Finite Group Databasebeta

← #31790959

Subgroups — #31790959

Order 17536.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22577932C211yesmaximal, characteristic
2C4×Q8×D137C211yesmaximal, characteristic, complemented
3#22577737C211yesmaximal, characteristic, complemented
4#22577720C211yesmaximal, characteristic
5#22577711C211yesmaximal, characteristic, complemented
6#22577710C211yesmaximal, characteristic, complemented
7#22577622C211yesmaximal, characteristic, complemented
8#22577557C211yesmaximal, characteristic, complemented
9#22577540C211yesmaximal, characteristic
10#22577510C211yesmaximal, characteristic, complemented
11#22577507C211yesmaximal, characteristic, complemented
12#22577504C211yesmaximal, characteristic, complemented
13#22577500C211yesmaximal, characteristic, complemented
14#22577459C211yesmaximal, characteristic, complemented
15#464×C137C211yesmaximal, characteristic, complemented
16C2×Q8×D137C2211yescharacteristic
17C2×#14555500C2211yescharacteristic
18C2×#14555498C2211yescharacteristic
19#17984081—21no—
20#17984080C2211yescharacteristic
21#17984070—21no—
22#17984064C2211yescharacteristic
23#17984059C2211yescharacteristic
24Q8×Dic137C2211yescharacteristic
25Q8×Dic137C2211yescharacteristic
26C4×Dic274—21no—
27C4×Dic274C2211yescharacteristic
28#17984053—21no—
29#17984053C2211yescharacteristic
30D4×Dic137C2211yescharacteristic
31#17984048—21no—
32#17984044—21no—
33C4⋊C4×D137—21no—
34C4⋊C4×D137C2211yescharacteristic
35C42×D137—21no—
36C42×D137C2211yescharacteristic
37#17984022C2211yescharacteristic
38#17984021C2211yescharacteristic
39#17984020C2211yescharacteristic
40#17984019C2211yescharacteristic
41C4⋊#11764040C2211yescharacteristic
42#17984013C2211yescharacteristic
43#17984013C2211yescharacteristic
44#17984012C2211yescharacteristic
45#17984012C2211yescharacteristic
46#17984012C2211yescharacteristic
47#17984009C2211yescharacteristic
48#17984000C2211yescharacteristic
49#17984000C2211yescharacteristic
50#17983998C2211yescharacteristic
51#17983998C2211yescharacteristic
52C2×Q16×C137C2211yescharacteristic
53C2×SD16×C137C2211yescharacteristic
54#124×C137C2211yescharacteristic
55C4×Q8×C137C2211yescharacteristic
56C4⋊C8×C137C2211yescharacteristic
57#103×C137C2211yescharacteristic
58#102×C137C2211yescharacteristic
59C2×Dic274—21no—
60C2×Dic274C2311yescharacteristic
61Q8×D137—22no—
62Q8×D137—22no—
63C2×C22⋊D137—21no—
64C22×Dic137—21no—
65C2×C4×D137—21no—
66C2×C4×D137—21no—
67C2×C4×D137D411yescharacteristic
68C2×C4×D137D411yescharacteristic
69C2×C4×D137C2311yescharacteristic
70#14555500—22no—
71#14555498—22no—
72C2×#11764040C2311yescharacteristic
73C2×#11764040C2311yescharacteristic
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76C4⋊Dic137C2311yescharacteristic
77C4⋊Dic137C2311yescharacteristic
78C4⋊Dic137C2311yescharacteristic
79#14555489—21no—
80#14555489—21no—
81#14555489—21no—
82#14555489—21no—
83#14555489—21no—
84#14555489—21no—
85C22⋊Dic137—21no—
86C22⋊Dic137—21no—
87#14555487—21no—
88C4×Dic137—21no—
