Finite Group Databasebeta

← #31791187

Subgroups — #31791187

Order 17536.

392 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 125 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×#17984079C211yesmaximal, characteristic, complemented
3#22577770C212yesmaximal, complemented
4#22577766C212yesmaximal, complemented
5#22577759C212yesmaximal, complemented
6#22577756C212yesmaximal, complemented
7#22577405C211yesmaximal, characteristic, complemented
8#22577404C211yesmaximal, characteristic, complemented
9#22577402C211yesmaximal, characteristic, complemented
10M16×Dic137C211yesmaximal, characteristic, complemented
11#417×C137C211yesmaximal, characteristic, complemented
12C2×#14555510C2211yescharacteristic
13C2×#14555494C2211yescharacteristic
14#17984086C2211yescharacteristic
15#17984084C2211yescharacteristic
16#17984079—21no—
17#17984079—21no—
18#17984079C2212yes—
19#17984079C2212yes—
20#17984077C2211yescharacteristic
21#17984075C2211yescharacteristic
22#17984060—21no—
23#17984060—21no—
24C4×C22⋊D137—21no—
25C4×C22⋊D137—21no—
26C2×C4⋊Dic137—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×Dic274C2311yescharacteristic
46C2×D548C2311yescharacteristic
47C2×C22⋊D137—21no—
48C2×C22⋊D137—21no—
49C2×C22⋊D137—21no—
50C22×Dic137—21no—
51C22×Dic137D411yescharacteristic
52C22×Dic137D411yescharacteristic
53C2×C4×D137—21no—
54#14555494—22no—
55C2×#11764040C2312yes—
56C4⋊Dic137—22no—
57C4⋊Dic137—22no—
58C4⋊Dic137—21no—
59C4⋊Dic137C2312yes—
60#14555489—22no—
61#14555489—22no—
62#14555489—21no—
63#14555489—21no—
64C22⋊Dic137—21no—
65C22⋊Dic137—21no—
66#14555487—21no—
67#14555487—21no—
68C4×Dic137D411yescharacteristic
69C4×Dic137D411yescharacteristic
70C4×Dic137C2311yescharacteristic
71C4×Dic137C2311yescharacteristic
72#41×C137—22no—
73#41×C137—22no—
74C2×Q8×C137C2311yescharacteristic
75C2×D4×C137—21no—
76C2×D4×C137C2311yescharacteristic
77C22×C4×C137—21no—
78C22×C4×C137—21no—
79C22×C4×C137C2311yescharacteristic
80M16×C137—22no—
81C2×C8×C137C2312yes—
82C4⋊C4×C137—21no—
83C4⋊C4×C137C2312yes—
84C22×D137—21no—
85Dic274—21no—
86Dic274—21no—
87D548—21no—
88D548—21no—
89C22⋊D137—42no—
90C22⋊D137—42no—
91C22⋊D137—42no—
92C2×Dic137—21no—
93C2×Dic137—22no—
94C2×Dic137—41no—
95C2×Dic137—41no—
96C2×Dic137—42no—
97C2×Dic137C2×D411yescharacteristic
98C2×Dic137C2×D411yescharacteristic
99C2×Dic137C2×D411yescharacteristic
100C2×Dic137C2×D411yescharacteristic
101C4×D137—42no—
102#11764040—22no—
103C23×C137—21no—
104C23×C137C2×D411yescharacteristic
105Q8×C137—21no—
106Q8×C137—21no—
107D4×C137—21no—
108D4×C137—21no—
109D4×C137—42no—
110C2×C4×C137—22no—
111C2×C4×C137—21no—
112C2×C4×C137—42no—
113C2×C4×C137—42no—
114C2×C4×C137C2×D411yescharacteristic
115C2×C4×C137#4112yes—
116C2×C4×C137#4112yes—
117C2×C4×C137C2411yescharacteristic
118C8×C137—22no—
119C22×C137—21no—
120C22×C137—21no—
121C22×C137—41no—
122C22×C137—41no—
123C22×C137—41no—
124C22×C137#12412yes—
125C22×C137C22×D411yescharacteristic
126C2×D137—41no—
127C2×D137—41no—
128C2×D137—41no—
129C4×C137—41no—
130C4×C137—42no—
131C4×C137#12412yes—
132C4×C137C2×#4111yescharacteristic
133C4×C137C2×#4111yescharacteristic
134Dic137—21no—
135Dic137—21no—
136Dic137—21no—
137Dic137—21no—
138Dic137—41no—
139Dic137—42no—
140C2×C137—22no—
141C2×C137—81no—
142C2×C137C2×#12411yescharacteristic
143C2×C137#57511yescharacteristic
144C2×C137#57511yescharacteristic
145D137—81no—
146C137#314211yescharacteristic, complemented, Sylow
147#3142—1371nomaximal, Sylow
