Finite Group Databasebeta

← C137×#33618 #46883004

Subgroups — C137×#33618 #46883004

Order 35072.

656 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#509×C137C211yesmaximal, characteristic, complemented
2#3407×C137C211yesmaximal, characteristic, complemented
3#3067×C137C211yesmaximal, characteristic, complemented
4C2×#472×C137C211yesmaximal, characteristic, complemented
5#2223×C137C214yesmaximal, complemented
6#2221×C137C214yesmaximal, complemented
7#2161×C137C211yesmaximal, characteristic, complemented
8#2148×C137C211yesmaximal, characteristic, complemented
9C2×#361×C137C211yesmaximal, characteristic, complemented
10#575×C137—21no—
11C2×AGL(1,ℤ/8)×C137—21no—
12C2×#135×C137C2211yescharacteristic
13C22×D8×C137C2211yescharacteristic
14C2×#131×C137C2211yescharacteristic
15#509×C137—22no—
16#509×C137—22no—
17C2×SD32×C137C2212yescomplemented
18C2×D16×C137C2212yescomplemented
19#504×C137—22no—
20#504×C137C2214yescomplemented
21C2×M32×C137—22no—
22C2×M32×C137C2211yescharacteristic
23C2×M32×C137C2211yescharacteristic
24#473×C137C2212yescomplemented
25#472×C137—22no—
26#472×C137C2214yes—
27#472×C137C2212yescomplemented
28#471×C137—22no—
29C2×#108×C137C2211yescharacteristic
30C2×#108×C137C2211yescharacteristic
31#413×C137—21no—
32C2×#100×C137—21no—
33#368×C137C2214yescomplemented
34#362×C137C2212yescomplemented
35#361×C137C2214yescomplemented
36#361×C137C2212yescomplemented
37C2×#41×C137—21no—
38C2×#41×C137D411yescharacteristic
39C22×D4×C137—21no—
40#137×C137—22no—
41#137×C137—41no—
42AGL(1,ℤ/8)×C137—22no—
43AGL(1,ℤ/8)×C137—24no—
44AGL(1,ℤ/8)×C137—42no—
45AGL(1,ℤ/8)×C137—41no—
46#135×C137—21no—
47#135×C137—21no—
48#135×C137—21no—
49#135×C137—21no—
50#135×C137—42no—
51C2×Q16×C137C2311yescharacteristic
52C2×SD16×C137—21no—
53C2×SD16×C137—41no—
54C2×D8×C137—21no—
55C2×D8×C137—21no—
56C2×D8×C137—21no—
57C2×D8×C137—21no—
58C2×D8×C137—21no—
59C2×D8×C137—41no—
60C2×D8×C137C2312yes—
61C2×D8×C137C2311yescharacteristic
62#131×C137—21no—
63#131×C137—21no—
64#131×C137D412yes—
65#131×C137C2312yes—
66C2×M16×C137—21no—
67C2×M16×C137—21no—
68C2×M16×C137—21no—
69C2×M16×C137D411yescharacteristic
70C22×C8×C137—21no—
71C22×C8×C137C2311yescharacteristic
72SD32×C137—24no—
73D16×C137—24no—
74M32×C137—21no—
75M32×C137—24no—
76M32×C137D412yes—
77M32×C137D412yes—
78M32×C137C2312yes—
79C2×C16×C137—21no—
80C2×C16×C137—24no—
81C2×C16×C137C2312yes—
82#108×C137—21no—
83#108×C137—21no—
84#108×C137C2312yes—
85#108×C137C2312yes—
86#101×C137—22no—
87#100×C137—24no—
88#100×C137—22no—
89C24×C137—41no—
90#41×C137—21no—
91#41×C137—21no—
92#41×C137—41no—
93#41×C137—41no—
94#41×C137—41no—
95#41×C137—41no—
96#41×C137D812yes—
97#41×C137C2×D412yes—
98C2×Q8×C137—21no—
99C2×Q8×C137D811yescharacteristic
100C2×D4×C137—21no—
101C2×D4×C137—22no—
102C2×D4×C137—21no—
103C2×D4×C137—41no—
104C2×D4×C137—41no—
105C2×D4×C137—41no—
