Finite Group Databasebeta

← C137×#10064 #46859450

Subgroups — C137×#10064 #46859450

Order 35072.

614 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#487×C137C211yesmaximal, characteristic, complemented
2C2×#461×C137C211yesmaximal, characteristic
3#2563×C137C211yesmaximal, characteristic
4#1988×C137C211yesmaximal, characteristic, complemented
5C2×#333×C137C211yesmaximal, characteristic, complemented
6C2×D4⋊C8×C137C211yesmaximal, characteristic
7#1279×C137C211yesmaximal, characteristic, complemented
8C22×SD16×C137—21no—
9C2×C4⋊Q8×C137C2211yescharacteristic
10C2×C4⋊Q8×C137C2211yescharacteristic
11C2×C4×D4×C137C2211yescharacteristic
12#487×C137—22no—
13#487×C137—22no—
14#461×C137—22no—
15#461×C137—22no—
16C2×C4⋊C8×C137C2211yescharacteristic
17C2×C4⋊C8×C137C2211yescharacteristic
18C2×#103×C137—22no—
19C2×#103×C137—21no—
20C2×C4×C8×C137C2211yescharacteristic
21#393×C137—22no—
22#390×C137—21no—
23#382×C137—21no—
24#380×C137—22no—
25C4×C4⋊C4×C137C2211yescharacteristic
26#340×C137—22no—
27#333×C137—22no—
28D4⋊C8×C137—22no—
29C22×Q8×C137—21no—
30C22×Q8×C137D412yes—
31C22×D4×C137D411yescharacteristic
32C23×C4×C137—21no—
33C2×SD16×C137—24no—
34C2×SD16×C137—42no—
35C2×SD16×C137—42no—
36C22×C8×C137—21no—
37C22×C8×C137—21no—
38C22×C8×C137—21no—
39C4⋊Q8×C137—21no—
40C4⋊Q8×C137—21no—
41C4⋊Q8×C137—21no—
42C4⋊Q8×C137—21no—
43C4⋊Q8×C137—21no—
44C4⋊Q8×C137—21no—
45C4⋊Q8×C137—21no—
46C4⋊Q8×C137D412yes—
47C4×D4×C137—21no—
48C4×D4×C137—21no—
49C4×D4×C137—21no—
50C4×D4×C137—21no—
51C2×C4⋊C4×C137—21no—
52C2×C4⋊C4×C137—21no—
53C2×C4⋊C4×C137—21no—
54C2×C4⋊C4×C137—21no—
55C2×C4⋊C4×C137D412yes—
56C2×C4⋊C4×C137D411yescharacteristic
57C2×C22⋊C4×C137—21no—
58C2×C22⋊C4×C137—24no—
59C2×C42×C137—21no—
60C2×C42×C137C2311yescharacteristic
61C4⋊C8×C137—21no—
62C4⋊C8×C137—21no—
63C4⋊C8×C137—21no—
64C4⋊C8×C137D412yes—
65#103×C137—24no—
66#103×C137—28no—
67C4×C8×C137—21no—
68C4×C8×C137D412yes—
69#95×C137—24no—
70#95×C137—21no—
71C24×C137—21no—
72C2×Q8×C137—22no—
73C2×Q8×C137—22no—
74C2×Q8×C137—22no—
75C2×Q8×C137—44no—
76C2×Q8×C137—41no—
77C2×Q8×C137—41no—
78C2×Q8×C137—41no—
79C2×Q8×C137—41no—
80C2×Q8×C137—41no—
81C2×D4×C137—21no—
82C2×D4×C137—21no—
83C2×D4×C137—21no—
84C2×D4×C137—21no—
85C2×D4×C137—21no—
86C2×D4×C137SD1612yes—
87C22×C4×C137—21no—
88C22×C4×C137—22no—
89C22×C4×C137—22no—
90C22×C4×C137—41no—
91C22×C4×C137—41no—
92C22×C4×C137—41no—
93C22×C4×C137—42no—
94C22×C4×C137—42no—
95C22×C4×C137C2×D411yescharacteristic
96C22×C4×C137C2×D411yescharacteristic
97C22×C4×C137C2×D411yescharacteristic
98SD16×C137—48no—
99C2×C8×C137—22no—
100C2×C8×C137—22no—
101C2×C8×C137—42no—
102C2×C8×C137—41no—
103C2×C8×C137—41no—
104C2×C8×C137—41no—
105C2×C8×C137—42no—
106C4⋊C4×C137—22no—
107C4⋊C4×C137—21no—
108C4⋊C4×C137—22no—
109C4⋊C4×C137—21no—
110C4⋊C4×C137—22no—
111C4⋊C4×C137—22no—
112C4⋊C4×C137—41no—
113C4⋊C4×C137—41no—
114C4⋊C4×C137—41no—
115C4⋊C4×C137—41no—
116C4⋊C4×C137—41no—
117C4⋊C4×C137—41no—
118C22⋊C4×C137—41no—
119C22⋊C4×C137—41no—
120C22⋊C4×C137—84no—
121C42×C137—41no—
122C42×C137—41no—
123C42×C137C2×D411yescharacteristic
124C42×C137C2×D411yescharacteristic
125C42×C137C2×D411yescharacteristic
126C42×C137#4111yescharacteristic
