Finite Group Databasebeta

← #31790067

Subgroups — #31790067

Order 17536.

752 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 752 conjugacy classes of subgroups — too many to draw. Only its 149 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×#17984133C211yesmaximal, characteristic
2C2×#17984121C211yesmaximal, characteristic, complemented
3C22×#14555498C211yesmaximal, characteristic, complemented
4#22577979C211yesmaximal, characteristic, complemented
5#22577566C214yesmaximal, complemented
6#22577555C214yesmaximal, complemented
7#22577397C211yesmaximal, characteristic, complemented
8#22577371C211yesmaximal, characteristic, complemented
9#418×C137C211yesmaximal, characteristic, complemented
10#17984133—24no—
11#17984133—24no—
12C2×#14555510C2211yescharacteristic
13C2×#14555510C2211yescharacteristic
14C2×#14555509—21no—
15C2×#14555509C2212yes—
16C22×Dic274C2211yescharacteristic, complemented
17C22×C22⋊D137—22no—
18C2×D4×D137—21no—
19C2×D4×D137C2212yes—
20#17984121—24no—
21C2×Dic548C2212yescomplemented
22C2×#14555499C2212yescomplemented
23C2×#14555498—24no—
24C2×#14555498C2214yescomplemented
25C22×#11764040C2211yescharacteristic, complemented
26#17984088—21no—
27#17984076—21no—
28#17984072—22no—
29#17984068C2214yescomplemented
30#17984011C2212yescomplemented
31#17984008C2212yescomplemented
32#17984000C2214yescomplemented
33C22×D4×C137C2211yescharacteristic
34C2×M16×C137C2211yescharacteristic, complemented
35#117×C137C2211yescharacteristic, complemented
36#102×C137C2214yescomplemented
37C23×D137—22no—
38#14555510—22no—
39#14555510—22no—
40#14555510—22no—
41#14555510D414yes—
42#14555509—28no—
43#14555509—24no—
44#14555509—24no—
45C2×Dic274—21no—
46C2×Dic274—22no—
47C2×Dic274—22no—
48C2×Dic274D411yescharacteristic
49C2×Dic274C2312yes—
50C2×Dic274C2311yescharacteristic
51C2×D548D411yescharacteristic
52C2×D548C2311yescharacteristic
53C2×C22⋊D137—216no—
54C2×C22⋊D137—22no—
55C2×C22⋊D137—22no—
56C2×C22⋊D137—21no—
57C2×C22⋊D137D412yes—
58D4×D137—28no—
59D4×D137—24no—
60D4×D137—24no—
61C22×Dic137—22no—
62C22×Dic137—21no—
63C2×C4×D137—21no—
64C2×C4×D137C2312yes—
65Dic548—24no—
66#14555499—24no—
67#14555498—28no—
68C2×#11764040—22no—
69C2×#11764040C2312yes—
70C4⋊Dic137—24no—
71#14555487—24no—
72C24×C137—21no—
73C2×D4×C137—21no—
74C2×D4×C137—22no—
75C2×D4×C137D414yes—
76C2×D4×C137C2312yes—
77C22×C4×C137C2311yescharacteristic
78M16×C137—22no—
79C2×C8×C137C2312yes—
80C4⋊C4×C137C2312yes—
81C22⋊C4×C137—21no—
82C42×C137—21no—
83C22×D137—21no—
84C22×D137—28no—
85C22×D137—84no—
86C22×D137C2×D412yes—
87Dic274—22no—
88Dic274—21no—
89Dic274—22no—
90Dic274—22no—
91Dic274—42no—
92Dic274C2×D412yes—
93D548—22no—
94D548—21no—
95D548C2×D412yes—
96C22⋊D137—28no—
97C22⋊D137—28no—
98C22⋊D137—24no—
99C22⋊D137—24no—
100C2×Dic137—28no—
101C2×Dic137—21no—
102C2×Dic137—22no—
103C2×Dic137—42no—
104C2×Dic137C2×D412yes—
105C4×D137—24no—
106C4×D137—22no—
107C4×D137—22no—
108#11764040—22no—
109C23×C137—22no—
110C23×C137—24no—
111C23×C137—82no—
112C23×C137C2×D411yescharacteristic
113D4×C137—24no—
114D4×C137—22no—
115D4×C137C2×D414yes—
116C2×C4×C137—22no—
117C2×C4×C137C2×D411yescharacteristic
118C2×C4×C137C2×D412yes—
119C2×C4×C137C2×D412yes—
120C2×C4×C137C2411yescharacteristic
121C8×C137—22no—
122C22×C137—22no—
123C22×C137—24no—
124C22×C137—44no—
125C22×C137—84no—
126C22×C137#12012yes—
127C22×C137C22×D411yescharacteristic
128C2×D137—22no—
129C2×D137—22no—
130C2×D137—24no—
131C2×D137—42no—
132C2×D137—88no—
133C4×C137—42no—
134C4×C137#12012yes—
135C4×C137C22×D411yescharacteristic
136C4×C137C22×D411yescharacteristic
137Dic137—22no—
138Dic137—22no—
139Dic137—42no—
140C2×C137—22no—
141C2×C137—44no—
142C2×C137C2×#12011yescharacteristic
143C2×C137#57712yes—
144D137—42no—
145D137—42no—
146C137#300911yescharacteristic, complemented, Sylow
147#3009—1371nomaximal, Sylow
148C2×H(5,𝔽2)—1371no—
