Finite Group Databasebeta

← #28372483

Subgroups — #28372483

Order 13952.

496 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812719C211yesmaximal, characteristic, complemented
2#20295572C211yesmaximal, characteristic, complemented
3#41×Dic109C211yesmaximal, characteristic, complemented
4#20295250C211yesmaximal, characteristic, complemented
5#20295246C211yesmaximal, characteristic, complemented
6#20295243C211yesmaximal, characteristic, complemented
7#20295239C211yesmaximal, characteristic, complemented
8#20295234C211yesmaximal, characteristic, complemented
9#20295229C211yesmaximal, characteristic, complemented
10#20295224C211yesmaximal, characteristic, complemented
11#20295219C211yesmaximal, characteristic, complemented
12C2×#16812628C211yesmaximal, characteristic, complemented
13#20295023C211yesmaximal, characteristic, complemented
14#20295022C211yesmaximal, characteristic, complemented
15C2×#135×C109C211yesmaximal, characteristic, complemented
16C2×#13182029C2211yescharacteristic
17#16812719—24no—
18C2×#13182019C2211yescharacteristic
19C2×#13182017C2211yescharacteristic
20C2×#13182016C2211yescharacteristic
21C2×#13182014C2211yescharacteristic
22C22×#11724784C2211yescharacteristic
23#16812697C2211yescharacteristic
24#16812695C2211yescharacteristic
25#16812688C2211yescharacteristic
26#16812686C2211yescharacteristic
27#16812671—21no—
28#16812671—21no—
29Q8×Dic109C2211yescharacteristic
30Q8×Dic109C2211yescharacteristic
31D4×Dic109—21no—
32D4×Dic109—21no—
33D4×Dic109C2211yescharacteristic
34D4×Dic109C2211yescharacteristic
35C4×C22⋊D109—21no—
36C4×C22⋊D109—21no—
37#16812658—21no—
38#16812658C2211yescharacteristic
39C2×C4×Dic109—21no—
40C2×C4×Dic109C2211yescharacteristic
41#16812628C2211yescharacteristic
42#16812628C2211yescharacteristic
43#16812628C2211yescharacteristic
44#16812628C2211yescharacteristic
45#16812624C2211yescharacteristic
46#16812624C2211yescharacteristic
47#16812623C2211yescharacteristic
48#16812623C2211yescharacteristic
49#16812621C2211yescharacteristic
50#16812621C2211yescharacteristic
51#16812620C2211yescharacteristic
52#16812620C2211yescharacteristic
53C2×#41×C109C2211yescharacteristic
54C2×#41×C109C2211yescharacteristic
55#135×C109—24no—
56C2×Q16×C109C2211yescharacteristic
57C2×SD16×C109C2211yescharacteristic
58C2×SD16×C109C2211yescharacteristic
59C2×D8×C109C2211yescharacteristic
60C22×C8×C109C2211yescharacteristic
61#13182029—22no—
62#13182029—22no—
63C2×Dic218C2311yescharacteristic
64C2×D436C2311yescharacteristic
65C2×C22⋊D109—21no—
66C2×C22⋊D109—21no—
67C2×C22⋊D109—21no—
68C22×Dic109—21no—
69C22×Dic109—21no—
70C22×Dic109D411yescharacteristic
71C22×Dic109D411yescharacteristic
72C2×C4×D109—21no—
73#13182019—22no—
74#13182017—22no—
75#13182016—22no—
76#13182014—22no—
77C2×#11724784—22no—
78C2×#11724784C2311yescharacteristic
79C2×#11724784C2311yescharacteristic
80C4⋊Dic109—21no—
81C4⋊Dic109—21no—
82C4⋊Dic109C2311yescharacteristic
83C4⋊Dic109C2311yescharacteristic
84#13182008—21no—
85#13182008—21no—
86C22⋊Dic109—21no—
87C22⋊Dic109—21no—
88C22⋊Dic109—21no—
89C22⋊Dic109—21no—
90C22⋊Dic109—21no—
91#13182006—21no—
92#13182006—21no—
93C4×Dic109—21no—
94C4×Dic109—21no—
95C4×Dic109—21no—
96C4×Dic109D411yescharacteristic
97C4×Dic109D411yescharacteristic
98C4×Dic109C2311yescharacteristic
99C4×Dic109C2311yescharacteristic
100#41×C109—22no—
101#41×C109—22no—
102#41×C109—22no—
