Finite Group Databasebeta

← #28376098

Subgroups — #28376098

Order 13952.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20295761C211yesmaximal, characteristic, complemented
2#20295418C211yesmaximal, characteristic, complemented
3#20295416C211yesmaximal, characteristic
4#20295408C211yesmaximal, characteristic, complemented
5#20295248C211yesmaximal, characteristic, complemented
6#20295245C211yesmaximal, characteristic, complemented
7#20295241C211yesmaximal, characteristic, complemented
8#20295238C211yesmaximal, characteristic, complemented
9#20295228C211yesmaximal, characteristic, complemented
10#20295226C211yesmaximal, characteristic, complemented
11#20295157C211yesmaximal, characteristic, complemented
12#20295136C211yesmaximal, characteristic, complemented
13#20295131C211yesmaximal, characteristic, complemented
14C4×#13182016C211yesmaximal, characteristic, complemented
15#556×C109C211yesmaximal, characteristic, complemented
16C2×#13182019C2211yescharacteristic
17C2×#13182019C2211yescharacteristic
18C2×#13182017C2211yescharacteristic
19C2×#13182016C2211yescharacteristic
20C2×#13182016C2211yescharacteristic
21C2×#13182016C2211yescharacteristic
22C2×#13182016C2211yescharacteristic
23C2×#13182014C2211yescharacteristic
24C2×#13182014C2211yescharacteristic
25#16812696C2211yescharacteristic
26#16812688—21no—
27#16812687C2211yescharacteristic
28#16812687C2211yescharacteristic
29#16812684—21no—
30#16812684—21no—
31C4×Dic218C2211yescharacteristic
32C4×D436C2211yescharacteristic
33C4×D436C2211yescharacteristic
34#16812661—21no—
35#16812630C2211yescharacteristic
36C4⋊#11724784C2211yescharacteristic
37C4⋊#11724784C2211yescharacteristic
38C4⋊#11724784C2211yescharacteristic
39#16812623C2211yescharacteristic
40#16812623C2211yescharacteristic
41#16812622C2211yescharacteristic
42#16812620C2211yescharacteristic
43#16812619C2211yescharacteristic
44#16812619C2211yescharacteristic
45#16812610C2211yescharacteristic
46#16812610C2211yescharacteristic
47C4×#11724784C2211yescharacteristic
48C4⋊Q8×C109C2211yescharacteristic
49#126×C109—21no—
50#126×C109—21no—
51#124×C109C2211yescharacteristic
52#124×C109C2211yescharacteristic
53C22⋊Q8×C109—21no—
54C4×Q8×C109C2211yescharacteristic
55C4×Q8×C109C2211yescharacteristic
56C4×Q8×C109C2211yescharacteristic
57C4×D4×C109C2211yescharacteristic
58#117×C109—21no—
59C2×Dic218C2311yescharacteristic
60C2×D436—21no—
61C2×D436—21no—
62C2×D436C2311yescharacteristic
63C2×D436C2311yescharacteristic
64C2×C4×D109—21no—
65C2×C4×D109—21no—
66#13182019—22no—
67#13182019—41no—
68#13182017—41no—
69#13182016—22no—
70#13182016—41no—
71#13182016—41no—
72#13182016—41no—
73#13182014—22no—
74#13182014—41no—
75C2×#11724784C2311yescharacteristic
76C2×#11724784C2311yescharacteristic
77C2×#11724784C2311yescharacteristic
78C2×#11724784C2311yescharacteristic
79C4⋊Dic109C2311yescharacteristic
80#13182008—21no—
81#13182006—21no—
82#13182006—21no—
83#13182006—21no—
84#13182006—21no—
85#13182006—21no—
86#13182006—21no—
87C4×Dic109—21no—
88C2×Q8×C109D411yescharacteristic
89C2×Q8×C109C2311yescharacteristic