89C4×Dic137—21no—
90C4×Dic137—21no—
91C4×Dic137—21no—
92C4×Dic137D411yescharacteristic
93C4×Dic137D411yescharacteristic
94C4×Dic137C2311yescharacteristic
95C2×Q8×C137C2311yescharacteristic
96C2×Q8×C137C2311yescharacteristic
97C2×D4×C137C2311yescharacteristic
98Q16×C137—22no—
99SD16×C137—22no—
100C2×C8×C137C2311yescharacteristic
101C2×C8×C137C2311yescharacteristic
102C4⋊C4×C137—21no—
103C4⋊C4×C137C2311yescharacteristic
104C22⋊C4×C137—21no—
105C22⋊C4×C137—21no—
106C42×C137—21no—
107C42×C137C2311yescharacteristic
108C22×D137C2×D411yescharacteristic
109Dic274—24no—
110Dic274—21no—
111Dic274#4112yes—
112C22⋊D137—42no—
113C2×Dic137—21no—
114C2×Dic137—21no—
115C2×Dic137—21no—
116C2×Dic137—21no—
117C2×Dic137—42no—
118C2×Dic137C2×D411yescharacteristic
119C2×Dic137C2×D411yescharacteristic
120C2×Dic137C2×D411yescharacteristic
121C4×D137—24no—
122C4×D137—22no—
123C4×D137—21no—
124C4×D137—21no—
125C4×D137—41no—
126C4×D137—42no—
127#11764040—21no—
128#11764040—21no—
129C23×C137—21no—
130Q8×C137—21no—
131Q8×C137—21no—
132Q8×C137—21no—
133Q8×C137#4112yes—
134D4×C137—21no—
135D4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137C2×D411yescharacteristic
140C2×C4×C137C2×D411yescharacteristic
141C2×C4×C137C2411yescharacteristic
142C8×C137—21no—
143C8×C137—21no—
144C22×C137—41no—
145C22×C137—41no—
146C22×C137—41no—
147C22×C137C22×D411yescharacteristic
148C2×D137—21no—
149C2×D137—21no—
150C2×D137#13712yes—
151C4×C137—22no—
152C4×C137—21no—
153C4×C137—21no—
154C4×C137—41no—
155C4×C137—41no—
156C4×C137C2×#4111yescharacteristic
157C4×C137C2×#4111yescharacteristic
158Dic137—22no—
159Dic137—21no—
160Dic137—21no—
161Dic137—21no—
162Dic137—41no—
163Dic137—41no—
164Dic137—41no—
165C2×C137—81no—
166C2×C137#52511yescharacteristic
167C2×C137C2×#13711yescharacteristic
168C2×C137#57511yescharacteristic
169D137—22no—
170C137#311711yescharacteristic, complemented, Sylow
171#3117—1371nomaximal, Sylow
172#529—1371no—
173C2×C4×Q8—1371no—
174#484—1371no—
175#481—1371no—
176#479—1371no—
177#479—1371no—
178#464—1371no—
179#464D13711yescharacteristic, complemented
180#451—1371no—
181#448—1371no—
182#441—1371no—
183#441—1371no—
184#440—1371no—
185#440—1371no—
186#432—1371no—
187C22×Q8—1371no—
188C2×Q16—1371no—
189C2×Q16C2×D13711yescharacteristic
190C2×SD16—1371no—
191C2×SD16C2×D13711yescharacteristic
192#126—2741no—
193#125—1371no—
194#124C2×D13711yescharacteristic
195#123—2741no—
196C22⋊Q8—1371no—
197#121—1371no—
198C4×Q8—2741no—
199C4×Q8—2741no—
200C4×Q8—1371no—
201C4×Q8—1371no—
202C4×Q8—1371no—
203C4×Q8—1371no—
204C4×Q8C2×D13711yescharacteristic
205C4×D4—2741no—
206C4×D4—1371no—
207#117—2741no—
208C2×C4⋊C4—2741no—
209C2×C4⋊C4—1371no—
210C2×C42—2741no—
211C2×C42—1371no—
212#107—1371no—
213#107—1371no—
214#106—1371no—
215#106—1371no—
216C4⋊C8—1371no—
217C4⋊C8C2×D13711yescharacteristic
218#103—1371no—
219#103—1371no—
220#103—1371no—
221#103—1371no—
222#103—1371no—
223#103C2×D13711yescharacteristic
224#102—1371no—
225#102C2×D13711yescharacteristic
226C22⋊C8—1371no—