149C2×#137—1371no—
150C22×SD16—1371no—
151#536—1371no—
152#452—1374no—
153#450—1374no—
154#418D13711yescharacteristic, complemented
155#416—1371no—
156#408—1371no—
157H(5,𝔽2)—2744no—
158H(5,𝔽2)—2744no—
159C2×#41—2741no—
160C2×#41—1372no—
161C2×#41—1371no—
162C2×#41—1371no—
163C22×Q8—1371no—
164C22×D4—2741no—
165C22×D4—2742no—
166C22×D4—1372no—
167C22×D4C2×D13711yescharacteristic
168#137—2744no—
169C2×Q16—1372no—
170C2×SD16—2744no—
171C2×SD16—1374no—
172C2×SD16—1372no—
173C2×M16C2×D13711yescharacteristic, complemented
174C22×C8—1371no—
175C4⋊Q8—2741no—
176#124—2741no—
177#123—2742no—
178C22⋊Q8—1374no—
179#117C2×D13711yescharacteristic, complemented
180#103—1372no—
181#102—1372no—
182#102C2×D13714yescomplemented
183C22⋊C8—1374no—
184C24—21no—
185C24—2742no—
186#41—2748no—
187#41—2744no—
188#41—2744no—
189#41—2742no—
190#41—2742no—
191#41—2742no—
192#41—1374no—
193C2×Q8—2741no—
194C2×Q8—2742no—
195C2×Q8—2742no—
196C2×Q8—1372no—
197C2×Q8—1371no—
198C2×Q8—1371no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—2748no—
202C2×D4—2744no—
203C2×D4—27416no—
204C2×D4—2742no—
205C2×D4—2741no—
206C2×D4—2744no—
207C2×D4—2742no—
208C2×D4—1372no—
209C2×D4—1371no—
210C2×D4—1371no—
211C2×D4D54814yes—
212C2×D4C22×D13712yes—
213C22×C4—2742no—
214C22×C4—2741no—
215C22×C4—2741no—
216C22×C4—1372no—
217C22×C4C22×D13711yescharacteristic
218Q16—2744no—
219SD16—2744no—
220SD16—2748no—
221M16—22no—
222C2×C8—2742no—
223C2×C8—1372no—
224C2×C8C22×D13712yes—
225C4⋊C4—2744no—
226C4⋊C4C22×D13712yes—
227C22⋊C4—21no—
228C22⋊C4—2744no—
229C42—21no—
230C23—22no—
231C23—24no—
232C23—82no—
233C23—2748no—
234C23—2741no—
235C23—10964no—
236C23—1372no—
237C23D4×D13711yescharacteristic
238Q8—2742no—
239Q8—2741no—
240Q8—2742no—
241Q8—2742no—
242Q8—5482no—
243Q8—1372no—
244D4—24no—
245D4—22no—
246D4—2748no—
247D4—2748no—
248D4—2744no—
249D4—2744no—
250D4—2742no—
251D4—2741no—
252D4—1372no—
253D4C2×D54814yes—
254C2×C4—22no—
255C2×C4—2748no—
256C2×C4—2741no—
257C2×C4—2744no—
258C2×C4—2742no—
259C2×C4—2742no—
260C2×C4—2742no—
261C2×C4—5482no—
262C2×C4—1372no—
263C2×C4D4×D13711yescharacteristic
264C2×C4D4×D13712yes—
265C2×C4C2×D54812yes—
266C2×C4C23×D13711yescharacteristic
267C8—22no—
268C8—2742no—
269C22—22no—
270C22—24no—
271C22—44no—
272C22—84no—
273C22—2744no—
274C22—2742no—
275C22—2742no—
276C22—5482no—
277C22—10968no—
278C22#1798405812yes—
279C22C2×D4×D13711yescharacteristic
280C4—42no—
281C4—2742no—
282C4—2742no—
283C4—5482no—
284C4#1798405812yes—
285C4C2×D4×D13711yescharacteristic
286C4C22×D54811yescharacteristic
287C2—22no—
288C2—44no—
289C2—5482no—
290C2—5482no—
291C2C2×#1798405811yescharacteristic
292C2#2257821312yes—
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#418×C137C2
socleC22×C137C22×D4
prime radicalC2×#17984133C2
hypercenter#418D137
hypocenterC137#3009
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #418

Complements

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

NH ≅ G/Ntype
2. C2×#17984121C2semidirect
3. C22×#14555498C2semidirect
4. #22577979C2semidirect
5. #22577566C2semidirect
6. #22577555C2semidirect
7. #22577397C2semidirect
8. #22577371C2semidirect
9. #418×C137C2semidirect
16. C22×Dic274C22semidirect
21. C2×Dic548C22semidirect
22. C2×#14555499C22semidirect
24. C2×#14555498C22semidirect
25. C22×#11764040C22semidirect
29. #17984068C22semidirect
30. #17984011C22semidirect
31. #17984008C22semidirect
32. #17984000C22semidirect
34. C2×M16×C137C22semidirect
35. #117×C137C22semidirect
36. #102×C137C22semidirect
146. C137#3009semidirect
154. #418D137semidirect
173. C2×M16C2×D137semidirect
179. #117C2×D137semidirect
182. #102C2×D137semidirect

Maximal subgroups

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

subgroupindex
C2×#179841332
C2×#179841212
C22×#145554982
#225779792
#225775662
#225775552
#225773972
#225773712
#418×C1372
#3009137