Group #30712549 — subgroups

Subgroups — C₆₇×#33298 #30712549

Order 17152.

694 conjugacy classes of subgroups; only the 286 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 286 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×#2862	C₂	1	2	yes	maximal
2	C₆₇×#1993	C₂	1	8	yes	maximal
3	C₆₇×#1835	C₂	1	4	yes	maximal
4	C₄×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₇	C₂²	1	4	yes	—
6	C₂×#97×C₆₇	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
8	#391×C₆₇	—	2	8	no	—
9	#382×C₆₇	C₂²	1	4	yes	—
10	#382×C₆₇	C₂²	1	4	yes	—
11	C₄³×C₆₇	C₂²	1	1	yes	characteristic
12	#340×C₆₇	C₂²	1	16	yes	—
13	C₂²×Q₈×C₆₇	—	2	4	no	—
14	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
15	C₄⋊Q₈×C₆₇	—	2	16	no	—
16	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
17	C₂×C₄⋊C₄×C₆₇	C₂³	1	8	yes	—
18	C₂×C₄²×C₆₇	D₄	1	4	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
21	#97×C₆₇	—	2	8	no	—
22	C₄×C₈×C₆₇	Q₈	1	8	yes	complemented
23	#95×C₆₇	—	2	8	no	—
24	#95×C₆₇	—	2	8	no	—
25	C₂×Q₈×C₆₇	—	4	16	no	—
26	C₂×Q₈×C₆₇	—	4	8	no	—
27	C₂²×C₄×C₆₇	—	2	8	no	—
28	C₂²×C₄×C₆₇	C₂×D₄	1	4	yes	—
29	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
30	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
31	C₂×C₈×C₆₇	—	2	8	no	—
32	C₂×C₈×C₆₇	C₂×Q₈	1	8	yes	—
33	C₄⋊C₄×C₆₇	—	2	16	no	—
34	C₄⋊C₄×C₆₇	—	4	8	no	—
35	C₄²×C₆₇	—	4	4	no	—
36	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
37	C₄²×C₆₇	#41	1	8	yes	—
38	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
39	Q₈×C₆₇	—	8	16	no	—
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	—	2	8	no	—
42	C₂×C₄×C₆₇	—	4	8	no	—
43	C₂×C₄×C₆₇	—	4	16	no	—
44	C₂×C₄×C₆₇	#124	1	8	yes	—
45	C₂×C₄×C₆₇	C₄⋊Q₈	1	4	yes	—
46	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
47	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
48	C₈×C₆₇	—	2	8	no	—
49	C₂²×C₆₇	C₂×#124	1	2	yes	—
50	C₂²×C₆₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₆₇	#575	1	4	yes	—
52	C₄×C₆₇	—	2	8	no	—
53	C₄×C₆₇	—	2	4	no	—
54	C₄×C₆₇	—	8	8	no	—
55	C₄×C₆₇	#533	1	4	yes	—
56	#33298	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×C₆₇	#2862	1	1	yes	characteristic
58	C₂×C₆₇	#3140	1	4	yes	—
59	C₂×C₆₇	#3162	1	2	yes	—
60	#2862	C₂×C₆₇	1	2	yes	—
61	#1993	C₂×C₆₇	1	8	yes	—
62	#1835	C₂×C₆₇	1	4	yes	—
63	C₄×#97	C₂×C₆₇	1	1	yes	characteristic
64	C₆₇	#33298	1	1	yes	characteristic, direct product factor, Sylow
65	C₂×C₄⋊Q₈	C₂²×C₆₇	1	4	yes	—
66	C₂×#97	C₂²×C₆₇	1	4	yes	—
67	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
68	#391	—	2	8	no	—
69	#382	C₂²×C₆₇	1	4	yes	—
70	#382	C₂²×C₆₇	1	4	yes	—
71	C₄³	C₂²×C₆₇	1	1	yes	characteristic
72	#340	C₂²×C₆₇	1	16	yes	—
73	C₂²×Q₈	—	2	4	no	—
74	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
75	C₄⋊Q₈	—	2	16	no	—
76	C₂×C₄⋊C₄	—	2	4	no	—
77	C₂×C₄⋊C₄	C₂³×C₆₇	1	8	yes	—
78	C₂×C₄²	D₄×C₆₇	1	4	yes	—
79	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
80	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
81	#97	—	2	8	no	—
82	C₄×C₈	Q₈×C₆₇	1	8	yes	complemented
83	#95	—	2	8	no	—
84	#95	—	2	8	no	—
85	C₂×Q₈	—	4	16	no	—
86	C₂×Q₈	—	4	8	no	—
87	C₂²×C₄	—	2	8	no	—
88	C₂²×C₄	C₂×D₄×C₆₇	1	4	yes	—
89	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
90	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
91	C₂×C₈	—	2	8	no	—
92	C₂×C₈	C₂×Q₈×C₆₇	1	8	yes	—
93	C₄⋊C₄	—	2	16	no	—
94	C₄⋊C₄	—	4	8	no	—
95	C₄²	—	4	4	no	—
96	C₄²	C₂×Q₈×C₆₇	1	4	yes	—
97	C₄²	#41×C₆₇	1	8	yes	—
98	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
99	Q₈	—	8	16	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	8	no	—
103	C₂×C₄	—	4	16	no	—
104	C₂×C₄	#124×C₆₇	1	8	yes	—
105	C₂×C₄	C₄⋊Q₈×C₆₇	1	4	yes	—
106	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
107	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
108	C₈	—	2	8	no	—
109	C₂²	C₂×#124×C₆₇	1	2	yes	—
110	C₂²	C₂×C₄⋊Q₈×C₆₇	1	1	yes	characteristic
111	C₂²	#575×C₆₇	1	4	yes	—
112	C₄	—	2	8	no	—
113	C₄	—	2	4	no	—
114	C₄	—	8	8	no	—
115	C₄	#533×C₆₇	1	4	yes	—
116	C₂	C₆₇×#2862	1	1	yes	characteristic
117	C₂	C₆₇×#3140	1	4	yes	—
118	C₂	C₆₇×#3162	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₆₇×#2862

