Finite Group Databasebeta

← #31792702

Subgroups — #31792702

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578121C211yesmaximal, characteristic, complemented
2#22578052C211yesmaximal, characteristic
3#22577752C211yesmaximal, characteristic, complemented
4#22577744C211yesmaximal, characteristic, complemented
5#22577743C211yesmaximal, characteristic, complemented
6#22577681C211yesmaximal, characteristic, complemented
7#22577644C211yesmaximal, characteristic, complemented
8#22577643C211yesmaximal, characteristic, complemented
9#22577612C211yesmaximal, characteristic, complemented
10#22577566C211yesmaximal, characteristic, complemented
11#22577565C211yesmaximal, characteristic, complemented
12#22577494C211yesmaximal, characteristic, complemented
13C8×C22⋊D137C211yesmaximal, characteristic, complemented
14#22577396C211yesmaximal, characteristic, complemented
15#416×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×Dic548C2211yescharacteristic
18C2×#14555500C2211yescharacteristic
19C2×#14555498C2211yescharacteristic
20C2×C8×D137C2211yescharacteristic
21#17984087C2211yescharacteristic
22#17984086C2211yescharacteristic
23#17984082—21no—
24#17984081—21no—
25#17984074—21no—
26#17984071—21no—
27#17984069—21no—
28#17984068C2211yescharacteristic
29#17984067C2211yescharacteristic
30#17984067C2211yescharacteristic
31#17984064C2211yescharacteristic
32#17984060—21no—
33C4×Dic274—21no—
34#17984053C2211yescharacteristic
35D4×Dic137C2211yescharacteristic
36C4×C22⋊D137C2211yescharacteristic
37#17984046C2211yescharacteristic
38C2×#14555489—21no—
39#17984022C2211yescharacteristic
40#17984021C2211yescharacteristic
41#17984019C2211yescharacteristic
42#17984017C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984013C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984011C2211yescharacteristic
47#17984011C2211yescharacteristic
48#17984009C2211yescharacteristic
49C22⋊#11764040C2211yescharacteristic
50#17984000C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×#41×C137C2211yescharacteristic
53C22×C8×C137C2211yescharacteristic
54#117×C137C2211yescharacteristic
55#103×C137C2211yescharacteristic
56#103×C137C2211yescharacteristic
57#102×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—41no—
61C2×Dic274—21no—
62C2×Dic274C2311yescharacteristic
63C2×Dic274C2311yescharacteristic
64C2×C22⋊D137—21no—
65C2×C22⋊D137D411yescharacteristic
66C22×Dic137—21no—
67C22×Dic137—21no—
68C2×C4×D137C2311yescharacteristic
69Dic548—41no—
70#14555500—41no—
71#14555498—22no—
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C8×D137—41no—
75C4⋊Dic137—21no—
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85#14555489D411yescharacteristic
86C22⋊Dic137—21no—
87C22⋊Dic137—21no—
88C22⋊Dic137—21no—
89C22⋊Dic137D411yescharacteristic
90#14555487—21no—
91#14555487—21no—
92#14555487—21no—
93#14555487D411yescharacteristic
94C4×Dic137—21no—
95C4×Dic137—21no—
96C4×Dic137C2311yescharacteristic
97#41×C137—22no—
98#41×C137—41no—
