Finite Group Databasebeta

← #34103418

Subgroups — #34103418

Order 20096.

568 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×Q8×Dic157C211yesmaximal, characteristic, complemented
2C2×#19097731C211yesmaximal, characteristic
3#23757215C212yesmaximal, complemented
4#23757200C211yesmaximal, characteristic, complemented
5#23757198C211yesmaximal, characteristic, complemented
6#23757189C212yesmaximal
7#23757184C211yesmaximal, characteristic, complemented
8C22⋊C4×Dic157C212yesmaximal, complemented
9#23757120C212yesmaximal, complemented
10#23757118C211yesmaximal, characteristic, complemented
11#393×C157C211yesmaximal, characteristic, complemented
12C22×C22⋊D157C2211yescharacteristic
13C23×Dic157C2211yescharacteristic
14C22×C4×D157C2211yescharacteristic
15Q8×Dic157—24no—
16#19097731—24no—
17C2×C4⋊Dic157C2212yes—
18C2×C4⋊Dic157C2211yescharacteristic
19C2×#14578933C2211yescharacteristic
20C2×C22⋊Dic157C2212yes—
21C2×#14578931C2212yes—
22C2×#14578931C2212yes—
23C2×#14578931C2211yescharacteristic
24C2×C4×Dic157C2212yes—
25C2×C4×Dic157C2212yes—
26C2×C4×Dic157C2211yescharacteristic
27#19097678C2211yescharacteristic
28#19097678C2212yes—
29#19097678C2212yes—
30#19097678C2212yes—
31#19097678C2212yes—
32C22×Q8×C157C2211yescharacteristic
33C2×C22⋊C4×C157C2212yes—
34C2×C22⋊C4×C157C2211yescharacteristic
35#95×C157C2211yescharacteristic
36#95×C157C2212yes—
37C23×D157C2311yescharacteristic
38C2×C22⋊D157—22no—
39C2×C22⋊D157—22no—
40C2×C22⋊D157—22no—
41C22×Dic157—22no—
42C22×Dic157—24no—
43C22×Dic157C2312yes—
44C22×Dic157C2311yescharacteristic
45C22×Dic157C2312yes—
46C22×Dic157C2311yescharacteristic
47C22×Dic157C2311yescharacteristic
48C2×C4×D157—22no—
49C2×C4×D157—24no—
50C4⋊Dic157—24no—
51C4⋊Dic157—22no—
52#14578933—22no—
53C22⋊Dic157—24no—
54#14578931—24no—
55#14578931—22no—
56#14578931—24no—
57C4×Dic157—24no—
58C4×Dic157—24no—
59C4×Dic157D414yes—
60C24×C157C2311yescharacteristic
61C2×Q8×C157—22no—
62C2×Q8×C157—24no—
63C22×C4×C157C2312yes—
64C22×C4×C157C2311yescharacteristic
65C22×C4×C157C2311yescharacteristic
66C22×C4×C157C2312yes—
67C22⋊C4×C157—22no—
68C22⋊C4×C157—24no—
69C22×D157—21no—
70C22×D157—22no—
71C22×D157—21no—
72C22×D157—22no—
73C22×D157#4112yes—
74C22⋊D157—28no—
75C2×Dic157—21no—
76C2×Dic157—21no—
77C2×Dic157—21no—
78C2×Dic157—22no—
79C2×Dic157—22no—
80C2×Dic157—22no—
81C2×Dic157—22no—
82C2×Dic157—22no—
83C2×Dic157—22no—
84C2×Dic157—22no—
85C2×Dic157—44no—
86C2×Dic157C2×D412yes—
87C2×Dic157C2×D412yes—
88C2×Dic157#4114yes—
89C4×D157—44no—
90C23×C157—21no—
91C23×C157—22no—
92C23×C157—21no—
93C23×C157—22no—
94C23×C157#4112yes—
95C23×C157C2411yescharacteristic
96Q8×C157—28no—
97C2×C4×C157—22no—
98C2×C4×C157—22no—
99C2×C4×C157—22no—
100C2×C4×C157—21no—
101C2×C4×C157—22no—
102C2×C4×C157—22no—
103C2×C4×C157—22no—
104C2×C4×C157—22no—
105C2×C4×C157C2×D412yes—
106C2×C4×C157#4114yes—
107C22×C157—22no—
108C22×C157—24no—
109C22×C157—41no—
110C22×C157—41no—
111C22×C157—42no—
112C22×C157C22×D411yescharacteristic
113C22×C157C2×#4111yescharacteristic
114C22×C157C2×#4112yes—
115C22×C157C2×#4112yes—
116C22×C157C2×#4111yescharacteristic
117C2×D157—22no—
118C2×D157—24no—
119C2×D157—41no—
120C2×D157—41no—
121C2×D157—42no—
122C4×C157—22no—
123C4×C157—24no—
124C4×C157—41no—
125C4×C157—42no—
126Dic157—22no—
127Dic157—42no—
128Dic157—41no—
129Dic157—42no—
130Dic157#12414yes—
131C2×C157—42no—
132C2×C157#52511yescharacteristic
133C2×C157#52511yescharacteristic
134C2×C157C2×#12411yescharacteristic
