Finite Group Databasebeta

← #31792660

Subgroups — #31792660

Order 17536.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578111C211yesmaximal, characteristic
2#22578066C211yesmaximal, characteristic
3#22577752C211yesmaximal, characteristic, complemented
4#22577740C211yesmaximal, characteristic, complemented
5#22577736C211yesmaximal, characteristic, complemented
6#22577697C211yesmaximal, characteristic, complemented
7#22577643C211yesmaximal, characteristic, complemented
8#22577611C211yesmaximal, characteristic, complemented
9#22577558C211yesmaximal, characteristic, complemented
10#22577557C211yesmaximal, characteristic, complemented
11#22577495C211yesmaximal, characteristic, complemented
12C8×C22⋊D137C211yesmaximal, characteristic
13C2×#17984013C211yesmaximal, characteristic, complemented
14#103×D137C211yesmaximal, characteristic, complemented
15#467×C137C211yesmaximal, characteristic, complemented
16C22×Dic274C2211yescharacteristic
17C2×Q8×D137C2211yescharacteristic
18C2×#14555500C2211yescharacteristic
19C2×#14555500C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic
21#17984087—21no—
22#17984087C2211yescharacteristic
23#17984086C2211yescharacteristic
24#17984078—21no—
25#17984068—21no—
26#17984068C2211yescharacteristic
27#17984066—21no—
28#17984065—21no—
29#17984065—21no—
30#17984064C2211yescharacteristic
31Q8×Dic137C2211yescharacteristic
32#17984053C2211yescharacteristic
33C4×C22⋊D137C2211yescharacteristic
34#17984048—21no—
35#17984048—21no—
36C2×C4⋊Dic137C2211yescharacteristic
37C4⋊C4×D137C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984021C2211yescharacteristic
40#17984017C2211yescharacteristic
41#17984013C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984011C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×Q16×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54C22⋊Q8×C137C2211yescharacteristic
55C22⋊Q8×C137C2211yescharacteristic
56#107×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#103×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274—21no—
61C2×Dic274—22no—
62C2×Dic274—21no—
63C2×Dic274—41no—
64C2×Dic274C2311yescharacteristic
65C2×Dic274C2311yescharacteristic
66Q8×D137—22no—
67Q8×D137—41no—
68C2×C22⋊D137D411yescharacteristic
69C22×Dic137—21no—
70C22×Dic137—21no—
71C2×C4×D137—21no—
72C2×C4×D137—21no—
73C2×C4×D137C2311yescharacteristic
74#14555500—22no—
75#14555500—41no—
76C2×#11764040C2311yescharacteristic
77C2×#11764040C2311yescharacteristic
78C8×D137—41no—
79C4⋊Dic137—21no—
80C4⋊Dic137—21no—
81C4⋊Dic137—21no—
82C4⋊Dic137C2311yescharacteristic
83C4⋊Dic137C2311yescharacteristic
84#14555489—21no—
85#14555489—21no—
86#14555489—21no—
87#14555489—21no—
88#14555489—21no—
89#14555489—21no—
90#14555489D411yescharacteristic
91C22⋊Dic137D411yescharacteristic
92#14555487—21no—
93#14555487—21no—
94#14555487D411yescharacteristic
95C4×Dic137—21no—
96C4×Dic137—21no—
97C4×Dic137C2311yescharacteristic
98C2×Q8×C137C2311yescharacteristic
99C2×Q8×C137C2311yescharacteristic
100C22×C4×C137C2311yescharacteristic
101Q16×C137—41no—
102C2×C8×C137—41no—
103C2×C8×C137C2311yescharacteristic
104C2×C8×C137C2311yescharacteristic
105C4⋊C4×C137—21no—
106C4⋊C4×C137—21no—
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C22⋊C4×C137—21no—
110C22⋊C4×C137—21no—
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—42no—
116Dic274—42no—
117Dic274—81no—
118Dic274#4112yes—
119C22⋊D137Q1614yescomplemented
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—21no—
124C2×Dic137—42no—
125C2×Dic137—81no—
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C2×Dic137C2×D411yescharacteristic
129C4×D137—21no—
130C4×D137—21no—
131C4×D137—42no—
132C4×D137—81no—
133#11764040—21no—
134#11764040—21no—
135C23×C137C2×D411yescharacteristic
136Q8×C137—21no—
137Q8×C137—21no—
138Q8×C137—21no—
139Q8×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137C2×D411yescharacteristic
147C2×C4×C137C2411yescharacteristic
148C8×C137—21no—
149C8×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137C2×Q1612yes—
153C22×C137C22×D411yescharacteristic
154C2×D137—21no—
155C2×D137—21no—
156C2×D137C2×Q1612yes—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137C2×#4111yescharacteristic
163C4×C137H(5,𝔽2)11yescharacteristic
164Dic137—21no—
