Finite Group Databasebeta

← #34100096

Subgroups — #34100096

Order 20096.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#19097818C211yesmaximal, characteristic
2C2×#19097699C211yesmaximal, characteristic, complemented
3#23757275C211yesmaximal, characteristic, complemented
4#23757274C211yesmaximal, characteristic, complemented
5C2×#19097689C211yesmaximal, characteristic, complemented
6#23757179C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic, complemented
8#19097818—22no—
9#19097818—22no—
10#19097818—22no—
11#19097818C414yescomplemented
12C2×#14578954—21no—
13C2×#14578954C2211yescharacteristic
14C2×#14578953—21no—
15C2×#14578953—22no—
16C22×Dic314—21no—
17C22×Dic314C2211yescharacteristic
18C2×Q8×D157—21no—
19C2×#14578938C2211yescharacteristic
20C2×C4⋊Dic157C2211yescharacteristic
21#19097699—22no—
22#19097696—21no—
23#19097695—21no—
24#19097693—21no—
25#19097692—21no—
26#19097689—22no—
27#19097678—22no—
28C2×#41×C157C2211yescharacteristic
29C2×M16×C157C2211yescharacteristic
30C2×C42×C157C2211yescharacteristic
31#104×C157—22no—
32#14578954—24no—
33#14578954—24no—
34#14578954—21no—
35#14578954—21no—
36#14578954C2×C412yes—
37#14578954D412yes—
38#14578953—28no—
39#14578953—24no—
40#14578953—28no—
41#14578953—24no—
42C2×Dic314—22no—
43C2×Dic314—24no—
44C2×Dic314—24no—
45C2×Dic314—21no—
46C2×Dic314—21no—
47C2×Dic314—21no—
48C2×Dic314—21no—
49C2×Dic314C2×C412yes—
50C2×Dic314D411yescharacteristic
51C2×Dic314D412yes—
52Q8×D157—24no—
53Q8×D157—24no—
54C2×D628D411yescharacteristic
55C2×C22⋊D157—22no—
56C2×C22⋊D157—21no—
57C22×Dic157—22no—
58C22×Dic157—21no—
59C22×Dic157D412yes—
60C2×C4×D157—22no—
61C2×C4×D157—21no—
62#14578938—21no—
63#14578938—21no—
64C2×#11929434—21no—
65C4⋊Dic157—21no—
66C4⋊Dic157—21no—
67#41×C157—21no—
68#41×C157—21no—
69#41×C157C2×C412yes—
70#41×C157D412yes—
71C2×Q8×C157D411yescharacteristic
72C2×D4×C157—21no—
73C2×D4×C157D411yescharacteristic
74C22×C4×C157—21no—
75C22×C4×C157—21no—
76C22×C4×C157C2311yescharacteristic
77M16×C157—21no—
78M16×C157—21no—
79C2×C8×C157—21no—
80C42×C157—21no—
81C42×C157—21no—
82C22×D157—21no—
83Dic314—28no—
84Dic314—22no—
85Dic314—22no—
86Dic314—22no—
87Dic314—21no—
88Dic314—22no—
89Dic314C22⋊C412yes—
90D628—21no—
91D628C22⋊C412yes—
92C22⋊D157—28no—
93C22⋊D157—22no—
94C22⋊D157—22no—
95C2×Dic157—28no—
96C2×Dic157—21no—
97C2×Dic157—22no—
98C2×Dic157—22no—
99C2×Dic157—42no—
100C2×Dic157—41no—
101C2×Dic157C22⋊C414yes—
102C4×D157—28no—
103C4×D157—22no—
104C4×D157—22no—
105#11929434—41no—
106C23×C157—21no—
107C23×C157C2×D411yescharacteristic
108Q8×C157—21no—
109Q8×C157C22⋊C412yes—
110D4×C157—22no—
111D4×C157—22no—
112D4×C157—21no—
113D4×C157C22⋊C412yes—
114C2×C4×C157—21no—
115C2×C4×C157—22no—
116C2×C4×C157—22no—
117C2×C4×C157—22no—
118C2×C4×C157—41no—
119C2×C4×C157—41no—
120C2×C4×C157C22×C411yescharacteristic
121C2×C4×C157C2×D411yescharacteristic
122C2×C4×C157C2×D411yescharacteristic
123C2×C4×C157C2×D411yescharacteristic
124C2×C4×C157C2×D411yescharacteristic
125C2×C4×C157C2×D411yescharacteristic
126C8×C157—41no—
127C22×C157—21no—
128C22×C157—21no—
129C22×C157—22no—
130C22×C157—41no—
131C22×C157—41no—
132C22×C157C2×C22⋊C411yescharacteristic
133C22×C157#12011yescharacteristic
134C22×C157#12011yescharacteristic
135C2×D157—22no—
136C2×D157—41no—
137C2×D157—41no—
138C4×C157—22no—
139C4×C157—42no—
140C4×C157C2×C22⋊C411yescharacteristic
141C4×C157C2×C22⋊C411yescharacteristic
142C4×C157#12011yescharacteristic
143C4×C157#12011yescharacteristic
144Dic157—22no—
145Dic157—24no—
146Dic157—81no—
147C2×C157—21no—
148C2×C157—21no—
149C2×C157—42no—
150C2×C157#37911yescharacteristic
151C2×C157#45411yescharacteristic
