Finite Group Databasebeta

← C137×#9215 #46858601

Subgroups — C137×#9215 #46858601

Order 35072.

578 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#476×C137C211yesmaximal, characteristic
2C2×C8×D4×C137C211yesmaximal, characteristic
3#1990×C137C211yesmaximal, characteristic, complemented
4#1802×C137C211yesmaximal, characteristic
5C2×#333×C137C211yesmaximal, characteristic, complemented
6#1296×C137C211yesmaximal, characteristic, complemented
7#1277×C137C211yesmaximal, characteristic, complemented
8C23×C8×C137—21no—
9C2×C4⋊Q8×C137C2211yescharacteristic
10C2×C4×D4×C137C2211yescharacteristic
11#476×C137—22no—
12#476×C137C414yescomplemented
13C8×D4×C137—22no—
14C8×D4×C137—22no—
15C2×#106×C137—21no—
16C2×C4⋊C8×C137C2211yescharacteristic
17C2×C4⋊C8×C137C2211yescharacteristic
18C2×C4⋊C8×C137C2211yescharacteristic
19C2×#102×C137—21no—
20C2×#102×C137—21no—
21C2×C22⋊C8×C137—21no—
22C2×C4×C8×C137C2211yescharacteristic
23#384×C137C2211yescharacteristic
24#340×C137—21no—
25#340×C137—21no—
26#340×C137—21no—
27#333×C137—22no—
28C22×Q8×C137—21no—
29C22×D4×C137D411yescharacteristic
30C23×C4×C137—21no—
31C22×C8×C137—24no—
32C22×C8×C137—21no—
33C22×C8×C137—21no—
34C22×C8×C137—21no—
35C22×C8×C137—42no—
36C22×C8×C137—42no—
37C22×C8×C137D411yescharacteristic
38C22×C8×C137D411yescharacteristic
39C4⋊Q8×C137—21no—
40C4⋊Q8×C137—21no—
41C4⋊Q8×C137—21no—
42C4⋊Q8×C137C2×C412yes—
43C4×D4×C137—21no—
44C4×D4×C137—21no—
45C4×D4×C137C2×C412yes—
46C4×D4×C137D412yes—
47C2×C4⋊C4×C137—21no—
48C2×C4⋊C4×C137—21no—
49C2×C4⋊C4×C137—21no—
50C2×C4⋊C4×C137—21no—
51C2×C4⋊C4×C137—21no—
52C2×C4⋊C4×C137D411yescharacteristic
53C2×C22⋊C4×C137—21no—
54C2×C42×C137C2311yescharacteristic
55#106×C137—24no—
56C4⋊C8×C137—21no—
57C4⋊C8×C137—21no—
58C4⋊C8×C137—21no—
59C4⋊C8×C137—21no—
60C4⋊C8×C137C2×C412yes—
61C4⋊C8×C137D412yes—
62#102×C137—24no—
63#102×C137—42no—
64C22⋊C8×C137—24no—
65C4×C8×C137—21no—
66C4×C8×C137—21no—
67C24×C137—21no—
68C2×Q8×C137—22no—
69C2×Q8×C137—41no—
70C2×Q8×C137—81no—
71C2×Q8×C137—81no—
72C2×D4×C137—21no—
73C2×D4×C137—21no—
74C2×D4×C137—21no—
75C2×D4×C137C22⋊C412yes—
76C2×D4×C137SD1612yes—
77C2×D4×C137Q1612yes—
78C22×C4×C137—21no—
79C22×C4×C137—22no—
80C22×C4×C137—22no—
81C22×C4×C137—41no—
82C22×C4×C137—41no—
83C22×C4×C137—41no—
84C22×C4×C137—41no—
85C22×C4×C137—41no—
86C22×C4×C137—41no—
87C22×C4×C137C2×D411yescharacteristic
88C22×C4×C137C2×D411yescharacteristic
89C22×C4×C137#4111yescharacteristic
90C2×C8×C137—21no—
91C2×C8×C137—21no—
92C2×C8×C137—22no—
93C2×C8×C137—21no—
94C2×C8×C137—22no—
95C2×C8×C137—21no—
96C2×C8×C137—21no—
97C2×C8×C137—48no—
98C2×C8×C137—41no—
99C2×C8×C137—41no—
100C2×C8×C137—41no—
101C2×C8×C137—41no—
102C2×C8×C137—41no—
103C2×C8×C137—41no—
104C2×C8×C137—41no—
105C2×C8×C137SD1612yes—
106C4⋊C4×C137—21no—
107C4⋊C4×C137—22no—
108C4⋊C4×C137—22no—
109C4⋊C4×C137—41no—
110C4⋊C4×C137—41no—
111C4⋊C4×C137—41no—
112C4⋊C4×C137—41no—
113C4⋊C4×C137—41no—
114C4⋊C4×C137—41no—
115C4⋊C4×C137—81no—
116C4⋊C4×C137C22⋊C412yes—
117C22⋊C4×C137—22no—
118C22⋊C4×C137—22no—
119C42×C137C22×C411yescharacteristic
120C42×C137C2×D411yescharacteristic
121C42×C137C2×D411yescharacteristic
122C42×C137#4111yescharacteristic
123C23×C137—22no—
124C23×C137—22no—
125C23×C137—22no—
126C23×C137—41no—
127C23×C137—41no—
128C23×C137—41no—
129C23×C137—41no—
130C23×C137#12111yescharacteristic
131Q8×C137—82no—
132Q8×C137—82no—
133D4×C137—22no—
134D4×C137—22no—
