Finite Group Databasebeta

← C137×#9209 #46858595

Subgroups — C137×#9209 #46858595

Order 35072.

456 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3076×C137C211yesmaximal, characteristic
2#2965×C137C211yesmaximal, characteristic
3#1806×C137C211yesmaximal, characteristic
4#1503×C137C211yesmaximal, characteristic, complemented
5#1493×C137C211yesmaximal, characteristic, complemented
6#1301×C137C211yesmaximal, characteristic, complemented
7#1279×C137C211yesmaximal, characteristic, complemented
8#533×C137C2211yescharacteristic
9C4×#41×C137C2211yescharacteristic
10#479×C137—21no—
11#478×C137—21no—
12#477×C137C411yescharacteristic, complemented
13#476×C137C411yescharacteristic, complemented
14#475×C137C411yescharacteristic, complemented
15#474×C137C411yescharacteristic, complemented
16#446×C137—21no—
17#436×C137—21no—
18#435×C137—22no—
19#431×C137—21no—
20#431×C137C2211yescharacteristic
21#427×C137—21no—
22#423×C137C2211yescharacteristic
23C2×C4⋊C8×C137C2211yescharacteristic
24C2×#104×C137—21no—
25#416×C137—21no—
26C4×M16×C137—21no—
27C2×C4×C8×C137C2211yescharacteristic
28C4×C4⋊C4×C137C2211yescharacteristic
29#339×C137—21no—
30#328×C137—22no—
31Q8⋊C8×C137—21no—
32Q8⋊C8×C137—21no—
33C2×#41×C137D411yescharacteristic
34C2×M16×C137—22no—
35C2×M16×C137—21no—
36C22×C8×C137—21no—
37C22×C8×C137—21no—
38C4⋊Q8×C137C2×C411yescharacteristic
39C4⋊Q8×C137C2×C411yescharacteristic
40#125×C137—21no—
41C22⋊Q8×C137—21no—
42C22⋊Q8×C137—21no—
43C4×Q8×C137—21no—
44C4×Q8×C137—21no—
45C4×Q8×C137C2×C411yescharacteristic
46C4×Q8×C137D411yescharacteristic
47C4×D4×C137—21no—
48C4×D4×C137—21no—
49C4×D4×C137C2×C411yescharacteristic
50C4×D4×C137D411yescharacteristic
51#117×C137—21no—
52#117×C137—21no—
53#117×C137D411yescharacteristic
54C2×C4⋊C4×C137—21no—
55C2×C42×C137—21no—
56C2×C42×C137—21no—
57C2×C42×C137C2311yescharacteristic
58#107×C137—21no—
59#107×C137—21no—
60#106×C137—21no—
61#106×C137—21no—
62C4⋊C8×C137—22no—
63C4⋊C8×C137—21no—
64C4⋊C8×C137—21no—
65C4⋊C8×C137—21no—
66C4⋊C8×C137—21no—
67C4⋊C8×C137C2×C411yescharacteristic
68C4⋊C8×C137C2×C411yescharacteristic
69#104×C137—42no—
70#103×C137—21no—
71#103×C137—21no—
72#102×C137—21no—
73#102×C137—21no—
74C22⋊C8×C137—22no—
75C22⋊C8×C137D412yes—
76#97×C137—22no—
77C4×C8×C137—21no—
78C4×C8×C137—21no—
79C4×C8×C137D411yescharacteristic
80C4×C8×C137D411yescharacteristic
81#95×C137—21no—
82#41×C137—21no—
83#41×C137—21no—
84#41×C137—21no—
85#41×C137—21no—
86C2×Q8×C137—21no—
87C2×Q8×C137—21no—
88C2×Q8×C137C22⋊C411yescharacteristic
89C2×D4×C137—21no—
90C2×D4×C137C22⋊C411yescharacteristic
91C22×C4×C137—21no—
92C22×C4×C137—21no—
93C22×C4×C137—21no—
94C22×C4×C137—41no—
95C22×C4×C137C2×D411yescharacteristic
96C22×C4×C137C2×D411yescharacteristic
97C22×C4×C137#4111yescharacteristic
98M16×C137—44no—
99M16×C137—41no—
100M16×C137—41no—
101C2×C8×C137—21no—
102C2×C8×C137—21no—
103C2×C8×C137—21no—
104C2×C8×C137—21no—
105C2×C8×C137—22no—
106C2×C8×C137—42no—
107C2×C8×C137—41no—
108C2×C8×C137—41no—
109C2×C8×C137—41no—
110C4⋊C4×C137—21no—
111C4⋊C4×C137—21no—
112C4⋊C4×C137—22no—
113C4⋊C4×C137—21no—
114C4⋊C4×C137—21no—
115C4⋊C4×C137—21no—
116C4⋊C4×C137—21no—
117C4⋊C4×C137—41no—
118C4⋊C4×C137—41no—
119C4⋊C4×C137—41no—
120C4⋊C4×C137C22⋊C411yescharacteristic
121C4⋊C4×C137C22⋊C411yescharacteristic
122C22⋊C4×C137—21no—
123C22⋊C4×C137—21no—
124C22⋊C4×C137—21no—
125C22⋊C4×C137—41no—
126C42×C137—21no—
127C42×C137—21no—
128C42×C137—21no—
129C42×C137—41no—
130C42×C137—41no—
131C42×C137—41no—
132C42×C137C22×C411yescharacteristic
