Group #33006881 — subgroups

Subgroups — #33006881

Order 18688.

164 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 67 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	#22612206	C₂	1	1	yes	maximal, characteristic
2	#22608130	C₂	1	2	yes	maximal
3	C₄⋊#14556497	C₂	1	1	yes	maximal, characteristic
4	#22608112	C₂	1	2	yes	maximal
5	C₄×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
6	#17988992	—	2	1	no	—
7	C₂×#14556497	C₂²	1	2	yes	—
8	#17988413	—	2	1	no	—
9	#17988413	C₂²	1	2	yes	—
10	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
11	C₄×Q₁₆×C₇₃	—	2	1	no	—
12	#366×C₇₃	C₂²	1	1	yes	characteristic
13	#358×C₇₃	—	2	1	no	—
14	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	#14556591	—	4	1	no	—
16	#14556525	—	2	2	no	—
17	#14556525	—	4	1	no	—
18	#14556497	—	2	2	no	—
19	C₂×Q₁₆×C₇₃	—	2	1	no	—
20	C₄×Q₈×C₇₃	—	4	1	no	—
21	Q₃₂×C₇₃	—	2	1	no	—
22	Q₃₂×C₇₃	Q₈	1	2	yes	complemented
23	C₂×C₁₆×C₇₃	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
25	#107×C₇₃	—	2	1	no	—
26	#103×C₇₃	—	4	1	no	—
27	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂×Dic₁₄₆	—	8	1	no	—
29	C₄⋊Dic₇₃	—	4	2	no	—
30	C₄⋊Dic₇₃	—	8	1	no	—
31	C₂×Q₈×C₇₃	—	4	1	no	—
32	Q₁₆×C₇₃	—	2	2	no	—
33	Q₁₆×C₇₃	—	4	1	no	—
34	C₂×C₈×C₇₃	D₈	1	1	yes	characteristic
35	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₇₃	—	4	1	no	—
37	C₄⋊C₄×C₇₃	—	8	1	no	—
38	C₄²×C₇₃	—	8	1	no	—
39	C₄²×C₇₃	D₈	1	1	yes	characteristic
40	C₁₆×C₇₃	—	2	1	no	—
41	C₁₆×C₇₃	C₂×Q₈	1	1	yes	characteristic
42	C₁₆×C₇₃	#41	1	1	yes	characteristic
43	Q₈×C₇₃	—	4	2	no	—
44	Q₈×C₇₃	—	8	1	no	—
45	C₂×C₄×C₇₃	—	8	1	no	—
46	C₂×C₄×C₇₃	—	8	1	no	—
47	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
48	C₂×C₄×C₇₃	D₁₆	1	1	yes	characteristic
49	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₇₃	—	8	2	no	—
51	Dic₁₄₆	—	16	2	no	—
52	C₈×C₇₃	—	2	1	no	—
53	C₈×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
54	C₈×C₇₃	#137	1	1	yes	characteristic
55	C₂²×C₇₃	C₂×D₁₆	1	1	yes	characteristic
56	C₄×C₇₃	—	2	1	no	—
57	C₄×C₇₃	—	8	2	no	—
58	C₄×C₇₃	—	16	1	no	—
59	C₄×C₇₃	Q₆₄	1	2	yes	—
60	C₄×C₇₃	#474	1	1	yes	characteristic
61	C₄×C₇₃	#510	1	1	yes	characteristic
