Finite Group Databasebeta

← #31791180

Subgroups — #31791180

Order 17536.

424 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 129 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
1#22577949C211yesmaximal, characteristic
2C2×#17984087C211yesmaximal, characteristic, complemented
3#22577764C212yesmaximal, complemented
4#22577761C212yesmaximal, complemented
5#22577752C212yesmaximal, complemented
6#22577748C212yesmaximal, complemented
7#22577405C211yesmaximal, characteristic, complemented
8#22577404C211yesmaximal, characteristic, complemented
9#22577402C211yesmaximal, characteristic, complemented
10M16×Dic137C211yesmaximal, characteristic, complemented
11#417×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C22×Dic274—21no—
14C2×#14555494C2211yescharacteristic
15#17984087—21no—
16#17984087—21no—
17#17984087C2212yes—
18#17984087C2212yes—
19#17984086C2211yescharacteristic
20#17984084C2211yescharacteristic
21#17984077C2211yescharacteristic
22#17984075C2211yescharacteristic
23#17984060—21no—
24#17984060—21no—
25C4×C22⋊D137—21no—
26C4×C22⋊D137—21no—
27C2×C4⋊Dic137C2211yescharacteristic
28C2×#14555489—21no—
29C2×C4×Dic137C2211yescharacteristic
30#17984013C2212yes—
31#17984012C2212yes—
32#17984011C2212yes—
33#17984010C2212yes—
34#17984009C2212yes—
35#17984008C2212yes—
36#17983998C2212yes—
37C8×Dic137C2212yes—
38C2×#41×C137C2211yescharacteristic
39C2×M16×C137C2211yescharacteristic
40C2×C4⋊C4×C137C2211yescharacteristic
41#103×C137C2212yes—
42#102×C137C2212yes—
43#14555510—22no—
44#14555510—22no—
45C2×Dic274—22no—
46C2×Dic274—22no—
47C2×Dic274—44no—
48C2×Dic274C2311yescharacteristic
49C2×D548C2311yescharacteristic
50C2×C22⋊D137—21no—
51C2×C22⋊D137—21no—
52C2×C22⋊D137—21no—
53C22×Dic137—21no—
54C22×Dic137D411yescharacteristic
55C22×Dic137D411yescharacteristic
56C2×C4×D137—21no—
57#14555494—22no—
58C2×#11764040C2312yes—
59C4⋊Dic137—21no—
60C4⋊Dic137C2312yes—
61#14555489—22no—
62#14555489—22no—
63#14555489—21no—
64#14555489—21no—
65C22⋊Dic137—21no—
66C22⋊Dic137—21no—
67#14555487—21no—
68#14555487—21no—
69C4×Dic137D411yescharacteristic
70C4×Dic137D411yescharacteristic
71C4×Dic137C2311yescharacteristic
72C4×Dic137C2311yescharacteristic
73#41×C137—22no—
74#41×C137—22no—
75C2×Q8×C137C2311yescharacteristic
76C2×D4×C137—21no—
77C2×D4×C137C2311yescharacteristic
78C22×C4×C137—21no—
79C22×C4×C137—21no—
80C22×C4×C137C2311yescharacteristic
81M16×C137—22no—
82C2×C8×C137C2312yes—
83C4⋊C4×C137—21no—
84C4⋊C4×C137C2312yes—
85C22×D137—21no—
86Dic274—21no—
87Dic274—21no—
88Dic274—44no—
89Dic274—44no—
90D548—21no—
91D548—21no—
92C22⋊D137—42no—
93C22⋊D137—42no—
94C22⋊D137—42no—
95C2×Dic137—22no—
96C2×Dic137—22no—
97C2×Dic137—21no—
98C2×Dic137—22no—
99C2×Dic137—42no—
100C2×Dic137C2×D411yescharacteristic
101C2×Dic137C2×D411yescharacteristic
102C2×Dic137C2×D411yescharacteristic
103C2×Dic137C2×D411yescharacteristic
104C4×D137—42no—
105#11764040—22no—
106C23×C137—21no—
107C23×C137C2×D411yescharacteristic
108Q8×C137—21no—
109Q8×C137—21no—
110D4×C137—21no—
111D4×C137—21no—
112D4×C137—42no—
113C2×C4×C137—22no—
114C2×C4×C137—21no—
115C2×C4×C137—42no—
116C2×C4×C137—42no—
117C2×C4×C137C2×D411yescharacteristic
118C2×C4×C137#4112yes—
119C2×C4×C137#4112yes—
120C2×C4×C137C2411yescharacteristic
121C8×C137—22no—
122C22×C137—21no—
123C22×C137—21no—
124C22×C137—41no—
125C22×C137—41no—
126C22×C137—41no—
127C22×C137#12412yes—
128C22×C137C22×D411yescharacteristic
129C2×D137—41no—
130C2×D137—41no—
131C2×D137—41no—
132C4×C137—41no—
133C4×C137—42no—
134C4×C137#12412yes—
135C4×C137C2×#4111yescharacteristic
136C4×C137C2×#4111yescharacteristic
137Dic137—21no—
138Dic137—21no—
139Dic137—41no—
140Dic137—42no—
141Dic137AGL(1,ℤ/8)12yes—
142Dic137#13712yes—
143C2×C137—22no—
