Finite Group Databasebeta

← C137×#8954 #46858340

Subgroups — C137×#8954 #46858340

Order 35072.

566 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2988×C137C211yesmaximal, characteristic, complemented
2C4×AGL(1,ℤ/8)×C137C211yesmaximal, characteristic
3#1954×C137C211yesmaximal, characteristic, complemented
4#1478×C137C211yesmaximal, characteristic, complemented
5#1476×C137C211yesmaximal, characteristic, complemented
6#1283×C137C211yesmaximal, characteristic, complemented
7#1278×C137C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C137C411yescharacteristic
9C4×#41×C137C2211yescharacteristic
10C2×C4×D4×C137C2211yescharacteristic
11C8×Q8×C137—21no—
12#442×C137—21no—
13#440×C137—21no—
14C4×SD16×C137—21no—
15C4×D8×C137—21no—
16C8×D4×C137—22no—
17C8×D4×C137—21no—
18#431×C137—21no—
19#431×C137C2211yescharacteristic
20#431×C137C2211yescharacteristic
21#428×C137C411yescharacteristic
22C2×C4⋊C8×C137—21no—
23C2×C4⋊C8×C137C2211yescharacteristic
24#418×C137C411yescharacteristic
25#417×C137C411yescharacteristic
26C2×C22⋊C8×C137—22no—
27C4×M16×C137C2211yescharacteristic
28C2×C4×C8×C137C2211yescharacteristic
29Q8⋊C8×C137—21no—
30D4⋊C8×C137—21no—
31D4⋊C8×C137—21no—
32D4⋊C8×C137—21no—
33C2×#41×C137C2×C411yescharacteristic
34C22×D4×C137C2×C411yescharacteristic
35C23×C4×C137—21no—
36AGL(1,ℤ/8)×C137—22no—
37AGL(1,ℤ/8)×C137C814yescomplemented
38C2×SD16×C137—21no—
39C2×D8×C137—21no—
40C2×M16×C137C2×C411yescharacteristic
41C2×M16×C137C2×C411yescharacteristic
42C22×C8×C137—21no—
43C22×C8×C137—22no—
44C22×C8×C137D412yes—
45C4×Q8×C137D411yescharacteristic
46C4×Q8×C137D411yescharacteristic
47C4×D4×C137—21no—
48C4×D4×C137—21no—
49C4×D4×C137—21no—
50C4×D4×C137—21no—
51C4×D4×C137—21no—
52C4×D4×C137—21no—
53C4×D4×C137D411yescharacteristic
54C4×D4×C137D411yescharacteristic
55#117×C137—21no—
56#117×C137C2×C411yescharacteristic
57C2×C4⋊C4×C137C2×C411yescharacteristic
58C2×C22⋊C4×C137—21no—
59C2×C42×C137—21no—
60C2×C42×C137C2311yescharacteristic
61#107×C137—21no—
62#106×C137—21no—
63C4⋊C8×C137—22no—
64C4⋊C8×C137—21no—
65C4⋊C8×C137—21no—
66C4⋊C8×C137—21no—
67C4⋊C8×C137D411yescharacteristic
68C4⋊C8×C137D411yescharacteristic
69#103×C137—21no—
70#102×C137—21no—
71#102×C137—21no—
72#102×C137—21no—
73C22⋊C8×C137—22no—
74C22⋊C8×C137—21no—
75C22⋊C8×C137—21no—
76C22⋊C8×C137—44no—
77#97×C137—21no—
78C4×C8×C137—22no—
79C4×C8×C137—21no—
80C4×C8×C137—21no—
81C4×C8×C137—21no—
82C4×C8×C137—21no—
83C4×C8×C137—21no—
84C24×C137—21no—
85#41×C137—21no—
86#41×C137—21no—
87#41×C137C2×C812yes—
88#41×C137M1612yes—
89C2×Q8×C137C22⋊C411yescharacteristic
90C2×D4×C137—21no—
91C2×D4×C137—21no—
92C2×D4×C137—21no—
93C2×D4×C137—21no—
94C2×D4×C137—21no—
95C2×D4×C137—21no—
96C2×D4×C137C22⋊C411yescharacteristic
97C2×D4×C137C2×C812yescomplemented
98C22×C4×C137—21no—
99C22×C4×C137—21no—
100C22×C4×C137—21no—
101C22×C4×C137—21no—
102C22×C4×C137—42no—
103C22×C4×C137—42no—
104C22×C4×C137—41no—
105C22×C4×C137—41no—
