Finite Group Databasebeta

← #36356716

Subgroups — #36356716

Order 22144.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149652C211yesmaximal, characteristic
2#26011393C211yesmaximal, characteristic, complemented
3#26011391C211yesmaximal, characteristic, complemented
4#26011383C211yesmaximal, characteristic, complemented
5#26011378C211yesmaximal, characteristic, complemented
6#26011375C211yesmaximal, characteristic, complemented
7#26011369C211yesmaximal, characteristic, complemented
8#26011368C211yesmaximal, characteristic, complemented
9#26011368C211yesmaximal, characteristic, complemented
10#26011367C211yesmaximal, characteristic, complemented
11#26011366C211yesmaximal, characteristic, complemented
12#26011362C211yesmaximal, characteristic, complemented
13#26011353C211yesmaximal, characteristic, complemented
14#26011348C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×C22⋊D173C2211yescharacteristic
18C22×C22⋊D173C2211yescharacteristic
19C23×Dic173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22#19149652C418yescomplemented
23C2×C4⋊Dic173C2211yescharacteristic
24C2×C4⋊Dic173C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×C4×Dic173C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45C2×C4⋊C4×C173C2211yescharacteristic
46C2×C22⋊C4×C173C2211yescharacteristic
47C2×C22⋊C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C42×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52C23×D173C2311yescharacteristic
53C23×D173C2311yescharacteristic
54C2×D692—41no—
55C2×D692—41no—
56C2×D692C2×C414yescomplemented
57C2×C22⋊D173—41no—
58C2×C22⋊D173—41no—
59C2×C22⋊D173—41no—
60C2×C22⋊D173—41no—
61C2×C22⋊D173C2×C414yes—
62C2×C22⋊D173C2×C414yescomplemented
63C22×Dic173—41no—
64C22×Dic173—41no—
65C22×Dic173—41no—
66C22×Dic173C2311yescharacteristic
67C22×Dic173C2311yescharacteristic
68C22×Dic173C2311yescharacteristic
69C22×Dic173C2311yescharacteristic
70C22×Dic173C2311yescharacteristic
71C22×Dic173C2311yescharacteristic
72C2×C4×D173—41no—
73C2×C4×D173—41no—
74C2×C4×D173—41no—
75C2×C4×D173—41no—
76C2×C4×D173—41no—
77C2×C4×D173C2×C414yescomplemented
78C4⋊Dic173—41no—
79C4⋊Dic173—41no—
80#14583805—41no—
81#14583805C2×C414yescomplemented
82C22⋊Dic173—41no—
83#14583803—41no—
84#14583803—41no—
85#14583803—41no—
86#14583803—41no—
87#14583803—41no—
88#14583803—41no—
89#14583803—41no—
90#14583803C2×C414yescomplemented
91C4×Dic173—41no—
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—41no—
101C22⋊C4×C173—41no—
102C22⋊C4×C173C2×C414yescomplemented
103C42×C173—41no—
104C22×D173—21no—
105C22×D173—21no—
106C22×D173—21no—
107C22×D173—21no—
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—21no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173C22×C412yes—
117C22×D173C22×C412yes—
118D692—44no—
119C22⋊D173—44no—
120C22⋊D173—44no—
121C2×Dic173—21no—
122C2×Dic173—21no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C2×Dic173—44no—
138C2×Dic173C22×C412yes—
139C2×Dic173C22×C412yes—
140C4×D173—44no—
141C4×D173—44no—
142C23×C173—21no—
143C23×C173—21no—
144C23×C173—21no—
145C23×C173—21no—
146C23×C173—21no—
147C23×C173—21no—
