Finite Group Databasebeta

← #31790960

Subgroups — #31790960

Order 17536.

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

normal subgroups only

This group has 386 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#22577937C211yesmaximal, characteristic
2#22577738C211yesmaximal, characteristic, complemented
3#22577721C211yesmaximal, characteristic
4#22577710C212yesmaximal, complemented
5#22577625C212yesmaximal, complemented
6#22577559C211yesmaximal, characteristic, complemented
7#22577542C211yesmaximal, characteristic
8#22577508C212yesmaximal, complemented
9#22577501C212yesmaximal, complemented
10#22577460C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C137C211yesmaximal, characteristic, complemented
12C2×#14555500C2212yes—
13C2×#14555497C2212yes—
14#17984082—21no—
15#17984080C2211yescharacteristic
16#17984076C2211yescharacteristic
17#17984071—21no—
18#17984067C2211yescharacteristic
19#17984062C2211yescharacteristic
20C4×Dic274C2211yescharacteristic
21C4×D548C2211yescharacteristic
22C4×C22⋊D137—21no—
23#17984045—21no—
24#17984021C2212yes—
25#17984019C2212yes—
26C4⋊#11764040C2212yes—
27#17984012C2212yes—
28#17984012C2212yes—
29#17984011C2212yes—
30#17984008C2212yes—
31C22⋊#11764040C2211yescharacteristic
32C22⋊#11764040C2211yescharacteristic
33#17983999C2212yes—
34C22×Q8×C137C2211yescharacteristic
35C4×Q8×C137—21no—
36C4×Q8×C137—21no—
37C4×Q8×C137C2212yes—
38C4×Q8×C137C2212yes—
39C2×C4⋊C4×C137—21no—
40C2×C4⋊C4×C137C2211yescharacteristic
41C2×C42×C137—21no—
42C2×C42×C137C2211yescharacteristic
43C2×Dic274C2311yescharacteristic
44C2×D548C2311yescharacteristic
45C2×C22⋊D137—21no—
46C2×C4×D137—21no—
47#14555500—24no—
48#14555497—24no—
49C2×#11764040C2312yes—
50C2×#11764040C2312yes—
51C4⋊Dic137C2311yescharacteristic
52C4⋊Dic137C2311yescharacteristic
53#14555489—21no—
54#14555489—21no—
55#14555489—21no—
56#14555489—21no—
57C22⋊Dic137—21no—
58#14555487—21no—
59#14555487—21no—
60#14555487—21no—
61#14555487—21no—
62C4×Dic137—21no—
63C2×Q8×C137—24no—
64C2×Q8×C137—21no—
65C2×Q8×C137C2312yes—
66C22×C4×C137—21no—
67C22×C4×C137—21no—
68C22×C4×C137D411yescharacteristic
69C22×C4×C137D411yescharacteristic
70C22×C4×C137C2311yescharacteristic
71C4⋊C4×C137—22no—
72C4⋊C4×C137—22no—
73C4⋊C4×C137—21no—
74C4⋊C4×C137C2312yes—
75C42×C137—22no—
76C42×C137—22no—
77C42×C137D411yescharacteristic
78C42×C137D411yescharacteristic
79C42×C137C2311yescharacteristic
80C42×C137C2311yescharacteristic
81C22×D137—21no—
82Dic274—22no—
83D548—22no—
84C22⋊D137—42no—
85C2×Dic137—21no—
86C2×Dic137—21no—
87C2×Dic137—21no—
88C4×D137—42no—
89#11764040—22no—
90#11764040—22no—
91C23×C137C2×D411yescharacteristic
92Q8×C137—24no—
93Q8×C137—22no—
94Q8×C137#4114yes—
95C2×C4×C137—24no—
96C2×C4×C137—22no—
97C2×C4×C137—22no—
98C2×C4×C137—22no—
99C2×C4×C137—22no—
100C2×C4×C137—41no—
101C2×C4×C137—42no—
102C2×C4×C137C2×D411yescharacteristic
103C2×C4×C137C2×D411yescharacteristic
104C2×C4×C137C2×D411yescharacteristic
105C2×C4×C137C2×D411yescharacteristic
106C2×C4×C137C2×D411yescharacteristic
107C2×C4×C137C2411yescharacteristic
108C22×C137—21no—
109C22×C137—21no—
110C22×C137#13712yes—
111C22×C137C22×D411yescharacteristic
112C2×D137—41no—
113C2×D137—41no—
114C2×D137—41no—
115C4×C137—24no—
116C4×C137—21no—
117C4×C137—21no—
118C4×C137—21no—
119C4×C137—21no—
120C4×C137—21no—
121C4×C137—42no—
122C4×C137C2×#4112yes—
123Dic137—41no—
124Dic137—41no—
125Dic137—41no—
126C2×C137—22no—
127C2×C137#52511yescharacteristic