90C2×Q8×C109C2311yescharacteristic
91C2×D4×C109C2311yescharacteristic
92C22×C4×C109—21no—
93C4⋊C4×C109—21no—
94C4⋊C4×C109—21no—
95C4⋊C4×C109—21no—
96C4⋊C4×C109—21no—
97C4⋊C4×C109D411yescharacteristic
98C4⋊C4×C109D411yescharacteristic
99C4⋊C4×C109D411yescharacteristic
100C4⋊C4×C109C2311yescharacteristic
101C22⋊C4×C109—21no—
102C22⋊C4×C109—21no—
103C22⋊C4×C109—21no—
104C22⋊C4×C109—21no—
105C22⋊C4×C109—21no—
106C42×C109—21no—
107C42×C109—21no—
108C42×C109C2311yescharacteristic
109C42×C109C2311yescharacteristic
110C42×C109C2311yescharacteristic
111C23×C109—21no—
112C22×D109—21no—
113C22×D109—21no—
114Q8×C109—21no—
115Q8×C109—21no—
116Q8×C109—21no—
117Q8×C109—21no—
118Q8×C109—41no—
119D4×C109—21no—
120D4×C109—21no—
121C2×C4×C109—21no—
122C2×C4×C109—21no—
123C2×C4×C109—21no—
124C2×C4×C109—42no—
125C2×C4×C109C2×D411yescharacteristic
126C2×C4×C109C2×D411yescharacteristic
127C2×C4×C109C2×D411yescharacteristic
128C2×C4×C109C2×D411yescharacteristic
129C2×C4×C109C2×D411yescharacteristic
130C2×C4×C109C2×D411yescharacteristic
131C2×C4×C109C2411yescharacteristic
132C2×Dic109—21no—
133C2×Dic109—21no—
134D436—21no—
135D436—21no—
136D436—21no—
137D436—21no—
138D436—42no—
139D436—81no—
140C4×D109—81no—
141C4×D109—81no—
142Dic218—21no—
143Dic218—21no—
144#11724784—21no—
145#11724784—21no—
146#11724784—21no—
147#11724784#4112yes—
148C22×C109—41no—
149C22×C109—41no—
150C22×C109—41no—
151C22×C109C22×D411yescharacteristic
152C2×D109—41no—
153C2×D109—41no—
154C2×D109—41no—
155C2×D109—41no—
156C2×D109—41no—
157C2×D109—41no—
158C4×C109—21no—
159C4×C109—21no—
160C4×C109—21no—
161C4×C109—21no—
162C4×C109—21no—
163C4×C109—21no—
164C4×C109—41no—
165C4×C109—41no—
166C4×C109—41no—
167C4×C109C2×#4111yescharacteristic
168C4×C109H(5,𝔽2)11yescharacteristic
169Dic109—41no—
170Dic109—41no—
171C2×C109—81no—
172C2×C109#55011yescharacteristic
173C2×C109#57511yescharacteristic
174C2×C109#57711yescharacteristic
175D109—81no—
176D109—81no—
177#3401—1091nomaximal, Sylow
178C109#340111yescharacteristic, complemented, Sylow
179#559—1091no—
180#556D10911yescharacteristic, complemented
181#497—1091no—
182#496—1091no—
183#495—1091no—
184#464—1091no—
185#464—1091no—
186#463—1091no—
187#463—1091no—
188#461—1091no—
189#460—1091no—
190#446—1091no—
191#442—1091no—
192#441—1091no—
193C4×SD16—1091no—
194C2×Q16—1091no—
195C2×Q16—1091no—
196C2×SD16—1091no—
197C2×SD16—1091no—
198C2×SD16—1091no—
199C2×SD16—1091no—
200C2×SD16—1091no—
201C2×D8—1091no—
202C2×D8—1091no—
203C4⋊Q8C2×D10911yescharacteristic
204#127—1091no—
205#126—21no—
206#126—21no—
207#124—1091no—
208#124—1091no—
209#124—2181no—
210#124C2×D10911yescharacteristic
211#124C2×D10911yescharacteristic
212#123—2181no—
213#123—2181no—
214C22⋊Q8—21no—
215C4×Q8—1091no—
216C4×Q8C2×D10911yescharacteristic
217C4×Q8C2×D10911yescharacteristic
218C4×Q8C2×D10911yescharacteristic
219C4×D4—1091no—
220C4×D4—1091no—
221C4×D4—2181no—
222C4×D4C2×D10911yescharacteristic
223#117—21no—
224#106—1091no—
225C4⋊C8—1091no—
226C4⋊C8—1091no—
227C4⋊C8—1091no—
228#103—1091no—
229#103—1091no—
230#103—1091no—
