Finite Group Databasebeta

← #34100091

Subgroups — #34100091

Order 20096.

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

normal subgroups only

This group has 620 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×#19097817C211yesmaximal, characteristic
2C2×#19097699C211yesmaximal, characteristic, complemented
3#23757275C211yesmaximal, characteristic, complemented
4#23757274C211yesmaximal, characteristic, complemented
5C2×#19097687C211yesmaximal, characteristic, complemented
6#23757175C211yesmaximal, characteristic, complemented
7C2×#104×C157C211yesmaximal, characteristic, complemented
8#19097817—22no—
9#19097817—22no—
10#19097817—22no—
11#19097817C414yescomplemented
12C2×#14578955—21no—
13C2×#14578954—21no—
14C2×#14578954C2211yescharacteristic
15C22×D628—21no—
16C22×D628C2211yescharacteristic
17C2×D4×D157—21no—
18C2×D4×D157—22no—
19C2×#14578938C2211yescharacteristic, complemented
20C2×C4⋊Dic157C2211yescharacteristic, complemented
21C2×#14578931—22no—
22#19097699—22no—
23#19097696—21no—
24#19097695—21no—
25#19097693—21no—
26#19097692—21no—
27#19097687—22no—
28C2×#41×C157C2211yescharacteristic
29C2×M16×C157C2211yescharacteristic, complemented
30C2×C42×C157C2211yescharacteristic, complemented
31#104×C157—22no—
32C23×D157—22no—
33C23×D157D412yes—
34#14578955—24no—
35#14578955—24no—
36#14578954—24no—
37#14578954—24no—
38#14578954—21no—
39#14578954—21no—
40#14578954C2×C412yes—
41#14578954D412yes—
42C2×Dic314D411yescharacteristic
43C2×D628—22no—
44C2×D628—24no—
45C2×D628—24no—
46C2×D628—21no—
47C2×D628—21no—
48C2×D628—21no—
49C2×D628—21no—
50C2×D628C2×C412yes—
51C2×D628D412yes—
52C2×D628D411yescharacteristic
53C2×C22⋊D157—22no—
54C2×C22⋊D157—21no—
55D4×D157—28no—
56D4×D157—24no—
57D4×D157—28no—
58D4×D157—24no—
59C22×Dic157—21no—
60C2×C4×D157—22no—
61C2×C4×D157—21no—
62#14578938—21no—
63#14578938—21no—
64C2×#11929434—21no—
65C4⋊Dic157—21no—
66C4⋊Dic157—21no—
67#14578931—44no—
68#41×C157—21no—
69#41×C157—21no—
70#41×C157C2×C412yes—
71#41×C157D412yes—
72C2×Q8×C157D411yescharacteristic
73C2×D4×C157—21no—
74C2×D4×C157D411yescharacteristic
75C22×C4×C157—21no—
76C22×C4×C157—21no—
77C22×C4×C157C2311yescharacteristic
78M16×C157—21no—
79M16×C157—21no—
80C2×C8×C157—21no—
81C42×C157—21no—
82C42×C157—21no—
83C22×D157—22no—
84C22×D157—21no—
85C22×D157—28no—
86C22×D157—22no—
87C22×D157—42no—
88C22×D157—42no—
89C22×D157—82no—
90C22×D157—82no—
91C22×D157C22⋊C414yes—
92Dic314—21no—
93Dic314C22⋊C412yescomplemented
94D628—28no—
95D628—22no—
96D628—22no—
97D628—22no—
98D628—22no—
99D628—21no—
100D628C22⋊C412yescomplemented
101C22⋊D157—28no—
102C22⋊D157—22no—
103C22⋊D157—22no—
104C2×Dic157—21no—
105C2×Dic157—42no—
106C2×Dic157—41no—
107C4×D157—28no—
108C4×D157—22no—
109C4×D157—22no—
110#11929434—41no—
111C23×C157—21no—
112C23×C157C2×D411yescharacteristic
113Q8×C157—21no—
114Q8×C157C22⋊C412yescomplemented
115D4×C157—22no—
116D4×C157—22no—
117D4×C157—21no—
118D4×C157C22⋊C412yescomplemented
119C2×C4×C157—21no—
120C2×C4×C157—22no—
121C2×C4×C157—22no—
122C2×C4×C157—22no—
123C2×C4×C157—41no—
124C2×C4×C157—41no—
125C2×C4×C157C22×C411yescharacteristic
126C2×C4×C157C2×D411yescharacteristic
127C2×C4×C157C2×D411yescharacteristic
128C2×C4×C157C2×D411yescharacteristic
129C2×C4×C157C2×D411yescharacteristic
130C2×C4×C157C2×D411yescharacteristic
131C8×C157—41no—
132C22×C157—21no—
133C22×C157—21no—
134C22×C157—22no—
135C22×C157—41no—
136C22×C157—41no—
137C22×C157C2×C22⋊C411yescharacteristic
138C22×C157#12011yescharacteristic
139C22×C157#12011yescharacteristic
140C2×D157—24no—
141C2×D157—22no—
142C2×D157—42no—
143C2×D157—41no—
144C2×D157—42no—
145C2×D157—41no—
146C2×D157—44no—
147C2×D157—82no—
148C2×D157—88no—
149C4×C157—22no—
150C4×C157—42no—
151C4×C157C2×C22⋊C411yescharacteristic
152C4×C157C2×C22⋊C411yescharacteristic
153C4×C157#12011yescharacteristic
154C4×C157#12011yescharacteristic
155Dic157—22no—
156Dic157—81no—
157C2×C157—21no—
158C2×C157—21no—
159C2×C157—42no—
160C2×C157#37911yescharacteristic
161C2×C157#45311yescharacteristic
162C2×C157#45511yescharacteristic
163D157—44no—
164D157—42no—
