Finite Group Databasebeta

← C137×#18551 #46867937

Subgroups — C137×#18551 #46867937

Order 35072.

918 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 918 conjugacy classes of subgroups — too many to draw. Only its 246 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
1C2×Q82×C137C211yesmaximal, characteristic
2#3139×C137C211yesmaximal, characteristic
3#3113×C137C212yesmaximal
4#1645×C137C218yesmaximal, complemented
5#1558×C137C211yesmaximal, characteristic
6#1491×C137C212yesmaximal
7Q82×C137—22no—
8Q82×C137—24no—
9Q82×C137C2214yes—
10#533×C137C2211yescharacteristic
11C2×C4⋊Q8×C137—22no—
12C2×C4⋊Q8×C137—22no—
13C2×C4⋊Q8×C137C2211yescharacteristic
14C2×C4×Q8×C137—22no—
15C2×C4×Q8×C137C2212yes—
16#491×C137—22no—
17#487×C137C2214yes—
18#483×C137—22no—
19#475×C137C2214yes—
20#462×C137C2214yes—
21#451×C137—22no—
22#441×C137—24no—
23#432×C137C2212yes—
24#419×C137—22no—
25C4×M16×C137C2211yescharacteristic
26#333×C137C2214yes—
27Q8⋊C8×C137C2218yes—
28C22×Q8×C137—22no—
29C22×Q8×C137D412yes—
30C22×Q8×C137D412yes—
31C2×Q16×C137—24no—
32C2×M16×C137—21no—
33C4⋊Q8×C137—28no—
34C4⋊Q8×C137—28no—
35C4⋊Q8×C137—28no—
36C4⋊Q8×C137—28no—
37C4⋊Q8×C137—22no—
38C4⋊Q8×C137D412yes—
39C4⋊Q8×C137C2312yes—
40C4⋊Q8×C137C2312yes—
41#125×C137—21no—
42C22⋊Q8×C137—22no—
43C4×Q8×C137—28no—
44C4×Q8×C137—28no—
45C4×Q8×C137—22no—
46C4×Q8×C137—22no—
47C4×Q8×C137—22no—
48C4×Q8×C137D414yes—
49C4×Q8×C137C2314yes—
50#117×C137—21no—
51C2×C4⋊C4×C137—24no—
52C2×C4⋊C4×C137—22no—
53C2×C4⋊C4×C137—22no—
54C2×C4⋊C4×C137D412yes—
55C2×C42×C137—22no—
56C2×C42×C137—22no—
57C2×C42×C137C2311yescharacteristic
58#106×C137—24no—
59C4⋊C8×C137C2314yes—
60#103×C137—28no—
61#103×C137—24no—
62#103×C137—24no—
63C22⋊C8×C137—22no—
64#97×C137—21no—
65#97×C137—22no—
66C4×C8×C137C2312yes—
67C2×Q8×C137—28no—
68C2×Q8×C137—28no—
69C2×Q8×C137—22no—
70C2×Q8×C137—22no—
71C2×Q8×C137—22no—
72C2×Q8×C137—44no—
73C2×Q8×C137—84no—
74C2×Q8×C137C2×D414yes—
75C2×Q8×C137C2×D414yes—
76C22×C4×C137—22no—
77C22×C4×C137—22no—
78C22×C4×C137—22no—
79C22×C4×C137C2×D412yes—
80C22×C4×C137C2×D411yescharacteristic
81Q16×C137—48no—
82M16×C137—42no—
83C2×C8×C137—22no—
84C2×C8×C137—24no—
85C4⋊C4×C137—216no—
86C4⋊C4×C137—24no—
87C4⋊C4×C137—24no—
88C4⋊C4×C137—22no—
89C4⋊C4×C137—24no—
90C4⋊C4×C137—24no—
91C4⋊C4×C137—22no—
92C4⋊C4×C137—22no—
93C4⋊C4×C137—42no—
94C4⋊C4×C137C2×D414yes—
95C22⋊C4×C137—41no—
96C42×C137—28no—
97C42×C137—24no—
98C42×C137—24no—
99C42×C137—41no—
100C42×C137C2×D412yes—
101C42×C137C2×D411yescharacteristic
102C42×C137C2411yescharacteristic
103C23×C137#12011yescharacteristic
104Q8×C137—28no—
105Q8×C137—24no—
106Q8×C137—44no—
107Q8×C137—44no—
108Q8×C137—82no—
109Q8×C137—88no—
110Q8×C137#13718yes—
111C2×C4×C137—28no—
112C2×C4×C137—24no—
113C2×C4×C137—22no—
114C2×C4×C137—24no—
115C2×C4×C137—24no—
116C2×C4×C137—24no—
117C2×C4×C137—44no—
118C2×C4×C137—44no—
119C2×C4×C137—42no—
120C2×C4×C137#12012yes—
121C2×C4×C137#12011yescharacteristic
122C2×C4×C137C22×D411yescharacteristic
123C2×C4×C137C22×D412yes—
124C8×C137—42no—
125C8×C137—44no—
126C22×C137—22no—
127C22×C137#45512yes—
128C22×C137C2×#12011yescharacteristic
129C4×C137—28no—
130C4×C137—22no—
131C4×C137—21no—
132C4×C137—21no—
133C4×C137—44no—
134C4×C137—44no—
135C4×C137—82no—