152C2×C157#45611yescharacteristic
153D157—42no—
154C157#179311yescharacteristic, complemented, Sylow
155#1793—1571nomaximal, Sylow
156C2×2−1+4—1571no—
157C2×#104—1571no—
158C2×#104D15711yescharacteristic, complemented
159#417—1571no—
160#417—1571no—
161C2×#101—1571no—
162#391—1571no—
1632−1+4—3142no—
1642−1+4—3142no—
1652−1+4—3142no—
1662−1+4—1574no—
167C2×#41—3141no—
168C2×#41—3141no—
169C2×#41—3142no—
170C2×#41—1571no—
171C2×#41C2×D15711yescharacteristic
172C22×Q8—3141no—
173C22×Q8—3141no—
174C22×Q8—1571no—
175C2×M16—1571no—
176C2×M16C2×D15711yescharacteristic
177C2×C4⋊C4—1571no—
178C2×C42C2×D15711yescharacteristic
179#104—22no—
180#104—3142no—
181#103—3141no—
182#103—3141no—
183#102—3141no—
184#102—3141no—
185#101—3142no—
186#95—3142no—
187#41—21no—
188#41—21no—
189#41—3144no—
190#41—3144no—
191#41—3148no—
192#41—3144no—
193#41—3144no—
194#41—3148no—
195#41—3141no—
196#41—3141no—
197#41—1572no—
198#41—1572no—
199#41C4×D15712yes—
200#41C22⋊D15712yes—
201C2×Q8—3142no—
202C2×Q8—3144no—
203C2×Q8—3144no—
204C2×Q8—3144no—
205C2×Q8—3144no—
206C2×Q8—3141no—
207C2×Q8—3141no—
208C2×Q8—3141no—
209C2×Q8—3141no—
210C2×Q8—1571no—
211C2×Q8—1572no—
212C2×Q8—1572no—
213C2×Q8D62811yescharacteristic
214C2×D4—21no—
215C2×D4—3142no—
216C2×D4—3141no—
217C2×D4—1571no—
218C2×D4D62811yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—3142no—
222C22×C4—3142no—
223C22×C4—3141no—
224C22×C4—3141no—
225C22×C4—1572no—
226C22×C4C22×D15711yescharacteristic
227M16—21no—
228M16—21no—
229M16—3141no—
230M16—3141no—
231C2×C8—21no—
232C2×C8—3141no—
233C4⋊C4—3141no—
234C4⋊C4—3141no—
235C42—21no—
236C42—21no—
237C23—21no—
238C23—3141no—
239C23D4×D15711yescharacteristic
240Q8—21no—
241Q8—3148no—
242Q8—3142no—
243Q8—3142no—
244Q8—3141no—
245Q8—3142no—
246Q8—3142no—
247Q8—1572no—
248Q8#1457893112yes—
249D4—22no—
250D4—22no—
251D4—21no—
252D4—3148no—
253D4—3142no—
254D4—3142no—
255D4—3141no—
256D4—1572no—
257D4#1457893112yes—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—3148no—
265C2×C4—3148no—
266C2×C4—3142no—
267C2×C4—3141no—
268C2×C4—3142no—
269C2×C4—3142no—
270C2×C4—3142no—
271C2×C4—6282no—
272C2×C4—6281no—
273C2×C4—1574no—
274C2×C4C2×C4×D15711yescharacteristic
275C2×C4D4×D15711yescharacteristic
276C2×C4D4×D15711yescharacteristic
277C2×C4D4×D15711yescharacteristic
278C2×C4C2×C22⋊D15711yescharacteristic
279C2×C4C2×D62811yescharacteristic
280C8—41no—
281C8—6281no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—3142no—
288C22—6281no—
289C22—6281no—
290C22C22⋊C4×D15711yescharacteristic
291C22#1909773911yescharacteristic
292C22#1909774111yescharacteristic
293C4—22no—
294C4—42no—
295C4—3142no—
296C4—3144no—
297C4—12561no—
298C4C22⋊C4×D15711yescharacteristic
299C4C2×#1457893111yescharacteristic
300C4#1909773911yescharacteristic
301C4#1909774111yescharacteristic
302C2—21no—
303C2—21no—
304C2—42no—
305C2—6282no—
306C2#2375712811yescharacteristic
307C2#2375744711yescharacteristic
308C2#2375745711yescharacteristic
subgroupG/H
centerC22#19097741
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157#120
prime radicalC2×#19097818C2
cosocleC22×C4×C157C23
hypercenterC2×#104D157
hypocenterC157#1793
LCSG ▷ C2×C4×C157 ▷ C2×C157 ▷ C157
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#19097699C2semidirect
3. #23757275C2semidirect
4. #23757274C2semidirect
5. C2×#19097689C2semidirect
6. #23757179C2semidirect
7. C2×#104×C157C2semidirect
11. #19097818C4semidirect
154. C157#1793semidirect
158. C2×#104D157semidirect

Maximal subgroups

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

subgroupindex
C2×#190978182
C2×#190976992
#237572752
#237572742
C2×#190976892
#237571792
C2×#104×C1572
#1793157