Finite Group Databasebeta

← #34116388

Subgroups — #34116388

Order 20096.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#23758260C211yesmaximal, characteristic, complemented
2#23758253C211yesmaximal, characteristic, complemented
3#23758176C212yesmaximal, complemented
4#23758173C212yesmaximal, complemented
5#23757779C211yesmaximal, characteristic
6C2×#14578963—21no—
7#19097850—21no—
8#19097832—21no—
9#19097831C2212yes—
10#19097829—21no—
11#19097829C2212yes—
12C2×#14578953C2211yescharacteristic
13C23×Dic157C411yescharacteristic, complemented
14#19097755C2211yescharacteristic
15#19097752C411yescharacteristic, complemented
16#19097748—21no—
17#19097743—21no—
18#19097740C2211yescharacteristic
19D4×Dic157—21no—
20#19097731—21no—
21#19097727C411yescharacteristic, complemented
22#120×C157C411yescharacteristic, complemented
23#14578963—22no—
24#14578963—41no—
25C2×#11929441—41no—
26C4⋊#10597845—41no—
27C22⋊#10597845—22no—
28C4×#10597845—41no—
29#14578953—41no—
30#14578953—41no—
31C2×Dic314D411yescharacteristic
32C2×C22⋊D157—21no—
33C2×C22⋊D157D411yescharacteristic
34C2×C22⋊D157C2311yescharacteristic
35C22×Dic157—21no—
36C22×Dic157—21no—
37C22×Dic157—41no—
38C22×Dic157—41no—
39C22×Dic157C2×C411yescharacteristic
40C22×Dic157C2×C411yescharacteristic
41C22×Dic157D411yescharacteristic
42C2×C4×D157D411yescharacteristic
43C4⋊Dic157—21no—
44#14578933—21no—
45#14578933—21no—
46C22⋊Dic157—21no—
47C22⋊Dic157—21no—
48C22⋊Dic157C2×C411yescharacteristic
49#14578931—21no—
50C4×Dic157—21no—
51C24×C157C2×C411yescharacteristic
52C2×D4×C157—21no—
53C2×D4×C157C2×C411yescharacteristic
54C22⋊C4×C157—21no—
55C22⋊C4×C157C2×C411yescharacteristic
56C2×#10597845—42no—
57#11929441—42no—
58C22×D157C2×D411yescharacteristic
59Dic314—41no—
60C22⋊D157—41no—
61C22⋊D157—41no—
62C22⋊D157—42no—
63C2×Dic157—21no—
64C2×Dic157—21no—
65C2×Dic157—21no—
66C2×Dic157—21no—
67C2×Dic157—21no—
68C2×Dic157—21no—
69C2×Dic157—42no—
70C2×Dic157—41no—
71C2×Dic157—41no—
72C2×Dic157—41no—
73C2×Dic157C22⋊C411yescharacteristic
74C2×Dic157C22⋊C411yescharacteristic
75C2×Dic157C2×D411yescharacteristic
76C4×D157—21no—
77C4×D157—21no—
78C23×C157—21no—
79C23×C157—41no—
80C23×C157—42no—
81C23×C157C22⋊C411yescharacteristic
82C23×C157C22×C411yescharacteristic
83D4×C157—41no—
84D4×C157—42no—
85C2×C4×C157—21no—
86C2×C4×C157C22⋊C411yescharacteristic
87C22×C157—22no—
88C22×C157—21no—
89C22×C157—21no—
90C22×C157—41no—
91C22×C157—41no—
92C22×C157—41no—
93C22×C157—42no—
94C22×C157—42no—
95C22×C157—41no—
96C22×C157C2×C22⋊C411yescharacteristic
97C2×D157—21no—
98C2×D157—41no—
99#10597845—82no—
100C4×C157—21no—
101C4×C157—41no—
102Dic157—21no—
103Dic157—21no—
104Dic157—41no—
105Dic157—41no—
106Dic157#9911yescharacteristic
107Dic157#9911yescharacteristic
108C2×C157—21no—
109C2×C157—41no—
110C2×C157—42no—
