Finite Group Databasebeta

← #36356244

Subgroups — #36356244

Order 22144.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149657C211yesmaximal, characteristic
2C2×C4×Dic346C211yesmaximal, characteristic, complemented
3C2×#19149637C211yesmaximal, characteristic
4#26011395C211yesmaximal, characteristic, complemented
5#26011390C211yesmaximal, characteristic, complemented
6#26011388C211yesmaximal, characteristic, complemented
7#26011388C211yesmaximal, characteristic, complemented
8#26011383C211yesmaximal, characteristic
9#26011383C211yesmaximal, characteristic
10#26011382C211yesmaximal, characteristic
11#26011380C211yesmaximal, characteristic
12#26011372C211yesmaximal, characteristic, complemented
13#26011350C211yesmaximal, characteristic
14#26011342C211yesmaximal, characteristic, complemented
15#387×C173C211yesmaximal, characteristic, complemented
16C22×Dic346C2211yescharacteristic
17C22×C4×D173C2211yescharacteristic
18#19149657C418yescomplemented
19C4×Dic346—24no—
20#19149637—24no—
21C2×C4⋊Dic173C2211yescharacteristic
22C2×C4⋊Dic173C2211yescharacteristic
23C2×C4⋊Dic173C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×#14583805C2211yescharacteristic
29C2×#14583805C2211yescharacteristic
30C2×#14583803C2211yescharacteristic
31C2×#14583803C2211yescharacteristic
32C2×#14583803C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×C4×Dic173C2211yescharacteristic
37C2×C4×Dic173C2211yescharacteristic
38C2×C4×Dic173C2211yescharacteristic
39C2×C4×Dic173C2211yescharacteristic
40C2×C4×Dic173C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47C2×C4⋊C4×C173C2211yescharacteristic
48C2×C4⋊C4×C173C2211yescharacteristic
49C2×C42×C173C2211yescharacteristic
50#95×C173C2211yescharacteristic
51#95×C173C2211yescharacteristic
52#95×C173C2211yescharacteristic
53#95×C173C2211yescharacteristic
54C23×D173C2311yescharacteristic
55C2×Dic346—22no—
56C2×Dic346—22no—
57C2×Dic346C2×C414yes—
58C22×Dic173C2311yescharacteristic
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C2×C4×D173—22no—
66C2×C4×D173—22no—
67C2×C4×D173C2×C414yes—
68C4⋊Dic173—22no—
69C4⋊Dic173—22no—
70C4⋊Dic173—41no—
71#14583805—22no—
72#14583805—22no—
73#14583805—22no—
74#14583805—41no—
75#14583805C2×C414yes—
76#14583805C2×C414yes—
77#14583803—22no—
78#14583803—22no—
79#14583803—41no—
80#14583803—41no—
81#14583803C2×C414yes—
82#14583803C2×C414yes—
83C4×Dic173—22no—
84C4×Dic173—22no—
85C4×Dic173—41no—
86C4×Dic173—41no—
87C4×Dic173D414yes—
88C22×C4×C173C2311yescharacteristic
89C22×C4×C173C2311yescharacteristic
90C22×C4×C173C2311yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C22×C4×C173C2311yescharacteristic
94C22×C4×C173C2311yescharacteristic
95C4⋊C4×C173—22no—
96C4⋊C4×C173C2×C414yes—
97C42×C173—22no—
98C22×D173—21no—
99C22×D173—21no—
100C22×D173—21no—
101C22×D173—21no—
102C22×D173—21no—
103C22×D173—21no—
104C22×D173C22×C412yes—
105Dic346—28no—
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—21no—
121C2×Dic173C22×C412yes—
122C2×Dic173C22×C412yes—
123C2×Dic173C22×C412yes—
124C2×Dic173C2×D412yes—
125C2×Dic173C2×D412yes—
126C2×Dic173#4112yes—
127C4×D173—28no—
128C23×C173C2411yescharacteristic
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×C173—21no—
142C2×C4×C173—21no—
143C2×C4×C173—21no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173C22×C412yes—
147C2×C4×C173C22×C412yes—
148C2×C4×C173C22×C412yes—
149C2×C4×C173C2×D412yes—
150C22×C173C23×C411yescharacteristic
151C22×C173C22×D411yescharacteristic
152C22×C173C2×#4111yescharacteristic
153C22×C173H(5,𝔽2)11yescharacteristic
154C22×C1732−1+411yescharacteristic
155C22×C1732−1+411yescharacteristic
156C22×C1732−1+411yescharacteristic
157C2×D173—22no—
158C2×D173—22no—
159C2×D173—22no—
160C2×D173—41no—
161C2×D173—41no—
162C2×D173—41no—
163C2×D173—41no—
164C4×C173—22no—
165C4×C173—22no—
166C4×C173—22no—
167C4×C173—22no—
168C4×C173—41no—
169C4×C173—41no—
170C4×C173—41no—
171Dic173—22no—
172Dic173—22no—
173Dic173—22no—
174Dic173—41no—
175Dic173—41no—
176Dic173—41no—
177Dic173C4×D414yes—
178C2×C173C2×C4×D411yescharacteristic
179C2×C173#51911yescharacteristic
180C2×C173#52011yescharacteristic
181C2×C173#53611yescharacteristic
182C2×C173#53711yescharacteristic
183C2×C173#54111yescharacteristic
184C2×C173#54211yescharacteristic
