Finite Group Databasebeta

← C137×#12092 #46861478

Subgroups — C137×#12092 #46861478

Order 35072.

544 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 74 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#2600×C137C211yesmaximal, characteristic, complemented
2#2022×C137C211yesmaximal, characteristic, complemented
3#1607×C137C211yesmaximal, characteristic, complemented
4#1595×C137C211yesmaximal, characteristic
5#1344×C137C211yesmaximal, characteristic, complemented
6#1343×C137C211yesmaximal, characteristic, complemented
7#1317×C137C211yesmaximal, characteristic, complemented
8#534×C137C2211yescharacteristic
9C2×#123×C137—21no—
10C2×C4×D4×C137—21no—
11#480×C137—21no—
12#447×C137—21no—
13C2×#99×C137—21no—
14#397×C137C2211yescharacteristic
15#394×C137—21no—
16#392×C137C2211yescharacteristic
17#388×C137—21no—
18#387×C137—21no—
19#386×C137—21no—
20#385×C137—21no—
21#381×C137C2211yescharacteristic
22#342×C137C2211yescharacteristic
23#327×C137—21no—
24#327×C137—21no—
25#323×C137C2211yescharacteristic
26#323×C137C2211yescharacteristic
27C22×D4×C137—21no—
28C22×D4×C137—21no—
29C22×D4×C137D411yescharacteristic
30C23×C4×C137—21no—
31C23×C4×C137D411yescharacteristic
32C2×D8×C137—41no—
33#123×C137—42no—
34#123×C137—41no—
35#123×C137—42no—
36#121×C137—21no—
37#121×C137—21no—
38#120×C137—41no—
39C4×D4×C137—42no—
40C4×D4×C137—42no—
41C2×C4⋊C4×C137—21no—
42C2×C4⋊C4×C137—21no—
43C2×C4⋊C4×C137—41no—
44C2×C4⋊C4×C137D411yescharacteristic
45C2×C22⋊C4×C137—21no—
46C2×C22⋊C4×C137—21no—
47C2×C22⋊C4×C137—21no—
48C2×C22⋊C4×C137—21no—
49C2×C22⋊C4×C137—21no—
50C2×C22⋊C4×C137—21no—
51C2×C22⋊C4×C137—41no—
52C2×C22⋊C4×C137D411yescharacteristic
53C2×C22⋊C4×C137C2311yescharacteristic
54C2×C42×C137—21no—
55#107×C137—41no—
56#102×C137—41no—
57#102×C137—41no—
58#99×C137—42no—
59C22⋊C8×C137—21no—
60C22⋊C8×C137—21no—
61#95×C137—21no—
62#95×C137—21no—
63#95×C137—21no—
64#95×C137—21no—
65#95×C137D411yescharacteristic
66#95×C137D411yescharacteristic
67C24×C137—21no—
68C24×C137—21no—
69C24×C137C2×D411yescharacteristic
70C2×D4×C137—21no—
71C2×D4×C137—21no—
72C2×D4×C137—42no—
73C2×D4×C137—41no—
74C2×D4×C137—42no—
75C2×D4×C137—41no—
76C2×D4×C137—41no—
77C2×D4×C137—42no—
78C2×D4×C137—41no—
79C2×D4×C137—41no—
80C2×D4×C137—41no—
81C2×D4×C137—41no—
82C2×D4×C137—41no—
83C2×D4×C137—41no—
84C22×C4×C137—21no—
85C22×C4×C137—21no—
86C22×C4×C137—21no—
87C22×C4×C137—21no—
88C22×C4×C137—21no—
89C22×C4×C137—21no—
90C22×C4×C137—21no—
91C22×C4×C137—42no—
92C22×C4×C137—41no—
93C22×C4×C137—41no—
94C22×C4×C137—41no—
95C22×C4×C137—81no—
96C22×C4×C137—81no—
97C22×C4×C137D812yescomplemented
98C22×C4×C137C2×D411yescharacteristic
99C22×C4×C137C2×D411yescharacteristic
100D8×C137—82no—
101C2×C8×C137—41no—
102C2×C8×C137—41no—
103C4⋊C4×C137—21no—
104C4⋊C4×C137—21no—
105C4⋊C4×C137—41no—
106C4⋊C4×C137—42no—
107C4⋊C4×C137—42no—
108C4⋊C4×C137—44no—
109C22⋊C4×C137—22no—
110C22⋊C4×C137—41no—
111C22⋊C4×C137—42no—
112C22⋊C4×C137—42no—
113C22⋊C4×C137—41no—
114C22⋊C4×C137—41no—
115C22⋊C4×C137—42no—
116C22⋊C4×C137—44no—
117C22⋊C4×C137—41no—
118C22⋊C4×C137—41no—
119C22⋊C4×C137—41no—
120C22⋊C4×C137—41no—
121C22⋊C4×C137—81no—