62	Dic₇₃	—	16	2	no	—
63	#13706	—	73	1	no	maximal, Sylow
64	C₂×C₇₃	#2224	1	1	yes	characteristic
65	C₂×C₇₃	C₂×Q₆₄	1	1	yes	characteristic
66	C₂×C₇₃	#2263	1	1	yes	characteristic
67	#2252	—	73	1	no	—
68	C₄×Q₃₂	D₇₃	1	1	yes	characteristic
69	#1423	—	73	2	no	—
70	C₄⋊C₃₂	—	73	1	no	—
71	#1416	—	73	2	no	—
72	C₇₃	#13706	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
74	#500	—	146	1	no	—
75	C₄×Q₁₆	—	2	1	no	—
76	C₂×C₃₂	—	73	2	no	—
77	#366	—	73	2	no	—
78	#366	—	146	1	no	—
79	#366	C₂×D₇₃	1	1	yes	characteristic
80	#358	—	2	1	no	—
81	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
82	C₂×Q₁₆	—	2	1	no	—
83	C₄⋊Q₈	—	292	1	no	—
84	C₄×Q₈	—	4	1	no	—
85	Q₃₂	—	2	1	no	—
86	Q₃₂	Dic₁₄₆	1	2	yes	complemented
87	C₂×C₁₆	C₂²⋊D₇₃	1	1	yes	characteristic
88	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
89	#107	—	2	1	no	—
90	#107	—	292	1	no	—
91	#107	—	146	2	no	—
92	#103	—	4	1	no	—
93	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
94	C₃₂	—	146	2	no	—
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	584	1	no	—
97	Q₁₆	—	2	2	no	—
98	Q₁₆	—	4	1	no	—
99	C₂×C₈	#11924013	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
101	C₄⋊C₄	—	4	1	no	—
102	C₄⋊C₄	—	8	1	no	—
103	C₄⋊C₄	—	292	2	no	—
104	C₄⋊C₄	—	584	1	no	—
105	C₄²	—	8	1	no	—
106	C₄²	#11924013	1	1	yes	characteristic
107	C₁₆	—	2	1	no	—
108	C₁₆	C₂×Dic₁₄₆	1	1	yes	characteristic
109	C₁₆	#11924028	1	1	yes	characteristic
110	Q₈	—	4	2	no	—
111	Q₈	—	8	1	no	—
112	Q₈	—	1168	2	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	584	2	no	—
116	C₂×C₄	#14556532	1	1	yes	characteristic
117	C₂×C₄	#14556532	1	1	yes	characteristic
118	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	#14556570	1	1	yes	characteristic
121	C₈	#14556621	1	1	yes	characteristic
122	C₂²	C₂×#14556532	1	1	yes	characteristic
123	C₄	—	2	1	no	—
124	C₄	—	8	2	no	—
125	C₄	—	16	1	no	—
126	C₄	—	1168	2	no	—
127	C₄	#17988427	1	2	yes	—
128	C₄	#17988867	1	1	yes	characteristic
129	C₄	#17989027	1	1	yes	characteristic
130	C₂	#22612068	1	1	yes	characteristic
131	C₂	C₂×#17988427	1	1	yes	characteristic
132	C₂	#22612243	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22612206