103#41×C109D414yescomplemented
104C2×Q8×C109C2311yescharacteristic
105C2×Q8×C109C2311yescharacteristic
106C2×D4×C109—21no—
107C2×D4×C109—21no—
108C2×D4×C109C2311yescharacteristic
109C2×D4×C109C2311yescharacteristic
110C22×C4×C109—21no—
111C22×C4×C109—21no—
112C22×C4×C109C2311yescharacteristic
113Q16×C109—22no—
114SD16×C109—22no—
115SD16×C109—22no—
116D8×C109—22no—
117C2×C8×C109—22no—
118C2×C8×C109C2311yescharacteristic
119C2×C8×C109C2311yescharacteristic
120C23×C109—21no—
121C23×C109—21no—
122C23×C109C2×D411yescharacteristic
123C22×D109—21no—
124Q8×C109—21no—
125Q8×C109—21no—
126Q8×C109—21no—
127Q8×C109C2×D412yes—
128D4×C109—24no—
129D4×C109—21no—
130D4×C109—21no—
131D4×C109—21no—
132D4×C109—42no—
133D4×C109C2×D412yes—
134C2×C4×C109—21no—
135C2×C4×C109—24no—
136C2×C4×C109—21no—
137C2×C4×C109—42no—
138C2×C4×C109C2×D411yescharacteristic
139C2×C4×C109C2×D412yes—
140C2×C4×C109#4112yes—
141C2×C4×C109C2411yescharacteristic
142C22⋊D109—42no—
143C22⋊D109—42no—
144C22⋊D109—42no—
145C2×Dic109—22no—
146C2×Dic109—21no—
147C2×Dic109—21no—
148C2×Dic109—21no—
149C2×Dic109—41no—
150C2×Dic109—42no—
151C2×Dic109—42no—
152C2×Dic109C2×D411yescharacteristic
153C2×Dic109C2×D411yescharacteristic
154C2×Dic109C2×D411yescharacteristic
155C2×Dic109C2×D411yescharacteristic
156D436—21no—
157D436—21no—
158C4×D109—42no—
159Dic218—21no—
160Dic218—21no—
161C8×C109—21no—
162C8×C109—21no—
163#11724784—21no—
164#11724784—21no—
165C22×C109—21no—
166C22×C109—21no—
167C22×C109—22no—
168C22×C109—41no—
169C22×C109—41no—
170C22×C109—41no—
171C22×C109—41no—
172C22×C109—41no—
173C22×C109#12112yes—
174C22×C109C22×D411yescharacteristic
175C2×D109—41no—
176C2×D109—41no—
177C2×D109—41no—
178C4×C109—22no—
179C4×C109—41no—
180C4×C109#12112yes—
181C4×C109C22×D411yescharacteristic
182C4×C109C2×#4111yescharacteristic
183Dic109—21no—
184Dic109—21no—
185Dic109—21no—
186Dic109—41no—
187Dic109—41no—
188Dic109—41no—
189Dic109#13512yescomplemented
190C2×C109—22no—
191C2×C109—42no—
192C2×C109—81no—
193C2×C109C2×#12111yescharacteristic
194C2×C109C2×#13511yescharacteristic
195C2×C109#57511yescharacteristic
196D109—81no—
197#3035—1091nomaximal, Sylow
198C109#303511yescharacteristic, complemented, Sylow
199C2×#135—1091no—
200C2×#135D10911yescharacteristic, complemented
201#532—1091no—
202C4×#41—1091no—
203#464—1091no—
204#464—1091no—
205#463—1091no—
206#463—1091no—
207#462—1091no—
208#461—1091no—
209#460—1091no—
210#459—1091no—
211C2×C4⋊C8—1091no—
212#416—1091no—
213#416—1091no—
214C2×#41—1091no—
215C2×#41C2×D10911yescharacteristic
216C2×#41C2×D10911yescharacteristic
217#135—24no—
218#135—2184no—
219C2×Q16—1091no—
220C2×Q16C2×D10911yescharacteristic
221C2×SD16—1091no—
222C2×SD16—1091no—
223C2×SD16C2×D10911yescharacteristic
224C2×SD16C2×D10911yescharacteristic
225C2×D8—1091no—
226C2×D8C2×D10911yescharacteristic
227C22×C8—1091no—
228C22×C8C2×D10911yescharacteristic
229C4⋊Q8—1091no—
230#127—1091no—
231#124—1091no—
232#124—1091no—
233#121—2181no—
234#121—2181no—
235C4×Q8—1091no—
236C4×Q8—1091no—
237C4×D4—1091no—
238C4×D4—1091no—
239C4×D4—2181no—
240C4×D4—2181no—
241C4×D4—2181no—
242C4×D4—2181no—
243#117—1091no—
244#117—2181no—
245C2×C42—1091no—
246C2×C42—2181no—
247C4⋊C8—1091no—
248C4⋊C8—1091no—
249C4⋊C8—1091no—
250C4⋊C8—1091no—
251#103—1091no—
252#103—1091no—
253#103—1091no—
254#103—1091no—
255#102—1091no—
256#102—1091no—
257#102—1091no—
