Finite Group Databasebeta

← #28374882

Subgroups — #28374882

Order 13952.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295722C211yesmaximal, characteristic, complemented
2#20295380C211yesmaximal, characteristic, complemented
3#20295359C211yesmaximal, characteristic, complemented
4#20295347C211yesmaximal, characteristic, complemented
5#20295326C211yesmaximal, characteristic, complemented
6#20295260C211yesmaximal, characteristic, complemented
7#20295257C211yesmaximal, characteristic, complemented
8#20295238C211yesmaximal, characteristic, complemented
9#20295171C211yesmaximal, characteristic, complemented
10#20295160C211yesmaximal, characteristic, complemented
11C4×#13182014C211yesmaximal, characteristic, complemented
12D4×#11724784C211yesmaximal, characteristic, complemented
13#20295022C211yesmaximal, characteristic, complemented
14C2×#16812620C211yesmaximal, characteristic, complemented
15#546×C109C211yesmaximal, characteristic, complemented
16C2×#13182016C2211yescharacteristic, complemented
17C2×#13182014C2211yescharacteristic
18C2×#13182014C2211yescharacteristic
19C22×#11724784C2211yescharacteristic
20C22×#11724784C2211yescharacteristic, complemented
21#16812694—21no—
22#16812691C2211yescharacteristic, complemented
23#16812681—21no—
24#16812673C2211yescharacteristic
25#16812673C2211yescharacteristic, complemented
26#16812671—21no—
27C4×D436C2211yescharacteristic, complemented
28#16812659—21no—
29#16812658C2211yescharacteristic, complemented
30C2×C4⋊Dic109C2211yescharacteristic
31#16812632C2211yescharacteristic, complemented
32#16812632C2211yescharacteristic, complemented
33#16812630C2211yescharacteristic, complemented
34C4⋊#11724784C2211yescharacteristic, complemented
35#16812623C2211yescharacteristic, complemented
36#16812623C2211yescharacteristic, complemented
37#16812623C2211yescharacteristic, complemented
38#16812620C2211yescharacteristic
39#16812620C2211yescharacteristic
40#16812620C2211yescharacteristic
41#16812620C2211yescharacteristic
42#16812620C2211yescharacteristic
43#16812620C2211yescharacteristic
44#16812619C2211yescharacteristic, complemented
45C22⋊#11724784C2211yescharacteristic
46C22⋊#11724784C2211yescharacteristic, complemented
47C4×#11724784C2211yescharacteristic, complemented
48C2×#41×C109C2211yescharacteristic
49C22×D4×C109C2211yescharacteristic
50#124×C109C2211yescharacteristic
51#123×C109—21no—
52C22⋊Q8×C109C2211yescharacteristic
53#121×C109—21no—
54#121×C109C2211yescharacteristic
55#120×C109—21no—
56C4×D4×C109C2211yescharacteristic
57C4×D4×C109C2211yescharacteristic
58C2×C22⋊C4×C109—21no—
59C2×D436C2311yescharacteristic
60C2×C22⋊D109—21no—
61C2×C22⋊D109—21no—
62C22×Dic109—21no—
63C2×C4×D109—21no—
64#13182016—41no—
65#13182014—22no—
66#13182014—41no—
67C2×#11724784—41no—
68C2×#11724784—41no—
69C2×#11724784C2311yescharacteristic
70C2×#11724784C2311yescharacteristic
71C2×#11724784C2311yescharacteristic
72C2×#11724784C2311yescharacteristic
73C4⋊Dic109—21no—
74C4⋊Dic109C2311yescharacteristic
75C4⋊Dic109C2311yescharacteristic
76C4⋊Dic109C2311yescharacteristic
77#13182008—21no—
78#13182008—21no—
79C22⋊Dic109—21no—
80#13182006—21no—
81#13182006—21no—
82#13182006—21no—
83C4×Dic109—21no—
84C24×C109—21no—
85#41×C109—22no—
86#41×C109—41no—
87C2×Q8×C109C2311yescharacteristic
88C2×D4×C109—22no—
89C2×D4×C109—21no—
90C2×D4×C109—21no—
91C2×D4×C109—21no—
92C2×D4×C109—41no—
93C2×D4×C109D411yescharacteristic
94C2×D4×C109C2311yescharacteristic
95C2×D4×C109C2311yescharacteristic
