Finite Group Databasebeta

← #36361890

Subgroups — #36361890

Order 22144.

688 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149664C211yesmaximal, characteristic, complemented
2C2×#19149661C211yesmaximal, characteristic, complemented
3C2×#19149652C211yesmaximal, characteristic
4C2×C42×D173C211yesmaximal, characteristic, complemented
5#26011391C211yesmaximal, characteristic, complemented
6#26011391C211yesmaximal, characteristic, complemented
7#26011388C211yesmaximal, characteristic, complemented
8#26011386C211yesmaximal, characteristic, complemented
9#26011383C211yesmaximal, characteristic, complemented
10#26011382C211yesmaximal, characteristic, complemented
11#26011379C211yesmaximal, characteristic, complemented
12#26011369C211yesmaximal, characteristic, complemented
13#26011367C211yesmaximal, characteristic, complemented
14#26011350C211yesmaximal, characteristic, complemented
15#388×C173C211yesmaximal, characteristic, complemented
16C22×D692C2211yescharacteristic
17C22×C22⋊D173C2211yescharacteristic
18C22×C22⋊D173C2211yescharacteristic
19C22×C4×D173C2211yescharacteristic
20C22×C4×D173C2211yescharacteristic
21C22×C4×D173C2211yescharacteristic
22#19149664—24no—
23#19149661—24no—
24#19149652—24no—
25C2×C4⋊Dic173C2211yescharacteristic
26C2×#14583805C2211yescharacteristic
27C2×#14583805C2211yescharacteristic
28C2×#14583805C2211yescharacteristic
29C2×#14583805C2211yescharacteristic
30C2×#14583805C2211yescharacteristic
31C2×C22⋊Dic173C2211yescharacteristic
32C2×C22⋊Dic173C2211yescharacteristic
33C2×#14583803C2211yescharacteristic
34C2×#14583803C2211yescharacteristic
35C2×#14583803C2211yescharacteristic
36C2×#14583803C2211yescharacteristic
37C2×#14583803C2211yescharacteristic
38C2×#14583803C2211yescharacteristic
39C2×#14583803C2211yescharacteristic
40C2×C4×Dic173C2211yescharacteristic
41C2×C4×Dic173C2211yescharacteristic
42C2×C4×Dic173C2211yescharacteristic
43C42×D173—24no—
44#19149586C2211yescharacteristic
45#19149586C2211yescharacteristic
46#19149586C2211yescharacteristic
47#19149586C2211yescharacteristic
48C2×C4⋊C4×C173C2211yescharacteristic
49C2×C22⋊C4×C173C2211yescharacteristic
50C2×C22⋊C4×C173C2211yescharacteristic
51C2×C22⋊C4×C173C2211yescharacteristic
52C2×C42×C173C2211yescharacteristic
53#95×C173C2211yescharacteristic
54#95×C173C2211yescharacteristic
55C23×D173C2311yescharacteristic
56C23×D173C2311yescharacteristic
57C2×D692—22no—
58C2×D692—22no—
59C2×D692—22no—
60C2×C22⋊D173—22no—
61C2×C22⋊D173—22no—
62C2×C22⋊D173—22no—
63C2×C22⋊D173—22no—
64C2×C22⋊D173—22no—
65C2×C22⋊D173—22no—
66C22×Dic173C2311yescharacteristic
67C22×Dic173C2311yescharacteristic
68C22×Dic173C2311yescharacteristic
69C22×Dic173C2311yescharacteristic
70C22×Dic173C2311yescharacteristic
71C22×Dic173C2311yescharacteristic
72C2×C4×D173—22no—
73C2×C4×D173—22no—
74C2×C4×D173—22no—
75C2×C4×D173—22no—
76C2×C4×D173—22no—
77C2×C4×D173—22no—
78C2×C4×D173—22no—
79C2×C4×D173D414yes—
80C2×C4×D173D414yescomplemented
81C4⋊Dic173—22no—
82#14583805—22no—
83#14583805—22no—
84#14583805—22no—
85#14583805—22no—
86#14583805—22no—
87C22⋊Dic173—22no—
88C22⋊Dic173—22no—
89#14583803—22no—
90#14583803—22no—
91#14583803—22no—
92#14583803—22no—
93#14583803—22no—
94#14583803—22no—
95#14583803—22no—
96C4×Dic173—22no—
97C4×Dic173—22no—
98C4×Dic173—22no—
99C24×C173C2311yescharacteristic
100C22×C4×C173C2311yescharacteristic
101C22×C4×C173C2311yescharacteristic
102C22×C4×C173C2311yescharacteristic
103C22×C4×C173C2311yescharacteristic
104C22×C4×C173C2311yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C4⋊C4×C173—22no—
