Finite Group Databasebeta

← C137×#32972 #46882358

Subgroups — C137×#32972 #46882358

Order 35072.

1210 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×2−1+4×C137C211yesmaximal, characteristic
2#3079×C137C212yesmaximal, complemented
3#2567×C137C211yesmaximal, characteristic, complemented
4#1859×C137C214yesmaximal, complemented
5#1857×C137C212yesmaximal, complemented
6#1806×C137C212yesmaximal, complemented
7#1801×C137C211yesmaximal, characteristic, complemented
8#1793×C137C212yesmaximal
9C2×2−1+4×C137C2211yescharacteristic
10C2×C4⋊Q8×C137C2211yescharacteristic
11#520×C137—22no—
12#520×C137—22no—
13#520×C137C2212yes—
14#519×C137—22no—
15#519×C137C2211yescharacteristic
16C4×#41×C137—22no—
17C4×#41×C137—22no—
18C4×#41×C137C2212yes—
19C2×C4×Q8×C137—22no—
20C2×C4×Q8×C137C2211yescharacteristic
21#477×C137—22no—
22#476×C137—22no—
23#475×C137—22no—
24#474×C137—22no—
25#428×C137C2214yes—
26#423×C137C2212yes—
27C2×#104×C137C2214yes—
28#417×C137C2214yes—
29C2×#101×C137C2212yes—
30#395×C137—22no—
31#391×C137C2212yes—
32#389×C137—22no—
33#384×C137C2212yes—
34C4×C4⋊C4×C137C2211yescharacteristic
35#343×C137C2212yes—
36#339×C137C2214yes—
372−1+4×C137—24no—
382−1+4×C137—22no—
392−1+4×C137Q814yescomplemented
40C2×#41×C137—22no—
41C2×#41×C137—22no—
42C2×#41×C137C2312yes—
43C22×Q8×C137—22no—
44C22×Q8×C137—21no—
45C22×Q8×C137C2311yescharacteristic
46C2×M16×C137C2314yes—
47C4⋊Q8×C137—22no—
48C4⋊Q8×C137—22no—
49C4×Q8×C137—22no—
50C4×Q8×C137—24no—
51C4×Q8×C137—22no—
52C4×Q8×C137—22no—
53C4×Q8×C137—22no—
54C4×Q8×C137—22no—
55C4×Q8×C137—22no—
56C4×Q8×C137—22no—
57C4×Q8×C137D412yes—
58C4×Q8×C137D412yes—
59C4×D4×C137—22no—
60C4×D4×C137—24no—
61C4×D4×C137—22no—
62C4×D4×C137—22no—
63C4×D4×C137—22no—
64C4×D4×C137—24no—
65C4×D4×C137—22no—
66C4×D4×C137D412yes—
67C4×D4×C137D412yes—
68#117×C137—24no—
69#117×C137—24no—
70#117×C137—22no—
71#117×C137—22no—
72#117×C137—22no—
73#117×C137C2312yes—
74C2×C4⋊C4×C137—24no—
75C2×C4⋊C4×C137—21no—
76C2×C4⋊C4×C137—22no—
77C2×C4⋊C4×C137—21no—
78C2×C4⋊C4×C137—24no—
79C2×C4⋊C4×C137D412yes—
80C2×C4⋊C4×C137C2312yes—
81C2×C4⋊C4×C137C2311yescharacteristic
82C2×C42×C137—24no—
83C2×C42×C137—22no—
84C2×C42×C137D412yes—
85C2×C42×C137C2312yes—
86C2×C42×C137C2311yescharacteristic
87#107×C137—24no—
88#106×C137—24no—
89C4⋊C8×C137—22no—
90C4⋊C8×C137—22no—
91#104×C137—28no—
92#103×C137—24no—
93#102×C137—24no—
94#101×C137—24no—
95#95×C137—21no—
96#95×C137—24no—
97#41×C137—28no—
98#41×C137—28no—
99#41×C137—28no—
100#41×C137—28no—
101#41×C137—22no—
102#41×C137—22no—
103#41×C137C2×Q814yes—
104#41×C137#4114yes—
105C2×Q8×C137—22no—
