Finite Group Databasebeta

← C137×#7825 #46857211

Subgroups — C137×#7825 #46857211

Order 35072.

516 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2682×C137C211yesmaximal, characteristic
2#2100×C137C211yesmaximal, characteristic, complemented
3#2094×C137C211yesmaximal, characteristic, complemented
4#2094×C137C211yesmaximal, characteristic, complemented
5#1271×C137C211yesmaximal, characteristic, complemented
6#1270×C137C211yesmaximal, characteristic, complemented
7#1270×C137C211yesmaximal, characteristic, complemented
8C2×#123×C137—21no—
9C2×#121×C137—21no—
10C2×#106×C137—21no—
11C2×#106×C137—21no—
12C2×#102×C137—21no—
13C2×#102×C137—21no—
14#396×C137—21no—
15#394×C137—21no—
16#392×C137—21no—
17#390×C137C2211yescharacteristic
18#390×C137C2211yescharacteristic
19#389×C137C2211yescharacteristic
20#385×C137—21no—
21#384×C137C2211yescharacteristic
22#340×C137—21no—
23#340×C137—21no—
24#336×C137C2211yescharacteristic
25#336×C137C2211yescharacteristic
26#336×C137C2211yescharacteristic
27C22×D4×C137—21no—
28C22×D4×C137—21no—
29C22×D4×C137D411yescharacteristic
30C22×D4×C137D411yescharacteristic
31C23×C4×C137—21no—
32C22×C8×C137—21no—
33C22×C8×C137—21no—
34C22×C8×C137—21no—
35#123×C137—42no—
36#123×C137—42no—
37#121×C137—42no—
38#121×C137—42no—
39C2×C4⋊C4×C137—21no—
40C2×C4⋊C4×C137—21no—
41C2×C4⋊C4×C137—21no—
42C2×C4⋊C4×C137—21no—
43C2×C4⋊C4×C137—21no—
44C2×C4⋊C4×C137D411yescharacteristic
45C2×C4⋊C4×C137D411yescharacteristic
46C2×C4⋊C4×C137D411yescharacteristic
47C2×C4⋊C4×C137D411yescharacteristic
48C2×C22⋊C4×C137—21no—
49C2×C22⋊C4×C137—21no—
50C2×C22⋊C4×C137—21no—
51C2×C22⋊C4×C137—21no—
52C2×C22⋊C4×C137—21no—
53C2×C22⋊C4×C137—21no—
54C2×C22⋊C4×C137—21no—
55C2×C42×C137C2311yescharacteristic
56#106×C137—42no—
57#106×C137—42no—
58#102×C137—42no—
59#102×C137—42no—
60#95×C137—21no—
61#95×C137—21no—
62C24×C137—21no—
63C24×C137—21no—
64C2×D4×C137—21no—
65C2×D4×C137—21no—
66C2×D4×C137—21no—
67C2×D4×C137—41no—
68C2×D4×C137—41no—
69C2×D4×C137—41no—
70C2×D4×C137—41no—
71C2×D4×C137—42no—
72C2×D4×C137—42no—
73C2×D4×C137—42no—
74C2×D4×C137—81no—
75C2×D4×C137—81no—
76C2×D4×C137—81no—
77C2×D4×C137SD1612yes—
78C22×C4×C137—21no—
79C22×C4×C137—21no—
80C22×C4×C137—21no—
81C22×C4×C137—21no—
82C22×C4×C137—42no—
83C22×C4×C137—42no—
84C22×C4×C137—41no—
85C22×C4×C137—81no—
86C22×C4×C137C2×D411yescharacteristic
87C22×C4×C137C2×D411yescharacteristic
88C22×C4×C137C2×D411yescharacteristic
89C2×C8×C137—22no—
90C2×C8×C137—22no—
91C2×C8×C137—41no—
92C2×C8×C137—41no—
93C2×C8×C137—41no—
94C2×C8×C137—41no—
95C2×C8×C137—41no—
96C2×C8×C137—41no—
97C2×C8×C137—41no—
98C4⋊C4×C137—21no—
99C4⋊C4×C137—41no—
100C4⋊C4×C137—41no—
101C4⋊C4×C137—41no—
102C4⋊C4×C137—42no—
103C4⋊C4×C137—41no—
104C4⋊C4×C137—41no—
105C4⋊C4×C137—41no—
106C4⋊C4×C137—41no—
107C4⋊C4×C137—42no—
108C4⋊C4×C137—41no—
109C4⋊C4×C137—41no—
110C4⋊C4×C137SD1612yes—
111C22⋊C4×C137—42no—
112C22⋊C4×C137—42no—
113C22⋊C4×C137—41no—
114C22⋊C4×C137—41no—
115C22⋊C4×C137—41no—
116C22⋊C4×C137—41no—
117C22⋊C4×C137—41no—
118C22⋊C4×C137—41no—
119C22⋊C4×C137—41no—
