Finite Group Databasebeta

← #34103410

Subgroups — #34103410

Order 20096.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic314C211yesmaximal, characteristic, complemented
2C2×#19097730C211yesmaximal, characteristic
3#23757215C211yesmaximal, characteristic, complemented
4#23757209C211yesmaximal, characteristic
5#23757207C211yesmaximal, characteristic, complemented
6#23757205C211yesmaximal, characteristic, complemented
7#23757196C211yesmaximal, characteristic
8#23757196C211yesmaximal, characteristic
9#23757194C211yesmaximal, characteristic, complemented
10#23757185C211yesmaximal, characteristic
11C4×#14578933C211yesmaximal, characteristic, complemented
12C4×#14578933C211yesmaximal, characteristic, complemented
13C4×C4⋊Dic157C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic157C211yesmaximal, characteristic
15C4×C22⋊C4×C157C211yesmaximal, characteristic
16C22×Dic314C2211yescharacteristic
17C4×Dic314—24no—
18#19097730—24no—
19C2×C4⋊Dic157C2211yescharacteristic
20C2×C4⋊Dic157C2211yescharacteristic
21C2×C4⋊Dic157C2211yescharacteristic
22C2×C4⋊Dic157C2211yescharacteristic
23C2×C4⋊Dic157C2211yescharacteristic
24C2×#14578933C2211yescharacteristic
25C2×#14578933C2211yescharacteristic
26C2×#14578933C2211yescharacteristic
27C2×#14578933C2211yescharacteristic
28C2×#14578933C2211yescharacteristic
29C2×#14578933C2211yescharacteristic
30C2×C22⋊Dic157C2211yescharacteristic
31C2×C22⋊Dic157C2211yescharacteristic
32C2×C22⋊Dic157C2211yescharacteristic
33C2×C22⋊Dic157C2211yescharacteristic
34C2×C4×Dic157C2211yescharacteristic
35C2×C4×Dic157C2211yescharacteristic
36C2×C4×Dic157C2211yescharacteristic
37C2×C4×Dic157C2211yescharacteristic
38#19097678C2211yescharacteristic
39#19097678C2211yescharacteristic
40#19097678C2211yescharacteristic
41#19097678C2211yescharacteristic
42#19097678C2211yescharacteristic
43#19097678C2211yescharacteristic
44#19097678C2211yescharacteristic
45#19097678C2211yescharacteristic
46C23×C4×C157C2211yescharacteristic
47C2×C22⋊C4×C157C2211yescharacteristic
48C2×C22⋊C4×C157C2211yescharacteristic
49C2×C42×C157C2211yescharacteristic
50C2×C42×C157C2211yescharacteristic
51#95×C157C2211yescharacteristic
52#95×C157C2211yescharacteristic
53C2×Dic314—22no—
54C2×Dic314—22no—
55C2×Dic314—22no—
56C22×Dic157C2311yescharacteristic
57C22×Dic157C2311yescharacteristic
58C22×Dic157C2311yescharacteristic
59C22×Dic157C2311yescharacteristic
60C22×Dic157C2311yescharacteristic
61C22×Dic157C2311yescharacteristic
62C22×Dic157C2311yescharacteristic
63C22×Dic157C2311yescharacteristic
64C4⋊Dic157—22no—
65C4⋊Dic157—22no—
66C4⋊Dic157—22no—
67C4⋊Dic157—22no—
68C4⋊Dic157—22no—
69#14578933—22no—
70#14578933—22no—
71#14578933—22no—
72#14578933—22no—
73#14578933—22no—
74#14578933—22no—
75C22⋊Dic157—22no—
76C22⋊Dic157—22no—
77C22⋊Dic157—22no—
78C22⋊Dic157—22no—
79C4×Dic157—22no—
80C4×Dic157—22no—
81C4×Dic157—22no—
82C4×Dic157D414yes—
83C24×C157C2311yescharacteristic
84C22×C4×C157—22no—
85C22×C4×C157—22no—
86C22×C4×C157Q814yescomplemented
87C22×C4×C157C2311yescharacteristic
88C22×C4×C157C2311yescharacteristic
89C22×C4×C157C2311yescharacteristic
90C22×C4×C157C2311yescharacteristic
91C22×C4×C157C2311yescharacteristic
92C22×C4×C157C2311yescharacteristic
