Finite Group Databasebeta

← D137⋊#327 #31796155

Subgroups — D137⋊#327 #31796155

Order 17536.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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#22578371C211yesmaximal, characteristic, complemented
2C2×(C2×#10587739)⋊C4C211yesmaximal, characteristic, complemented
3#22578255C212yesmaximal, complemented
4#22578254C212yesmaximal, complemented
5#121×D137C211yesmaximal, characteristic
6C23×#10587739—21no—
7C2×#14555518—21no—
8#17984145C2212yescomplemented
9C22⋊#11764047—21no—
10(C2×#10587739)⋊C4—21no—
11(C2×#10587739)⋊C4C2212yescomplemented
12C2×D4×D137—21no—
13C2×D4×D137C411yescharacteristic, complemented
14C22×C4×D137C2211yescharacteristic
15#17984063C2211yescharacteristic
16#17984062C411yescharacteristic, complemented
17#17984061—21no—
18#17984060—21no—
19#17984059C2211yescharacteristic
20C4⋊C4×D137C411yescharacteristic, complemented
21C22⋊C4×D137—21no—
22#121×C137C411yescharacteristic, complemented
23C22×#10587739—24no—
24C22×#10587739—22no—
25C22×#10587739—42no—
26C22×#10587739—42no—
27#14555518—42no—
28C2×#11764047—22no—
29C23×D137—21no—
30C23×D137D411yescharacteristic
31C2×D548—21no—
32C2×D548C2×C411yescharacteristic, complemented
33C2×C22⋊D137—21no—
34C2×C22⋊D137—21no—
35C2×C22⋊D137—21no—
36D4×D137—42no—
37D4×D137—42no—
38D4×D137—42no—
39C22×Dic137D411yescharacteristic
40C2×C4×D137—21no—
41C2×C4×D137—42no—
42C2×C4×D137C2×C411yescharacteristic
43C2×C4×D137D411yescharacteristic
44C2×C4×D137D411yescharacteristic
45C2×C4×D137C2311yescharacteristic
46C4⋊Dic137C2×C411yescharacteristic, complemented
47#14555489—21no—
48C22⋊Dic137—21no—
49#14555487—21no—
50#14555487—21no—
51C2×D4×C137—21no—
52C2×D4×C137C2×C411yescharacteristic, complemented
53C22×C4×C137C2×C411yescharacteristic
54C4⋊C4×C137C2×C411yescharacteristic, complemented
55C22⋊C4×C137—21no—
56C2×#10587739—24no—
57C2×#10587739—42no—
58C2×#10587739—48no—
59C2×#10587739—42no—
60#11764047—42no—
61C22×D137—22no—
62C22×D137—22no—
63C22×D137—21no—
64C22×D137—42no—
65C22×D137—42no—
66C22×D137—42no—
67C22×D137C22⋊C411yescharacteristic
68C22×D137D812yes—
69C22×D137SD1612yes—
70C22×D137C2×D411yescharacteristic
71D548—41no—
72D548—41no—
73D548—41no—
74C22⋊D137—42no—
75C22⋊D137—42no—
76C22⋊D137—41no—
77C22⋊D137—41no—
78C2×Dic137—21no—
79C2×Dic137—41no—
80C2×Dic137C22⋊C411yescharacteristic
81C2×Dic137C2×D411yescharacteristic
82C4×D137—22no—
83C4×D137—22no—
84C4×D137—41no—
85C4×D137—41no—
86C4×D137—42no—
87C23×C137—21no—
88C23×C137C22⋊C411yescharacteristic
89D4×C137—41no—
90D4×C137—42no—
91C2×C4×C137—21no—
92C2×C4×C137—41no—
93C2×C4×C137C22⋊C411yescharacteristic
94C2×C4×C137C22×C411yescharacteristic
95C22×C137—21no—
96C22×C137—21no—
97C22×C137—41no—
98C22×C137—41no—
99C22×C137—41no—
100C22×C137#10212yes—
101C22×C137C2×C22⋊C411yescharacteristic
102C2×D137—21no—
103C2×D137—21no—
104C2×D137—24no—
105C2×D137—24no—
106C2×D137—41no—
107C2×D137—41no—
108C2×D137—41no—
109C2×D137—84no—
110C2×D137#9912yes—
111C2×D137#10212yes—
112C2×D137#10412yes—
113C2×D137C2×D811yescharacteristic
114C2×D137C2×SD1611yescharacteristic
115#10587739—44no—
116C4×C137—21no—
117C4×C137—21no—
118C4×C137—41no—
119Dic137—21no—
120Dic137—21no—
121Dic137—41no—
122C2×C137—22no—
123C2×C137—81no—
124C2×C137C2×#9911yescharacteristic
125C2×C137C2×#10211yescharacteristic
126C2×C137C2×#10411yescharacteristic
127D137—21no—
128D137—21no—
129D137—81no—
130D137#32714yescomplemented
