Group #34014479 — subgroups

Subgroups — #34014479

Order 18688.

206 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 112 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	C₂×#17989519	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556641	C₂	1	1	yes	maximal, characteristic
3	#22625089	C₂	1	4	yes	maximal, complemented
4	#22611530	C₂	1	1	yes	maximal, characteristic
5	#17989519	—	2	2	no	—
6	C₂×#14556641	—	2	2	no	—
7	C₂×#14556641	C₂²	1	2	yes	—
8	C₂×#14556641	C₂²	1	2	yes	—
9	C₂²×#11924003	C₂²	1	1	yes	characteristic
10	#17988542	C₄	1	1	yes	characteristic
11	#17988362	C₄	1	2	yes	—
12	C₁₆×Dic₇₃	C₂²	1	2	yes	—
13	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic
14	#14556641	—	2	2	no	—
15	#14556641	—	2	2	no	—
16	C₂×#11924012	C₈	1	1	yes	characteristic
17	#14556550	C₈	1	1	yes	characteristic
18	C₂×#11924003	C₂×C₄	1	1	yes	characteristic
19	C₂×#11924003	D₄	1	2	yes	—
20	C₂×#11924003	Q₈	1	2	yes	—
21	C₂×#11924003	C₂³	1	1	yes	characteristic
22	#14556501	C₈	1	2	yes	—
23	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
25	M₃₂×C₇₃	—	2	2	no	—
26	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
27	#11924012	—	2	2	no	—
28	C₂×#10596831	C₂×C₈	1	2	yes	—
29	C₄⋊Dic₇₃	C₁₆	1	2	yes	—
30	C₂²⋊Dic₇₃	C₁₆	1	2	yes	—
31	C₄×Dic₇₃	C₂×C₈	1	2	yes	—
32	#11924003	C₄⋊C₄	1	2	yes	—
33	#11924003	C₂×D₄	1	1	yes	characteristic
34	#11924003	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
36	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₇₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
39	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₁₆×C₇₃	—	2	2	no	—
41	C₂³×C₇₃	C₂×C₁₆	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₇₃	C₂×C₁₆	1	1	yes	characteristic
44	C₂×C₄×C₇₃	M₃₂	1	2	yes	—
45	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
46	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	—
47	C₈×C₇₃	C₄⋊C₈	1	2	yes	—
48	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₈×C₇₃	C₂×M₁₆	1	1	yes	characteristic
50	#10596831	—	2	2	no	—
51	C₂²×C₇₃	—	2	2	no	—
52	C₂²×C₇₃	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₇₃	C₄⋊C₁₆	1	2	yes	—
55	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₄×C₇₃	C₂×M₃₂	1	1	yes	characteristic
57	Dic₇₃	—	2	4	no	—
58	#13628	—	73	1	no	maximal, Sylow
59	C₂×C₇₃	—	2	2	no	—
60	C₂×C₇₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
61	C₂×C₇₃	#2258	1	2	yes	—
62	C₂×M₆₄	—	73	1	no	—
63	C₂²×C₃₂	—	73	1	no	—
64	#2108	—	73	1	no	—
65	C₄⋊C₃₂	—	73	4	no	—
66	C₇₃	#13628	1	1	yes	characteristic, complemented, Sylow
67	C₂×M₃₂	#64408	1	1	yes	characteristic
68	C₂²×C₁₆	—	73	1	no	—
69	#405	—	73	1	no	—
70	M₆₄	—	146	2	no	—
71	C₂×C₃₂	—	73	2	no	—
72	C₂×C₃₂	—	73	2	no	—
73	C₂×C₃₂	—	146	2	no	—
74	#346	—	73	2	no	—
75	C₄×C₁₆	—	73	2	no	—
76	C₂×M₁₆	—	73	1	no	—
77	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
78	#117	—	73	1	no	—
79	M₃₂	—	2	2	no	—
80	C₂×C₁₆	—	73	2	no	—
81	C₂×C₁₆	—	73	2	no	—
82	C₂×C₁₆	—	73	1	no	—
83	C₂×C₁₆	—	73	1	no	—
84	C₂×C₁₆	C₂×#64408	1	2	yes	—
85	#97	—	73	2	no	—
86	C₄×C₈	—	73	2	no	—
87	C₃₂	—	146	2	no	—
88	C₃₂	—	146	2	no	—
89	C₂²×C₄	#11924036	1	1	yes	characteristic
90	M₁₆	—	146	2	no	—
91	C₂×C₈	—	73	2	no	—
92	C₂×C₈	C₄⋊#64408	1	2	yes	—
93	C₂×C₈	#11924036	1	1	yes	characteristic
94	C₂×C₈	#11924037	1	2	yes	—
95	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
96	C₄⋊C₄	—	73	2	no	—
97	C₂²⋊C₄	—	73	2	no	—
98	C₄²	—	73	2	no	—
99	C₁₆	—	2	2	no	—
100	C₁₆	—	73	2	no	—
101	C₁₆	—	73	1	no	—
102	C₁₆	—	73	1	no	—
103	C₂³	#14556664	1	1	yes	characteristic
104	C₂×C₄	—	73	4	no	—
105	C₂×C₄	#14556658	1	2	yes	—
106	C₂×C₄	#14556664	1	1	yes	characteristic
107	C₂×C₄	#14556666	1	2	yes	—
108	C₂×C₄	C₂×#11924036	1	1	yes	characteristic
109	C₈	—	146	2	no	—
110	C₈	#14556658	1	2	yes	—
111	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
112	C₈	C₂×#11924037	1	1	yes	characteristic
113	C₂²	—	2	2	no	—
114	C₂²	#17989508	1	2	yes	—
115	C₂²	C₂×#14556664	1	1	yes	characteristic
116	C₄	—	146	4	no	—
117	C₄	#17989508	1	2	yes	—
118	C₄	C₂×#14556658	1	1	yes	characteristic
119	C₄	C₂×#14556666	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×#17989508	1	1	yes	characteristic
122	C₂	#22626355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17989519