165C157#179211yescharacteristic, complemented, Sylow
166#1792—1571nomaximal, Sylow
167C2×H(5,𝔽2)—1571no—
168C2×#104—1571no—
169C2×#104D15711yescharacteristic, complemented
170#417—1571no—
171#417—1571no—
172C2×#100—1571no—
173#390—1571no—
174H(5,𝔽2)—3142no—
175H(5,𝔽2)—3142no—
176H(5,𝔽2)—3142no—
177H(5,𝔽2)—1574no—
178C2×#41—3141no—
179C2×#41—3141no—
180C2×#41—1571no—
181C2×#41C2×D15711yescharacteristic
182C22×D4—3141no—
183C22×D4—3142no—
184C22×D4—3141no—
185C22×D4—1571no—
186C2×M16—1571no—
187C2×M16C2×D15711yescharacteristic, complemented
188C2×C4⋊C4—1571no—
189C2×C22⋊C4—3142no—
190C2×C42C2×D15711yescharacteristic, complemented
191#104—22no—
192#104—3142no—
193#103—3141no—
194#103—3141no—
195#102—3141no—
196#102—3141no—
197#100—3142no—
198C24—3142no—
199C24—1572no—
200#41—21no—
201#41—21no—
202#41—3144no—
203#41—3144no—
204#41—3144no—
205#41—3144no—
206#41—3141no—
207#41—3141no—
208#41—1572no—
209#41—1572no—
210#41C4×D15712yes—
211#41C22⋊D15712yes—
212C2×Q8—1571no—
213C2×Q8D62811yescharacteristic
214C2×D4—21no—
215C2×D4—3142no—
216C2×D4—3144no—
217C2×D4—3144no—
218C2×D4—3148no—
219C2×D4—3144no—
220C2×D4—3148no—
221C2×D4—3142no—
222C2×D4—3141no—
223C2×D4—3144no—
224C2×D4—3141no—
225C2×D4—3141no—
226C2×D4—3141no—
227C2×D4—3141no—
228C2×D4—1572no—
229C2×D4—1572no—
230C2×D4—1571no—
231C2×D4D62811yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—3142no—
235C22×C4—3141no—
236C22×C4—3141no—
237C22×C4C22×D15711yescharacteristic
238M16—21no—
239M16—21no—
240M16—3141no—
241M16—3141no—
242C2×C8—21no—
243C2×C8—3141no—
244C4⋊C4—3141no—
245C4⋊C4—3141no—
246C22⋊C4—6284no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—3148no—
251C23—3142no—
252C23—3141no—
253C23—3142no—
254C23—6282no—
255C23—6282no—
256C23—1574no—
257C23—12562no—
258C23—12562no—
259C23D4×D15711yescharacteristic
260Q8—21no—
261Q8—3141no—
262Q8—1572no—
263Q8#1457893112yescomplemented
264D4—22no—
265D4—22no—
266D4—21no—
267D4—3148no—
268D4—3148no—
269D4—3142no—
270D4—3142no—
271D4—3142no—
272D4—3142no—
273D4—3142no—
274D4—3141no—
275D4—3142no—
276D4—1572no—
277D4#1457893112yescomplemented
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—3148no—
285C2×C4—3141no—
286C2×C4—3142no—
287C2×C4—3142no—
288C2×C4—6282no—
289C2×C4—6281no—
290C2×C4C2×C4×D15711yescharacteristic
291C2×C4D4×D15711yescharacteristic
292C2×C4D4×D15711yescharacteristic
293C2×C4D4×D15711yescharacteristic
294C2×C4C2×C22⋊D15711yescharacteristic
295C2×C4C2×D62811yescharacteristic
296C8—41no—
297C8—6281no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—3144no—
304C22—3142no—
305C22—6284no—
306C22—6282no—
307C22—6281no—
308C22—6282no—
309C22—6281no—
310C22—12562no—
311C22—12568no—
312C22C22⋊C4×D15711yescharacteristic
313C22#1909773911yescharacteristic
314C22#1909774111yescharacteristic
315C4—22no—
316C4—42no—
317C4—3142no—
318C4—12561no—
319C4C22⋊C4×D15711yescharacteristic
320C4C2×#1457893111yescharacteristic
321C4#1909773911yescharacteristic
322C4#1909774111yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2—6284no—
327C2—6282no—
328C2#2375712811yescharacteristic
329C2#2375744211yescharacteristic
330C2#2375745211yescharacteristic
subgroupG/H
centerC22#19097741
commutatorC2×C4×C157C22×C4
FrattiniC22×C4C22×D157
FittingC2×#104×C157C2
socleC22×C157#120
prime radicalC2×#19097817C2
cosocleC22×C4×C157C23
hypercenterC2×#104D157
hypocenterC157#1792
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×#19097687C2semidirect
6. #23757175C2semidirect
7. C2×#104×C157C2semidirect
11. #19097817C4semidirect
19. C2×#14578938C22semidirect
20. C2×C4⋊Dic157C22semidirect
29. C2×M16×C157C22semidirect
30. C2×C42×C157C22semidirect
93. Dic314C22⋊C4semidirect
100. D628C22⋊C4semidirect
114. Q8×C157C22⋊C4semidirect
118. D4×C157C22⋊C4semidirect
165. C157#1792semidirect
169. C2×#104D157semidirect
187. C2×M16C2×D157semidirect
190. C2×C42C2×D157semidirect
263. Q8#14578931semidirect
277. D4#14578931semidirect

Maximal subgroups

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

subgroupindex
C2×#190978172
C2×#190976992
#237572752
#237572742
C2×#190976872
#237571752
C2×#104×C1572
#1792157