Finite Group Databasebeta

← C137×#18319 #46867705

Subgroups — C137×#18319 #46867705

Order 35072.

1058 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#543×C137C211yesmaximal, characteristic
2#3195×C137C212yesmaximal, complemented
3#2561×C137C211yesmaximal, characteristic
4#1608×C137C214yesmaximal, complemented
5#1602×C137C214yesmaximal, complemented
6#1508×C137C212yesmaximal, complemented
7#1462×C137C211yesmaximal, characteristic
8#543×C137—24no—
9#543×C137—22no—
10#543×C137C2214yes—
11C2×C22⋊Q8×C137—22no—
12C2×C22⋊Q8×C137—21no—
13C2×C22⋊Q8×C137—22no—
14C2×C22⋊Q8×C137C2212yes—
15C2×C22⋊Q8×C137C2211yescharacteristic
16C22×C22⋊C4×C137—21no—
17C22×C22⋊C4×C137C2211yescharacteristic
18#483×C137—22no—
19#483×C137C2214yescomplemented
20#470×C137—24no—
21#451×C137—22no—
22#451×C137C2214yescomplemented
23#407×C137C2212yes—
24#397×C137C2212yes—
25#393×C137C2212yes—
26#388×C137—22no—
27#381×C137—21no—
28#379×C137—21no—
29C4×C22⋊C4×C137C2211yescharacteristic
30#328×C137C2218yescomplemented
31#323×C137C2214yes—
32C25×C137D411yescharacteristic
33C22×Q8×C137—21no—
34C22×Q8×C137D412yes—
35C22×Q8×C137C2311yescharacteristic
36C23×C4×C137—21no—
37C23×C4×C137—21no—
38C23×C4×C137D411yescharacteristic
39C23×C4×C137C2311yescharacteristic
40C2×Q16×C137—24no—
41C2×M16×C137—22no—
42C22⋊Q8×C137—28no—
43C22⋊Q8×C137—28no—
44C22⋊Q8×C137—24no—
45C22⋊Q8×C137—28no—
46C22⋊Q8×C137—24no—
47C22⋊Q8×C137—22no—
48C22⋊Q8×C137—22no—
49C22⋊Q8×C137—28no—
50C22⋊Q8×C137—22no—
51C22⋊Q8×C137—22no—
52C22⋊Q8×C137D414yes—
53C22⋊Q8×C137C2314yes—
54C2×C4⋊C4×C137—21no—
55C2×C4⋊C4×C137—22no—
56C2×C4⋊C4×C137—22no—
57C2×C4⋊C4×C137—21no—
58C2×C4⋊C4×C137D412yes—
59C2×C4⋊C4×C137C2311yescharacteristic
60C2×C22⋊C4×C137—22no—
61C2×C22⋊C4×C137—24no—
62C2×C22⋊C4×C137—22no—
63C2×C22⋊C4×C137—24no—
64C2×C22⋊C4×C137—22no—
65C2×C22⋊C4×C137—22no—
66C2×C22⋊C4×C137—21no—
67C2×C22⋊C4×C137—21no—
68C2×C22⋊C4×C137—22no—
69C2×C22⋊C4×C137—22no—
70C2×C22⋊C4×C137D412yes—
71C2×C22⋊C4×C137C2312yes—
72C2×C42×C137—21no—
73#106×C137—24no—
74#103×C137—24no—
75#103×C137—24no—
76C22⋊C8×C137—22no—
77C22⋊C8×C137C2314yes—
78#95×C137—22no—
79#95×C137—22no—
80#95×C137C2312yes—
81C24×C137—21no—
82C24×C137—21no—
83C24×C137—24no—
84C24×C137—42no—
85C24×C137—81no—
86C24×C137C2×D411yescharacteristic
87C24×C137C2×D412yes—
88C2×Q8×C137—24no—
89C2×Q8×C137—22no—
90C2×Q8×C137—22no—
91C2×Q8×C137—81no—
92C2×Q8×C137—82no—
93C2×Q8×C137—82no—
94C2×Q8×C137C2×D414yes—
95C22×C4×C137—22no—
96C22×C4×C137—21no—
97C22×C4×C137—22no—
98C22×C4×C137—24no—
99C22×C4×C137—24no—
100C22×C4×C137—21no—
101C22×C4×C137—22no—
102C22×C4×C137—21no—
103C22×C4×C137—21no—
104C22×C4×C137—42no—
105C22×C4×C137—81no—
106C22×C4×C137—82no—
107C22×C4×C137—82no—
108C22×C4×C137C2×D412yes—
109C22×C4×C137C2×D411yescharacteristic
110C22×C4×C137C2×D412yes—
111C22×C4×C137C2×D412yes—
112C22×C4×C137C2411yescharacteristic
113Q16×C137—84no—
114M16×C137—82no—
115C2×C8×C137—24no—
116C4⋊C4×C137—24no—
117C4⋊C4×C137—28no—
118C4⋊C4×C137—24no—
119C4⋊C4×C137—22no—
120C4⋊C4×C137—24no—
121C4⋊C4×C137—22no—
122C4⋊C4×C137—42no—
123C4⋊C4×C137C2×D414yes—
124C22⋊C4×C137—28no—
125C22⋊C4×C137—24no—
126C22⋊C4×C137—24no—