111C2×C157—41no—
112C2×C157C2×#9911yescharacteristic
113D157—41no—
114C157#212011yescharacteristic, complemented, Sylow
115#2120—1571nomaximal, Sylow
116#525—1571no—
117#413—1571no—
118#410—1571no—
119#352—1572no—
120AΣL(1,𝔽16)—1572no—
121C2×#41—1571no—
122C23×C4—1571no—
123C2×M16—3141no—
124#123—1571no—
125#123—1571no—
126C22⋊Q8—3141no—
127#121—3141no—
128#120—1571no—
129#120#1059784511yescharacteristic, complemented
130C4×D4—3141no—
131C4×D4—3141no—
132#117—3141no—
133#117—1571no—
134#101—3141no—
135#100—1572no—
136#99—3141no—
137#99—1572no—
138C24C2×#1059784511yescharacteristic
139#41—6281no—
140#41—6281no—
141C2×Q8—1571no—
142C2×D4—21no—
143C2×D4—3141no—
144C2×D4—1571no—
145C2×D4—1571no—
146C2×D4C2×#1059784511yescharacteristic
147C22×C4—3141no—
148C22×C4—3141no—
149C22×C4—6281no—
150C22×C4—6281no—
151C22×C4—1571no—
152C22×C4—1571no—
153C22×C4—1571no—
154C22×C4—1571no—
155M16—3142no—
156M16—6281no—
157C2×C8—6281no—
158C4⋊C4—3141no—
159C4⋊C4—3141no—
160C4⋊C4—3141no—
161C4⋊C4—6281no—
162C22⋊C4—21no—
163C22⋊C4—3141no—
164C22⋊C4—3141no—
165C22⋊C4—3141no—
166C22⋊C4—3142no—
167C22⋊C4—1571no—
168C22⋊C4C2×#1059784511yescharacteristic
169C42—3141no—
170C42—6281no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—1571no—
175C23C22⋊#1059784511yescharacteristic
176C23C22×#1059784511yescharacteristic
177Q8—6281no—
178D4—41no—
179D4—42no—
180D4—6281no—
181D4—6281no—
182D4—6282no—
183C2×C4—21no—
184C2×C4—3141no—
185C2×C4—3141no—
186C2×C4—3141no—
187C2×C4—3141no—
188C2×C4—3141no—
189C2×C4—3141no—
190C2×C4—3141no—
191C2×C4—3141no—
192C2×C4—6282no—
193C2×C4—6281no—
194C2×C4—6281no—
195C2×C4—6281no—
196C2×C4—6282no—
197C2×C4—1571no—
198C2×C4—1571no—
199C2×C4—1571no—
200C2×C4C22⋊#1059784511yescharacteristic
201C8—6282no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—42no—
210C22—41no—
211C22—3141no—
212C22—6281no—
213C22C2×C22⋊#1059784511yescharacteristic
214C4—21no—
215C4—41no—
216C4—3141no—
217C4—3141no—
218C4—6281no—
219C4—6281no—
220C4—1571no—
221C4—1571no—
222C4—12562no—
223C2—21no—
224C2—42no—
225C2—41no—
226C2—41no—
227C2—6281no—
228C2D157⋊#9911yescharacteristic
subgroupG/H
centerC2D157⋊#99
commutatorC23×C157C22×C4
FrattiniC23C22×#10597845
Fitting#120×C157C4
socleC2×C157C2×#99
prime radical#23757779C2
cosocleC2×C22⋊D157C23
hypercenter#120#10597845
hypocenterC157#2120
LCSG ▷ C23×C157 ▷ C22×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #23758260C2semidirect
2. #23758253C2semidirect
3. #23758176C2semidirect
4. #23758173C2semidirect
13. C23×Dic157C4semidirect
15. #19097752C4semidirect
21. #19097727C4semidirect
22. #120×C157C4semidirect
114. C157#2120semidirect
129. #120#10597845semidirect

Maximal subgroups

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

subgroupindex
#237582602
#237582532
#237581762
#237581732
#237577792
#2120157