Finite Group Databasebeta

← #36356727

Subgroups — #36356727

Order 22144.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149659C211yesmaximal, characteristic
2#26011395C211yesmaximal, characteristic, complemented
3#26011395C211yesmaximal, characteristic, complemented
4#26011389C211yesmaximal, characteristic, complemented
5#26011387C211yesmaximal, characteristic, complemented
6#26011383C211yesmaximal, characteristic
7#26011382C211yesmaximal, characteristic
8#26011381C211yesmaximal, characteristic
9#26011380C211yesmaximal, characteristic
10#26011377C211yesmaximal, characteristic, complemented
11#26011376C211yesmaximal, characteristic, complemented
12#26011374C211yesmaximal, characteristic, complemented
13#26011353C211yesmaximal, characteristic
14#26011350C211yesmaximal, characteristic
15#387×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18#19149659C418yescomplemented
19C2×C4⋊Dic173C2211yescharacteristic
20C2×C4⋊Dic173C2211yescharacteristic
21C2×C4⋊Dic173C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583803C2211yescharacteristic
27C2×#14583803C2211yescharacteristic
28C2×#14583803C2211yescharacteristic
29C2×#14583803C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33C2×C4×Dic173C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35#19149586C2211yescharacteristic
36#19149586C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45C2×C4⋊C4×C173C2211yescharacteristic
46C2×C4⋊C4×C173C2211yescharacteristic
47C2×C42×C173C2211yescharacteristic
48#95×C173C2211yescharacteristic
49#95×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52C23×D173C2311yescharacteristic
53C2×Dic346—41no—
54C2×Dic346—41no—
55C2×Dic346C2×C414yes—
56C22×Dic173C2311yescharacteristic
57C22×Dic173C2311yescharacteristic
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C2×C4×D173—41no—
64C2×C4×D173—41no—
65C2×C4×D173C2×C414yes—
66C4⋊Dic173—41no—
67C4⋊Dic173C2×C414yes—
68C4⋊Dic173C2×C414yes—
69#14583805—41no—
70#14583805—41no—
71#14583805—41no—
72#14583805—41no—
73#14583803—41no—
74#14583803—41no—
75#14583803—41no—
76#14583803—41no—
77#14583803C2×C414yes—
78#14583803C2×C414yes—
79C4×Dic173—41no—
80C4×Dic173—41no—
81C4×Dic173—41no—
82C22×C4×C173C2311yescharacteristic
83C22×C4×C173C2311yescharacteristic
84C22×C4×C173C2311yescharacteristic
85C22×C4×C173C2311yescharacteristic
86C22×C4×C173C2311yescharacteristic
87C22×C4×C173C2311yescharacteristic
88C22×C4×C173C2311yescharacteristic
89C4⋊C4×C173—41no—
90C4⋊C4×C173C2×C414yes—
91C42×C173—41no—
92C22×D173—21no—
93C22×D173—21no—
94C22×D173—21no—
95C22×D173—21no—
96C22×D173—21no—
97C22×D173—21no—
98C22×D173C22×C412yes—
99Dic346—44no—
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×Dic173C22×C412yes—
119C2×Dic173C22×C412yes—
120C2×Dic173C22×C412yes—
121C4×D173—44no—
122C23×C173C2411yescharacteristic
123C2×C4×C173—21no—
124C2×C4×C173—21no—
125C2×C4×C173—21no—
126C2×C4×C173—21no—
127C2×C4×C173—21no—
128C2×C4×C173—21no—
129C2×C4×C173—21no—
130C2×C4×C173—21no—
131C2×C4×C173—21no—
132C2×C4×C173—21no—
133C2×C4×C173—21no—
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173—21no—
137C2×C4×C173—21no—
138C2×C4×C173—21no—
139C2×C4×C173—21no—
140C2×C4×C173—21no—
141C2×C4×C173C22×C412yes—
142C2×C4×C173C22×C412yes—
143C2×C4×C173C22×C412yes—
144C22×C173C23×C411yescharacteristic
145C22×C173H(5,𝔽2)11yescharacteristic
146C22×C173H(5,𝔽2)11yescharacteristic
147C22×C173H(5,𝔽2)11yescharacteristic
148C22×C1732−1+411yescharacteristic
149C22×C1732−1+411yescharacteristic
150C22×C1732−1+411yescharacteristic
151C2×D173—22no—
152C2×D173—22no—
153C2×D173—22no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C4×C173—22no—
159C4×C173—22no—
160C4×C173—22no—
161C4×C173—41no—
162C4×C173—41no—
163C4×C173—41no—
164C4×C173—41no—
165Dic173—22no—
166Dic173—22no—
167Dic173—22no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171Dic173—41no—
172C2×C173#51811yescharacteristic
173C2×C173#51911yescharacteristic
174C2×C173#52011yescharacteristic
175C2×C173#56211yescharacteristic
176C2×C173#56211yescharacteristic
