Finite Group Databasebeta

← #31797663

Subgroups — #31797663

Order 17536.

422 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 148 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#22578435C212yesmaximal, complemented
2#22578433C212yesmaximal, complemented
3#22578431C212yesmaximal, complemented
4#22578430C212yesmaximal, complemented
5#22578427C212yesmaximal, complemented
6#22578377C211yesmaximal, characteristic, complemented
7#22578376C211yesmaximal, characteristic, complemented
8#22578371C212yesmaximal, complemented
9#22577911C211yesmaximal, characteristic
10C2×#14555520C2211yescharacteristic, complemented
11C2×#14555519C2211yescharacteristic, complemented
12C2×#14555518C2212yes—
13C2×#14555518C2212yes—
14C2×#14555518C2211yescharacteristic, complemented
15C22×#11764047C2211yescharacteristic, complemented
16#17984155C2212yescomplemented
17C4⋊#11764047C2212yescomplemented
18#17984145C2212yes—
19#17984145C2212yes—
20#17984145C2212yescomplemented
21C22⋊#11764047C2212yes—
22C22⋊#11764047C2212yes—
23C22⋊#11764047C2212yescomplemented
24#17984143C2212yescomplemented
25C4×#11764047C2212yescomplemented
26C2×#14555509C2211yescharacteristic
27C22×C4×D137C2211yescharacteristic
28#17984071C411yescharacteristic
29#17984066C412yes—
30#17984061C412yes—
31#17984058C411yescharacteristic
32#17984056C411yescharacteristic
33#17984050C2212yes—
34#17984048C2212yes—
35#17984044C2211yescharacteristic
36#123×C137C411yescharacteristic
37#14555520—41no—
38#14555519—41no—
39#14555518—42no—
40#14555518—42no—
41#14555518—41no—
42C2×#11764047—41no—
43C2×#11764047C2312yes—
44C2×#11764047C2312yes—
45C2×#11764047C2312yes—
46C2×#11764047C2312yes—
47C23×D137C2×C411yescharacteristic
48#14555509—41no—
49#14555509—41no—
50C2×Dic274C2×C411yescharacteristic
51C2×D548C2×C412yes—
52C2×C22⋊D137C2×C412yes—
53C2×C22⋊D137C2×C411yescharacteristic
54C2×C22⋊D137C2×C411yescharacteristic
55C22×Dic137C2311yescharacteristic
56C22×Dic137C2311yescharacteristic
57C2×C4×D137—41no—
58C2×C4×D137—41no—
59C2×C4×D137C2×C412yes—
60C2×C4×D137C2312yes—
61C2×C4×D137C2311yescharacteristic
62#14555489C2×C412yes—
63#14555489C2×C412yes—
64C22⋊Dic137C2×C411yescharacteristic
65#14555487C2×C412yes—
66#14555487C2×C412yes—
67#14555487C2×C412yes—
68C4×Dic137C2312yes—
69C2×D4×C137C2×C411yescharacteristic
70C22×C4×C137C2×C411yescharacteristic
71C4⋊C4×C137C2×C412yes—
72C22⋊C4×C137C2×C411yescharacteristic
73C22⋊C4×C137C2×C412yes—
74#11764047—22no—
75#11764047—22no—
76#11764047—22no—
77#11764047—22no—
78C22×D137—24no—
79C22×D137—41no—
80C22×D137C22×C411yescharacteristic
81C22×D137C22×C412yes—
82Dic274—41no—
83D548—24no—
84C22⋊D137—24no—
85C22⋊D137—41no—
86C22⋊D137—41no—
87C2×Dic137—21no—
88C2×Dic137—21no—
89C2×Dic137—21no—
90C2×Dic137—21no—
91C2×Dic137C22×C412yes—
92C2×Dic137C22×C411yescharacteristic
93C2×Dic137C22×C411yescharacteristic
94C2×Dic137C2411yescharacteristic
95C4×D137—22no—
96C4×D137—22no—
97C4×D137—21no—
98C4×D137—21no—
99C4×D137—42no—
100C23×C137C22×C411yescharacteristic
101C23×C137C22×C411yescharacteristic
102D4×C137—41no—
103C2×C4×C137—41no—
104C2×C4×C137C22×C412yes—
105C2×C4×C137C22×C412yes—
106C2×C4×C137C22×C411yescharacteristic
107C22×C137—21no—
108C22×C137—21no—
109C22×C137—22no—
110C22×C137—22no—
111C22×C137C23×C411yescharacteristic
112C2×D137—22no—
113C2×D137—22no—
114C2×D137—21no—
115C2×D137—22no—
116C2×D137—44no—
117C2×D137#13114yes—
118C4×C137—22no—
119C4×C137—21no—
120C4×C137—22no—
121Dic137—21no—
122Dic137—21no—
123Dic137—22no—
124Dic137H(5,𝔽2)11yescharacteristic
125Dic137H(5,𝔽2)11yescharacteristic
126C2×C137—41no—
127C2×C137—41no—
