Finite Group Databasebeta

← #31792340

Subgroups — #31792340

Order 17536.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984118C211yesmaximal, characteristic, complemented
2C2×#17984108C211yesmaximal, characteristic, complemented
3#22578062C211yesmaximal, characteristic, complemented
4#22577795C211yesmaximal, characteristic, complemented
5#22577651C211yesmaximal, characteristic, complemented
6#22577642C211yesmaximal, characteristic, complemented
7#22577618C211yesmaximal, characteristic, complemented
8#22577610C211yesmaximal, characteristic, complemented
9#22577565C211yesmaximal, characteristic, complemented
10#22577563C211yesmaximal, characteristic, complemented
11#22577558C211yesmaximal, characteristic, complemented
12#22577548C211yesmaximal, characteristic, complemented
13#22577503C211yesmaximal, characteristic, complemented
14#22577475C211yesmaximal, characteristic, complemented
15#547×C137C211yesmaximal, characteristic, complemented
16C2×#14555510C2211yescharacteristic
17C22×Dic274C2211yescharacteristic
18#17984118—24no—
19#17984108—24no—
20C2×#14555500C2211yescharacteristic
21C2×#14555500C2211yescharacteristic
22C2×#14555500C2211yescharacteristic
23C2×#14555500C2211yescharacteristic
24C2×#14555498C2211yescharacteristic
25C2×#14555498C2211yescharacteristic
26C2×#14555497C2211yescharacteristic
27C2×#14555497C2211yescharacteristic
28C2×#14555495C2211yescharacteristic
29C2×#14555494C2211yescharacteristic
30C22×#11764040C2211yescharacteristic
31#17984076C2211yescharacteristic
32#17984074—21no—
33#17984067C2211yescharacteristic
34#17984066—21no—
35#17984062C2211yescharacteristic
36#17984060—21no—
37C4×Dic274C2211yescharacteristic
38#17984048—21no—
39#17984021C2211yescharacteristic
40C4⋊#11764040C2211yescharacteristic
41#17984012C2211yescharacteristic
42#17984012C2211yescharacteristic
43#17984011C2211yescharacteristic
44#17984011C2211yescharacteristic
45#17984009C2211yescharacteristic
46#17984008C2211yescharacteristic
47C22⋊#11764040C2211yescharacteristic
48C22⋊#11764040C2211yescharacteristic
49#17983999C2211yescharacteristic
50C2×#41×C137C2211yescharacteristic
51C2×#41×C137C2211yescharacteristic
52C4⋊Q8×C137C2211yescharacteristic
53#123×C137—21no—
54#123×C137—21no—
55C22⋊Q8×C137C2211yescharacteristic
56C22⋊Q8×C137C2211yescharacteristic
57#121×C137—21no—
58C4×D4×C137C2211yescharacteristic
59C4×D4×C137C2211yescharacteristic
60C2×C4⋊C4×C137—21no—
61#14555510—22no—
62#14555510—41no—
63C2×Dic274—22no—
64C2×Dic274—21no—
65C2×Dic274—41no—
66C2×Dic274C2311yescharacteristic
67C2×Dic274C2311yescharacteristic
68C2×D548C2311yescharacteristic
69C2×C22⋊D137—21no—
70C2×C22⋊D137—21no—
71C22×Dic137—21no—
72C2×C4×D137—21no—
73#14555500—22no—
74#14555500—22no—
75#14555500—22no—
76#14555500—41no—
77#14555498—22no—
78#14555498D414yescomplemented
79#14555497—22no—
80#14555497—41no—
81#14555495—22no—
82#14555494—22no—
83C2×#11764040—22no—
84C2×#11764040C2311yescharacteristic
85C2×#11764040C2311yescharacteristic
86C2×#11764040C2311yescharacteristic
87C2×#11764040C2311yescharacteristic
88C4⋊Dic137C2311yescharacteristic
89#14555489—21no—
90#14555489—21no—
91C22⋊Dic137—21no—
92#14555487—21no—
93#14555487—21no—
94#14555487—21no—
95C4×Dic137—21no—
96#41×C137—22no—
97#41×C137—22no—
98#41×C137—41no—
99#41×C137—41no—
100C2×Q8×C137C2311yescharacteristic
