Finite Group Databasebeta

← #36356819

Subgroups — #36356819

Order 22144.

560 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic173C211yesmaximal, characteristic
2C2×#19149635C211yesmaximal, characteristic, complemented
3#26011433C211yesmaximal, characteristic, complemented
4#26011427C211yesmaximal, characteristic, complemented
5#26011424C211yesmaximal, characteristic, complemented
6#26011419C211yesmaximal, characteristic, complemented
7#26011419C211yesmaximal, characteristic, complemented
8#26011416C211yesmaximal, characteristic, complemented
9#26011405C211yesmaximal, characteristic
10C4⋊C4×Dic173C211yesmaximal, characteristic, complemented
11#26011342C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
14C22⋊C4×Dic173C211yesmaximal, characteristic, complemented
15#394×C173C211yesmaximal, characteristic
16C23×Dic173C2211yescharacteristic
17C23×Dic173C2211yescharacteristic
18D4×Dic173—24no—
19#19149635—24no—
20C2×C4⋊Dic173C2211yescharacteristic
21C2×C4⋊Dic173C2211yescharacteristic
22C2×#14583805C2211yescharacteristic
23C2×#14583805C2211yescharacteristic
24C2×#14583805C2211yescharacteristic
25C2×#14583805C2211yescharacteristic
26C2×C22⋊Dic173C2211yescharacteristic
27C2×C22⋊Dic173C2211yescharacteristic
28C2×C22⋊Dic173C2211yescharacteristic
29C2×C22⋊Dic173C2211yescharacteristic
30C2×C22⋊Dic173C2211yescharacteristic
31C2×C22⋊Dic173C2211yescharacteristic
32C2×C4×Dic173C2211yescharacteristic
33C2×C4×Dic173C2211yescharacteristic
34C2×C4×Dic173C2211yescharacteristic
35C2×C4×Dic173C2211yescharacteristic
36C2×C4×Dic173C2211yescharacteristic
37#19149586C2211yescharacteristic
38#19149586C2211yescharacteristic
39#19149586C2211yescharacteristic
40#19149586C2211yescharacteristic
41#19149586C2211yescharacteristic
42#19149586C2211yescharacteristic
43#19149586C2211yescharacteristic
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46C22×D4×C173C2211yescharacteristic
47C2×C4⋊C4×C173C2211yescharacteristic
48C2×C22⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52#95×C173C2211yescharacteristic
53C22×Dic173—22no—
54C22×Dic173—22no—
55C22×Dic173—22no—
56C22×Dic173—22no—
57C22×Dic173—22no—
58C22×Dic173D414yes—
59C22×Dic173C2311yescharacteristic
60C22×Dic173C2311yescharacteristic
61C22×Dic173C2311yescharacteristic
62C22×Dic173C2311yescharacteristic
63C22×Dic173C2311yescharacteristic
64C22×Dic173C2311yescharacteristic
65C22×Dic173C2311yescharacteristic
66C22×Dic173C2311yescharacteristic
67C4⋊Dic173—22no—
68C4⋊Dic173—22no—
69#14583805—22no—
70#14583805—22no—
71#14583805—22no—
72#14583805—22no—
73C22⋊Dic173—22no—
74C22⋊Dic173—22no—
75C22⋊Dic173—22no—
76C22⋊Dic173—22no—
77C22⋊Dic173—22no—
78C22⋊Dic173—22no—
79C4×Dic173—22no—
80C4×Dic173—22no—
81C4×Dic173—22no—
82C4×Dic173—22no—
83C4×Dic173—22no—
84C24×C173C2311yescharacteristic
85C24×C173C2311yescharacteristic
86C2×D4×C173—22no—
87C2×D4×C173—22no—
88C2×D4×C173—22no—
89C22×C4×C173C2311yescharacteristic
90C22×C4×C173C2311yescharacteristic
91C22×C4×C173C2311yescharacteristic
92C22×C4×C173C2311yescharacteristic
93C22×C4×C173C2311yescharacteristic
94C4⋊C4×C173—22no—
95C22⋊C4×C173—22no—
96C22⋊C4×C173—22no—
97C22⋊C4×C173—22no—
98C22⋊C4×C173—22no—
99C2×Dic173—21no—
100C2×Dic173—21no—
101C2×Dic173—21no—
102C2×Dic173—21no—
103C2×Dic173—28no—
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—44no—
120C2×Dic173C2×D412yes—
121C2×Dic173C2×D412yes—
122C2×Dic173#4112yes—
123C2×Dic173#4112yes—
124C2×Dic173#4112yes—
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×C173—21no—
136C23×C173—21no—
137C23×C173C2×D412yes—
138C23×C173#4112yes—
139C23×C173C2411yescharacteristic
140D4×C173—28no—
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—21no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173#4112yes—
155C2×C4×C173#4112yes—
156C22×C173—22no—
157C22×C173—22no—
158C22×C173—22no—
159C22×C173—22no—
160C22×C173—22no—
161C22×C173—22no—
162C22×C173—41no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173—41no—
169C22×C173—41no—
170C22×C173C22×D411yescharacteristic
171C22×C173C2×#4111yescharacteristic
172C22×C173C2×#4111yescharacteristic
173C22×C173C2×#4111yescharacteristic
174C22×C173C2×#4111yescharacteristic
175C22×C173C2×#4111yescharacteristic
176C22×C173C2×#4111yescharacteristic
177C4×C173—22no—
178C4×C173—22no—
179C4×C173—41no—
180C4×C173—41no—
181C4×C173—41no—
182Dic173—22no—
183Dic173—22no—
184Dic173—41no—
185Dic173—41no—
186Dic173—41no—
