Finite Group Databasebeta

← C137×#11825 #46861211

Subgroups — C137×#11825 #46861211

Order 35072.

476 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 98 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#2631×C137C211yesmaximal, characteristic
2#1799×C137C211yesmaximal, characteristic, complemented
3#1465×C137C211yesmaximal, characteristic, complemented
4#1351×C137C212yesmaximal, complemented
5#1343×C137C212yesmaximal, complemented
6C2×#124×C137C411yescharacteristic, complemented
7C2×C4×D4×C137C2211yescharacteristic
8C2×#99×C137—21no—
9#407×C137—21no—
10#397×C137—21no—
11#393×C137C411yescharacteristic, complemented
12#393×C137C2211yescharacteristic
13#392×C137—21no—
14#392×C137C411yescharacteristic, complemented
15#392×C137C2211yescharacteristic
16#386×C137—21no—
17#386×C137—21no—
18#383×C137C411yescharacteristic, complemented
19#380×C137—21no—
20#342×C137C2212yes—
21#324×C137—21no—
22#323×C137C2212yes—
23C22×Q8×C137C2×C411yescharacteristic
24C22×D4×C137—21no—
25C22×D4×C137—21no—
26C22×D4×C137C2×C411yescharacteristic
27C22×D4×C137D411yescharacteristic
28C23×C4×C137D411yescharacteristic
29C23×C4×C137D411yescharacteristic
30C2×M16×C137—41no—
31#124×C137—41no—
32#124×C137—41no—
33C4×D4×C137—41no—
34C4×D4×C137—41no—
35C2×C4⋊C4×C137—41no—
36C2×C4⋊C4×C137D411yescharacteristic
37C2×C22⋊C4×C137—21no—
38C2×C22⋊C4×C137—21no—
39C2×C22⋊C4×C137—22no—
40C2×C22⋊C4×C137—21no—
41C2×C22⋊C4×C137—21no—
42C2×C22⋊C4×C137—21no—
43C2×C22⋊C4×C137—21no—
44C2×C22⋊C4×C137C2×C411yescharacteristic
45C2×C22⋊C4×C137C2311yescharacteristic
46C2×C42×C137C2×C411yescharacteristic
47#99×C137—42no—
48C22⋊C8×C137—22no—
49C22⋊C8×C137—41no—
50#95×C137—21no—
51#95×C137—21no—
52#95×C137—21no—
53#95×C137—21no—
54#95×C137—41no—
55#95×C137C2×C411yescharacteristic
56#95×C137C2×C411yescharacteristic
57C24×C137—21no—
58C24×C137C22⋊C411yescharacteristic
59C24×C137C2×D411yescharacteristic
60C2×Q8×C137—41no—
61C2×Q8×C137—81no—
62C2×D4×C137—21no—
63C2×D4×C137—21no—
64C2×D4×C137—41no—
65C2×D4×C137—41no—
66C2×D4×C137—41no—
67C2×D4×C137—42no—
68C2×D4×C137—41no—
69C2×D4×C137—41no—
70C2×D4×C137—81no—
71C2×D4×C137—81no—
72C2×D4×C137—81no—
73C2×D4×C137—81no—
74C2×D4×C137—81no—
75C22×C4×C137—21no—
76C22×C4×C137—21no—
77C22×C4×C137—21no—
78C22×C4×C137—21no—
79C22×C4×C137—21no—
80C22×C4×C137—21no—
81C22×C4×C137—21no—
82C22×C4×C137—42no—
83C22×C4×C137—81no—
84C22×C4×C137—81no—
85C22×C4×C137C22⋊C411yescharacteristic
86C22×C4×C137C22⋊C411yescharacteristic
87C22×C4×C137C22⋊C411yescharacteristic
88C22×C4×C137C22×C411yescharacteristic
89C22×C4×C137C2×D411yescharacteristic
90M16×C137—82no—
91C2×C8×C137—42no—
92C4⋊C4×C137—41no—
93C4⋊C4×C137—82no—
94C22⋊C4×C137—41no—
95C22⋊C4×C137—41no—
96C22⋊C4×C137—42no—
97C22⋊C4×C137—41no—
98C22⋊C4×C137—41no—
99C22⋊C4×C137—42no—
100C22⋊C4×C137—42no—
101C22⋊C4×C137—81no—
102C22⋊C4×C137—81no—
103C22⋊C4×C137—81no—
104C22⋊C4×C137—81no—
105C42×C137—21no—
106C42×C137—21no—
107C23×C137—21no—
108C23×C137—21no—
109C23×C137—21no—