185D173—42no—
186C173#232311yescharacteristic, complemented, Sylow
187#2323—1731nomaximal, Sylow
188C2×C22⋊Q8—1731no—
189C2×C4×Q8—1731no—
190C2×#117—1731no—
191#391—1731no—
192#389—1731no—
193#389—1731no—
194#389—1731no—
195#388—1731no—
196#388—1731no—
197#388—1731no—
198#388—1731no—
199#387—1731no—
200#387D17311yescharacteristic, complemented
201#380—1731no—
202C4×C4⋊C4—1731no—
203C22×Q8—1731no—
204C23×C4—1731no—
205C22⋊Q8—1738no—
206C4×Q8—3464no—
207#117—3464no—
208C2×C4⋊C4—1731no—
209C2×C4⋊C4—1731no—
210C2×C4⋊C4—1731no—
211C2×C4⋊C4—1731no—
212C2×C4⋊C4—1731no—
213C2×C4⋊C4—1731no—
214C2×C4⋊C4—1731no—
215C2×C4⋊C4—1731no—
216C2×C4⋊C4—1731no—
217C2×C4⋊C4C2×D17311yescharacteristic
218C2×C4⋊C4C2×D17311yescharacteristic
219C2×C22⋊C4—1731no—
220C2×C22⋊C4—1731no—
221C2×C22⋊C4—1731no—
222C2×C22⋊C4—1731no—
223C2×C22⋊C4—1731no—
224C2×C22⋊C4—1731no—
225C2×C42—1731no—
226C2×C42—1731no—
227C2×C42—1731no—
228C2×C42—1731no—
229C2×C42—1731no—
230C2×C42C2×D17311yescharacteristic
231#95—1731no—
232#95—1731no—
233#95—1731no—
234#95—1731no—
235#95—1731no—
236#95—1731no—
237#95C2×D17311yescharacteristic
238#95C2×D17311yescharacteristic
239#95C2×D17311yescharacteristic
240#95C2×D17311yescharacteristic
241C24—1731no—
242C2×Q8—3462no—
243C2×Q8—3462no—
244C2×Q8—1734no—
245C22×C4—3462no—
246C22×C4—3462no—
247C22×C4—1731no—
248C22×C4—1731no—
249C22×C4—1731no—
250C22×C4—1731no—
251C22×C4—1731no—
252C22×C4—1731no—
253C22×C4—1734no—
254C22×C4—1731no—
255C22×C4C22×D17311yescharacteristic
256C22×C4C22×D17311yescharacteristic
257C22×C4C22×D17311yescharacteristic
258C22×C4C22×D17311yescharacteristic
259C22×C4C22×D17311yescharacteristic
260C22×C4C22×D17311yescharacteristic
261C22×C4C22×D17311yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—3462no—
264C4⋊C4—3462no—
265C4⋊C4—3462no—
266C4⋊C4—3462no—
267C4⋊C4—3462no—
268C4⋊C4—1734no—
269C4⋊C4—1734no—
270C4⋊C4—6921no—
271C4⋊C4—6921no—
272C4⋊C4C4×D17314yes—
273C22⋊C4—3462no—
274C22⋊C4—3462no—
275C22⋊C4—1734no—
276C22⋊C4—1734no—
277C22⋊C4—6921no—
278C22⋊C4—6921no—
279C42—22no—
280C42—3462no—
281C42—3462no—
282C42—1734no—
283C42—6921no—
284C42—6921no—
285C23—3461no—
286C23—3461no—
287C23—3461no—
288C23—3461no—
289C23—3461no—
290C23—3461no—
291C23—1732no—
292C23C23×D17311yescharacteristic
293Q8—3468no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—3468no—
312C2×C4—3461no—
313C2×C4—3461no—
314C2×C4—3461no—
315C2×C4—3461no—
316C2×C4—3461no—
317C2×C4—3461no—
318C2×C4—3461no—
319C2×C4—3461no—
320C2×C4—3461no—
321C2×C4—3461no—
322C2×C4—3461no—
323C2×C4—3461no—
324C2×C4—3461no—
325C2×C4—3461no—
326C2×C4—3461no—
327C2×C4—1732no—
328C2×C4—1732no—
329C2×C4—1732no—
330C2×C4—1732no—
331C2×C4—1732no—
332C2×C4—1732no—
333C2×C4C2×C4×D17312yes—
334C2×C4C2×C4×D17312yes—
335C2×C4C2×C4×D17312yes—
336C2×C4D4×D17312yes—
337C22—3462no—
338C22—3462no—
339C22—3462no—
340C22—6921no—
341C22—6921no—
342C22—6921no—
343C22—6921no—
344C22C22×C4×D17311yescharacteristic
345C22C2×D4×D17311yescharacteristic
346C22#41×D17311yescharacteristic
347C22#1914972511yescharacteristic
348C22#1914972611yescharacteristic
349C22#1914972611yescharacteristic
350C22#1914972711yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—3462no—
359C4—3462no—
360C4—3462no—
361C4—1734no—
362C4—6921no—
363C4—6921no—
364C4—6921no—
365C2—6922no—
366C2C4×D4×D17311yescharacteristic
367C2#2601196711yescharacteristic
368C2#2601197111yescharacteristic
369C2#2601206311yescharacteristic
370C2#2601207111yescharacteristic
371C2#2601210011yescharacteristic
372C2#2601210711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutatorC22×C173C23×C4
Fitting#387×C173C2
socle, cosocleC23×C173C24
prime radicalC23×D173C23
hypercenter#387D173
hypocenterC173#2323
LCSG ▷ C22×C173 ▷ C173
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic346C2semidirect
4. #26011395C2semidirect
5. #26011390C2semidirect
6. #26011388C2semidirect
7. #26011388C2semidirect
12. #26011372C2semidirect
14. #26011342C2semidirect
15. #387×C173C2semidirect
18. #19149657C4semidirect
186. C173#2323semidirect
200. #387D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496572
C2×C4×Dic3462
C2×#191496372
#260113952
#260113902
#260113882
#260113832
#260113822
#260113802
#260113722
#260113502
#260113422
#387×C1732
#2323173