C₄²×C₆₇	—	4	4	no	—
38	C₄²×C₆₇	C₂×D₄	1	4	yes	—
39	C₄²×C₆₇	C₂×Q₈	1	8	yes	—
40	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	8	no	—
42	C₂×C₄×C₆₇	—	2	4	no	—
43	C₂×C₄×C₆₇	—	4	8	no	—
44	C₂×C₄×C₆₇	—	4	16	no	—
45	C₂×C₄×C₆₇	#127	1	4	yes	—
46	C₂×C₄×C₆₇	C₄⋊Q₈	1	8	yes	—
47	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
48	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
49	C₈×C₆₇	C₄⋊Q₈	1	16	yes	—
50	C₂²×C₆₇	C₂×#127	1	1	yes	characteristic
51	C₂²×C₆₇	C₂×C₄⋊Q₈	1	2	yes	—
52	C₂²×C₆₇	#575	1	4	yes	—
53	C₄×C₆₇	—	2	4	no	—
54	C₄×C₆₇	—	2	8	no	—
55	C₄×C₆₇	—	8	8	no	—
56	C₄×C₆₇	C₂×C₄⋊Q₈	1	4	yes	—
57	#33315	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂×C₆₇	#2868	1	1	yes	characteristic
59	C₂×C₆₇	#3150	1	2	yes	—
60	C₂×C₆₇	#3166	1	4	yes	—
61	C₂×#499	C₂×C₆₇	1	4	yes	—
62	#2859	C₂×C₆₇	1	2	yes	—
63	#1847	C₂×C₆₇	1	8	yes	—
64	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
65	C₆₇	#33315	1	1	yes	characteristic, direct product factor, Sylow
66	C₂×#125	C₂²×C₆₇	1	4	yes	—
67	#499	—	2	16	no	—
68	C₂×#107	C₂²×C₆₇	1	8	yes	—
69	C₂×#106	C₂²×C₆₇	1	8	yes	—
70	C₂×#97	C₂²×C₆₇	1	4	yes	—
71	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
72	#389	—	2	8	no	—
73	#384	C₂²×C₆₇	1	4	yes	—
74	#382	C₂²×C₆₇	1	4	yes	—
75	C₄³	C₂²×C₆₇	1	1	yes	characteristic
76	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
77	#125	—	2	16	no	—
78	C₂×C₄⋊C₄	—	2	8	no	—
79	C₂×C₄⋊C₄	—	2	8	no	—
80	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
81	C₂×C₄²	D₄×C₆₇	1	4	yes	—
82	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
83	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
84	#107	—	2	16	no	—
85	#106	—	2	16	no	—
86	#97	Q₈×C₆₇	1	16	yes	—
87	C₄×C₈	D₄×C₆₇	1	8	yes	—
88	#95	—	2	8	no	—
89	C₂²×C₄	—	2	8	no	—
90	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
91	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
92	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
93	C₂×C₈	C₂×D₄×C₆₇	1	8	yes	—
94	C₂×C₈	C₂×Q₈×C₆₇	1	16	yes	—
95	C₄⋊C₄	—	2	16	no	—
96	C₄⋊C₄	—	4	16	no	—
97	C₄⋊C₄	—	4	16	no	—
98	C₄²	—	4	4	no	—
99	C₄²	C₂×D₄×C₆₇	1	4	yes	—
100	C₄²	C₂×Q₈×C₆₇	1	8	yes	—
101	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	4	8	no	—
105	C₂×C₄	—	4	16	no	—
106	C₂×C₄	#127×C₆₇	1	4	yes	—
107	C₂×C₄	C₄⋊Q₈×C₆₇	1	8	yes	—
108	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
109	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
110	C₈	C₄⋊Q₈×C₆₇	1	16	yes	—
111	C₂²	C₂×#127×C₆₇	1	1	yes	characteristic
112	C₂²	C₂×C₄⋊Q₈×C₆₇	1	2	yes	—
113	C₂²	#575×C₆₇	1	4	yes	—
114	C₄	—	2	4	no	—
115	C₄	—	2	8	no	—
116	C₄	—	8	8	no	—
117	C₄	C₂×C₄⋊Q₈×C₆₇	1	4	yes	—
118	C₂	C₆₇×#2868	1	1	yes	characteristic
119	C₂	C₆₇×#3150	1	2	yes	—
120	C₂	C₆₇×#3166	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#499×C₆₇