106C2×Q8×C137—28no—
107C2×Q8×C137—28no—
108C2×Q8×C137—22no—
109C2×Q8×C137—22no—
110C2×Q8×C137—42no—
111C2×Q8×C137—82no—
112C2×Q8×C137C2×D412yes—
113C2×Q8×C137C2×Q812yes—
114C2×Q8×C137#4112yes—
115C2×D4×C137—22no—
116C2×D4×C137—22no—
117C2×D4×C137C2×D412yes—
118C22×C4×C137—24no—
119C22×C4×C137—22no—
120C22×C4×C137—22no—
121C22×C4×C137—24no—
122C22×C4×C137—22no—
123C22×C4×C137—22no—
124C22×C4×C137—22no—
125C22×C4×C137C2×D411yescharacteristic
126C22×C4×C137C2×D412yes—
127C22×C4×C137C2×D412yes—
128C22×C4×C137C2×D411yescharacteristic
129C22×C4×C137C2411yescharacteristic
130M16×C137—24no—
131M16×C137—24no—
132C2×C8×C137—24no—
133C4⋊C4×C137—24no—
134C4⋊C4×C137—24no—
135C4⋊C4×C137—22no—
136C4⋊C4×C137—22no—
137C4⋊C4×C137—22no—
138C4⋊C4×C137—24no—
139C4⋊C4×C137—22no—
140C4⋊C4×C137—22no—
141C4⋊C4×C137—24no—
142C4⋊C4×C137—41no—
143C4⋊C4×C137—41no—
144C4⋊C4×C137—84no—
145C4⋊C4×C137C2×D412yes—
146C4⋊C4×C137C2×D412yes—
147C22⋊C4×C137—24no—
148C22⋊C4×C137—22no—
149C22⋊C4×C137—22no—
150C22⋊C4×C137—22no—
151C42×C137—24no—
152C42×C137—24no—
153C42×C137—22no—
154C42×C137—22no—
155C42×C137—22no—
156C42×C137—22no—
157C42×C137—22no—
158C42×C137—24no—
159C42×C137C2×D412yes—
160C42×C137C2×D412yes—
161C23×C137—22no—
162C23×C137C22×D411yescharacteristic
163Q8×C137—28no—
164Q8×C137—24no—
165Q8×C137—22no—
166Q8×C137—24no—
167Q8×C137—84no—
168Q8×C137C22⋊Q814yes—
169D4×C137—28no—
170D4×C137—24no—
171D4×C137—24no—
172D4×C137—22no—
173D4×C137C22⋊Q814yes—
174C2×C4×C137—216no—
175C2×C4×C137—22no—
176C2×C4×C137—22no—
177C2×C4×C137—22no—
178C2×C4×C137—24no—
179C2×C4×C137—24no—
180C2×C4×C137—22no—
181C2×C4×C137—24no—
182C2×C4×C137—22no—
183C2×C4×C137—22no—
184C2×C4×C137—22no—
185C2×C4×C137—41no—
186C2×C4×C137—42no—
187C2×C4×C137—42no—
188C2×C4×C137—44no—
189C2×C4×C137—42no—
190C2×C4×C137—44no—
191C2×C4×C137—42no—
192C2×C4×C137—82no—
193C2×C4×C137—84no—
194C2×C4×C137#12011yescharacteristic
195C2×C4×C137#12012yes—
196C2×C4×C137#12011yescharacteristic
197C2×C4×C137C22⋊Q814yes—
198C2×C4×C137C22×D412yes—
199C2×C4×C137C22×Q811yescharacteristic
200C2×C4×C137C2×#4112yes—
201C2×C4×C137C2×#4111yescharacteristic
202C8×C137—44no—
203C22×C137—21no—
204C22×C137—21no—
205C22×C137—24no—
206C22×C137—42no—
207C22×C137—42no—
208C22×C137C2×#12011yescharacteristic
209C22×C137C2×C22⋊Q811yescharacteristic
210C22×C137#52511yescharacteristic
211C4×C137—24no—
212C4×C137—24no—
213C4×C137—21no—
214C4×C137—22no—
215C4×C137—44no—
216C4×C137—42no—
217C4×C137—42no—
218C4×C137—42no—
219C4×C137—82no—
220C4×C137#45612yes—
221C4×C137C2×C22⋊Q812yes—
222C4×C137#52512yes—
223C2×C137—21no—
