Finite Group Databasebeta

← C137×#11530 #46860916

Subgroups — C137×#11530 #46860916

Order 35072.

444 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 110 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
1C4×#124×C137C211yesmaximal, characteristic
2#1467×C137C212yesmaximal, complemented
3#1324×C137C212yesmaximal, complemented
4#1318×C137C212yesmaximal
5C2×#124×C137C411yescharacteristic
6C2×C4×Q8×C137C2211yescharacteristic
7C2×C4×D4×C137C411yescharacteristic
8#432×C137—22no—
9#391×C137C411yescharacteristic
10#390×C137C2211yescharacteristic
11#388×C137—22no—
12#382×C137C2211yescharacteristic
13C4×C22⋊C4×C137—22no—
14C43×C137C411yescharacteristic
15#324×C137—22no—
16#324×C137C2214yes—
17C22×Q8×C137C2×C411yescharacteristic
18C22×D4×C137C2×C411yescharacteristic
19C23×C4×C137—21no—
20#124×C137—22no—
21#124×C137—22no—
22C4×Q8×C137—41no—
23C4×Q8×C137—41no—
24C4×D4×C137—41no—
25C4×D4×C137—41no—
26C2×C4⋊C4×C137C2×C411yescharacteristic
27C2×C4⋊C4×C137D412yes—
28C2×C4⋊C4×C137C2311yescharacteristic
29C2×C22⋊C4×C137—22no—
30C2×C22⋊C4×C137—21no—
31C2×C22⋊C4×C137—22no—
32C2×C42×C137—21no—
33C2×C42×C137—21no—
34C2×C42×C137C2×C411yescharacteristic
35C2×C42×C137C2×C411yescharacteristic
36C2×C42×C137C2×C411yescharacteristic
37C2×C42×C137D412yes—
38C4⋊C8×C137—42no—
39C22⋊C8×C137—24no—
40#97×C137—42no—
41#95×C137—22no—
42#95×C137—22no—
43C24×C137—21no—
44C2×Q8×C137—21no—
45C2×Q8×C137—21no—
46C2×Q8×C137—41no—
47C2×Q8×C137—41no—
48C2×D4×C137—21no—
49C2×D4×C137—41no—
50C2×D4×C137—41no—
51C2×D4×C137M1612yes—
52C22×C4×C137—21no—
53C22×C4×C137—21no—
54C22×C4×C137—21no—
55C22×C4×C137—44no—
56C22×C4×C137—41no—
57C22×C4×C137—41no—
58C22×C4×C137C22⋊C412yes—
59C22×C4×C137C22⋊C411yescharacteristic
60C22×C4×C137C22⋊C411yescharacteristic
61C22×C4×C137C22×C411yescharacteristic
62C22×C4×C137C2×D412yes—
63C2×C8×C137—44no—
64C4⋊C4×C137—42no—
65C4⋊C4×C137—41no—
66C4⋊C4×C137—41no—
67C22⋊C4×C137—28no—
68C22⋊C4×C137—44no—
69C22⋊C4×C137—42no—
70C42×C137—22no—
71C42×C137—22no—
72C42×C137—21no—
73C42×C137—21no—
74C42×C137—21no—
75C42×C137—41no—
76C42×C137—42no—
77C42×C137—42no—
78C42×C137M1612yes—
79C23×C137—22no—
80C23×C137—42no—
81C23×C137—42no—
82C23×C137—41no—
83C23×C137—41no—
84C23×C137C2×C22⋊C411yescharacteristic
85Q8×C137—42no—
86Q8×C137—42no—
87D4×C137—42no—
88D4×C137—42no—
89C2×C4×C137—21no—
90C2×C4×C137—21no—
91C2×C4×C137—22no—
92C2×C4×C137—22no—
93C2×C4×C137—22no—
94C2×C4×C137—22no—
95C2×C4×C137—21no—
96C2×C4×C137—22no—
97C2×C4×C137—22no—
98C2×C4×C137—22no—
99C2×C4×C137—41no—
100C2×C4×C137—41no—
101C2×C4×C137—41no—
102C2×C4×C137—41no—
103C2×C4×C137—41no—
104C2×C4×C137—42no—
105C2×C4×C137—42no—
106C2×C4×C137—41no—
107C2×C4×C137—82no—
108C2×C4×C137—82no—
109C2×C4×C137#9911yescharacteristic
110C2×C4×C137#9911yescharacteristic
111C2×C4×C137C2×M1611yescharacteristic
112C2×C4×C137C2×M1611yescharacteristic
113C8×C137—84no—
114C22×C137—21no—
115C22×C137—21no—
116C22×C137—44no—
117C22×C137—42no—
118C22×C137—82no—
119C22×C137—81no—
120C22×C137—81no—
121C22×C137#40711yescharacteristic
122C22×C137C2×#9911yescharacteristic
123C22×C137#41311yescharacteristic
124C4×C137—21no—
125C4×C137—21no—
126C4×C137—21no—
127C4×C137—42no—
128C4×C137—42no—
129C4×C137—42no—
130C4×C137—41no—
131C4×C137—42no—
132C4×C137—81no—
133C4×C137#35512yes—
134C2×C137—21no—
135C2×C137—21no—
136C2×C137—82no—
137C2×C137#146511yescharacteristic
