Group #16828624 — subgroups

Subgroups — #16828624

Order 3712.

394 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×#6597	C₂	1	2	yes	maximal, complemented
2	#14271241	C₂	1	6	yes	maximal, complemented
3	#14271216	C₂	1	3	yes	maximal, complemented
4	#14271187	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₂₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#41×C₂₉	C₄	1	1	yes	characteristic
7	C₄×Q₈×C₂₉	C₄	1	1	yes	characteristic
8	C₄×Q₈×C₂₉	C₂²	1	1	yes	characteristic
9	C₄×D₄×C₂₉	C₄	1	3	yes	—
10	C₄×D₄×C₂₉	C₂²	1	3	yes	—
11	C₂₉×#117	C₄	1	3	yes	—
12	C₂×C₄²×C₂₉	C₂²	1	3	yes	—
13	C₂×#90777	C₂²	1	6	yes	—
14	C₂²⋊#6597	C₂²	1	6	yes	—
15	C₄⋊#6597	C₂²	1	6	yes	—
16	C₄⋊#6597	C₂²	1	3	yes	—
17	C₄⋊#6597	C₂²	1	1	yes	characteristic
18	C₄×#6597	C₂²	1	6	yes	—
19	#41×C₂₉	—	2	4	no	—
20	C₂×Q₈×C₂₉	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₂₉	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₂₉	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₂₉	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₂₉	C₂³	1	3	yes	—
25	C₄⋊C₄×C₂₉	C₂×C₄	1	3	yes	—
26	C₄⋊C₄×C₂₉	C₂×C₄	1	3	yes	—
27	C₂²⋊C₄×C₂₉	C₂×C₄	1	6	yes	—
28	C₄²×C₂₉	C₂×C₄	1	3	yes	—
29	C₄²×C₂₉	C₂×C₄	1	3	yes	—
30	C₄²×C₂₉	C₂³	1	1	yes	characteristic
31	C₄²×C₂₉	C₂³	1	3	yes	—
32	#90777	—	4	6	no	—
33	C₂×#6597	C₂³	1	6	yes	—
34	C₂×#6597	C₂³	1	2	yes	—
35	C₂³×C₂₉	C₂²×C₄	1	3	yes	—
36	Q₈×C₂₉	M₁₆	1	4	yes	complemented
37	D₄×C₂₉	—	2	6	no	—
38	C₂×C₄×C₂₉	—	2	6	no	—
39	C₂×C₄×C₂₉	—	2	6	no	—
40	C₂×C₄×C₂₉	—	2	3	no	—
41	C₂×C₄×C₂₉	—	2	3	no	—
42	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₂₉	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₂₉	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₂₉	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₂₉	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₂₉	C₂⁴	1	1	yes	characteristic
48	#6597	—	2	6	no	—
49	#6597	—	2	2	no	—
50	#2991	—	29	1	no	maximal, Sylow
51	C₂²×C₂₉	—	2	3	no	—
52	C₂²×C₂₉	—	2	3	no	—
53	C₂²×C₂₉	—	2	3	no	—
54	C₂²×C₂₉	C₂³×C₄	1	1	yes	characteristic
55	C₄×C₂₉	—	2	3	no	—
56	C₄×C₂₉	—	2	1	no	—
57	C₄×C₂₉	—	2	3	no	—
58	C₄×C₂₉	C₂×M₁₆	1	6	yes	—
59	C₄×C₂₉	#131	1	2	yes	—
60	C₄×C₂₉	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₂₉	2₋¹⁺⁴	1	1	yes	characteristic
62	C₄×#41	D₂₉	1	1	yes	characteristic
63	C₈×Q₈	—	29	2	no	—
64	#436	—	29	6	no	—
65	#431	—	29	3	no	—
66	#423	—	29	3	no	—
67	C₂×C₂₉	—	4	3	no	—
68	C₂×C₂₉	#520	1	1	yes	characteristic
69	C₂×C₂₉	C₂²×M₁₆	1	1	yes	characteristic
70	C₂×C₂₉	C₂×#131	1	1	yes	characteristic
71	C₂×#41	Dic₂₉	1	1	yes	characteristic
72	C₂×M₁₆	—	29	6	no	—
73	C₄×Q₈	Dic₂₉	1	1	yes	characteristic
74	C₄×Q₈	C₂×D₂₉	1	1	yes	characteristic
75	C₄×D₄	Dic₂₉	1	3	yes	—
76	C₄×D₄	C₂×D₂₉	1	3	yes	—
77	#117	Dic₂₉	1	3	yes	—
78	C₂×C₄²	C₂×D₂₉	1	3	yes	—
79	C₄⋊C₈	—	29	3	no	—
80	C₄⋊C₈	—	29	6	no	—
81	C₄⋊C₈	—	29	1	no	—
82	C₂²⋊C₈	—	29	6	no	—
83	C₄×C₈	—	29	6	no	—
84	C₂₉	#2991	1	1	yes	characteristic, complemented, Sylow
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×Dic₂₉	1	1	yes	characteristic
87	C₂×D₄	C₂×Dic₂₉	1	3	yes	—
88	C₂²×C₄	C₂×Dic₂₉	1	3	yes	—
89	C₂²×C₄	C₂×Dic₂₉	1	3	yes	—
90	C₂²×C₄	C₂²×D₂₉	1	3	yes	—
91	M₁₆	—	116	6	no	—
92	C₂×C₈	—	29	2	no	—
93	C₂×C₈	—	29	6	no	—
94	C₄⋊C₄	C₂×Dic₂₉	1	3	yes	—
95	C₄⋊C₄	C₂×Dic₂₉	1	3	yes	—
96	C₂²⋊C₄	C₂×Dic₂₉	1	6	yes	—
97	C₄²	C₂×Dic₂₉	1	3	yes	—
98	C₄²	C₂×Dic₂₉	1	3	yes	—
99	C₄²	C₂²×D₂₉	1	3	yes	—
100	C₄²	C₂²×D₂₉	1	1	yes	characteristic
101	C₂³	C₂²×Dic₂₉	1	3	yes	—
102	Q₈	#90777	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₂₉	1	3	yes	—
110	C₂×C₄	C₂²×Dic₂₉	1	3	yes	—
111	C₂×C₄	C₂²×Dic₂₉	1	3	yes	—
112	C₂×C₄	C₂²×Dic₂₉	1	1	yes	characteristic
113	C₂×C₄	C₂³×D₂₉	1	1	yes	characteristic
114	C₈	—	58	6	no	—
115	C₈	—	58	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×Dic₂₉	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×#90777	1	6	yes	—
124	C₄	#11745310	1	2	yes	—
125	C₄	#11745364	1	1	yes	characteristic
126	C₄	#11745365	1	1	yes	characteristic
127	C₂	—	4	3	no	—
128	C₂	#14271646	1	1	yes	characteristic
129	C₂	C₂²×#90777	1	1	yes	characteristic
130	C₂	C₂×#11745310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#6597