224C2×C137—21no—
225C2×C137—44no—
226C2×C137#284811yescharacteristic
227C2×C137C2×#45611yescharacteristic
228C2×C137#302111yescharacteristic
229#32972C13711yesmaximal, characteristic, direct product factor, Sylow
230C137#3297211yescharacteristic, direct product factor, Sylow
231C4×2−1+4C2×C13711yescharacteristic
232#3079C2×C13712yescomplemented
233#2567C2×C13711yescharacteristic, complemented
234#1859C2×C13714yescomplemented
235#1857C2×C13712yescomplemented
236#1806C2×C13712yescomplemented
237#1801C2×C13711yescharacteristic, complemented
238#1793C2×C13712yes—
239C2×2−1+4C22×C13711yescharacteristic
240C2×C4⋊Q8C22×C13711yescharacteristic
241#520—22no—
242#520—22no—
243#520C22×C13712yes—
244#519—22no—
245#519C22×C13711yescharacteristic
246C4×#41—22no—
247C4×#41—22no—
248C4×#41C22×C13712yes—
249C2×C4×Q8—22no—
250C2×C4×Q8C22×C13711yescharacteristic
251#477—22no—
252#476—22no—
253#475—22no—
254#474—22no—
255#428C22×C13714yes—
256#423C22×C13712yes—
257C2×#104C22×C13714yes—
258#417C22×C13714yes—
259C2×#101C22×C13712yes—
260#395—22no—
261#391C22×C13712yes—
262#389—22no—
263#384C22×C13712yes—
264C4×C4⋊C4C22×C13711yescharacteristic
265#343C22×C13712yes—
266#339C22×C13714yes—
2672−1+4—24no—
2682−1+4—22no—
2692−1+4Q8×C13714yescomplemented
270C2×#41—22no—
271C2×#41—22no—
272C2×#41C23×C13712yes—
273C22×Q8—22no—
274C22×Q8—21no—
275C22×Q8C23×C13711yescharacteristic
276C2×M16C23×C13714yes—
277C4⋊Q8—22no—
278C4⋊Q8—22no—
279C4×Q8—22no—
280C4×Q8—24no—
281C4×Q8—22no—
282C4×Q8—22no—
283C4×Q8—22no—
284C4×Q8—22no—
285C4×Q8—22no—
286C4×Q8—22no—
287C4×Q8D4×C13712yes—
288C4×Q8D4×C13712yes—
289C4×D4—22no—
290C4×D4—24no—
291C4×D4—22no—
292C4×D4—22no—
293C4×D4—22no—
294C4×D4—24no—
295C4×D4—22no—
296C4×D4D4×C13712yes—
297C4×D4D4×C13712yes—
298#117—24no—
299#117—24no—
300#117—22no—
301#117—22no—
302#117—22no—
303#117C23×C13712yes—
304C2×C4⋊C4—24no—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—24no—
309C2×C4⋊C4D4×C13712yes—
310C2×C4⋊C4C23×C13712yes—
311C2×C4⋊C4C23×C13711yescharacteristic
312C2×C42—24no—
313C2×C42—22no—
314C2×C42D4×C13712yes—
315C2×C42C23×C13712yes—
316C2×C42C23×C13711yescharacteristic
317#107—24no—
318#106—24no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321#104—28no—
322#103—24no—
323#102—24no—
324#101—24no—
325#95—21no—
326#95—24no—
327#41—28no—
328#41—28no—
329#41—28no—
330#41—28no—
331#41—22no—
332#41—22no—
333#41C2×Q8×C13714yes—
334#41#41×C13714yes—
335C2×Q8—22no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—22no—
339C2×Q8—22no—
340C2×Q8—42no—
341C2×Q8—82no—
342C2×Q8C2×D4×C13712yes—
343C2×Q8C2×Q8×C13712yes—