127C23×C137—22no—
128C23×C137—41no—
129C23×C137—41no—
130C23×C137—42no—
131C23×C137—42no—
132C23×C137#12011yescharacteristic
133Q8×C137—44no—
134Q8×C137—42no—
135Q8×C137—42no—
136Q8×C137—82no—
137Q8×C137—81no—
138Q8×C137—81no—
139D4×C137—22no—
140D4×C137—22no—
141D4×C137—41no—
142D4×C137—41no—
143C2×C4×C137—22no—
144C2×C4×C137—41no—
145C2×C4×C137—42no—
146C2×C4×C137—42no—
147C2×C4×C137—42no—
148C2×C4×C137—42no—
149C2×C4×C137—42no—
150C2×C4×C137—42no—
151C2×C4×C137—44no—
152C2×C4×C137—81no—
153C2×C4×C137—81no—
154C2×C4×C137—82no—
155C2×C4×C137—81no—
156C2×C4×C137—81no—
157C2×C4×C137#12011yescharacteristic
158C2×C4×C137#12011yescharacteristic
159C2×C4×C137#12111yescharacteristic
160C2×C4×C137#12111yescharacteristic
161C2×C4×C137#12111yescharacteristic
162C2×C4×C137#12711yescharacteristic
163C2×C4×C137C2×SD1611yescharacteristic
164C2×C4×C137#13511yescharacteristic
165C2×C4×C137#13512yes—
166C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
167C2×C4×C137#13711yescharacteristic
168C2×C4×C137#13712yes—
169C8×C137—42no—
170C8×C137—42no—
171C8×C137—81no—
172C22×C137—44no—
173C22×C137—42no—
174C22×C137—82no—
175C22×C137—81no—
176C22×C137—81no—
177C22×C137#39211yescharacteristic
178C22×C137#44811yescharacteristic
179C22×C137#44911yescharacteristic
180C22×C137#45212yes—
181C22×C137#45512yes—
182C4×C137—21no—
183C4×C137—21no—
184C4×C137—82no—
185C4×C137—82no—
186C4×C137—82no—
187C4×C137—81no—
188C4×C137#45011yescharacteristic
189C4×C137#45211yescharacteristic
190C4×C137#46412yes—
191C4×C137#46511yescharacteristic
192C4×C137#46811yescharacteristic
193C4×C137#49511yescharacteristic
194C4×C137#49711yescharacteristic
195C2×C137—82no—
196C2×C137#164312yes—
197C2×C137#164612yes—
198C2×C137#200011yescharacteristic
199C2×C137#200411yescharacteristic
200C2×C137#201511yescharacteristic
201#10064C13711yesmaximal, characteristic, direct product factor, Sylow
202C137#1006411yescharacteristic, direct product factor, Sylow
203C2×#487C2×C13711yescharacteristic, complemented
204C2×#461C2×C13711yescharacteristic
205#2563C2×C13711yescharacteristic
206#1988C2×C13711yescharacteristic, complemented
207C2×#333C2×C13711yescharacteristic, complemented
208C2×D4⋊C8C2×C13711yescharacteristic
209#1279C2×C13711yescharacteristic, complemented
210C22×SD16—21no—
211C2×C4⋊Q8C22×C13711yescharacteristic
212C2×C4⋊Q8C22×C13711yescharacteristic
213C2×C4×D4C22×C13711yescharacteristic
214#487—22no—
215#487—22no—
216#461—22no—
217#461—22no—
218C2×C4⋊C8C22×C13711yescharacteristic
219C2×C4⋊C8C22×C13711yescharacteristic
220C2×#103—22no—
221C2×#103—21no—
222C2×C4×C8C22×C13711yescharacteristic
223#393—22no—
224#390—21no—
225#382—21no—
226#380—22no—
227C4×C4⋊C4C22×C13711yescharacteristic
228#340—22no—
229#333—22no—
230D4⋊C8—22no—
231C22×Q8—21no—
232C22×Q8D4×C13712yes—
233C22×D4D4×C13711yescharacteristic
234C23×C4—21no—
235C2×SD16—24no—
236C2×SD16—42no—
237C2×SD16—42no—
238C22×C8—21no—
239C22×C8—21no—
240C22×C8—21no—
241C4⋊Q8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—21no—
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8D4×C13712yes—
249C4×D4—21no—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C13711yescharacteristic