C₂

yes

maximal, complemented

#14271241

C₂

yes

maximal, complemented

#14271216

C₂

yes

maximal, complemented

#14271187

C₂

yes

maximal, complemented

C₄×#41×C₂₉

C₂

yes

maximal, characteristic

C₂×#41×C₂₉

C₄

yes

characteristic

C₄×Q₈×C₂₉

C₄

yes

characteristic

C₄×Q₈×C₂₉

C₂²

yes

characteristic

C₄×D₄×C₂₉

C₄

yes

—

C₄×D₄×C₂₉

C₂²

yes

—

C₂₉×#117

C₄

yes

—

C₂×C₄²×C₂₉

C₂²

yes

—

C₂×#90777

C₂²

yes

—

C₂²⋊#6597

C₂²

yes

—

C₄⋊#6597

C₂²

yes

—

C₄⋊#6597

C₂²

yes

—

C₄⋊#6597

C₂²

yes

characteristic

C₄×#6597

C₂²

yes

—

#41×C₂₉

—

C₂×Q₈×C₂₉

C₂×C₄

yes

characteristic

C₂×D₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂×C₄

yes

—

C₂²×C₄×C₂₉

C₂³

yes

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₄⋊C₄×C₂₉

C₂×C₄

yes

—

C₂²⋊C₄×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂×C₄

yes

—

C₄²×C₂₉

C₂³

yes

characteristic

C₄²×C₂₉

C₂³

yes

—

#90777

—

C₂×#6597

C₂³

yes

—

C₂×#6597

C₂³

yes

—

C₂³×C₂₉

C₂²×C₄

yes

—

Q₈×C₂₉

M₁₆

yes

complemented

D₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

—

C₂×C₄×C₂₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₉

C₂⁴

yes

—

—

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

—

C₂²×C₂₉

C₂³×C₄

yes

characteristic

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

—

C₄×C₂₉

C₂×M₁₆

yes

—

C₄×C₂₉

#131

yes

—

C₄×C₂₉

H(5,𝔽₂)

