Finite Group Databasebeta

← C89×#10426 #36388274

Subgroups — C89×#10426 #36388274

Order 22784.

444 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C89×#2820C211yesmaximal, characteristic
2C89×#1992C211yesmaximal, characteristic, complemented
3C89×#1959C211yesmaximal, characteristic
4C89×#1836C211yesmaximal, characteristic, complemented
5C2×#330×C89C211yesmaximal, characteristic, complemented
6C2×#329×C89C211yesmaximal, characteristic
7C89×#1295C211yesmaximal, characteristic, complemented
8C2×#125×C89C2211yescharacteristic
9C2×#124×C89C2211yescharacteristic
10C2×#103×C89—21no—
11C2×#102×C89—21no—
12C2×#97×C89C2211yescharacteristic
13C2×#97×C89C2211yescharacteristic
14C2×#97×C89C2211yescharacteristic
15#399×C89—21no—
16#397×C89—21no—
17#396×C89—21no—
18#396×C89—21no—
19#395×C89—21no—
20#394×C89—21no—
21#384×C89C2211yescharacteristic
22#382×C89C2211yescharacteristic
23#340×C89—21no—
24#340×C89—21no—
25#340×C89—21no—
26#340×C89—21no—
27#330×C89—22no—
28#329×C89—22no—
29C22×Q8×C89D411yescharacteristic
30C22×D4×C89D411yescharacteristic
31C22×C8×C89—21no—
32C22×C8×C89—21no—
33C22×C8×C89—21no—
34#125×C89—21no—
35#125×C89—21no—
36#125×C89—21no—
37#125×C89—21no—
38#124×C89—21no—
39#124×C89—21no—
40#124×C89—21no—
41#124×C89—21no—
42C2×C4⋊C4×C89—21no—
43C2×C4⋊C4×C89—21no—
44C2×C4⋊C4×C89—21no—
45C2×C4⋊C4×C89—21no—
46C2×C4⋊C4×C89—21no—
47C2×C4⋊C4×C89—21no—
48C2×C4⋊C4×C89—21no—
49C2×C4⋊C4×C89D411yescharacteristic
50C2×C4⋊C4×C89D411yescharacteristic
51C2×C4⋊C4×C89D411yescharacteristic
52C2×C4⋊C4×C89D411yescharacteristic
53C2×C22⋊C4×C89—21no—
54C2×C22⋊C4×C89—21no—
55C2×C22⋊C4×C89—21no—
56C2×C22⋊C4×C89—21no—
57C2×C22⋊C4×C89—21no—
58C2×C22⋊C4×C89—21no—
59C2×C42×C89C2311yescharacteristic
60#103×C89—42no—
61#102×C89—42no—
62#97×C89—21no—
63#97×C89—21no—
64#97×C89—21no—
65#97×C89—21no—
66#97×C89—21no—
67#97×C89D412yes—
68#95×C89—21no—
69#95×C89—21no—
70#95×C89—21no—
71#95×C89—21no—
72C24×C89—21no—
73C2×Q8×C89—21no—
74C2×Q8×C89—21no—
75C2×Q8×C89—41no—
76C2×Q8×C89—41no—
77C2×D4×C89—21no—
78C2×D4×C89—21no—
79C2×D4×C89—41no—
80C2×D4×C89—41no—
81C22×C4×C89—21no—
82C22×C4×C89—21no—
83C22×C4×C89—21no—
84C22×C4×C89—21no—
85C22×C4×C89—21no—
86C22×C4×C89—41no—
87C22×C4×C89C2×D411yescharacteristic
88C22×C4×C89C2×D411yescharacteristic
89C22×C4×C89C2×D411yescharacteristic
90C2×C8×C89—22no—
91C2×C8×C89—22no—
92C2×C8×C89—22no—
93C2×C8×C89—41no—
94C2×C8×C89—41no—
95C2×C8×C89—41no—
96C2×C8×C89—41no—
97C2×C8×C89—41no—
98C2×C8×C89—41no—
99C4⋊C4×C89—21no—
100C4⋊C4×C89—21no—
101C4⋊C4×C89—21no—
102C4⋊C4×C89—21no—
103C4⋊C4×C89—41no—
104C4⋊C4×C89—41no—
105C4⋊C4×C89—41no—
106C4⋊C4×C89—41no—
107C4⋊C4×C89—42no—
108C4⋊C4×C89—42no—
109C4⋊C4×C89—41no—
110C4⋊C4×C89—41no—
111C4⋊C4×C89—41no—
112C4⋊C4×C89—41no—
113C4⋊C4×C89—81no—
114C4⋊C4×C89—81no—
115C22⋊C4×C89—41no—
116C22⋊C4×C89—41no—
117C22⋊C4×C89—41no—
118C22⋊C4×C89—41no—
119C22⋊C4×C89—81no—
120C22⋊C4×C89—81no—
121C22⋊C4×C89—81no—
122C22⋊C4×C89—81no—
123C42×C89C2×D411yescharacteristic