138C2×C137#212611yescharacteristic
139C2×C137C2×#35511yescharacteristic
140#11530C13711yesmaximal, characteristic, direct product factor, Sylow
141C137#1153011yescharacteristic, direct product factor, Sylow
142C4×#124C2×C13711yescharacteristic
143#1467C2×C13712yescomplemented
144#1324C2×C13712yescomplemented
145#1318C2×C13712yes—
146C2×#124C4×C13711yescharacteristic
147C2×C4×Q8C22×C13711yescharacteristic
148C2×C4×D4C4×C13711yescharacteristic
149#432—22no—
150#391C4×C13711yescharacteristic
151#390C22×C13711yescharacteristic
152#388—22no—
153#382C22×C13711yescharacteristic
154C4×C22⋊C4—22no—
155C43C4×C13711yescharacteristic
156#324—22no—
157#324C22×C13714yes—
158C22×Q8C2×C4×C13711yescharacteristic
159C22×D4C2×C4×C13711yescharacteristic
160C23×C4—21no—
161#124—22no—
162#124—22no—
163C4×Q8—41no—
164C4×Q8—41no—
165C4×D4—41no—
166C4×D4—41no—
167C2×C4⋊C4C2×C4×C13711yescharacteristic
168C2×C4⋊C4D4×C13712yes—
169C2×C4⋊C4C23×C13711yescharacteristic
170C2×C22⋊C4—22no—
171C2×C22⋊C4—21no—
172C2×C22⋊C4—22no—
173C2×C42—21no—
174C2×C42—21no—
175C2×C42C2×C4×C13711yescharacteristic
176C2×C42C2×C4×C13711yescharacteristic
177C2×C42C2×C4×C13711yescharacteristic
178C2×C42D4×C13712yes—
179C4⋊C8—42no—
180C22⋊C8—24no—
181#97—42no—
182#95—22no—
183#95—22no—
184C24—21no—
185C2×Q8—21no—
186C2×Q8—21no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×D4—21no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4M16×C13712yes—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—44no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4C22⋊C4×C13712yes—
200C22×C4C22⋊C4×C13711yescharacteristic
201C22×C4C22⋊C4×C13711yescharacteristic
202C22×C4C22×C4×C13711yescharacteristic
203C22×C4C2×D4×C13712yes—
204C2×C8—44no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—41no—
208C22⋊C4—28no—
209C22⋊C4—44no—
210C22⋊C4—42no—
211C42—22no—
212C42—22no—
213C42—21no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—42no—
218C42—42no—
219C42M16×C13712yes—
220C23—22no—
221C23—42no—
222C23—42no—
223C23—41no—
224C23—41no—
225C23C2×C22⋊C4×C13711yescharacteristic
226Q8—42no—
227Q8—42no—
228D4—42no—
229D4—42no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—22no—
235C2×C4—22no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—42no—
247C2×C4—41no—
248C2×C4—82no—
249C2×C4—82no—
250C2×C4#99×C13711yescharacteristic
251C2×C4#99×C13711yescharacteristic
252C2×C4C2×M16×C13711yescharacteristic
253C2×C4C2×M16×C13711yescharacteristic
254C8—84no—
255C22—21no—
256C22—21no—
257C22—44no—
258C22—42no—
259C22—82no—
260C22—81no—
261C22—81no—
262C22#407×C13711yescharacteristic
263C22C2×#99×C13711yescharacteristic
264C22#413×C13711yescharacteristic
265C4—21no—
266C4—21no—
267C4—21no—
268C4—42no—
269C4—42no—
270C4—42no—
271C4—41no—
272C4—42no—
273C4—81no—
274C4#355×C13712yes—
275C2—21no—
276C2—21no—
277C2—82no—
278C2#1465×C13711yescharacteristic
279C2#2126×C13711yescharacteristic
280C2C2×#355×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×Q8×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1467×C137C2semidirect
3. #1324×C137C2semidirect
140. #11530C137direct (N × H)
141. C137#11530direct (N × H)
143. #1467C2×C137semidirect
144. #1324C2×C137semidirect

Maximal subgroups

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

subgroupindex
C4×#124×C1372
#1467×C1372
#1324×C1372
#1318×C1372
#11530137