Group #34011821 — subgroups

Subgroups — #34011821

Order 18688.

180 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 68 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	#22626306	C₂	1	2	yes	maximal
2	#22624967	C₂	1	2	yes	maximal, complemented
3	#22624959	C₂	1	2	yes	maximal, complemented
4	#22611529	C₂	1	1	yes	maximal, characteristic
5	#17989509	C₂²	1	4	yes	—
6	#17989484	—	2	2	no	—
7	#17989479	—	2	2	no	—
8	C₂×#14556530	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₇₃	C₄	1	1	yes	characteristic
10	#17988541	C₂²	1	1	yes	characteristic
11	#17988362	C₂²	1	1	yes	characteristic
12	C₄×#11924003	C₄	1	1	yes	characteristic
13	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
14	#346×C₇₃	C₄	1	1	yes	characteristic
15	#14556665	—	2	4	no	—
16	C₂×#11924011	D₄	1	1	yes	characteristic
17	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
18	#14556548	Q₈	1	1	yes	characteristic
19	#14556530	—	2	2	no	—
20	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
21	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
22	C₁₆×D₇₃	—	2	2	no	—
23	#14556502	D₄	1	1	yes	characteristic
24	#14556501	Q₈	1	1	yes	characteristic
25	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
29	#11924031	—	4	4	no	—
30	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
31	#11924011	—	2	2	no	—
32	C₂×#10596831	C₄⋊C₄	1	1	yes	characteristic
33	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
34	C₈×D₇₃	—	2	1	no	—
35	C₈×D₇₃	—	2	1	no	—
36	#11924007	—	2	1	no	—
37	#11924005	—	2	1	no	—
38	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
39	#11924003	M₁₆	1	2	yes	—
40	#11924003	M₁₆	1	2	yes	—
41	C₂×C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₇₃	—	2	1	no	—
45	C₁₆×C₇₃	—	2	1	no	—
46	C₂²×D₇₃	#108	1	1	yes	characteristic
47	C₂×C₄×C₇₃	—	2	1	no	—
48	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×Dic₇₃	—	2	1	no	—
50	C₂×Dic₇₃	#108	1	1	yes	characteristic
51	C₄×D₇₃	—	2	1	no	—
52	C₄×D₇₃	—	2	1	no	—
53	C₈×C₇₃	—	2	1	no	—
54	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
56	#10596831	—	2	1	no	—
57	#10596831	#135	1	2	yes	—
58	C₂²×C₇₃	C₂×#108	1	1	yes	characteristic
59	C₂×D₇₃	—	2	1	no	—
60	C₂×D₇₃	—	4	1	no	—
61	C₄×C₇₃	—	4	1	no	—
62	C₄×C₇₃	#423	1	1	yes	characteristic
63	C₄×C₇₃	#427	1	1	yes	characteristic
64	Dic₇₃	—	2	1	no	—
65	Dic₇₃	—	4	1	no	—
66	#12027	—	73	1	no	maximal, Sylow
67	C₂×C₇₃	—	2	1	no	—
68	C₂×C₇₃	#1568	1	1	yes	characteristic
69	D₇₃	—	4	1	no	—
70	#2166	—	73	2	no	—
71	#2108	—	73	1	no	—
72	#1373	—	73	2	no	—
73	#1369	—	73	2	no	—
74	C₇₃	#12027	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₃₂	—	73	1	no	—
76	C₂²×C₁₆	—	73	1	no	—
77	#405	—	73	1	no	—
78	#364	—	73	4	no	—
79	#349	—	146	2	no	—
80	#347	—	146	2	no	—
81	#346	—	73	1	no	—
82	#346	#64408	1	1	yes	characteristic
83	C₄×C₁₆	—	73	1	no	—
84	C₄×C₁₆	—	73	1	no	—
85	C₂×M₁₆	—	73	1	no	—
86	C₂²×C₈	—	73	1	no	—
87	#117	—	73	1	no	—
88	M₃₂	—	146	2	no	—
89	M₃₂	—	146	4	no	—
90	C₂×C₁₆	—	73	1	no	—
91	C₂×C₁₆	—	73	1	no	—
92	C₂×C₁₆	—	146	2	no	—
93	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
94	C₂×C₁₆	C₂×#64408	1	1	yes	characteristic
95	#97	—	73	1	no	—
96	#97	—	73	1	no	—
97	C₄×C₈	—	73	1	no	—
98	C₄×C₈	C₂×#64408	1	1	yes	characteristic
99	C₂²×C₄	—	73	1	no	—
100	M₁₆	—	146	2	no	—
101	C₂×C₈	—	73	1	no	—
102	C₂×C₈	—	73	1	no	—
103	C₂×C₈	—	146	1	no	—
104	C₂×C₈	—	146	1	no	—
105	C₂×C₈	C₄⋊#64408	1	1	yes	characteristic
106	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
107	C₄⋊C₄	—	146	1	no	—
108	C₂²⋊C₄	—	146	1	no	—
109	C₄²	—	73	1	no	—
110	C₄²	C₄⋊#64408	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	292	4	no	—
114	C₁₆	—	73	2	no	—
115	C₁₆	—	73	2	no	—
116	C₂³	—	73	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	146	1	no	—
120	C₂×C₄	—	146	1	no	—
121	C₂×C₄	—	146	1	no	—
122	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	73	2	no	—
125	C₈	—	146	1	no	—
126	C₈	D₇₃⋊M₁₆	1	1	yes	characteristic
127	C₈	D₇₃⋊M₁₆	1	1	yes	characteristic
128	C₂²	—	292	1	no	—
129	C₂²	—	146	1	no	—
130	C₂²	D₇₃⋊#108	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	292	1	no	—
133	C₄	—	146	1	no	—
134	C₄	#17989592	1	1	yes	characteristic
135	C₄	#17989597	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	292	1	no	—
138	C₂	#22625494	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22626306

