Finite Group Databasebeta

← #31791000

Subgroups — #31791000

Order 17536.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22578014C211yesmaximal, characteristic, complemented
2#22577882C211yesmaximal, characteristic
3#22577737C211yesmaximal, characteristic, complemented
4#22577732C211yesmaximal, characteristic, complemented
5#22577711C211yesmaximal, characteristic, complemented
6#22577703C211yesmaximal, characteristic, complemented
7#22577610C211yesmaximal, characteristic, complemented
8#22577562C211yesmaximal, characteristic
9#22577562C211yesmaximal, characteristic
10#22577507C211yesmaximal, characteristic, complemented
11#22577504C211yesmaximal, characteristic, complemented
12#22577495C211yesmaximal, characteristic, complemented
13SD16×Dic137C211yesmaximal, characteristic, complemented
14#22577463C211yesmaximal, characteristic, complemented
15#461×C137C211yesmaximal, characteristic, complemented
16C2×#14555509C2211yescharacteristic
17C2×#14555500C2211yescharacteristic
18C2×#14555500C2211yescharacteristic
19#17984083—21no—
20#17984083—21no—
21#17984080C2211yescharacteristic
22#17984078—21no—
23#17984078—21no—
24#17984064C2211yescharacteristic
25#17984064C2211yescharacteristic
26Q8×Dic137C2211yescharacteristic
27Q8×Dic137C2211yescharacteristic
28C4×Dic274C2211yescharacteristic
29#17984053C2211yescharacteristic
30D4×Dic137C2211yescharacteristic
31C4×C22⋊D137—21no—
32#17984048—21no—
33#17984045C2211yescharacteristic
34#17984044—21no—
35C2×#14555489—21no—
36C4⋊C4×D137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984019C2211yescharacteristic
41C4⋊#11764040C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984009C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984000C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17983998C2211yescharacteristic
51C8×Dic137C2211yescharacteristic
52C2×SD16×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54C4⋊Q8×C137C2211yescharacteristic
55C4×D4×C137C2211yescharacteristic
56C4⋊C8×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#103×C137C2211yescharacteristic
59#14555509—22no—
60#14555509—22no—
61C2×Dic274C2311yescharacteristic
62C2×C22⋊D137—21no—
63C22×Dic137—21no—
64C22×Dic137—21no—
65C2×C4×D137—21no—
66C2×C4×D137C2311yescharacteristic
67#14555500—22no—
68#14555500—22no—
69C2×#11764040C2311yescharacteristic
70C2×#11764040C2311yescharacteristic
71C4⋊Dic137—21no—
72C4⋊Dic137—21no—
73C4⋊Dic137C2311yescharacteristic
74C4⋊Dic137C2311yescharacteristic
75C4⋊Dic137C2311yescharacteristic
76#14555489—21no—
77#14555489—21no—
78#14555489—21no—
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489D411yescharacteristic
84#14555489D411yescharacteristic
85C22⋊Dic137—21no—
86#14555487—21no—
87#14555487—21no—
88#14555487D411yescharacteristic
89#14555487D411yescharacteristic
90C4×Dic137—21no—
91C4×Dic137—21no—
92C4×Dic137—21no—
93C4×Dic137—21no—
94C4×Dic137C2311yescharacteristic
95C2×Q8×C137C2311yescharacteristic
96C2×Q8×C137C2311yescharacteristic
97C2×D4×C137C2311yescharacteristic
98C22×C4×C137—21no—
99SD16×C137—22no—
100SD16×C137—22no—
101C2×C8×C137C2311yescharacteristic
102C2×C8×C137C2311yescharacteristic
103C4⋊C4×C137—21no—
104C4⋊C4×C137—21no—
105C4⋊C4×C137C2311yescharacteristic
106C22⋊C4×C137—21no—
107C42×C137C2311yescharacteristic
108C22×D137C2×D411yescharacteristic
109Dic274—21no—
110Dic274#4112yes—
111C22⋊D137—24no—
112C2×Dic137—24no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—81no—
118C2×Dic137C2×D411yescharacteristic
119C2×Dic137C2×D411yescharacteristic
120C2×Dic137C2×D411yescharacteristic
121C4×D137—21no—
122C4×D137—21no—
123C4×D137—81no—
124#11764040—21no—
125#11764040—21no—
126C23×C137—21no—
127Q8×C137—21no—
128Q8×C137—21no—
129Q8×C137—21no—
130Q8×C137—21no—
131D4×C137—21no—
132D4×C137#4112yes—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—81no—
137C2×C4×C137C2×D411yescharacteristic
138C2×C4×C137C2×D411yescharacteristic
139C2×C4×C137C2411yescharacteristic
140C8×C137—21no—
141C8×C137—21no—
142C22×C137—22no—
143C22×C137—41no—
144C22×C137—41no—
145C22×C137C22×D411yescharacteristic
146C2×D137—21no—
147C2×D137—21no—
148C2×D137—21no—
149C4×C137—21no—
150C4×C137—21no—
151C4×C137—41no—
152C4×C137—41no—
153C4×C137—41no—
154C4×C137C2×#4111yescharacteristic
155C4×C137C2×#4111yescharacteristic
156Dic137—21no—
