Finite Group Databasebeta

← #31790420

Subgroups — #31790420

Order 17536.

496 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17984121C211yesmaximal, characteristic, complemented
2C2×M16×D137C211yesmaximal, characteristic, complemented
3#22577915C211yesmaximal, characteristic
4C2×#17984068C211yesmaximal, characteristic, complemented
5#22577662C212yesmaximal, complemented
6#22577650C212yesmaximal, complemented
7#22577645C212yesmaximal, complemented
8#22577633C212yesmaximal, complemented
9#22577411C211yesmaximal, characteristic, complemented
10#22577402C211yesmaximal, characteristic, complemented
11#428×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C22×Dic274C2211yescharacteristic
14C22×C4×D137C2211yescharacteristic
15#17984121—24no—
16C2×Dic548C2212yescomplemented
17C2×#14555499C2212yescomplemented
18C2×#14555494C2211yescharacteristic
19C2×#14555493C2212yes—
20M16×D137—24no—
21C2×C8×D137C2212yes—
22#17984072—21no—
23#17984068—21no—
24#17984068—21no—
25#17984068C2212yes—
26#17984068C2211yescharacteristic
27#17984068C2212yes—
28#17984068C2211yescharacteristic
29#17984063C2211yescharacteristic
30#17984063C2211yescharacteristic
31#17984058—21no—
32C4×D548—21no—
33C4×D548—21no—
34C2×C4⋊Dic137—21no—
35C2×#14555487—21no—
36#17984011C2212yescomplemented
37#17984011C2212yescomplemented
38#17984011C2212yescomplemented
39#17984008C2212yescomplemented
40C2×M16×C137C2211yescharacteristic
41#117×C137C2211yescharacteristic
42C2×C4⋊C4×C137C2211yescharacteristic
43#107×C137C2212yescomplemented
44#106×C137C2212yescomplemented
45C23×D137D411yescharacteristic
46#14555510—22no—
47#14555510—22no—
48C2×Dic274—21no—
49C2×Dic274—22no—
50C2×Dic274—22no—
51C2×Dic274C2312yes—
52C2×Dic274C2311yescharacteristic
53C2×D548—21no—
54C2×D548—21no—
55C2×D548C2311yescharacteristic
56C2×C22⋊D137—21no—
57C22×Dic137—21no—
58C22×Dic137D411yescharacteristic
59C2×C4×D137—22no—
60C2×C4×D137—21no—
61C2×C4×D137—22no—
62C2×C4×D137D411yescharacteristic
63C2×C4×D137D411yescharacteristic
64C2×C4×D137C2311yescharacteristic
65C2×C4×D137C2311yescharacteristic
66Dic548—24no—
67#14555499—24no—
68#14555494—22no—
69#14555493—24no—
70C2×#11764040C2312yes—
71C8×D137—24no—
72C4⋊Dic137—22no—
73C4⋊Dic137—22no—
74C4⋊Dic137—21no—
75C4⋊Dic137—21no—
76#14555487—22no—
77#14555487—22no—
78#14555487—21no—
79#14555487—21no—
80#14555487—21no—
81#14555487—21no—
82C22×C4×C137—21no—
83C22×C4×C137C2311yescharacteristic
84M16×C137D414yescomplemented
85C2×C8×C137C2312yes—
86C4⋊C4×C137—21no—
87C4⋊C4×C137C2312yes—
88C4⋊C4×C137C2311yescharacteristic
89C4⋊C4×C137C2311yescharacteristic
90C22⋊C4×C137—21no—
91C42×C137—21no—
92C22×D137—22no—
93C22×D137—22no—
94C22×D137—21no—
95C22×D137—41no—
96C22×D137C2×D411yescharacteristic
97C22×D137C2×D411yescharacteristic
98Dic274—22no—
