Finite Group Databasebeta

← C137×#12621 #46862007

Subgroups — C137×#12621 #46862007

Order 35072.

548 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2322×C137C211yesmaximal, characteristic
2C2×#352×C137C212yesmaximal
3C4×#100×C137C211yesmaximal, characteristic
4C4×#99×C137C211yesmaximal, characteristic
5#1390×C137C212yesmaximal
6C2×#123×C137C2211yescharacteristic
7C2×C4×D4×C137C2211yescharacteristic
8C22×C4⋊C4×C137C411yescharacteristic, complemented
9C2×#100×C137C2212yes—
10C2×#99×C137C2212yes—
11C4×M16×C137—21no—
12#389×C137—21no—
13#388×C137—21no—
14#387×C137—21no—
15#386×C137C2211yescharacteristic
16#385×C137—21no—
17#385×C137C411yescharacteristic, complemented
18#381×C137C411yescharacteristic, complemented
19C4×C22⋊C4×C137—21no—
20C4×C22⋊C4×C137C411yescharacteristic, complemented
21#352×C137C418yescomplemented
22#343×C137—21no—
23#342×C137—21no—
24C22×D4×C137C2311yescharacteristic
25C23×C4×C137C2×C411yescharacteristic
26C23×C4×C137C2×C411yescharacteristic
27C23×C4×C137C2×C411yescharacteristic
28C23×C4×C137D411yescharacteristic
29C2×M16×C137—22no—
30#123×C137—22no—
31#123×C137C2×C414yes—
32C4×D4×C137—41no—
33C4×D4×C137—41no—
34C2×C4⋊C4×C137—21no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137—21no—
37C2×C4⋊C4×C137—21no—
38C2×C4⋊C4×C137—41no—
39C2×C4⋊C4×C137—41no—
40C2×C4⋊C4×C137D411yescharacteristic
41C2×C22⋊C4×C137—21no—
42C2×C22⋊C4×C137—22no—
43C2×C22⋊C4×C137—21no—
44C2×C22⋊C4×C137C2×C411yescharacteristic
45C2×C22⋊C4×C137C2×C411yescharacteristic
46C2×C22⋊C4×C137C2×C411yescharacteristic
47C2×C22⋊C4×C137D411yescharacteristic
48C2×C42×C137—21no—
49C2×C42×C137—41no—
50C2×C42×C137D411yescharacteristic
51#100×C137—22no—
52#100×C137C2×C414yes—
53#99×C137—22no—
54#99×C137C2×C414yes—
55#97×C137—41no—
56C4×C8×C137—41no—
57#95×C137—21no—
58#95×C137—21no—
59#95×C137—21no—
60#95×C137—21no—
61#95×C137—21no—
62#95×C137—41no—
63C24×C137C22×C411yescharacteristic
64C24×C137C2×D411yescharacteristic
65C2×D4×C137—41no—
66C2×D4×C137—41no—
67C2×D4×C137C22×C412yes—
68C2×D4×C137#4112yes—
69C22×C4×C137—21no—
70C22×C4×C137—21no—
71C22×C4×C137—21no—
72C22×C4×C137—21no—
73C22×C4×C137—21no—
74C22×C4×C137—21no—
75C22×C4×C137—21no—
76C22×C4×C137—21no—
77C22×C4×C137—21no—
78C22×C4×C137—41no—
79C22×C4×C137—41no—
80C22×C4×C137—42no—
81C22×C4×C137—41no—
82C22×C4×C137—81no—
83C22×C4×C137—81no—
84C22×C4×C137—81no—
85C22×C4×C137C4212yescomplemented
86C22×C4×C137C22⋊C411yescharacteristic
87C22×C4×C137C22⋊C411yescharacteristic
88C22×C4×C137C22⋊C411yescharacteristic
89C22×C4×C137C22⋊C411yescharacteristic
90C22×C4×C137C2×D411yescharacteristic
91M16×C137—24no—
92M16×C137—42no—
93C2×C8×C137—42no—
94C4⋊C4×C137—24no—
95C4⋊C4×C137—41no—
96C4⋊C4×C137—42no—
97C4⋊C4×C137—41no—
98C4⋊C4×C137—41no—
99C4⋊C4×C137—81no—
100C22⋊C4×C137—21no—
101C22⋊C4×C137—24no—
102C22⋊C4×C137—24no—
103C22⋊C4×C137—41no—
104C22⋊C4×C137—41no—
105C22⋊C4×C137—42no—
106C22⋊C4×C137—41no—
107C22⋊C4×C137—81no—
108C22⋊C4×C137C4212yescomplemented
109C42×C137—21no—
110C42×C137—21no—
111C42×C137—81no—
112C42×C137—82no—
