Group #30689567 — subgroups

Subgroups — C₆₇×#10316 #30689567

Order 17152.

586 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 186 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×#498×C₆₇	C₂	1	3	yes	maximal
2	C₆₇×#1956	C₂	1	3	yes	maximal
3	C₆₇×#1273	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₆₇	C₂²	1	3	yes	—
5	#498×C₆₇	—	2	12	no	—
6	C₂×#106×C₆₇	—	2	6	no	—
7	C₂×#103×C₆₇	—	2	6	no	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	#384×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×Q₈×C₆₇	—	2	3	no	—
11	C₂²×C₈×C₆₇	D₄	1	6	yes	—
12	C₄⋊Q₈×C₆₇	—	2	6	no	—
13	C₄⋊Q₈×C₆₇	—	2	3	no	—
14	C₄⋊Q₈×C₆₇	D₄	1	6	yes	—
15	C₂×C₄⋊C₄×C₆₇	—	2	6	no	—
16	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
17	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
18	#106×C₆₇	—	2	24	no	—
19	#103×C₆₇	—	4	12	no	—
20	C₄×C₈×C₆₇	—	2	6	no	—
21	C₂×Q₈×C₆₇	—	2	6	no	—
22	C₂×Q₈×C₆₇	—	4	3	no	—
23	C₂×Q₈×C₆₇	—	4	6	no	—
24	C₂×Q₈×C₆₇	—	4	6	no	—
25	C₂²×C₄×C₆₇	—	4	3	no	—
26	C₂²×C₄×C₆₇	—	4	1	no	—
27	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
28	C₂×C₈×C₆₇	—	2	6	no	—
29	C₂×C₈×C₆₇	—	2	6	no	—
30	C₂×C₈×C₆₇	SD₁₆	1	12	yes	—
31	C₄⋊C₄×C₆₇	—	2	12	no	—
32	C₄⋊C₄×C₆₇	—	4	6	no	—
33	C₄⋊C₄×C₆₇	—	4	6	no	—
34	C₄²×C₆₇	C₂×D₄	1	3	yes	—
35	C₄²×C₆₇	#41	1	1	yes	characteristic
36	C₂³×C₆₇	#127	1	1	yes	characteristic
37	Q₈×C₆₇	—	4	12	no	—
38	Q₈×C₆₇	—	8	6	no	—
39	C₂×C₄×C₆₇	—	4	6	no	—
40	C₂×C₄×C₆₇	—	8	3	no	—
41	C₂×C₄×C₆₇	—	8	3	no	—
42	C₂×C₄×C₆₇	—	8	3	no	—
43	C₂×C₄×C₆₇	#120	1	3	yes	—
44	C₂×C₄×C₆₇	#121	1	3	yes	—
45	C₂×C₄×C₆₇	C₂×SD₁₆	1	6	yes	—
46	C₂×C₄×C₆₇	#137	1	6	yes	—
47	C₈×C₆₇	—	2	12	no	—
48	C₂²×C₆₇	#392	1	1	yes	characteristic
49	C₂²×C₆₇	#492	1	3	yes	—
50	C₂²×C₆₇	#497	1	3	yes	—
51	C₄×C₆₇	—	8	6	no	—
52	C₄×C₆₇	—	16	1	no	—
53	C₄×C₆₇	#448	1	6	yes	—
54	C₄×C₆₇	#461	1	6	yes	—
55	#10316	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×C₆₇	#1705	1	1	yes	characteristic
57	C₂×C₆₇	#1716	1	3	yes	—
58	C₂×C₆₇	#2013	1	3	yes	—
59	C₂×#498	C₂×C₆₇	1	3	yes	—
60	#1956	C₂×C₆₇	1	3	yes	—
61	#1273	C₂×C₆₇	1	1	yes	characteristic
62	C₆₇	#10316	1	1	yes	characteristic, direct product factor, Sylow
63	C₂×C₄⋊Q₈	C₂²×C₆₇	1	3	yes	—
64	#498	—	2	12	no	—
65	C₂×#106	—	2	6	no	—
66	C₂×#103	—	2	6	no	—
67	C₂×C₄×C₈	C₂²×C₆₇	1	3	yes	—
68	#384	C₂²×C₆₇	1	1	yes	characteristic
69	C₂²×Q₈	—	2	3	no	—
70	C₂²×C₈	D₄×C₆₇	1	6	yes	—
71	C₄⋊Q₈	—	2	6	no	—
72	C₄⋊Q₈	—	2	3	no	—
73	C₄⋊Q₈	D₄×C₆₇	1	6	yes	—
74	C₂×C₄⋊C₄	—	2	6	no	—
75	C₂×C₄⋊C₄	—	2	3	no	—
76	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
77	#106	—	2	24	no	—
78	#103	—	4	12	no	—
79	C₄×C₈	—	2	6	no	—
80	C₂×Q₈	—	2	6	no	—
81	C₂×Q₈	—	4	3	no	—
82	C₂×Q₈	—	4	6	no	—
83	C₂×Q₈	—	4	6	no	—
84	C₂²×C₄	—	4	3	no	—
85	C₂²×C₄	—	4	1	no	—
86	C₂²×C₄	C₂×D₄×C₆₇	1	3	yes	—
87	C₂×C₈	—	2	6	no	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	SD₁₆×C₆₇	1	12	yes	—
90	C₄⋊C₄	—	2	12	no	—
91	C₄⋊C₄	—	4	6	no	—
92	C₄⋊C₄	—	4	6	no	—
93	C₄²	C₂×D₄×C₆₇	1	3	yes	—
94	C₄²	#41×C₆₇	1	1	yes	characteristic
95	C₂³	#127×C₆₇	1	1	yes	characteristic
96	Q₈	—	4	12	no	—
97	Q₈	—	8	6	no	—
98	C₂×C₄	—	4	6	no	—
99	C₂×C₄	—	8	3	no	—
100	C₂×C₄	—	8	3	no	—
101	C₂×C₄	—	8	3	no	—
102	C₂×C₄	#120×C₆₇	1	3	yes	—
103	C₂×C₄	#121×C₆₇	1	3	yes	—
104	C₂×C₄	C₂×SD₁₆×C₆₇	1	6	yes	—
105	C₂×C₄	#137×C₆₇	1	6	yes	—
106	C₈	—	2	12	no	—
107	C₂²	#392×C₆₇	1	1	yes	characteristic
108	C₂²	#492×C₆₇	1	3	yes	—
109	C₂²	#497×C₆₇	1	3	yes	—
110	C₄	—	8	6	no	—
111	C₄	—	16	1	no	—
112	C₄	#448×C₆₇	1	6	yes	—
113	C₄	#461×C₆₇	1	6	yes	—
114	C₂	C₆₇×#1705	1	1	yes	characteristic
115	C₂	C₆₇×#1716	1	3	yes	—
116	C₂	C₆₇×#2013	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#498×C₆₇

