Finite Group Databasebeta

← #31796930

Subgroups — #31796930

Order 17536.

408 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 72 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×#17984147C212yesmaximal, complemented
2C2×#17984146C212yesmaximal, complemented
3#22578345C211yesmaximal, characteristic, complemented
4C4×C22⋊#10587739C211yesmaximal, characteristic, complemented
5C2×#17984061C211yesmaximal, characteristic
6C2×C4⋊#10587739—21no—
7C2×C22⋊#10587739C2212yes—
8C2×C22⋊#10587739C2212yes—
9C2×C4×#10587739—21no—
10#17984147—24no—
11#17984146—24no—
12(C2×#10587739)⋊C4—21no—
13(C2×#10587739)⋊C4—21no—
14C22×D548C2211yescharacteristic
15C22×C22⋊D137C411yescharacteristic, complemented
16C22×C22⋊D137C2211yescharacteristic
17C22×C4×D137C2211yescharacteristic
18#17984061—21no—
19#17984061—21no—
20#17984061—21no—
21#17984061—21no—
22C2×#14555489C411yescharacteristic, complemented
23C2×#14555487C411yescharacteristic, complemented
24C2×C22⋊C4×C137C411yescharacteristic, complemented
25C22×#10587739—22no—
26C22×#10587739—22no—
27C4⋊#10587739—42no—
28C22⋊#10587739—22no—
29C22⋊#10587739—22no—
30C22⋊#10587739—22no—
31C22⋊#10587739D414yescomplemented
32C4×#10587739—42no—
33C23×D137C2×C411yescharacteristic
34C23×D137C2311yescharacteristic
35C2×D548—21no—
36C2×D548—21no—
37C2×D548—22no—
38C2×D548—22no—
39C2×C22⋊D137—21no—
40C2×C22⋊D137—21no—
41C2×C22⋊D137—22no—
42C2×C22⋊D137—22no—
43C2×C22⋊D137—21no—
44C2×C22⋊D137—21no—
45C2×C22⋊D137—42no—
46C22×Dic137C2×C411yescharacteristic
47C22×Dic137C2×C411yescharacteristic
48C2×C4×D137—42no—
49C2×C4×D137D411yescharacteristic
50C2×C4×D137D411yescharacteristic
51C2×C4×D137D411yescharacteristic
52C2×C4×D137D411yescharacteristic
53#14555489—21no—
54#14555489—21no—
55#14555487—21no—
56#14555487—21no—
57C24×C137C2×C411yescharacteristic
58C22×C4×C137C2×C411yescharacteristic
59C22×C4×C137C2×C411yescharacteristic
60C22⋊C4×C137—21no—
61C22⋊C4×C137—21no—
62C2×#10587739—24no—
63C2×#10587739—42no—
64C2×#10587739—42no—
65C2×#10587739—42no—
66C2×#10587739—42no—
67C2×#10587739—42no—
68C22×D137—21no—
69C22×D137—22no—
70C22×D137—41no—
71C22×D137—41no—
72C22×D137—41no—
73C22×D137—41no—
74C22×D137C2×D411yescharacteristic
75C22×D137C2×D411yescharacteristic
76C22×D137C2×D412yes—
77C22×D137#4112yes—
78D548—42no—
79D548—41no—
80D548—41no—
81C22⋊D137—42no—
82C22⋊D137—42no—
83C22⋊D137—41no—
84C22⋊D137—41no—
85C22⋊D137—42no—
86C2×Dic137—22no—
87C2×Dic137—21no—
88C2×Dic137—22no—
89C2×Dic137C22⋊C411yescharacteristic
90C2×Dic137C22⋊C411yescharacteristic
91C4×D137—22no—
92C4×D137—22no—
93C4×D137—41no—
94C4×D137—41no—
95C23×C137—21no—
96C23×C137—22no—
97C23×C137—41no—
98C23×C137—41no—
99C23×C137C22×C411yescharacteristic
100C2×C4×C137—21no—
101C2×C4×C137—22no—
102C2×C4×C137—41no—
103C2×C4×C137C22⋊C411yescharacteristic
104C2×C4×C137C22⋊C411yescharacteristic
105C22×C137—21no—
106C22×C137—21no—
107C22×C137—41no—
108C22×C137—41no—
109C22×C137—41no—
110C22×C137—82no—
111C22×C137C2×C22⋊C411yescharacteristic
112C22×C137C4×D412yes—
113C2×D137—21no—
114C2×D137—21no—
115C2×D137—21no—
116C2×D137—21no—
117C2×D137—42no—
118C2×D137—42no—
119C2×D137—41no—
120C2×D137—41no—
121C2×D137—41no—
122C2×D137—82no—
123C2×D137#12112yes—
124C2×D137#12411yescharacteristic
125C2×D137#12711yescharacteristic
126#10587739—44no—
127#10587739—82no—
128C4×C137—21no—
129C4×C137—21no—
130C4×C137—41no—
131Dic137—21no—
132Dic137—41no—
133Dic137#9912yescomplemented
134C2×C137—22no—
135C2×C137—81no—
136C2×C137#39011yescharacteristic
