Finite Group Databasebeta

← #34099270

Subgroups — #34099270

Order 20096.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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#23757652C211yesmaximal, characteristic, complemented
2#23757568C211yesmaximal, characteristic, complemented
3#23757563C211yesmaximal, characteristic
4#23757022C211yesmaximal, characteristic, complemented
5Q8⋊#11929434C211yesmaximal, characteristic, complemented
6D4⋊#11929434C211yesmaximal, characteristic
7#554×C157C211yesmaximal, characteristic
8C2×#14578944—22no—
9#19097771C2211yescharacteristic
10#19097704—21no—
11#19097702—21no—
12C4⋊#11929434C2211yescharacteristic
13C4⋊#11929434C2211yescharacteristic
14#19097695—21no—
15#19097692—21no—
16C4×#11929434C2211yescharacteristic
17C4⋊Q8×C157C2211yescharacteristic
18#125×C157—21no—
19C22⋊Q8×C157—21no—
20C22⋊Q8×C157—22no—
21C4×Q8×C157C2211yescharacteristic
22C4×D4×C157—21no—
23C4×D4×C157C2211yescharacteristic
24C2×C4⋊C4×C157—21no—
25C2×Dic314—21no—
26#14578944—44no—
27C2×#11929434—21no—
28C2×#11929434—21no—
29C2×#11929434—21no—
30C4⋊Dic157—21no—
31C4⋊Dic157—21no—
32C2×Q8×C157D411yescharacteristic
33C2×Q8×C157D412yes—
34C2×D4×C157D411yescharacteristic
35C22×C4×C157—22no—
36C22×C4×C157—21no—
37C4⋊C4×C157—22no—
38C4⋊C4×C157—22no—
39C4⋊C4×C157—21no—
40C4⋊C4×C157—21no—
41C4⋊C4×C157—21no—
42C4⋊C4×C157D411yescharacteristic
43C4⋊C4×C157D411yescharacteristic
44C22⋊C4×C157—22no—
45C22⋊C4×C157—21no—
46C42×C157—21no—
47C42×C157C2311yescharacteristic
48Dic314—42no—
49C2×Dic157—41no—
50#11929434—22no—
51#11929434—41no—
52#11929434—41no—
53C23×C157—21no—
54Q8×C157—21no—
55Q8×C157—21no—
56Q8×C157—42no—
57D4×C157—21no—
58D4×C157Q1612yescomplemented
59C2×C4×C157—21no—
60C2×C4×C157—21no—
61C2×C4×C157—21no—
62C2×C4×C157—22no—
63C2×C4×C157—41no—
64C2×C4×C157—41no—
65C2×C4×C157—82no—
66C2×C4×C157C2×D411yescharacteristic
67C2×C4×C157C2×D411yescharacteristic
68C2×C4×C157C2×D411yescharacteristic
69C22×C157—22no—
70C22×C157—41no—
71C22×C157—41no—
72C22×C157#12011yescharacteristic
73C4×C157—21no—
74C4×C157—41no—
75C4×C157—41no—
76C4×C157—41no—
77C4×C157—42no—
78C4×C157C2×Q1611yescharacteristic
79C4×C157#13511yescharacteristic
80C4×C157#13711yescharacteristic
81C4×C157#13711yescharacteristic
82Dic157—81no—
83C2×C157—42no—
84C2×C157#45111yescharacteristic
85C2×C157#45211yescharacteristic
86C2×C157#45311yescharacteristic
87C157#163211yescharacteristic, complemented, Sylow
88#1632—1571nomaximal, Sylow
89#554D15711yescharacteristic
90#494—1571no—
91#477—1571no—
92#476—1571no—
93#333—1571no—
94Q8⋊C8—1571no—
95D4⋊C8—1571no—
96C2×Q16—3142no—
97C4⋊Q8—1571no—
98C4⋊Q8C2×D15711yescharacteristic
99#125—21no—
100C22⋊Q8—21no—
101C22⋊Q8—22no—
102C4×Q8C2×D15711yescharacteristic
103C4×D4—21no—
104C4×D4C2×D15711yescharacteristic
105C2×C4⋊C4—21no—
106#107—3141no—
107#106—3141no—
108C4⋊C8—1571no—
109C4⋊C8—1571no—
110#103—3141no—
111#102—3141no—
112C4×C8—1571no—
113C2×Q8—3141no—
114C2×Q8C22⋊D15711yescharacteristic
115C2×Q8C22⋊D15712yes—
116C2×D4C22⋊D15711yescharacteristic
117C22×C4—22no—
118C22×C4—21no—
119Q16—6284no—
120C2×C8—3141no—
121C2×C8—3141no—
122C2×C8—3141no—
123C4⋊C4—22no—
124C4⋊C4—22no—
125C4⋊C4—21no—
126C4⋊C4—21no—
127C4⋊C4—21no—
128C4⋊C4—3141no—
129C4⋊C4—3141no—
130C4⋊C4C22⋊D15711yescharacteristic
131C4⋊C4C22⋊D15711yescharacteristic
132C22⋊C4—22no—
133C22⋊C4—21no—
134C42—21no—
135C42C22×D15711yescharacteristic
136C23—21no—
137Q8—21no—
138Q8—21no—
139Q8—42no—
140Q8—6282no—
141D4—21no—
142D4#1457894412yescomplemented
143C2×C4—21no—
144C2×C4—21no—
145C2×C4—21no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—82no—
150C2×C4—6281no—
151C2×C4C2×C22⋊D15711yescharacteristic
152C2×C4C2×C22⋊D15711yescharacteristic
153C2×C4C2×C22⋊D15711yescharacteristic
154C8—3142no—
155C8—6281no—
156C8—6281no—
157C22—22no—
158C22—41no—
159C22—41no—
160C22#1909774011yescharacteristic
161C4—21no—
162C4—41no—
163C4—41no—
164C4—41no—
165C4—42no—
166C4—12561no—
167C4C2×#1457894411yescharacteristic
168C4#1909779111yescharacteristic
169C4#1909780111yescharacteristic
170C4#1909780211yescharacteristic
171C2—42no—
172C2#2375742911yescharacteristic
173C2#2375743511yescharacteristic
174C2#2375744111yescharacteristic
subgroupG/H
centerC22#19097740
commutator, cosocleC42×C157C23
FrattiniC42C22×D157
Fitting#554×C157C2
socleC22×C157#120
prime radicalC2×D4×C157D4
hypercenter#554D157
hypocenterC157#1632
LCSG ▷ C42×C157 ▷ C22×C157 ▷ C157
UCSC1 ◁ C22 ◁ C42 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #23757652C2semidirect
2. #23757568C2semidirect
4. #23757022C2semidirect
5. Q8⋊#11929434C2semidirect
58. D4×C157Q16semidirect
87. C157#1632semidirect
142. D4#14578944semidirect

Maximal subgroups

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

subgroupindex
#237576522
#237575682
#237575632
#237570222
Q8⋊#119294342
D4⋊#119294342
#554×C1572
#1632157