Finite Group Databasebeta

← #28374876

Subgroups — #28374876

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#20295672C211yesmaximal, characteristic, complemented
2#20295383C211yesmaximal, characteristic, complemented
3#20295361C211yesmaximal, characteristic, complemented
4#20295350C211yesmaximal, characteristic, complemented
5#20295323C211yesmaximal, characteristic, complemented
6#20295260C211yesmaximal, characteristic, complemented
7#20295253C211yesmaximal, characteristic, complemented
8#20295241C211yesmaximal, characteristic, complemented
9#20295169C211yesmaximal, characteristic, complemented
10#20295159C211yesmaximal, characteristic, complemented
11C4×#13182014C211yesmaximal, characteristic, complemented
12D4×#11724784C211yesmaximal, characteristic, complemented
13#20295020C211yesmaximal, characteristic, complemented
14C2×#16812619C211yesmaximal, characteristic, complemented
15#553×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17C22×D436C2211yescharacteristic
18C2×#13182017C2211yescharacteristic, complemented
19C2×#13182014C2211yescharacteristic
20C2×#13182014C2211yescharacteristic
21C22×#11724784C2211yescharacteristic
22C22×#11724784C2211yescharacteristic, complemented
23#16812687C2211yescharacteristic
24#16812684—21no—
25#16812678C2211yescharacteristic
26#16812673C2211yescharacteristic
27#16812672—21no—
28#16812667—21no—
29C4×D436C2211yescharacteristic
30C22⋊C4×D109—21no—
31#16812632C2211yescharacteristic, complemented
32#16812632C2211yescharacteristic, complemented
33#16812630C2211yescharacteristic, complemented
34C4⋊#11724784C2211yescharacteristic, complemented
35#16812622C2211yescharacteristic, complemented
36#16812622C2211yescharacteristic, complemented
37#16812622C2211yescharacteristic, complemented
38#16812620C2211yescharacteristic, complemented
39#16812619C2211yescharacteristic
40#16812619C2211yescharacteristic
41#16812619C2211yescharacteristic
42#16812619C2211yescharacteristic
43#16812619C2211yescharacteristic
44#16812619C2211yescharacteristic
45C22⋊#11724784C2211yescharacteristic
46C22⋊#11724784C2211yescharacteristic, complemented
47C4×#11724784C2211yescharacteristic, complemented
48#126×C109—21no—
49#125×C109C2211yescharacteristic, complemented
50#123×C109—21no—
51#121×C109—21no—
52#121×C109C2211yescharacteristic
53#121×C109C2211yescharacteristic, complemented
54C4×D4×C109—21no—
55C4×D4×C109C2211yescharacteristic
56C4×D4×C109C2211yescharacteristic, complemented
57#117×C109C2211yescharacteristic, complemented
58C2×C4⋊C4×C109C2211yescharacteristic
59C23×D109—21no—
60#13182029—22no—
61#13182029—41no—
62C2×Dic218C2311yescharacteristic
63C2×D436—22no—
64C2×D436—21no—
65C2×D436—41no—
66C2×D436C2311yescharacteristic
67C2×D436C2311yescharacteristic
68C2×C22⋊D109—21no—
69C2×C22⋊D109—21no—
70C2×C4×D109—21no—
71C2×C4×D109—21no—
72#13182017—41no—
73#13182014—22no—
74#13182014—41no—
75C2×#11724784—41no—
76C2×#11724784—41no—
77C2×#11724784C2311yescharacteristic
78C2×#11724784C2311yescharacteristic
79C2×#11724784C2311yescharacteristic
80C2×#11724784C2311yescharacteristic
81C4⋊Dic109C2311yescharacteristic
82#13182008—21no—
83C22⋊Dic109—21no—
84#13182006—21no—