177C2×C173#56311yescharacteristic
178C2×C173#56311yescharacteristic
179D173—42no—
180C173#237811yescharacteristic, complemented, Sylow
181#2378—1731nomaximal, Sylow
182C2×C22⋊Q8—1731no—
183#391—1731no—
184#391—1731no—
185#389—1731no—
186#389—1731no—
187#388—1731no—
188#388—1731no—
189#388—1731no—
190#388—1731no—
191#387—1731no—
192#387—1731no—
193#387—1731no—
194#387D17311yescharacteristic, complemented
195#381—1731no—
196#380—1731no—
197C22×Q8—1731no—
198C23×C4—1731no—
199C22⋊Q8—1738no—
200C2×C4⋊C4—1731no—
201C2×C4⋊C4—1731no—
202C2×C4⋊C4—1731no—
203C2×C4⋊C4—1731no—
204C2×C4⋊C4—1731no—
205C2×C4⋊C4—1731no—
206C2×C4⋊C4—1731no—
207C2×C4⋊C4C2×D17311yescharacteristic
208C2×C4⋊C4C2×D17311yescharacteristic
209C2×C22⋊C4—1731no—
210C2×C22⋊C4—1731no—
211C2×C22⋊C4—1731no—
212C2×C22⋊C4—1731no—
213C2×C22⋊C4—1731no—
214C2×C22⋊C4—1731no—
215C2×C42—1731no—
216C2×C42—1731no—
217C2×C42—1731no—
218C2×C42C2×D17311yescharacteristic
219#95—1731no—
220#95—1731no—
221#95—1731no—
222#95—1731no—
223#95—1731no—
224#95—1731no—
225#95—1731no—
226#95—1731no—
227#95—1731no—
228#95—1731no—
229#95C2×D17311yescharacteristic
230#95C2×D17311yescharacteristic
231#95C2×D17311yescharacteristic
232#95C2×D17311yescharacteristic
233C24—1731no—
234C2×Q8—1734no—
235C2×Q8—6921no—
236C2×Q8—6921no—
237C22×C4—1731no—
238C22×C4—1731no—
239C22×C4—1731no—
240C22×C4—1731no—
241C22×C4—1731no—
242C22×C4—1731no—
243C22×C4—1731no—
244C22×C4—1734no—
245C22×C4—6921no—
246C22×C4—6921no—
247C22×C4C22×D17311yescharacteristic
248C22×C4C22×D17311yescharacteristic
249C22×C4C22×D17311yescharacteristic
250C22×C4C22×D17311yescharacteristic
251C22×C4C22×D17311yescharacteristic
252C22×C4C22×D17311yescharacteristic
253C22×C4C22×D17311yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—1734no—
256C4⋊C4—1734no—
257C4⋊C4—6921no—
258C4⋊C4—6921no—
259C4⋊C4—6921no—
260C4⋊C4—6921no—
261C4⋊C4—6921no—
262C4⋊C4C4×D17314yes—
263C22⋊C4—1734no—
264C22⋊C4—1734no—
265C22⋊C4—6921no—
266C22⋊C4—6921no—
267C22⋊C4—6921no—
268C22⋊C4—6921no—
269C42—41no—
270C42—6921no—
271C42—6921no—
272C42—6921no—
273C23—3461no—
274C23—3461no—
275C23—3461no—
276C23—3461no—
277C23—3461no—
278C23—3461no—
279C23—1732no—
280C23C23×D17311yescharacteristic
281Q8—6924no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—3461no—
301C2×C4—3461no—
302C2×C4—3461no—
303C2×C4—3461no—
304C2×C4—3461no—
305C2×C4—3461no—
306C2×C4—3461no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—1732no—
319C2×C4—1732no—
320C2×C4—1732no—
321C2×C4—6924no—
322C2×C4C2×C4×D17312yes—
323C2×C4C2×C4×D17312yes—
324C2×C4C2×C4×D17312yes—
325C22—3462no—
326C22—3462no—
327C22—3462no—
328C22—6921no—
329C22—6921no—
330C22—6921no—
331C22—6921no—
332C22C22×C4×D17311yescharacteristic
333C22#1914972411yescharacteristic
334C22#1914972511yescharacteristic
335C22#1914972511yescharacteristic
336C22#1914972611yescharacteristic
337C22#1914972611yescharacteristic
338C22#1914972711yescharacteristic
339C4—22no—
340C4—22no—
341C4—22no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3462no—
347C4—3462no—
348C4—3462no—
349C4—6921no—
350C4—6921no—
351C4—6921no—
352C4—6921no—
353C2—6922no—
354C2#2601196511yescharacteristic
355C2#2601196711yescharacteristic
356C2#2601197011yescharacteristic
357C2#2601223711yescharacteristic
358C2#2601223811yescharacteristic
359C2#2601224311yescharacteristic
360C2#2601224511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#387×C173C2
socle, cosocleC23×C173C24
prime radicalC23×D173C23
hypercenter#387D173
hypocenterC173#2378
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. #26011395C2semidirect
3. #26011395C2semidirect
4. #26011389C2semidirect
5. #26011387C2semidirect
10. #26011377C2semidirect
11. #26011376C2semidirect
12. #26011374C2semidirect
15. #387×C173C2semidirect
18. #19149659C4semidirect
180. C173#2378semidirect
194. #387D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496592
#260113952
#260113892
#260113872
#260113832
#260113822
#260113812
#260113802
#260113772
#260113762
#260113742
#260113532
#260113502
#387×C1732
#2378173