106C2×D4×C137—41no—
107C2×D4×C137—41no—
108C2×D4×C137—41no—
109C2×D4×C137D811yescharacteristic
110C22×C4×C137—21no—
111C22×C4×C137—41no—
112C22×C4×C137C2×D411yescharacteristic
113Q16×C137—21no—
114Q16×C137—21no—
115Q16×C137—42no—
116SD16×C137—44no—
117SD16×C137—42no—
118SD16×C137—42no—
119SD16×C137—41no—
120SD16×C137—41no—
121D8×C137—21no—
122D8×C137—22no—
123D8×C137—22no—
124D8×C137—21no—
125D8×C137—44no—
126D8×C137—42no—
127D8×C137—41no—
128D8×C137—41no—
129M16×C137—21no—
130M16×C137—22no—
131M16×C137—22no—
132M16×C137—22no—
133M16×C137—22no—
134M16×C137—41no—
135M16×C137—41no—
136M16×C137C2×D412yes—
137C2×C8×C137—21no—
138C2×C8×C137—22no—
139C2×C8×C137—22no—
140C2×C8×C137—41no—
141C2×C8×C137—41no—
142C2×C8×C137C2×D411yescharacteristic
143C2×C8×C137C2×D411yescharacteristic
144C2×C8×C137C2×D411yescharacteristic
145C2×C8×C137#4111yescharacteristic
146C2×C8×C137#4111yescharacteristic
147C2×C8×C137C2411yescharacteristic
148C16×C137—22no—
149C16×C137C2×D414yes—
150C23×C137—21no—
151C23×C137—41no—
152C23×C137—42no—
153C23×C137—81no—
154C23×C137—81no—
155C23×C137—162no—
156C23×C137#13511yescharacteristic
157Q8×C137—21no—
158Q8×C137—41no—
159Q8×C137—41no—
160Q8×C137C2×D812yes—
161D4×C137—22no—
162D4×C137—22no—
163D4×C137—21no—
164D4×C137—42no—
165D4×C137—41no—
166D4×C137—42no—
167D4×C137—41no—
168D4×C137—81no—
169D4×C137—81no—
170D4×C137—81no—
171D4×C137—81no—
172D4×C137C2×D812yes—
173C2×C4×C137—21no—
174C2×C4×C137—22no—
175C2×C4×C137—22no—
176C2×C4×C137—41no—
177C2×C4×C137—81no—
178C2×C4×C137—81no—
179C2×C4×C137#12111yescharacteristic
180C2×C4×C137#12111yescharacteristic
181C2×C4×C137C2×D811yescharacteristic
182C2×C4×C137C2×D811yescharacteristic
183C2×C4×C137#13511yescharacteristic
184C2×C4×C137C22×D411yescharacteristic
185C8×C137—22no—
186C8×C137—42no—
187C8×C137—42no—
188C8×C137#12111yescharacteristic
189C8×C137#12111yescharacteristic
190C8×C137C22×D411yescharacteristic
191C8×C137C2×#4111yescharacteristic
192C22×C137—22no—
193C22×C137—41no—
194C22×C137—81no—
195C22×C137—81no—
196C22×C137—82no—
197C22×C137—161no—
198C22×C137—162no—
199C22×C137—162no—
200C22×C137—162no—
201C22×C137#46611yescharacteristic
202C22×C137#46611yescharacteristic
203C22×C137C2×#13511yescharacteristic
204C4×C137—22no—
205C4×C137—81no—
206C4×C137#46611yescharacteristic
207C4×C137#46611yescharacteristic
208C4×C137C2×#12111yescharacteristic
209C4×C137C22×D811yescharacteristic
210C2×C137—21no—
211C2×C137—21no—
212C2×C137—21no—
213C2×C137—42no—
214C2×C137—161no—
215C2×C137—162no—
216C2×C137C2×#46611yescharacteristic
217#33618C13711yesmaximal, characteristic, direct product factor, Sylow
218C137#3361811yescharacteristic, direct product factor, Sylow
219C2×#509C2×C13711yescharacteristic, complemented
220#3407C2×C13711yescharacteristic, complemented
221#3067C2×C13711yescharacteristic, complemented
222C2×#472C2×C13711yescharacteristic, complemented