93C22⋊C4×C157—22no—
94C22⋊C4×C157—22no—
95C42×C157—22no—
96C42×C157—22no—
97Dic314—28no—
98C2×Dic157—21no—
99C2×Dic157—21no—
100C2×Dic157—21no—
101C2×Dic157—21no—
102C2×Dic157—21no—
103C2×Dic157—21no—
104C2×Dic157—21no—
105C2×Dic157—21no—
106C2×Dic157—21no—
107C2×Dic157—21no—
108C2×Dic157—21no—
109C2×Dic157—21no—
110C2×Dic157—21no—
111C2×Dic157—21no—
112C2×Dic157—21no—
113C2×Dic157—21no—
114C2×Dic157—21no—
115C2×Dic157—21no—
116C2×Dic157—21no—
117C2×Dic157—21no—
118C2×Dic157C2×D412yes—
119C2×Dic157C2×D412yes—
120C2×Dic157#4112yes—
121C2×Dic157#4112yes—
122C23×C157—21no—
123C23×C157—21no—
124C23×C157—21no—
125C23×C157—21no—
126C23×C157—21no—
127C23×C157—21no—
128C23×C157C2×Q812yes—
129C23×C157C2411yescharacteristic
130C2×C4×C157—21no—
131C2×C4×C157—21no—
132C2×C4×C157—21no—
133C2×C4×C157—21no—
134C2×C4×C157—21no—
135C2×C4×C157—21no—
136C2×C4×C157—28no—
137C2×C4×C157—21no—
138C2×C4×C157—21no—
139C2×C4×C157—21no—
140C2×C4×C157—21no—
141C2×C4×C157—21no—
142C2×C4×C157—21no—
143C2×C4×C157C2×D412yes—
144C2×C4×C157C2×Q812yes—
145C2×C4×C157C2×Q812yes—
146C2×C4×C157#4112yes—
147C2×C4×C157#4112yes—
148C2×C4×C157#4112yes—
149C22×C157—22no—
150C22×C157—22no—
151C22×C157—22no—
152C22×C157—41no—
153C22×C157—41no—
154C22×C157—41no—
155C22×C157—41no—
156C22×C157C22×D411yescharacteristic
157C22×C157C22×Q811yescharacteristic
158C22×C157C2×#4111yescharacteristic
159C22×C157C2×#4111yescharacteristic
160C22×C157C2×#4111yescharacteristic
161C22×C157C2×#4111yescharacteristic
162C22×C157C2×#4111yescharacteristic
163C4×C157—22no—
164C4×C157—22no—
165C4×C157—22no—
166C4×C157—41no—
167C4×C157—41no—
168C4×C157C22⋊Q814yes—
169Dic157—22no—
170Dic157—22no—
171Dic157—22no—
172Dic157—22no—
173Dic157—41no—
174Dic157—41no—
175Dic157—41no—
176Dic157—41no—
177C2×C157—42no—
178C2×C157C2×C22⋊Q811yescharacteristic
179C2×C157#52511yescharacteristic
180C2×C157#52911yescharacteristic
181C2×C157#52911yescharacteristic
182C2×C157#53211yescharacteristic
183C2×C157#53311yescharacteristic
184C2×C157#53311yescharacteristic
185C157#239611yescharacteristic, complemented, Sylow
186#2396—1571nomaximal, Sylow
187C2×C4×Q8—1571no—
188C2×#117—1571no—
189#400—1571no—
190#399—1571no—
191#398—1571no—
192#398—1571no—
193#396—1571no—
194#396—1571no—
195#395—1571no—
196#393—1571no—
197C4×C4⋊C4—1571no—
198C4×C4⋊C4—1571no—
199C4×C4⋊C4—1571no—
200C4×C22⋊C4—1571no—
201C4×C22⋊C4D15711yescharacteristic
202C22×Q8—1571no—
203C23×C4C2×D15711yescharacteristic
204C4×Q8—3144no—
205#117—3144no—
206C2×C4⋊C4—1571no—
207C2×C4⋊C4—1571no—
208C2×C4⋊C4—1571no—
209C2×C4⋊C4—1571no—
210C2×C4⋊C4—1571no—
211C2×C4⋊C4—1571no—
212C2×C4⋊C4—1571no—
213C2×C4⋊C4—1571no—
214C2×C4⋊C4—1571no—
215C2×C4⋊C4—1571no—
216C2×C4⋊C4—1571no—
217C2×C22⋊C4—1571no—
218C2×C22⋊C4—1571no—
219C2×C22⋊C4—1571no—
220C2×C22⋊C4—1571no—
221C2×C22⋊C4C2×D15711yescharacteristic
222C2×C22⋊C4C2×D15711yescharacteristic
223C2×C42—1571no—
224C2×C42—1571no—
225C2×C42—1571no—
226C2×C42—1571no—
227C2×C42C2×D15711yescharacteristic
228C2×C42C2×D15711yescharacteristic
229#95—1571no—
230#95—1571no—
231#95—1571no—
232#95—1571no—
233#95—1571no—
234#95—1571no—
235#95—1571no—
236#95—1571no—
237#95C2×D15711yescharacteristic
