Finite Group Databasebeta

← C137×#12622 #46862008

Subgroups — C137×#12622 #46862008

Order 35072.

636 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2321×C137C211yesmaximal, characteristic
2C2×AΣL(1,𝔽16)×C137C212yesmaximal
3C4×#100×C137C211yesmaximal, characteristic, complemented
4C4×#99×C137C211yesmaximal, characteristic, complemented
5#1390×C137C212yesmaximal, complemented
6C2×#120×C137C2211yescharacteristic
7C2×C4×D4×C137C2211yescharacteristic
8C22×C22⋊C4×C137C411yescharacteristic, complemented
9C2×#100×C137C2212yes—
10C2×#99×C137C2212yes—
11C4×M16×C137—21no—
12#390×C137—21no—
13#388×C137—21no—
14#388×C137—21no—
15#386×C137—21no—
16#386×C137C2211yescharacteristic
17#385×C137C411yescharacteristic, complemented
18#379×C137C411yescharacteristic, complemented
19C4×C22⋊C4×C137—21no—
20C4×C22⋊C4×C137C411yescharacteristic, complemented
21AΣL(1,𝔽16)×C137C418yescomplemented
22#343×C137—21no—
23#342×C137—21no—
24C25×C137C2×C411yescharacteristic
25C22×D4×C137—21no—
26C22×D4×C137C2311yescharacteristic
27C23×C4×C137C2×C411yescharacteristic
28C23×C4×C137C2×C411yescharacteristic
29C23×C4×C137D411yescharacteristic
30C2×M16×C137—22no—
31#120×C137—22no—
32#120×C137C2×C414yes—
33C4×D4×C137—41no—
34C4×D4×C137—41no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137D411yescharacteristic
37C2×C22⋊C4×C137—21no—
38C2×C22⋊C4×C137—22no—
39C2×C22⋊C4×C137—21no—
40C2×C22⋊C4×C137—21no—
41C2×C22⋊C4×C137—21no—
42C2×C22⋊C4×C137—21no—
43C2×C22⋊C4×C137—21no—
44C2×C22⋊C4×C137—41no—
45C2×C22⋊C4×C137—41no—
46C2×C22⋊C4×C137C2×C411yescharacteristic
47C2×C22⋊C4×C137C2×C411yescharacteristic
48C2×C22⋊C4×C137C2×C411yescharacteristic
49C2×C22⋊C4×C137D411yescharacteristic
50C2×C42×C137—21no—
51C2×C42×C137—41no—
52C2×C42×C137D411yescharacteristic
53#100×C137—22no—
54#100×C137C2×C414yes—
55#99×C137—22no—
56#99×C137C2×C414yes—
57#97×C137—41no—
58C4×C8×C137—41no—
59#95×C137—21no—
60#95×C137—21no—
61#95×C137—21no—
62#95×C137—41no—
63C24×C137—21no—
64C24×C137—21no—
65C24×C137—41no—
66C24×C137—41no—
67C24×C137—81no—
68C24×C137—81no—
69C24×C137C4212yescomplemented
70C24×C137C22×C411yescharacteristic
71C24×C137C2×D411yescharacteristic
72C2×D4×C137—24no—
73C2×D4×C137—41no—
74C2×D4×C137—41no—
75C2×D4×C137—81no—
76C2×D4×C137—81no—
77C2×D4×C137C22×C412yes—
78C2×D4×C137#4112yes—
79C22×C4×C137—21no—
80C22×C4×C137—21no—
81C22×C4×C137—21no—
82C22×C4×C137—21no—
83C22×C4×C137—21no—
84C22×C4×C137—21no—
85C22×C4×C137—21no—
86C22×C4×C137—42no—
87C22×C4×C137—41no—
88C22×C4×C137—81no—
89C22×C4×C137—81no—
90C22×C4×C137—81no—
91C22×C4×C137C22⋊C411yescharacteristic
92C22×C4×C137C22⋊C411yescharacteristic
93C22×C4×C137C22⋊C411yescharacteristic
94C22×C4×C137C22⋊C411yescharacteristic
95C22×C4×C137C2×D411yescharacteristic
96M16×C137—24no—
97M16×C137—42no—
98C2×C8×C137—42no—
99C4⋊C4×C137—41no—
100C4⋊C4×C137—81no—
101C22⋊C4×C137—21no—
102C22⋊C4×C137—24no—
103C22⋊C4×C137—24no—
104C22⋊C4×C137—41no—
105C22⋊C4×C137—41no—
106C22⋊C4×C137—42no—
107C22⋊C4×C137—42no—
108C22⋊C4×C137—41no—
109C22⋊C4×C137—41no—
110C22⋊C4×C137—41no—
111C22⋊C4×C137—81no—
112C22⋊C4×C137—81no—
113C22⋊C4×C137—81no—
114C22⋊C4×C137C4212yescomplemented
115C42×C137—21no—
116C42×C137—21no—
117C42×C137—81no—
118C42×C137—82no—
119C23×C137—21no—
120C23×C137—21no—
121C23×C137—21no—
122C23×C137—21no—
123C23×C137—22no—
124C23×C137—41no—
125C23×C137—41no—
126C23×C137—41no—
127C23×C137—41no—
128C23×C137—41no—
129C23×C137—41no—
130C23×C137—41no—
131C23×C137—41no—
132C23×C137—42no—