127C22⋊C4×C137—24no—
128C22⋊C4×C137—24no—
129C22⋊C4×C137—42no—
130C22⋊C4×C137—41no—
131C22⋊C4×C137—41no—
132C22⋊C4×C137—44no—
133C22⋊C4×C137—82no—
134C42×C137—81no—
135C23×C137—24no—
136C23×C137—24no—
137C23×C137—22no—
138C23×C137—22no—
139C23×C137—22no—
140C23×C137—22no—
141C23×C137—41no—
142C23×C137—42no—
143C23×C137—42no—
144C23×C137—42no—
145C23×C137—42no—
146C23×C137—81no—
147C23×C137—81no—
148C23×C137—88no—
149C23×C137#12011yescharacteristic
150C23×C137#12011yescharacteristic
151C23×C137#13714yes—
152C23×C137C22×D411yescharacteristic
153Q8×C137—44no—
154Q8×C137—44no—
155Q8×C137—82no—
156Q8×C137—84no—
157C2×C4×C137—24no—
158C2×C4×C137—22no—
159C2×C4×C137—24no—
160C2×C4×C137—41no—
161C2×C4×C137—42no—
162C2×C4×C137—42no—
163C2×C4×C137—41no—
164C2×C4×C137—44no—
165C2×C4×C137—42no—
166C2×C4×C137—44no—
167C2×C4×C137—42no—
168C2×C4×C137—42no—
169C2×C4×C137—42no—
170C2×C4×C137—42no—
171C2×C4×C137—41no—
172C2×C4×C137—41no—
173C2×C4×C137—42no—
174C2×C4×C137—42no—
175C2×C4×C137—81no—
176C2×C4×C137—84no—
177C2×C4×C137—84no—
178C2×C4×C137#12012yes—
179C2×C4×C137C22×D412yes—
180C8×C137—44no—
181C22×C137—21no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—21no—
185C22×C137—21no—
186C22×C137—24no—
187C22×C137—21no—
188C22×C137—21no—
189C22×C137—44no—
190C22×C137—44no—
191C22×C137—48no—
192C22×C137—42no—
193C22×C137—42no—
194C22×C137—82no—
195C22×C137—84no—
196C22×C137#45512yes—
197C22×C137C2×#12011yescharacteristic
198C22×C137C2×#13712yes—
199C4×C137—22no—
200C4×C137—22no—
201C4×C137—42no—
202C4×C137—44no—
203C4×C137—44no—
204C4×C137—82no—
205C4×C137—81no—
206C4×C137—81no—
207C2×C137—22no—
208C2×C137—22no—
209C2×C137—41no—
210C2×C137—44no—
211C2×C137#300411yescharacteristic
212C2×C137C2×#45511yescharacteristic
213C2×C137#302611yescharacteristic
214#18319C13711yesmaximal, characteristic, direct product factor, Sylow
215C137#1831911yescharacteristic, direct product factor, Sylow
216C2×#543C2×C13711yescharacteristic
217#3195C2×C13712yescomplemented
218#2561C2×C13711yescharacteristic
219#1608C2×C13714yescomplemented
220#1602C2×C13714yescomplemented
221#1508C2×C13712yescomplemented
222#1462C2×C13711yescharacteristic
223#543—24no—
224#543—22no—
225#543C22×C13714yes—
226C2×C22⋊Q8—22no—
227C2×C22⋊Q8—21no—
228C2×C22⋊Q8—22no—
229C2×C22⋊Q8C22×C13712yes—
230C2×C22⋊Q8C22×C13711yescharacteristic
231C22×C22⋊C4—21no—
232C22×C22⋊C4C22×C13711yescharacteristic
233#483—22no—
234#483C22×C13714yescomplemented
235#470—24no—
236#451—22no—
237#451C22×C13714yescomplemented
238#407C22×C13712yes—
239#397C22×C13712yes—
240#393C22×C13712yes—
241#388—22no—
242#381—21no—
243#379—21no—
244C4×C22⋊C4C22×C13711yescharacteristic
245#328C22×C13718yescomplemented
246#323C22×C13714yes—
247C25D4×C13711yescharacteristic
248C22×Q8—21no—
249C22×Q8D4×C13712yes—
250C22×Q8C23×C13711yescharacteristic
251C23×C4—21no—
252C23×C4—21no—
253C23×C4D4×C13711yescharacteristic
254C23×C4C23×C13711yescharacteristic
255C2×Q16—24no—
256C2×M16—22no—
257C22⋊Q8—28no—
258C22⋊Q8—28no—
259C22⋊Q8—24no—
260C22⋊Q8—28no—
261C22⋊Q8—24no—
262C22⋊Q8—22no—
263C22⋊Q8—22no—
264C22⋊Q8—28no—
265C22⋊Q8—22no—
266C22⋊Q8—22no—
267C22⋊Q8D4×C13714yes—