107C22⋊C4×C173—22no—
108C22⋊C4×C173—22no—
109C22⋊C4×C173—22no—
110C42×C173—22no—
111C22×D173—21no—
112C22×D173—21no—
113C22×D173—21no—
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—21no—
118C22×D173—21no—
119C22×D173—21no—
120C22×D173—21no—
121C22×D173—21no—
122C22×D173C2×D412yes—
123C22×D173C2×D412yes—
124C22×D173#4112yes—
125D692—44no—
126C22⋊D173—44no—
127C22⋊D173—44no—
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—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173C2×D412yes—
142C2×Dic173C2×D412yes—
143C2×Dic173#4112yes—
144C2×Dic173#4112yes—
145C2×Dic173#4112yes—
146C4×D173—28no—
147C4×D173—28no—
148C4×D173—28no—
149C23×C173—21no—
150C23×C173—21no—
151C23×C173—21no—
152C23×C173—21no—
153C23×C173—21no—
154C23×C173—21no—
155C23×C173—21no—
156C23×C173C2411yescharacteristic
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×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—21no—
170C2×C4×C173—21no—
171C2×C4×C173—21no—
172C2×C4×C173C2×D412yes—
173C2×C4×C173C2×D412yes—
174C2×C4×C173#4112yes—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173—41no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173C22×D411yescharacteristic
183C22×C173C22×D411yescharacteristic
184C22×C173C2×#4111yescharacteristic
185C22×C173C2×#4111yescharacteristic
186C22×C173C2×#4111yescharacteristic
187C22×C173C2×#4111yescharacteristic
188C22×C173C2×#4111yescharacteristic
189C2×D173—22no—
190C2×D173—22no—
191C2×D173—22no—
192C2×D173—22no—
193C2×D173—22no—
194C2×D173—22no—
195C2×D173—41no—
196C2×D173—41no—
197C2×D173—41no—
198C2×D173—41no—
199C2×D173—41no—
200C2×D173—41no—
201C2×D173—41no—
202C2×D173#12114yes—
203C4×C173—22no—
204C4×C173—22no—
205C4×C173—22no—
206C4×C173—41no—
207C4×C173—41no—
208C4×C173—41no—
209Dic173—22no—
210Dic173—22no—
211Dic173—41no—
212Dic173—41no—
213Dic173—41no—
214Dic173#12314yescomplemented
215C2×C173—81no—
216C2×C173C2×#12111yescharacteristic
217C2×C173C2×#12311yescharacteristic
218C2×C173#52511yescharacteristic
219C2×C173#52911yescharacteristic
220C2×C173#52911yescharacteristic
221C2×C173#53211yescharacteristic
222C2×C173#53211yescharacteristic
223D173—24no—
224D173—81no—
225C173#285411yescharacteristic, complemented, Sylow
226#2854—1731nomaximal, Sylow
227C2×#123—1731no—
228C2×#123—1731no—
229C2×#121—1731no—
230C22×C42—1731no—
231#390—1731no—
232#390—1731no—
233#389—1731no—
234#389—1731no—
235#388—1731no—
236#388—1731no—
237#388—1731no—
238#388D17311yescharacteristic, complemented
239#386—1731no—
240#386—1731no—
241#380—1731no—
242C22×D4—1731no—
243C22×D4—1731no—
244C22×D4—1731no—
245C23×C4—1731no—
246C23×C4—1731no—
247C23×C4—1731no—
248#123—3464no—
249#123—3464no—
250#121—3464no—
251C2×C4⋊C4—1731no—
252C2×C4⋊C4—1731no—
253C2×C4⋊C4—1731no—
254C2×C4⋊C4—1731no—
255C2×C4⋊C4—1731no—
256C2×C4⋊C4—1731no—
257C2×C4⋊C4C2×D17311yescharacteristic
258C2×C22⋊C4—1731no—
259C2×C22⋊C4—1731no—
260C2×C22⋊C4—1731no—
261C2×C22⋊C4—1731no—
262C2×C22⋊C4—1731no—
263C2×C22⋊C4—1731no—
264C2×C22⋊C4—1731no—
265C2×C22⋊C4—1731no—
266C2×C22⋊C4—1731no—
267C2×C22⋊C4C2×D17311yescharacteristic
268C2×C22⋊C4C2×D17311yescharacteristic
269C2×C22⋊C4C2×D17311yescharacteristic
270C2×C42—3464no—
271C2×C42—1731no—
272C2×C42—1731no—
273C2×C42—1731no—
274C2×C42C2×D17311yescharacteristic
275#95—1731no—
276#95—1731no—
277#95—1731no—
278#95—1731no—
279#95C2×D17311yescharacteristic
280#95C2×D17311yescharacteristic
281C24—1731no—
282C24—1731no—
283C24C22×D17311yescharacteristic
284C2×D4—3462no—
285C2×D4—3462no—
286C2×D4—3462no—
287C2×D4—3462no—
288C2×D4—3462no—
289C2×D4—3462no—
290C2×D4—3462no—