85#13182006—21no—
86#13182006—21no—
87#13182006—21no—
88C2×D4×C109—21no—
89C2×D4×C109—21no—
90C2×D4×C109D411yescharacteristic
91C2×D4×C109C2311yescharacteristic
92C22×C4×C109—21no—
93C22×C4×C109—21no—
94C22×C4×C109C2311yescharacteristic
95C22×C4×C109C2311yescharacteristic
96C4⋊C4×C109—21no—
97C4⋊C4×C109—21no—
98C4⋊C4×C109—21no—
99C4⋊C4×C109D411yescharacteristic
100C4⋊C4×C109C2311yescharacteristic
101C4⋊C4×C109C2311yescharacteristic
102C4⋊C4×C109C2311yescharacteristic
103C22⋊C4×C109—21no—
104C22⋊C4×C109—21no—
105C22⋊C4×C109—21no—
106C22⋊C4×C109—21no—
107C22⋊C4×C109D411yescharacteristic
108C22⋊C4×C109D411yescharacteristic
109C42×C109—21no—
110C42×C109C2311yescharacteristic
111C23×C109—21no—
112C23×C109C2×D411yescharacteristic
113C23×C109C2×D411yescharacteristic
114C22×D109—21no—
115C22×D109—21no—
116C22×D109—42no—
117C22×D109—42no—
118C22×D109—81no—
119D4×C109—22no—
120D4×C109—21no—
121D4×C109—21no—
122D4×C109—42no—
123D4×C109—81no—
124C2×C4×C109—21no—
125C2×C4×C109—21no—
126C2×C4×C109—21no—
127C2×C4×C109—21no—
128C2×C4×C109—21no—
129C2×C4×C109—21no—
130C2×C4×C109—21no—
131C2×C4×C109—81no—
132C2×C4×C109—81no—
133C2×C4×C109C2×D411yescharacteristic
134C2×C4×C109C2×D411yescharacteristic
135C2×C4×C109C2×D411yescharacteristic
136C2×C4×C109C2×D411yescharacteristic
137C2×C4×C109C2411yescharacteristic
138C22⋊D109—42no—
139C22⋊D109—81no—
140C2×Dic109—21no—
141C2×Dic109—21no—
142D436—21no—
143D436—21no—
144D436—21no—
145D436—42no—
146D436#4112yes—
147C4×D109—42no—
148C4×D109—81no—
149Dic218—21no—
150Dic218—21no—
151#11724784—21no—
152#11724784—21no—
153#11724784—21no—
154#11724784—21no—
155C22×C109—21no—
156C22×C109—21no—
157C22×C109—21no—
158C22×C109—21no—
159C22×C109—21no—
160C22×C109—41no—
161C22×C109—41no—
162C22×C109—41no—
163C22×C109#13512yes—
164C22×C109C22×D411yescharacteristic
165C2×D109—22no—
166C2×D109—41no—
167C2×D109—41no—
168C2×D109—41no—
169C2×D109—41no—
170C2×D109—41no—
171C2×D109—84no—
172C4×C109—21no—
173C4×C109—21no—
174C4×C109—21no—
175C4×C109—21no—
176C4×C109—41no—
177C4×C109—41no—
178C4×C109—41no—
179C4×C109C2×#4111yescharacteristic
180C4×C109H(5,𝔽2)11yescharacteristic
181Dic109—41no—
182Dic109—41no—
183C2×C109—22no—
184C2×C109—41no—
185C2×C109—81no—
186C2×C109#54611yescharacteristic
187C2×C109C2×#13511yescharacteristic
188C2×C109#57611yescharacteristic
189D109—42no—
190D109—81no—
191#3297—1091nomaximal, Sylow
192C109#329711yescharacteristic, complemented, Sylow
193#553D10911yescharacteristic, complemented
194#546—1091no—
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×#41—1091no—
209C22×D4—1091no—
210C2×SD16—1091no—
211C2×D8—1091no—
212C2×D8—1091no—
213C22×C8—1091no—
214C22×C8—1091no—
215#126—21no—
216#125C2×D10911yescharacteristic, complemented
217#124—1091no—
218#123—21no—
219#123—2181no—
220C22⋊Q8—1091no—
221#121—21no—
222#121—1091no—
223#121—2181no—
224#121C2×D10911yescharacteristic
225#121C2×D10911yescharacteristic, complemented
226#120—2181no—
227C4×D4—21no—
228C4×D4—1091no—
229C4×D4C2×D10911yescharacteristic
230C4×D4C2×D10911yescharacteristic, complemented
231#117C2×D10911yescharacteristic, complemented
232C2×C4⋊C4C2×D10911yescharacteristic
233C2×C22⋊C4—2181no—