148C23×C173C22×C412yes—
149C23×C173C2411yescharacteristic
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×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173C22×C412yes—
167C2×C4×C173C22×C412yes—
168C22×C173—22no—
169C22×C173—22no—
170C22×C173—22no—
171C22×C173—41no—
172C22×C173—41no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173C23×C411yescharacteristic
176C22×C173H(5,𝔽2)11yescharacteristic
177C22×C173H(5,𝔽2)11yescharacteristic
178C22×C173H(5,𝔽2)11yescharacteristic
179C22×C173H(5,𝔽2)11yescharacteristic
180C22×C173H(5,𝔽2)11yescharacteristic
181C22×C1732−1+411yescharacteristic
182C2×D173—22no—
183C2×D173—22no—
184C2×D173—22no—
185C2×D173—22no—
186C2×D173—22no—
187C2×D173—22no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—41no—
195C2×D173—41no—
196C4×C173—22no—
197C4×C173—22no—
198C4×C173—41no—
199C4×C173—41no—
200C4×C173—41no—
201C4×C173—41no—
202Dic173—22no—
203Dic173—22no—
204Dic173—41no—
205Dic173—41no—
206Dic173—41no—
207Dic173—41no—
208C2×C173—42no—
209C2×C173#51811yescharacteristic
210C2×C173#51811yescharacteristic
211C2×C173#52011yescharacteristic
212C2×C173#56011yescharacteristic
213C2×C173#56011yescharacteristic
214C2×C173#56211yescharacteristic
215C2×C173#56211yescharacteristic
216D173—42no—
217D173—42no—
218C173#237711yescharacteristic, complemented, Sylow
219#2377—1731nomaximal, Sylow
220C2×#121—1731no—
221#390—1731no—
222#390—1731no—
223#388—1731no—
224#388D17311yescharacteristic, complemented
225#387—1731no—
226#387—1731no—
227#386—1731no—
228#386—1731no—
229#386—1731no—
230#386—1731no—
231#385—1731no—
232#385—1731no—
233#381—1731no—
234#380—1731no—
235C22×D4—1731no—
236C22×D4—1731no—
237C22×D4—1731no—
238C23×C4—1731no—
239C23×C4—1731no—
240C23×C4—1731no—
241#121—1738no—
242C2×C4⋊C4—1731no—
243C2×C4⋊C4—1731no—
244C2×C4⋊C4—1731no—
245C2×C4⋊C4—1731no—
246C2×C4⋊C4C2×D17311yescharacteristic
247C2×C22⋊C4—1731no—
248C2×C22⋊C4—1731no—
249C2×C22⋊C4—1731no—
250C2×C22⋊C4—1731no—
251C2×C22⋊C4—1731no—
252C2×C22⋊C4—1731no—
253C2×C22⋊C4—1731no—
254C2×C22⋊C4—1731no—
255C2×C22⋊C4—1731no—
256C2×C22⋊C4C2×D17311yescharacteristic
257C2×C22⋊C4C2×D17311yescharacteristic
258C2×C22⋊C4C2×D17311yescharacteristic
259C2×C42—1731no—
260C2×C42C2×D17311yescharacteristic
261#95—1731no—
262#95—1731no—
263#95—1731no—
264#95—1731no—
265#95—1731no—
266#95—1731no—
267#95—1731no—
268#95—1731no—
269#95C2×D17311yescharacteristic
270#95C2×D17311yescharacteristic
271C24—1731no—
272C24—1731no—
273C24C22×D17311yescharacteristic
274C2×D4—1734no—
275C2×D4—1734no—
276C2×D4—1734no—
277C2×D4—6921no—
278C2×D4—6921no—
279C2×D4—6921no—
280C2×D4—6921no—
281C2×D4—6921no—
282C2×D4—6921no—
283C22×C4—1731no—
284C22×C4—1731no—
285C22×C4—1731no—
286C22×C4—1731no—
287C22×C4—1731no—
288C22×C4—1731no—
289C22×C4—1734no—
290C22×C4—6921no—
291C22×C4—6921no—
292C22×C4—6921no—
293C22×C4—6921no—
294C22×C4—6921no—
295C22×C4—6921no—
296C22×C4—6921no—
297C22×C4—6921no—
298C22×C4C22×D17311yescharacteristic
299C22×C4C22×D17311yescharacteristic
300C22×C4C22×D17311yescharacteristic
301C22×C4C22×D17311yescharacteristic
302C22×C4C22×D17311yescharacteristic
