Group #34012326 — subgroups

Subgroups — #34012326

Order 18688.

338 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 148 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	#22624972	C₂	1	4	yes	maximal
3	#22609618	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×#14556502	C₄	1	1	yes	characteristic
8	C₂×#14556502	C₂²	1	2	yes	—
9	C₂×C₄×#10596831	C₄	1	2	yes	—
10	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
11	#376×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	2	yes	—
15	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
16	C₂²×#10596831	C₂³	1	1	yes	characteristic
17	#14556502	—	2	4	no	—
18	#14556502	—	2	2	no	—
19	C₄×#10596831	C₈	1	8	yes	—
20	C₄×#10596831	C₂×C₄	1	2	yes	—
21	C₄×#10596831	C₂×C₄	1	2	yes	—
22	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
24	#95×C₇₃	—	2	2	no	—
25	#11924031	—	2	8	no	—
26	C₂×#10596831	—	2	2	no	—
27	C₂×#10596831	—	2	2	no	—
28	C₂×#10596831	C₄²	1	1	yes	characteristic
29	C₂×#10596831	C₄²	1	1	yes	characteristic
30	C₂×#10596831	C₂×C₈	1	4	yes	—
31	C₂×#10596831	C₂×C₈	1	4	yes	—
32	C₂×#10596831	C₂²×C₄	1	2	yes	—
33	C₂×#10596831	#41	1	4	yes	—
34	C₂²×C₄×C₇₃	—	2	2	no	—
35	C₂²×C₄×C₇₃	M₁₆	1	2	yes	—
36	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₄²×C₇₃	—	2	2	no	—
38	C₄²×C₇₃	C₄²	1	2	yes	—
39	C₄²×C₇₃	C₂×C₈	1	4	yes	—
40	C₄²×C₇₃	M₁₆	1	2	yes	—
41	C₂³×C₇₃	C₂×M₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₃	—	2	4	no	—
43	C₂×C₄×C₇₃	—	4	2	no	—
44	C₂×C₄×C₇₃	—	4	4	no	—
45	C₂×C₄×C₇₃	#97	1	4	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	#10596831	—	2	4	no	—
51	#10596831	C₄×C₈	1	2	yes	—
52	#10596831	C₄×C₈	1	2	yes	—
53	#10596831	#117	1	4	yes	—
54	C₂²×C₇₃	—	2	2	no	—
55	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
56	C₂²×C₇₃	#431	1	2	yes	—
57	C₄×C₇₃	—	2	4	no	—
58	C₄×C₇₃	—	4	4	no	—
59	C₄×C₇₃	#376	1	1	yes	characteristic
60	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
61	C₄×C₇₃	#431	1	2	yes	—
62	#12593	—	73	1	no	maximal, Sylow
63	C₂×C₇₃	—	2	2	no	—
64	C₂×C₇₃	#1744	1	1	yes	characteristic
65	C₂×C₇₃	#2110	1	2	yes	—
66	C₄×M₃₂	—	73	2	no	—
67	#1726	—	73	1	no	—
68	#1375	—	73	4	no	—
69	C₇₃	#12593	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₃₂	—	73	4	no	—
71	C₂×#97	—	73	2	no	—
72	C₂×#97	—	73	1	no	—
73	C₂×C₄×C₈	—	73	1	no	—
74	C₂×C₄×C₈	—	73	2	no	—
75	#376	#64408	1	1	yes	characteristic
76	#346	—	73	4	no	—
77	C₄×C₁₆	—	73	4	no	—
78	C₂²×C₈	—	73	1	no	—
79	C₂²×C₈	—	73	1	no	—
80	C₂²×C₈	—	73	2	no	—
81	C₂×C₄²	C₂×#64408	1	2	yes	—
82	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
83	M₃₂	—	146	8	no	—
84	C₂×C₁₆	—	73	8	no	—
85	#97	—	146	4	no	—
86	#97	—	146	2	no	—
87	C₄×C₈	—	73	2	no	—
88	C₄×C₈	—	73	2	no	—
89	C₄×C₈	—	73	8	no	—
90	#95	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	#11924037	1	2	yes	—
93	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
94	C₂×C₈	—	73	4	no	—
95	C₂×C₈	—	73	1	no	—
96	C₂×C₈	—	73	1	no	—
97	C₂×C₈	—	73	2	no	—
98	C₂×C₈	—	73	4	no	—
99	C₂×C₈	—	73	4	no	—
100	C₂×C₈	—	146	2	no	—
101	C₂×C₈	—	146	2	no	—
102	C₄²	—	2	2	no	—
103	C₄²	C₄×#64408	1	2	yes	—
104	C₄²	#11924036	1	4	yes	—
105	C₄²	#11924038	1	2	yes	—
106	C₁₆	—	146	8	no	—
107	C₂³	C₂×#11924037	1	1	yes	characteristic
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	4	no	—
111	C₂×C₄	#14556646	1	4	yes	—
112	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
113	C₂×C₄	#14556670	1	2	yes	—
114	C₂×C₄	C₂×#11924036	1	2	yes	—
115	C₂×C₄	C₂×#11924038	1	1	yes	characteristic
116	C₈	—	73	4	no	—
117	C₈	—	73	2	no	—
118	C₈	—	73	2	no	—
119	C₈	—	146	4	no	—
120	C₂²	—	2	2	no	—
121	C₂²	C₂×#14556646	1	1	yes	characteristic
122	C₂²	#17989604	1	2	yes	—
123	C₄	—	2	4	no	—
124	C₄	—	4	4	no	—
125	C₄	#17989523	1	1	yes	characteristic
126	C₄	C₄×#11924036	1	1	yes	characteristic
127	C₄	#17989606	1	2	yes	—
128	C₂	—	2	2	no	—
129	C₂	#22625646	1	1	yes	characteristic
130	C₂	#22626152	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14556665