99C2×Q8×C137C2311yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137C2311yescharacteristic
102C22×C4×C137—21no—
103C22×C4×C137C2311yescharacteristic
104C2×C8×C137—41no—
105C2×C8×C137C2311yescharacteristic
106C2×C8×C137C2311yescharacteristic
107C4⋊C4×C137C2311yescharacteristic
108C4⋊C4×C137C2311yescharacteristic
109C22⋊C4×C137—21no—
110C42×C137—21no—
111C22×D137C2×D411yescharacteristic
112Dic274—21no—
113Dic274—21no—
114Dic274—21no—
115Dic274—21no—
116Dic274—42no—
117C22⋊D137—22no—
118C22⋊D137—42no—
119C2×Dic137—21no—
120C2×Dic137—21no—
121C2×Dic137—21no—
122C2×Dic137—21no—
123C2×Dic137—42no—
124C2×Dic137—42no—
125C2×Dic137C2×D411yescharacteristic
126C2×Dic137C2×D411yescharacteristic
127C2×Dic137C2×D411yescharacteristic
128C4×D137—21no—
129C4×D137—21no—
130#11764040—21no—
131#11764040—21no—
132C23×C137—21no—
133C23×C137C2×D411yescharacteristic
134Q8×C137—21no—
135Q8×C137—21no—
136D4×C137—21no—
137D4×C137—42no—
138D4×C137#4112yes—
139C2×C4×C137—21no—
140C2×C4×C137—21no—
141C2×C4×C137—21no—
142C2×C4×C137—21no—
143C2×C4×C137—21no—
144C2×C4×C137—42no—
145C2×C4×C137C2×D411yescharacteristic
146C2×C4×C137C2411yescharacteristic
147C8×C137—21no—
148C8×C137—21no—
149C22×C137—21no—
150C22×C137—22no—
151C22×C137—21no—
152C22×C137—21no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137C22×D411yescharacteristic
156C2×D137—21no—
157C2×D137—21no—
158C2×D137—21no—
159C4×C137—21no—
160C4×C137—41no—
161C4×C137—41no—
162C4×C137—41no—
163C4×C137C2×#4111yescharacteristic
164C4×C137H(5,𝔽2)11yescharacteristic
165Dic137—21no—
166Dic137—21no—
167Dic137—41no—
168Dic137—41no—
169Dic137—41no—
170Dic137—41no—
171Dic137#13512yes—
172C2×C137—41no—
173C2×C137—42no—
174C2×C137#54611yescharacteristic
175C2×C137C2×#13511yescharacteristic
176C2×C137#57811yescharacteristic
177D137—41no—
178C137#330211yescharacteristic, complemented, Sylow
179#3302—1371nomaximal, Sylow
180#556—1371no—
181#547—1371no—
182#487—1371no—
183#485—1371no—
184#485—1371no—
185#474—1371no—
186#467—1371no—
187#467—1371no—
188#462—1371no—
189#452—1371no—
190#452—1371no—
191C4×SD16—1371no—
192C8×D4—1371no—
193#416—1371no—
194#416D13711yescharacteristic, complemented
195C2×#41—1371no—
196C2×#41C2×D13711yescharacteristic
197C2×Q16—1371no—
198C2×Q16—1371no—
199C2×SD16—1371no—
200C22×C8—1371no—
201C22×C8C2×D13711yescharacteristic
202C4⋊Q8—1371no—
203C4⋊Q8—1371no—
204#126—2741no—
205#126—2741no—
206#124—2741no—
207#123—2741no—
208#123—2741no—
209C22⋊Q8—1371no—
210C22⋊Q8—1371no—
211C22⋊Q8—1371no—
212C22⋊Q8—1371no—
213#121—2741no—
214C4×Q8—2741no—
215C4×Q8—1371no—
216C4×D4—1371no—
217C4×D4—1371no—
218#117—1371no—
219#117C2×D13711yescharacteristic
220C2×C4⋊C4—2741no—
221#107—1371no—
222#107—1371no—
223#106—1371no—
224C4⋊C8—1371no—
225#103—1371no—
226#103—1371no—
227#103—1371no—
228#103—1371no—
229#103—1371no—
230#103C2×D13711yescharacteristic
231#103C2×D13711yescharacteristic
232#102—1371no—
233#102C2×D13711yescharacteristic
234#102C2×D13711yescharacteristic
235C22⋊C8—1371no—
