Finite Group Databasebeta

← #99×#10587739 #31796537

Subgroups — #99×#10587739 #31796537

Order 17536.

412 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 90 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
1C2×D4×#10587739C211yesmaximal, characteristic
2D137⋊#99C211yesmaximal, characteristic
3C22⋊C4×#10587739C212yesmaximal, complemented
4#22578280C212yesmaximal, complemented
5#99×D137C211yesmaximal, characteristic
6C23×#10587739C411yescharacteristic, complemented
7C23×#10587739C2211yescharacteristic
8D4×#10587739—22no—
9D4×#10587739—22no—
10C2×C4⋊#10587739C411yescharacteristic, complemented
11C2×C22⋊#10587739C411yescharacteristic, complemented
12C2×C22⋊#10587739C2211yescharacteristic
13C2×C22⋊#10587739C2212yes—
14C2×C4×#10587739—22no—
15C2×C4×#10587739C411yescharacteristic, complemented
16#17984147C412yescomplemented
17#17984146C412yescomplemented
18(C2×#10587739)⋊C4—22no—
19C2×D4×D137C2211yescharacteristic
20C22⋊C4×D137C2212yes—
21#17984003C411yescharacteristic, complemented
22#17984002C412yescomplemented
23#99×C137C411yescharacteristic, complemented
24C22×#10587739—21no—
25C22×#10587739—21no—
26C22×#10587739—22no—
27C22×#10587739—42no—
28C22×#10587739—42no—
29C22×#10587739C2×C411yescharacteristic
30C22×#10587739C2×C411yescharacteristic
31C22×#10587739C2×C411yescharacteristic
32C22×#10587739C2×C411yescharacteristic
33C22×#10587739D412yes—
34C22×#10587739D412yes—
35C4⋊#10587739—22no—
36C22⋊#10587739—22no—
37C22⋊#10587739—22no—
38C22⋊#10587739—22no—
39C22⋊#10587739C2×C414yescomplemented
40C4×#10587739—21no—
41C4×#10587739—21no—
42C4×#10587739—44no—
43C23×D137C2×C411yescharacteristic
44C23×D137C2311yescharacteristic
45C2×D548C2×C411yescharacteristic
46C2×C22⋊D137C2×C411yescharacteristic
47C2×C22⋊D137C2×C411yescharacteristic
48D4×D137—22no—
49D4×D137—22no—
50C2×C4×D137—22no—
51C2×C4×D137C2×C411yescharacteristic
52C22⋊Dic137C2×C412yescomplemented
53#14555487—22no—
54C2×D4×C137C2×C411yescharacteristic
55C22⋊C4×C137C2×C412yescomplemented
56C2×#10587739—24no—
57C2×#10587739—22no—
58C2×#10587739—21no—
59C2×#10587739—21no—
60C2×#10587739—24no—
61C2×#10587739—41no—
62C2×#10587739—42no—
63C2×#10587739—42no—
64C2×#10587739—42no—
65C2×#10587739—42no—
66C2×#10587739—42no—
67C2×#10587739—41no—
68C2×#10587739—41no—
69C2×#10587739C22⋊C411yescharacteristic
70C2×#10587739C22⋊C412yes—
71C2×#10587739C22⋊C411yescharacteristic
72C22×D137—21no—
73C22×D137—21no—
74C22×D137—41no—
75C22×D137—41no—
76C22×D137—41no—
77C22×D137—41no—
78C22×D137C4211yescharacteristic, complemented
79C22×D137C22×C411yescharacteristic
80C22×D137C22×C411yescharacteristic
81C22×D137C2×D412yes—
82C22×D137#4112yes—
83D548—22no—
84C22⋊D137—22no—
85C22⋊D137—22no—
86C2×Dic137—22no—
87C2×Dic137C4211yescharacteristic, complemented
88C4×D137—21no—
89C4×D137—21no—
90C4×D137—42no—
91C4×D137—42no—
92C23×C137C4211yescharacteristic, complemented
93C23×C137C22×C411yescharacteristic
94D4×C137—22no—
95C2×C4×C137—22no—
96C2×C4×C137C4211yescharacteristic, complemented
97C22×C137—21no—
98C22×C137—41no—
99C22×C137—41no—
100C22×C137C2×C4211yescharacteristic
101C22×C137C4×D412yes—
102C2×D137—22no—
103C2×D137—21no—
104C2×D137—22no—
105C2×D137—21no—
106C2×D137—21no—
107C2×D137—41no—
108C2×D137—42no—
109C2×D137—42no—
110C2×D137—41no—
111C2×D137—41no—
112C2×D137—41no—
113C2×D137C2×C22⋊C411yescharacteristic
114C2×D137#11711yescharacteristic
115C2×D137C4×D412yes—
116#10587739—21no—
117#10587739—22no—
118#10587739—44no—
119#10587739—41no—
120#10587739—41no—
121#10587739#9912yesdirect product factor
122C4×C137—21no—
123C4×C137—42no—
124Dic137—21no—
125Dic137—42no—
126C2×C137—22no—
127C2×C137—21no—
128C2×C137—41no—
129C2×C137C4×C22⋊C411yescharacteristic
130D137—22no—
131D137—21no—
132D137—41no—
