Finite Group Databasebeta

← C137×#9453 #46858839

Subgroups — C137×#9453 #46858839

Order 35072.

498 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 126 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#2973×C137C211yesmaximal, characteristic
2#2563×C137C211yesmaximal, characteristic
3#1556×C137C212yesmaximal, complemented
4#1466×C137C211yesmaximal, characteristic
5#1285×C137C212yesmaximal, complemented
6C2×C4⋊Q8×C137C2211yescharacteristic
7C2×C4×D4×C137C2211yescharacteristic
8#435×C137—24no—
9#432×C137C2212yes—
10#432×C137C2212yes—
11#407×C137—21no—
12C2×C22⋊C8×C137—21no—
13#393×C137—21no—
14#393×C137—21no—
15#390×C137C412yes—
16#382×C137C412yes—
17#380×C137—21no—
18#380×C137—21no—
19C4×C4⋊C4×C137C2211yescharacteristic
20#332×C137—24no—
21#323×C137—21no—
22#323×C137—21no—
23C22×Q8×C137—21no—
24C22×D4×C137C2×C411yescharacteristic
25C23×C4×C137D411yescharacteristic
26C23×C4×C137D411yescharacteristic
27C2×M16×C137—22no—
28C22×C8×C137—22no—
29C4⋊Q8×C137—22no—
30C4⋊Q8×C137—22no—
31C4×D4×C137—22no—
32C4×D4×C137—22no—
33C2×C4⋊C4×C137—21no—
34C2×C4⋊C4×C137—21no—
35C2×C4⋊C4×C137C2×C412yes—
36C2×C4⋊C4×C137C2×C411yescharacteristic
37C2×C22⋊C4×C137—22no—
38C2×C22⋊C4×C137—22no—
39C2×C22⋊C4×C137D411yescharacteristic
40C2×C22⋊C4×C137D411yescharacteristic
41C2×C42×C137C2×C412yes—
42C2×C42×C137C2311yescharacteristic
43C4⋊C8×C137—22no—
44C4⋊C8×C137D414yes—
45C22⋊C8×C137—22no—
46C22⋊C8×C137—22no—
47C22⋊C8×C137—22no—
48C22⋊C8×C137—22no—
49#97×C137—22no—
50#97×C137—22no—
51#95×C137—22no—
52#95×C137—22no—
53#95×C137—21no—
54C24×C137C22⋊C411yescharacteristic
55C24×C137C22⋊C411yescharacteristic
56C2×Q8×C137—42no—
57C2×Q8×C137—82no—
58C2×D4×C137—22no—
59C2×D4×C137—22no—
60C2×D4×C137—41no—
61C22×C4×C137—21no—
62C22×C4×C137—21no—
63C22×C4×C137—21no—
64C22×C4×C137—21no—
65C22×C4×C137—22no—
66C22×C4×C137—22no—
67C22×C4×C137—42no—
68C22×C4×C137—42no—
69C22×C4×C137C22⋊C411yescharacteristic
70C22×C4×C137C22⋊C411yescharacteristic
71C22×C4×C137C22×C411yescharacteristic
72C22×C4×C137C2×D411yescharacteristic
73C22×C4×C137C2×D411yescharacteristic
74M16×C137—82no—
75C2×C8×C137—24no—
76C2×C8×C137—42no—
77C2×C8×C137—82no—
78C4⋊C4×C137—22no—
79C4⋊C4×C137—24no—
80C4⋊C4×C137—42no—
81C4⋊C4×C137—41no—
82C4⋊C4×C137—41no—
83C22⋊C4×C137—22no—
84C22⋊C4×C137—22no—
85C22⋊C4×C137—82no—
86C22⋊C4×C137—82no—
87C42×C137—42no—
88C42×C137C2×D412yes—
89C42×C137#4112yes—
90C23×C137—21no—
91C23×C137—21no—
92C23×C137—21no—
93C23×C137—21no—
94C23×C137—21no—
95C23×C137—41no—
96C23×C137—41no—
97C23×C137—41no—
98C23×C137—41no—
99C23×C137#10112yes—
100C23×C137C2×C22⋊C411yescharacteristic
101Q8×C137—84no—
102D4×C137—44no—
103C2×C4×C137—21no—
104C2×C4×C137—21no—
105C2×C4×C137—22no—
106C2×C4×C137—21no—
107C2×C4×C137—21no—
108C2×C4×C137—41no—
109C2×C4×C137—42no—
110C2×C4×C137—42no—
111C2×C4×C137—41no—
112C2×C4×C137—41no—
113C2×C4×C137—42no—
114C2×C4×C137—42no—
115C2×C4×C137—42no—
116C2×C4×C137—42no—
117C2×C4×C137—42no—
118C2×C4×C137—42no—
119C2×C4×C137—81no—
120C2×C4×C137—81no—
121C2×C4×C137C4×D412yes—
122C2×C4×C137#12011yescharacteristic
123C2×C4×C137#12112yes—
124C2×C4×C137#12311yescharacteristic
125C8×C137—44no—
126C8×C137—82no—
127C22×C137—21no—
128C22×C137—21no—
129C22×C137—22no—
130C22×C137—42no—
131C22×C137—42no—
132C22×C137—41no—
133C22×C137—41no—
134C22×C137—41no—
135C22×C137—41no—
136C22×C137—41no—
137C22×C137—81no—
138C22×C137—81no—
139C22×C137#38611yescharacteristic
140C22×C137C2×#10111yescharacteristic
141C22×C137#41311yescharacteristic
142C4×C137—21no—
143C4×C137—22no—
144C4×C137—22no—
145C4×C137—41no—
