Finite Group Databasebeta

← #31781619

Subgroups — #31781619

Order 17536.

320 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 83 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
2#22577426C211yesmaximal, characteristic, complemented
3C2×#17984016C211yesmaximal, characteristic, complemented
4C2×#17984015C211yesmaximal, characteristic, complemented
5C2×#17984014C211yesmaximal, characteristic, complemented
6#22577249C211yesmaximal, characteristic, complemented
7C2×#104×C137C211yesmaximal, characteristic, complemented
8C2×#14555510C2211yescharacteristic
9C2×#14555494C2211yescharacteristic
10#17984086C411yescharacteristic, complemented
11#17984084C411yescharacteristic, complemented
12#17984077C411yescharacteristic, complemented
13#17984075C411yescharacteristic, complemented
14#17984060—22no—
15C4×C22⋊D137—22no—
16C2×C4×Dic137C2211yescharacteristic
17C2×C4×Dic137C2211yescharacteristic
18#17984017—22no—
19#17984016—22no—
20#17984015—22no—
21#17984014—22no—
22#17983995—22no—
23C2×#41×C137C2211yescharacteristic
24C2×M16×C137C2211yescharacteristic
25C2×C42×C137C2211yescharacteristic
26#104×C137—22no—
27#14555510—21no—
28#14555510—21no—
29#14555510—21no—
30#14555510—21no—
31C2×Dic274C2×C411yescharacteristic
32C2×D548C2×C411yescharacteristic
33C2×C22⋊D137—22no—
34C2×C22⋊D137—21no—
35C22×Dic137—21no—
36C22×Dic137D412yes—
37C2×C4×D137—21no—
38#14555494—21no—
39#14555494—21no—
40C2×#11764040—21no—
41#14555489—22no—
42C22⋊Dic137—22no—
43#14555487—22no—
44C4×Dic137—21no—
45C4×Dic137C2×C411yescharacteristic
46C4×Dic137C2×C411yescharacteristic
47C4×Dic137D412yes—
48C4×Dic137D412yescomplemented
49#41×C137—21no—
50#41×C137—21no—
51#41×C137—21no—
52#41×C137—21no—
53C2×Q8×C137C2×C411yescharacteristic
54C2×D4×C137—21no—
55C2×D4×C137C2×C411yescharacteristic
56C22×C4×C137—21no—
57C22×C4×C137—21no—
58C22×C4×C137C2311yescharacteristic
59M16×C137—21no—
60M16×C137—21no—
61C2×C8×C137—21no—
62C42×C137—21no—
63C42×C137D412yescomplemented
64C22×D137—21no—
65Dic274—21no—
66Dic274—21no—
67D548—21no—
68D548—21no—
69C22⋊D137—44no—
70C22⋊D137—41no—
71C22⋊D137—41no—
72C2×Dic137—21no—
73C2×Dic137—22no—
74C2×Dic137—42no—
75C2×Dic137—42no—
76C2×Dic137C22⋊C412yes—
77C2×Dic137C22⋊C412yes—
78C4×D137—41no—
79C4×D137—41no—
80#11764040—41no—
81C23×C137—21no—
82C23×C137C2×D411yescharacteristic
83Q8×C137—21no—
84Q8×C137—21no—
85D4×C137—21no—
86D4×C137—21no—
87D4×C137—41no—
88D4×C137—41no—
89C2×C4×C137—21no—
90C2×C4×C137—22no—
91C2×C4×C137—42no—
92C2×C4×C137—41no—
93C2×C4×C137—41no—
94C2×C4×C137C22×C411yescharacteristic
95C2×C4×C137C2×D411yescharacteristic
96C2×C4×C137C2×D411yescharacteristic
97C2×C4×C137C2×D411yescharacteristic
98C2×C4×C137#4111yescharacteristic
99C2×C4×C137#4111yescharacteristic
100C8×C137—41no—
101C22×C137—22no—
102C22×C137—41no—
103C22×C137—42no—
104C22×C137C2×C22⋊C411yescharacteristic
105C22×C137#12111yescharacteristic
106C22×C137#12111yescharacteristic
107C2×D137—41no—
108C2×D137—42no—
109C4×C137—42no—
110C4×C137—41no—
111C4×C137C4×D411yescharacteristic
112C4×C137C4×D411yescharacteristic
113C4×C137#12411yescharacteristic
114C4×C137#12711yescharacteristic
115Dic137—22no—
116Dic137—22no—
117Dic137—42no—
118Dic137—41no—
119C2×C137—21no—
120C2×C137—21no—
121C2×C137—81no—
122C2×C137#39011yescharacteristic
123C2×C137#42112yes—
124D137—81no—