133C42×C137C2×D411yescharacteristic
134C42×C137C2×D411yescharacteristic
135C42×C137#4111yescharacteristic
136C23×C137—21no—
137C23×C137#12111yescharacteristic
138Q8×C137—21no—
139Q8×C137—21no—
140Q8×C137—41no—
141Q8×C137—41no—
142Q8×C137—81no—
143D4×C137—21no—
144D4×C137—21no—
145D4×C137—41no—
146D4×C137—41no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—21no—
151C2×C4×C137—21no—
152C2×C4×C137—41no—
153C2×C4×C137—42no—
154C2×C4×C137—41no—
155C2×C4×C137—41no—
156C2×C4×C137—41no—
157C2×C4×C137—41no—
158C2×C4×C137—42no—
159C2×C4×C137—41no—
160C2×C4×C137—41no—
161C2×C4×C137—81no—
162C2×C4×C137—81no—
163C2×C4×C137C2×C22⋊C411yescharacteristic
164C2×C4×C137C4×D411yescharacteristic
165C2×C4×C137C4×D411yescharacteristic
166C2×C4×C137#12011yescharacteristic
167C2×C4×C137#12111yescharacteristic
168C2×C4×C137#12311yescharacteristic
169C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C137#13711yescharacteristic
171C2×C4×C137#13711yescharacteristic
172C2×C4×C137#13711yescharacteristic
173C8×C137—22no—
174C8×C137—41no—
175C8×C137—42no—
176C8×C137—41no—
177C8×C137—41no—
178C8×C137—81no—
179C22×C137—21no—
180C22×C137—21no—
181C22×C137—41no—
182C22×C137—41no—
183C22×C137—41no—
184C22×C137#38611yescharacteristic
185C22×C137#46811yescharacteristic
186C22×C137#47011yescharacteristic
187C4×C137—21no—
188C4×C137—21no—
189C4×C137—21no—
190C4×C137—41no—
191C4×C137—41no—
192C4×C137—41no—
193C4×C137—82no—
194C4×C137—81no—
195C4×C137—81no—
196C4×C137#41711yescharacteristic
197C4×C137#41911yescharacteristic
198C4×C137#44311yescharacteristic
199C4×C137#44411yescharacteristic
200C4×C137#46811yescharacteristic
201C4×C137#47011yescharacteristic
202C2×C137—21no—
203C2×C137—21no—
204C2×C137—81no—
205C2×C137#187611yescharacteristic
206C2×C137#189411yescharacteristic
207C2×C137#190611yescharacteristic
208#9209C13711yesmaximal, characteristic, direct product factor, Sylow
209C137#920911yescharacteristic, direct product factor, Sylow
210#3076C2×C13711yescharacteristic
211#2965C2×C13711yescharacteristic
212#1806C2×C13711yescharacteristic
213#1503C2×C13711yescharacteristic, complemented
214#1493C2×C13711yescharacteristic, complemented
215#1301C2×C13711yescharacteristic, complemented
216#1279C2×C13711yescharacteristic, complemented
217#533C22×C13711yescharacteristic
218C4×#41C22×C13711yescharacteristic
219#479—21no—
220#478—21no—
221#477C4×C13711yescharacteristic, complemented
222#476C4×C13711yescharacteristic, complemented
223#475C4×C13711yescharacteristic, complemented
224#474C4×C13711yescharacteristic, complemented
225#446—21no—
226#436—21no—
227#435—22no—
228#431—21no—
229#431C22×C13711yescharacteristic
230#427—21no—
231#423C22×C13711yescharacteristic
232C2×C4⋊C8C22×C13711yescharacteristic
233C2×#104—21no—
234#416—21no—
235C4×M16—21no—
236C2×C4×C8C22×C13711yescharacteristic
237C4×C4⋊C4C22×C13711yescharacteristic
238#339—21no—
239#328—22no—
240Q8⋊C8—21no—
241Q8⋊C8—21no—
242C2×#41D4×C13711yescharacteristic
243C2×M16—22no—
244C2×M16—21no—
245C22×C8—21no—
246C22×C8—21no—
247C4⋊Q8C2×C4×C13711yescharacteristic
248C4⋊Q8C2×C4×C13711yescharacteristic
249#125—21no—
250C22⋊Q8—21no—
251C22⋊Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C2×C4×C13711yescharacteristic
255C4×Q8D4×C13711yescharacteristic
256C4×D4—21no—
257C4×D4—21no—
258C4×D4C2×C4×C13711yescharacteristic
259C4×D4D4×C13711yescharacteristic
260#117—21no—
261#117—21no—
262#117D4×C13711yescharacteristic
263C2×C4⋊C4—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42C23×C13711yescharacteristic
