Finite Group Databasebeta

← C137×#9201 #46858587

Subgroups — C137×#9201 #46858587

Order 35072.

612 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 612 conjugacy classes of subgroups — too many to draw. Only its 134 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#2626×C137C211yesmaximal, characteristic
2#2312×C137C211yesmaximal, characteristic
3#1520×C137C212yesmaximal, complemented
4C2×#323×C137C211yesmaximal, characteristic
5#1289×C137C212yesmaximal, complemented
6C2×C22⋊Q8×C137C2211yescharacteristic
7C22×C22⋊C4×C137C2211yescharacteristic
8C2×C22⋊C8×C137C2212yes—
9#399×C137C412yescomplemented
10#396×C137—22no—
11#393×C137C412yescomplemented
12#388×C137—24no—
13#385×C137—21no—
14#380×C137C2211yescharacteristic
15#380×C137C2212yes—
16#379×C137—21no—
17#379×C137—21no—
18C4×C22⋊C4×C137—21no—
19#328×C137—24no—
20#323×C137—22no—
21C25×C137D411yescharacteristic
22C22×Q8×C137C2×C411yescharacteristic
23C23×C4×C137D411yescharacteristic
24C23×C4×C137C2311yescharacteristic
25C22×C8×C137—22no—
26C22⋊Q8×C137—22no—
27C22⋊Q8×C137—22no—
28C2×C4⋊C4×C137—21no—
29C2×C4⋊C4×C137—22no—
30C2×C4⋊C4×C137—22no—
31C2×C4⋊C4×C137—21no—
32C2×C4⋊C4×C137C2×C411yescharacteristic
33C2×C22⋊C4×C137—22no—
34C2×C22⋊C4×C137—22no—
35C2×C22⋊C4×C137—22no—
36C2×C22⋊C4×C137—21no—
37C2×C22⋊C4×C137—21no—
38C2×C22⋊C4×C137—22no—
39C2×C22⋊C4×C137—22no—
40C2×C22⋊C4×C137—22no—
41C2×C22⋊C4×C137—22no—
42C2×C22⋊C4×C137C2×C412yes—
43C2×C22⋊C4×C137D412yes—
44C2×C42×C137—22no—
45C22⋊C8×C137—22no—
46C22⋊C8×C137—22no—
47#95×C137—22no—
48#95×C137—22no—
49#95×C137C2×C412yes—
50#95×C137D414yes—
51C24×C137—21no—
52C24×C137—21no—
53C24×C137—22no—
54C24×C137—22no—
55C24×C137—81no—
56C24×C137—81no—
57C24×C137C22⋊C412yes—
58C24×C137C2×D411yescharacteristic
59C2×Q8×C137—22no—
60C2×Q8×C137—81no—
61C22×C4×C137—21no—
62C22×C4×C137—21no—
63C22×C4×C137—21no—
64C22×C4×C137—22no—
65C22×C4×C137—21no—
66C22×C4×C137—24no—
67C22×C4×C137—81no—
68C22×C4×C137—81no—
69C22×C4×C137C22⋊C412yes—
70C22×C4×C137C22×C411yescharacteristic
71C22×C4×C137C2×D412yes—
72C22×C4×C137C2×D411yescharacteristic
73C22×C4×C137#4112yes—
74C2×C8×C137—42no—
75C2×C8×C137—82no—
76C4⋊C4×C137—22no—
77C4⋊C4×C137—42no—
78C4⋊C4×C137—82no—
79C4⋊C4×C137—81no—
80C4⋊C4×C137—82no—
81C22⋊C4×C137—24no—
82C22⋊C4×C137—24no—
83C22⋊C4×C137—41no—
84C22⋊C4×C137—41no—
85C22⋊C4×C137—42no—
86C22⋊C4×C137—42no—
87C22⋊C4×C137—42no—
88C22⋊C4×C137—82no—
89C22⋊C4×C137—82no—
90C22⋊C4×C137—82no—
91C22⋊C4×C137—82no—
92C42×C137—82no—
93C23×C137—22no—
94C23×C137—22no—
95C23×C137—22no—
96C23×C137—41no—
97C23×C137—41no—
98C23×C137—42no—
99C23×C137—42no—
100C23×C137—42no—
101C23×C137—42no—
102C23×C137—42no—
103C23×C137—42no—
104C23×C137—81no—
105C23×C137—81no—
106C23×C137—84no—
107C23×C137—84no—
108C23×C137C2×C22⋊C411yescharacteristic
109C23×C137#12011yescharacteristic
110C23×C137#12311yescharacteristic
111C23×C137#13512yes—
112C23×C137#13712yes—
113Q8×C137—82no—
114C2×C4×C137—22no—
115C2×C4×C137—41no—
116C2×C4×C137—42no—
117C2×C4×C137—42no—
118C2×C4×C137—41no—
119C2×C4×C137—42no—
120C2×C4×C137—42no—
121C2×C4×C137—44no—
122C2×C4×C137—42no—
123C2×C4×C137—42no—
124C2×C4×C137—44no—
125C2×C4×C137—42no—
126C2×C4×C137—42no—
127C2×C4×C137—41no—
128C2×C4×C137—41no—
129C2×C4×C137—42no—
130C2×C4×C137—44no—
131C2×C4×C137—81no—
132C2×C4×C137C4×D412yes—
133C2×C4×C137#12112yes—
134C8×C137—82no—
135C22×C137—21no—
136C22×C137—21no—
137C22×C137—21no—
138C22×C137—21no—
139C22×C137—21no—
140C22×C137—21no—
141C22×C137—42no—
142C22×C137—42no—
143C22×C137—44no—
144C22×C137—42no—
145C22×C137—42no—
146C22×C137—42no—
147C22×C137—82no—
148C22×C137—84no—
149C22×C137—84no—
150C22×C137#38611yescharacteristic
151C22×C137#41611yescharacteristic
152C22×C137#41911yescharacteristic
153C22×C137#45212yes—
154C22×C137#48512yes—
155C4×C137—22no—