C₂

yes

maximal

C₆₇×#2859

C₂

yes

maximal

C₆₇×#1847

C₂

yes

maximal

C₄×#97×C₆₇

C₂

yes

maximal, characteristic

C₂×#125×C₆₇

C₂²

yes

—

#499×C₆₇

—

C₂×#107×C₆₇

C₂²

yes

—

C₂×#106×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

#389×C₆₇

—

#384×C₆₇

C₂²

yes

—

#382×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

—

#125×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂³

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

C₂³

yes

—

#107×C₆₇

—

#106×C₆₇

—

#97×C₆₇

Q₈

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#95×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²×D₄

yes

—

C₂×C₄×C₆₇

C₂²×Q₈

yes

—

C₈×C₆₇

C₄⋊Q₈

yes

—

C₂²×C₆₇

C₂×#127

yes

characteristic

C₂²×C₆₇

C₂×C₄⋊Q₈

yes

—

C₂²×C₆₇

#575

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×C₄⋊Q₈

yes

—

#33315

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

#2868

yes

characteristic

C₂×C₆₇

#3150

yes

—

C₂×C₆₇

#3166

yes

—

C₂×#499

C₂×C₆₇

yes

—

#2859

C₂×C₆₇

yes

—

#1847

C₂×C₆₇

yes

—

C₄×#97

C₂×C₆₇

yes

characteristic

C₆₇

#33315

yes

characteristic, direct product factor, Sylow

C₂×#125

C₂²×C₆₇

yes

—

#499

—

C₂×#107

C₂²×C₆₇

yes

—

C₂×#106

C₂²×C₆₇

yes

—

C₂×#97

C₂²×C₆₇

yes

—

C₂×C₄×C₈

C₂²×C₆₇

yes

—

#389

—

#384

C₂²×C₆₇

yes

—

#382

C₂²×C₆₇

yes

—

C₄³

C₂²×C₆₇

yes

characteristic

C₂²×C₈

C₂³×C₆₇

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³×C₆₇

yes

—

C₂×C₄²

D₄×C₆₇

yes

—

C₂×C₄²

C₂³×C₆₇

yes

characteristic

C₂×C₄²

C₂³×C₆₇

yes

—

#107

—

#106

—

#97

Q₈×C₆₇

yes

—

C₄×C₈

D₄×C₆₇

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₆₇

yes

—

C₂²×C₄

C₂×D₄×C₆₇

yes

—

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

C₂×C₈

C₂×D₄×C₆₇

yes

—

C₂×C₈

C₂×Q₈×C₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×D₄×C₆₇

yes

—

100

C₄²

C₂×Q₈×C₆₇

yes

—

101

C₂³

C₂²×D₄×C₆₇

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

#127×C₆₇

yes

—

107

C₂×C₄

C₄⋊Q₈×C₆₇

yes

—

108

C₂×C₄

C₂²×D₄×C₆₇

yes

—

109

C₂×C₄

C₂²×Q₈×C₆₇

yes

—

110

C₈

C₄⋊Q₈×C₆₇

yes

—

111

C₂²

C₂×#127×C₆₇

yes

characteristic

112

C₂²

C₂×C₄⋊Q₈×C₆₇

yes

—

113

C₂²

#575×C₆₇

yes

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

C₂×C₄⋊Q₈×C₆₇

yes

—

118

C₂

C₆₇×#2868

yes

characteristic

119

C₂

C₆₇×#3150

yes

—

120

C₂

C₆₇×#3166

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂⁴×C₆₇

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

N	H ≅ G/N	type
57. #33315	C₆₇	direct (N × H)
65. C₆₇	#33315	direct (N × H)

H ≅ G/N

type

57. #33315

C₆₇

direct (N × H)

65. C₆₇

#33315

direct (N × H)

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

subgroup	index
C₂×#499×C₆₇	2
C₆₇×#2859	2
C₆₇×#1847	2
C₄×#97×C₆₇	2
#33315	67

subgroup

index

C₂×#499×C₆₇

C₆₇×#2859

C₆₇×#1847

C₄×#97×C₆₇

#33315

Subgroups — C₆₇×#33315 #30712566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#33315 #30712566

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33315 #30712566