268C22⋊Q8C23×C13714yes—
269C2×C4⋊C4—21no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4D4×C13712yes—
274C2×C4⋊C4C23×C13711yescharacteristic
275C2×C22⋊C4—22no—
276C2×C22⋊C4—24no—
277C2×C22⋊C4—22no—
278C2×C22⋊C4—24no—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—22no—
281C2×C22⋊C4—21no—
282C2×C22⋊C4—21no—
283C2×C22⋊C4—22no—
284C2×C22⋊C4—22no—
285C2×C22⋊C4D4×C13712yes—
286C2×C22⋊C4C23×C13712yes—
287C2×C42—21no—
288#106—24no—
289#103—24no—
290#103—24no—
291C22⋊C8—22no—
292C22⋊C8C23×C13714yes—
293#95—22no—
294#95—22no—
295#95C23×C13712yes—
296C24—21no—
297C24—21no—
298C24—24no—
299C24—42no—
300C24—81no—
301C24C2×D4×C13711yescharacteristic
302C24C2×D4×C13712yes—
303C2×Q8—24no—
304C2×Q8—22no—
305C2×Q8—22no—
306C2×Q8—81no—
307C2×Q8—82no—
308C2×Q8—82no—
309C2×Q8C2×D4×C13714yes—
310C22×C4—22no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—81no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4C2×D4×C13712yes—
324C22×C4C2×D4×C13711yescharacteristic
325C22×C4C2×D4×C13712yes—
326C22×C4C2×D4×C13712yes—
327C22×C4C24×C13711yescharacteristic
328Q16—84no—
329M16—82no—
330C2×C8—24no—
331C4⋊C4—24no—
332C4⋊C4—28no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—22no—
337C4⋊C4—42no—
338C4⋊C4C2×D4×C13714yes—
339C22⋊C4—28no—
340C22⋊C4—24no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—42no—
345C22⋊C4—41no—
346C22⋊C4—41no—
347C22⋊C4—44no—
348C22⋊C4—82no—
349C42—81no—
350C23—24no—
351C23—24no—
352C23—22no—
353C23—22no—
354C23—22no—
355C23—22no—
356C23—41no—
357C23—42no—
358C23—42no—
359C23—42no—
360C23—42no—
361C23—81no—
362C23—81no—
363C23—88no—
364C23#120×C13711yescharacteristic
365C23#120×C13711yescharacteristic
366C23#137×C13714yes—
367C23C22×D4×C13711yescharacteristic
368Q8—44no—
369Q8—44no—
370Q8—82no—
371Q8—84no—
372C2×C4—24no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—81no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4#120×C13712yes—
394C2×C4C22×D4×C13712yes—
395C8—44no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—24no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—44no—
406C22—48no—
407C22—42no—
408C22—42no—
409C22—82no—
410C22—84no—
411C22#455×C13712yes—
412C22C2×#120×C13711yescharacteristic
413C22C2×#137×C13712yes—
414C4—22no—
415C4—22no—
416C4—42no—
417C4—44no—
418C4—44no—
419C4—82no—
420C4—81no—
421C4—81no—
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2#3004×C13711yescharacteristic
427C2C2×#455×C13711yescharacteristic
428C2#3026×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×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
2. #3195×C137C2semidirect
4. #1608×C137C2semidirect
5. #1602×C137C2semidirect
6. #1508×C137C2semidirect
19. #483×C137C22semidirect
22. #451×C137C22semidirect
30. #328×C137C22semidirect
214. #18319C137direct (N × H)
215. C137#18319direct (N × H)
217. #3195C2×C137semidirect
219. #1608C2×C137semidirect
220. #1602C2×C137semidirect
221. #1508C2×C137semidirect
234. #483C22×C137semidirect
237. #451C22×C137semidirect
245. #328C22×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#543×C1372
#3195×C1372
#2561×C1372
#1608×C1372
#1602×C1372
#1508×C1372
#1462×C1372
#18319137