Finite Group Databasebeta

← #36356509

Subgroups — #36356509

Order 22144.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149639C211yesmaximal, characteristic
2C2×#19149636C211yesmaximal, characteristic, complemented
3C2×#19149635C211yesmaximal, characteristic, complemented
4#26011391C211yesmaximal, characteristic
5#26011391C211yesmaximal, characteristic
6#26011384C211yesmaximal, characteristic, complemented
7#26011383C211yesmaximal, characteristic, complemented
8#26011381C211yesmaximal, characteristic, complemented
9#26011360C211yesmaximal, characteristic, complemented
10#26011350C211yesmaximal, characteristic, complemented
11#26011349C211yesmaximal, characteristic, complemented
12C4×#14583805C211yesmaximal, characteristic, complemented
13#26011342C211yesmaximal, characteristic, complemented
14#26011342C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C22×C22⋊D173C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18C22×C4×D173C2211yescharacteristic
19#19149639C418yescomplemented
20#19149636—24no—
21#19149635—24no—
22C2×C4⋊Dic173C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×#14583805C2211yescharacteristic
29C2×#14583805C2211yescharacteristic
30C2×C22⋊Dic173C2211yescharacteristic
31C2×C22⋊Dic173C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×C4×Dic173C2211yescharacteristic
38C2×C4×Dic173C2211yescharacteristic
39C2×C4×Dic173C2211yescharacteristic
40C2×C4×Dic173C2211yescharacteristic
41C2×C4×Dic173C2211yescharacteristic
42C2×C4×Dic173C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47C2×C4⋊C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C42×C173C2211yescharacteristic
52#95×C173C2211yescharacteristic
53#95×C173C2211yescharacteristic
54C23×D173C2311yescharacteristic
55C2×C22⋊D173—22no—
56C2×C22⋊D173—22no—
57C2×C22⋊D173C2×C414yes—
58C22×Dic173—22no—
59C22×Dic173—22no—
60C22×Dic173C2×C414yes—
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C22×Dic173C2311yescharacteristic
66C22×Dic173C2311yescharacteristic
67C22×Dic173C2311yescharacteristic
68C2×C4×D173—22no—
69C2×C4×D173—22no—
70C2×C4×D173C2×C414yes—
71C4⋊Dic173—41no—
72#14583805—22no—
73#14583805—22no—
74#14583805—22no—
75#14583805—22no—
76#14583805—22no—
77#14583805C2×C414yes—
78#14583805D414yescomplemented
79C22⋊Dic173—22no—
80C22⋊Dic173—41no—
81#14583803—22no—
82#14583803—22no—
83#14583803—41no—
84#14583803—41no—
85#14583803C2×C414yes—
86C4×Dic173—22no—
87C4×Dic173—22no—
88C4×Dic173—22no—
89C4×Dic173—22no—
90C4×Dic173—22no—
91C4×Dic173C2×C414yes—
92C24×C173C2311yescharacteristic
93C22×C4×C173C2311yescharacteristic
94C22×C4×C173C2311yescharacteristic
95C22×C4×C173C2311yescharacteristic
96C22×C4×C173C2311yescharacteristic
97C22×C4×C173C2311yescharacteristic
98C22×C4×C173C2311yescharacteristic
99C4⋊C4×C173—41no—
100C22⋊C4×C173—22no—
101C22⋊C4×C173—41no—
102C22⋊C4×C173C2×C414yes—
103C42×C173—22no—
104C22×D173—21no—
105C22×D173—21no—
106C22×D173—21no—
107C22×D173—21no—
108C22×D173—21no—
109C22×D173—21no—
110C22×D173C22×C412yes—
111C22⋊D173—28no—
112C2×Dic173—21no—
113C2×Dic173—21no—
114C2×Dic173—21no—
115C2×Dic173—21no—
116C2×Dic173—28no—