135D4×C137—22no—
136D4×C137#10314yes—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—22no—
140C2×C4×C137—41no—
141C2×C4×C137—41no—
142C2×C4×C137—42no—
143C2×C4×C137—42no—
144C2×C4×C137—42no—
145C2×C4×C137—42no—
146C2×C4×C137—42no—
147C2×C4×C137—81no—
148C2×C4×C137—81no—
149C2×C4×C137—81no—
150C2×C4×C137—81no—
151C2×C4×C137—81no—
152C2×C4×C137C2×C22⋊C411yescharacteristic
153C2×C4×C137C4×D411yescharacteristic
154C2×C4×C137C4×D411yescharacteristic
155C2×C4×C137#12011yescharacteristic
156C2×C4×C137#12111yescharacteristic
157C2×C4×C137#12311yescharacteristic
158C2×C4×C137C2×SD1611yescharacteristic
159C2×C4×C137C2×SD1611yescharacteristic
160C2×C4×C137C2×Q1611yescharacteristic
161C2×C4×C137#13511yescharacteristic
162C2×C4×C137#13511yescharacteristic
163C2×C4×C137#13511yescharacteristic
164C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
165C2×C4×C137#13711yescharacteristic
166C8×C137—22no—
167C8×C137—42no—
168C8×C137—42no—
169C8×C137—41no—
170C8×C137—81no—
171C22×C137—24no—
172C22×C137—42no—
173C22×C137—42no—
174C22×C137—42no—
175C22×C137—42no—
176C22×C137—42no—
177C22×C137—42no—
178C22×C137#38611yescharacteristic
179C22×C137#46111yescharacteristic
180C22×C137#46311yescharacteristic
181C22×C137#46511yescharacteristic
182C22×C137#46711yescharacteristic
183C22×C137#47111yescharacteristic
184C22×C137#47211yescharacteristic
185C4×C137—21no—
186C4×C137—21no—
187C4×C137—42no—
188C4×C137—82no—
189C4×C137—161no—
190C4×C137C2×#10311yescharacteristic
191C4×C137#41611yescharacteristic
192C4×C137C4×SD1611yescharacteristic
193C4×C137#44011yescharacteristic
194C4×C137#45011yescharacteristic
195C4×C137#45211yescharacteristic
196C4×C137#46511yescharacteristic
197C4×C137#46711yescharacteristic
198C2×C137—44no—
199C2×C137#167111yescharacteristic
200C2×C137#167511yescharacteristic
201C2×C137#167711yescharacteristic
202C2×C137#168311yescharacteristic
203C2×C137#187711yescharacteristic
204C2×C137#189211yescharacteristic
205C2×C137#190711yescharacteristic
206#9215C13711yesmaximal, characteristic, direct product factor, Sylow
207C137#921511yescharacteristic, direct product factor, Sylow
208C2×#476C2×C13711yescharacteristic
209C2×C8×D4C2×C13711yescharacteristic
210#1990C2×C13711yescharacteristic, complemented
211#1802C2×C13711yescharacteristic
212C2×#333C2×C13711yescharacteristic, complemented
213#1296C2×C13711yescharacteristic, complemented
214#1277C2×C13711yescharacteristic, complemented
215C23×C8—21no—
216C2×C4⋊Q8C22×C13711yescharacteristic
217C2×C4×D4C22×C13711yescharacteristic
218#476—22no—
219#476C4×C13714yescomplemented
220C8×D4—22no—
221C8×D4—22no—
222C2×#106—21no—
223C2×C4⋊C8C22×C13711yescharacteristic
224C2×C4⋊C8C22×C13711yescharacteristic
225C2×C4⋊C8C22×C13711yescharacteristic
226C2×#102—21no—
227C2×#102—21no—
228C2×C22⋊C8—21no—
229C2×C4×C8C22×C13711yescharacteristic
230#384C22×C13711yescharacteristic
231#340—21no—
232#340—21no—
233#340—21no—
234#333—22no—
235C22×Q8—21no—
236C22×D4D4×C13711yescharacteristic
237C23×C4—21no—
238C22×C8—24no—
239C22×C8—21no—
240C22×C8—21no—
241C22×C8—21no—
242C22×C8—42no—
243C22×C8—42no—
244C22×C8D4×C13711yescharacteristic
245C22×C8D4×C13711yescharacteristic
246C4⋊Q8—21no—
247C4⋊Q8—21no—
248C4⋊Q8—21no—
249C4⋊Q8C2×C4×C13712yes—
250C4×D4—21no—
251C4×D4—21no—
252C4×D4C2×C4×C13712yes—
253C4×D4D4×C13712yes—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4D4×C13711yescharacteristic