128C2×C137C2×#13711yescharacteristic
129C2×C137#57511yescharacteristic
130D137—81no—
131C137#311711yescharacteristic, complemented, Sylow
132#3117—1371nomaximal, Sylow
133#529—1371no—
134C2×C4×Q8D13711yescharacteristic, complemented
135#484—1371no—
136#481—1371no—
137#479—1372no—
138#464—1372no—
139#451—1371no—
140#448—1371no—
141#441—1372no—
142#440—1372no—
143#432—1371no—
144C22×Q8C2×D13711yescharacteristic
145C2×Q16—1372no—
146C2×SD16—1372no—
147#126—2741no—
148#125—1371no—
149#124—1371no—
150#123—2741no—
151C22⋊Q8—1371no—
152#121—1371no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—1371no—
156C4×Q8C2×D13712yes—
157C4×Q8C2×D13712yes—
158C4×D4—2741no—
159C4×D4—1371no—
160#117—2741no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×D13711yescharacteristic
163C2×C42—21no—
164C2×C42C2×D13711yescharacteristic
165#107—1372no—
166#106—1372no—
167C4⋊C8—1372no—
168#103—1372no—
169#103—1372no—
170#103—1372no—
171#102—1372no—
172C22⋊C8—1371no—
173C22⋊C8—1371no—
174#97—1372no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1371no—
178C2×Q8C22×D13712yes—
179C2×D4—2741no—
180C2×D4—1371no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—2741no—
184C22×C4C22⋊D13711yescharacteristic
185C22×C4C22⋊D13711yescharacteristic
186C22×C4C22×D13711yescharacteristic
187Q16—2744no—
188SD16—2744no—
189C2×C8—1372no—
190C2×C8—1372no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—2741no—
195C4⋊C4—2741no—
196C4⋊C4—2741no—
197C4⋊C4—2741no—
198C4⋊C4—1371no—
199C4⋊C4—1371no—
200C4⋊C4C22×D13712yes—
201C22⋊C4—2741no—
202C22⋊C4—2741no—
203C22⋊C4—2741no—
204C22⋊C4—2741no—
205C22⋊C4—2741no—
206C42—22no—
207C42—22no—
208C42—2741no—
209C42C22⋊D13711yescharacteristic
210C42C22⋊D13711yescharacteristic
211C42C22×D13711yescharacteristic
212C42C22×D13711yescharacteristic
213C23—2741no—
214C23C2×C22⋊D13711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—2742no—
218Q8#1455551014yes—
219D4—2742no—
220D4—5482no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—2741no—
229C2×C4—2741no—
230C2×C4—2741no—
231C2×C4—5482no—
232C2×C4C2×C22⋊D13711yescharacteristic
233C2×C4C2×C22⋊D13711yescharacteristic
234C2×C4C2×C22⋊D13711yescharacteristic
235C2×C4C2×C22⋊D13711yescharacteristic
236C2×C4C2×C22⋊D13711yescharacteristic
237C2×C4C23×D13711yescharacteristic
238C8—2742no—
239C8—2742no—
240C22—21no—
241C22—21no—
242C22—5481no—
243C22—5481no—
244C22—5481no—
245C22#1798411912yes—
246C22C22×C22⋊D13711yescharacteristic
247C4—21no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—24no—
252C4—21no—
253C4—42no—
254C4—5481no—
255C4—5481no—
256C4—5481no—
257C4C2×#1455551012yes—
258C2—22no—
259C2—10961no—
260C2#2257791411yescharacteristic
261C2C2×#1798411911yescharacteristic
262C2#2257820111yescharacteristic
subgroupG/H
centerC22C22×C22⋊D137
commutator, cosocleC2×C4×C137C24
FrattiniC2×C4C23×D137
FittingC2×C4×Q8×C137C2
socleC22×C137C22×D4
prime radical#17984062C22
hypercenterC2×C4×Q8D137
hypocenterC137#3117
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #22577738C2semidirect
4. #22577710C2semidirect
5. #22577625C2semidirect
6. #22577559C2semidirect
8. #22577508C2semidirect
9. #22577501C2semidirect
10. #22577460C2semidirect
11. C2×C4×Q8×C137C2semidirect
131. C137#3117semidirect
134. C2×C4×Q8D137semidirect

Maximal subgroups

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

subgroupindex
#225779372
#225777382
#225777212
#225777102
#225776252
#225775592
#225775422
#225775082
#225775012
#225774602
C2×C4×Q8×C1372
#3117137