Finite Group Databasebeta

← C137×#11535 #46860921

Subgroups — C137×#11535 #46860921

Order 35072.

440 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 106 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#2326×C137C211yesmaximal, characteristic
2#1465×C137C212yesmaximal
3#1325×C137C212yesmaximal, complemented
4#1319×C137C212yesmaximal, complemented
5C2×#124×C137C411yescharacteristic
6C2×C4×Q8×C137C411yescharacteristic
7C2×C4×D4×C137C2211yescharacteristic
8#407×C137—22no—
9#391×C137C2211yescharacteristic
10#390×C137C411yescharacteristic
11#388×C137—22no—
12#385×C137—22no—
13#384×C137C2211yescharacteristic
14#376×C137C411yescharacteristic
15#324×C137C2214yes—
16#323×C137—22no—
17C22×Q8×C137C2×C411yescharacteristic
18C22×D4×C137C2×C411yescharacteristic
19C23×C4×C137D412yes—
20C2×M16×C137—42no—
21#124×C137—22no—
22#124×C137—22no—
23C4×Q8×C137—42no—
24C4×D4×C137—42no—
25C2×C4⋊C4×C137—21no—
26C2×C4⋊C4×C137—22no—
27C2×C4⋊C4×C137C2×C411yescharacteristic
28C2×C4⋊C4×C137C2311yescharacteristic
29C2×C22⋊C4×C137—22no—
30C2×C22⋊C4×C137—22no—
31C2×C22⋊C4×C137D412yes—
32C2×C42×C137—21no—
33C2×C42×C137C2×C411yescharacteristic
34C2×C42×C137C2×C411yescharacteristic
35C2×C42×C137C2×C411yescharacteristic
36C22⋊C8×C137—24no—
37C22⋊C8×C137—42no—
38#95×C137—21no—
39#95×C137—21no—
40#95×C137—22no—
41C24×C137C22⋊C412yes—
42C2×Q8×C137—21no—
43C2×Q8×C137—41no—
44C2×Q8×C137—41no—
45C2×Q8×C137M1612yescomplemented
46C2×D4×C137—21no—
47C2×D4×C137—21no—
48C2×D4×C137—41no—
49C2×D4×C137—41no—
50C22×C4×C137—21no—
51C22×C4×C137—21no—
52C22×C4×C137—22no—
53C22×C4×C137—22no—
54C22×C4×C137—21no—
55C22×C4×C137—21no—
56C22×C4×C137—82no—
57C22×C4×C137C22⋊C411yescharacteristic
58C22×C4×C137C22⋊C411yescharacteristic
59C22×C4×C137C22×C411yescharacteristic
60C22×C4×C137C2×D412yes—
61M16×C137—84no—
62C2×C8×C137—44no—
63C4⋊C4×C137—41no—
64C4⋊C4×C137—41no—
65C4⋊C4×C137—42no—
66C4⋊C4×C137—82no—
67C22⋊C4×C137—28no—
68C22⋊C4×C137—42no—
69C22⋊C4×C137—82no—
70C42×C137—22no—
71C42×C137—21no—
72C42×C137—41no—
73C42×C137—42no—
74C42×C137M1612yescomplemented
75C23×C137—22no—
76C23×C137—22no—
77C23×C137—22no—
78C23×C137—42no—
79C23×C137—42no—
80C23×C137C2×C22⋊C411yescharacteristic
81Q8×C137—42no—
82Q8×C137—42no—
83D4×C137—42no—
84D4×C137—42no—
85C2×C4×C137—21no—
86C2×C4×C137—21no—
87C2×C4×C137—22no—
88C2×C4×C137—22no—
89C2×C4×C137—21no—
90C2×C4×C137—22no—
91C2×C4×C137—41no—
92C2×C4×C137—44no—
93C2×C4×C137—41no—
94C2×C4×C137—41no—
95C2×C4×C137—41no—
96C2×C4×C137—41no—
97C2×C4×C137—42no—
98C2×C4×C137—41no—
99C2×C4×C137—41no—
100C2×C4×C137—42no—
101C2×C4×C137—41no—
102C2×C4×C137—41no—
103C2×C4×C137—41no—
104C2×C4×C137—82no—
105C2×C4×C137#9912yes—
106C2×C4×C137C2×M1611yescharacteristic
107C2×C4×C137C2×M1611yescharacteristic
108C8×C137—84no—
109C22×C137—21no—
110C22×C137—21no—
111C22×C137—44no—
112C22×C137—42no—
113C22×C137—42no—
114C22×C137—42no—
115C22×C137—82no—
116C22×C137#40711yescharacteristic
117C22×C137C2×#9911yescharacteristic
118C22×C137#41311yescharacteristic
119C4×C137—22no—
120C4×C137—21no—
121C4×C137—21no—
122C4×C137—42no—
123C4×C137—42no—
124C4×C137—41no—
125C4×C137—42no—
126C4×C137—81no—
127C4×C137—81no—
128C2×C137—21no—
129C2×C137—21no—
130C2×C137—82no—
131C2×C137#146511yescharacteristic
132C2×C137#212711yescharacteristic
133C2×C137#213511yescharacteristic
134#11535C13711yesmaximal, characteristic, direct product factor, Sylow