106C22×C4×C137C22×C411yescharacteristic
107C22×C4×C137C2×D411yescharacteristic
108C22×C4×C137#4111yescharacteristic
109SD16×C137—24no—
110D8×C137—24no—
111M16×C137—21no—
112M16×C137—21no—
113M16×C137—21no—
114M16×C137C2×C812yescomplemented
115C2×C8×C137—21no—
116C2×C8×C137—22no—
117C2×C8×C137—21no—
118C2×C8×C137—21no—
119C2×C8×C137—22no—
120C2×C8×C137—21no—
121C2×C8×C137—41no—
122C2×C8×C137—41no—
123C2×C8×C137—42no—
124C2×C8×C137—41no—
125C2×C8×C137—82no—
126C4⋊C4×C137—21no—
127C4⋊C4×C137—21no—
128C4⋊C4×C137—21no—
129C4⋊C4×C137—21no—
130C4⋊C4×C137C22⋊C411yescharacteristic
131C4⋊C4×C137C22⋊C411yescharacteristic
132C22⋊C4×C137—21no—
133C22⋊C4×C137—21no—
134C22⋊C4×C137—21no—
135C22⋊C4×C137—41no—
136C22⋊C4×C137—41no—
137C42×C137—21no—
138C42×C137—21no—
139C42×C137—21no—
140C42×C137C2×D411yescharacteristic
141C42×C137C2×D411yescharacteristic
142C42×C137C2×D411yescharacteristic
143C42×C137#4111yescharacteristic
144C23×C137—21no—
145C23×C137—22no—
146C23×C137—42no—
147C23×C137—42no—
148C23×C137—41no—
149C23×C137—41no—
150C23×C137C4×D411yescharacteristic
151Q8×C137—21no—
152Q8×C137C22⋊C812yescomplemented
153D4×C137—22no—
154D4×C137—22no—
155D4×C137—22no—
156D4×C137—21no—
157D4×C137—22no—
158D4×C137—41no—
159D4×C137—41no—
160D4×C137C22⋊C812yescomplemented
161C2×C4×C137—21no—
162C2×C4×C137—22no—
163C2×C4×C137—22no—
164C2×C4×C137—21no—
165C2×C4×C137—21no—
166C2×C4×C137—21no—
167C2×C4×C137—21no—
168C2×C4×C137—41no—
169C2×C4×C137—41no—
170C2×C4×C137—41no—
171C2×C4×C137—42no—
172C2×C4×C137—41no—
173C2×C4×C137—81no—
174C2×C4×C137—82no—
175C2×C4×C137—81no—
176C2×C4×C137—81no—
177C2×C4×C137C2×C22⋊C411yescharacteristic
178C2×C4×C137C4×D411yescharacteristic
179C2×C4×C137#12011yescharacteristic
180C2×C4×C137#12111yescharacteristic
181C2×C4×C137#12111yescharacteristic
182C2×C4×C137#12311yescharacteristic
183C2×C4×C137C22×C811yescharacteristic
184C2×C4×C137C2×M1611yescharacteristic
185C2×C4×C137#13111yescharacteristic
186C2×C4×C137#13111yescharacteristic
187C8×C137—22no—
188C8×C137—42no—
189C8×C137—41no—
190C8×C137—42no—
191C8×C137—41no—
192C8×C137—41no—
193C8×C137—81no—
194C22×C137—21no—
195C22×C137—21no—
196C22×C137—22no—
197C22×C137—44no—
198C22×C137—41no—
199C22×C137—41no—
200C22×C137—42no—
201C22×C137—82no—
202C22×C137—81no—
203C22×C137—81no—
204C22×C137#38611yescharacteristic
205C22×C137C8×D411yescharacteristic
206C22×C137#43511yescharacteristic
207C4×C137—22no—
208C4×C137—21no—
209C4×C137—21no—
210C4×C137—21no—
211C4×C137—41no—
212C4×C137—41no—
213C4×C137—81no—
214C4×C137C2×C22⋊C811yescharacteristic
215C4×C137#40811yescharacteristic
216C4×C137C8×D411yescharacteristic
217C4×C137#43511yescharacteristic
218C4×C137#45311yescharacteristic
219C4×C137#45411yescharacteristic
220C2×C137—21no—
221C2×C137—21no—
222C2×C137—42no—
223C2×C137—82no—
224C2×C137#186911yescharacteristic
225C2×C137#188111yescharacteristic
226C2×C137#188311yescharacteristic
227#8954C13711yesmaximal, characteristic, direct product factor, Sylow