133D137C2×#9911yescharacteristic
134D137#41011yescharacteristic
135C137C4×#9911yescharacteristic, complemented, Sylow
136C4×#99—1371nomaximal, Sylow
137C2×C4×D4—1371no—
138C2×#99—1371no—
139C2×#99—1371no—
140C4×C22⋊C4—1372no—
141#342—1372no—
142C22×D4—1371no—
143C23×C4—1371no—
144C23×C4—1371no—
145C4×D4—2742no—
146C4×D4—2742no—
147C2×C4⋊C4—1371no—
148C2×C22⋊C4—1371no—
149C2×C22⋊C4—1371no—
150C2×C22⋊C4—1372no—
151C2×C22⋊C4—1372no—
152C2×C42—2742no—
153C2×C42—1371no—
154#99—1372no—
155#99—1372no—
156#99—1371no—
157#99—1372no—
158#99#1058773911yescharacteristic, direct product factor
159#95—2742no—
160C24—1371no—
161C24—1371no—
162C2×D4—2742no—
163C2×D4—2742no—
164C2×D4—1371no—
165C2×D4—1371no—
166C2×D4—1371no—
167C2×D4C2×#1058773911yescharacteristic
168C22×C4—2742no—
169C22×C4—2742no—
170C22×C4—2741no—
171C22×C4—2741no—
172C22×C4—5482no—
173C22×C4—5482no—
174C22×C4—1372no—
175C22×C4—1371no—
176C22×C4—1371no—
177C22×C4—1371no—
178C22×C4—1371no—
179C22×C4—1372no—
180C22×C4—1371no—
181C4⋊C4—2742no—
182C22⋊C4—2742no—
183C22⋊C4—2742no—
184C22⋊C4—2742no—
185C22⋊C4—2742no—
186C22⋊C4—1374no—
187C22⋊C4—1372no—
188C22⋊C4C2×#1058773912yescomplemented
189C42—2741no—
190C42—2741no—
191C42—5484no—
192C23—2741no—
193C23—2741no—
194C23—5481no—
195C23—5481no—
196C23—5481no—
197C23—5481no—
198C23—1371no—
199C23—1372no—
200C23—1371no—
201C23—1371no—
202C23—1372no—
203C23C4×#1058773911yescharacteristic, complemented
204C23C22×#1058773911yescharacteristic
205D4—22no—
206D4—2742no—
207D4—2742no—
208D4—2742no—
209C2×C4—22no—
210C2×C4—2741no—
211C2×C4—2741no—
212C2×C4—2744no—
213C2×C4—2742no—
214C2×C4—2741no—
215C2×C4—2741no—
216C2×C4—2744no—
217C2×C4—2742no—
218C2×C4—5482no—
219C2×C4—5482no—
220C2×C4—5482no—
221C2×C4—5481no—
222C2×C4—5482no—
223C2×C4—5482no—
224C2×C4—5482no—
225C2×C4—5482no—
226C2×C4—5481no—
227C2×C4—5481no—
228C2×C4—1371no—
229C2×C4—1372no—
230C2×C4—1371no—
231C2×C4—1371no—
232C2×C4C4×#1058773911yescharacteristic, complemented
233C22—21no—
234C22—41no—
235C22—41no—
236C22—2742no—
237C22—2741no—
238C22—2742no—
239C22—2741no—
240C22—2741no—
241C22—5482no—
242C22—5481no—
243C22—5481no—
244C22—5482no—
245C22—5481no—
246C22—5481no—
247C22—1372no—
248C22—1371no—
249C22—1371no—
250C22C2×C4×#1058773911yescharacteristic
251C22D4×#1058773912yes—
252C4—21no—
253C4—42no—
254C4—2741no—
255C4—2742no—
256C4—2741no—
257C4—5482no—
258C4—5484no—
259C4—5481no—
260C4—5481no—
261C4—1372no—
262C2—22no—
263C2—21no—
264C2—41no—
265C2—2742no—
266C2—2741no—
267C2—5481no—
268C2—1371no—
269C2—1371no—
270C2C22⋊C4×#1058773911yescharacteristic
subgroupG/H
centerC2C22⋊C4×#10587739
commutatorC22×C137C2×C42
FrattiniC23C22×#10587739
Fitting#99×C137C4
socleC2×C137C4×C22⋊C4
prime radicalC2×D4×D137C22
cosocleC23×D137C23
hypercenter#99#10587739
hypocenterC137C4×#99
LCSG ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C23 ◁ #99

Complements

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

NH ≅ G/Ntype
3. C22⋊C4×#10587739C2semidirect
4. #22578280C2semidirect
6. C23×#10587739C4semidirect
10. C2×C4⋊#10587739C4semidirect
11. C2×C22⋊#10587739C4semidirect
15. C2×C4×#10587739C4semidirect
16. #17984147C4semidirect
17. #17984146C4semidirect
21. #17984003C4semidirect
22. #17984002C4semidirect
23. #99×C137C4semidirect
39. C22⋊#10587739C2×C4semidirect
52. C22⋊Dic137C2×C4semidirect
55. C22⋊C4×C137C2×C4semidirect
78. C22×D137C42semidirect
87. C2×Dic137C42semidirect
92. C23×C137C42semidirect
96. C2×C4×C137C42semidirect
121. #10587739#99direct (N × H)
135. C137C4×#99semidirect
158. #99#10587739direct (N × H)
188. C22⋊C4C2×#10587739semidirect
203. C23C4×#10587739semidirect
232. C2×C4C4×#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#105877392
D137⋊#992
C22⋊C4×#105877392
#225782802
#99×D1372
C4×#99137