C₂

yes

maximal

C₆₇×#1956

C₂

yes

maximal

C₆₇×#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#498×C₆₇

—

C₂×#106×C₆₇

—

C₂×#103×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

#384×C₆₇

C₂²

yes

characteristic

C₂²×Q₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#106×C₆₇

—

#103×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

yes

C₂³×C₆₇

yes

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#120

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₈×C₆₇

—

C₂²×C₆₇

#392

yes

characteristic

C₂²×C₆₇

#492

yes

—

C₂²×C₆₇

#497

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#448

yes

—

C₄×C₆₇

#461

yes

—

#10316

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₆₇

#1705

yes

characteristic

C₂×C₆₇

#1716

yes

—

C₂×C₆₇

#2013

yes

—

C₂×#498

C₂×C₆₇

yes

—

#1956

C₂×C₆₇

yes

—

#1273

C₂×C₆₇

yes

characteristic

C₆₇

#10316

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂²×C₆₇

yes

—

#498

—

C₂×#106

—

C₂×#103

—

C₂×C₄×C₈

C₂²×C₆₇

yes

—

#384

C₂²×C₆₇

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

D₄×C₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

D₄×C₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³×C₆₇

yes

characteristic

#106

—

#103

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

SD₁₆×C₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×D₄×C₆₇

yes

—

C₄²

#41×C₆₇

yes

characteristic

C₂³

#127×C₆₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#120×C₆₇

yes

—

103

C₂×C₄

#121×C₆₇

yes

—

104

C₂×C₄

C₂×SD₁₆×C₆₇

yes

—

105

C₂×C₄

#137×C₆₇

yes

—

106

C₈

—

107

C₂²

#392×C₆₇

yes

characteristic

108

C₂²

#492×C₆₇

yes

—

109

C₂²

#497×C₆₇

yes

—

110

C₄

—

111

C₄

—

112

C₄

#448×C₆₇

yes

—

113

C₄

#461×C₆₇

yes

—

114

C₂

C₆₇×#1705

yes

characteristic

115

C₂

C₆₇×#1716

yes

—

116

C₂

C₆₇×#2013

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₆₇	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₆₇
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₆₇

#127

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₆₇

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
55. #10316	C₆₇	direct (N × H)
62. C₆₇	#10316	direct (N × H)

H ≅ G/N

type

55. #10316

C₆₇

direct (N × H)

62. C₆₇

#10316

direct (N × H)

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

subgroup	index
C₂×#498×C₆₇	2
C₆₇×#1956	2
C₆₇×#1273	2
#10316	67

subgroup

index

C₂×#498×C₆₇

C₆₇×#1956

C₆₇×#1273

#10316

Subgroups — C₆₇×#10316 #30689567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#10316 #30689567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#10316 #30689567