Finite Group Databasebeta

← #36356357

Subgroups — #36356357

Order 22144.

532 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26011394C211yesmaximal, characteristic
2#26011390C211yesmaximal, characteristic, complemented
3#26011384C211yesmaximal, characteristic, complemented
4#26011384C211yesmaximal, characteristic, complemented
5#26011384C211yesmaximal, characteristic, complemented
6#26011384C211yesmaximal, characteristic, complemented
7#26011377C211yesmaximal, characteristic, complemented
8#26011377C211yesmaximal, characteristic, complemented
9#26011371C211yesmaximal, characteristic
10#26011366C211yesmaximal, characteristic, complemented
11#26011351C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic173C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic173C211yesmaximal, characteristic, complemented
15C2×#123×C173C211yesmaximal, characteristic
16C23×Dic173C2211yescharacteristic
17C2×C4⋊Dic173C2211yescharacteristic
18C2×C4⋊Dic173C2211yescharacteristic
19C2×C4⋊Dic173C2211yescharacteristic
20C2×C4⋊Dic173C2211yescharacteristic
21C2×C4⋊Dic173C2211yescharacteristic
22C2×C22⋊Dic173C2211yescharacteristic
23C2×C22⋊Dic173C2211yescharacteristic
24C2×C22⋊Dic173C2211yescharacteristic
25C2×C22⋊Dic173C2211yescharacteristic
26C2×C22⋊Dic173C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×C4×Dic173C2211yescharacteristic
30C2×C4×Dic173C2211yescharacteristic
31C2×C4×Dic173C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33C2×C4×Dic173C2211yescharacteristic
34#19149586C2211yescharacteristic
35#19149586C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44C22×D4×C173C2211yescharacteristic
45C23×C4×C173C2211yescharacteristic
46#123×C173C418yescomplemented
47C2×C4⋊C4×C173C2211yescharacteristic
48C2×C4⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52C22×Dic173—22no—
53C22×Dic173—41no—
54C22×Dic173—41no—
55C22×Dic173C2311yescharacteristic
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C4⋊Dic173—22no—
64C4⋊Dic173—41no—
65C4⋊Dic173—41no—
66C4⋊Dic173—41no—
67C4⋊Dic173—41no—
68C22⋊Dic173—22no—
69C22⋊Dic173—22no—
70C22⋊Dic173—22no—
71C22⋊Dic173—41no—
72C22⋊Dic173—41no—
73C22⋊Dic173—41no—
74C22⋊Dic173—41no—
75C4×Dic173—22no—
76C4×Dic173—41no—
77C4×Dic173—41no—
78C4×Dic173—41no—
79C4×Dic173—41no—
80C24×C173C2311yescharacteristic
81C24×C173C2311yescharacteristic
82C2×D4×C173—41no—
83C2×D4×C173—41no—
84C2×D4×C173C2×C414yes—
85C22×C4×C173—41no—
86C22×C4×C173—41no—
87C22×C4×C173C2×C414yescomplemented
88C22×C4×C173C2311yescharacteristic
89C22×C4×C173C2311yescharacteristic
90C22×C4×C173C2311yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C4⋊C4×C173C2×C414yescomplemented
94C4⋊C4×C173C2×C414yescomplemented
95C22⋊C4×C173C2×C414yes—
96C22⋊C4×C173C2×C414yes—
97C22⋊C4×C173C2×C414yescomplemented
98C2×Dic173—21no—
99C2×Dic173—21no—
100C2×Dic173—21no—
101C2×Dic173—21no—
102C2×Dic173—21no—
103C2×Dic173—21no—
104C2×Dic173—21no—
105C2×Dic173—21no—
106C2×Dic173—21no—
107C2×Dic173—21no—
108C2×Dic173—21no—
109C2×Dic173—21no—
110C2×Dic173—21no—