133C23×C137—42no—
134C23×C137—41no—
135C23×C137—41no—
136C23×C137—82no—
137C23×C137—82no—
138C23×C137—82no—
139C23×C137—82no—
140C23×C137—84no—
141C23×C137C2×C4211yescharacteristic
142C23×C137C2×C22⋊C411yescharacteristic
143C23×C137#11711yescharacteristic
144C23×C137C4×D412yes—
145D4×C137—42no—
146D4×C137—42no—
147D4×C137—84no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—22no—
151C2×C4×C137—21no—
152C2×C4×C137—21no—
153C2×C4×C137—21no—
154C2×C4×C137—41no—
155C2×C4×C137—42no—
156C2×C4×C137—41no—
157C2×C4×C137—41no—
158C2×C4×C137—41no—
159C2×C4×C137—41no—
160C2×C4×C137—41no—
161C2×C4×C137—41no—
162C2×C4×C137—44no—
163C2×C4×C137—41no—
164C2×C4×C137—41no—
165C2×C4×C137—41no—
166C2×C4×C137—41no—
167C2×C4×C137—41no—
168C2×C4×C137—81no—
169C2×C4×C137—81no—
170C2×C4×C137—81no—
171C2×C4×C137—82no—
172C2×C4×C137—82no—
173C2×C4×C137—82no—
174C2×C4×C137#9911yescharacteristic
175C2×C4×C137#9911yescharacteristic
176C2×C4×C137C4×D412yes—
177C8×C137—44no—
178C22×C137—21no—
179C22×C137—21no—
180C22×C137—22no—
181C22×C137—21no—
182C22×C137—21no—
183C22×C137—42no—
184C22×C137—41no—
185C22×C137—41no—
186C22×C137—42no—
187C22×C137—42no—
188C22×C137—42no—
189C22×C137—41no—
190C22×C137—41no—
191C22×C137—41no—
192C22×C137—41no—
193C22×C137—42no—
194C22×C137—81no—
195C22×C137—81no—
196C22×C137—81no—
197C22×C137—84no—
198C22×C137—84no—
199C22×C137—81no—
200C22×C137—81no—
201C22×C137C4×C22⋊C411yescharacteristic
202C22×C137C2×#9911yescharacteristic
203C22×C137#41011yescharacteristic
204C4×C137—21no—
205C4×C137—21no—
206C4×C137—22no—
207C4×C137—42no—
208C4×C137—41no—
209C4×C137—41no—
210C4×C137—41no—
211C4×C137—81no—
212C4×C137—84no—
213C4×C137—81no—
214C2×C137—21no—
215C2×C137—21no—
216C2×C137—42no—
217C2×C137—41no—
218C2×C137—41no—
219C2×C137—82no—
220C2×C137C4×#9911yescharacteristic
221C2×C137#212111yescharacteristic
222C2×C137#212211yescharacteristic
223#12622C13711yesmaximal, characteristic, direct product factor, Sylow
224C137#1262211yescharacteristic, direct product factor, Sylow
225#2321C2×C13711yescharacteristic
226C2×AΣL(1,𝔽16)C2×C13712yes—
227C4×#100C2×C13711yescharacteristic, complemented
228C4×#99C2×C13711yescharacteristic, complemented
229#1390C2×C13712yescomplemented
230C2×#120C22×C13711yescharacteristic
231C2×C4×D4C22×C13711yescharacteristic
232C22×C22⋊C4C4×C13711yescharacteristic, complemented
233C2×#100C22×C13712yes—
234C2×#99C22×C13712yes—
235C4×M16—21no—
236#390—21no—
237#388—21no—
238#388—21no—
239#386—21no—
240#386C22×C13711yescharacteristic
241#385C4×C13711yescharacteristic, complemented
242#379C4×C13711yescharacteristic, complemented
243C4×C22⋊C4—21no—
244C4×C22⋊C4C4×C13711yescharacteristic, complemented
245AΣL(1,𝔽16)C4×C13718yescomplemented
246#343—21no—
247#342—21no—
248C25C2×C4×C13711yescharacteristic
249C22×D4—21no—
250C22×D4C23×C13711yescharacteristic
251C23×C4C2×C4×C13711yescharacteristic
252C23×C4C2×C4×C13711yescharacteristic
253C23×C4D4×C13711yescharacteristic
254C2×M16—22no—
255#120—22no—
256#120C2×C4×C13714yes—
257C4×D4—41no—
258C4×D4—41no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4D4×C13711yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—21no—
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—41no—
269C2×C22⋊C4—41no—
270C2×C22⋊C4C2×C4×C13711yescharacteristic
271C2×C22⋊C4C2×C4×C13711yescharacteristic
272C2×C22⋊C4C2×C4×C13711yescharacteristic
273C2×C22⋊C4D4×C13711yescharacteristic
274C2×C42—21no—
275C2×C42—41no—