128C2×C137#51811yescharacteristic
129C2×C137C2×#13112yes—
130D137—24no—
131D137—41no—
132C137#297711yescharacteristic, complemented, Sylow
133#2977—1371nomaximal, Sylow
134#525—1371no—
135#435—1372no—
136#435—1372no—
137C8×D4—1372no—
138C8×D4—1372no—
139#433—1372no—
140#408—1371no—
141#408—1371no—
142C2×C22⋊C8—1372no—
143C2×#41—1371no—
144C23×C4—1371no—
145C2×M16—1371no—
146C2×M16—1371no—
147C22×C8—1372no—
148C22×C8—1372no—
149C22×C8—1371no—
150C22×C8—1371no—
151#123—1371no—
152#123#1058773911yescharacteristic
153C22⋊Q8—1372no—
154#121—1372no—
155#120—1371no—
156#120—1371no—
157C4×D4—1372no—
158C4×D4—1372no—
159#117—1371no—
160C4⋊C8—1372no—
161C4⋊C8—1372no—
162C22⋊C8—1372no—
163C22⋊C8—1372no—
164C22⋊C8—1372no—
165C22⋊C8—1372no—
166C22⋊C8—1372no—
167C22⋊C8—1372no—
168#97—1372no—
169C4×C8—1372no—
170C24—1371no—
171#41—5481no—
172#41—5481no—
173C2×Q8—1371no—
174C2×D4—1372no—
175C2×D4—1371no—
176C2×D4—1371no—
177C2×D4—1372no—
178C2×D4C2×#1058773911yescharacteristic
179C22×C4—5481no—
180C22×C4—5481no—
181C22×C4—1371no—
182C22×C4—1371no—
183C22×C4—1372no—
184C22×C4—1371no—
185C22×C4—1372no—
186C22×C4C2×#1058773911yescharacteristic
187M16—5481no—
188M16—5481no—
189C2×C8—5481no—
190C2×C8—5481no—
191C2×C8—5482no—
192C2×C8—5482no—
193C2×C8—1372no—
194C2×C8—1372no—
195C2×C8—1372no—
196C2×C8—1372no—
197C4⋊C4—1372no—
198C4⋊C4—1372no—
199C4⋊C4C2×#1058773912yes—
200C22⋊C4—1371no—
201C22⋊C4—1372no—
202C22⋊C4—1372no—
203C22⋊C4—1372no—
204C22⋊C4C2×#1058773911yescharacteristic
205C22⋊C4C2×#1058773912yes—
206C42—1372no—
207C23—2744no—
208C23—5481no—
209C23—1371no—
210C23—1372no—
211C23C22×#1058773911yescharacteristic
212C23C22×#1058773911yescharacteristic
213Q8—5481no—
214D4—41no—
215D4—2744no—
216D4—2744no—
217D4—5481no—
218D4—5481no—
219C2×C4—41no—
220C2×C4—2742no—
221C2×C4—2742no—
222C2×C4—2741no—
223C2×C4—2741no—
224C2×C4—2741no—
225C2×C4—2741no—
226C2×C4—2741no—
227C2×C4—2741no—
228C2×C4—5482no—
229C2×C4—1372no—
230C2×C4—1371no—
231C2×C4—1371no—
232C2×C4—1371no—
233C2×C4C22×#1058773912yes—
234C2×C4C22×#1058773911yescharacteristic
235C2×C4C22×#1058773912yes—
236C8—2742no—
237C8—2742no—
238C8—2742no—
239C8—2742no—
240C22—21no—
241C22—21no—
242C22—22no—
243C22—22no—
244C22—2742no—
245C22—2742no—
246C22—2741no—
247C22—2742no—
248C22—5484no—
249C22—1374no—
250C22C23×#1058773911yescharacteristic
251C4—22no—
252C4—21no—
253C4—22no—
254C4—2741no—
255C4—2741no—
256C4—2742no—
257C4—1371no—
258C4—1371no—
259C2—41no—
260C2—41no—
261C2—2744no—
262C2—5481no—
263C2#2257847011yescharacteristic
264C2D137⋊#13112yes—
subgroupG/H
center, FrattiniC22C23×#10587739
commutator, socleC22×C137C23×C4
Fitting#123×C137C4
prime radical#22577911C2
cosocleC2×Dic137C24
hypercenter#123#10587739
hypocenterC137#2977
LCSG ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ #123

Complements

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

NH ≅ G/Ntype
1. #22578435C2semidirect
2. #22578433C2semidirect
3. #22578431C2semidirect
4. #22578430C2semidirect
5. #22578427C2semidirect
6. #22578377C2semidirect
7. #22578376C2semidirect
8. #22578371C2semidirect
10. C2×#14555520C22semidirect
11. C2×#14555519C22semidirect
14. C2×#14555518C22semidirect
15. C22×#11764047C22semidirect
16. #17984155C22semidirect
17. C4⋊#11764047C22semidirect
20. #17984145C22semidirect
23. C22⋊#11764047C22semidirect
24. #17984143C22semidirect
25. C4×#11764047C22semidirect
132. C137#2977semidirect

Maximal subgroups

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

subgroupindex
#225784352
#225784332
#225784312
#225784302
#225784272
#225783772
#225783762
#225783712
#225779112
#2977137