122C42×C137—42no—
123C23×C137—21no—
124C23×C137—21no—
125C23×C137—22no—
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—81no—
138C23×C137—81no—
139C23×C137#12011yescharacteristic
140C23×C137C2×D811yescharacteristic
141C23×C137AGL(1,ℤ/8)11yescharacteristic
142D4×C137—42no—
143D4×C137—44no—
144D4×C137—44no—
145D4×C137—81no—
146D4×C137—81no—
147D4×C137—81no—
148D4×C137—81no—
149D4×C137—81no—
150D4×C137—81no—
151D4×C137—81no—
152C2×C4×C137—22no—
153C2×C4×C137—22no—
154C2×C4×C137—21no—
155C2×C4×C137—21no—
156C2×C4×C137—41no—
157C2×C4×C137—41no—
158C2×C4×C137—41no—
159C2×C4×C137—41no—
160C2×C4×C137—42no—
161C2×C4×C137—41no—
162C2×C4×C137—41no—
163C2×C4×C137—42no—
164C2×C4×C137—41no—
165C2×C4×C137—41no—
166C2×C4×C137—41no—
167C2×C4×C137—41no—
168C2×C4×C137—41no—
169C2×C4×C137—41no—
170C2×C4×C137—41no—
171C2×C4×C137—41no—
172C2×C4×C137—41no—
173C2×C4×C137—41no—
174C2×C4×C137—41no—
175C2×C4×C137—41no—
176C2×C4×C137—81no—
177C2×C4×C137—84no—
178C2×C4×C137—81no—
179C2×C4×C137—81no—
180C8×C137—81no—
181C8×C137—81no—
182C22×C137—21no—
183C22×C137—21no—
184C22×C137—21no—
185C22×C137—21no—
186C22×C137—41no—
187C22×C137—41no—
188C22×C137—42no—
189C22×C137—42no—
190C22×C137—41no—
191C22×C137—41no—
192C22×C137—42no—
193C22×C137—81no—
194C22×C137—81no—
195C22×C137—81no—
196C22×C137—82no—
197C22×C137—81no—
198C22×C137—81no—
199C22×C137—81no—
200C22×C137—81no—
201C22×C137—81no—
202C22×C137—161no—
203C22×C137#44711yescharacteristic
204C22×C137#45311yescharacteristic
205C22×C137B(4,𝔽2)11yescharacteristic
206C4×C137—42no—
207C4×C137—42no—
208C4×C137—41no—
209C4×C137—41no—
210C4×C137—81no—
211C4×C137—81no—
212C4×C137—82no—
213C4×C137—81no—
214C4×C137—81no—
215C2×C137—21no—
216C2×C137—21no—
217C2×C137—41no—
218C2×C137—41no—
219C2×C137—82no—
220C2×C137—161no—
221C2×C137#159511yescharacteristic
222C2×C137#219711yescharacteristic
223C2×C137#219811yescharacteristic
224#12092C13711yesmaximal, characteristic, direct product factor, Sylow
225C137#1209211yescharacteristic, direct product factor, Sylow
226#2600C2×C13711yescharacteristic, complemented
227#2022C2×C13711yescharacteristic, complemented
228#1607C2×C13711yescharacteristic, complemented
229#1595C2×C13711yescharacteristic
230#1344C2×C13711yescharacteristic, complemented
231#1343C2×C13711yescharacteristic, complemented
232#1317C2×C13711yescharacteristic, complemented
233#534C22×C13711yescharacteristic
234C2×#123—21no—
235C2×C4×D4—21no—
236#480—21no—
237#447—21no—
238C2×#99—21no—
239#397C22×C13711yescharacteristic
240#394—21no—
241#392C22×C13711yescharacteristic
242#388—21no—
243#387—21no—
244#386—21no—
245#385—21no—
246#381C22×C13711yescharacteristic
247#342C22×C13711yescharacteristic
248#327—21no—
249#327—21no—
250#323C22×C13711yescharacteristic
251#323C22×C13711yescharacteristic
252C22×D4—21no—
253C22×D4—21no—
254C22×D4D4×C13711yescharacteristic
255C23×C4—21no—
256C23×C4D4×C13711yescharacteristic
257C2×D8—41no—
258#123—42no—
259#123—41no—
260#123—42no—
261#121—21no—
262#121—21no—
263#120—41no—
264C4×D4—42no—
265C4×D4—42no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—41no—
269C2×C4⋊C4D4×C13711yescharacteristic