96C22×C4×C109—21no—
97C22×C4×C109—21no—
98C22×C4×C109C2311yescharacteristic
99C22×C4×C109C2311yescharacteristic
100C4⋊C4×C109—21no—
101C4⋊C4×C109D411yescharacteristic
102C4⋊C4×C109C2311yescharacteristic
103C22⋊C4×C109—21no—
104C22⋊C4×C109—21no—
105C22⋊C4×C109—21no—
106C22⋊C4×C109—21no—
107C22⋊C4×C109—21no—
108C22⋊C4×C109D411yescharacteristic
109C22⋊C4×C109D411yescharacteristic
110C42×C109C2311yescharacteristic
111C23×C109—21no—
112C23×C109—21no—
113C23×C109—42no—
114C23×C109—42no—
115C23×C109—81no—
116C23×C109C2×D411yescharacteristic
117C23×C109C2×D411yescharacteristic
118C22×D109—21no—
119Q8×C109—21no—
120Q8×C109—21no—
121D4×C109—22no—
122D4×C109—21no—
123D4×C109—21no—
124D4×C109—21no—
125D4×C109—42no—
126D4×C109—42no—
127D4×C109—81no—
128D4×C109#4112yes—
129C2×C4×C109—21no—
130C2×C4×C109—21no—
131C2×C4×C109—21no—
132C2×C4×C109—21no—
133C2×C4×C109—21no—
134C2×C4×C109—21no—
135C2×C4×C109—42no—
136C2×C4×C109—81no—
137C2×C4×C109C2×D411yescharacteristic
138C2×C4×C109C2×D411yescharacteristic
139C2×C4×C109C2×D411yescharacteristic
140C2×C4×C109C2×D411yescharacteristic
141C2×C4×C109C2411yescharacteristic
142C22⋊D109—42no—
143C22⋊D109—81no—
144C2×Dic109—21no—
145C2×Dic109—21no—
146C2×Dic109—21no—
147C2×Dic109—81no—
148D436—21no—
149D436—21no—
150C4×D109—81no—
151#11724784—21no—
152#11724784—21no—
153#11724784—21no—
154#11724784—21no—
155C22×C109—21no—
156C22×C109—21no—
157C22×C109—21no—
158C22×C109—22no—
159C22×C109—21no—
160C22×C109—21no—
161C22×C109—41no—
162C22×C109—41no—
163C22×C109—41no—
164C22×C109—41no—
165C22×C109—41no—
166C22×C109—84no—
167C22×C109#13512yes—
168C22×C109C22×D411yescharacteristic
169C2×D109—41no—
170C2×D109—41no—
171C2×D109—41no—
172C4×C109—21no—
173C4×C109—21no—
174C4×C109—21no—
175C4×C109—21no—
176C4×C109—41no—
177C4×C109—41no—
178C4×C109C2×#4111yescharacteristic
179C4×C109H(5,𝔽2)11yescharacteristic
180Dic109—41no—
181Dic109—41no—
182Dic109—41no—
183C2×C109—22no—
184C2×C109—41no—
185C2×C109—42no—
186C2×C109—81no—
187C2×C109#54611yescharacteristic
188C2×C109C2×#13511yescharacteristic
189C2×C109#57611yescharacteristic
190D109—81no—
191#3297—1091nomaximal, Sylow
192C109#329711yescharacteristic, complemented, Sylow
193#553—1091no—
194#546D10911yescharacteristic, complemented
195#488—1091no—
196#484—1091no—
197#482—1091no—
198#478—1091no—
199#466—1091no—
200#465—1091no—
201#463—1091no—
202#449—1091no—
203#447—1091no—
204C4×D8—1091no—
205C8×D4—1091no—
206#416—1091no—
207C2×#102—1091no—
208C2×#41C2×D10911yescharacteristic
209C22×D4C2×D10911yescharacteristic
210C2×SD16—1091no—
211C2×D8—1091no—
212C2×D8—1091no—
213C22×C8—1091no—
214C22×C8—1091no—
215#126—2181no—
216#125—1091no—
217#124C2×D10911yescharacteristic
218#123—21no—
219#123—2181no—
220C22⋊Q8C2×D10911yescharacteristic
221#121—21no—
222#121—1091no—
223#121—1091no—
224#121—2181no—
225#121C2×D10911yescharacteristic
226#120—21no—
227C4×D4—1091no—
228C4×D4—2181no—
229C4×D4C2×D10911yescharacteristic
230C4×D4C2×D10911yescharacteristic
231#117—1091no—
232C2×C4⋊C4—1091no—
233C2×C22⋊C4—21no—
234#107—1091no—
235#107—1091no—
236#106—1091no—
237C4⋊C8—1091no—
238#103—1091no—
239#103—1091no—
240#103—1091no—
241#102—1091no—
242#102—1091no—
243#102—1091no—
244#102—1091no—
245#102—1091no—
246#102—1091no—
247#102—1091no—
248C22⋊C8—1091no—
249C22⋊C8—1091no—
250C4×C8—1091no—
251C24—21no—
252#41—22no—
253#41—41no—
254C2×Q8C22×D10911yescharacteristic
255C2×D4—22no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—41no—