99Dic274—21no—
100Dic274—22no—
101Dic274—21no—
102Dic274—42no—
103D548—21no—
104D548—21no—
105D548—42no—
106D548—42no—
107C22⋊D137—42no—
108C2×Dic137—22no—
109C2×Dic137—21no—
110C2×Dic137—22no—
111C2×Dic137—42no—
112C2×Dic137C2×D411yescharacteristic
113C2×Dic137C2×D411yescharacteristic
114C4×D137—22no—
115C4×D137—22no—
116C4×D137—21no—
117C4×D137—21no—
118C4×D137—21no—
119C4×D137—21no—
120C4×D137—42no—
121#11764040—22no—
122C23×C137C2×D411yescharacteristic
123C2×C4×C137—21no—
124C2×C4×C137—21no—
125C2×C4×C137—22no—
126C2×C4×C137—42no—
127C2×C4×C137C2×D411yescharacteristic
128C2×C4×C137C2×D412yes—
129C2×C4×C137#4112yes—
130C2×C4×C137C2411yescharacteristic
131C8×C137C2×D414yes—
132C22×C137—21no—
133C22×C137—21no—
134C22×C137#12112yes—
135C22×C137C22×D411yescharacteristic
136C2×D137—21no—
137C2×D137—21no—
138C2×D137—21no—
139C2×D137—21no—
140C2×D137—21no—
141C2×D137—44no—
142C2×D137—41no—
143C2×D137—41no—
144C2×D137—41no—
145C2×D137#13712yescomplemented
146C4×C137—41no—
147C4×C137—42no—
148C4×C137—41no—
149C4×C137#12112yes—
150C4×C137C22×D411yescharacteristic
151C4×C137C2×#4111yescharacteristic
152Dic137—21no—
153Dic137—21no—
154Dic137—41no—
155Dic137—42no—
156C2×C137—22no—
157C2×C137C2×#12111yescharacteristic
158C2×C137C2×#13711yescharacteristic
159C2×C137#57511yescharacteristic
160D137—22no—
161D137—41no—
162D137—81no—
163C137#305611yescharacteristic, complemented, Sylow
164#3056—1371nomaximal, Sylow
165C2×#137—1371no—
166C22×M16—1371no—
167#525—1371no—
168C2×C22⋊Q8—1371no—
169#470—1372no—
170#468—1372no—
171#467—1372no—
172#465—1372no—
173#428D13711yescharacteristic, complemented
174#419—1371no—
175#417—1371no—
176C2×#41—1371no—
177C22×Q8—1371no—
178C23×C4—1371no—
179#137—2744no—
180C2×Q16—1372no—
181C2×SD16—1372no—
182C2×M16—2744no—
183C2×M16—1371no—
184C2×M16—1372no—
185C2×M16C2×D13711yescharacteristic
186C22×C8—1372no—
187#123—2741no—
188C22⋊Q8—2741no—
189C22⋊Q8—2741no—
190C22⋊Q8—1372no—
191C22⋊Q8—1372no—
192C22⋊Q8—1371no—
193C22⋊Q8—1371no—
194#121—1371no—
195#121—1371no—
196#120—2741no—
197C4×D4—2741no—
198C4×D4—2741no—
199#117C2×D13711yescharacteristic
200C2×C4⋊C4—2741no—
201C2×C4⋊C4C2×D13711yescharacteristic
202C2×C22⋊C4—2741no—
203#107C2×D13712yescomplemented
204#106C2×D13712yescomplemented
205#103—1372no—
206#103—1372no—
207#103—1372no—
208#102—1372no—
209C24—1371no—
210#41—2742no—
211#41—2742no—
212C2×Q8—2741no—
213C2×Q8—2742no—
214C2×Q8—2742no—
215C2×Q8—1372no—
216C2×Q8—1371no—
217C2×D4—2741no—
218C2×D4—2741no—
219C2×D4—2741no—
220C2×D4—1371no—
221C22×C4—21no—
222C22×C4—2742no—
223C22×C4—2742no—
224C22×C4—2741no—
225C22×C4—2741no—
226C22×C4—1371no—
227C22×C4—1371no—
228C22×C4—1371no—
229C22×C4—1371no—
230C22×C4—1371no—
231C22×C4C22×D13711yescharacteristic
232Q16—2744no—
233SD16—2744no—