148#532—1371no—
149C2×#125—1371no—
150#491—1372no—
151#490—1372no—
152#488—1372no—
153#488—1372no—
154#417—1371no—
155#417—1371no—
156#417—1371no—
157#417D13711yescharacteristic, complemented
158C4×M16—1371no—
159C2×#41—1371no—
160C2×#41C2×D13711yescharacteristic
161C2×M16—1371no—
162C2×M16C2×D13711yescharacteristic
163C4⋊Q8—1371no—
164#127—1371no—
165#125—2741no—
166#125—2741no—
167#125—1372no—
168#125—1372no—
169#124—1371no—
170#124—1371no—
171#121—2741no—
172#121—2741no—
173C4×D4—2741no—
174C4×D4—2741no—
175C2×C4⋊C4—2741no—
176C2×C4⋊C4—2741no—
177C2×C4⋊C4—1371no—
178C2×C4⋊C4C2×D13711yescharacteristic
179C2×C42—1371no—
180#103—1372no—
181#103—1372no—
182#103—1372no—
183#103C2×D13712yes—
184#102—1372no—
185#102—1372no—
186#102—1372no—
187#102C2×D13712yes—
188#97—1372no—
189C4×C8—1372no—
190#41—22no—
191#41—22no—
192#41—2742no—
193#41—2742no—
194C2×Q8—1371no—
195C2×Q8C22×D13711yescharacteristic
196C2×D4—21no—
197C2×D4—2741no—
198C2×D4—2741no—
199C2×D4—2741no—
200C2×D4—1371no—
201C2×D4C22×D13711yescharacteristic
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—2741no—
205C22×C4—2741no—
206C22×C4—1371no—
207C22×C4—1371no—
208C22×C4C22×D13711yescharacteristic
209M16—22no—
210M16—2742no—
211C2×C8—1372no—
212C2×C8C22×D13712yes—
213C4⋊C4—21no—
214C4⋊C4—2742no—
215C4⋊C4—2742no—
216C4⋊C4—2741no—
217C4⋊C4—2741no—
218C4⋊C4—2742no—
219C4⋊C4—2742no—
220C4⋊C4—2741no—
221C4⋊C4—1372no—
222C4⋊C4C22×D13712yes—
223C22⋊C4—2741no—
224C22⋊C4—2741no—
225C22⋊C4—2741no—
226C22⋊C4—2741no—
227C42—1371no—
228C42—1371no—
229C42—1371no—
230C42—1371no—
231C23—21no—
232C23—2741no—
233C23D4×D13711yescharacteristic
234Q8—21no—
235Q8—21no—
236Q8—2741no—
237Q8—2741no—
238D4—21no—
239D4—21no—
240D4—42no—
241D4—2741no—
242D4—2741no—
243D4—5482no—
244D4—5482no—
245D4—5482no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—2741no—
251C2×C4—2742no—
252C2×C4—5481no—
253C2×C4—5481no—
254C2×C4—5482no—
255C2×C4—5482no—
256C2×C4—1371no—
257C2×C4—1371no—
258C2×C4—1371no—
259C2×C4—1371no—
260C2×C4D4×D13711yescharacteristic
261C2×C4#1455550912yes—
262C2×C4#1455551012yes—
263C2×C4C23×D13711yescharacteristic
264C8—22no—
265C8—2742no—
266C22—21no—
267C22—21no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—5481no—
272C22—5481no—
273C22—5481no—
274C22#1798407412yes—
275C22C2×D4×D13711yescharacteristic
276C4—42no—
277C4—41no—
278C4—2741no—
279C4—2741no—
280C4—2741no—
281C4—2741no—
282C4—5482no—
283C4—5481no—
284C4#1798407412yes—
285C4C2×#1455550911yescharacteristic
286C4C2×#1455551011yescharacteristic
287C2—22no—
288C2—81no—
289C2—10961no—
290C2C2×#1798407411yescharacteristic
291C2#2257819911yescharacteristic
292C2#2257820211yescharacteristic
subgroupG/H
centerC22C2×D4×D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
Fitting#417×C137C2
socleC22×C137C22×D4
prime radical#22577949C2
hypercenter#417D137
hypocenterC137#3142
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×#17984079C2semidirect
3. #22577770C2semidirect
4. #22577766C2semidirect
5. #22577759C2semidirect
6. #22577756C2semidirect
7. #22577405C2semidirect
8. #22577404C2semidirect
9. #22577402C2semidirect
10. M16×Dic137C2semidirect
11. #417×C137C2semidirect
146. C137#3142semidirect
157. #417D137semidirect

Maximal subgroups

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

subgroupindex
#225779492
C2×#179840792
#225777702
#225777662
#225777592
#225777562
#225774052
#225774042
#225774022
M16×Dic1372
#417×C1372
#3142137