Finite Group Databasebeta

← C137×#9990 #46859376

Subgroups — C137×#9990 #46859376

Order 35072.

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

normal subgroups only

This group has 826 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#3195×C137C211yesmaximal, characteristic, complemented
2C2×#450×C137C212yesmaximal
3#1783×C137C211yesmaximal, characteristic
4#1472×C137C212yesmaximal, complemented
5#1298×C137C211yesmaximal, characteristic, complemented
6C23×D4×C137C2211yescharacteristic
7C22×SD16×C137—22no—
8C2×C22⋊Q8×C137C2212yes—
9#483×C137—22no—
10#470×C137—22no—
11#451×C137—22no—
12#450×C137—24no—
13#450×C137—24no—
14C2×#102×C137—22no—
15C2×#100×C137—22no—
16#407×C137C2211yescharacteristic
17#407×C137C2211yescharacteristic
18C2×C22⋊C8×C137C2212yes—
19#324×C137—24no—
20C25×C137—21no—
21C22×Q8×C137D412yes—
22C22×D4×C137—22no—
23C22×D4×C137—21no—
24C22×D4×C137—22no—
25C22×D4×C137—44no—
26C22×D4×C137D412yes—
27C23×C4×C137C2311yescharacteristic
28C2×Q16×C137—22no—
29C2×SD16×C137—28no—
30C2×SD16×C137—48no—
31C2×M16×C137—21no—
32C2×M16×C137D412yes—
33C22×C8×C137—22no—
34C22⋊Q8×C137—22no—
35C22⋊Q8×C137—22no—
36C22⋊Q8×C137—22no—
37C22⋊Q8×C137—22no—
38C2×C4⋊C4×C137—22no—
39C2×C4⋊C4×C137D412yes—
40C2×C22⋊C4×C137—22no—
41#106×C137—22no—
42#103×C137—24no—
43#102×C137—28no—
44#100×C137—44no—
45C22⋊C8×C137—22no—
46C22⋊C8×C137—22no—
47C22⋊C8×C137—22no—
48C22⋊C8×C137D414yes—
49C24×C137—22no—
50C24×C137—44no—
51C24×C137—42no—
52C24×C137—84no—
53C24×C137#4111yescharacteristic
54C2×Q8×C137—22no—
55C2×Q8×C137—22no—
56C2×Q8×C137—44no—
57C2×D4×C137—28no—
58C2×D4×C137—24no—
59C2×D4×C137—24no—
60C2×D4×C137—44no—
61C2×D4×C137—42no—
62C2×D4×C137—44no—
63C2×D4×C137—48no—
64C2×D4×C137—41no—
65C2×D4×C137—41no—
66C22×C4×C137—22no—
67C22×C4×C137—22no—
68C22×C4×C137—42no—
69C22×C4×C137C2×D412yes—
70C22×C4×C137C2×D411yescharacteristic
71C22×C4×C137C2×D411yescharacteristic
72C22×C4×C137C2×D412yes—
73Q16×C137—82no—
74SD16×C137—416no—
75M16×C137—42no—
76M16×C137—42no—
77C2×C8×C137—22no—
78C2×C8×C137—24no—
79C2×C8×C137—44no—
80C2×C8×C137—41no—
81C4⋊C4×C137—22no—
82C4⋊C4×C137—22no—
83C4⋊C4×C137—42no—
84C4⋊C4×C137—42no—
85C22⋊C4×C137—42no—
86C22⋊C4×C137—42no—
87C23×C137—24no—
88C23×C137—24no—
89C23×C137—41no—
90C23×C137—48no—
91C23×C137—42no—
92C23×C137—42no—
93C23×C137—84no—
94C23×C137—84no—
95C23×C137—816no—
96C23×C137—84no—
97C23×C137#12111yescharacteristic
98C23×C137#12112yes—
99Q8×C137—44no—
100Q8×C137—44no—
101D4×C137—28no—
102D4×C137—44no—
103D4×C137—44no—
104D4×C137—44no—
105C2×C4×C137—24no—
106C2×C4×C137—22no—
107C2×C4×C137—41no—
108C2×C4×C137—42no—
109C2×C4×C137—42no—
110C2×C4×C137—42no—
111C2×C4×C137—44no—
112C2×C4×C137—82no—
113C2×C4×C137#12012yes—
114C2×C4×C137#12011yescharacteristic
115C2×C4×C137#12711yescharacteristic
116C8×C137—44no—
117C8×C137—42no—
118C8×C137—81no—
119C22×C137—22no—
120C22×C137—22no—
121C22×C137—22no—
122C22×C137—21no—
123C22×C137—44no—
124C22×C137—44no—
125C22×C137—816no—
126C22×C137—84no—
127C22×C137—84no—
128C22×C137—84no—
129C22×C137#39211yescharacteristic
130C22×C137#47114yes—
131C4×C137—22no—
132C4×C137—21no—
133C4×C137—82no—
134C4×C137—82no—
135C4×C137B(4,𝔽2)12yes—
136C2×C137—24no—