135C137#1153511yescharacteristic, direct product factor, Sylow
136#2326C2×C13711yescharacteristic
137#1465C2×C13712yes—
138#1325C2×C13712yescomplemented
139#1319C2×C13712yescomplemented
140C2×#124C4×C13711yescharacteristic
141C2×C4×Q8C4×C13711yescharacteristic
142C2×C4×D4C22×C13711yescharacteristic
143#407—22no—
144#391C22×C13711yescharacteristic
145#390C4×C13711yescharacteristic
146#388—22no—
147#385—22no—
148#384C22×C13711yescharacteristic
149#376C4×C13711yescharacteristic
150#324C22×C13714yes—
151#323—22no—
152C22×Q8C2×C4×C13711yescharacteristic
153C22×D4C2×C4×C13711yescharacteristic
154C23×C4D4×C13712yes—
155C2×M16—42no—
156#124—22no—
157#124—22no—
158C4×Q8—42no—
159C4×D4—42no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4C2×C4×C13711yescharacteristic
163C2×C4⋊C4C23×C13711yescharacteristic
164C2×C22⋊C4—22no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4D4×C13712yes—
167C2×C42—21no—
168C2×C42C2×C4×C13711yescharacteristic
169C2×C42C2×C4×C13711yescharacteristic
170C2×C42C2×C4×C13711yescharacteristic
171C22⋊C8—24no—
172C22⋊C8—42no—
173#95—21no—
174#95—21no—
175#95—22no—
176C24C22⋊C4×C13712yes—
177C2×Q8—21no—
178C2×Q8—41no—
179C2×Q8—41no—
180C2×Q8M16×C13712yescomplemented
181C2×D4—21no—
182C2×D4—21no—
183C2×D4—41no—
184C2×D4—41no—
185C22×C4—21no—
186C22×C4—21no—
187C22×C4—22no—
188C22×C4—22no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—82no—
192C22×C4C22⋊C4×C13711yescharacteristic
193C22×C4C22⋊C4×C13711yescharacteristic
194C22×C4C22×C4×C13711yescharacteristic
195C22×C4C2×D4×C13712yes—
196M16—84no—
197C2×C8—44no—
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C22⋊C4—28no—
203C22⋊C4—42no—
204C22⋊C4—82no—
205C42—22no—
206C42—21no—
207C42—41no—
208C42—42no—
209C42M16×C13712yescomplemented
210C23—22no—
211C23—22no—
212C23—22no—
213C23—42no—
214C23—42no—
215C23C2×C22⋊C4×C13711yescharacteristic
216Q8—42no—
217Q8—42no—
218D4—42no—
219D4—42no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—21no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—44no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—42no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—42no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—82no—
240C2×C4#99×C13712yes—
241C2×C4C2×M16×C13711yescharacteristic
242C2×C4C2×M16×C13711yescharacteristic
243C8—84no—
244C22—21no—
245C22—21no—
246C22—44no—
247C22—42no—
248C22—42no—
249C22—42no—
250C22—82no—
251C22#407×C13711yescharacteristic
252C22C2×#99×C13711yescharacteristic
253C22#413×C13711yescharacteristic
254C4—22no—
255C4—21no—
256C4—21no—
257C4—42no—
258C4—42no—
259C4—41no—
260C4—42no—
261C4—81no—
262C4—81no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2#1465×C13711yescharacteristic
267C2#2127×C13711yescharacteristic
268C2#2135×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C4⋊C4C23×C137
prime radicalC22×D4×C137C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×C4×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1325×C137C2semidirect
4. #1319×C137C2semidirect
45. C2×Q8×C137M16semidirect
74. C42×C137M16semidirect
134. #11535C137direct (N × H)
135. C137#11535direct (N × H)
138. #1325C2×C137semidirect
139. #1319C2×C137semidirect
180. C2×Q8M16×C137semidirect
209. C42M16×C137semidirect

Maximal subgroups

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

subgroupindex
#2326×C1372
#1465×C1372
#1325×C1372
#1319×C1372
#11535137