165Dic137—22no—
166Dic137—21no—
167Dic137—41no—
168Dic137—41no—
169Dic137—41no—
170Dic137C2×Q1612yes—
171C2×C137—22no—
172C2×C137#54611yescharacteristic
173C2×C137C22×Q1611yescharacteristic
174C2×C137#57711yescharacteristic
175D137—22no—
176C137#329911yescharacteristic, complemented, Sylow
177#3299—1371nomaximal, Sylow
178#554—1371no—
179D4×Q8—1371no—
180#487—1371no—
181#484—1371no—
182#484—1371no—
183#477—1371no—
184#467—1371no—
185#467D13711yescharacteristic, complemented
186#462—1371no—
187#451—1371no—
188#451—1371no—
189C4×Q16—1371no—
190C8×D4—1371no—
191C2×#103—1371no—
192C2×#103—1371no—
193C22×Q8—1371no—
194C22×Q8—1371no—
195C2×Q16—1371no—
196C2×Q16—1371no—
197C2×Q16C2×D13711yescharacteristic
198C22×C8—1371no—
199C22×C8C2×D13711yescharacteristic
200C4⋊Q8—2741no—
201C4⋊Q8—1371no—
202C4⋊Q8—1371no—
203#125—2741no—
204C22⋊Q8—2741no—
205C22⋊Q8—2741no—
206C22⋊Q8—2741no—
207C22⋊Q8—2741no—
208C22⋊Q8—1371no—
209C22⋊Q8—1371no—
210C22⋊Q8C2×D13711yescharacteristic
211C22⋊Q8C2×D13711yescharacteristic
212C4×Q8—1371no—
213C4×Q8—1371no—
214C4×D4—2741no—
215C4×D4—2741no—
216C4×D4—1371no—
217C2×C4⋊C4—1371no—
218C2×C4⋊C4—1371no—
219#107—1371no—
220#107—1371no—
221#107C2×D13711yescharacteristic
222C4⋊C8—1371no—
223#103—1371no—
224#103—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103—1371no—
231#103C2×D13711yescharacteristic
232#103C2×D13711yescharacteristic
233C22⋊C8—1371no—
234C22⋊C8—1371no—
235C4×C8—1371no—
236C2×Q8—2741no—
237C2×Q8—2741no—
238C2×Q8—2742no—
239C2×Q8—2741no—
240C2×Q8—2742no—
241C2×Q8—5481no—
242C2×Q8—5481no—
243C2×Q8—1371no—
244C2×Q8—1371no—
245C2×Q8C22×D13711yescharacteristic
246C2×Q8C22×D13711yescharacteristic
247C2×D4—1371no—
248C22×C4—2741no—
249C22×C4—2741no—
250C22×C4—2741no—
251C22×C4—2741no—
252C22×C4—1371no—
253C22×C4C22×D13711yescharacteristic
254Q16—41no—
255Q16—2742no—
256Q16—5481no—
257C2×C8—41no—
258C2×C8—5481no—
259C2×C8—1371no—
260C2×C8—1371no—
261C2×C8C22×D13711yescharacteristic
262C2×C8C22×D13711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—2741no—
273C4⋊C4—2741no—
274C4⋊C4—1371no—
275C4⋊C4—1371no—
276C4⋊C4—1371no—
277C4⋊C4C22×D13711yescharacteristic
278C4⋊C4C22×D13711yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—21no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—1371no—
284C22⋊C4—1371no—
285C42—2741no—
286C42—2741no—
287C42—1371no—
288C23—1371no—
289C23D4×D13711yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—21no—
293Q8—21no—
294Q8—2741no—
295Q8—2741no—
296Q8—2741no—
297Q8—5482no—
298Q8—5482no—
299Q8—10961no—
300Q8—1372no—
301D4—1374no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—2741no—
309C2×C4—2741no—
310C2×C4—2741no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—5482no—
315C2×C4—5482no—
316C2×C4—10961no—
317C2×C4—10961no—
318C2×C4—1371no—
319C2×C4—1371no—
320C2×C4—1371no—
321C2×C4D4×D13711yescharacteristic
322C2×C4C23×D13711yescharacteristic
323C8—21no—
324C8—21no—
325C8—2741no—
326C8—2741no—
327C22—21no—
328C22—21no—
329C22—2741no—
330C22—2741no—
331C22—1372no—
332C22Q16×D13712yes—
333C22C2×D4×D13711yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—2741no—
340C4—2741no—
341C4—2742no—
342C4—5481no—
343C4—5481no—
344C4—5481no—
345C4—1372no—
346C4#41×D13711yescharacteristic
347C4#1798413311yescharacteristic
348C2—22no—
349C2—2742no—
350C2#2257804211yescharacteristic
351C2C2×Q16×D13711yescharacteristic
352C2#2257821011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#467×C137C2
socleC22×C137C22×D4
prime radicalC2×C22⋊D137D4
hypercenter#467D137
hypocenterC137#3299
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
3. #22577752C2semidirect
4. #22577740C2semidirect
5. #22577736C2semidirect
6. #22577697C2semidirect
7. #22577643C2semidirect
8. #22577611C2semidirect
9. #22577558C2semidirect
10. #22577557C2semidirect
11. #22577495C2semidirect
13. C2×#17984013C2semidirect
14. #103×D137C2semidirect
15. #467×C137C2semidirect
119. C22⋊D137Q16semidirect
176. C137#3299semidirect
185. #467D137semidirect

Maximal subgroups

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

subgroupindex
#225781112
#225780662
#225777522
#225777402
#225777362
#225776972
#225776432
#225776112
#225775582
#225775572
#225774952
C8×C22⋊D1372
C2×#179840132
#103×D1372
#467×C1372
#3299137