C₂

yes

maximal, characteristic

#22608130

C₂

yes

maximal

C₄⋊#14556497

C₂

yes

maximal, characteristic

#22608112

C₂

yes

maximal

C₄×Q₃₂×C₇₃

C₂

yes

maximal, characteristic

#17988992

—

C₂×#14556497

C₂²

yes

—

#17988413

—

#17988413

C₂²

yes

—

C₂×Q₃₂×C₇₃

C₂²

yes

characteristic

C₄×Q₁₆×C₇₃

—

#366×C₇₃

C₂²

yes

characteristic

#358×C₇₃

—

C₄×C₁₆×C₇₃

C₂²

yes

—

—

—

—

C₂×Q₁₆×C₇₃

—

C₄×Q₈×C₇₃

—

Q₃₂×C₇₃

—

Q₃₂×C₇₃

Q₈

yes

complemented

C₂×C₁₆×C₇₃

D₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂³

yes

characteristic

#107×C₇₃

—

#103×C₇₃

—

C₄×C₈×C₇₃

D₄

yes

characteristic

C₂×Dic₁₄₆

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₈

yes

characteristic

C₂×C₈×C₇₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

D₈

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

C₂×Q₈

yes

characteristic

C₁₆×C₇₃

#41

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

D₁₆

yes

characteristic

C₂×C₄×C₇₃

C₂×D₈

yes

characteristic

C₂×Dic₇₃

—

Dic₁₄₆

—

C₈×C₇₃

—

C₈×C₇₃

C₂²⋊Q₈

yes

characteristic

C₈×C₇₃

#137

yes

characteristic

C₂²×C₇₃

C₂×D₁₆

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

Q₆₄

yes

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

—

C₂×C₇₃

yes

C₂×C₇₃

C₂×Q₆₄

yes

C₂×C₇₃

yes

—

C₄×Q₃₂

D₇₃

yes

characteristic

#1423

—

C₄⋊C₃₂

—

#1416

—

C₇₃

#13706

yes

characteristic, complemented, Sylow

C₂×Q₃₂

C₂×D₇₃

yes

characteristic

#500

—

146

—

C₄×Q₁₆

—

C₂×C₃₂

—

#366

—

#366

—

146

—

#366

C₂×D₇₃

yes

characteristic

#358

—

C₄×C₁₆

C₂×D₇₃

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

292

—

C₄×Q₈

—

Q₃₂

—

Q₃₂

Dic₁₄₆

yes

complemented

C₂×C₁₆

C₂²⋊D₇₃

yes

characteristic

C₂×C₁₆

C₂²×D₇₃

yes

characteristic

#107

—

#107

—

292

—

#107

—

146

—

#103

—

C₄×C₈

C₂²⋊D₇₃

yes

characteristic

C₃₂

—

146

—

C₂×Q₈

—

C₂×Q₈

—

584

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#11924013

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₇₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

292

—

104

C₄⋊C₄

—

584

—

105

C₄²

—

106

C₄²

#11924013

yes

characteristic

107

C₁₆

—

108

C₁₆

C₂×Dic₁₄₆

yes

characteristic

109

C₁₆

#11924028

yes

characteristic

110

Q₈

—

111

Q₈

—

112

Q₈

—

1168

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

584

—

116

C₂×C₄

#14556532

yes

characteristic

117

C₂×C₄

#14556532

yes

characteristic

118

C₂×C₄

C₂×#11924013

yes

characteristic

119

C₈

—

120

C₈

#14556570

yes

characteristic

121

C₈

#14556621

yes

characteristic

122

C₂²

C₂×#14556532

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

1168

—

127

C₄

#17988427

yes

—

128

C₄

#17988867

yes

characteristic

129

C₄

#17989027

yes

characteristic

130

C₂

#22612068

yes

characteristic

131

C₂

C₂×#17988427

yes

characteristic

132

C₂

#22612243

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14556532
commutator, cosocle	C₂×C₁₆×C₇₃	C₂³
Frattini	C₂×C₁₆	C₂²×D₇₃
Fitting	C₄×Q₃₂×C₇₃	C₂
socle, prime radical	C₂²×C₇₃	C₂×D₁₆
hypercenter	C₄×Q₃₂	D₇₃
hypocenter	C₇₃	#13706
LCS	G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

subgroup

G/H

center

C₂²

C₂×#14556532

commutator, cosocle

C₂×C₁₆×C₇₃

C₂³

Frattini

C₂×C₁₆

C₂²×D₇₃

Fitting

C₄×Q₃₂×C₇₃

C₂

socle, prime radical

C₂²×C₇₃

C₂×D₁₆

hypercenter

C₄×Q₃₂

D₇₃

hypocenter

C₇₃

#13706

LCS

G ▷ C₂×C₁₆×C₇₃ ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×Q₃₂

N	H ≅ G/N	type
22. Q₃₂×C₇₃	Q₈	semidirect
72. C₇₃	#13706	semidirect
86. Q₃₂	Dic₁₄₆	semidirect

H ≅ G/N

type

22. Q₃₂×C₇₃

Q₈

semidirect

72. C₇₃

#13706

semidirect

86. Q₃₂

Dic₁₄₆

semidirect

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

subgroup	index
#22612206	2
#22608130	2
C₄⋊#14556497	2
#22608112	2
C₄×Q₃₂×C₇₃	2
#13706	73

subgroup

index

C₄×Q₃₂×C₇₃

Subgroups — #33006881

Conjugacy classes of subgroups

Complements

Maximal subgroups