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—21no—
273C2×C22⋊C4—21no—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—21no—
276C2×C22⋊C4—41no—
277C2×C22⋊C4D4×C13711yescharacteristic
278C2×C22⋊C4C23×C13711yescharacteristic
279C2×C42—21no—
280#107—41no—
281#102—41no—
282#102—41no—
283#99—42no—
284C22⋊C8—21no—
285C22⋊C8—21no—
286#95—21no—
287#95—21no—
288#95—21no—
289#95—21no—
290#95D4×C13711yescharacteristic
291#95D4×C13711yescharacteristic
292C24—21no—
293C24—21no—
294C24C2×D4×C13711yescharacteristic
295C2×D4—21no—
296C2×D4—21no—
297C2×D4—42no—
298C2×D4—41no—
299C2×D4—42no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4D8×C13712yescomplemented
323C22×C4C2×D4×C13711yescharacteristic
324C22×C4C2×D4×C13711yescharacteristic
325D8—82no—
326C2×C8—41no—
327C2×C8—41no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—44no—
334C22⋊C4—22no—
335C22⋊C4—41no—
336C22⋊C4—42no—
337C22⋊C4—42no—
338C22⋊C4—41no—
339C22⋊C4—41no—
340C22⋊C4—42no—
341C22⋊C4—44no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—41no—
346C22⋊C4—81no—
347C42—42no—
348C23—21no—
349C23—21no—
350C23—22no—
351C23—41no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—81no—
363C23—81no—
364C23#120×C13711yescharacteristic
365C23C2×D8×C13711yescharacteristic
366C23AGL(1,ℤ/8)×C13711yescharacteristic
367D4—42no—
368D4—44no—
369D4—44no—
370D4—81no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—81no—
376D4—81no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—42no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—81no—
402C2×C4—84no—
403C2×C4—81no—
404C2×C4—81no—
405C8—81no—
406C8—81no—
407C22—21no—
408C22—21no—
409C22—21no—
410C22—21no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—42no—
415C22—41no—
416C22—41no—
417C22—42no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—82no—
422C22—81no—
423C22—81no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—161no—
428C22#447×C13711yescharacteristic
429C22#453×C13711yescharacteristic
430C22B(4,𝔽2)×C13711yescharacteristic
431C4—42no—
432C4—42no—
433C4—41no—
434C4—41no—
435C4—81no—
436C4—81no—
437C4—82no—
438C4—81no—
439C4—81no—
440C2—21no—
441C2—21no—
442C2—41no—
443C2—41no—
444C2—82no—
445C2—161no—
446C2#1595×C13711yescharacteristic
447C2#2197×C13711yescharacteristic
448C2#2198×C13711yescharacteristic
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C137
prime radical#1595×C137C2
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C2×C22⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2600×C137C2semidirect
2. #2022×C137C2semidirect
3. #1607×C137C2semidirect
5. #1344×C137C2semidirect
6. #1343×C137C2semidirect
7. #1317×C137C2semidirect
97. C22×C4×C137D8semidirect
224. #12092C137direct (N × H)
225. C137#12092direct (N × H)
226. #2600C2×C137semidirect
227. #2022C2×C137semidirect
228. #1607C2×C137semidirect
230. #1344C2×C137semidirect
231. #1343C2×C137semidirect
232. #1317C2×C137semidirect
322. C22×C4D8×C137semidirect

Maximal subgroups

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

subgroupindex
#2600×C1372
#2022×C1372
#1607×C1372
#1595×C1372
#1344×C1372
#1343×C1372
#1317×C1372
#12092137