157Dic137—22no—
158Dic137—21no—
159Dic137—21no—
160Dic137—41no—
161Dic137—41no—
162Dic137—41no—
163C2×C137—42no—
164C2×C137#52511yescharacteristic
165C2×C137#57711yescharacteristic
166C2×C137#57811yescharacteristic
167D137—41no—
168C137#312011yescharacteristic, complemented, Sylow
169#3120—1371nomaximal, Sylow
170#541—1371no—
171#520—1371no—
172#484—1371no—
173#483—1371no—
174#479—1371no—
175#478—1371no—
176#462—1371no—
177#461D13711yescharacteristic, complemented
178#452—1371no—
179#452—1371no—
180#441—1371no—
181#440—1371no—
182C4×Q16—1371no—
183C4×SD16—1371no—
184#433—1371no—
185C2×#41—1371no—
186C2×Q16—1371no—
187C2×Q16—1371no—
188C2×SD16C2×D13711yescharacteristic
189C2×SD16C2×D13711yescharacteristic
190C4⋊Q8C2×D13711yescharacteristic
191#126—2741no—
192#126—2741no—
193#125—2741no—
194#125—2741no—
195#125—1371no—
196C22⋊Q8—1371no—
197C22⋊Q8—1371no—
198C4×Q8—1371no—
199C4×Q8—1371no—
200C4×Q8—1371no—
201C4×Q8—1371no—
202C4×D4—2741no—
203C4×D4—2741no—
204C4×D4—1371no—
205C4×D4C2×D13711yescharacteristic
206#117—2741no—
207#117—1371no—
208C2×C4⋊C4—2741no—
209C2×C4⋊C4—1371no—
210#107—1371no—
211#107—1371no—
212#106—1371no—
213#106—1371no—
214C4⋊C8—1371no—
215C4⋊C8C2×D13711yescharacteristic
216#103—1371no—
217#103—1371no—
218#103—1371no—
219#103—1371no—
220#103C2×D13711yescharacteristic
221#103C2×D13711yescharacteristic
222#102—1371no—
223#102—1371no—
224C22⋊C8—1371no—
225C22⋊C8—1371no—
226#97—1371no—
227C4×C8—1371no—
228#41—2742no—
229#41—2742no—
230C2×Q8—1371no—
231C2×Q8C22×D13711yescharacteristic
232C2×Q8C22×D13711yescharacteristic
233C2×D4—2741no—
234C2×D4C22×D13711yescharacteristic
235C22×C4—21no—
236C22×C4—2741no—
237C22×C4—2741no—
238C22×C4—2741no—
239C22×C4—1371no—
240Q16—2742no—
241Q16—2742no—
242SD16—22no—
243SD16—22no—
244C2×C8—1371no—
245C2×C8—1371no—
246C2×C8C22×D13711yescharacteristic
247C2×C8C22×D13711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—21no—
250C4⋊C4—2741no—
251C4⋊C4—2741no—
252C4⋊C4—2741no—
253C4⋊C4—2741no—
254C4⋊C4—2741no—
255C4⋊C4—2741no—
256C4⋊C4—2741no—
257C4⋊C4—2741no—
258C4⋊C4—2741no—
259C4⋊C4—1371no—
260C4⋊C4—1371no—
261C4⋊C4—1371no—
262C4⋊C4—1371no—
263C4⋊C4—1371no—
264C4⋊C4C22×D13711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—2741no—
267C22⋊C4—2741no—
268C22⋊C4—2741no—
269C22⋊C4—1371no—
270C22⋊C4—1371no—
271C42—2741no—
272C42—2741no—
273C42—2741no—
274C42—2741no—
275C42—1371no—
276C42C22×D13711yescharacteristic
277C23—21no—
278C23—1371no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—2741no—
284Q8—1372no—
285D4—21no—
286D4—2744no—
287D4#1455550912yes—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—81no—
292C2×C4—2744no—
293C2×C4—2741no—
294C2×C4—2741no—
295C2×C4—2741no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—10961no—
300C2×C4—10961no—
301C2×C4—1371no—
302C2×C4—1371no—
303C2×C4—1371no—
304C2×C4D4×D13711yescharacteristic
305C2×C4D4×D13711yescharacteristic
306C2×C4C23×D13711yescharacteristic
307C8—21no—
308C8—21no—
309C8—2741no—
310C8—2741no—
311C22—22no—
312C22—41no—
313C22—41no—
314C22—2741no—
315C22—2741no—
316C22—2741no—
317C22C2×D4×D13711yescharacteristic
318C4—21no—
319C4—21no—
320C4—41no—
321C4—41no—
322C4—41no—
323C4—2741no—
324C4—2741no—
325C4—2741no—
326C4—2742no—
327C4—5481no—
328C4—5481no—
329C4—5481no—
330C4#41×D13711yescharacteristic
331C4C2×#1455550911yescharacteristic
332C2—42no—
333C2—5481no—
334C2#2257791311yescharacteristic
335C2#2257820911yescharacteristic
336C2#2257821911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#461×C137C2
socleC22×C137C22×D4
prime radicalC2×#14555509C22
hypercenter#461D137
hypocenterC137#3120
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #22578014C2semidirect
3. #22577737C2semidirect
4. #22577732C2semidirect
5. #22577711C2semidirect
6. #22577703C2semidirect
7. #22577610C2semidirect
10. #22577507C2semidirect
11. #22577504C2semidirect
12. #22577495C2semidirect
13. SD16×Dic137C2semidirect
14. #22577463C2semidirect
15. #461×C137C2semidirect
168. C137#3120semidirect
177. #461D137semidirect

Maximal subgroups

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

subgroupindex
#225780142
#225778822
#225777372
#225777322
#225777112
#225777032
#225776102
#225775622
#225775072
#225775042
#225774952
SD16×Dic1372
#225774632
#461×C1372
#3120137