117C2×Dic173—21no—
118C2×Dic173—21no—
119C2×Dic173—21no—
120C2×Dic173—21no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173C22×C412yes—
127C2×Dic173C22×C412yes—
128C2×Dic173C22×C412yes—
129C2×Dic173C2×D412yes—
130C2×Dic173C2×D412yes—
131C2×Dic173#4112yes—
132C2×Dic173#4112yes—
133C2×Dic173#4112yes—
134C4×D173—28no—
135C23×C173—21no—
136C23×C173—21no—
137C23×C173—21no—
138C23×C173—21no—
139C23×C173—21no—
140C23×C173—21no—
141C23×C173C22×C412yes—
142C23×C173C2411yescharacteristic
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—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—21no—
157C2×C4×C173—21no—
158C2×C4×C173C22×C412yes—
159C2×C4×C173C22×C412yes—
160C2×C4×C173C2×D412yes—
161C22×C173—22no—
162C22×C173—22no—
163C22×C173—22no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173C23×C411yescharacteristic
169C22×C173C22×D411yescharacteristic
170C22×C173C2×#4111yescharacteristic
171C22×C173C2×#4111yescharacteristic
172C22×C173C2×#4111yescharacteristic
173C22×C173H(5,𝔽2)11yescharacteristic
174C22×C1732−1+411yescharacteristic
175C2×D173—22no—
176C2×D173—22no—
177C2×D173—22no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C4×C173—22no—
183C4×C173—22no—
184C4×C173—22no—
185C4×C173—41no—
186C4×C173—41no—
187C4×C173—41no—
188Dic173—22no—
189Dic173—22no—
190Dic173—41no—
191Dic173—41no—
192Dic173—41no—
193Dic173#11714yescomplemented
194Dic173C4×D414yes—
195C2×C173—42no—
196C2×C173C2×#11711yescharacteristic
197C2×C173C2×C4×D411yescharacteristic
198C2×C173#52011yescharacteristic
199C2×C173#54711yescharacteristic
200C2×C173#55011yescharacteristic
201C2×C173#55511yescharacteristic
202C2×C173#55611yescharacteristic
203D173—42no—
204C173#235411yescharacteristic, complemented, Sylow
205#2354—1731nomaximal, Sylow
206C2×C4×D4—1731no—
207C2×#117—1731no—
208C2×#117—1731no—
209#390—1731no—
210#390—1731no—
211#388—1731no—
212#388—1731no—
213#388—1731no—
214#388D17311yescharacteristic, complemented
215#384—1731no—
216#380—1731no—
217#380—1731no—
218C4×C4⋊C4—1731no—
219C4×C4⋊C4—1731no—
220C4×C4⋊C4—1731no—
221C22×D4—1731no—
222C23×C4—1731no—
223C23×C4—1731no—
224C4×D4—1738no—
225#117—3464no—
226#117—3464no—
227C2×C4⋊C4—1731no—
228C2×C4⋊C4—1731no—
229C2×C4⋊C4—1731no—
230C2×C4⋊C4—1731no—
231C2×C4⋊C4—1731no—
232C2×C4⋊C4—1731no—
233C2×C4⋊C4—1731no—
234C2×C4⋊C4—1731no—
235C2×C4⋊C4C2×D17311yescharacteristic
236C2×C22⋊C4—1731no—
237C2×C22⋊C4—1731no—
238C2×C22⋊C4—1731no—
239C2×C22⋊C4—1731no—
240C2×C22⋊C4—1731no—
241C2×C22⋊C4—1731no—
242C2×C22⋊C4—1731no—
243C2×C22⋊C4C2×D17311yescharacteristic
244C2×C22⋊C4C2×D17311yescharacteristic
245C2×C22⋊C4C2×D17311yescharacteristic
246C2×C42—1731no—
247C2×C42—1731no—
248C2×C42—1731no—
249C2×C42—1731no—
250C2×C42—1731no—
251C2×C42—1731no—
252C2×C42C2×D17311yescharacteristic
253#95—1731no—
254#95—1731no—
255#95—1731no—
256#95—1731no—
257#95C2×D17311yescharacteristic
258#95C2×D17311yescharacteristic
259C24—1731no—
260C24C22×D17311yescharacteristic
261C2×D4—3462no—
262C2×D4—3462no—
263C2×D4—1734no—
264C22×C4—3462no—
265C22×C4—3462no—
266C22×C4—3462no—
267C22×C4—3462no—
268C22×C4—1731no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1734no—