258#102—1091no—
259#41—22no—
260#41—22no—
261#41—22no—
262#41—2182no—
263#41—2182no—
264#41C22⋊D10914yescomplemented
265C2×Q8—1091no—
266C2×Q8C22×D10911yescharacteristic
267C2×Q8C22×D10911yescharacteristic
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—1091no—
271C2×D4—2181no—
272C2×D4—2181no—
273C2×D4—2181no—
274C2×D4C22×D10911yescharacteristic
275C2×D4C22×D10911yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—1091no—
279C22×C4—1091no—
280C22×C4—2181no—
281C22×C4—2181no—
282C22×C4—2181no—
283C22×C4C22×D10911yescharacteristic
284Q16—22no—
285Q16—2182no—
286SD16—22no—
287SD16—22no—
288SD16—2182no—
289SD16—2182no—
290D8—22no—
291D8—2182no—
292C2×C8—22no—
293C2×C8—1091no—
294C2×C8—1091no—
295C2×C8—2182no—
296C2×C8C22×D10911yescharacteristic
297C2×C8C22×D10911yescharacteristic
298C4⋊C4—1091no—
299C4⋊C4—1091no—
300C4⋊C4—2181no—
301C4⋊C4—2181no—
302C4⋊C4—2181no—
303C4⋊C4—2181no—
304C22⋊C4—2181no—
305C22⋊C4—2181no—
306C22⋊C4—2181no—
307C22⋊C4—2181no—
308C22⋊C4—2181no—
309C22⋊C4—2181no—
310C22⋊C4—2181no—
311C42—1091no—
312C42—1091no—
313C42—1091no—
314C42—1091no—
315C42—2181no—
316C42—2181no—
317C42—2181no—
318C23—21no—
319C23—21no—
320C23—2181no—
321C23D4×D10911yescharacteristic
322Q8—21no—
323Q8—21no—
324Q8—21no—
325Q8—2181no—
326Q8—2181no—
327Q8C2×C22⋊D10912yes—
328D4—24no—
329D4—21no—
330D4—21no—
331D4—21no—
332D4—42no—
333D4—4362no—
334D4—4362no—
335D4—4362no—
336D4—2181no—
337D4—2181no—
338D4C2×C22⋊D10912yes—
339C2×C4—21no—
340C2×C4—24no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—1091no—
344C2×C4—1091no—
345C2×C4—1091no—
346C2×C4—1091no—
347C2×C4—4362no—
348C2×C4—4362no—
349C2×C4—4362no—
350C2×C4—4361no—
351C2×C4—2182no—
352C2×C4—2181no—
353C2×C4—2181no—
354C2×C4—2181no—
355C2×C4D4×D10911yescharacteristic
356C2×C4C2×C22⋊D10912yes—
357C2×C4#1318202812yes—
358C2×C4C23×D10911yescharacteristic
359C8—21no—
360C8—21no—
361C8—2181no—
362C8—2181no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—4361no—
372C22—4361no—
373C22—4361no—
374C22#1681267112yes—
375C22C2×D4×D10911yescharacteristic
376C4—22no—
377C4—41no—
378C4—1092no—
379C4—4361no—
380C4—4361no—
381C4—4361no—
382C4—2181no—
383C4—2181no—
384C4—2181no—
385C4#1681267112yes—
386C4C22×C22⋊D10911yescharacteristic
387C4C2×#1318202811yescharacteristic
388C2—22no—
389C2—42no—
390C2—81no—
391C2—8721no—
392C2C2×#1681267111yescharacteristic
393C2#135×D10911yescharacteristic
394C2#2029582311yescharacteristic
subgroupG/H
centerC2×C4D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×#135×C109C2
socleC22×C109C22×D4
hypercenterC2×#135D109
hypocenterC109#3035
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#16812719C2semidirect
2. #20295572C2semidirect
3. #41×Dic109C2semidirect
4. #20295250C2semidirect
5. #20295246C2semidirect
6. #20295243C2semidirect
7. #20295239C2semidirect
8. #20295234C2semidirect
9. #20295229C2semidirect
10. #20295224C2semidirect
11. #20295219C2semidirect
12. C2×#16812628C2semidirect
13. #20295023C2semidirect
14. #20295022C2semidirect
15. C2×#135×C109C2semidirect
103. #41×C109D4semidirect
189. Dic109#135semidirect
198. C109#3035semidirect
200. C2×#135D109semidirect
264. #41C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127192
#202955722
#41×Dic1092
#202952502
#202952462
#202952432
#202952392
#202952342
#202952292
#202952242
#202952192
C2×#168126282
#202950232
#202950222
C2×#135×C1092
#3035109