120C22⋊C4×C137—41no—
121C22⋊C4×C137—81no—
122C42×C137—41no—
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—41no—
133C23×C137—41no—
134C23×C137—41no—
135C23×C137—41no—
136C23×C137—41no—
137C23×C137#12011yescharacteristic
138D4×C137—42no—
139D4×C137—42no—
140D4×C137—81no—
141D4×C137—81no—
142D4×C137—84no—
143D4×C137—84no—
144C2×C4×C137—21no—
145C2×C4×C137—21no—
146C2×C4×C137—21no—
147C2×C4×C137—21no—
148C2×C4×C137—21no—
149C2×C4×C137—21no—
150C2×C4×C137—22no—
151C2×C4×C137—41no—
152C2×C4×C137—41no—
153C2×C4×C137—41no—
154C2×C4×C137—41no—
155C2×C4×C137—41no—
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—84no—
163C2×C4×C137—81no—
164C2×C4×C137—81no—
165C2×C4×C137—81no—
166C2×C4×C137C2×SD1611yescharacteristic
167C2×C4×C137C2×SD1611yescharacteristic
168C2×C4×C137#13511yescharacteristic
169C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
170C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
171C2×C4×C137#13711yescharacteristic
172C8×C137—42no—
173C8×C137—42no—
174C8×C137—81no—
175C22×C137—42no—
176C22×C137—42no—
177C22×C137—42no—
178C22×C137—81no—
179C22×C137—81no—
180C22×C137—81no—
181C22×C137—81no—
182C22×C137—81no—
183C22×C137—81no—
184C22×C137—81no—
185C22×C137—81no—
186C22×C137—81no—
187C22×C137—81no—
188C22×C137—81no—
189C22×C137#45011yescharacteristic
190C22×C137#45011yescharacteristic
191C22×C137#45211yescharacteristic
192C22×C137#45311yescharacteristic
193C22×C137#45311yescharacteristic
194C22×C137#45411yescharacteristic
195C22×C137#45811yescharacteristic
196C4×C137—21no—
197C4×C137—21no—
198C4×C137—21no—
199C4×C137—21no—
200C4×C137—21no—
201C4×C137—21no—
202C4×C137—42no—
203C4×C137—81no—
204C4×C137—81no—
205C4×C137—81no—
206C4×C137—161no—
207C2×C137—82no—
208C2×C137—161no—
209C2×C137#161311yescharacteristic
210C2×C137#161611yescharacteristic
211C2×C137#161811yescharacteristic
212C2×C137#162911yescharacteristic
213C2×C137#163111yescharacteristic
214C2×C137#163511yescharacteristic
215C2×C137#165611yescharacteristic
216#7825C13711yesmaximal, characteristic, direct product factor, Sylow
217C137#782511yescharacteristic, direct product factor, Sylow
218#2682C2×C13711yescharacteristic
219#2100C2×C13711yescharacteristic, complemented
220#2094C2×C13711yescharacteristic, complemented
221#2094C2×C13711yescharacteristic, complemented
222#1271C2×C13711yescharacteristic, complemented
223#1270C2×C13711yescharacteristic, complemented
224#1270C2×C13711yescharacteristic, complemented
225C2×#123—21no—
226C2×#121—21no—
227C2×#106—21no—
228C2×#106—21no—
229C2×#102—21no—
230C2×#102—21no—
231#396—21no—
232#394—21no—
233#392—21no—
234#390C22×C13711yescharacteristic
235#390C22×C13711yescharacteristic
236#389C22×C13711yescharacteristic
237#385—21no—
238#384C22×C13711yescharacteristic
239#340—21no—
240#340—21no—
241#336C22×C13711yescharacteristic
242#336C22×C13711yescharacteristic
243#336C22×C13711yescharacteristic
244C22×D4—21no—
245C22×D4—21no—
246C22×D4D4×C13711yescharacteristic
247C22×D4D4×C13711yescharacteristic
248C23×C4—21no—
249C22×C8—21no—
250C22×C8—21no—
251C22×C8—21no—