223#2223C2×C13714yescomplemented
224#2221C2×C13714yescomplemented
225#2161C2×C13711yescharacteristic, complemented
226#2148C2×C13711yescharacteristic, complemented
227C2×#361C2×C13711yescharacteristic, complemented
228#575—21no—
229C2×AGL(1,ℤ/8)—21no—
230C2×#135C22×C13711yescharacteristic
231C22×D8C22×C13711yescharacteristic
232C2×#131C22×C13711yescharacteristic
233#509—22no—
234#509—22no—
235C2×SD32C22×C13712yescomplemented
236C2×D16C22×C13712yescomplemented
237#504—22no—
238#504C22×C13714yescomplemented
239C2×M32—22no—
240C2×M32C22×C13711yescharacteristic
241C2×M32C22×C13711yescharacteristic
242#473C22×C13712yescomplemented
243#472—22no—
244#472C22×C13714yes—
245#472C22×C13712yescomplemented
246#471—22no—
247C2×#108C22×C13711yescharacteristic
248C2×#108C22×C13711yescharacteristic
249#413—21no—
250C2×#100—21no—
251#368C22×C13714yescomplemented
252#362C22×C13712yescomplemented
253#361C22×C13714yescomplemented
254#361C22×C13712yescomplemented
255C2×#41—21no—
256C2×#41D4×C13711yescharacteristic
257C22×D4—21no—
258#137—22no—
259#137—41no—
260AGL(1,ℤ/8)—24no—
261AGL(1,ℤ/8)—22no—
262AGL(1,ℤ/8)—42no—
263AGL(1,ℤ/8)—41no—
264#135—21no—
265#135—21no—
266#135—21no—
267#135—21no—
268#135—42no—
269C2×Q16C23×C13711yescharacteristic
270C2×SD16—21no—
271C2×SD16—41no—
272C2×D8—21no—
273C2×D8—21no—
274C2×D8—21no—
275C2×D8—21no—
276C2×D8—21no—
277C2×D8—41no—
278C2×D8C23×C13712yes—
279C2×D8C23×C13711yescharacteristic
280#131—21no—
281#131—21no—
282#131D4×C13712yes—
283#131C23×C13712yes—
284C2×M16—21no—
285C2×M16—21no—
286C2×M16—21no—
287C2×M16D4×C13711yescharacteristic
288C22×C8—21no—
289C22×C8C23×C13711yescharacteristic
290SD32—24no—
291D16—24no—
292M32—21no—
293M32—24no—
294M32D4×C13712yes—
295M32D4×C13712yes—
296M32C23×C13712yes—
297C2×C16—21no—
298C2×C16—24no—
299C2×C16C23×C13712yes—
300#108—21no—
301#108—21no—
302#108C23×C13712yes—
303#108C23×C13712yes—
304#101—22no—
305#100—24no—
306#100—22no—
307C24—41no—
308#41—21no—
309#41—21no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41D8×C13712yes—
315#41C2×D4×C13712yes—
316C2×Q8—21no—
317C2×Q8D8×C13711yescharacteristic
318C2×D4—21no—
319C2×D4—22no—
320C2×D4—21no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4D8×C13711yescharacteristic
328C22×C4—21no—
329C22×C4—41no—
330C22×C4C2×D4×C13711yescharacteristic
331Q16—21no—
332Q16—21no—
333Q16—42no—
334SD16—44no—
335SD16—42no—
336SD16—42no—
337SD16—41no—
338SD16—41no—
339D8—21no—
340D8—22no—
341D8—22no—
342D8—21no—
343D8—44no—
344D8—42no—
345D8—41no—
346D8—41no—
347M16—21no—
348M16—22no—
349M16—22no—
350M16—22no—
351M16—22no—
352M16—41no—
353M16—41no—
354M16C2×D4×C13712yes—
355C2×C8—21no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8C2×D4×C13711yescharacteristic
361C2×C8C2×D4×C13711yescharacteristic
362C2×C8C2×D4×C13711yescharacteristic
363C2×C8#41×C13711yescharacteristic