260C2×D4—1091no—
261C2×D4—2181no—
262C2×D4—2181no—
263C2×D4C22⋊D10911yescharacteristic
264C2×D4C22×D10911yescharacteristic
265C2×D4C22×D10911yescharacteristic
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—2181no—
269C22×C4—2181no—
270C22×C4C22×D10911yescharacteristic
271C22×C4C22×D10911yescharacteristic
272SD16—4361no—
273D8—4361no—
274D8—2182no—
275C2×C8—1091no—
276C2×C8—1091no—
277C2×C8—1091no—
278C2×C8—1091no—
279C2×C8—4361no—
280C2×C8—4361no—
281C4⋊C4—21no—
282C4⋊C4—1091no—
283C4⋊C4—1091no—
284C4⋊C4—1091no—
285C4⋊C4—2181no—
286C4⋊C4—2181no—
287C4⋊C4—2181no—
288C4⋊C4C22⋊D10911yescharacteristic
289C4⋊C4C22×D10911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C22⋊C4—2181no—
296C22⋊C4—2181no—
297C22⋊C4—2181no—
298C22⋊C4—2181no—
299C22⋊C4C22⋊D10911yescharacteristic
300C22⋊C4C22⋊D10911yescharacteristic
301C42—2181no—
302C42C22×D10911yescharacteristic
303C23—21no—
304C23—21no—
305C23—42no—
306C23—42no—
307C23—81no—
308C23—2181no—
309C23C2×C22⋊D10911yescharacteristic
310C23C2×C22⋊D10911yescharacteristic
311Q8—21no—
312Q8—21no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—42no—
319D4—81no—
320D4—8721no—
321D4—4362no—
322D4—2181no—
323D4—2181no—
324D4#1318202812yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—42no—
332C2×C4—81no—
333C2×C4—8721no—
334C2×C4—8721no—
335C2×C4—2181no—
336C2×C4—2181no—
337C2×C4—2181no—
338C2×C4C2×C22⋊D10911yescharacteristic
339C2×C4C2×C22⋊D10911yescharacteristic
340C2×C4C2×C22⋊D10911yescharacteristic
341C2×C4C2×C22⋊D10911yescharacteristic
342C2×C4C23×D10911yescharacteristic
343C8—2181no—
344C8—2181no—
345C8—2181no—
346C8—2181no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—22no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—84no—
359C22—4361no—
360C22—4361no—
361C22—4361no—
362C22#1681271912yes—
363C22C22×C22⋊D10911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—4361no—
371C4—4361no—
372C4—4361no—
373C4C2×#1318202811yescharacteristic
374C4#1681274411yescharacteristic
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—8721no—
380C2#2029567011yescharacteristic
381C2C2×#1681271911yescharacteristic
382C2#2029582911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#546×C109C2
socleC22×C109C22×D4
hypercenter#546D109
hypocenterC109#3297
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. #20295722C2semidirect
2. #20295380C2semidirect
3. #20295359C2semidirect
4. #20295347C2semidirect
5. #20295326C2semidirect
6. #20295260C2semidirect
7. #20295257C2semidirect
8. #20295238C2semidirect
9. #20295171C2semidirect
10. #20295160C2semidirect
11. C4×#13182014C2semidirect
12. D4×#11724784C2semidirect
13. #20295022C2semidirect
14. C2×#16812620C2semidirect
15. #546×C109C2semidirect
16. C2×#13182016C22semidirect
20. C22×#11724784C22semidirect
22. #16812691C22semidirect
25. #16812673C22semidirect
27. C4×D436C22semidirect
29. #16812658C22semidirect
31. #16812632C22semidirect
32. #16812632C22semidirect
33. #16812630C22semidirect
34. C4⋊#11724784C22semidirect
35. #16812623C22semidirect
36. #16812623C22semidirect
37. #16812623C22semidirect
44. #16812619C22semidirect
46. C22⋊#11724784C22semidirect
47. C4×#11724784C22semidirect
192. C109#3297semidirect
194. #546D109semidirect

Maximal subgroups

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

subgroupindex
#202957222
#202953802
#202953592
#202953472
#202953262
#202952602
#202952572
#202952382
#202951712
#202951602
C4×#131820142
D4×#117247842
#202950222
C2×#168126202
#546×C1092
#3297109