C₂

yes

maximal, characteristic

C₂²×#14556641

C₂

yes

maximal, characteristic

#22625089

C₂

yes

maximal, complemented

#22611530

C₂

yes

maximal, characteristic

#17989519

—

C₂×#14556641

—

C₂×#14556641

C₂²

yes

—

C₂×#14556641

C₂²

yes

—

C₂²×#11924003

C₂²

yes

characteristic

#17988542

C₄

yes

characteristic

#17988362

C₄

yes

—

C₁₆×Dic₇₃

C₂²

yes

—

C₂×M₃₂×C₇₃

C₄

yes

—

—

C₈

yes

characteristic

#14556550

C₈

yes

characteristic

C₂×#11924003

C₂×C₄

yes

characteristic

C₂×#11924003

D₄

yes

—

C₂×#11924003

Q₈

yes

—

C₂×#11924003

C₂³

yes

characteristic

#14556501

C₈

yes

—

C₈×Dic₇₃

C₂×C₄

yes

—

C₂²×C₈×C₇₃

C₂×C₄

yes

characteristic

M₃₂×C₇₃

—

C₂×C₁₆×C₇₃

C₂×C₄

yes

—

#11924012

—

C₂×#10596831

C₂×C₈

yes

—

C₄⋊Dic₇₃

C₁₆

yes

—

C₂²⋊Dic₇₃

C₁₆

yes

—

C₄×Dic₇₃

C₂×C₈

yes

—

#11924003

C₄⋊C₄

yes

—

#11924003

C₂×D₄

yes

characteristic

#11924003

C₂×Q₈

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

C₄⋊C₄

yes

—

C₂×C₈×C₇₃

C₂×C₈

yes

characteristic

C₂×C₈×C₇₃

M₁₆

yes

—

C₂×C₈×C₇₃

C₂²×C₄

yes

characteristic

C₁₆×C₇₃

—

C₂³×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

C₄⋊C₈

yes

—

C₂×C₄×C₇₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₇₃

M₃₂

yes

—

C₂×C₄×C₇₃

C₂²×C₈

yes

characteristic

C₂×Dic₇₃

C₂×C₁₆

yes

—

C₈×C₇₃

C₄⋊C₈

yes

—

C₈×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₂×M₁₆

yes

characteristic

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄⋊C₁₆

yes

—

C₂²×C₇₃

C₂²×C₁₆

yes

characteristic

C₄×C₇₃

C₄⋊C₁₆

yes

—

C₄×C₇₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₇₃

C₂×M₃₂

yes

characteristic

Dic₇₃

—

#13628

—

maximal, Sylow

C₂×C₇₃

—

C₂×C₇₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₇₃

#2258

yes

—

C₂×M₆₄

—

C₂²×C₃₂

—

#2108

—

C₄⋊C₃₂

—

C₇₃

#13628

yes

characteristic, complemented, Sylow

C₂×M₃₂

#64408

yes

characteristic

C₂²×C₁₆

—

#405

—

M₆₄

—

146

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

146

—

#346

—

C₄×C₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×#64408

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#64408

yes

—

#97

—

C₄×C₈

—

C₃₂

—

146

—

C₃₂

—

146

—

C₂²×C₄

#11924036

yes

characteristic

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

C₄⋊#64408

yes

—

C₂×C₈

#11924036

yes

characteristic

C₂×C₈

#11924037

yes

—

C₂×C₈

C₂²×#64408

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

100

C₁₆

—

101

C₁₆

—

102

C₁₆

—

103

C₂³

#14556664

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#14556658

yes

—

106

C₂×C₄

#14556664

yes

characteristic

107

C₂×C₄

#14556666

yes

—

108

C₂×C₄

C₂×#11924036

yes

characteristic

109

C₈

—

146

—

110

C₈

#14556658

yes

—

111

C₈

C₂×C₄⋊#64408

yes

characteristic

112

C₈

C₂×#11924037

yes

characteristic

113

C₂²

—

114

C₂²

#17989508

yes

—

115

C₂²

C₂×#14556664

yes

characteristic

116

C₄

—

146

—

117

C₄

#17989508

yes

—

118

C₄

C₂×#14556658

yes

characteristic

119

C₄

C₂×#14556666

yes

characteristic

120

C₂

—

121

C₂

C₂×#17989508

yes

characteristic

122

C₂

#22626355

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#64408
commutator, socle	C₂²×C₇₃	C₂²×C₁₆
Fitting	C₂×M₃₂×C₇₃	C₄
prime radical	C₂³×C₇₃	C₂×C₁₆
cosocle	C₂×#11924003	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	#13628
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#64408

commutator, socle

C₂²×C₇₃

C₂²×C₁₆

Fitting

C₂×M₃₂×C₇₃

C₄

prime radical

C₂³×C₇₃

C₂×C₁₆

C₂³

C₂×M₃₂

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
3. #22625089	C₂	semidirect
66. C₇₃	#13628	semidirect

H ≅ G/N

type

3. #22625089

C₂

semidirect

66. C₇₃

#13628

semidirect

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

subgroup	index
C₂×#17989519	2
C₂²×#14556641	2
#22625089	2
#22611530	2
#13628	73

subgroup

index

Subgroups — #34014479

Conjugacy classes of subgroups

Complements

Maximal subgroups