C₂

yes

maximal

#22624972

C₂

yes

maximal

#22609618

C₂

yes

maximal, characteristic

C₂×#14556665

C₂²

yes

—

#17989475

C₄

yes

complemented

C₄×#11924031

C₄

yes

complemented

C₂×#14556502

C₄

yes

characteristic

C₂×#14556502

C₂²

yes

—

C₂×C₄×#10596831

C₄

yes

—

C₂×C₄×#10596831

C₂²

yes

characteristic

#376×C₇₃

C₄

yes

characteristic

#14556665

—

C₂×#11924031

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

—

C₂²×#10596831

C₂×C₄

yes

characteristic

C₂²×#10596831

C₂³

yes

—

—

C₈

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₄×#10596831

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#95×C₇₃

—

—

—

—

C₄²

yes

characteristic

C₂×#10596831

C₄²

yes

characteristic

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂×C₈

yes

—

C₂×#10596831

C₂²×C₄

yes

—

C₂×#10596831

#41

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

M₁₆

yes

—

C₂²×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₄²

yes

—

C₄²×C₇₃

C₂×C₈

yes

—

C₄²×C₇₃

M₁₆

yes

—

C₂³×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

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

#10596831

—

#10596831

C₄×C₈

yes

—

#10596831

C₄×C₈

yes

—

#10596831

#117

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×#97

yes

characteristic

C₂²×C₇₃

#431

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#376

yes

characteristic

C₄×C₇₃

C₂×C₄×C₈

yes

characteristic

C₄×C₇₃

#431

yes

—

#12593

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

#1744

yes

characteristic

C₂×C₇₃

#2110

yes

—

C₄×M₃₂

—

#1726

—

#1375

—

C₇₃

#12593

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

yes

—

C₄×C₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₂×#64408

yes

—

C₂×C₄²

C₂×#64408

yes

characteristic

M₃₂

—

146

—

C₂×C₁₆

—

#97

—

146

—

#97

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11924037

yes

—

C₂²×C₄

C₂²×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

146

—

101

C₂×C₈

—

146

—

102

C₄²

—

103

C₄²

C₄×#64408

yes

—

104

C₄²

#11924036

yes

—

105

C₄²

#11924038

yes

—

106

C₁₆

—

146

—

107

C₂³

C₂×#11924037

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

#14556646

yes

—

112

C₂×C₄

C₂×C₄×#64408

yes

characteristic

113

C₂×C₄

#14556670

yes

—

114

C₂×C₄

C₂×#11924036

yes

—

115

C₂×C₄

C₂×#11924038

yes

characteristic

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

146

—

120

C₂²

—

121

C₂²

C₂×#14556646

yes

characteristic

122

C₂²

#17989604

yes

—

123

C₄

—

124

C₄

—

125

C₄

#17989523

yes

characteristic

126

C₄

C₄×#11924036

yes

characteristic

127

C₄

#17989606

yes

—

128

C₂

—

129

C₂

#22625646

yes

characteristic

130

C₂

#22626152

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂²×C₄	C₂²×#64408
Fitting	#376×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×C₇₃	C₂×M₁₆
cosocle	C₂²×#10596831	C₂³
hypercenter	#376	#64408
hypocenter	C₇₃	#12593
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×C₄×#64408

commutator

C₄×C₇₃

C₂×C₄×C₈

Frattini

C₂²×C₄

C₂²×#64408

Fitting

#376×C₇₃

C₄

socle

C₂²×C₇₃

C₂×#97

prime radical

C₂³×C₇₃

C₂×M₁₆

C₂³

C₇₃

G ▷ C₄×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₄ ◁ #376

N	H ≅ G/N	type
5. #17989475	C₄	semidirect
6. C₄×#11924031	C₄	semidirect
69. C₇₃	#12593	semidirect

H ≅ G/N

type

5. #17989475

C₄

semidirect

6. C₄×#11924031

C₄

semidirect

69. C₇₃

#12593

semidirect

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

subgroup	index
C₄×#14556665	2
#22624972	2
#22609618	2
#12593	73

subgroup

index

Subgroups — #34012326

Conjugacy classes of subgroups

Complements

Maximal subgroups