156C4×C137—22no—
157C4×C137—42no—
158C4×C137—82no—
159C4×C137—84no—
160C4×C137—81no—
161C4×C137—81no—
162C2×C137—21no—
163C2×C137—22no—
164C2×C137—21no—
165C2×C137—22no—
166C2×C137—82no—
167C2×C137#187311yescharacteristic
168C2×C137#191111yescharacteristic
169C2×C137#191311yescharacteristic
170#9201C13711yesmaximal, characteristic, direct product factor, Sylow
171C137#920111yescharacteristic, direct product factor, Sylow
172#2626C2×C13711yescharacteristic
173#2312C2×C13711yescharacteristic
174#1520C2×C13712yescomplemented
175C2×#323C2×C13711yescharacteristic
176#1289C2×C13712yescomplemented
177C2×C22⋊Q8C22×C13711yescharacteristic
178C22×C22⋊C4C22×C13711yescharacteristic
179C2×C22⋊C8C22×C13712yes—
180#399C4×C13712yescomplemented
181#396—22no—
182#393C4×C13712yescomplemented
183#388—24no—
184#385—21no—
185#380C22×C13711yescharacteristic
186#380C22×C13712yes—
187#379—21no—
188#379—21no—
189C4×C22⋊C4—21no—
190#328—24no—
191#323—22no—
192C25D4×C13711yescharacteristic
193C22×Q8C2×C4×C13711yescharacteristic
194C23×C4D4×C13711yescharacteristic
195C23×C4C23×C13711yescharacteristic
196C22×C8—22no—
197C22⋊Q8—22no—
198C22⋊Q8—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4C2×C4×C13711yescharacteristic
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4C2×C4×C13712yes—
214C2×C22⋊C4D4×C13712yes—
215C2×C42—22no—
216C22⋊C8—22no—
217C22⋊C8—22no—
218#95—22no—
219#95—22no—
220#95C2×C4×C13712yes—
221#95D4×C13714yes—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—81no—
227C24—81no—
228C24C22⋊C4×C13712yes—
229C24C2×D4×C13711yescharacteristic
230C2×Q8—22no—
231C2×Q8—81no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—24no—
238C22×C4—81no—
239C22×C4—81no—
240C22×C4C22⋊C4×C13712yes—
241C22×C4C22×C4×C13711yescharacteristic
242C22×C4C2×D4×C13712yes—
243C22×C4C2×D4×C13711yescharacteristic
244C22×C4#41×C13712yes—
245C2×C8—42no—
246C2×C8—82no—
247C4⋊C4—22no—
248C4⋊C4—42no—
249C4⋊C4—82no—
250C4⋊C4—81no—
251C4⋊C4—82no—
252C22⋊C4—24no—
253C22⋊C4—24no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—82no—
260C22⋊C4—82no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—82no—
264C23—22no—
265C23—22no—
266C23—22no—
267C23—41no—
268C23—41no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—81no—
276C23—81no—
277C23—84no—
278C23—84no—
279C23C2×C22⋊C4×C13711yescharacteristic
280C23#120×C13711yescharacteristic
281C23#123×C13711yescharacteristic
282C23#135×C13712yes—
283C23#137×C13712yes—
284Q8—82no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—44no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—44no—
302C2×C4—81no—
303C2×C4C4×D4×C13712yes—
304C2×C4#121×C13712yes—
305C8—82no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—44no—
315C22—42no—
316C22—42no—
317C22—42no—
318C22—82no—
319C22—84no—
320C22—84no—
321C22#386×C13711yescharacteristic
322C22#416×C13711yescharacteristic
323C22#419×C13711yescharacteristic
324C22#452×C13712yes—
325C22#485×C13712yes—
326C4—22no—
327C4—22no—
328C4—42no—
329C4—82no—
330C4—84no—
331C4—81no—
332C4—81no—
333C2—21no—
334C2—22no—
335C2—21no—
336C2—22no—
337C2—82no—
338C2#1873×C13711yescharacteristic
339C2#1911×C13711yescharacteristic
340C2#1913×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#386
commutatorC22×C4C22×C4×C137
Frattini, cosocleC23×C4C23×C137
prime radicalC25×C137D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1520×C137C2semidirect
5. #1289×C137C2semidirect
9. #399×C137C4semidirect
11. #393×C137C4semidirect
170. #9201C137direct (N × H)
171. C137#9201direct (N × H)
174. #1520C2×C137semidirect
176. #1289C2×C137semidirect
180. #399C4×C137semidirect
182. #393C4×C137semidirect

Maximal subgroups

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

subgroupindex
#2626×C1372
#2312×C1372
#1520×C1372
C2×#323×C1372
#1289×C1372
#9201137