234M16—2742no—
235M16—2744no—
236M16D54814yescomplemented
237C2×C8—2744no—
238C2×C8—1372no—
239C2×C8C22×D13712yes—
240C4⋊C4—21no—
241C4⋊C4—2742no—
242C4⋊C4—2742no—
243C4⋊C4—2741no—
244C4⋊C4—2741no—
245C4⋊C4C22×D13712yes—
246C4⋊C4C22×D13711yescharacteristic
247C4⋊C4C22×D13711yescharacteristic
248C22⋊C4—21no—
249C22⋊C4—2742no—
250C22⋊C4—2742no—
251C22⋊C4—2741no—
252C22⋊C4—2741no—
253C22⋊C4—2741no—
254C22⋊C4—2741no—
255C42—21no—
256C23—2742no—
257C23—2742no—
258C23—2741no—
259C23—5481no—
260C23—1371no—
261C23—1371no—
262C23D4×D13711yescharacteristic
263Q8—2742no—
264Q8—2741no—
265Q8—2742no—
266Q8—2741no—
267Q8—5482no—
268D4—2741no—
269D4—2741no—
270D4—5482no—
271D4—5482no—
272D4—5482no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—2742no—
278C2×C4—2742no—
279C2×C4—2742no—
280C2×C4—2741no—
281C2×C4—2741no—
282C2×C4—2741no—
283C2×C4—2741no—
284C2×C4—2741no—
285C2×C4—2742no—
286C2×C4—5482no—
287C2×C4—5482no—
288C2×C4—1371no—
289C2×C4—1371no—
290C2×C4D4×D13711yescharacteristic
291C2×C4C2×D54812yes—
292C2×C4#1455551112yes—
293C2×C4C23×D13711yescharacteristic
294C8—2742no—
295C8C2×D54814yes—
296C22—21no—
297C22—21no—
298C22—2741no—
299C22—2741no—
300C22—2741no—
301C22—2741no—
302C22—2741no—
303C22—5484no—
304C22—5481no—
305C22—5481no—
306C22—5481no—
307C22—1372no—
308C22#1798406312yes—
309C22C2×D4×D13711yescharacteristic
310C4—41no—
311C4—42no—
312C4—41no—
313C4—2741no—
314C4—2741no—
315C4—5481no—
316C4—5482no—
317C4#1798406312yes—
318C4C22×D54811yescharacteristic
319C4C2×#1455551111yescharacteristic
320C2—22no—
321C2—2742no—
322C2—5481no—
323C2—10961no—
324C2C2×#1798406311yescharacteristic
325C2#137×D13711yescharacteristic
326C2#2257819911yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#428×C137C2
socleC22×C137C22×D4
prime radical#22577915C2
hypercenter#428D137
hypocenterC137#3056
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
1. C2×#17984121C2semidirect
2. C2×M16×D137C2semidirect
4. C2×#17984068C2semidirect
5. #22577662C2semidirect
6. #22577650C2semidirect
7. #22577645C2semidirect
8. #22577633C2semidirect
9. #22577411C2semidirect
10. #22577402C2semidirect
11. #428×C137C2semidirect
16. C2×Dic548C22semidirect
17. C2×#14555499C22semidirect
36. #17984011C22semidirect
37. #17984011C22semidirect
38. #17984011C22semidirect
39. #17984008C22semidirect
43. #107×C137C22semidirect
44. #106×C137C22semidirect
84. M16×C137D4semidirect
145. C2×D137#137semidirect
163. C137#3056semidirect
173. #428D137semidirect
203. #107C2×D137semidirect
204. #106C2×D137semidirect
236. M16D548semidirect

Maximal subgroups

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

subgroupindex
C2×#179841212
C2×M16×D1372
#225779152
C2×#179840682
#225776622
#225776502
#225776452
#225776332
#225774112
#225774022
#428×C1372
#3056137