101C2×Q8×C137C2311yescharacteristic
102C2×D4×C137—21no—
103C2×D4×C137—21no—
104C2×D4×C137D411yescharacteristic
105C2×D4×C137C2311yescharacteristic
106C22×C4×C137—21no—
107C22×C4×C137—21no—
108C22×C4×C137—21no—
109C22×C4×C137C2311yescharacteristic
110C22×C4×C137C2311yescharacteristic
111C4⋊C4×C137—21no—
112C4⋊C4×C137—21no—
113C4⋊C4×C137—21no—
114C4⋊C4×C137—21no—
115C4⋊C4×C137D411yescharacteristic
116C4⋊C4×C137C2311yescharacteristic
117C22⋊C4×C137—21no—
118C22⋊C4×C137—21no—
119C22⋊C4×C137—21no—
120C22⋊C4×C137D411yescharacteristic
121C22⋊C4×C137D411yescharacteristic
122C42×C137C2311yescharacteristic
123C22×D137—21no—
124Dic274—21no—
125Dic274—21no—
126Dic274—21no—
127Dic274—42no—
128Dic274C2×D412yes—
129D548—21no—
130D548—21no—
131C22⋊D137—42no—
132C22⋊D137—42no—
133C2×Dic137—21no—
134C2×Dic137—21no—
135C2×Dic137—21no—
136C2×Dic137—42no—
137C4×D137—42no—
138#11764040—21no—
139#11764040—21no—
140#11764040—21no—
141#11764040C2×D412yes—
142C23×C137—21no—
143C23×C137C2×D411yescharacteristic
144C23×C137C2×D411yescharacteristic
145Q8×C137—21no—
146Q8×C137—21no—
147Q8×C137—21no—
148Q8×C137—21no—
149D4×C137—21no—
150D4×C137—41no—
151D4×C137—42no—
152D4×C137—42no—
153D4×C137C2×D412yes—
154C2×C4×C137—21no—
155C2×C4×C137—21no—
156C2×C4×C137—21no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—42no—
162C2×C4×C137—42no—
163C2×C4×C137—42no—
164C2×C4×C137C2×D411yescharacteristic
165C2×C4×C137C2×D411yescharacteristic
166C2×C4×C137C2×D411yescharacteristic
167C2×C4×C137C2×D411yescharacteristic
168C2×C4×C137C2411yescharacteristic
169C22×C137—21no—
170C22×C137—21no—
171C22×C137—22no—
172C22×C137—21no—
173C22×C137—21no—
174C22×C137—21no—
175C22×C137—21no—
176C22×C137—41no—
177C22×C137—41no—
178C22×C137C22×D411yescharacteristic
179C2×D137—41no—
180C2×D137—41no—
181C2×D137—41no—
182C4×C137—21no—
183C4×C137—21no—
184C4×C137—21no—
185C4×C137—21no—
186C4×C137—41no—
187C4×C137—41no—
188C4×C137—41no—
189C4×C137C22×D411yescharacteristic
190C4×C137H(5,𝔽2)11yescharacteristic
191Dic137—22no—
192Dic137—41no—
193Dic137—41no—
194C2×C137—41no—
195C2×C137—41no—
196C2×C137—42no—
197C2×C137D4211yescharacteristic
198C2×C137#57511yescharacteristic
199C2×C137#57811yescharacteristic
200D137—81no—
201C137#326911yescharacteristic, complemented, Sylow
202#3269—1371nomaximal, Sylow
203C2×#137—1371no—
204C2×#135—1371no—
205#548—1371no—
206#547D13711yescharacteristic, complemented
207#497—1371no—
208#468—1371no—
209#467—1371no—
210#463—1371no—
211#462—1371no—
212#452—1371no—
213#452—1371no—
214#451—1371no—
215#449—1371no—
216#440—1371no—
217#435—1371no—
218C2×#41—1371no—
219C2×#41C2×D13711yescharacteristic
220C2×#41C2×D13711yescharacteristic
221C22×Q8—1371no—
222#137—2744no—
223#135—2744no—
224C2×Q16—1371no—
225C2×Q16—1371no—
226C2×Q16—1371no—
227C2×Q16—1371no—
228C2×SD16—1371no—
229C2×SD16—1371no—
230C2×SD16—1371no—
231C2×SD16—1371no—
232C2×D8—1371no—
233C2×M16—1371no—
234C22×C8—1371no—
235C4⋊Q8C2×D13711yescharacteristic
236#124—2741no—
237#124—1371no—
238#123—21no—
239#123—21no—
240C22⋊Q8—2741no—
241C22⋊Q8—1371no—
242C22⋊Q8C2×D13711yescharacteristic
243C22⋊Q8C2×D13711yescharacteristic
244#121—21no—
245#121—2741no—
246#121—1371no—
247C4×Q8—1371no—
248C4×D4—2741no—
249C4×D4C2×D13711yescharacteristic
250C4×D4C2×D13711yescharacteristic
251C2×C4⋊C4—21no—
252#107—1371no—
253C4⋊C8—1371no—