231#102—1091no—
232#102—1091no—
233#102—1091no—
234#97—1091no—
235#97—1091no—
236C4×C8—1091no—
237C2×Q8—1091no—
238C2×Q8C22⋊D10911yescharacteristic
239C2×Q8C22×D10911yescharacteristic
240C2×Q8C22×D10911yescharacteristic
241C2×D4—1091no—
242C2×D4—1091no—
243C2×D4—2181no—
244C2×D4—2181no—
245C2×D4C22×D10911yescharacteristic
246C22×C4—21no—
247C22×C4—2181no—
248C22×C4—2181no—
249Q16—4361no—
250Q16—2182no—
251SD16—4361no—
252SD16—4361no—
253SD16—4361no—
254SD16—4361no—
255SD16—2182no—
256D8—4361no—
257D8—2182no—
258C2×C8—1091no—
259C2×C8—1091no—
260C2×C8—1091no—
261C2×C8—1091no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—1091no—
267C4⋊C4—2181no—
268C4⋊C4C22⋊D10911yescharacteristic
269C4⋊C4C22⋊D10911yescharacteristic
270C4⋊C4C22⋊D10911yescharacteristic
271C4⋊C4C22×D10911yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2181no—
278C22⋊C4—2181no—
279C22⋊C4—2181no—
280C22⋊C4—2181no—
281C22⋊C4—2181no—
282C22⋊C4—2181no—
283C42—21no—
284C42—21no—
285C42—2181no—
286C42C22×D10911yescharacteristic
287C42C22×D10911yescharacteristic
288C42C22×D10911yescharacteristic
289C23—21no—
290C23—2181no—
291C23—2181no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—2181no—
298Q8—2181no—
299D4—21no—
300D4—21no—
301D4—8721no—
302D4—4362no—
303D4—2181no—
304D4—2181no—
305D4—2181no—
306D4—2181no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—8721no—
312C2×C4—8721no—
313C2×C4—2181no—
314C2×C4—2181no—
315C2×C4C2×C22⋊D10911yescharacteristic
316C2×C4C2×C22⋊D10911yescharacteristic
317C2×C4C2×C22⋊D10911yescharacteristic
318C2×C4C2×C22⋊D10911yescharacteristic
319C2×C4C2×C22⋊D10911yescharacteristic
320C2×C4C2×C22⋊D10911yescharacteristic
321C2×C4C23×D10911yescharacteristic
322C8—1092no—
323C8—2181no—
324C8—2181no—
325C8—2181no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—4361no—
330C22—4361no—
331C22—4361no—
332C22—4361no—
333C22—4361no—
334C22—4361no—
335C22C22×C22⋊D10911yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—4361no—
346C4—4361no—
347C4#41×D10911yescharacteristic
348C4#1681274511yescharacteristic
349C2—81no—
350C2—8721no—
351C2—8721no—
352C2#2029569511yescharacteristic
353C2#2029582411yescharacteristic
354C2#2029583411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#556×C109C2
socleC22×C109C22×D4
prime radical#20295416C2
hypercenter#556D109
hypocenterC109#3401
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20295761C2semidirect
2. #20295418C2semidirect
4. #20295408C2semidirect
5. #20295248C2semidirect
6. #20295245C2semidirect
7. #20295241C2semidirect
8. #20295238C2semidirect
9. #20295228C2semidirect
10. #20295226C2semidirect
11. #20295157C2semidirect
12. #20295136C2semidirect
13. #20295131C2semidirect
14. C4×#13182016C2semidirect
15. #556×C109C2semidirect
178. C109#3401semidirect
180. #556D109semidirect

Maximal subgroups

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

subgroupindex
#202957612
#202954182
#202954162
#202954082
#202952482
#202952452
#202952412
#202952382
#202952282
#202952262
#202951572
#202951362
#202951312
C4×#131820162
#556×C1092
#3401109