131C137C2×#32711yescharacteristic, complemented, Sylow
132C2×#327—1371nomaximal, Sylow
133C2×#121—1371no—
134C2×C22⋊C8—1371no—
135C2×#95—1371no—
136#327—1372no—
137#327—1372no—
138C22×D4—2741no—
139C22×D4—1371no—
140C23×C4—2741no—
141C23×C4—1371no—
142C22×C8—2741no—
143#121—2741no—
144#121—2741no—
145#121—1371no—
146#121—1371no—
147#121—1371no—
148#121#1058773911yescharacteristic, complemented
149C2×C4⋊C4—1371no—
150C2×C22⋊C4—2741no—
151C22⋊C8—2741no—
152C22⋊C8—1372no—
153#95—2741no—
154#95—1372no—
155C24—2741no—
156C24—1371no—
157C2×D4—21no—
158C2×D4—2741no—
159C2×D4—2741no—
160C2×D4—2741no—
161C2×D4—2741no—
162C2×D4—5482no—
163C2×D4—5482no—
164C2×D4—5482no—
165C2×D4—1371no—
166C2×D4C2×#1058773911yescharacteristic, complemented
167C22×C4—2741no—
168C22×C4—2744no—
169C22×C4—2742no—
170C22×C4—5482no—
171C22×C4—5482no—
172C22×C4—5482no—
173C22×C4—1371no—
174C22×C4—1371no—
175C22×C4—1371no—
176C22×C4—1371no—
177C22×C4—1371no—
178C22×C4C2×#1058773911yescharacteristic
179C2×C8—2742no—
180C2×C8—5482no—
181C4⋊C4—2741no—
182C4⋊C4—1371no—
183C4⋊C4C2×#1058773911yescharacteristic, complemented
184C22⋊C4—21no—
185C22⋊C4—2741no—
186C22⋊C4—2741no—
187C22⋊C4—2741no—
188C23—21no—
189C23—2742no—
190C23—2742no—
191C23—2741no—
192C23—5482no—
193C23—5482no—
194C23—5482no—
195C23—1372no—
196C23—1372no—
197C23—1371no—
198C23—1371no—
199C23C22⋊#1058773911yescharacteristic
200D4—41no—
201D4—42no—
202D4—5481no—
203D4—5481no—
204D4—5481no—
205D4—5482no—
206D4—5481no—
207D4—5481no—
208D4—5482no—
209C2×C4—21no—
210C2×C4—41no—
211C2×C4—2742no—
212C2×C4—2742no—
213C2×C4—2744no—
214C2×C4—2741no—
215C2×C4—5481no—
216C2×C4—5481no—
217C2×C4—5482no—
218C2×C4—5481no—
219C2×C4—5488no—
220C2×C4—5482no—
221C2×C4—5482no—
222C2×C4—1371no—
223C2×C4—1371no—
224C2×C4C22⋊#1058773911yescharacteristic
225C2×C4C22×#1058773911yescharacteristic
226C8—5482no—
227C22—21no—
228C22—21no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—2744no—
233C22—2744no—
234C22—2741no—
235C22—2741no—
236C22—5481no—
237C22—5481no—
238C22—5481no—
239C22—10964no—
240C22—1371no—
241C22—1371no—
242C22—1372no—
243C22—1372no—
244C22—1372no—
245C22#1798415012yes—
246C22C2×C22⋊#1058773911yescharacteristic
247C4—21no—
248C4—21no—
249C4—41no—
250C4—2741no—
251C4—2741no—
252C4—5484no—
253C4—5481no—
254C2—22no—
255C2—81no—
256C2—2741no—
257C2—2741no—
258C2—10961no—
259C2—1374no—
260C2D137⋊#9911yescharacteristic
261C2C2×#1798415011yescharacteristic
262C2D137⋊#10411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#10587739
commutatorC2×C4×C137C22×C4
FrattiniC2×C4C22×#10587739
Fitting#121×C137C4
socleC22×C137C2×C22⋊C4
prime radical#121×D137C2
cosocleC2×C4×D137C23
hypercenter#121#10587739
hypocenterC137C2×#327
LCSG ▷ C2×C4×C137 ▷ C22×C137 ▷ C137
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. #22578371C2semidirect
2. C2×(C2×#10587739)⋊C4C2semidirect
3. #22578255C2semidirect
4. #22578254C2semidirect
8. #17984145C22semidirect
11. (C2×#10587739)⋊C4C22semidirect
13. C2×D4×D137C4semidirect
16. #17984062C4semidirect
20. C4⋊C4×D137C4semidirect
22. #121×C137C4semidirect
32. C2×D548C2×C4semidirect
46. C4⋊Dic137C2×C4semidirect
52. C2×D4×C137C2×C4semidirect
54. C4⋊C4×C137C2×C4semidirect
130. D137#327semidirect
131. C137C2×#327semidirect
148. #121#10587739semidirect
166. C2×D4C2×#10587739semidirect
183. C4⋊C4C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
#225783712
C2×(C2×#10587739)⋊C42
#225782552
#225782542
#121×D1372
C2×#327137