146C4×C137—41no—
147C4×C137—82no—
148C4×C137—82no—
149C4×C137#45512yes—
150C2×C137—21no—
151C2×C137—21no—
152C2×C137—42no—
153C2×C137—81no—
154C2×C137#188511yescharacteristic
155C2×C137#190111yescharacteristic
156C2×C137#191011yescharacteristic
157#9453C13711yesmaximal, characteristic, direct product factor, Sylow
158C137#945311yescharacteristic, direct product factor, Sylow
159#2973C2×C13711yescharacteristic
160#2563C2×C13711yescharacteristic
161#1556C2×C13712yescomplemented
162#1466C2×C13711yescharacteristic
163#1285C2×C13712yescomplemented
164C2×C4⋊Q8C22×C13711yescharacteristic
165C2×C4×D4C22×C13711yescharacteristic
166#435—24no—
167#432C22×C13712yes—
168#432C22×C13712yes—
169#407—21no—
170C2×C22⋊C8—21no—
171#393—21no—
172#393—21no—
173#390C4×C13712yes—
174#382C4×C13712yes—
175#380—21no—
176#380—21no—
177C4×C4⋊C4C22×C13711yescharacteristic
178#332—24no—
179#323—21no—
180#323—21no—
181C22×Q8—21no—
182C22×D4C2×C4×C13711yescharacteristic
183C23×C4D4×C13711yescharacteristic
184C23×C4D4×C13711yescharacteristic
185C2×M16—22no—
186C22×C8—22no—
187C4⋊Q8—22no—
188C4⋊Q8—22no—
189C4×D4—22no—
190C4×D4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×C4×C13712yes—
194C2×C4⋊C4C2×C4×C13711yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4D4×C13711yescharacteristic
198C2×C22⋊C4D4×C13711yescharacteristic
199C2×C42C2×C4×C13712yes—
200C2×C42C23×C13711yescharacteristic
201C4⋊C8—22no—
202C4⋊C8D4×C13714yes—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207#97—22no—
208#97—22no—
209#95—22no—
210#95—22no—
211#95—21no—
212C24C22⋊C4×C13711yescharacteristic
213C24C22⋊C4×C13711yescharacteristic
214C2×Q8—42no—
215C2×Q8—82no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—41no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4C22⋊C4×C13711yescharacteristic
228C22×C4C22⋊C4×C13711yescharacteristic
229C22×C4C22×C4×C13711yescharacteristic
230C22×C4C2×D4×C13711yescharacteristic
231C22×C4C2×D4×C13711yescharacteristic
232M16—82no—
233C2×C8—24no—
234C2×C8—42no—
235C2×C8—82no—
236C4⋊C4—22no—
237C4⋊C4—24no—
238C4⋊C4—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C22⋊C4—22no—
242C22⋊C4—22no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C42—42no—
246C42C2×D4×C13712yes—
247C42#41×C13712yes—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—41no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23#101×C13712yes—
258C23C2×C22⋊C4×C13711yescharacteristic
259Q8—84no—
260D4—44no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4C4×D4×C13712yes—
280C2×C4#120×C13711yescharacteristic
281C2×C4#121×C13712yes—
282C2×C4#123×C13711yescharacteristic
283C8—44no—
284C8—82no—
285C22—21no—
286C22—21no—
287C22—22no—
288C22—42no—
289C22—42no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—81no—
297C22#386×C13711yescharacteristic
298C22C2×#101×C13711yescharacteristic
299C22#413×C13711yescharacteristic
300C4—21no—
301C4—22no—
302C4—22no—
303C4—41no—
304C4—41no—
305C4—82no—
306C4—82no—
307C4#455×C13712yes—
308C2—21no—
309C2—21no—
310C2—42no—
311C2—81no—
312C2#1885×C13711yescharacteristic
313C2#1901×C13711yescharacteristic
314C2#1910×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#386
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C42C23×C137
prime radicalC22×D4×C137C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1556×C137C2semidirect
5. #1285×C137C2semidirect
157. #9453C137direct (N × H)
158. C137#9453direct (N × H)
161. #1556C2×C137semidirect
163. #1285C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2973×C1372
#2563×C1372
#1556×C1372
#1466×C1372
#1285×C1372
#9453137