137C2×C137—41no—
138C2×C137—84no—
139C2×C137#201912yes—
140C2×C137#202111yescharacteristic
141#9990C13711yesmaximal, characteristic, direct product factor, Sylow
142C137#999011yescharacteristic, direct product factor, Sylow
143#3195C2×C13711yescharacteristic, complemented
144C2×#450C2×C13712yes—
145#1783C2×C13711yescharacteristic
146#1472C2×C13712yescomplemented
147#1298C2×C13711yescharacteristic, complemented
148C23×D4C22×C13711yescharacteristic
149C22×SD16—22no—
150C2×C22⋊Q8C22×C13712yes—
151#483—22no—
152#470—22no—
153#451—22no—
154#450—24no—
155#450—24no—
156C2×#102—22no—
157C2×#100—22no—
158#407C22×C13711yescharacteristic
159#407C22×C13711yescharacteristic
160C2×C22⋊C8C22×C13712yes—
161#324—24no—
162C25—21no—
163C22×Q8D4×C13712yes—
164C22×D4—22no—
165C22×D4—21no—
166C22×D4—22no—
167C22×D4—44no—
168C22×D4D4×C13712yes—
169C23×C4C23×C13711yescharacteristic
170C2×Q16—22no—
171C2×SD16—28no—
172C2×SD16—48no—
173C2×M16—21no—
174C2×M16D4×C13712yes—
175C22×C8—22no—
176C22⋊Q8—22no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4D4×C13712yes—
182C2×C22⋊C4—22no—
183#106—22no—
184#103—24no—
185#102—28no—
186#100—44no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8D4×C13714yes—
191C24—22no—
192C24—44no—
193C24—42no—
194C24—84no—
195C24#41×C13711yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—44no—
199C2×D4—28no—
200C2×D4—24no—
201C2×D4—24no—
202C2×D4—44no—
203C2×D4—42no—
204C2×D4—44no—
205C2×D4—48no—
206C2×D4—41no—
207C2×D4—41no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4C2×D4×C13712yes—
212C22×C4C2×D4×C13711yescharacteristic
213C22×C4C2×D4×C13711yescharacteristic
214C22×C4C2×D4×C13712yes—
215Q16—82no—
216SD16—416no—
217M16—42no—
218M16—42no—
219C2×C8—22no—
220C2×C8—24no—
221C2×C8—44no—
222C2×C8—41no—
223C4⋊C4—22no—
224C4⋊C4—22no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C23—24no—
230C23—24no—
231C23—41no—
232C23—48no—
233C23—42no—
234C23—42no—
235C23—84no—
236C23—84no—
237C23—816no—
238C23—84no—
239C23#121×C13711yescharacteristic
240C23#121×C13712yes—
241Q8—44no—
242Q8—44no—
243D4—28no—
244D4—44no—
245D4—44no—
246D4—44no—
247C2×C4—24no—
248C2×C4—22no—
249C2×C4—41no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—44no—
254C2×C4—82no—
255C2×C4#120×C13712yes—
256C2×C4#120×C13711yescharacteristic
257C2×C4#127×C13711yescharacteristic
258C8—44no—
259C8—42no—
260C8—81no—
261C22—22no—
262C22—22no—
263C22—22no—
264C22—21no—
265C22—44no—
266C22—44no—
267C22—816no—
268C22—84no—
269C22—84no—
270C22—84no—
271C22#392×C13711yescharacteristic
272C22#471×C13714yes—
273C4—22no—
274C4—21no—
275C4—82no—
276C4—82no—
277C4B(4,𝔽2)×C13712yes—
278C2—24no—
279C2—41no—
280C2—84no—
281C2#2019×C13712yes—
282C2#2021×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#392
commutator, Frattini, cosocleC23×C4C23×C137
prime radicalC23×D4×C137C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #3195×C137C2semidirect
4. #1472×C137C2semidirect
5. #1298×C137C2semidirect
141. #9990C137direct (N × H)
142. C137#9990direct (N × H)
143. #3195C2×C137semidirect
146. #1472C2×C137semidirect
147. #1298C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3195×C1372
C2×#450×C1372
#1783×C1372
#1472×C1372
#1298×C1372
#9990137