252#123—42no—
253#123—42no—
254#121—42no—
255#121—42no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4D4×C13711yescharacteristic
262C2×C4⋊C4D4×C13711yescharacteristic
263C2×C4⋊C4D4×C13711yescharacteristic
264C2×C4⋊C4D4×C13711yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C42C23×C13711yescharacteristic
273#106—42no—
274#106—42no—
275#102—42no—
276#102—42no—
277#95—21no—
278#95—21no—
279C24—21no—
280C24—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—41no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—81no—
292C2×D4—81no—
293C2×D4—81no—
294C2×D4SD16×C13712yes—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—81no—
303C22×C4C2×D4×C13711yescharacteristic
304C22×C4C2×D4×C13711yescharacteristic
305C22×C4C2×D4×C13711yescharacteristic
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C4⋊C4—21no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—42no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—42no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4SD16×C13712yes—
328C22⋊C4—42no—
329C22⋊C4—42no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—81no—
339C42—41no—
340C23—22no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23#120×C13711yescharacteristic
355D4—42no—
356D4—42no—
357D4—81no—
358D4—81no—
359D4—84no—
360D4—84no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—84no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4C2×SD16×C13711yescharacteristic
384C2×C4C2×SD16×C13711yescharacteristic
385C2×C4#135×C13711yescharacteristic
386C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
387C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
388C2×C4#137×C13711yescharacteristic
389C8—42no—
390C8—42no—
391C8—81no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—81no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#450×C13711yescharacteristic
407C22#450×C13711yescharacteristic
408C22#452×C13711yescharacteristic
409C22#453×C13711yescharacteristic
410C22#453×C13711yescharacteristic
411C22#454×C13711yescharacteristic
412C22#458×C13711yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—42no—
420C4—81no—
421C4—81no—
422C4—81no—
423C4—161no—
424C2—82no—
425C2—161no—
426C2#1613×C13711yescharacteristic
427C2#1616×C13711yescharacteristic
428C2#1618×C13711yescharacteristic
429C2#1629×C13711yescharacteristic
430C2#1631×C13711yescharacteristic
431C2#1635×C13711yescharacteristic
432C2#1656×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#120
commutator, Frattini, cosocleC2×C42C23×C137
prime radical#2682×C137C2
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2100×C137C2semidirect
3. #2094×C137C2semidirect
4. #2094×C137C2semidirect
5. #1271×C137C2semidirect
6. #1270×C137C2semidirect
7. #1270×C137C2semidirect
216. #7825C137direct (N × H)
217. C137#7825direct (N × H)
219. #2100C2×C137semidirect
220. #2094C2×C137semidirect
221. #2094C2×C137semidirect
222. #1271C2×C137semidirect
223. #1270C2×C137semidirect
224. #1270C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2682×C1372
#2100×C1372
#2094×C1372
#1271×C1372
#1270×C1372
#7825137