C₂

yes

maximal

#22624967

C₂

yes

maximal, complemented

#22624959

C₂

yes

maximal, complemented

#22611529

C₂

yes

maximal, characteristic

#17989509

C₂²

yes

—

#17989484

—

#17989479

—

C₂×#14556530

C₄

yes

characteristic

C₂×C₁₆×D₇₃

C₄

yes

characteristic

#17988541

C₂²

yes

characteristic

#17988362

C₂²

yes

characteristic

C₄×#11924003

C₄

yes

characteristic

C₁₆×Dic₇₃

C₂²

yes

characteristic

#346×C₇₃

C₄

yes

characteristic

#14556665

—

C₂×#11924011

D₄

yes

characteristic

C₂×C₈×D₇₃

C₂×C₄

yes

characteristic

#14556548

Q₈

yes

characteristic

#14556530

—

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₁₆×D₇₃

—

#14556502

D₄

yes

characteristic

#14556501

Q₈

yes

characteristic

C₈×Dic₇₃

C₂³

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

C₂×C₄

yes

characteristic

#11924031

—

C₂×C₄×D₇₃

C₄⋊C₄

yes

characteristic

#11924011

—

C₂×#10596831

C₄⋊C₄

yes

characteristic

C₂×#10596831

C₂×D₄

yes

characteristic

C₈×D₇₃

—

C₈×D₇₃

—

#11924007

—

#11924005

—

C₄×Dic₇₃

C₂×Q₈

yes

characteristic

#11924003

M₁₆

yes

—

#11924003

M₁₆

yes

—

C₂×C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

C₄⋊C₄

yes

characteristic

C₁₆×C₇₃

—

C₁₆×C₇₃

—

C₂²×D₇₃

#108

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

#108

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×M₁₆

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

—

yes

—

C₂²×C₇₃

C₂×#108

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

Dic₇₃

—

#12027

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1568

yes

characteristic

D₇₃

—

—

—

—

—

C₇₃

#12027

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#364

—

#349

—

146

—

#347

—

146

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

146

—

M₃₂

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×#64408

yes

characteristic

C₂²×C₄

—

100

M₁₆

—

146

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

146

—

104

C₂×C₈

—

146

—

105

C₂×C₈

C₄⋊#64408

yes

characteristic

106

C₂×C₈

C₂²×#64408

yes

characteristic

107

C₄⋊C₄

—

146

—

108

C₂²⋊C₄

—

146

—

109

C₄²

—

110

C₄²

C₄⋊#64408

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

292

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

146

—

120

C₂×C₄

—

146

—

121

C₂×C₄

—

146

—

122

C₂×C₄

C₂×C₄⋊#64408

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

146

—

126

C₈

D₇₃⋊M₁₆

yes

characteristic

127

C₈

D₇₃⋊M₁₆

yes

characteristic

128

C₂²

—

292

—

129

C₂²

—

146

—

130

C₂²

D₇₃⋊#108

yes

characteristic

131

C₄

—

132

C₄

—

292

—

133

C₄

—

146

—

134

C₄

#17989592

yes

characteristic

135

C₄

#17989597

yes

characteristic

136

C₂

—

137

C₂

—

292

—

138

C₂

#22625494

yes

characteristic

	subgroup	G/H
center	C₄	#17989597
commutator	C₂×C₈×C₇₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#64408
Fitting	#346×C₇₃	C₄
socle	C₂×C₇₃	#1568
prime radical	C₂²×D₇₃	#108
cosocle	C₈×Dic₇₃	C₂³
hypercenter	#346	#64408
hypocenter	C₇₃	#12027
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

#17989597

commutator

C₂×C₈×C₇₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#64408

Fitting

#346×C₇₃

C₄

C₂×C₇₃

C₂²×D₇₃

C₈×Dic₇₃

C₂³

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #22624967	C₂	semidirect
3. #22624959	C₂	semidirect
74. C₇₃	#12027	semidirect

H ≅ G/N

type

2. #22624967

C₂

semidirect

3. #22624959

C₂

semidirect

74. C₇₃

#12027

semidirect

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

subgroup	index
#22626306	2
#22624967	2
#22624959	2
#22611529	2
#12027	73

subgroup

index

Subgroups — #34011821

Conjugacy classes of subgroups

Complements

Maximal subgroups