Group #34012332 — subgroups

Subgroups — #34012332

Order 18688.

354 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 156 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	C₄×#14556665	C₂	1	2	yes	maximal
2	#22624973	C₂	1	4	yes	maximal
3	C₄×#14556501	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556665	C₂²	1	4	yes	—
5	#17989475	C₄	1	4	yes	complemented
6	C₄×#11924031	C₄	1	4	yes	complemented
7	C₂×#14556501	C₄	1	2	yes	—
8	C₂×#14556501	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
10	C₄²×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
12	#14556665	—	2	8	no	—
13	C₂×#11924031	C₂×C₄	1	8	yes	—
14	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
15	C₂²×#10596831	C₂³	1	1	yes	characteristic
16	C₂×C₄×Dic₇₃	—	2	2	no	—
17	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
18	#14556501	—	2	4	no	—
19	#14556501	C₈	1	8	yes	—
20	C₄×#10596831	C₂×C₄	1	2	yes	—
21	C₄×#10596831	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
24	C₄×C₈×C₇₃	—	2	2	no	—
25	#11924031	—	2	8	no	—
26	C₂²×Dic₇₃	—	2	2	no	—
27	C₂×#10596831	C₄²	1	2	yes	—
28	C₂×#10596831	C₂×C₈	1	4	yes	—
29	C₂×#10596831	C₂²×C₄	1	2	yes	—
30	C₂×#10596831	#41	1	4	yes	—
31	C₄×Dic₇₃	—	2	2	no	—
32	C₄×Dic₇₃	—	4	4	no	—
33	C₄×Dic₇₃	C₂×C₈	1	4	yes	—
34	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
35	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₇₃	—	2	2	no	—
37	C₂×C₈×C₇₃	—	2	2	no	—
38	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	—
39	C₄²×C₇₃	C₄²	1	1	yes	characteristic
40	C₄²×C₇₃	C₄²	1	1	yes	characteristic
41	C₄²×C₇₃	M₁₆	1	2	yes	—
42	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
43	C₂×C₄×C₇₃	—	2	4	no	—
44	C₂×C₄×C₇₃	#97	1	2	yes	—
45	C₂×C₄×C₇₃	#97	1	2	yes	—
46	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₇₃	#117	1	2	yes	—
48	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
49	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
50	C₂×Dic₇₃	—	2	4	no	—
51	C₂×Dic₇₃	—	4	4	no	—
52	C₈×C₇₃	—	2	4	no	—
53	#10596831	C₄×C₈	1	4	yes	—
54	#10596831	#117	1	4	yes	—
55	C₂²×C₇₃	—	2	2	no	—
56	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
57	C₂²×C₇₃	#431	1	2	yes	—
58	C₄×C₇₃	—	2	2	no	—
59	C₄×C₇₃	#347	1	4	yes	—
60	C₄×C₇₃	#376	1	1	yes	characteristic
61	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
62	C₄×C₇₃	#431	1	2	yes	—
63	Dic₇₃	—	4	4	no	—
64	#12594	—	73	1	no	maximal, Sylow
65	C₂×C₇₃	—	2	2	no	—
66	C₂×C₇₃	#1744	1	1	yes	characteristic
67	C₂×C₇₃	C₂×#347	1	2	yes	—
68	C₄×M₃₂	—	73	2	no	—
69	C₄×#97	—	73	1	no	—
70	#1375	—	73	4	no	—
71	C₇₃	#12594	1	1	yes	characteristic, complemented, Sylow
72	C₂×M₃₂	—	73	4	no	—
73	C₂×#97	—	73	2	no	—
74	C₂×#97	—	73	2	no	—
75	C₂×C₄×C₈	—	73	1	no	—
76	C₂×C₄×C₈	#64408	1	1	yes	characteristic
77	C₄³	—	73	1	no	—
78	#346	—	73	4	no	—
79	C₄×C₁₆	—	73	4	no	—
80	C₂²×C₈	—	73	1	no	—
81	C₂²×C₈	—	73	1	no	—
82	C₂²×C₈	C₂×#64408	1	2	yes	—
83	C₂×C₄²	—	73	2	no	—
84	C₂×C₄²	—	146	2	no	—
85	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
86	M₃₂	—	146	8	no	—
87	C₂×C₁₆	—	73	8	no	—
88	#97	—	73	8	no	—
89	#97	—	146	4	no	—
90	C₄×C₈	—	2	2	no	—
91	C₄×C₈	—	73	2	no	—
92	C₄×C₈	—	73	2	no	—
93	C₂²×C₄	—	146	2	no	—
94	C₂²×C₄	#11924038	1	2	yes	—
95	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	73	4	no	—
99	C₂×C₈	—	73	2	no	—
100	C₂×C₈	—	73	2	no	—
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	#11924036	1	4	yes	—
103	C₄²	—	292	4	no	—
104	C₄²	—	73	4	no	—
105	C₄²	—	146	2	no	—
106	C₄²	C₄×#64408	1	1	yes	characteristic
107	C₄²	C₄×#64408	1	1	yes	characteristic
108	C₄²	#11924037	1	2	yes	—
109	C₁₆	—	146	8	no	—
110	C₂³	C₂×#11924038	1	1	yes	characteristic
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	292	4	no	—
113	C₂×C₄	—	146	4	no	—
114	C₂×C₄	#14556646	1	2	yes	—
115	C₂×C₄	#14556646	1	2	yes	—
116	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
117	C₂×C₄	#14556670	1	2	yes	—
118	C₂×C₄	C₂×#11924036	1	2	yes	—
119	C₂×C₄	C₂×#11924037	1	1	yes	characteristic
120	C₈	—	2	4	no	—
121	C₈	—	73	4	no	—
122	C₈	—	73	4	no	—
123	C₂²	—	2	2	no	—
124	C₂²	C₂×#14556646	1	1	yes	characteristic
125	C₂²	#17989606	1	2	yes	—
126	C₄	—	2	2	no	—
127	C₄	—	292	4	no	—
128	C₄	#17989480	1	4	yes	—
129	C₄	#17989523	1	1	yes	characteristic
130	C₄	C₄×#11924036	1	1	yes	characteristic
131	C₄	#17989604	1	2	yes	—
132	C₂	—	2	2	no	—
133	C₂	#22625646	1	1	yes	characteristic
134	C₂	C₂×#17989480	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556665