344C2×Q8#41×C13712yes—
345C2×D4—22no—
346C2×D4—22no—
347C2×D4C2×D4×C13712yes—
348C22×C4—24no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4C2×D4×C13711yescharacteristic
356C22×C4C2×D4×C13712yes—
357C22×C4C2×D4×C13712yes—
358C22×C4C2×D4×C13711yescharacteristic
359C22×C4C24×C13711yescharacteristic
360M16—24no—
361M16—24no—
362C2×C8—24no—
363C4⋊C4—24no—
364C4⋊C4—24no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—24no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—84no—
375C4⋊C4C2×D4×C13712yes—
376C4⋊C4C2×D4×C13712yes—
377C22⋊C4—24no—
378C22⋊C4—22no—
379C22⋊C4—22no—
380C22⋊C4—22no—
381C42—24no—
382C42—24no—
383C42—22no—
384C42—22no—
385C42—22no—
386C42—22no—
387C42—22no—
388C42—24no—
389C42C2×D4×C13712yes—
390C42C2×D4×C13712yes—
391C23—22no—
392C23C22×D4×C13711yescharacteristic
393Q8—28no—
394Q8—24no—
395Q8—22no—
396Q8—24no—
397Q8—84no—
398Q8C22⋊Q8×C13714yes—
399D4—28no—
400D4—24no—
401D4—24no—
402D4—22no—
403D4C22⋊Q8×C13714yes—
404C2×C4—216no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—82no—
423C2×C4—84no—
424C2×C4#120×C13711yescharacteristic
425C2×C4#120×C13712yes—
426C2×C4#120×C13711yescharacteristic
427C2×C4C22⋊Q8×C13714yes—
428C2×C4C22×D4×C13712yes—
429C2×C4C22×Q8×C13711yescharacteristic
430C2×C4C2×#41×C13712yes—
431C2×C4C2×#41×C13711yescharacteristic
432C8—44no—
433C22—21no—
434C22—21no—
435C22—24no—
436C22—42no—
437C22—42no—
438C22C2×#120×C13711yescharacteristic
439C22C2×C22⋊Q8×C13711yescharacteristic
440C22#525×C13711yescharacteristic
441C4—24no—
442C4—24no—
443C4—21no—
444C4—22no—
445C4—44no—
446C4—42no—
447C4—42no—
448C4—42no—
449C4—82no—
450C4#456×C13712yes—
451C4C2×C22⋊Q8×C13712yes—
452C4#525×C13712yes—
453C2—21no—
454C2—21no—
455C2—44no—
456C2#2848×C13711yescharacteristic
457C2C2×#456×C13711yescharacteristic
458C2#3021×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC22×C4C24×C137
prime radicalC2×2−1+4×C137C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3079×C137C2semidirect
3. #2567×C137C2semidirect
4. #1859×C137C2semidirect
5. #1857×C137C2semidirect
6. #1806×C137C2semidirect
7. #1801×C137C2semidirect
39. 2−1+4×C137Q8semidirect
229. #32972C137direct (N × H)
230. C137#32972direct (N × H)
232. #3079C2×C137semidirect
233. #2567C2×C137semidirect
234. #1859C2×C137semidirect
235. #1857C2×C137semidirect
236. #1806C2×C137semidirect
237. #1801C2×C137semidirect
269. 2−1+4Q8×C137semidirect

Maximal subgroups

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

subgroupindex
C4×2−1+4×C1372
#3079×C1372
#2567×C1372
#1859×C1372
#1857×C1372
#1806×C1372
#1801×C1372
#1793×C1372
#32972137