Finite Group Databasebeta

← D137⋊#352 #31797038

Subgroups — D137⋊#352 #31797038

Order 17536.

278 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 54 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
1D137⋊#100C211yesmaximal, characteristic
2D137⋊#99C211yesmaximal, characteristic
3#22578306C212yesmaximal, complemented
4#22578305C212yesmaximal, complemented
5#123×D137C211yesmaximal, characteristic
6D137⋊M16—21no—
7C2×C22⋊#10587739—21no—
8#17984148—21no—
9#17984148C2212yes—
10#17984147—21no—
11#17984146C2212yes—
12C2×D4×D137C2211yescharacteristic
13C22×C4×D137C411yescharacteristic, complemented
14#17984073—21no—
15#17984072C2211yescharacteristic
16#17984071C2211yescharacteristic
17#17984070—21no—
18#17984069C411yescharacteristic, complemented
19C4⋊C4×D137—21no—
20C22⋊C4×D137—21no—
21C22⋊C4×D137C411yescharacteristic, complemented
22#123×C137C411yescharacteristic, complemented
23C22×#10587739—41no—
24#14555520—41no—
25#14555519—22no—
26#14555518—41no—
27C22⋊#10587739—22no—
28C22⋊#10587739—41no—
29C23×D137C2×C411yescharacteristic
30C23×D137D411yescharacteristic
31C2×D548D411yescharacteristic
32C2×C22⋊D137D411yescharacteristic
33C2×C22⋊D137C2311yescharacteristic
34D4×D137—41no—
35D4×D137—41no—
36C22×Dic137C2×C411yescharacteristic, complemented
37C2×C4×D137—21no—
38C2×C4×D137—21no—
39C2×C4×D137—21no—
40C2×C4×D137—41no—
41C2×C4×D137—41no—
42C2×C4×D137D411yescharacteristic
43C4⋊Dic137—21no—
44#14555489—21no—
45#14555489—21no—
46C22⋊Dic137—21no—
47C22⋊Dic137C2×C411yescharacteristic, complemented
48#14555487—21no—
49#14555487—21no—
50#14555487—21no—
51C2×D4×C137C2×C411yescharacteristic
52C22×C4×C137C2×C411yescharacteristic, complemented
53C4⋊C4×C137—21no—
54C22⋊C4×C137—21no—
55C22⋊C4×C137C2×C411yescharacteristic, complemented
56C2×#10587739—42no—
57C2×#10587739—82no—
58#11764047—42no—
59C22×D137—21no—
60C22×D137—21no—
61C22×D137—21no—
62C22×D137—21no—
63C22×D137—42no—
64C22×D137—42no—
65C22×D137C22⋊C411yescharacteristic
66C22×D137C22⋊C411yescharacteristic
67C22×D137C2×D411yescharacteristic
68D548—41no—
69C22⋊D137—41no—
70C22⋊D137—41no—
71C2×Dic137—21no—
72C2×Dic137—21no—
73C2×Dic137—41no—
74C2×Dic137C2×D411yescharacteristic
75C4×D137—21no—
76C4×D137—21no—
77C4×D137—42no—
78C4×D137—42no—
79C4×D137—41no—
80C4×D137—41no—
81C23×C137C22⋊C411yescharacteristic
82C23×C137C22×C411yescharacteristic
83D4×C137—41no—
84C2×C4×C137—21no—
85C2×C4×C137—21no—
86C2×C4×C137—41no—
87C2×C4×C137C22⋊C411yescharacteristic
88C22×C137—21no—
89C22×C137—21no—
90C22×C137—41no—
91C22×C137—41no—
92C22×C137C2×C22⋊C411yescharacteristic
93C2×D137—22no—
94C2×D137—21no—
95C2×D137—21no—
96C2×D137—42no—
97C2×D137—42no—
98C2×D137—42no—
99C2×D137—41no—
100C2×D137—41no—
101C2×D137—81no—
102C2×D137#9911yescharacteristic
103C2×D137#9911yescharacteristic
104#10587739—82no—
105C4×C137—21no—
106C4×C137—41no—
107C4×C137—41no—
108Dic137—21no—
109Dic137—41no—
110Dic137—41no—
111C2×C137—21no—
112C2×C137—41no—
113C2×C137—41no—
114C2×C137C2×#9911yescharacteristic
115D137—21no—
116D137—41no—
117D137—41no—
118D137#35212yescomplemented
119C137C2×#35211yescharacteristic, complemented, Sylow
120C2×#352—1371nomaximal, Sylow