C₂

yes

maximal

#22624973

C₂

yes

maximal

C₄×#14556501

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

#17989475

C₄

yes

complemented

C₄×#11924031

C₄

yes

complemented

C₂×#14556501

C₄

yes

—

C₂×#14556501

C₂²

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

C₄²×Dic₇₃

C₄

yes

characteristic

C₂×C₄×C₈×C₇₃

C₄

yes

characteristic

#14556665

—

C₂×#11924031

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

characteristic

C₂×C₄×Dic₇₃

—

C₂×C₄×Dic₇₃

C₂×C₄

yes

—

#14556501

—

#14556501

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄×C₈×C₇₃

—

#11924031

—

C₂²×Dic₇₃

—

C₂×#10596831

C₄²

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₂×#10596831

#41

yes

—

C₄×Dic₇₃

—

C₄×Dic₇₃

—

C₄×Dic₇₃

C₂×C₈

yes

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

C₄²

yes

characteristic

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

#97

yes

—

C₂×C₄×C₇₃

#97

yes

—

C₂×C₄×C₇₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₇₃

#117

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

—

C₂×C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

#10596831

C₄×C₈

yes

—

#10596831

#117

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#97

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₄×C₇₃

—

C₄×C₇₃

#347

yes

—

C₄×C₇₃

#376

yes

characteristic

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

#431

yes

—

Dic₇₃

—

#12594

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1744

yes

characteristic

C₂×C₇₃

C₂×#347

yes

—

C₄×M₃₂

—

C₄×#97

—

#1375

—

C₇₃

#12594

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

#64408

yes

characteristic

C₄³

—

#346

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×#64408

yes

—

C₂×C₄²

—

C₂×C₄²

—

146

—

C₂×C₄²

C₂×#64408

yes

characteristic

M₃₂

—

146

—

C₂×C₁₆

—

#97

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

146

—

C₂²×C₄

#11924038

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

#11924036

yes

—

103

C₄²

—

292

—

104

C₄²

—

105

C₄²

—

146

—

106

C₄²

C₄×#64408

yes

characteristic

107

C₄²

C₄×#64408

yes

characteristic

108

C₄²

#11924037

yes

—

109

C₁₆

—

146

—

110

C₂³

C₂×#11924038

yes

characteristic

111

C₂×C₄

—

112

C₂×C₄

—

292

—

113

C₂×C₄

—

146

—

114

C₂×C₄

#14556646

yes

—

115

C₂×C₄

#14556646

yes

—

116

C₂×C₄

C₂×C₄×#64408

yes

characteristic

117

C₂×C₄

#14556670

yes

—

118

C₂×C₄

C₂×#11924036

yes

—

119

C₂×C₄

C₂×#11924037

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

C₂×#14556646

yes

characteristic

125

C₂²

#17989606

yes

—

126

C₄

—

127

C₄

—

292

—

128

C₄

#17989480

yes

—

129

C₄

#17989523

yes

characteristic

130

C₄

C₄×#11924036

yes

characteristic

131

C₄

#17989604

yes

—

132

C₂

—

133

C₂

#22625646

yes

characteristic

134

C₂

C₂×#17989480

yes

—

	subgroup	G/H
center	C₄²	C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	#12594
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₄²

C₄×#64408

commutator

C₄×C₇₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

C₂×C₄×C₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#97

prime radical

C₂³×C₇₃

C₂×M₁₆

C₂³

C₂×C₄×C₈

C₇₃

G ▷ C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. #17989475	C₄	semidirect
6. C₄×#11924031	C₄	semidirect
71. C₇₃	#12594	semidirect

H ≅ G/N

type

5. #17989475

C₄

semidirect

6. C₄×#11924031

C₄

semidirect

71. C₇₃

#12594

semidirect

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

subgroup	index
C₄×#14556665	2
#22624973	2
C₄×#14556501	2
#12594	73

subgroup

index

Subgroups — #34012332

Conjugacy classes of subgroups

Complements

Maximal subgroups