110C23×C137—21no—
111C23×C137—21no—
112C23×C137—41no—
113C23×C137—41no—
114C23×C137—41no—
115C23×C137—41no—
116C23×C137—41no—
117C23×C137—41no—
118C23×C137—41no—
119C23×C137—41no—
120C23×C137—41no—
121C23×C137—41no—
122C23×C137—41no—
123C23×C137C2×C22⋊C411yescharacteristic
124C23×C137AGL(1,ℤ/8)11yescharacteristic
125C23×C137#13711yescharacteristic
126Q8×C137—82no—
127D4×C137—42no—
128D4×C137—42no—
129D4×C137—82no—
130D4×C137—82no—
131D4×C137—82no—
132D4×C137—84no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—41no—
140C2×C4×C137—42no—
141C2×C4×C137—42no—
142C2×C4×C137—41no—
143C2×C4×C137—41no—
144C2×C4×C137—41no—
145C2×C4×C137—41no—
146C2×C4×C137—41no—
147C2×C4×C137—41no—
148C2×C4×C137—41no—
149C2×C4×C137—41no—
150C2×C4×C137—41no—
151C2×C4×C137—41no—
152C2×C4×C137—41no—
153C2×C4×C137—41no—
154C2×C4×C137—81no—
155C2×C4×C137—81no—
156C2×C4×C137—82no—
157C2×C4×C137—82no—
158C2×C4×C137—82no—
159C2×C4×C137#9911yescharacteristic
160C2×C4×C137#9911yescharacteristic
161C8×C137—82no—
162C22×C137—21no—
163C22×C137—21no—
164C22×C137—21no—
165C22×C137—21no—
166C22×C137—41no—
167C22×C137—41no—
168C22×C137—42no—
169C22×C137—41no—
170C22×C137—41no—
171C22×C137—41no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—81no—
175C22×C137—81no—
176C22×C137—81no—
177C22×C137—81no—
178C22×C137—81no—
179C22×C137—81no—
180C22×C137—81no—
181C22×C137—81no—
182C22×C137—81no—
183C22×C137C2×#9911yescharacteristic
184C22×C137#41711yescharacteristic
185C22×C137#42111yescharacteristic
186C4×C137—21no—
187C4×C137—21no—
188C4×C137—41no—
189C4×C137—41no—
190C4×C137—41no—
191C4×C137—41no—
192C4×C137—81no—
193C4×C137—81no—
194C4×C137—81no—
195C4×C137—162no—
196C2×C137—21no—
197C2×C137—21no—
198C2×C137—41no—
199C2×C137—41no—
200C2×C137—81no—
201C2×C137—161no—
202C2×C137#151311yescharacteristic
203C2×C137#212211yescharacteristic
204C2×C137#212711yescharacteristic
205#11825C13711yesmaximal, characteristic, direct product factor, Sylow
206C137#1182511yescharacteristic, direct product factor, Sylow
207#2631C2×C13711yescharacteristic
208#1799C2×C13711yescharacteristic, complemented
209#1465C2×C13711yescharacteristic, complemented
210#1351C2×C13712yescomplemented
211#1343C2×C13712yescomplemented
212C2×#124C4×C13711yescharacteristic, complemented
213C2×C4×D4C22×C13711yescharacteristic
214C2×#99—21no—
215#407—21no—
216#397—21no—
217#393C4×C13711yescharacteristic, complemented
218#393C22×C13711yescharacteristic
219#392—21no—
220#392C4×C13711yescharacteristic, complemented
221#392C22×C13711yescharacteristic
222#386—21no—
223#386—21no—
224#383C4×C13711yescharacteristic, complemented
225#380—21no—
226#342C22×C13712yes—
227#324—21no—
228#323C22×C13712yes—
229C22×Q8C2×C4×C13711yescharacteristic
230C22×D4—21no—
231C22×D4—21no—
232C22×D4C2×C4×C13711yescharacteristic
233C22×D4D4×C13711yescharacteristic
234C23×C4D4×C13711yescharacteristic
235C23×C4D4×C13711yescharacteristic
236C2×M16—41no—
237#124—41no—
238#124—41no—
239C4×D4—41no—
