Group #33999998 — subgroups

Subgroups — #33999998

Order 18688.

188 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 46 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	#22624867	C₂	1	2	yes	maximal
2	#22610872	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11924031	—	2	2	no	—
4	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
5	C₂×#14556502	C₄	1	1	yes	characteristic
6	#17988342	—	2	2	no	—
7	C₂×C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
8	C₂×#11924031	—	4	2	no	—
9	C₂²×#10596831	—	2	1	no	—
10	C₂²×#10596831	D₄	1	2	yes	—
11	C₄⋊#10596831	—	4	1	no	—
12	#14556502	—	4	1	no	—
13	C₂²×Q₈×C₇₃	C₈	1	1	yes	characteristic
14	C₄×Q₈×C₇₃	—	4	1	no	—
15	C₄×Q₈×C₇₃	—	4	1	no	—
16	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₇₃	C₈	1	1	yes	characteristic
18	C₂×C₄²×C₇₃	—	2	1	no	—
19	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
20	#11924031	—	8	2	no	—
21	C₂×#10596831	—	2	2	no	—
22	C₂×#10596831	—	2	2	no	—
23	C₂×#10596831	—	4	1	no	—
24	C₂×#10596831	—	4	1	no	—
25	C₂×#10596831	—	4	2	no	—
26	C₂×Q₈×C₇₃	—	4	2	no	—
27	C₂×Q₈×C₇₃	—	4	1	no	—
28	C₂²×C₄×C₇₃	—	2	1	no	—
29	C₂²×C₄×C₇₃	—	2	1	no	—
30	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	—	4	1	no	—
34	C₄⋊C₄×C₇₃	—	4	2	no	—
35	C₄²×C₇₃	—	2	1	no	—
36	C₄²×C₇₃	—	2	1	no	—
37	C₄²×C₇₃	—	4	1	no	—
38	C₄²×C₇₃	—	4	1	no	—
39	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
40	Q₈×C₇₃	—	4	2	no	—
41	Q₈×C₇₃	—	4	2	no	—
42	C₂×C₄×C₇₃	—	2	1	no	—
43	C₂×C₄×C₇₃	—	2	1	no	—
44	C₂×C₄×C₇₃	—	2	2	no	—
45	C₂×C₄×C₇₃	—	2	1	no	—
46	C₂×C₄×C₇₃	—	2	1	no	—
47	C₂×C₄×C₇₃	—	2	2	no	—
48	C₂×C₄×C₇₃	—	4	1	no	—
49	C₂×C₄×C₇₃	—	4	1	no	—
50	C₂×C₄×C₇₃	—	4	1	no	—
51	C₂×C₄×C₇₃	—	4	1	no	—
52	C₂×C₄×C₇₃	—	4	2	no	—
53	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
54	C₂×C₄×C₇₃	#100	1	1	yes	characteristic
55	#10596831	—	4	2	no	—
56	#10596831	—	4	2	no	—
57	C₂²×C₇₃	—	2	1	no	—
58	C₂²×C₇₃	—	2	1	no	—
59	C₂²×C₇₃	#323	1	1	yes	characteristic
60	C₂²×C₇₃	#350	1	2	yes	—
61	C₄×C₇₃	—	2	1	no	—
62	C₄×C₇₃	—	4	1	no	—
63	C₄×C₇₃	—	4	2	no	—
64	C₄×C₇₃	—	4	1	no	—
65	C₄×C₇₃	—	8	1	no	—
66	C₄×C₇₃	#354	1	1	yes	characteristic
67	C₄×C₇₃	#356	1	1	yes	characteristic
68	#7061	—	73	1	no	maximal, Sylow
69	C₂×C₇₃	—	2	2	no	—
70	C₂×C₇₃	#1315	1	1	yes	characteristic
71	C₂×C₇₃	#1319	1	1	yes	characteristic
72	C₂×C₇₃	#1320	1	1	yes	characteristic
73	#1985	—	73	1	no	—
74	#1322	—	73	2	no	—
75	C₇₃	#7061	1	1	yes	characteristic, complemented, Sylow
76	C₂×C₄×Q₈	#64408	1	1	yes	characteristic
77	C₂×C₄⋊C₈	—	73	1	no	—
78	C₂×#97	—	73	1	no	—
79	C₂²⋊C₁₆	—	146	2	no	—
80	#336	—	146	2	no	—
81	C₂²×Q₈	#10596833	1	1	yes	characteristic
82	C₂²×C₈	—	73	2	no	—
83	C₂²×C₈	—	146	1	no	—
84	C₄×Q₈	—	4	1	no	—
85	C₄×Q₈	—	4	1	no	—
86	C₂×C₄⋊C₄	—	2	1	no	—
87	C₂×C₄⋊C₄	#10596833	1	1	yes	characteristic
88	C₂×C₄²	—	2	1	no	—
89	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
90	C₂×C₁₆	—	292	2	no	—
91	C₄⋊C₈	—	292	1	no	—
92	#97	—	292	1	no	—
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	4	1	no	—
95	C₂²×C₄	—	2	1	no	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
98	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
99	C₂²×C₄	#11924037	1	1	yes	characteristic
100	C₂×C₈	—	292	1	no	—
101	C₂×C₈	—	292	1	no	—
102	C₂×C₈	—	292	2	no	—
103	C₂×C₈	—	146	2	no	—
104	C₂×C₈	—	146	2	no	—
105	C₄⋊C₄	—	4	1	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄²	—	2	1	no	—
108	C₄²	—	2	1	no	—
109	C₄²	—	4	1	no	—
110	C₄²	—	4	1	no	—
111	C₁₆	—	584	2	no	—
112	C₂³	C₂²⋊#10596833	1	1	yes	characteristic
113	Q₈	—	4	2	no	—
114	Q₈	—	4	2	no	—
115	C₂×C₄	—	2	1	no	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	1	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	#14556649	1	1	yes	characteristic
127	C₂×C₄	#14556651	1	1	yes	characteristic
128	C₈	—	292	2	no	—
129	C₈	—	292	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	#17989430	1	1	yes	characteristic
133	C₂²	#17989485	1	2	yes	—
134	C₄	—	2	1	no	—
135	C₄	—	4	1	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	1	no	—
138	C₄	—	8	1	no	—
139	C₄	#17989493	1	1	yes	characteristic
140	C₄	#17989497	1	1	yes	characteristic
141	C₂	—	2	2	no	—
142	C₂	#22624853	1	1	yes	characteristic
143	C₂	#22624861	1	1	yes	characteristic
144	C₂	#22624863	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624867