113C23×C137—21no—
114C23×C137—21no—
115C23×C137—21no—
116C23×C137—21no—
117C23×C137—41no—
118C23×C137—41no—
119C23×C137—41no—
120C23×C137—41no—
121C23×C137C2×C4211yescharacteristic
122C23×C137C2×C22⋊C411yescharacteristic
123C23×C137#11711yescharacteristic
124C23×C137C4×D412yes—
125D4×C137—42no—
126D4×C137—42no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—22no—
130C2×C4×C137—22no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—41no—
135C2×C4×C137—42no—
136C2×C4×C137—41no—
137C2×C4×C137—41no—
138C2×C4×C137—42no—
139C2×C4×C137—41no—
140C2×C4×C137—41no—
141C2×C4×C137—41no—
142C2×C4×C137—41no—
143C2×C4×C137—42no—
144C2×C4×C137—41no—
145C2×C4×C137—41no—
146C2×C4×C137—44no—
147C2×C4×C137—41no—
148C2×C4×C137—41no—
149C2×C4×C137—41no—
150C2×C4×C137—41no—
151C2×C4×C137—41no—
152C2×C4×C137—81no—
153C2×C4×C137—81no—
154C2×C4×C137—81no—
155C2×C4×C137—82no—
156C2×C4×C137—82no—
157C2×C4×C137—82no—
158C2×C4×C137#9911yescharacteristic
159C2×C4×C137#9911yescharacteristic
160C2×C4×C137C4×D412yes—
161C8×C137—44no—
162C22×C137—21no—
163C22×C137—21no—
164C22×C137—21no—
165C22×C137—21no—
166C22×C137—22no—
167C22×C137—41no—
168C22×C137—41no—
169C22×C137—42no—
170C22×C137—42no—
171C22×C137—41no—
172C22×C137—41no—
173C22×C137—41no—
174C22×C137—41no—
175C22×C137—81no—
176C22×C137C4×C22⋊C411yescharacteristic
177C22×C137C2×#9911yescharacteristic
178C22×C137#41011yescharacteristic
179C4×C137—21no—
180C4×C137—21no—
181C4×C137—22no—
182C4×C137—42no—
183C4×C137—41no—
184C4×C137—41no—
185C4×C137—41no—
186C4×C137—42no—
187C4×C137—81no—
188C4×C137—84no—
189C4×C137—81no—
190C2×C137—21no—
191C2×C137—21no—
192C2×C137—42no—
193C2×C137—41no—
194C2×C137—41no—
195C2×C137C4×#9911yescharacteristic
196C2×C137#212111yescharacteristic
197C2×C137#212311yescharacteristic
198#12621C13711yesmaximal, characteristic, direct product factor, Sylow
199C137#1262111yescharacteristic, direct product factor, Sylow
200#2322C2×C13711yescharacteristic
201C2×#352C2×C13712yes—
202C4×#100C2×C13711yescharacteristic
203C4×#99C2×C13711yescharacteristic
204#1390C2×C13712yes—
205C2×#123C22×C13711yescharacteristic
206C2×C4×D4C22×C13711yescharacteristic
207C22×C4⋊C4C4×C13711yescharacteristic, complemented
208C2×#100C22×C13712yes—
209C2×#99C22×C13712yes—
210C4×M16—21no—
211#389—21no—
212#388—21no—
213#387—21no—
214#386C22×C13711yescharacteristic
215#385—21no—
216#385C4×C13711yescharacteristic, complemented
217#381C4×C13711yescharacteristic, complemented
218C4×C22⋊C4—21no—
219C4×C22⋊C4C4×C13711yescharacteristic, complemented
220#352C4×C13718yescomplemented
221#343—21no—
222#342—21no—
223C22×D4C23×C13711yescharacteristic
224C23×C4C2×C4×C13711yescharacteristic
225C23×C4C2×C4×C13711yescharacteristic
226C23×C4C2×C4×C13711yescharacteristic
227C23×C4D4×C13711yescharacteristic
228C2×M16—22no—
229#123—22no—
230#123C2×C4×C13714yes—
231C4×D4—41no—
232C4×D4—41no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—41no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4D4×C13711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4C2×C4×C13711yescharacteristic