258C2×C4⋊C4D4×C13712yes—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—24no—
261C2×C42—21no—
262C2×C42C23×C13711yescharacteristic
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8D4×C13712yes—
267#103—24no—
268#103—28no—
269C4×C8—21no—
270C4×C8D4×C13712yes—
271#95—24no—
272#95—21no—
273C24—21no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—44no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4SD16×C13712yes—
289C22×C4—21no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4C2×D4×C13711yescharacteristic
298C22×C4C2×D4×C13711yescharacteristic
299C22×C4C2×D4×C13711yescharacteristic
300SD16—48no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—42no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—42no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—22no—
313C4⋊C4—22no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C22⋊C4—41no—
321C22⋊C4—41no—
322C22⋊C4—84no—
323C42—41no—
324C42—41no—
325C42C2×D4×C13711yescharacteristic
326C42C2×D4×C13711yescharacteristic
327C42C2×D4×C13711yescharacteristic
328C42#41×C13711yescharacteristic
329C23—22no—
330C23—41no—
331C23—41no—
332C23—42no—
333C23—42no—
334C23#120×C13711yescharacteristic
335Q8—44no—
336Q8—42no—
337Q8—42no—
338Q8—82no—
339Q8—81no—
340Q8—81no—
341D4—22no—
342D4—22no—
343D4—41no—
344D4—41no—
345C2×C4—22no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—82no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4#120×C13711yescharacteristic
360C2×C4#120×C13711yescharacteristic
361C2×C4#121×C13711yescharacteristic
362C2×C4#121×C13711yescharacteristic
363C2×C4#121×C13711yescharacteristic
364C2×C4#127×C13711yescharacteristic
365C2×C4C2×SD16×C13711yescharacteristic
366C2×C4#135×C13711yescharacteristic
367C2×C4#135×C13712yes—
368C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
369C2×C4#137×C13711yescharacteristic
370C2×C4#137×C13712yes—
371C8—42no—
372C8—42no—
373C8—81no—
374C22—44no—
375C22—42no—
376C22—81no—
377C22—82no—
378C22—81no—
379C22#392×C13711yescharacteristic
380C22#448×C13711yescharacteristic
381C22#449×C13711yescharacteristic
382C22#452×C13712yes—
383C22#455×C13712yes—
384C4—21no—
385C4—21no—
386C4—81no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4#450×C13711yescharacteristic
391C4#452×C13711yescharacteristic
392C4#464×C13712yes—
393C4#465×C13711yescharacteristic
394C4#468×C13711yescharacteristic
395C4#495×C13711yescharacteristic
396C4#497×C13711yescharacteristic
397C2—82no—
398C2#1643×C13712yes—
399C2#1646×C13712yes—
400C2#2000×C13711yescharacteristic
401C2#2004×C13711yescharacteristic
402C2#2015×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#120
commutator, Frattini, cosocleC2×C42C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#487×C137C2semidirect
4. #1988×C137C2semidirect
5. C2×#333×C137C2semidirect
7. #1279×C137C2semidirect
201. #10064C137direct (N × H)
202. C137#10064direct (N × H)
203. C2×#487C2×C137semidirect
206. #1988C2×C137semidirect
207. C2×#333C2×C137semidirect
209. #1279C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#487×C1372
C2×#461×C1372
#2563×C1372
#1988×C1372
C2×#333×C1372
C2×D4⋊C8×C1372
#1279×C1372
#10064137