238#95C2×D15711yescharacteristic
239C24C22×D15711yescharacteristic
240C2×Q8—3142no—
241C2×Q8—3142no—
242C2×Q8—3142no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—1571no—
246C22×C4—1571no—
247C22×C4—1571no—
248C22×C4—1571no—
249C22×C4—1571no—
250C22×C4—1571no—
251C22×C4—1571no—
252C22×C4—1571no—
253C22×C4Dic31414yescomplemented
254C22×C4C22×D15711yescharacteristic
255C22×C4C22×D15711yescharacteristic
256C22×C4C22×D15711yescharacteristic
257C22×C4C22×D15711yescharacteristic
258C22×C4C22×D15711yescharacteristic
259C22×C4C22×D15711yescharacteristic
260C4⋊C4—3142no—
261C4⋊C4—3142no—
262C4⋊C4—3142no—
263C4⋊C4—3142no—
264C4⋊C4—3142no—
265C4⋊C4—3142no—
266C4⋊C4—3142no—
267C4⋊C4—3142no—
268C4⋊C4—3142no—
269C4⋊C4—3142no—
270C4⋊C4—3142no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—3142no—
274C22⋊C4—3142no—
275C22⋊C4—3142no—
276C22⋊C4—3142no—
277C42—22no—
278C42—22no—
279C42—3142no—
280C42—3142no—
281C42—3142no—
282C42—1574no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C2×Dic31412yes—
290C23C23×D15711yescharacteristic
291Q8—3148no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—28no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—3141no—
306C2×C4—3141no—
307C2×C4—3141no—
308C2×C4—3141no—
309C2×C4—3141no—
310C2×C4—3141no—
311C2×C4—3141no—
312C2×C4—3141no—
313C2×C4—3141no—
314C2×C4—3141no—
315C2×C4—3141no—
316C2×C4—3141no—
317C2×C4—3141no—
318C2×C4—3141no—
319C2×C4—3141no—
320C2×C4—3141no—
321C2×C4—3141no—
322C2×C4—3141no—
323C2×C4—3141no—
324C2×C4—3141no—
325C2×C4—1572no—
326C2×C4—1572no—
327C2×C4—1572no—
328C2×C4—1572no—
329C2×C4D4×D15712yes—
330C2×C4C2×Dic31412yes—
331C2×C4C2×Dic31412yes—
332C2×C4#1457895312yes—
333C2×C4#1457895412yes—
334C2×C4#1457895412yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D15711yescharacteristic
343C22C22×Dic31411yescharacteristic
344C22#41×D15711yescharacteristic
345C22#41×D15711yescharacteristic
346C22C2×#1457895311yescharacteristic
347C22C2×#1457895411yescharacteristic
348C22C2×#1457895411yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—3142no—
355C4—3142no—
356C4—3142no—
357C4—3142no—
358C4—6281no—
359C4—6281no—
360C4—6281no—
361C4—6281no—
362C4#1909774814yes—
363C2—42no—
364C2C2×#1909774811yescharacteristic
365C2#2375778211yescharacteristic
366C2#2375780311yescharacteristic
367C2#2375780511yescharacteristic
368C2#2375782011yescharacteristic
369C2#2375782611yescharacteristic
370C2#2375783011yescharacteristic
subgroupG/H
centerC22×C4C22×D157
commutator, socle, cosocleC23×C157C24
FrattiniC23C23×D157
FittingC4×C22⋊C4×C157C2
prime radicalC24×C157C23
hypercenterC4×C22⋊C4D157
hypocenterC157#2396
LCSG ▷ C23×C157 ▷ C157
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic314C2semidirect
3. #23757215C2semidirect
5. #23757207C2semidirect
6. #23757205C2semidirect
9. #23757194C2semidirect
11. C4×#14578933C2semidirect
12. C4×#14578933C2semidirect
13. C4×C4⋊Dic157C2semidirect
86. C22×C4×C157Q8semidirect
185. C157#2396semidirect
253. C22×C4Dic314semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic3142
C2×#190977302
#237572152
#237572092
#237572072
#237572052
#237571962
#237571942
#237571852
C4×#145789332
C4×C4⋊Dic1572
C4×C22⋊Dic1572
C4×C22⋊C4×C1572
#2396157