228C137#895411yescharacteristic, direct product factor, Sylow
229#2988C2×C13711yescharacteristic, complemented
230C4×AGL(1,ℤ/8)C2×C13711yescharacteristic
231#1954C2×C13711yescharacteristic, complemented
232#1478C2×C13711yescharacteristic, complemented
233#1476C2×C13711yescharacteristic, complemented
234#1283C2×C13711yescharacteristic, complemented
235#1278C2×C13711yescharacteristic, complemented
236C2×AGL(1,ℤ/8)C4×C13711yescharacteristic
237C4×#41C22×C13711yescharacteristic
238C2×C4×D4C22×C13711yescharacteristic
239C8×Q8—21no—
240#442—21no—
241#440—21no—
242C4×SD16—21no—
243C4×D8—21no—
244C8×D4—22no—
245C8×D4—21no—
246#431—21no—
247#431C22×C13711yescharacteristic
248#431C22×C13711yescharacteristic
249#428C4×C13711yescharacteristic
250C2×C4⋊C8—21no—
251C2×C4⋊C8C22×C13711yescharacteristic
252#418C4×C13711yescharacteristic
253#417C4×C13711yescharacteristic
254C2×C22⋊C8—22no—
255C4×M16C22×C13711yescharacteristic
256C2×C4×C8C22×C13711yescharacteristic
257Q8⋊C8—21no—
258D4⋊C8—21no—
259D4⋊C8—21no—
260D4⋊C8—21no—
261C2×#41C2×C4×C13711yescharacteristic
262C22×D4C2×C4×C13711yescharacteristic
263C23×C4—21no—
264AGL(1,ℤ/8)—22no—
265AGL(1,ℤ/8)C8×C13714yescomplemented
266C2×SD16—21no—
267C2×D8—21no—
268C2×M16C2×C4×C13711yescharacteristic
269C2×M16C2×C4×C13711yescharacteristic
270C22×C8—21no—
271C22×C8—22no—
272C22×C8D4×C13712yes—
273C4×Q8D4×C13711yescharacteristic
274C4×Q8D4×C13711yescharacteristic
275C4×D4—21no—
276C4×D4—21no—
277C4×D4—21no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4D4×C13711yescharacteristic
282C4×D4D4×C13711yescharacteristic
283#117—21no—
284#117C2×C4×C13711yescharacteristic
285C2×C4⋊C4C2×C4×C13711yescharacteristic
286C2×C22⋊C4—21no—
287C2×C42—21no—
288C2×C42C23×C13711yescharacteristic
289#107—21no—
290#106—21no—
291C4⋊C8—22no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8D4×C13711yescharacteristic
296C4⋊C8D4×C13711yescharacteristic
297#103—21no—
298#102—21no—
299#102—21no—
300#102—21no—
301C22⋊C8—22no—
302C22⋊C8—21no—
303C22⋊C8—21no—
304C22⋊C8—44no—
305#97—21no—
306C4×C8—22no—
307C4×C8—21no—
308C4×C8—21no—
309C4×C8—21no—
310C4×C8—21no—
311C4×C8—21no—
312C24—21no—
313#41—21no—
314#41—21no—
315#41C2×C8×C13712yes—
316#41M16×C13712yes—
317C2×Q8C22⋊C4×C13711yescharacteristic
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4C22⋊C4×C13711yescharacteristic
325C2×D4C2×C8×C13712yescomplemented
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4C22×C4×C13711yescharacteristic
335C22×C4C2×D4×C13711yescharacteristic
336C22×C4#41×C13711yescharacteristic
337SD16—24no—
338D8—24no—
339M16—21no—
340M16—21no—
341M16—21no—
342M16C2×C8×C13712yescomplemented
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—22no—
348C2×C8—21no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—42no—
352C2×C8—41no—
353C2×C8—82no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4C22⋊C4×C13711yescharacteristic