227C22⋊C8—1371no—
228#97—1371no—
229#97—1371no—
230C2×Q8—2741no—
231C2×Q8—2742no—
232C2×Q8—2742no—
233C2×Q8—1371no—
234C2×Q8C22×D13711yescharacteristic
235C2×Q8C22×D13711yescharacteristic
236C2×D4—2741no—
237C2×D4C22×D13711yescharacteristic
238C22×C4—2741no—
239C22×C4—2741no—
240C22×C4—2741no—
241C22×C4—1371no—
242C22×C4—1371no—
243C22×C4—1371no—
244Q16—22no—
245Q16—2742no—
246SD16—22no—
247SD16—2742no—
248C2×C8—1371no—
249C2×C8—1371no—
250C2×C8C22×D13711yescharacteristic
251C2×C8C22×D13711yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—2741no—
254C4⋊C4—2741no—
255C4⋊C4—2741no—
256C4⋊C4—2741no—
257C4⋊C4—2741no—
258C4⋊C4—2741no—
259C4⋊C4—2741no—
260C4⋊C4—2741no—
261C4⋊C4—1371no—
262C4⋊C4—1371no—
263C4⋊C4—1371no—
264C4⋊C4C22×D13711yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—2741no—
268C22⋊C4—2741no—
269C22⋊C4—2741no—
270C42—21no—
271C42—2741no—
272C42—2741no—
273C42—2741no—
274C42—2741no—
275C42—1371no—
276C42—1371no—
277C42—1371no—
278C42C22×D13711yescharacteristic
279C23—21no—
280C23—1371no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—2744no—
285Q8—2741no—
286Q8—1372no—
287Q8#1455550912yes—
288D4—21no—
289D4—21no—
290D4—5482no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—2744no—
295C2×C4—2742no—
296C2×C4—2741no—
297C2×C4—2741no—
298C2×C4—2741no—
299C2×C4—2741no—
300C2×C4—2741no—
301C2×C4—2741no—
302C2×C4—5481no—
303C2×C4—5482no—
304C2×C4—5482no—
305C2×C4—1371no—
306C2×C4—1371no—
307C2×C4—1371no—
308C2×C4D4×D13711yescharacteristic
309C2×C4D4×D13711yescharacteristic
310C2×C4C23×D13711yescharacteristic
311C8—21no—
312C8—21no—
313C8—2741no—
314C8—2741no—
315C22—41no—
316C22—41no—
317C22—41no—
318C22—2741no—
319C22—2741no—
320C22—1372no—
321C22C2×D4×D13711yescharacteristic
322C4—21no—
323C4—21no—
324C4—22no—
325C4—41no—
326C4—41no—
327C4—2741no—
328C4—2741no—
329C4—2742no—
330C4—2741no—
331C4—5481no—
332C4—5481no—
333C4—5481no—
334C4#41×D13711yescharacteristic
335C4C2×#1455550911yescharacteristic
336C2—81no—
337C2—2742no—
338C2#2257791311yescharacteristic
339C2#137×D13711yescharacteristic
340C2#2257820011yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#464×C137C2
socleC22×C137C22×D4
prime radical#17984059C22
hypercenter#464D137
hypocenterC137#3117
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C42 ◁ #464

Complements

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

NH ≅ G/Ntype
2. C4×Q8×D137C2semidirect
3. #22577737C2semidirect
5. #22577711C2semidirect
6. #22577710C2semidirect
7. #22577622C2semidirect
8. #22577557C2semidirect
10. #22577510C2semidirect
11. #22577507C2semidirect
12. #22577504C2semidirect
13. #22577500C2semidirect
14. #22577459C2semidirect
15. #464×C137C2semidirect
170. C137#3117semidirect
179. #464D137semidirect

Maximal subgroups

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

subgroupindex
#225779322
C4×Q8×D1372
#225777372
#225777202
#225777112
#225777102
#225776222
#225775572
#225775402
#225775102
#225775072
#225775042
#225775002
#225774592
#464×C1372
#3117137