244C2×C22⋊C4C2×C4×C13711yescharacteristic
245C2×C22⋊C4C2×C4×C13711yescharacteristic
246C2×C22⋊C4D4×C13711yescharacteristic
247C2×C42—21no—
248C2×C42—41no—
249C2×C42D4×C13711yescharacteristic
250#100—22no—
251#100C2×C4×C13714yes—
252#99—22no—
253#99C2×C4×C13714yes—
254#97—41no—
255C4×C8—41no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262C24C22×C4×C13711yescharacteristic
263C24C2×D4×C13711yescharacteristic
264C2×D4—41no—
265C2×D4—41no—
266C2×D4C22×C4×C13712yes—
267C2×D4#41×C13712yes—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4—81no—
284C22×C4C42×C13712yescomplemented
285C22×C4C22⋊C4×C13711yescharacteristic
286C22×C4C22⋊C4×C13711yescharacteristic
287C22×C4C22⋊C4×C13711yescharacteristic
288C22×C4C22⋊C4×C13711yescharacteristic
289C22×C4C2×D4×C13711yescharacteristic
290M16—24no—
291M16—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—41no—
295C4⋊C4—42no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—81no—
299C22⋊C4—21no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—42no—
305C22⋊C4—41no—
306C22⋊C4—81no—
307C22⋊C4C42×C13712yescomplemented
308C42—21no—
309C42—21no—
310C42—81no—
311C42—82no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23C2×C42×C13711yescharacteristic
321C23C2×C22⋊C4×C13711yescharacteristic
322C23#117×C13711yescharacteristic
323C23C4×D4×C13712yes—
324D4—42no—
325D4—42no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—41no—
336C2×C4—42no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#99×C13711yescharacteristic
358C2×C4#99×C13711yescharacteristic
359C2×C4C4×D4×C13712yes—
360C8—44no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—22no—
366C22—41no—
367C22—41no—
368C22—42no—
369C22—42no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—81no—
375C22C4×C22⋊C4×C13711yescharacteristic
376C22C2×#99×C13711yescharacteristic
377C22#410×C13711yescharacteristic
378C4—21no—
379C4—21no—
380C4—22no—
381C4—42no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—42no—
386C4—81no—
387C4—84no—
388C4—81no—
389C2—21no—
390C2—21no—
391C2—42no—
392C2—41no—
393C2—41no—
394C2C4×#99×C13711yescharacteristic
395C2#2121×C13711yescharacteristic
396C2#2123×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC23C2×C42×C137
Frattini, cosocleC22×D4C23×C137
prime radicalC22×D4×C137C23
LCSG ▷ C23 ▷ C22 ▷ 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
8. C22×C4⋊C4×C137C4semidirect
17. #385×C137C4semidirect
18. #381×C137C4semidirect
20. C4×C22⋊C4×C137C4semidirect
21. #352×C137C4semidirect
85. C22×C4×C137C42semidirect
108. C22⋊C4×C137C42semidirect
198. #12621C137direct (N × H)
199. C137#12621direct (N × H)
207. C22×C4⋊C4C4×C137semidirect
216. #385C4×C137semidirect
217. #381C4×C137semidirect
219. C4×C22⋊C4C4×C137semidirect
220. #352C4×C137semidirect
284. C22×C4C42×C137semidirect
307. C22⋊C4C42×C137semidirect

Maximal subgroups

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

subgroupindex
#2322×C1372
C2×#352×C1372
C4×#100×C1372
C4×#99×C1372
#1390×C1372
#12621137