187Dic173—41no—
188Dic173—41no—
189Dic173#12314yes—
190C2×C173—42no—
191C2×C173—42no—
192C2×C173C2×#12311yescharacteristic
193C2×C173#52511yescharacteristic
194C2×C173#52911yescharacteristic
195C2×C173#52911yescharacteristic
196C2×C173#52911yescharacteristic
197C2×C173#52911yescharacteristic
198C2×C173#53211yescharacteristic
199C173#239511yescharacteristic, complemented, Sylow
200#2395—1731nomaximal, Sylow
201C2×C4×D4—1731no—
202C2×#117—1731no—
203#400—1731no—
204#399—1731no—
205#398—1731no—
206#397—1731no—
207#397—1731no—
208#397—1731no—
209#394—1731no—
210#394D17311yescharacteristic
211C4×C4⋊C4—1731no—
212C4×C4⋊C4—1731no—
213C4×C22⋊C4—1731no—
214C4×C22⋊C4—1731no—
215C4×C22⋊C4—1731no—
216C22×D4C2×D17311yescharacteristic
217C23×C4—1731no—
218C23×C4—1731no—
219C4×D4—3464no—
220#117—3464no—
221C2×C4⋊C4—1731no—
222C2×C4⋊C4—1731no—
223C2×C4⋊C4—1731no—
224C2×C4⋊C4—1731no—
225C2×C4⋊C4—1731no—
226C2×C4⋊C4—1731no—
227C2×C4⋊C4C2×D17311yescharacteristic
228C2×C22⋊C4—1731no—
229C2×C22⋊C4—1731no—
230C2×C22⋊C4—1731no—
231C2×C22⋊C4—1731no—
232C2×C22⋊C4—1731no—
233C2×C22⋊C4—1731no—
234C2×C22⋊C4C2×D17311yescharacteristic
235C2×C22⋊C4C2×D17311yescharacteristic
236C2×C22⋊C4C2×D17311yescharacteristic
237C2×C22⋊C4C2×D17311yescharacteristic
238C2×C42—1731no—
239C2×C42—1731no—
240C2×C42—1731no—
241C2×C42—1731no—
242C2×C42—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—
251#95—1731no—
252#95C2×D17311yescharacteristic
253C24C22×D17311yescharacteristic
254C24C22×D17311yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4—22no—
258C22×C4—3462no—
259C22×C4—3462no—
260C22×C4—3462no—
261C22×C4—3462no—
262C22×C4—3462no—
263C22×C4—1731no—
264C22×C4—1731no—
265C22×C4—1731no—
266C22×C4—1731no—
267C22×C4—1731no—
268C22×C4—1734no—
269C22×C4—1731no—
270C22×C4—1731no—
271C22×C4—1731no—
272C22×C4C22×D17311yescharacteristic
273C22×C4C22×D17311yescharacteristic
274C22×C4C22×D17311yescharacteristic
275C22×C4C22×D17311yescharacteristic
276C22×C4C22×D17311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—3462no—
279C4⋊C4—3462no—
280C4⋊C4—3462no—
281C4⋊C4—3462no—
282C4⋊C4—3462no—
283C4⋊C4—3462no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—3462no—
289C22⋊C4—3462no—
290C22⋊C4—3462no—
291C22⋊C4—3462no—
292C22⋊C4—3462no—
293C22⋊C4—3462no—
294C42—3462no—
295C42—3462no—
296C42—3462no—
297C42—3462no—
298C42—3462no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23D4×D17312yes—
312C23#1458382512yes—
313C23C23×D17311yescharacteristic
314D4—28no—
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—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—3461no—
329C2×C4—3461no—
330C2×C4—3461no—
331C2×C4—3461no—
332C2×C4—3468no—
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—3461no—
347C2×C4—3461no—
348C2×C4—1732no—
349C2×C4—1732no—
350C2×C4—1732no—
351C2×C4—1732no—
352C2×C4—1732no—
353C2×C4—6924no—
354C2×C4#1458382512yes—
355C2×C4#1458382512yes—
356C22—22no—
357C22—22no—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22C2×D4×D17311yescharacteristic
371C22#41×D17311yescharacteristic
372C22#41×D17311yescharacteristic
373C22#41×D17311yescharacteristic
374C22C2×#1458382511yescharacteristic
375C22C2×#1458382511yescharacteristic
376C22C2×#1458382511yescharacteristic
377C4—22no—
378C4—22no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—3462no—
383C4—3462no—
384C4—1734no—
385C4—6921no—
386C4—6921no—
387C4—6921no—
388C4—6921no—
389C4—6921no—
390C2—42no—
391C2—42no—
392C2#123×D17311yescharacteristic
393C2#2601199711yescharacteristic
394C2#2601201811yescharacteristic
395C2#2601202011yescharacteristic
396C2#2601202211yescharacteristic
397C2#2601202211yescharacteristic
398C2#2601203611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#394×C173C2
prime radicalC22×D4×C173C22
hypercenter#394D173
hypocenterC173#2395
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#19149635C2semidirect
3. #26011433C2semidirect
4. #26011427C2semidirect
5. #26011424C2semidirect
6. #26011419C2semidirect
7. #26011419C2semidirect
8. #26011416C2semidirect
10. C4⋊C4×Dic173C2semidirect
11. #26011342C2semidirect
12. C22⋊C4×Dic173C2semidirect
13. C22⋊C4×Dic173C2semidirect
14. C22⋊C4×Dic173C2semidirect
199. C173#2395semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1732
C2×#191496352
#260114332
#260114272
#260114242
#260114192
#260114162
#260114052
C4⋊C4×Dic1732
#260113422
C22⋊C4×Dic1732
#394×C1732
#2395173