303C22×C4C22×D17311yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—1734no—
306C4⋊C4—6921no—
307C4⋊C4—6921no—
308C4⋊C4—6921no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—1734no—
312C22⋊C4—6921no—
313C22⋊C4—6921no—
314C22⋊C4—6921no—
315C22⋊C4—6921no—
316C22⋊C4—6921no—
317C22⋊C4—6921no—
318C22⋊C4—6921no—
319C22⋊C4—6921no—
320C22⋊C4C4×D17314yescomplemented
321C42—41no—
322C42—6921no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—3461no—
330C23—3461no—
331C23—3461no—
332C23—3461no—
333C23—3461no—
334C23—3461no—
335C23—3461no—
336C23—3461no—
337C23—3461no—
338C23—3461no—
339C23—3461no—
340C23—3461no—
341C23—1732no—
342C23—1732no—
343C23C2×C4×D17312yes—
344C23C23×D17311yescharacteristic
345D4—6924no—
346D4—6924no—
347D4—6924no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—3461no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—3461no—
370C2×C4—3461no—
371C2×C4—3461no—
372C2×C4—3461no—
373C2×C4—3461no—
374C2×C4—3461no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—3461no—
378C2×C4—3461no—
379C2×C4—3461no—
380C2×C4—1732no—
381C2×C4—1732no—
382C2×C4—6924no—
383C2×C4—6924no—
384C2×C4—6924no—
385C2×C4C2×C4×D17312yes—
386C2×C4C2×C4×D17312yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—3462no—
395C22—3462no—
396C22—3462no—
397C22—3462no—
398C22—3462no—
399C22—3462no—
400C22—6921no—
401C22—6921no—
402C22—6921no—
403C22—6921no—
404C22—6921no—
405C22—6921no—
406C22—6921no—
407C22—6921no—
408C22C22×C4×D17311yescharacteristic
409C22#1914972411yescharacteristic
410C22#1914972411yescharacteristic
411C22#1914972511yescharacteristic
412C22#1914972511yescharacteristic
413C22#1914972511yescharacteristic
414C22#1914972611yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—3462no—
422C4—3462no—
423C4—6921no—
424C4—6921no—
425C4—6921no—
426C4—6921no—
427C2—42no—
428C2—6922no—
429C2—6922no—
430C2#2601196311yescharacteristic
431C2#2601196311yescharacteristic
432C2#2601197111yescharacteristic
433C2#2601222911yescharacteristic
434C2#2601223211yescharacteristic
435C2#2601223511yescharacteristic
436C2#2601223811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#388×C173C2
socle, cosocleC23×C173C24
prime radicalC2×#19149652C2
hypercenter#388D173
hypocenterC173#2377
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #26011393C2semidirect
3. #26011391C2semidirect
4. #26011383C2semidirect
5. #26011378C2semidirect
6. #26011375C2semidirect
7. #26011369C2semidirect
8. #26011368C2semidirect
9. #26011368C2semidirect
10. #26011367C2semidirect
11. #26011366C2semidirect
12. #26011362C2semidirect
13. #26011353C2semidirect
14. #26011348C2semidirect
15. #388×C173C2semidirect
22. #19149652C4semidirect
56. C2×D692C2×C4semidirect
62. C2×C22⋊D173C2×C4semidirect
77. C2×C4×D173C2×C4semidirect
81. #14583805C2×C4semidirect
90. #14583803C2×C4semidirect
102. C22⋊C4×C173C2×C4semidirect
218. C173#2377semidirect
224. #388D173semidirect
320. C22⋊C4C4×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496522
#260113932
#260113912
#260113832
#260113782
#260113752
#260113692
#260113682
#260113672
#260113662
#260113622
#260113532
#260113482
#388×C1732
#2377173