Group #78507 — subgroups

Subgroups — #78507

Order 384.

348 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 165 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	#5470	C₂	1	1	yes	maximal, characteristic
2	C₂×#870	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×M₁₆	C₂	1	1	yes	maximal, characteristic
4	#5121	C₂	1	4	yes	maximal
5	#4702	C₂	1	4	yes	maximal, complemented
6	#4683	C₂	1	4	yes	maximal, complemented
7	#3166	—	3	1	no	maximal, Sylow
8	C₂×C₃×M₁₆	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	#926	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	#924	—	2	2	no	—
13	C₂×Dic₃⋊C₄	—	2	2	no	—
14	#870	—	2	2	no	—
15	#870	C₂²	1	2	yes	—
16	#870	C₂²	1	2	yes	—
17	#870	C₂²	1	2	yes	—
18	#869	C₂²	1	2	yes	—
19	C₄×Dic₆	—	2	2	no	—
20	C₃×#97	C₂²	1	2	yes	—
21	C₃×C₄×C₈	C₂²	1	2	yes	—
22	#818	C₂²	1	8	yes	—
23	#817	C₂²	1	8	yes	—
24	#533	—	3	1	no	—
25	C₂×#125	—	3	1	no	—
26	#501	—	3	4	no	—
27	#499	—	3	4	no	—
28	#428	—	3	4	no	—
29	C₄×M₁₆	D₃	1	1	yes	characteristic
30	C₂²×C₃×C₄	D₄	1	2	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×Dic₃	—	2	2	no	—
33	C₂×Dic₆	—	2	2	no	—
34	C₃×M₁₆	Q₈	1	8	yes	complemented
35	C₂×C₃×C₈	C₂³	1	4	yes	—
36	C₃×C₄²	D₄	1	2	yes	—
37	C₃×C₄²	C₂³	1	1	yes	characteristic
38	C₃×C₄²	C₂³	1	1	yes	characteristic
39	#217	—	2	2	no	—
40	C₄⋊Dic₃	—	2	2	no	—
41	C₄⋊Dic₃	—	2	2	no	—
42	C₄⋊Dic₃	C₂³	1	4	yes	—
43	C₄⋊Dic₃	C₂³	1	4	yes	—
44	Dic₃⋊C₄	—	2	4	no	—
45	Dic₃⋊C₄	—	2	4	no	—
46	Dic₃⋊C₄	—	2	4	no	—
47	C₄×Dic₃	—	2	2	no	—
48	C₂×M₁₆	C₂×D₃	1	2	yes	—
49	C₄⋊Q₈	—	3	2	no	—
50	#125	—	3	2	no	—
51	#125	—	3	2	no	—
52	#125	—	3	2	no	—
53	#125	—	6	2	no	—
54	C₂²⋊Q₈	—	6	2	no	—
55	C₄×Q₈	—	6	2	no	—
56	#117	—	3	2	no	—
57	C₂×C₄⋊C₄	—	3	2	no	—
58	C₂×C₄⋊C₄	—	6	2	no	—
59	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
60	#107	—	3	8	no	—
61	#106	—	3	8	no	—
62	#97	C₂×D₃	1	2	yes	—
63	C₄×C₈	C₂×D₃	1	2	yes	—
64	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
65	C₂×C₃×C₄	—	4	2	no	—
66	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
67	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
68	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
70	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
71	C₂×Dic₃	—	2	4	no	—
72	C₂×Dic₃	—	2	4	no	—
73	C₂×Dic₃	—	4	4	no	—
74	Dic₆	—	4	4	no	—
75	C₃×C₈	C₂×Q₈	1	8	yes	—
76	C₂×Q₈	—	6	2	no	—
77	C₂²×C₄	—	6	2	no	—
78	C₂²×C₄	D₁₂	1	2	yes	—
79	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
80	M₁₆	Dic₆	1	8	yes	complemented
81	C₂×C₈	C₂²×D₃	1	4	yes	—
82	C₄⋊C₄	—	3	4	no	—
83	C₄⋊C₄	—	3	4	no	—
84	C₄⋊C₄	—	6	2	no	—
85	C₄⋊C₄	—	6	4	no	—
86	C₄⋊C₄	—	6	4	no	—
87	C₄⋊C₄	—	6	4	no	—
88	C₄⋊C₄	—	6	2	no	—
89	C₂²⋊C₄	—	6	2	no	—
90	C₄²	—	6	2	no	—
91	C₄²	D₁₂	1	2	yes	—
92	C₄²	C₂²×D₃	1	1	yes	characteristic
93	C₄²	C₂²×D₃	1	1	yes	characteristic
94	C₂²×C₃	—	2	2	no	—
95	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
96	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
97	C₃×C₄	—	2	2	no	—
98	C₃×C₄	—	2	2	no	—
99	C₃×C₄	C₄⋊Q₈	1	2	yes	—
100	C₃×C₄	C₂²×Q₈	1	2	yes	—
101	Dic₃	—	4	4	no	—
102	Dic₃	—	4	4	no	—
103	C₂³	C₂×D₁₂	1	1	yes	characteristic
104	Q₈	—	12	4	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	6	4	no	—
107	C₂×C₄	—	6	4	no	—
108	C₂×C₄	—	12	4	no	—
109	C₂×C₄	C₂×Dic₆	1	4	yes	—
110	C₂×C₄	C₂×D₁₂	1	2	yes	—
111	C₂×C₄	C₂×D₁₂	1	2	yes	—
112	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
113	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
114	C₈	C₂×Dic₆	1	8	yes	—
115	C₂×C₃	—	2	2	no	—
116	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
117	C₂×C₃	#575	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	#869	1	2	yes	—
120	C₂²	C₂²×D₁₂	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	2	no	—
123	C₄	—	12	4	no	—
124	C₄	—	12	4	no	—
125	C₄	#869	1	2	yes	—
126	C₄	C₂²×Dic₆	1	2	yes	—
127	C₃	#3166	1	1	yes	characteristic, complemented, Sylow
128	C₂	—	2	2	no	—
129	C₂	C₂×#869	1	1	yes	characteristic
130	C₂	#5748	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#5470