C₂

yes

maximal

#22610872

C₂

yes

maximal, characteristic

C₂²⋊#11924031

—

C₂×C₄⋊#10596831

C₂²

yes

characteristic

C₂×#14556502

C₄

yes

characteristic

#17988342

—

C₂×C₄×Q₈×C₇₃

C₄

yes

—

—

D₄

yes

—

C₄⋊#10596831

—

#14556502

—

C₂²×Q₈×C₇₃

C₈

yes

characteristic

C₄×Q₈×C₇₃

—

C₄×Q₈×C₇₃

—

C₂×C₄⋊C₄×C₇₃

—

C₂×C₄⋊C₄×C₇₃

C₈

yes

characteristic

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

—

—

—

—

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂²⋊C₈

yes

characteristic

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

yes

C₂×C₄×C₇₃

yes

—

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#323

yes

characteristic

C₂²×C₇₃

#350

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

—

C₇₃

#7061

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

#64408

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

C₂²⋊C₁₆

—

146

—

#336

—

146

—

C₂²×Q₈

#10596833

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

146

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#10596833

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

292

—

C₄⋊C₈

—

292

—

#97

—

292

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

100

C₂×C₈

—

292

—

101

C₂×C₈

—

292

—

102

C₂×C₈

—

292

—

103

C₂×C₈

—

146

—

104

C₂×C₈

—

146

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₁₆

—

584

—

112

C₂³

C₂²⋊#10596833

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

#14556649

yes

characteristic

127

C₂×C₄

#14556651

yes

characteristic

128

C₈

—

292

—

129

C₈

—

292

—

130

C₂²

—

131

C₂²

—

132

C₂²

#17989430

yes

characteristic

133

C₂²

#17989485

yes

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

#17989493

yes

characteristic

140

C₄

#17989497

yes

characteristic

141

C₂

—

142

C₂

#22624853

yes

characteristic

143

C₂

#22624861

yes

characteristic

144

C₂

#22624863

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556649
commutator	C₂²×C₄×C₇₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	C₂×C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	#323
prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄⋊#10596831	C₂²
hypercenter	C₂×C₄×Q₈	#64408
hypocenter	C₇₃	#7061
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂×C₄

#14556649

commutator

C₂²×C₄×C₇₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

C₂×C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

#323

prime radical

C₂³×C₇₃

C₂²⋊C₈

C₂²

C₂×C₄×Q₈

C₇₃

G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
75. C₇₃	#7061	semidirect

H ≅ G/N

type

75. C₇₃

#7061

semidirect

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

subgroup	index
#22624867	2
#22610872	2
#7061	73

subgroup

index

#22624867

#22610872

#7061

Subgroups — #33999998

Conjugacy classes of subgroups

Complements

Maximal subgroups