359C4⋊C4C22⋊C4×C13711yescharacteristic
360C22⋊C4—21no—
361C22⋊C4—21no—
362C22⋊C4—21no—
363C22⋊C4—41no—
364C22⋊C4—41no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42C2×D4×C13711yescharacteristic
369C42C2×D4×C13711yescharacteristic
370C42C2×D4×C13711yescharacteristic
371C42#41×C13711yescharacteristic
372C23—21no—
373C23—22no—
374C23—42no—
375C23—42no—
376C23—41no—
377C23—41no—
378C23C4×D4×C13711yescharacteristic
379Q8—21no—
380Q8C22⋊C8×C13712yescomplemented
381D4—22no—
382D4—22no—
383D4—22no—
384D4—21no—
385D4—22no—
386D4—41no—
387D4—41no—
388D4C22⋊C8×C13712yescomplemented
389C2×C4—21no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—82no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4C2×C22⋊C4×C13711yescharacteristic
406C2×C4C4×D4×C13711yescharacteristic
407C2×C4#120×C13711yescharacteristic
408C2×C4#121×C13711yescharacteristic
409C2×C4#121×C13711yescharacteristic
410C2×C4#123×C13711yescharacteristic
411C2×C4C22×C8×C13711yescharacteristic
412C2×C4C2×M16×C13711yescharacteristic
413C2×C4#131×C13711yescharacteristic
414C2×C4#131×C13711yescharacteristic
415C8—22no—
416C8—42no—
417C8—41no—
418C8—42no—
419C8—41no—
420C8—41no—
421C8—81no—
422C22—21no—
423C22—21no—
424C22—22no—
425C22—44no—
426C22—41no—
427C22—41no—
428C22—42no—
429C22—82no—
430C22—81no—
431C22—81no—
432C22#386×C13711yescharacteristic
433C22C8×D4×C13711yescharacteristic
434C22#435×C13711yescharacteristic
435C4—22no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—41no—
440C4—41no—
441C4—81no—
442C4C2×C22⋊C8×C13711yescharacteristic
443C4#408×C13711yescharacteristic
444C4C8×D4×C13711yescharacteristic
445C4#435×C13711yescharacteristic
446C4#453×C13711yescharacteristic
447C4#454×C13711yescharacteristic
448C2—21no—
449C2—21no—
450C2—42no—
451C2—82no—
452C2#1869×C13711yescharacteristic
453C2#1881×C13711yescharacteristic
454C2#1883×C13711yescharacteristic
subgroupG/H
centerC2×C4×C137#120
commutatorC2×C4C22×C8×C137
Frattini, cosocleC2×C42C23×C137
socleC22×C137#386
prime radicalC2×AGL(1,ℤ/8)×C137C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2988×C137C2semidirect
3. #1954×C137C2semidirect
4. #1478×C137C2semidirect
5. #1476×C137C2semidirect
6. #1283×C137C2semidirect
7. #1278×C137C2semidirect
37. AGL(1,ℤ/8)×C137C8semidirect
97. C2×D4×C137C2×C8semidirect
114. M16×C137C2×C8semidirect
152. Q8×C137C22⋊C8semidirect
160. D4×C137C22⋊C8semidirect
227. #8954C137direct (N × H)
228. C137#8954direct (N × H)
229. #2988C2×C137semidirect
231. #1954C2×C137semidirect
232. #1478C2×C137semidirect
233. #1476C2×C137semidirect
234. #1283C2×C137semidirect
235. #1278C2×C137semidirect
265. AGL(1,ℤ/8)C8×C137semidirect
325. C2×D4C2×C8×C137semidirect
342. M16C2×C8×C137semidirect
380. Q8C22⋊C8×C137semidirect
388. D4C22⋊C8×C137semidirect

Maximal subgroups

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

subgroupindex
#2988×C1372
C4×AGL(1,ℤ/8)×C1372
#1954×C1372
#1478×C1372
#1476×C1372
#1283×C1372
#1278×C1372
#8954137