254#103—1371no—
255#103—1371no—
256#103—1371no—
257#103—1371no—
258#102—1371no—
259#102—1371no—
260C22⋊C8—1371no—
261C22⋊C8—1371no—
262#97—1371no—
263#41—22no—
264#41—22no—
265#41—41no—
266#41—41no—
267#41—2742no—
268#41—5481no—
269C2×Q8—2742no—
270C2×Q8—2741no—
271C2×Q8—5481no—
272C2×Q8—1371no—
273C2×Q8—1371no—
274C2×Q8C22×D13711yescharacteristic
275C2×Q8C22×D13711yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—2741no—
279C2×D4—2741no—
280C2×D4—1371no—
281C2×D4C22⋊D13711yescharacteristic
282C2×D4C22×D13711yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—2741no—
287C22×C4—2741no—
288C22×C4C22×D13711yescharacteristic
289C22×C4C22×D13711yescharacteristic
290Q16—2742no—
291Q16—2742no—
292Q16—2742no—
293Q16—5481no—
294SD16—2742no—
295SD16—2742no—
296SD16—5481no—
297SD16—1374no—
298D8—2742no—
299M16—2742no—
300C2×C8—2742no—
301C2×C8—1371no—
302C2×C8—1371no—
303C2×C8—1371no—
304C2×C8—1371no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—2741no—
310C4⋊C4—2741no—
311C4⋊C4—1371no—
312C4⋊C4C22⋊D13711yescharacteristic
313C4⋊C4C22×D13711yescharacteristic
314C22⋊C4—21no—
315C22⋊C4—21no—
316C22⋊C4—21no—
317C22⋊C4—2741no—
318C22⋊C4—2741no—
319C22⋊C4—2741no—
320C22⋊C4—2741no—
321C22⋊C4C22⋊D13711yescharacteristic
322C22⋊C4C22⋊D13711yescharacteristic
323C42—2741no—
324C42C22×D13711yescharacteristic
325C23—21no—
326C23—2741no—
327C23C2×C22⋊D13711yescharacteristic
328C23C2×C22⋊D13711yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—2741no—
334Q8—2741no—
335Q8—2741no—
336Q8—5482no—
337Q8—1372no—
338D4—21no—
339D4—41no—
340D4—42no—
341D4—42no—
342D4—2741no—
343D4—2741no—
344D4—5482no—
345D4—5482no—
346D4D4×D13712yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—2741no—
358C2×C4—2741no—
359C2×C4—2741no—
360C2×C4—5482no—
361C2×C4—5482no—
362C2×C4C2×C22⋊D13711yescharacteristic
363C2×C4C2×C22⋊D13711yescharacteristic
364C2×C4C2×C22⋊D13711yescharacteristic
365C2×C4C2×C22⋊D13711yescharacteristic
366C2×C4C23×D13711yescharacteristic
367C8—2741no—
368C8—2741no—
369C8—2741no—
370C8—1372no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—41no—
379C22—41no—
380C22—5481no—
381C22—5481no—
382C22—5481no—
383C22C22×C22⋊D13711yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—2742no—
392C4—5481no—
393C4—5481no—
394C4C2×D4×D13711yescharacteristic
395C4#1798413311yescharacteristic
396C2—41no—
397C2—41no—
398C2—42no—
399C2—10961no—
400C2D4×C22⋊D13711yescharacteristic
401C2#2257820111yescharacteristic
402C2#2257821811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#547×C137C2
socleC22×C137C22×D4
hypercenter#547D137
hypocenterC137#3269
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C2×#17984118C2semidirect
2. C2×#17984108C2semidirect
3. #22578062C2semidirect
4. #22577795C2semidirect
5. #22577651C2semidirect
6. #22577642C2semidirect
7. #22577618C2semidirect
8. #22577610C2semidirect
9. #22577565C2semidirect
10. #22577563C2semidirect
11. #22577558C2semidirect
12. #22577548C2semidirect
13. #22577503C2semidirect
14. #22577475C2semidirect
15. #547×C137C2semidirect
78. #14555498D4semidirect
201. C137#3269semidirect
206. #547D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841182
C2×#179841082
#225780622
#225777952
#225776512
#225776422
#225776182
#225776102
#225775652
#225775632
#225775582
#225775482
#225775032
#225774752
#547×C1372
#3269137