111C2×Dic173—21no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—21no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—44no—
121C2×Dic173#4112yes—
122C2×Dic173#4112yes—
123C23×C173—21no—
124C23×C173—21no—
125C23×C173—21no—
126C23×C173—21no—
127C23×C173—21no—
128C23×C173—21no—
129C23×C173—21no—
130C23×C173—21no—
131C23×C173—21no—
132C23×C173—21no—
133C23×C173—21no—
134C23×C173—21no—
135C23×C173C22×C412yes—
136C23×C173C22×C412yes—
137C23×C173C2411yescharacteristic
138D4×C173—44no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—44no—
150C2×C4×C173C22×C412yes—
151C2×C4×C173C22×C412yes—
152C2×C4×C173C22×C412yes—
153C2×C4×C173C22×C412yes—
154C2×C4×C173C22×C412yes—
155C22×C173—22no—
156C22×C173—22no—
157C22×C173—22no—
158C22×C173—22no—
159C22×C173—22no—
160C22×C173—22no—
161C22×C173—41no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173C23×C411yescharacteristic
170C22×C173C2×#4111yescharacteristic
171C22×C173C2×#4111yescharacteristic
172C22×C173H(5,𝔽2)11yescharacteristic
173C22×C173H(5,𝔽2)11yescharacteristic
174C22×C173H(5,𝔽2)11yescharacteristic
175C22×C1732−1+411yescharacteristic
176C4×C173—22no—
177C4×C173—22no—
178C4×C173—22no—
179C4×C173—22no—
180C4×C173—22no—
181Dic173—22no—
182Dic173—22no—
183Dic173—41no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189C2×C173—42no—
190C2×C173—42no—
191C2×C173C4×#4111yescharacteristic
192C2×C173#51811yescharacteristic
193C2×C173#52011yescharacteristic
194C2×C173#53811yescharacteristic
195C2×C173#53911yescharacteristic
196C2×C173#54011yescharacteristic
197C2×C173#54211yescharacteristic
198C173#233511yescharacteristic, complemented, Sylow
199#2335—1731nomaximal, Sylow
200C2×#123D17311yescharacteristic
201#390—1731no—
202#389—1731no—
203#388—1731no—
204#388—1731no—
205#388—1731no—
206#388—1731no—
207#387—1731no—
208#387—1731no—
209#386—1731no—
210#385—1731no—
211#380—1731no—
212C4×C4⋊C4—1731no—
213C4×C22⋊C4—1731no—
214C4×C22⋊C4—1731no—
215C22×D4C2×D17311yescharacteristic
216C23×C4—1731no—
217C23×C4C2×D17311yescharacteristic
218#123Dic17318yescomplemented
219C2×C4⋊C4—1731no—
220C2×C4⋊C4—1731no—
221C2×C4⋊C4—1731no—
222C2×C4⋊C4—1731no—
223C2×C4⋊C4—1731no—
224C2×C4⋊C4C2×D17311yescharacteristic
225C2×C4⋊C4C2×D17311yescharacteristic
226C2×C22⋊C4—1731no—
227C2×C22⋊C4—1731no—
228C2×C22⋊C4—1731no—
229C2×C22⋊C4—1731no—
230C2×C22⋊C4—1731no—
231C2×C22⋊C4—1731no—
232C2×C22⋊C4—1731no—
233C2×C22⋊C4C2×D17311yescharacteristic
234C2×C22⋊C4C2×D17311yescharacteristic
235C2×C22⋊C4C2×D17311yescharacteristic
236C2×C42—1731no—
237C2×C42—1731no—
238C2×C42—1731no—
239C2×C42—1731no—
240C2×C42—1731no—
241#95—1731no—
242#95—1731no—
243#95—1731no—
244#95—1731no—
245#95—1731no—
246#95—1731no—
247#95—1731no—
248#95—1731no—
249#95—1731no—
250#95—1731no—
251C24C22×D17311yescharacteristic
252C24C22×D17311yescharacteristic
253C2×D4—41no—
254C2×D4—41no—
255C2×D4C2×Dic17314yes—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—3462no—
259C22×C4—1731no—
260C22×C4—1731no—
261C22×C4—1731no—
262C22×C4—1731no—
263C22×C4—1731no—
264C22×C4—1731no—
265C22×C4—1731no—
266C22×C4—1731no—
267C22×C4—6921no—
268C22×C4—6921no—