135C2×C157#52912yes—
136C2×C157#52912yes—
137D157—42no—
138C157#239811yescharacteristic, complemented, Sylow
139#2398—1571nomaximal, Sylow
140C2×C4×Q8—1571no—
141C2×C4×D4—1571no—
142#400—1572no—
143#397—1571no—
144#397—1571no—
145#394—1572no—
146#393—1571no—
147#393D15711yescharacteristic, complemented
148C4×C22⋊C4—1572no—
149C4×C22⋊C4—1572no—
150#376—1571no—
151C22×Q8C2×D15711yescharacteristic
152C22×D4—1571no—
153C23×C4—1571no—
154C23×C4—1571no—
155C4×Q8—3144no—
156C4×D4—3144no—
157C2×C4⋊C4—1571no—
158C2×C4⋊C4—1572no—
159C2×C4⋊C4—1571no—
160C2×C22⋊C4—1572no—
161C2×C22⋊C4—1571no—
162C2×C22⋊C4—1572no—
163C2×C22⋊C4—1572no—
164C2×C22⋊C4C2×D15711yescharacteristic
165C2×C22⋊C4C2×D15712yes—
166C2×C42—1572no—
167C2×C42—1572no—
168C2×C42—1571no—
169#95—1571no—
170#95—1572no—
171#95—1572no—
172#95—1572no—
173#95—1572no—
174#95C2×D15711yescharacteristic
175#95C2×D15712yes—
176C24—1571no—
177C24C22×D15711yescharacteristic
178C2×Q8—22no—
179C2×Q8—24no—
180C2×D4—3142no—
181C2×D4—3142no—
182C2×D4—3142no—
183C22×C4—3142no—
184C22×C4—3144no—
185C22×C4—3142no—
186C22×C4—3144no—
187C22×C4—1572no—
188C22×C4—1571no—
189C22×C4—1572no—
190C22×C4—1571no—
191C22×C4—1571no—
192C22×C4C22×D15712yes—
193C22×C4C22×D15711yescharacteristic
194C22×C4C22×D15711yescharacteristic
195C22×C4C22×D15712yes—
196C4⋊C4—3142no—
197C4⋊C4—3144no—
198C4⋊C4—3142no—
199C22⋊C4—22no—
200C22⋊C4—24no—
201C22⋊C4—3144no—
202C22⋊C4—3144no—
203C22⋊C4—3142no—
204C22⋊C4—3144no—
205C42—3144no—
206C42—3144no—
207C42—1574no—
208C23—21no—
209C23—22no—
210C23—21no—
211C23—22no—
212C23—3141no—
213C23—3142no—
214C23—3142no—
215C23—3141no—
216C23—1572no—
217C23#1457895512yes—
218C23C23×D15711yescharacteristic
219Q8—28no—
220D4—3148no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—3141no—
230C2×C4—3141no—
231C2×C4—3142no—
232C2×C4—3141no—
233C2×C4—3142no—
234C2×C4—3142no—
235C2×C4—3142no—
236C2×C4—3142no—
237C2×C4—3142no—
238C2×C4—3142no—
239C2×C4—6284no—
240C2×C4—6284no—
241C2×C4—1572no—
242C2×C4—1572no—
243C2×C4—1574no—
244C2×C4D4×D15712yes—
245C2×C4#1457895314yes—
246C22—22no—
247C22—24no—
248C22—41no—
249C22—41no—
250C22—42no—
251C22—3142no—
252C22—3144no—
253C22—6281no—
254C22—6281no—
255C22—6282no—
256C22C2×D4×D15711yescharacteristic
257C22#41×D15711yescharacteristic
258C22#41×D15712yes—
259C22C2×#1457895312yes—
260C22C2×#1457895511yescharacteristic
261C4—22no—
262C4—24no—
263C4—41no—
264C4—42no—
265C4—3142no—
266C4—6282no—
267C4—6281no—
268C4—6282no—
269C4—1574no—
270C2—42no—
271C2—6282no—
272C2#2375777911yescharacteristic
273C2#2375778611yescharacteristic
274C2#124×D15711yescharacteristic
275C2#2375780212yes—
276C2#2375780912yes—
subgroupG/H
center, FrattiniC23C23×D157
commutator, socle, cosocleC23×C157C24
Fitting#393×C157C2
prime radicalC22×C22⋊D157C22
hypercenter#393D157
hypocenterC157#2398
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C23 ◁ #393

Complements

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

NH ≅ G/Ntype
1. C2×Q8×Dic157C2semidirect
3. #23757215C2semidirect
4. #23757200C2semidirect
5. #23757198C2semidirect
7. #23757184C2semidirect
8. C22⋊C4×Dic157C2semidirect
9. #23757120C2semidirect
10. #23757118C2semidirect
11. #393×C157C2semidirect
138. C157#2398semidirect
147. #393D157semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1572
C2×#190977312
#237572152
#237572002
#237571982
#237571892
#237571842
C22⋊C4×Dic1572
#237571202
#237571182
#393×C1572
#2398157