137C2×C137C2×#9911yescharacteristic
138C2×C137#41011yescharacteristic
139D137—22no—
140D137—21no—
141D137—21no—
142D137—81no—
143C137#196411yescharacteristic, complemented, Sylow
144#1964—1371nomaximal, Sylow
145C2×#121—1371no—
146C2×#99—1372no—
147C2×#99—1372no—
148#380—1371no—
149C4×C22⋊C4—1371no—
150C22×D4—1371no—
151C22×D4—1371no—
152C22×D4—1371no—
153C23×C4—1371no—
154#121—2741no—
155#121—2741no—
156#121—2741no—
157#121—2741no—
158C2×C4⋊C4—2741no—
159C2×C4⋊C4—1371no—
160C2×C22⋊C4—1372no—
161C2×C22⋊C4—1371no—
162C2×C22⋊C4—1372no—
163C2×C22⋊C4#1058773911yescharacteristic, complemented
164C2×C42—2741no—
165#99—2744no—
166#99—2744no—
167#95—2741no—
168#95—2741no—
169C24—1371no—
170C24—1371no—
171C24C2×#1058773911yescharacteristic
172C2×D4—2742no—
173C2×D4—2741no—
174C2×D4—2741no—
175C2×D4—2742no—
176C2×D4—2742no—
177C2×D4—2741no—
178C2×D4—2741no—
179C2×D4—2742no—
180C2×D4—2741no—
181C2×D4—2741no—
182C2×D4—5482no—
183C22×C4—2742no—
184C22×C4—2742no—
185C22×C4—5482no—
186C22×C4—1371no—
187C22×C4—1371no—
188C22×C4—1371no—
189C22×C4—1371no—
190C22×C4—1371no—
191C22×C4—1371no—
192C22×C4C2×#1058773911yescharacteristic
193C22×C4C2×#1058773911yescharacteristic
194C4⋊C4—2741no—
195C4⋊C4—2741no—
196C4⋊C4—5482no—
197C22⋊C4—21no—
198C22⋊C4—21no—
199C22⋊C4—2742no—
200C22⋊C4—2741no—
201C22⋊C4—2741no—
202C22⋊C4—2742no—
203C22⋊C4—2742no—
204C22⋊C4—1374no—
205C42—5482no—
206C23—21no—
207C23—22no—
208C23—41no—
209C23—41no—
210C23—2741no—
211C23—2742no—
212C23—5481no—
213C23—5481no—
214C23—5481no—
215C23—5481no—
216C23—1371no—
217C23—1371no—
218C23—1372no—
219C23—1372no—
220C23C22×#1058773911yescharacteristic
221D4—5482no—
222D4—5481no—
223D4—5481no—
224D4—5482no—
225D4—5482no—
226D4—5481no—
227D4—5481no—
228D4—5482no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—2742no—
233C2×C4—2742no—
234C2×C4—2742no—
235C2×C4—2744no—
236C2×C4—2741no—
237C2×C4—2742no—
238C2×C4—5481no—
239C2×C4—5481no—
240C2×C4—5482no—
241C2×C4—5482no—
242C2×C4—5482no—
243C2×C4—5482no—
244C2×C4—5482no—
245C2×C4—1371no—
246C2×C4—1371no—
247C2×C4C22⋊#1058773911yescharacteristic
248C2×C4C22⋊#1058773911yescharacteristic
249C22—21no—
250C22—21no—
251C22—41no—
252C22—41no—
253C22—41no—
254C22—82no—
255C22—2741no—
256C22—2741no—
257C22—2741no—
258C22—2741no—
259C22—5482no—
260C22—5482no—
261C22—5481no—
262C22—5481no—
263C22—5481no—
264C22—10962no—
265C22—1372no—
266C22—1371no—
267C22—1371no—
268C22C2×C22⋊#1058773911yescharacteristic
269C22D4×#1058773912yes—
270C4—21no—
271C4—21no—
272C4—41no—
273C4—2741no—
274C4—5484no—
275C4—5481no—
276C4—10962no—
277C4—1372no—
278C2—22no—
279C2—81no—
280C2—2742no—
281C2—2741no—
282C2—2741no—
283C2—10961no—
284C2#2257835511yescharacteristic
285C2D137⋊#9911yescharacteristic
286C2#2257838211yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
FittingC2×C22⋊C4×C137C4
socleC22×C137C2×C22⋊C4
prime radicalC2×#17984061C2
cosocleC23×D137C23
hypercenterC2×C22⋊C4#10587739
hypocenterC137#1964
LCSG ▷ C23×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#17984147C2semidirect
2. C2×#17984146C2semidirect
3. #22578345C2semidirect
4. C4×C22⋊#10587739C2semidirect
15. C22×C22⋊D137C4semidirect
22. C2×#14555489C4semidirect
23. C2×#14555487C4semidirect
24. C2×C22⋊C4×C137C4semidirect
31. C22⋊#10587739D4semidirect
133. Dic137#99semidirect
143. C137#1964semidirect
163. C2×C22⋊C4#10587739semidirect

Maximal subgroups

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

subgroupindex
C2×#179841472
C2×#179841462
#225783452
C4×C22⋊#105877392
C2×#179840612
#1964137