121C2×#123—1371no—
122C2×#100—1371no—
123C2×#99—1371no—
124#352—1372no—
125#352—1372no—
126C22×D4—1371no—
127C23×C4—1371no—
128C2×M16—2741no—
129#123—2741no—
130#123—2741no—
131#123—1371no—
132#123—1371no—
133#123—1371no—
134#123#1058773911yescharacteristic, complemented
135C2×C4⋊C4—2741no—
136C2×C22⋊C4—2741no—
137C2×C22⋊C4—2741no—
138C2×C22⋊C4—1371no—
139#100—2741no—
140#100—1372no—
141#99—2741no—
142#99—1372no—
143C24—1371no—
144C24—1371no—
145C2×D4—5481no—
146C2×D4—5481no—
147C2×D4—1371no—
148C2×D4—1371no—
149C2×D4—1371no—
150C2×D4C2×#1058773911yescharacteristic
151C22×C4—2741no—
152C22×C4—2741no—
153C22×C4—2741no—
154C22×C4—5481no—
155C22×C4—5481no—
156C22×C4—5481no—
157C22×C4—1371no—
158C22×C4—1371no—
159C22×C4C2×#1058773911yescharacteristic, complemented
160M16—2742no—
161M16—5481no—
162C2×C8—5481no—
163C4⋊C4—21no—
164C4⋊C4—2741no—
165C4⋊C4—2741no—
166C4⋊C4—2741no—
167C22⋊C4—21no—
168C22⋊C4—2741no—
169C22⋊C4—2741no—
170C22⋊C4—2741no—
171C22⋊C4—2741no—
172C22⋊C4—2742no—
173C22⋊C4—5481no—
174C22⋊C4—1371no—
175C22⋊C4C2×#1058773911yescharacteristic, complemented
176C23—2741no—
177C23—2741no—
178C23—2741no—
179C23—2741no—
180C23—5482no—
181C23—5482no—
182C23—1371no—
183C23—1371no—
184C23—1371no—
185C23C22⋊#1058773911yescharacteristic
186C23C22×#1058773911yescharacteristic
187D4—41no—
188D4—5481no—
189D4—5481no—
190D4—5481no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—41no—
194C2×C4—2741no—
195C2×C4—2741no—
196C2×C4—2741no—
197C2×C4—2741no—
198C2×C4—5482no—
199C2×C4—5482no—
200C2×C4—5481no—
201C2×C4—5481no—
202C2×C4—5481no—
203C2×C4—5482no—
204C2×C4—10962no—
205C2×C4—1371no—
206C2×C4C22⋊#1058773911yescharacteristic
207C8—5482no—
208C22—21no—
209C22—21no—
210C22—41no—
211C22—41no—
212C22—2742no—
213C22—2741no—
214C22—2741no—
215C22—5481no—
216C22—5481no—
217C22—5482no—
218C22—5482no—
219C22—5482no—
220C22—10961no—
221C22—1371no—
222C22—1371no—
223C22C2×C22⋊#1058773911yescharacteristic
224C4—21no—
225C4—41no—
226C4—41no—
227C4—2741no—
228C4—5481no—
229C4—5481no—
230C4—10962no—
231C2—21no—
232C2—41no—
233C2—41no—
234C2—2741no—
235C2—5481no—
236C2—5481no—
237C2—1372no—
238C2D137⋊#9911yescharacteristic
subgroupG/H
centerC2D137⋊#99
commutatorC23×C137C22×C4
FrattiniC23C22×#10587739
Fitting#123×C137C4
socleC2×C137C2×#99
prime radicalC2×D4×D137C22
cosocleC2×C22⋊D137C23
hypercenter#123#10587739
hypocenterC137C2×#352
LCSG ▷ C23×C137 ▷ C22×C137 ▷ C2×C137 ▷ C137
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #123

Complements

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

NH ≅ G/Ntype
3. #22578306C2semidirect
4. #22578305C2semidirect
13. C22×C4×D137C4semidirect
18. #17984069C4semidirect
21. C22⋊C4×D137C4semidirect
22. #123×C137C4semidirect
36. C22×Dic137C2×C4semidirect
47. C22⋊Dic137C2×C4semidirect
52. C22×C4×C137C2×C4semidirect
55. C22⋊C4×C137C2×C4semidirect
118. D137#352semidirect
119. C137C2×#352semidirect
134. #123#10587739semidirect
159. C22×C4C2×#10587739semidirect
175. C22⋊C4C2×#10587739semidirect

Maximal subgroups

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

subgroupindex
D137⋊#1002
D137⋊#992
#225783062
#225783052
#123×D1372
C2×#352137