124C42×C89#4111yescharacteristic
125C42×C89#4111yescharacteristic
126C42×C89#4111yescharacteristic
127C23×C89—41no—
128C23×C89—41no—
129C23×C89—41no—
130C23×C89—41no—
131C23×C89—41no—
132C23×C89—41no—
133C23×C89—41no—
134C23×C89#12011yescharacteristic
135Q8×C89—42no—
136Q8×C89—81no—
137D4×C89—42no—
138D4×C89—81no—
139C2×C4×C89—21no—
140C2×C4×C89—21no—
141C2×C4×C89—21no—
142C2×C4×C89—21no—
143C2×C4×C89—21no—
144C2×C4×C89—21no—
145C2×C4×C89—41no—
146C2×C4×C89—41no—
147C2×C4×C89—41no—
148C2×C4×C89—41no—
149C2×C4×C89—41no—
150C2×C4×C89—41no—
151C2×C4×C89—41no—
152C2×C4×C89—41no—
153C2×C4×C89—41no—
154C2×C4×C89—41no—
155C2×C4×C89—41no—
156C2×C4×C89—41no—
157C2×C4×C89—41no—
158C2×C4×C89—41no—
159C2×C4×C89—41no—
160C2×C4×C89—81no—
161C2×C4×C89—81no—
162C2×C4×C89—81no—
163C2×C4×C89#12111yescharacteristic
164C2×C4×C89#12111yescharacteristic
165C2×C4×C89#12111yescharacteristic
166C2×C4×C89#12311yescharacteristic
167C2×C4×C89#12311yescharacteristic
168C2×C4×C89#12411yescharacteristic
169C8×C89—42no—
170C8×C89—42no—
171C8×C89—42no—
172C22×C89—81no—
173C22×C89—81no—
174C22×C89—81no—
175C22×C89—81no—
176C22×C89—81no—
177C22×C89—81no—
178C22×C89—81no—
179C22×C89#39411yescharacteristic
180C22×C89#45311yescharacteristic
181C22×C89#45411yescharacteristic
182C22×C89#45411yescharacteristic
183C22×C89#45511yescharacteristic
184C22×C89#45511yescharacteristic
185C22×C89#45611yescharacteristic
186C4×C89—21no—
187C4×C89—21no—
188C4×C89—21no—
189C4×C89—21no—
190C4×C89—21no—
191C4×C89—21no—
192C4×C89—81no—
193C4×C89—81no—
194C4×C89—81no—
195C4×C89—81no—
196C4×C89—81no—
197C4×C89—161no—
198#10426C8911yesmaximal, characteristic, direct product factor, Sylow
199C2×C89—161no—
200C2×C89#165711yescharacteristic
201C2×C89#166111yescharacteristic
202C2×C89#166211yescharacteristic
203C2×C89#166411yescharacteristic
204C2×C89#203711yescharacteristic
205C2×C89#203811yescharacteristic
206C2×C89#204011yescharacteristic
207#2820C2×C8911yescharacteristic
208#1992C2×C8911yescharacteristic, complemented
209#1959C2×C8911yescharacteristic
210#1836C2×C8911yescharacteristic, complemented
211C2×#330C2×C8911yescharacteristic, complemented
212C2×#329C2×C8911yescharacteristic
213#1295C2×C8911yescharacteristic, complemented
214C89#1042611yescharacteristic, direct product factor, Sylow
215C2×#125C22×C8911yescharacteristic
216C2×#124C22×C8911yescharacteristic
217C2×#103—21no—
218C2×#102—21no—
219C2×#97C22×C8911yescharacteristic
220C2×#97C22×C8911yescharacteristic
221C2×#97C22×C8911yescharacteristic
222#399—21no—
223#397—21no—
224#396—21no—
225#396—21no—
226#395—21no—
227#394—21no—
228#384C22×C8911yescharacteristic
229#382C22×C8911yescharacteristic
230#340—21no—
231#340—21no—
232#340—21no—
233#340—21no—
234#330—22no—
235#329—22no—
236C22×Q8D4×C8911yescharacteristic
237C22×D4D4×C8911yescharacteristic
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241#125—21no—
242#125—21no—
243#125—21no—
244#125—21no—
245#124—21no—