364C2×C8#41×C13711yescharacteristic
365C2×C8C24×C13711yescharacteristic
366C16—22no—
367C16C2×D4×C13714yes—
368C23—21no—
369C23—41no—
370C23—42no—
371C23—81no—
372C23—81no—
373C23—162no—
374C23#135×C13711yescharacteristic
375Q8—21no—
376Q8—41no—
377Q8—41no—
378Q8C2×D8×C13712yes—
379D4—22no—
380D4—22no—
381D4—21no—
382D4—42no—
383D4—41no—
384D4—42no—
385D4—41no—
386D4—81no—
387D4—81no—
388D4—81no—
389D4—81no—
390D4C2×D8×C13712yes—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—81no—
397C2×C4#121×C13711yescharacteristic
398C2×C4#121×C13711yescharacteristic
399C2×C4C2×D8×C13711yescharacteristic
400C2×C4C2×D8×C13711yescharacteristic
401C2×C4#135×C13711yescharacteristic
402C2×C4C22×D4×C13711yescharacteristic
403C8—22no—
404C8—42no—
405C8—42no—
406C8#121×C13711yescharacteristic
407C8#121×C13711yescharacteristic
408C8C22×D4×C13711yescharacteristic
409C8C2×#41×C13711yescharacteristic
410C22—22no—
411C22—41no—
412C22—81no—
413C22—81no—
414C22—82no—
415C22—162no—
416C22—161no—
417C22—162no—
418C22—162no—
419C22#466×C13711yescharacteristic
420C22#466×C13711yescharacteristic
421C22C2×#135×C13711yescharacteristic
422C4—22no—
423C4—81no—
424C4#466×C13711yescharacteristic
425C4#466×C13711yescharacteristic
426C4C2×#121×C13711yescharacteristic
427C4C22×D8×C13711yescharacteristic
428C2—21no—
429C2—21no—
430C2—21no—
431C2—42no—
432C2—162no—
433C2—161no—
434C2C2×#466×C13711yescharacteristic
subgroupG/H
center, socleC2×C137C2×#466
commutator, Frattini, cosocleC2×C8C24×C137
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C137 ◁ C22×C4×C137 ◁ C2×#131×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#509×C137C2semidirect
2. #3407×C137C2semidirect
3. #3067×C137C2semidirect
4. C2×#472×C137C2semidirect
5. #2223×C137C2semidirect
6. #2221×C137C2semidirect
7. #2161×C137C2semidirect
8. #2148×C137C2semidirect
9. C2×#361×C137C2semidirect
17. C2×SD32×C137C22semidirect
18. C2×D16×C137C22semidirect
20. #504×C137C22semidirect
24. #473×C137C22semidirect
27. #472×C137C22semidirect
33. #368×C137C22semidirect
34. #362×C137C22semidirect
35. #361×C137C22semidirect
36. #361×C137C22semidirect
217. #33618C137direct (N × H)
218. C137#33618direct (N × H)
219. C2×#509C2×C137semidirect
220. #3407C2×C137semidirect
221. #3067C2×C137semidirect
222. C2×#472C2×C137semidirect
223. #2223C2×C137semidirect
224. #2221C2×C137semidirect
225. #2161C2×C137semidirect
226. #2148C2×C137semidirect
227. C2×#361C2×C137semidirect
235. C2×SD32C22×C137semidirect
236. C2×D16C22×C137semidirect
238. #504C22×C137semidirect
242. #473C22×C137semidirect
245. #472C22×C137semidirect
251. #368C22×C137semidirect
252. #362C22×C137semidirect
253. #361C22×C137semidirect
254. #361C22×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#509×C1372
#3407×C1372
#3067×C1372
C2×#472×C1372
#2223×C1372
#2221×C1372
#2161×C1372
#2148×C1372
C2×#361×C1372
#33618137