Group #34000008 — subgroups

Subgroups — #34000008

Order 18688.

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

normal subgroups only

This group has 176 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	#22610085	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊#11924031	—	2	2	no	—
4	C₂×C₄⋊#10596831	—	2	2	no	—
5	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
6	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
7	#376×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	C₄⋊#10596831	—	4	4	no	—
13	C₄⋊#10596831	—	4	1	no	—
14	C₂×C₄²×C₇₃	—	2	1	no	—
15	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
16	#95×C₇₃	—	2	1	no	—
17	#95×C₇₃	C₈	1	2	yes	—
18	#11924031	—	8	2	no	—
19	C₂×#10596831	—	2	2	no	—
20	C₂×#10596831	—	4	2	no	—
21	C₂×#10596831	—	4	2	no	—
22	C₂×#10596831	—	4	1	no	—
23	C₂²×C₄×C₇₃	—	2	2	no	—
24	C₂²×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
26	C₂²×C₄×C₇₃	M₁₆	1	1	yes	characteristic
27	C₄²×C₇₃	—	2	2	no	—
28	C₄²×C₇₃	—	2	2	no	—
29	C₄²×C₇₃	—	2	1	no	—
30	C₄²×C₇₃	—	2	1	no	—
31	C₂³×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	2	4	no	—
35	C₂×C₄×C₇₃	—	2	1	no	—
36	C₂×C₄×C₇₃	—	2	1	no	—
37	C₂×C₄×C₇₃	—	2	2	no	—
38	C₂×C₄×C₇₃	—	4	2	no	—
39	C₂×C₄×C₇₃	—	4	2	no	—
40	C₂×C₄×C₇₃	—	4	1	no	—
41	C₂×C₄×C₇₃	#99	1	1	yes	characteristic
42	C₂×C₄×C₇₃	#101	1	1	yes	characteristic
43	#10596831	—	4	2	no	—
44	#10596831	—	8	1	no	—
45	C₂²×C₇₃	—	2	1	no	—
46	C₂²×C₇₃	—	2	1	no	—
47	C₂²×C₇₃	#324	1	1	yes	characteristic
48	C₂²×C₇₃	#350	1	2	yes	—
49	C₄×C₇₃	—	2	1	no	—
50	C₄×C₇₃	—	4	4	no	—
51	C₄×C₇₃	—	4	1	no	—
52	C₄×C₇₃	—	4	1	no	—
53	C₄×C₇₃	#353	1	1	yes	characteristic
54	C₄×C₇₃	#356	1	1	yes	characteristic
55	#7066	—	73	1	no	maximal, Sylow
56	C₂×C₇₃	—	2	2	no	—
57	C₂×C₇₃	#1323	1	1	yes	characteristic
58	C₂×C₇₃	#1325	1	1	yes	characteristic
59	C₂×C₇₃	#1328	1	1	yes	characteristic
60	#1843	—	73	1	no	—
61	#1322	—	73	2	no	—
62	C₇₃	#7066	1	1	yes	characteristic, complemented, Sylow
63	C₂×C₄⋊C₈	—	73	1	no	—
64	C₂×C₄⋊C₈	—	73	1	no	—
65	C₂×C₄⋊C₈	—	146	2	no	—
66	#376	#64408	1	1	yes	characteristic
67	C₂²⋊C₁₆	—	146	2	no	—
68	C₂²×C₈	—	73	2	no	—
69	C₂²×C₈	—	146	1	no	—
70	C₂×C₄²	—	2	1	no	—
71	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
72	C₂×C₁₆	—	292	2	no	—
73	C₄⋊C₈	—	292	1	no	—
74	C₄⋊C₈	—	292	4	no	—
75	C₄⋊C₈	—	292	1	no	—
76	#95	—	2	1	no	—
77	#95	#10596833	1	2	yes	—
78	C₂²×C₄	—	2	2	no	—
79	C₂²×C₄	C₂²⋊#64408	1	1	yes	characteristic
80	C₂²×C₄	C₂×#10596833	1	1	yes	characteristic
81	C₂²×C₄	#11924037	1	1	yes	characteristic
82	C₂×C₈	—	292	2	no	—
83	C₂×C₈	—	292	2	no	—
84	C₂×C₈	—	292	1	no	—
85	C₂×C₈	—	146	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	2	2	no	—
88	C₄²	—	2	1	no	—
89	C₄²	—	2	1	no	—
90	C₁₆	—	584	2	no	—
91	C₂³	C₂²⋊#10596833	1	1	yes	characteristic
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	2	1	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	1	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	#14556650	1	1	yes	characteristic
102	C₂×C₄	#14556652	1	1	yes	characteristic
103	C₈	—	292	2	no	—
104	C₈	—	584	1	no	—
105	C₂²	—	2	1	no	—
106	C₂²	—	2	1	no	—
107	C₂²	#17989432	1	1	yes	characteristic
108	C₂²	#17989485	1	2	yes	—
109	C₄	—	2	1	no	—
110	C₄	—	4	4	no	—
111	C₄	—	4	1	no	—
112	C₄	—	4	1	no	—
113	C₄	#17989492	1	1	yes	characteristic
114	C₄	#17989498	1	1	yes	characteristic
115	C₂	—	2	2	no	—
116	C₂	#22624869	1	1	yes	characteristic
117	C₂	#22624873	1	1	yes	characteristic
118	C₂	#22624879	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22624867