240C4×D4—41no—
241C2×C4⋊C4—41no—
242C2×C4⋊C4D4×C13711yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4C2×C4×C13711yescharacteristic
251C2×C22⋊C4C23×C13711yescharacteristic
252C2×C42C2×C4×C13711yescharacteristic
253#99—42no—
254C22⋊C8—22no—
255C22⋊C8—41no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95C2×C4×C13711yescharacteristic
262#95C2×C4×C13711yescharacteristic
263C24—21no—
264C24C22⋊C4×C13711yescharacteristic
265C24C2×D4×C13711yescharacteristic
266C2×Q8—41no—
267C2×Q8—81no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—81no—
277C2×D4—81no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—81no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—42no—
289C22×C4—81no—
290C22×C4—81no—
291C22×C4C22⋊C4×C13711yescharacteristic
292C22×C4C22⋊C4×C13711yescharacteristic
293C22×C4C22⋊C4×C13711yescharacteristic
294C22×C4C22×C4×C13711yescharacteristic
295C22×C4C2×D4×C13711yescharacteristic
296M16—82no—
297C2×C8—42no—
298C4⋊C4—41no—
299C4⋊C4—82no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—42no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—42no—
306C22⋊C4—42no—
307C22⋊C4—81no—
308C22⋊C4—81no—
309C22⋊C4—81no—
310C22⋊C4—81no—
311C42—21no—
312C42—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23C2×C22⋊C4×C13711yescharacteristic
330C23AGL(1,ℤ/8)×C13711yescharacteristic
331C23#137×C13711yescharacteristic
332Q8—82no—
333D4—42no—
334D4—42no—
335D4—82no—
336D4—82no—
337D4—82no—
338D4—84no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4#99×C13711yescharacteristic
366C2×C4#99×C13711yescharacteristic
367C8—82no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—41no—
373C22—41no—
374C22—42no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22—81no—
388C22—81no—
389C22C2×#99×C13711yescharacteristic
390C22#417×C13711yescharacteristic
391C22#421×C13711yescharacteristic
392C4—21no—
393C4—21no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—81no—
399C4—81no—
400C4—81no—
401C4—162no—
402C2—21no—
403C2—21no—
404C2—41no—
405C2—41no—
406C2—81no—
407C2—161no—
408C2#1513×C13711yescharacteristic
409C2#2122×C13711yescharacteristic
410C2#2127×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C22⋊C4C23×C137
prime radical#2631×C137C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×C4×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1799×C137C2semidirect
3. #1465×C137C2semidirect
4. #1351×C137C2semidirect
5. #1343×C137C2semidirect
6. C2×#124×C137C4semidirect
11. #393×C137C4semidirect
14. #392×C137C4semidirect
18. #383×C137C4semidirect
205. #11825C137direct (N × H)
206. C137#11825direct (N × H)
208. #1799C2×C137semidirect
209. #1465C2×C137semidirect
210. #1351C2×C137semidirect
211. #1343C2×C137semidirect
212. C2×#124C4×C137semidirect
217. #393C4×C137semidirect
220. #392C4×C137semidirect
224. #383C4×C137semidirect

Maximal subgroups

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

subgroupindex
#2631×C1372
#1799×C1372
#1465×C1372
#1351×C1372
#1343×C1372
#11825137