269C22×C4C2×Dic17314yescomplemented
270C22×C4C22×D17311yescharacteristic
271C22×C4C22×D17311yescharacteristic
272C22×C4C22×D17311yescharacteristic
273C22×C4C22×D17311yescharacteristic
274C22×C4C22×D17311yescharacteristic
275C4⋊C4—3462no—
276C4⋊C4—6921no—
277C4⋊C4—6921no—
278C4⋊C4—6921no—
279C4⋊C4—6921no—
280C4⋊C4C2×Dic17314yescomplemented
281C4⋊C4C2×Dic17314yescomplemented
282C22⋊C4—3462no—
283C22⋊C4—3462no—
284C22⋊C4—3462no—
285C22⋊C4—6921no—
286C22⋊C4—6921no—
287C22⋊C4—6921no—
288C22⋊C4—6921no—
289C22⋊C4C2×Dic17314yes—
290C22⋊C4C2×Dic17314yes—
291C22⋊C4C2×Dic17314yescomplemented
292C42—3462no—
293C42—6921no—
294C42—6921no—
295C42—6921no—
296C42—6921no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C22×Dic17312yes—
310C23C22×Dic17312yes—
311C23C23×D17311yescharacteristic
312D4—44no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—44no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—3461no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—3461no—
339C2×C4—3461no—
340C2×C4—3461no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3461no—
346C2×C4—1732no—
347C2×C4—1732no—
348C2×C4—6924no—
349C2×C4C22×Dic17312yes—
350C2×C4C22×Dic17312yes—
351C2×C4C22×Dic17312yes—
352C2×C4C22×Dic17312yes—
353C2×C4C22×Dic17312yes—
354C22—22no—
355C22—22no—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22C23×Dic17311yescharacteristic
369C22#41×D17311yescharacteristic
370C22#41×D17311yescharacteristic
371C22#1914972411yescharacteristic
372C22#1914972411yescharacteristic
373C22#1914972511yescharacteristic
374C22#1914972611yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—3462no—
381C4—3462no—
382C4—6921no—
383C4—6921no—
384C4—6921no—
385C4—6921no—
386C4—6921no—
387C4—6921no—
388C2—42no—
389C2—42no—
390C2#41×Dic17311yescharacteristic
391C2#2601196611yescharacteristic
392C2#2601197411yescharacteristic
393C2#2601207711yescharacteristic
394C2#2601208411yescharacteristic
395C2#2601209311yescharacteristic
396C2#2601210911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
FittingC2×#123×C173C2
socle, cosocleC23×C173C24
prime radicalC22×D4×C173C22
hypercenterC2×#123D173
hypocenterC173#2335
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. #26011390C2semidirect
3. #26011384C2semidirect
4. #26011384C2semidirect
5. #26011384C2semidirect
6. #26011384C2semidirect
7. #26011377C2semidirect
8. #26011377C2semidirect
10. #26011366C2semidirect
11. #26011351C2semidirect
12. C4⋊C4×Dic173C2semidirect
13. C22⋊C4×Dic173C2semidirect
14. C4×C22⋊Dic173C2semidirect
46. #123×C173C4semidirect
87. C22×C4×C173C2×C4semidirect
93. C4⋊C4×C173C2×C4semidirect
94. C4⋊C4×C173C2×C4semidirect
97. C22⋊C4×C173C2×C4semidirect
198. C173#2335semidirect
218. #123Dic173semidirect
269. C22×C4C2×Dic173semidirect
280. C4⋊C4C2×Dic173semidirect
281. C4⋊C4C2×Dic173semidirect
291. C22⋊C4C2×Dic173semidirect

Maximal subgroups

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

subgroupindex
#260113942
#260113902
#260113842
#260113772
#260113712
#260113662
#260113512
C4⋊C4×Dic1732
C22⋊C4×Dic1732
C4×C22⋊Dic1732
C2×#123×C1732
#2335173