267#107—21no—
268#107—21no—
269#106—21no—
270#106—21no—
271C4⋊C8—22no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8C2×C4×C13711yescharacteristic
277C4⋊C8C2×C4×C13711yescharacteristic
278#104—42no—
279#103—21no—
280#103—21no—
281#102—21no—
282#102—21no—
283C22⋊C8—22no—
284C22⋊C8D4×C13712yes—
285#97—22no—
286C4×C8—21no—
287C4×C8—21no—
288C4×C8D4×C13711yescharacteristic
289C4×C8D4×C13711yescharacteristic
290#95—21no—
291#41—21no—
292#41—21no—
293#41—21no—
294#41—21no—
295C2×Q8—21no—
296C2×Q8—21no—
297C2×Q8C22⋊C4×C13711yescharacteristic
298C2×D4—21no—
299C2×D4C22⋊C4×C13711yescharacteristic
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—41no—
304C22×C4C2×D4×C13711yescharacteristic
305C22×C4C2×D4×C13711yescharacteristic
306C22×C4#41×C13711yescharacteristic
307M16—44no—
308M16—41no—
309M16—41no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—22no—
315C2×C8—42no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4C22⋊C4×C13711yescharacteristic
330C4⋊C4C22⋊C4×C13711yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C42—21no—
336C42—21no—
337C42—21no—
338C42—41no—
339C42—41no—
340C42—41no—
341C42C22×C4×C13711yescharacteristic
342C42C2×D4×C13711yescharacteristic
343C42C2×D4×C13711yescharacteristic
344C42#41×C13711yescharacteristic
345C23—21no—
346C23#121×C13711yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—41no—
350Q8—41no—
351Q8—81no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—41no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—41no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4C2×C22⋊C4×C13711yescharacteristic
373C2×C4C4×D4×C13711yescharacteristic
374C2×C4C4×D4×C13711yescharacteristic
375C2×C4#120×C13711yescharacteristic
376C2×C4#121×C13711yescharacteristic
377C2×C4#123×C13711yescharacteristic
378C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
379C2×C4#137×C13711yescharacteristic
380C2×C4#137×C13711yescharacteristic
381C2×C4#137×C13711yescharacteristic
382C8—22no—
383C8—41no—
384C8—42no—
385C8—41no—
386C8—41no—
387C8—81no—
388C22—21no—
389C22—21no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22#386×C13711yescharacteristic
394C22#468×C13711yescharacteristic
395C22#470×C13711yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—41no—
402C4—82no—
403C4—81no—
404C4—81no—
405C4#417×C13711yescharacteristic
406C4#419×C13711yescharacteristic
407C4#443×C13711yescharacteristic
408C4#444×C13711yescharacteristic
409C4#468×C13711yescharacteristic
410C4#470×C13711yescharacteristic
411C2—21no—
412C2—21no—
413C2—81no—
414C2#1876×C13711yescharacteristic
415C2#1894×C13711yescharacteristic
416C2#1906×C13711yescharacteristic
subgroupG/H
centerC2×C4×C137#121
commutatorC42C22×C4×C137
Frattini, cosocleC2×C42C23×C137
socleC22×C137#386
prime radicalC2×#41×C137D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1503×C137C2semidirect
5. #1493×C137C2semidirect
6. #1301×C137C2semidirect
7. #1279×C137C2semidirect
12. #477×C137C4semidirect
13. #476×C137C4semidirect
14. #475×C137C4semidirect
15. #474×C137C4semidirect
208. #9209C137direct (N × H)
209. C137#9209direct (N × H)
213. #1503C2×C137semidirect
214. #1493C2×C137semidirect
215. #1301C2×C137semidirect
216. #1279C2×C137semidirect
221. #477C4×C137semidirect
222. #476C4×C137semidirect
223. #475C4×C137semidirect
224. #474C4×C137semidirect

Maximal subgroups

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

subgroupindex
#3076×C1372
#2965×C1372
#1806×C1372
#1503×C1372
#1493×C1372
#1301×C1372
#1279×C1372
#9209137