125C137#196311yescharacteristic, complemented, Sylow
126#1963—1371nomaximal, Sylow
127#532—1371no—
128#423—1371no—
129C2×#104—1371no—
130C2×#104—1371no—
131C2×#104—1371no—
132C2×#104D13711yescharacteristic, complemented
133#376—1371no—
134C2×#41—1371no—
135C2×#41C2×D13711yescharacteristic
136C2×M16—1371no—
137C2×M16C2×D13711yescharacteristic
138C4⋊Q8—1371no—
139#127—1371no—
140#124—1371no—
141#124—1371no—
142#121—2742no—
143C4×D4—2742no—
144C2×C42—1371no—
145C2×C42—1371no—
146C2×C42C2×D13711yescharacteristic
147C4⋊C8—2742no—
148#104—22no—
149#104—2742no—
150#104—2742no—
151#104—2742no—
152#95—2742no—
153#41—21no—
154#41—21no—
155#41—21no—
156#41—21no—
157#41—2741no—
158#41—2741no—
159#41—2741no—
160#41—2741no—
161C2×Q8—1371no—
162C2×Q8C4×D13711yescharacteristic
163C2×D4—21no—
164C2×D4—2741no—
165C2×D4—2742no—
166C2×D4—1371no—
167C2×D4C4×D13711yescharacteristic
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—2741no—
171C22×C4—2741no—
172C22×C4—1372no—
173C22×C4C22×D13711yescharacteristic
174M16—21no—
175M16—21no—
176M16—2741no—
177M16—2741no—
178C2×C8—21no—
179C2×C8—2741no—
180C4⋊C4—2742no—
181C22⋊C4—2742no—
182C22⋊C4—2742no—
183C42—21no—
184C42—2741no—
185C42—1372no—
186C42—1371no—
187C42—1371no—
188C42—1372no—
189C42C22⋊D13712yescomplemented
190C23—21no—
191C23—2741no—
192C23D4×D13711yescharacteristic
193Q8—21no—
194Q8—21no—
195Q8—2741no—
196Q8—2741no—
197D4—21no—
198D4—21no—
199D4—41no—
200D4—41no—
201D4—2741no—
202D4—2741no—
203D4—5484no—
204D4—5481no—
205D4—5481no—
206C2×C4—21no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—2741no—
212C2×C4—2742no—
213C2×C4—5482no—
214C2×C4—5482no—
215C2×C4—5481no—
216C2×C4—5481no—
217C2×C4—1372no—
218C2×C4—1372no—
219C2×C4C2×C4×D13711yescharacteristic
220C2×C4D4×D13711yescharacteristic
221C2×C4D4×D13711yescharacteristic
222C2×C4C2×C22⋊D13711yescharacteristic
223C2×C4#1455550911yescharacteristic
224C2×C4#1455551011yescharacteristic
225C8—41no—
226C8—5481no—
227C22—22no—
228C22—42no—
229C22—41no—
230C22—5482no—
231C22—5481no—
232C22C22⋊C4×D13711yescharacteristic
233C22#1798406011yescharacteristic
234C22#1798406111yescharacteristic
235C4—42no—
236C4—41no—
237C4—2742no—
238C4—2742no—
239C4—5482no—
240C4—5481no—
241C4#1798404811yescharacteristic
242C4C4×C22⋊D13711yescharacteristic
243C4#1798407411yescharacteristic
244C4#1798408411yescharacteristic
245C2—21no—
246C2—21no—
247C2—81no—
248C2—10961no—
249C2#2257730311yescharacteristic
250C2#2257741812yes—
subgroupG/H
centerC2×C4D4×D137
commutatorC2×C4×C137C22×C4
FrattiniC22×C4C22×D137
FittingC2×#104×C137C2
socleC22×C137C2×C22⋊C4
prime radical#22577949C2
cosocleC22×C4×C137C23
hypercenterC2×#104D137
hypocenterC137#1963
LCSG ▷ C2×C4×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. #22577426C2semidirect
3. C2×#17984016C2semidirect
4. C2×#17984015C2semidirect
5. C2×#17984014C2semidirect
6. #22577249C2semidirect
7. C2×#104×C137C2semidirect
10. #17984086C4semidirect
11. #17984084C4semidirect
12. #17984077C4semidirect
13. #17984075C4semidirect
48. C4×Dic137D4semidirect
63. C42×C137D4semidirect
125. C137#1963semidirect
132. C2×#104D137semidirect
189. C42C22⋊D137semidirect

Maximal subgroups

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

subgroupindex
#225779492
#225774262
C2×#179840162
C2×#179840152
C2×#179840142
#225772492
C2×#104×C1372
#1963137