236C22⋊C8—1371no—
237C4×C8—1371no—
238#41—22no—
239#41—41no—
240#41—2742no—
241#41—5481no—
242C2×Q8—2741no—
243C2×Q8—1371no—
244C2×Q8—1371no—
245C2×Q8C22×D13711yescharacteristic
246C2×D4—21no—
247C2×D4—2741no—
248C2×D4—1371no—
249C2×D4C22×D13711yescharacteristic
250C22×C4—21no—
251C22×C4—2741no—
252C22×C4—2741no—
253C22×C4—1371no—
254C22×C4C22×D13711yescharacteristic
255Q16—5481no—
256Q16—5481no—
257SD16—2742no—
258C2×C8—41no—
259C2×C8—5481no—
260C2×C8—1371no—
261C2×C8—1371no—
262C2×C8C22×D13711yescharacteristic
263C2×C8C22×D13711yescharacteristic
264C4⋊C4—2741no—
265C4⋊C4—2741no—
266C4⋊C4—2741no—
267C4⋊C4—2741no—
268C4⋊C4—2741no—
269C4⋊C4—2741no—
270C4⋊C4—2741no—
271C4⋊C4—2741no—
272C4⋊C4—1371no—
273C4⋊C4—1371no—
274C4⋊C4—1371no—
275C4⋊C4C22×D13711yescharacteristic
276C4⋊C4C22×D13711yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—2741no—
279C22⋊C4—2741no—
280C22⋊C4—2741no—
281C22⋊C4—2741no—
282C22⋊C4—2741no—
283C22⋊C4—2741no—
284C22⋊C4—1371no—
285C22⋊C4—1371no—
286C42—21no—
287C42—2741no—
288C42—2741no—
289C42—1371no—
290C23—21no—
291C23—1371no—
292C23D4×D13711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—2741no—
296Q8—2741no—
297Q8—2741no—
298Q8—2741no—
299Q8—5482no—
300D4—21no—
301D4—42no—
302D4—2742no—
303D4—5482no—
304D4#1455551012yes—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—2741no—
312C2×C4—2741no—
313C2×C4—2741no—
314C2×C4—2741no—
315C2×C4—2741no—
316C2×C4—2741no—
317C2×C4—5482no—
318C2×C4—5482no—
319C2×C4—1371no—
320C2×C4—1371no—
321C2×C4—1371no—
322C2×C4D4×D13711yescharacteristic
323C2×C4C23×D13711yescharacteristic
324C8—21no—
325C8—21no—
326C8—2741no—
327C8—2741no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—2741no—
335C22—2741no—
336C22—2741no—
337C22C2×D4×D13711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—2741no—
343C4—2741no—
344C4—5481no—
345C4—5481no—
346C4—5481no—
347C4—5481no—
348C4—1372no—
349C4C2×#1455551011yescharacteristic
350C4#1798413311yescharacteristic
351C2—41no—
352C2—42no—
353C2—5481no—
354C2#2257804311yescharacteristic
355C2#135×D13711yescharacteristic
356C2#2257821711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#416×C137C2
socleC22×C137C22×D4
prime radical#22578052C2
hypercenter#416D137
hypocenterC137#3302
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #22578121C2semidirect
3. #22577752C2semidirect
4. #22577744C2semidirect
5. #22577743C2semidirect
6. #22577681C2semidirect
7. #22577644C2semidirect
8. #22577643C2semidirect
9. #22577612C2semidirect
10. #22577566C2semidirect
11. #22577565C2semidirect
12. #22577494C2semidirect
13. C8×C22⋊D137C2semidirect
14. #22577396C2semidirect
15. #416×C137C2semidirect
178. C137#3302semidirect
194. #416D137semidirect

Maximal subgroups

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

subgroupindex
#225781212
#225780522
#225777522
#225777442
#225777432
#225776812
#225776442
#225776432
#225776122
#225775662
#225775652
#225774942
C8×C22⋊D1372
#225773962
#416×C1372
#3302137