C₂

yes

maximal

#22610085

C₂

yes

maximal, characteristic

C₂²⋊#11924031

—

C₂×C₄⋊#10596831

—

C₂×C₄⋊#10596831

C₄

yes

characteristic

C₂×C₄⋊#10596831

C₂²

yes

characteristic

#376×C₇₃

C₄

yes

—

—

D₄

yes

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₄⋊#10596831

—

C₂×C₄²×C₇₃

—

C₂×C₄²×C₇₃

C₂×C₄

yes

characteristic

#95×C₇₃

—

#95×C₇₃

C₈

yes

—

—

—

—

—

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₇₃

M₁₆

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂²⋊C₈

yes

characteristic

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

#324

yes

characteristic

C₂²×C₇₃

#350

yes

—

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

#7066

yes

characteristic, complemented, Sylow

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

146

—

#376

#64408

yes

characteristic

C₂²⋊C₁₆

—

146

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₄²

—

C₂×C₄²

C₂×#64408

yes

characteristic

C₂×C₁₆

—

292

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

292

—

C₄⋊C₈

—

292

—

#95

—

#95

#10596833

yes

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

C₂×C₈

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

292

—

C₂×C₈

—

146

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

584

—

C₂³

C₂²⋊#10596833

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#14556650

yes

characteristic

102

C₂×C₄

#14556652

yes

characteristic

103

C₈

—

292

—

104

C₈

—

584

—

105

C₂²

—

106

C₂²

—

107

C₂²

#17989432

yes

characteristic

108

C₂²

#17989485

yes

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

#17989492

yes

characteristic

114

C₄

#17989498

yes

characteristic

115

C₂

—

116

C₂

#22624869

yes

characteristic

117

C₂

#22624873

yes

characteristic

118

C₂

#22624879

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14556650
commutator	C₂²×C₄×C₇₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#64408
Fitting	#376×C₇₃	C₄
socle	C₂²×C₇₃	#324
prime radical	C₂³×C₇₃	C₂²⋊C₈
cosocle	C₂×C₄⋊#10596831	C₂²
hypercenter	#376	#64408
hypocenter	C₇₃	#7066
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

#14556650

commutator

C₂²×C₄×C₇₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#64408

Fitting

#376×C₇₃

C₄

socle

C₂²×C₇₃

#324

prime radical

C₂³×C₇₃

C₂²⋊C₈

C₂²

C₇₃

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

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
62. C₇₃	#7066	semidirect

H ≅ G/N

type

62. C₇₃

#7066

semidirect

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

subgroup	index
#22624867	2
#22610085	2
#7066	73

subgroup

index

#22624867

#22610085

#7066

Subgroups — #34000008

Conjugacy classes of subgroups

Complements

Maximal subgroups