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—2181no—
252#41—4361no—
253#41—2182no—
254C2×Q8—1091no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—1091no—
258C2×D4—1091no—
259C2×D4—4361no—
260C2×D4—2182no—
261C2×D4—2181no—
262C2×D4—2181no—
263C2×D4—2181no—
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—21no—
283C4⋊C4—21no—
284C4⋊C4—1091no—
285C4⋊C4—2181no—
286C4⋊C4C22⋊D10911yescharacteristic
287C4⋊C4C22×D10911yescharacteristic
288C4⋊C4C22×D10911yescharacteristic
289C4⋊C4C22×D10911yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—2181no—
295C22⋊C4—2181no—
296C22⋊C4—2181no—
297C22⋊C4—2181no—
298C22⋊C4—2181no—
299C22⋊C4C22⋊D10911yescharacteristic
300C22⋊C4C22⋊D10911yescharacteristic
301C42—21no—
302C42C22×D10911yescharacteristic
303C23—21no—
304C23—8721no—
305C23—4362no—
306C23—4362no—
307C23—2181no—
308C23—2181no—
309C23C2×C22⋊D10911yescharacteristic
310C23C2×C22⋊D10911yescharacteristic
311Q8—2181no—
312Q8—2181no—
313D4—22no—
314D4—21no—
315D4—21no—
316D4—42no—
317D4—81no—
318D4—8721no—
319D4—1092no—
320D4—4362no—
321D4—4362no—
322D4—2181no—
323D4—2181no—
324D4—2181no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—81no—
334C2×C4—8721no—
335C2×C4—4362no—
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—41no—
353C22—41no—
354C22—41no—
355C22—8724no—
356C22—4361no—
357C22—4361no—
358C22—4361no—
359C22—4361no—
360C22—4361no—
361C22—2182no—
362C22#1681271912yes—
363C22C22×C22⋊D10911yescharacteristic
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—4361no—
372C4—4361no—
373C4#41×D10911yescharacteristic
374C4#1681274511yescharacteristic
375C2—22no—
376C2—41no—
377C2—81no—
378C2—8721no—
379C2—4362no—
380C2#2029566411yescharacteristic
381C2C2×#1681271911yescharacteristic
382C2#2029582911yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#553×C109C2
socleC22×C109C22×D4
hypercenter#553D109
hypocenterC109#3297
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
1. #20295672C2semidirect
2. #20295383C2semidirect
3. #20295361C2semidirect
4. #20295350C2semidirect
5. #20295323C2semidirect
6. #20295260C2semidirect
7. #20295253C2semidirect
8. #20295241C2semidirect
9. #20295169C2semidirect
10. #20295159C2semidirect
11. C4×#13182014C2semidirect
12. D4×#11724784C2semidirect
13. #20295020C2semidirect
14. C2×#16812619C2semidirect
15. #553×C109C2semidirect
18. C2×#13182017C22semidirect
22. C22×#11724784C22semidirect
31. #16812632C22semidirect
32. #16812632C22semidirect
33. #16812630C22semidirect
34. C4⋊#11724784C22semidirect
35. #16812622C22semidirect
36. #16812622C22semidirect
37. #16812622C22semidirect
38. #16812620C22semidirect
46. C22⋊#11724784C22semidirect
47. C4×#11724784C22semidirect
49. #125×C109C22semidirect
53. #121×C109C22semidirect
56. C4×D4×C109C22semidirect
57. #117×C109C22semidirect
192. C109#3297semidirect
193. #553D109semidirect
216. #125C2×D109semidirect
225. #121C2×D109semidirect
230. C4×D4C2×D109semidirect
231. #117C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202956722
#202953832
#202953612
#202953502
#202953232
#202952602
#202952532
#202952412
#202951692
#202951592
C4×#131820142
D4×#117247842
#202950202
C2×#168126192
#553×C1092
#3297109