Finite Group Databasebeta

← #34102211

Subgroups — #34102211

Order 20096.

170 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23757711C211yesmaximal, characteristic, complemented
2#23757709C211yesmaximal, characteristic, complemented
3#23757648C211yesmaximal, characteristic, complemented
4#23757389C211yesmaximal, characteristic, complemented
5#23757093C211yesmaximal, characteristic, complemented
6#23757085C211yesmaximal, characteristic, complemented
7#361×C157C211yesmaximal, characteristic, complemented
8#19097793C2211yescharacteristic
9C2×D1256C2211yescharacteristic
10C2×#14578939—21no—
11#19097779—21no—
12#19097767—21no—
13#19097716—21no—
14#19097715—21no—
15D2512—21no—
16#19097712—21no—
17#19097711C2211yescharacteristic
18#19097697—21no—
19C8×Dic157C2211yescharacteristic
20C2×D8×C157C2211yescharacteristic
21M32×C157C2211yescharacteristic
22#108×C157C2211yescharacteristic
23#14578954—21no—
24C2×D628—21no—
25C2×C22⋊D157—42no—
26Dic628D411yescharacteristic, complemented
27#14578943—21no—
28D1256—21no—
29D1256—21no—
30D1256D411yescharacteristic, complemented
31#14578939—41no—
32#14578939—41no—
33#14578937—41no—
34C2×#11929434D411yescharacteristic
35C8×D157—41no—
36C4×Dic157D411yescharacteristic
37#14578929—21no—
38C2×D4×C157—21no—
39D8×C157—21no—
40D8×C157—21no—
41M16×C157—21no—
42C2×C8×C157C2311yescharacteristic
43C16×C157—21no—
44C22×D157—41no—
45Dic314—21no—
46D628—21no—
47D628—41no—
48D628—41no—
49C22⋊D157—41no—
50C22⋊D157—84no—
51C2×Dic157—21no—
52C4×D157—41no—
53#11929434D811yescharacteristic
54#11929434D811yescharacteristic
55C23×C157—41no—
56D4×C157—41no—
57D4×C157—41no—
58C2×C4×C157C2×D411yescharacteristic
59C8×C157—41no—
60C8×C157C2×D411yescharacteristic
61C8×C157#4111yescharacteristic
62C22×C157—82no—
63C22×C157—81no—
64C22×C157AGL(1,ℤ/8)11yescharacteristic
65C2×D157—41no—
66C2×D157—82no—
67C2×D157—81no—
68C4×C157#12111yescharacteristic
69C4×C157C2×D811yescharacteristic
70Dic157—22no—
71Dic157—41no—
72C2×C157—21no—
73C2×C157—161no—
74C2×C157#45911yescharacteristic
75D157—81no—
76D157—161no—
77C157#221311yescharacteristic, complemented, Sylow
78#2213—1571nomaximal, Sylow
79#509—1571no—
80#509—1571no—
81#493—1571no—
82#443—1571no—
83#364—1571no—
84#361—1571no—
85#361D15711yescharacteristic, complemented
86#135—1571no—
87C2×D8—3141no—
88C2×D8—1571no—
89C2×D8C2×D15711yescharacteristic
90#131—3141no—
91#127—3141no—
92SD32—3141no—
93SD32—3141no—
94D16—3141no—
95D16—3141no—
96M32—1571no—
97M32C2×D15711yescharacteristic
98#108C2×D15711yescharacteristic
99#104—3141no—
100C4×C8—1571no—
101#41—3141no—
102C2×D4—21no—
103C2×D4—3141no—
104C2×D4—6282no—
105Q16—1571no—
106SD16—3141no—
107D8—21no—
108D8—21no—
109D8—3141no—
110D8—3141no—
111D8—6281no—
112D8—6281no—
113D8—1571no—
114M16—21no—
115M16—6281no—
116C2×C8—6281no—
117C2×C8—1571no—
118C2×C8C22×D15711yescharacteristic
119C42—1571no—
120C16—21no—
121C16—3141no—
122C23—41no—
123C23—6281no—
124Q8—3141no—
125D4—41no—
126D4—41no—
127D4—3141no—
128D4—6281no—
129D4—6281no—
130D4—6281no—
131D4—12564no—
132C2×C4—3141no—
133C2×C4—6281no—
134C2×C4D4×D15711yescharacteristic
135C8—41no—
136C8—1571no—
137C8—1571no—
138C8D4×D15711yescharacteristic
139C8#1457895411yescharacteristic
140C22—82no—
141C22—81no—
142C22—6281no—
143C22—12562no—
144C22—12561no—
145C22#1909779811yescharacteristic
146C4—3142no—
147C4—6281no—
148C4#1909774411yescharacteristic
149C4D8×D15711yescharacteristic
150C2—21no—
151C2—161no—
152C2—25121no—
153C2—12561no—
154C2#2375746711yescharacteristic
subgroupG/H
centerC2#23757467
commutator, cosocleC2×C8×C157C23
FrattiniC2×C8C22×D157
Fitting#361×C157C2
socleC2×C157#459
hypercenter#361D157
hypocenterC157#2213
LCSG ▷ C2×C8×C157 ▷ C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #361

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #23757711C2semidirect
2. #23757709C2semidirect
3. #23757648C2semidirect
4. #23757389C2semidirect
5. #23757093C2semidirect
6. #23757085C2semidirect
7. #361×C157C2semidirect
26. Dic628D4semidirect
30. D1256D4semidirect
77. C157#2213semidirect
85. #361D157semidirect

Maximal subgroups

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

subgroupindex
#237577112
#237577092
#237576482
#237573892
#237570932
#237570852
#361×C1572
#2213157