272C22×C4—1731no—
273C22×C4—1731no—
274C22×C4—1734no—
275C22×C4—1731no—
276C22×C4—1731no—
277C22×C4C22×D17311yescharacteristic
278C22×C4C22×D17311yescharacteristic
279C22×C4C22×D17311yescharacteristic
280C22×C4C22×D17311yescharacteristic
281C22×C4C22×D17311yescharacteristic
282C22×C4C22×D17311yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—3462no—
285C4⋊C4—3462no—
286C4⋊C4—3462no—
287C4⋊C4—3462no—
288C4⋊C4—3462no—
289C4⋊C4—1734no—
290C4⋊C4—1734no—
291C4⋊C4—6921no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—3462no—
295C22⋊C4—3462no—
296C22⋊C4—3462no—
297C22⋊C4—1734no—
298C22⋊C4—6921no—
299C22⋊C4—6921no—
300C22⋊C4—6921no—
301C22⋊C4C4×D17314yes—
302C42—22no—
303C42—3462no—
304C42—3462no—
305C42—3462no—
306C42—3462no—
307C42—3462no—
308C42—1734no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—3461no—
316C23—3461no—
317C23—3461no—
318C23—3461no—
319C23—3461no—
320C23—3461no—
321C23—1732no—
322C23C2×C4×D17312yes—
323C23C23×D17311yescharacteristic
324D4—3468no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—3468no—
341C2×C4—3461no—
342C2×C4—3461no—
343C2×C4—3461no—
344C2×C4—3461no—
345C2×C4—3468no—
346C2×C4—3461no—
347C2×C4—3461no—
348C2×C4—3461no—
349C2×C4—3461no—
350C2×C4—3461no—
351C2×C4—3461no—
352C2×C4—3461no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—1732no—
356C2×C4—1732no—
357C2×C4—1732no—
358C2×C4—1732no—
359C2×C4—1732no—
360C2×C4—1732no—
361C2×C4—1732no—
362C2×C4—1732no—
363C2×C4C2×C4×D17312yes—
364C2×C4C2×C4×D17312yes—
365C2×C4D4×D17312yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—3462no—
374C22—3462no—
375C22—3462no—
376C22—6921no—
377C22—6921no—
378C22—6921no—
379C22—6921no—
380C22C22×C4×D17311yescharacteristic
381C22C2×D4×D17311yescharacteristic
382C22#41×D17311yescharacteristic
383C22#41×D17311yescharacteristic
384C22#41×D17311yescharacteristic
385C22#1914972411yescharacteristic
386C22#1914972611yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—3462no—
394C4—3462no—
395C4—1734no—
396C4—1734no—
397C4—6921no—
398C4—6921no—
399C4—6921no—
400C2—42no—
401C2—6922no—
402C2#117×D17311yescharacteristic
403C2C4×D4×D17311yescharacteristic
404C2#2601197011yescharacteristic
405C2#2601213911yescharacteristic
406C2#2601215911yescharacteristic
407C2#2601220311yescharacteristic
408C2#2601220811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC22×C22⋊D173C22
hypercenter#388D173
hypocenterC173#2354
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#19149636C2semidirect
3. C2×#19149635C2semidirect
6. #26011384C2semidirect
7. #26011383C2semidirect
8. #26011381C2semidirect
9. #26011360C2semidirect
10. #26011350C2semidirect
11. #26011349C2semidirect
12. C4×#14583805C2semidirect
13. #26011342C2semidirect
14. #26011342C2semidirect
15. #388×C173C2semidirect
19. #19149639C4semidirect
78. #14583805D4semidirect
193. Dic173#117semidirect
204. C173#2354semidirect
214. #388D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496392
C2×#191496362
C2×#191496352
#260113912
#260113842
#260113832
#260113812
#260113602
#260113502
#260113492
C4×#145838052
#260113422
#388×C1732
#2354173