C₂

yes

maximal, characteristic

C₂×#870

C₂

yes

maximal, characteristic

C₃×C₄×M₁₆

C₂

yes

maximal, characteristic

#5121

C₂

yes

maximal

#4702

C₂

yes

maximal, complemented

#4683

C₂

yes

maximal, complemented

#3166

—

maximal, Sylow

C₂×C₃×M₁₆

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

#926

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

#924

—

C₂×Dic₃⋊C₄

—

#870

—

#870

C₂²

yes

—

#870

C₂²

yes

—

#870

C₂²

yes

—

#869

C₂²

yes

—

C₄×Dic₆

—

C₃×#97

C₂²

yes

—

C₃×C₄×C₈

C₂²

yes

—

#818

C₂²

yes

—

#817

C₂²

yes

—

—

—

—

—

—

C₄×M₁₆

D₃

yes

characteristic

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₃×M₁₆

Q₈

yes

complemented

C₂×C₃×C₈

C₂³

yes

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂³

yes

—

C₄⋊Dic₃

C₂³

yes

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×M₁₆

C₂×D₃

yes

—

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

—

—

C₂×D₃

yes

—

C₄×C₈

C₂×D₃

yes

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃×C₈

C₂×Q₈

yes

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₁₂

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

Dic₆

yes

complemented

C₂×C₈

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

D₁₂

yes

—

C₄²

C₂²×D₃

yes

characteristic

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₄⋊Q₈

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄⋊Q₈

yes

—

100

C₃×C₄

C₂²×Q₈

yes

—

101

Dic₃

—

102

Dic₃

—

103

C₂³

C₂×D₁₂

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×Dic₆

yes

—

110

C₂×C₄

C₂×D₁₂

yes

—

111

C₂×C₄

C₂×D₁₂

yes

—

112

C₂×C₄

C₂×D₁₂

yes

characteristic

113

C₂×C₄

C₂³×D₃

yes

characteristic

114

C₈

C₂×Dic₆

yes

—

115

C₂×C₃

—

116

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

117

C₂×C₃

#575

yes

—

118

C₂²

—

119

C₂²

#869

yes

—

120

C₂²

C₂²×D₁₂

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#869

yes

—

126

C₄

C₂²×Dic₆

yes

—

127

C₃

#3166

yes

characteristic, complemented, Sylow

128

C₂

—

129

C₂

C₂×#869

yes

characteristic

130

C₂

#5748

yes

—

	subgroup	G/H
center	C₂²	C₂²×D₁₂
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×C₄×M₁₆	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₄×M₁₆	D₃
hypocenter	C₃	#3166
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂²×D₁₂

commutator, cosocle

C₂×C₃×C₄

C₂⁴

Frattini

C₂×C₄

C₂³×D₃

Fitting

C₃×C₄×M₁₆

C₂

socle

C₂²×C₃

C₂²×D₄

prime radical

C₂³×C₃

C₂×D₄

hypercenter

C₄×M₁₆

D₃

hypocenter

C₃

#3166

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
5. #4702	C₂	semidirect
6. #4683	C₂	semidirect
34. C₃×M₁₆	Q₈	semidirect
80. M₁₆	Dic₆	semidirect
127. C₃	#3166	semidirect

H ≅ G/N

type

5. #4702

C₂

semidirect

6. #4683

C₂

semidirect

34. C₃×M₁₆

Q₈

semidirect

80. M₁₆

Dic₆

semidirect

127. C₃

#3166

semidirect

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

subgroup	index
#5470	2
C₂×#870	2
C₃×C₄×M₁₆	2
#5121	2
#4702	2
#4683	2
#3166	3

subgroup

index

C₃×C₄×M₁₆

Subgroups — #78507

Conjugacy classes of subgroups

Complements

Maximal subgroups