291C2×D4—3462no—
292C2×D4—3462no—
293C22×C4—3462no—
294C22×C4—3462no—
295C22×C4—3462no—
296C22×C4—3462no—
297C22×C4—3462no—
298C22×C4—3462no—
299C22×C4—3462no—
300C22×C4—1731no—
301C22×C4—1731no—
302C22×C4—1731no—
303C22×C4—1731no—
304C22×C4—1731no—
305C22×C4—1734no—
306C22×C4—1734no—
307C22×C4—1731no—
308C22×C4C22×D17311yescharacteristic
309C22×C4C22×D17311yescharacteristic
310C22×C4C22×D17311yescharacteristic
311C22×C4C22×D17311yescharacteristic
312C22×C4C22×D17311yescharacteristic
313C22×C4C22×D17311yescharacteristic
314C4⋊C4—22no—
315C4⋊C4—3462no—
316C4⋊C4—3462no—
317C4⋊C4—3462no—
318C4⋊C4—3462no—
319C4⋊C4—3462no—
320C4⋊C4—3462no—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—22no—
324C22⋊C4—3462no—
325C22⋊C4—3462no—
326C22⋊C4—3462no—
327C22⋊C4—3462no—
328C22⋊C4—3462no—
329C22⋊C4—3462no—
330C22⋊C4—3462no—
331C22⋊C4—3462no—
332C22⋊C4—3462no—
333C42—22no—
334C42—3462no—
335C42—3462no—
336C42—3462no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—3461no—
345C23—3461no—
346C23—3461no—
347C23—3461no—
348C23—3461no—
349C23—3461no—
350C23—3461no—
351C23—3461no—
352C23—3461no—
353C23—3461no—
354C23—3461no—
355C23—1732no—
356C23—1732no—
357C23—1732no—
358C23C23×D17311yescharacteristic
359D4—6924no—
360D4—6924no—
361D4—6924no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—3468no—
378C2×C4—3468no—
379C2×C4—3468no—
380C2×C4—3461no—
381C2×C4—3461no—
382C2×C4—3461no—
383C2×C4—3461no—
384C2×C4—3461no—
385C2×C4—3461no—
386C2×C4—3461no—
387C2×C4—3461no—
388C2×C4—3461no—
389C2×C4—3461no—
390C2×C4—3461no—
391C2×C4—3461no—
392C2×C4—3461no—
393C2×C4—1732no—
394C2×C4—1732no—
395C2×C4—1732no—
396C2×C4—1732no—
397C2×C4—1732no—
398C2×C4D4×D17312yes—
399C2×C4D4×D17312yes—
400C2×C4#1458382612yes—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—3462no—
409C22—3462no—
410C22—3462no—
411C22—3462no—
412C22—3462no—
413C22—3462no—
414C22—1734no—
415C22—6921no—
416C22—6921no—
417C22—6921no—
418C22—6921no—
419C22—6921no—
420C22—6921no—
421C22—6921no—
422C22C2×D4×D17311yescharacteristic
423C22C2×D4×D17311yescharacteristic
424C22#41×D17311yescharacteristic
425C22#41×D17311yescharacteristic
426C22#41×D17311yescharacteristic
427C22#41×D17311yescharacteristic
428C22C2×#1458382611yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—3462no—
436C4—3462no—
437C4—1734no—
438C4—6921no—
439C4—6921no—
440C4—6921no—
441C2—81no—
442C2—3464no—
443C2—13841no—
444C2#121×D17311yescharacteristic
445C2#123×D17311yescharacteristic
446C2#2601200011yescharacteristic
447C2#2601201911yescharacteristic
448C2#2601202411yescharacteristic
449C2#2601203811yescharacteristic
450C2#2601204011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D173
commutator, socle, cosocleC23×C173C24
Fitting#388×C173C2
prime radicalC2×#19149652C2
hypercenter#388D173
hypocenterC173#2854
LCSG ▷ C23×C173 ▷ C173
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#19149664C2semidirect
2. C2×#19149661C2semidirect
4. C2×C42×D173C2semidirect
5. #26011391C2semidirect
6. #26011391C2semidirect
7. #26011388C2semidirect
8. #26011386C2semidirect
9. #26011383C2semidirect
10. #26011382C2semidirect
11. #26011379C2semidirect
12. #26011369C2semidirect
13. #26011367C2semidirect
14. #26011350C2semidirect
15. #388×C173C2semidirect
80. C2×C4×D173D4semidirect
214. Dic173#123semidirect
225. C173#2854semidirect
238. #388D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191496642
C2×#191496612
C2×#191496522
C2×C42×D1732
#260113912
#260113882
#260113862
#260113832
#260113822
#260113792
#260113692
#260113672
#260113502
#388×C1732
#2854173