246#124—21no—
247#124—21no—
248#124—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4D4×C8911yescharacteristic
257C2×C4⋊C4D4×C8911yescharacteristic
258C2×C4⋊C4D4×C8911yescharacteristic
259C2×C4⋊C4D4×C8911yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C42C23×C8911yescharacteristic
267#103—42no—
268#102—42no—
269#97—21no—
270#97—21no—
271#97—21no—
272#97—21no—
273#97—21no—
274#97D4×C8912yes—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—21no—
279C24—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—41no—
287C2×D4—41no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—41no—
294C22×C4C2×D4×C8911yescharacteristic
295C22×C4C2×D4×C8911yescharacteristic
296C22×C4C2×D4×C8911yescharacteristic
297C2×C8—22no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—81no—
321C4⋊C4—81no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C22⋊C4—41no—
326C22⋊C4—81no—
327C22⋊C4—81no—
328C22⋊C4—81no—
329C22⋊C4—81no—
330C42C2×D4×C8911yescharacteristic
331C42#41×C8911yescharacteristic
332C42#41×C8911yescharacteristic
333C42#41×C8911yescharacteristic
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23#120×C8911yescharacteristic
342Q8—42no—
343Q8—81no—
344D4—42no—
345D4—81no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—81no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4#121×C8911yescharacteristic
371C2×C4#121×C8911yescharacteristic
372C2×C4#121×C8911yescharacteristic
373C2×C4#123×C8911yescharacteristic
374C2×C4#123×C8911yescharacteristic
375C2×C4#124×C8911yescharacteristic
376C8—42no—
377C8—42no—
378C8—42no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#394×C8911yescharacteristic
387C22#453×C8911yescharacteristic
388C22#454×C8911yescharacteristic
389C22#454×C8911yescharacteristic
390C22#455×C8911yescharacteristic
391C22#455×C8911yescharacteristic
392C22#456×C8911yescharacteristic
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—21no—
398C4—21no—
399C4—81no—
400C4—81no—
401C4—81no—
402C4—81no—
403C4—81no—
404C4—161no—
405C2—161no—
406C2C89×#165711yescharacteristic
407C2C89×#166111yescharacteristic
408C2C89×#166211yescharacteristic
409C2C89×#166411yescharacteristic
410C2C89×#203711yescharacteristic
411C2C89×#203811yescharacteristic
412C2C89×#204011yescharacteristic
subgroupG/H
center, socleC23×C89#120
commutator, Frattini, cosocleC2×C42C23×C89
prime radicalC22×D4×C89D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C89 ◁ C2×C42×C89 ◁ G

Complements

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

NH ≅ G/Ntype
2. C89×#1992C2semidirect
4. C89×#1836C2semidirect
5. C2×#330×C89C2semidirect
7. C89×#1295C2semidirect
198. #10426C89direct (N × H)
208. #1992C2×C89semidirect
210. #1836C2×C89semidirect
211. C2×#330C2×C89semidirect
213. #1295C2×C89semidirect
214. C89#10426direct (N × H)

Maximal subgroups

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

subgroupindex
C89×#28202
C89×#19922
C89×#19592
C89×#18362
C2×#330×C892
C2×#329×C892
C89×#12952
#1042689