144C2×C137—81no—
145C2×C137C2×#12411yescharacteristic
146C2×C137C2×AGL(1,ℤ/8)11yescharacteristic
147C2×C137C2×#13711yescharacteristic
148D137—81no—
149C137#314111yescharacteristic, complemented, Sylow
150#3141—1371nomaximal, Sylow
151#532—1371no—
152C2×C4⋊Q8—1371no—
153#489—1372no—
154#489—1372no—
155#487—1372no—
156#486—1372no—
157#417—1371no—
158#417—1371no—
159#417—1371no—
160#417D13711yescharacteristic, complemented
161C4×M16—1371no—
162C2×#41—1371no—
163C2×#41C2×D13711yescharacteristic
164C22×Q8—2741no—
165C2×M16—1371no—
166C2×M16C2×D13711yescharacteristic
167C4⋊Q8—2741no—
168C4⋊Q8—2741no—
169C4⋊Q8—1371no—
170C4⋊Q8—1372no—
171C4⋊Q8—1372no—
172#127—1371no—
173#124—1371no—
174#124—1371no—
175#121—2741no—
176#121—2741no—
177C4×D4—2741no—
178C4×D4—2741no—
179C2×C4⋊C4—2741no—
180C2×C4⋊C4—1371no—
181C2×C4⋊C4C2×D13711yescharacteristic
182C2×C42—1371no—
183#103—1372no—
184#103—1372no—
185#103—1372no—
186#103C2×D13712yes—
187#102—1372no—
188#102—1372no—
189#102—1372no—
190#102C2×D13712yes—
191#97—1372no—
192C4×C8—1372no—
193#41—22no—
194#41—22no—
195#41—2742no—
196#41—2742no—
197C2×Q8—2742no—
198C2×Q8—2742no—
199C2×Q8—5484no—
200C2×Q8—1371no—
201C2×Q8C22×D13711yescharacteristic
202C2×D4—21no—
203C2×D4—2741no—
204C2×D4—2741no—
205C2×D4—2741no—
206C2×D4—1371no—
207C2×D4C22×D13711yescharacteristic
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—2741no—
211C22×C4—2741no—
212C22×C4—1371no—
213C22×C4—1371no—
214C22×C4C22×D13711yescharacteristic
215M16—22no—
216M16—2742no—
217C2×C8—1372no—
218C2×C8C22×D13712yes—
219C4⋊C4—21no—
220C4⋊C4—2742no—
221C4⋊C4—2742no—
222C4⋊C4—2741no—
223C4⋊C4—2741no—
224C4⋊C4—2741no—
225C4⋊C4—1372no—
226C4⋊C4C22×D13712yes—
227C22⋊C4—2741no—
228C22⋊C4—2741no—
229C22⋊C4—2741no—
230C22⋊C4—2741no—
231C42—1371no—
232C42—1371no—
233C42—1371no—
234C42—1371no—
235C23—21no—
236C23—2741no—
237C23D4×D13711yescharacteristic
238Q8—21no—
239Q8—21no—
240Q8—2741no—
241Q8—2741no—
242Q8—5484no—
243Q8—5484no—
244D4—21no—
245D4—21no—
246D4—42no—
247D4—2741no—
248D4—2741no—
249D4—5482no—
250D4—5482no—
251D4—5482no—
252C2×C4—21no—
253C2×C4—22no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—2742no—
257C2×C4—2742no—
258C2×C4—2741no—
259C2×C4—2742no—
260C2×C4—5482no—
261C2×C4—5482no—
262C2×C4—1371no—
263C2×C4—1371no—
264C2×C4—1371no—
265C2×C4—1371no—
266C2×C4D4×D13711yescharacteristic
267C2×C4#1455550912yes—
268C2×C4#1455551012yes—
269C2×C4C23×D13711yescharacteristic
270C8—22no—
271C8—2742no—
272C22—21no—
273C22—21no—
274C22—41no—
275C22—41no—
276C22—41no—
277C22—5481no—
278C22—5481no—
279C22—5481no—
280C22#1798407412yes—
281C22C2×D4×D13711yescharacteristic
282C4—42no—
283C4—41no—
284C4—2741no—
285C4—2741no—
286C4—5482no—
287C4—5481no—
288C4—1372no—
289C4—1372no—
290C4#1798407412yes—
291C4C2×#1455550911yescharacteristic
292C4C2×#1455551011yescharacteristic
293C2—22no—
294C2—81no—
295C2—10961no—
296C2C2×#1798407411yescharacteristic
297C2AGL(1,ℤ/8)×D13711yescharacteristic
298C2#137×D13711yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#417×C137C2
socleC22×C137C22×D4
prime radical#22577949C2
hypercenter#417D137
hypocenterC137#3141
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
2. C2×#17984087C2semidirect
3. #22577764C2semidirect
4. #22577761C2semidirect
5. #22577752C2semidirect
6. #22577748C2semidirect
7. #22577405C2semidirect
8. #22577404C2semidirect
9. #22577402C2semidirect
10. M16×Dic137C2semidirect
11. #417×C137C2semidirect
149. C137#3141semidirect
160. #417D137semidirect

Maximal subgroups

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

subgroupindex
#225779492
C2×#179840872
#225777642
#225777612
#225777522
#225777482
#225774052
#225774042
#225774022
M16×Dic1372
#417×C1372
#3141137