C₂

yes

maximal

C₆₇×#1993

C₂

yes

maximal

C₆₇×#1835

C₂

yes

maximal

C₄×#97×C₆₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

C₂×#97×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

#391×C₆₇

—

#382×C₆₇

C₂²

yes

—

#382×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#340×C₆₇

C₂²

yes

—

C₂²×Q₈×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

—

C₄⋊Q₈×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

—

#97×C₆₇

—

C₄×C₈×C₆₇

Q₈

yes

complemented

#95×C₆₇

—

#95×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×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×C₈×C₆₇

C₂×Q₈

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂²×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂²×Q₈

yes

—

C₂×C₄×C₆₇

C₂×#41

yes

—

C₈×C₆₇

—

C₂²×C₆₇

C₂×#124

yes

—

C₂²×C₆₇

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₆₇

#575

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#533

yes

—

#33298

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

#2862

yes

characteristic

C₂×C₆₇

#3140

yes

—

C₂×C₆₇

#3162

yes

—

#2862

C₂×C₆₇

yes

—

#1993

C₂×C₆₇

yes

—

#1835

C₂×C₆₇

yes

—

C₄×#97

C₂×C₆₇

yes

characteristic

C₆₇

#33298

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

C₂×#97

C₂²×C₆₇

yes

—

C₂×C₄×C₈

C₂²×C₆₇

yes

—

#391

—

#382

C₂²×C₆₇

yes

—

#382

C₂²×C₆₇

yes

—

C₄³

C₂²×C₆₇

yes

characteristic

#340

C₂²×C₆₇

yes

—

C₂²×Q₈

—

C₂²×C₈

C₂³×C₆₇

yes

—

C₄⋊Q₈

—

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

—

#97

—

C₄×C₈

Q₈×C₆₇

yes

complemented

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₆₇

yes

—

C₂²×C₄

C₂×D₄×C₆₇

yes

—

C₂²×C₄

C₂⁴×C₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂×Q₈×C₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×Q₈×C₆₇

yes

—

C₄²

#41×C₆₇

yes

—

C₂³

C₂²×D₄×C₆₇

yes

characteristic

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

#124×C₆₇

yes

—

105

C₂×C₄

C₄⋊Q₈×C₆₇

yes

—

106

C₂×C₄

C₂²×Q₈×C₆₇

yes

—

107

C₂×C₄

C₂×#41×C₆₇

yes

—

108

C₈

—

109

C₂²

C₂×#124×C₆₇

yes

—

110

C₂²

C₂×C₄⋊Q₈×C₆₇

yes

characteristic

111

C₂²

#575×C₆₇

yes

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

#533×C₆₇

yes

—

116

C₂

C₆₇×#2862

yes

characteristic

117

C₂

C₆₇×#3140

yes

—

118

C₂

C₆₇×#3162

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
22. C₄×C₈×C₆₇	Q₈	semidirect
56. #33298	C₆₇	direct (N × H)
64. C₆₇	#33298	direct (N × H)
82. C₄×C₈	Q₈×C₆₇	semidirect

H ≅ G/N

type

22. C₄×C₈×C₆₇

Q₈

semidirect

56. #33298

C₆₇

direct (N × H)

64. C₆₇

#33298

direct (N × H)

82. C₄×C₈

Q₈×C₆₇

semidirect

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

subgroup	index
C₆₇×#2862	2
C₆₇×#1993	2
C₆₇×#1835	2
C₄×#97×C₆₇	2
#33298	67

subgroup

index

C₆₇×#2862

C₆₇×#1993

C₆₇×#1835

C₄×#97×C₆₇

#33298

Subgroups — C₆₇×#33298 #30712549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#33298 #30712549

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33298 #30712549