136C4×C137C2×#13714yes—
137C2×C137—22no—
138C2×C137#300412yes—
139C2×C137C2×#45511yescharacteristic
140#18551C13711yesmaximal, characteristic, direct product factor, Sylow
141C137#1855111yescharacteristic, direct product factor, Sylow
142C2×Q82C2×C13711yescharacteristic
143#3139C2×C13711yescharacteristic
144#3113C2×C13712yes—
145#1645C2×C13718yescomplemented
146#1558C2×C13711yescharacteristic
147#1491C2×C13712yes—
148Q82—22no—
149Q82—24no—
150Q82C22×C13714yes—
151#533C22×C13711yescharacteristic
152C2×C4⋊Q8—22no—
153C2×C4⋊Q8—22no—
154C2×C4⋊Q8C22×C13711yescharacteristic
155C2×C4×Q8—22no—
156C2×C4×Q8C22×C13712yes—
157#491—22no—
158#487C22×C13714yes—
159#483—22no—
160#475C22×C13714yes—
161#462C22×C13714yes—
162#451—22no—
163#441—24no—
164#432C22×C13712yes—
165#419—22no—
166C4×M16C22×C13711yescharacteristic
167#333C22×C13714yes—
168Q8⋊C8C22×C13718yes—
169C22×Q8—22no—
170C22×Q8D4×C13712yes—
171C22×Q8D4×C13712yes—
172C2×Q16—24no—
173C2×M16—21no—
174C4⋊Q8—28no—
175C4⋊Q8—28no—
176C4⋊Q8—28no—
177C4⋊Q8—28no—
178C4⋊Q8—22no—
179C4⋊Q8D4×C13712yes—
180C4⋊Q8C23×C13712yes—
181C4⋊Q8C23×C13712yes—
182#125—21no—
183C22⋊Q8—22no—
184C4×Q8—28no—
185C4×Q8—28no—
186C4×Q8—22no—
187C4×Q8—22no—
188C4×Q8—22no—
189C4×Q8D4×C13714yes—
190C4×Q8C23×C13714yes—
191#117—21no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4D4×C13712yes—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42C23×C13711yescharacteristic
199#106—24no—
200C4⋊C8C23×C13714yes—
201#103—28no—
202#103—24no—
203#103—24no—
204C22⋊C8—22no—
205#97—21no—
206#97—22no—
207C4×C8C23×C13712yes—
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—44no—
214C2×Q8—84no—
215C2×Q8C2×D4×C13714yes—
216C2×Q8C2×D4×C13714yes—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4C2×D4×C13712yes—
221C22×C4C2×D4×C13711yescharacteristic
222Q16—48no—
223M16—42no—
224C2×C8—22no—
225C2×C8—24no—
226C4⋊C4—216no—
227C4⋊C4—24no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—24no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—22no—
234C4⋊C4—42no—
235C4⋊C4C2×D4×C13714yes—
236C22⋊C4—41no—
237C42—28no—
238C42—24no—
239C42—24no—
240C42—41no—
241C42C2×D4×C13712yes—
242C42C2×D4×C13711yescharacteristic
243C42C24×C13711yescharacteristic
244C23#120×C13711yescharacteristic
245Q8—28no—
246Q8—24no—
247Q8—44no—
248Q8—44no—
249Q8—82no—
250Q8—88no—
251Q8#137×C13718yes—
252C2×C4—28no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—24no—
256C2×C4—24no—
257C2×C4—24no—
258C2×C4—44no—
259C2×C4—44no—
260C2×C4—42no—
261C2×C4#120×C13712yes—
262C2×C4#120×C13711yescharacteristic
263C2×C4C22×D4×C13711yescharacteristic
264C2×C4C22×D4×C13712yes—
265C8—42no—
266C8—44no—
267C22—22no—
268C22#455×C13712yes—
269C22C2×#120×C13711yescharacteristic
270C4—28no—
271C4—22no—
272C4—21no—
273C4—21no—
274C4—44no—
275C4—44no—
276C4—82no—
277C4C2×#137×C13714yes—
278C2—22no—
279C2#3004×C13712yes—
280C2C2×#455×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC42C24×C137
prime radicalC23×C137#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1645×C137C2semidirect
140. #18551C137direct (N × H)
141. C137#18551direct (N × H)
145. #1645C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×Q82×C1372
#3139×C1372
#3113×C1372
#1645×C1372
#1558×C1372
#1491×C1372
#18551137