260C2×C22⋊C4—21no—
261C2×C42C23×C13711yescharacteristic
262#106—24no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8—21no—
266C4⋊C8—21no—
267C4⋊C8C2×C4×C13712yes—
268C4⋊C8D4×C13712yes—
269#102—24no—
270#102—42no—
271C22⋊C8—24no—
272C4×C8—21no—
273C4×C8—21no—
274C24—21no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×Q8—81no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4C22⋊C4×C13712yes—
283C2×D4SD16×C13712yes—
284C2×D4Q16×C13712yes—
285C22×C4—21no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×D4×C13711yescharacteristic
295C22×C4C2×D4×C13711yescharacteristic
296C22×C4#41×C13711yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—48no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8SD16×C13712yes—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—81no—
323C4⋊C4C22⋊C4×C13712yes—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C42C22×C4×C13711yescharacteristic
327C42C2×D4×C13711yescharacteristic
328C42C2×D4×C13711yescharacteristic
329C42#41×C13711yescharacteristic
330C23—22no—
331C23—22no—
332C23—22no—
333C23—41no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23#121×C13711yescharacteristic
338Q8—82no—
339Q8—82no—
340D4—22no—
341D4—22no—
342D4—22no—
343D4#103×C13714yes—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4C2×C22⋊C4×C13711yescharacteristic
360C2×C4C4×D4×C13711yescharacteristic
361C2×C4C4×D4×C13711yescharacteristic
362C2×C4#120×C13711yescharacteristic
363C2×C4#121×C13711yescharacteristic
364C2×C4#123×C13711yescharacteristic
365C2×C4C2×SD16×C13711yescharacteristic
366C2×C4C2×SD16×C13711yescharacteristic
367C2×C4C2×Q16×C13711yescharacteristic
368C2×C4#135×C13711yescharacteristic
369C2×C4#135×C13711yescharacteristic
370C2×C4#135×C13711yescharacteristic
371C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
372C2×C4#137×C13711yescharacteristic
373C8—22no—
374C8—42no—
375C8—42no—
376C8—41no—
377C8—81no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22#386×C13711yescharacteristic
386C22#461×C13711yescharacteristic
387C22#463×C13711yescharacteristic
388C22#465×C13711yescharacteristic
389C22#467×C13711yescharacteristic
390C22#471×C13711yescharacteristic
391C22#472×C13711yescharacteristic
392C4—21no—
393C4—21no—
394C4—42no—
395C4—82no—
396C4—161no—
397C4C2×#103×C13711yescharacteristic
398C4#416×C13711yescharacteristic
399C4C4×SD16×C13711yescharacteristic
400C4#440×C13711yescharacteristic
401C4#450×C13711yescharacteristic
402C4#452×C13711yescharacteristic
403C4#465×C13711yescharacteristic
404C4#467×C13711yescharacteristic
405C2—44no—
406C2#1671×C13711yescharacteristic
407C2#1675×C13711yescharacteristic
408C2#1677×C13711yescharacteristic
409C2#1683×C13711yescharacteristic
410C2#1877×C13711yescharacteristic
411C2#1892×C13711yescharacteristic
412C2#1907×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#121
commutatorC42C22×C4×C137
Frattini, cosocleC2×C42C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1990×C137C2semidirect
5. C2×#333×C137C2semidirect
6. #1296×C137C2semidirect
7. #1277×C137C2semidirect
12. #476×C137C4semidirect
206. #9215C137direct (N × H)
207. C137#9215direct (N × H)
210. #1990C2×C137semidirect
212. C2×#333C2×C137semidirect
213. #1296C2×C137semidirect
214. #1277C2×C137semidirect
219. #476C4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#476×C1372
C2×C8×D4×C1372
#1990×C1372
#1802×C1372
C2×#333×C1372
#1296×C1372
#1277×C1372
#9215137