yes

characteristic

C₄×C₂₉

2₋¹⁺⁴

yes

characteristic

C₄×#41

D₂₉

yes

characteristic

C₈×Q₈

—

#436

—

#431

—

#423

—

C₂×C₂₉

—

C₂×C₂₉

#520

yes

characteristic

C₂×C₂₉

C₂²×M₁₆

yes

characteristic

C₂×C₂₉

C₂×#131

yes

characteristic

C₂×#41

Dic₂₉

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

Dic₂₉

yes

characteristic

C₄×Q₈

C₂×D₂₉

yes

characteristic

C₄×D₄

Dic₂₉

yes

—

C₄×D₄

C₂×D₂₉

yes

—

#117

Dic₂₉

yes

—

C₂×C₄²

C₂×D₂₉

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂₉

#2991

yes

characteristic, complemented, Sylow

#41

—

C₂×Q₈

C₂×Dic₂₉

yes

characteristic

C₂×D₄

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂×Dic₂₉

yes

—

C₂²×C₄

C₂²×D₂₉

yes

—

M₁₆

—

116

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₂₉

yes

—

C₄⋊C₄

C₂×Dic₂₉

yes

—

C₂²⋊C₄

C₂×Dic₂₉

yes

—

C₄²

C₂×Dic₂₉

yes

—

C₄²

C₂×Dic₂₉

yes

—

C₄²

C₂²×D₂₉

yes

—

100

C₄²

C₂²×D₂₉

yes

characteristic

101

C₂³

C₂²×Dic₂₉

yes

—

102

Q₈

#90777

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₂₉

yes

characteristic

109

C₂×C₄

C₂²×Dic₂₉

yes

—

110

C₂×C₄

C₂²×Dic₂₉

yes

—

111

C₂×C₄

C₂²×Dic₂₉

yes

—

112

C₂×C₄

C₂²×Dic₂₉

yes

characteristic

113

C₂×C₄

C₂³×D₂₉

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×Dic₂₉

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×#90777

yes

—

124

C₄

#11745310

yes

—

125

C₄

#11745364

yes

characteristic

126

C₄

#11745365

yes

characteristic

127

C₂

—

128

C₂

#14271646

yes

characteristic

129

C₂

C₂²×#90777

yes

characteristic

130

C₂

C₂×#11745310

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₉
commutator, socle	C₂²×C₂₉	C₂³×C₄
Fitting	C₄×#41×C₂₉	C₂
prime radical	C₂×#41×C₂₉	C₄
cosocle	C₂×C₄×C₂₉	C₂⁴
hypercenter	C₄×#41	D₂₉
hypocenter	C₂₉	#2991
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₂₉

commutator, socle

C₂²×C₂₉

C₂³×C₄

Fitting

C₄×#41×C₂₉

C₂

prime radical

C₂×#41×C₂₉

C₄

cosocle

C₂×C₄×C₂₉

C₂⁴

hypercenter

C₄×#41

D₂₉

C₂₉

G ▷ C₂²×C₂₉ ▷ C₂₉

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×#6597	C₂	semidirect
2. #14271241	C₂	semidirect
3. #14271216	C₂	semidirect
4. #14271187	C₂	semidirect
36. Q₈×C₂₉	M₁₆	semidirect
84. C₂₉	#2991	semidirect
102. Q₈	#90777	semidirect

H ≅ G/N

type

1. Q₈×#6597

C₂

semidirect

2. #14271241

C₂

semidirect

3. #14271216

C₂

semidirect

4. #14271187

C₂

semidirect

36. Q₈×C₂₉

M₁₆

semidirect

84. C₂₉

#2991

semidirect

102. Q₈

#90777

semidirect

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

subgroup	index
Q₈×#6597	2
#14271241	2
#14271216	2
#14271187	2
C₄×#41×C₂₉	2
#2991	29

subgroup

index

C₄×#41×C₂₉

Subgroups — #16828624

Conjugacy classes of subgroups

Complements

Maximal subgroups