276C2×C42D4×C13711yescharacteristic
277#100—22no—
278#100C2×C4×C13714yes—
279#99—22no—
280#99C2×C4×C13714yes—
281#97—41no—
282C4×C8—41no—
283#95—21no—
284#95—21no—
285#95—21no—
286#95—41no—
287C24—21no—
288C24—21no—
289C24—41no—
290C24—41no—
291C24—81no—
292C24—81no—
293C24C42×C13712yescomplemented
294C24C22×C4×C13711yescharacteristic
295C24C2×D4×C13711yescharacteristic
296C2×D4—24no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4C22×C4×C13712yes—
302C2×D4#41×C13712yes—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4C22⋊C4×C13711yescharacteristic
316C22×C4C22⋊C4×C13711yescharacteristic
317C22×C4C22⋊C4×C13711yescharacteristic
318C22×C4C22⋊C4×C13711yescharacteristic
319C22×C4C2×D4×C13711yescharacteristic
320M16—24no—
321M16—42no—
322C2×C8—42no—
323C4⋊C4—41no—
324C4⋊C4—81no—
325C22⋊C4—21no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—41no—
329C22⋊C4—41no—
330C22⋊C4—42no—
331C22⋊C4—42no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—81no—
336C22⋊C4—81no—
337C22⋊C4—81no—
338C22⋊C4C42×C13712yescomplemented
339C42—21no—
340C42—21no—
341C42—81no—
342C42—82no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—22no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23—42no—
358C23—41no—
359C23—41no—
360C23—82no—
361C23—82no—
362C23—82no—
363C23—82no—
364C23—84no—
365C23C2×C42×C13711yescharacteristic
366C23C2×C22⋊C4×C13711yescharacteristic
367C23#117×C13711yescharacteristic
368C23C4×D4×C13712yes—
369D4—42no—
370D4—42no—
371D4—84no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—22no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—44no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—81no—
393C2×C4—81no—
394C2×C4—81no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—82no—
398C2×C4#99×C13711yescharacteristic
399C2×C4#99×C13711yescharacteristic
400C2×C4C4×D4×C13712yes—
401C8—44no—
402C22—21no—
403C22—21no—
404C22—22no—
405C22—21no—
406C22—21no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—42no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—84no—
422C22—84no—
423C22—81no—
424C22—81no—
425C22C4×C22⋊C4×C13711yescharacteristic
426C22C2×#99×C13711yescharacteristic
427C22#410×C13711yescharacteristic
428C4—21no—
429C4—21no—
430C4—22no—
431C4—42no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—84no—
437C4—81no—
438C2—21no—
439C2—21no—
440C2—42no—
441C2—41no—
442C2—41no—
443C2—82no—
444C2C4×#99×C13711yescharacteristic
445C2#2121×C13711yescharacteristic
446C2#2122×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC23C2×C42×C137
Frattini, cosocleC22×D4C23×C137
prime radicalC2×#120×C137C22
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
3. C4×#100×C137C2semidirect
4. C4×#99×C137C2semidirect
5. #1390×C137C2semidirect
8. C22×C22⋊C4×C137C4semidirect
17. #385×C137C4semidirect
18. #379×C137C4semidirect
20. C4×C22⋊C4×C137C4semidirect
21. AΣL(1,𝔽16)×C137C4semidirect
69. C24×C137C42semidirect
114. C22⋊C4×C137C42semidirect
223. #12622C137direct (N × H)
224. C137#12622direct (N × H)
227. C4×#100C2×C137semidirect
228. C4×#99C2×C137semidirect
229. #1390C2×C137semidirect
232. C22×C22⋊C4C4×C137semidirect
241. #385C4×C137semidirect
242. #379C4×C137semidirect
244. C4×C22⋊C4C4×C137semidirect
245. AΣL(1,𝔽16)C4×C137semidirect
293. C24C42×C137semidirect
338. C22⋊C4C42×C137semidirect

Maximal subgroups

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

subgroupindex
#2321×C1372
C2×AΣL(1,𝔽16)×C1372
C4×#100×C1372
C4×#99×C1372
#1390×C1372
#12622137