Finite Group Databasebeta

← C137×#23989 #46873375

Subgroups — C137×#23989 #46873375

Order 35072.

1210 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3491×C137C211yesmaximal, characteristic
2#2614×C137C212yesmaximal, complemented
3#2310×C137C211yesmaximal, characteristic, complemented
4#2030×C137C212yesmaximal, complemented
5#2022×C137C212yesmaximal
6#1899×C137C212yesmaximal, complemented
7#1890×C137C214yesmaximal, complemented
8#1820×C137C211yesmaximal, characteristic, complemented
9C23×Q8×C137C2211yescharacteristic
10#552×C137—24no—
11#551×C137—24no—
12#543×C137C2211yescharacteristic
13#539×C137—22no—
14#534×C137C2211yescharacteristic
15#529×C137—22no—
16C2×#123×C137C2214yes—
17C2×#123×C137C2212yes—
18#518×C137C2212yes—
19C2×#117×C137C2211yescharacteristic
20C2×#117×C137C2211yescharacteristic
21C2×#99×C137C2214yes—
22#395×C137—24no—
23#393×C137—24no—
24#391×C137—22no—
25#391×C137—22no—
26#381×C137C2212yes—
27#381×C137C2214yes—
28#381×C137C2212yes—
29#381×C137C2212yes—
30#342×C137C2212yes—
31#342×C137C2214yes—
32#342×C137C2212yes—
33C22×Q8×C137—21no—
34C22×Q8×C137—21no—
35C22×Q8×C137—24no—
36C22×Q8×C137—24no—
37C22×Q8×C137—42no—
38C22×D4×C137C2312yes—
39C23×C4×C137C2312yes—
40C23×C4×C137C2311yescharacteristic
41#126×C137—24no—
42#126×C137—24no—
43#125×C137—22no—
44#125×C137—24no—
45#124×C137—22no—
46#123×C137—24no—
47#123×C137—22no—
48#123×C137—22no—
49#123×C137—24no—
50#123×C137—22no—
51#123×C137—28no—
52#123×C137—44no—
53C22⋊Q8×C137—24no—
54C22⋊Q8×C137—22no—
55C22⋊Q8×C137—24no—
56#121×C137—22no—
57#120×C137—22no—
58C4×Q8×C137—22no—
59C4×D4×C137—22no—
60C4×D4×C137—24no—
61C4×D4×C137—24no—
62#117×C137—24no—
63#117×C137—24no—
64#117×C137—22no—
65#117×C137—42no—
66C2×C4⋊C4×C137—21no—
67C2×C4⋊C4×C137—24no—
68C2×C4⋊C4×C137D412yes—
69C2×C4⋊C4×C137D414yes—
70C2×C22⋊C4×C137—22no—
71C2×C22⋊C4×C137—24no—
72C2×C22⋊C4×C137D414yes—
73C2×C22⋊C4×C137C2312yes—
74C2×C22⋊C4×C137C2314yes—
75C2×C22⋊C4×C137C2311yescharacteristic
76C2×C22⋊C4×C137C2311yescharacteristic
77C2×C22⋊C4×C137C2312yes—
78C2×C42×C137—21no—
79C2×C42×C137D412yes—
80#99×C137—44no—
81#95×C137—24no—
82#95×C137—24no—
83#95×C137—28no—
84#95×C137—24no—
85C24×C137C2×D412yes—
86C24×C137C2411yescharacteristic
87C2×Q8×C137—24no—
88C2×Q8×C137—24no—
89C2×Q8×C137—22no—
90C2×Q8×C137—416no—
91C2×Q8×C137—42no—
92C2×Q8×C137—42no—
93C2×Q8×C137—42no—
94C2×Q8×C137—42no—
95C2×D4×C137—22no—
96C2×D4×C137—22no—
97C2×D4×C137—24no—
98C2×D4×C137—24no—
99C2×D4×C137—22no—
100C22×C4×C137—22no—
101C22×C4×C137—22no—
102C22×C4×C137—22no—
103C22×C4×C137—24no—
104C22×C4×C137—22no—
105C22×C4×C137—24no—
106C22×C4×C137—24no—
107C22×C4×C137—22no—
108C22×C4×C137—24no—
109C22×C4×C137C2×D412yes—
110C22×C4×C137C2×D414yes—
111C22×C4×C137C2×D411yescharacteristic
112C22×C4×C137C2×D411yescharacteristic
113C22×C4×C137C2×D412yes—
114C22×C4×C137C2×D412yes—
115C22×C4×C137C2×D414yes—
116C22×C4×C137#4114yes—
117C4⋊C4×C137—24no—
118C4⋊C4×C137—24no—
119C4⋊C4×C137—22no—
120C4⋊C4×C137—22no—
121C4⋊C4×C137—24no—
122C4⋊C4×C137—24no—
123C4⋊C4×C137—24no—
124C4⋊C4×C137—48no—
125C4⋊C4×C137—42no—
126C22⋊C4×C137—24no—
127C22⋊C4×C137—24no—
128C22⋊C4×C137—22no—
129C22⋊C4×C137—22no—
130C22⋊C4×C137—22no—
131C22⋊C4×C137—22no—
132C22⋊C4×C137—24no—
133C22⋊C4×C137—24no—
134C22⋊C4×C137—24no—
135C22⋊C4×C137—24no—
136C22⋊C4×C137—24no—
137C22⋊C4×C137—48no—
138C42×C137—24no—
139C42×C137—22no—
140C42×C137—22no—
141C42×C137—42no—
142C23×C137—22no—
143C23×C137—24no—
144C23×C137—42no—
145C23×C137—44no—
146C23×C137C22×D411yescharacteristic
147C23×C137C22×D412yes—
148C23×C137C2×#4112yes—
149C23×C137H(5,𝔽2)12yes—
150Q8×C137—48no—
151Q8×C137—48no—
152Q8×C137—82no—
153D4×C137—42no—
154D4×C137—42no—
155D4×C137—44no—
156D4×C137—82no—
157C2×C4×C137—21no—
158C2×C4×C137—21no—
159C2×C4×C137—21no—
160C2×C4×C137—21no—
161C2×C4×C137—28no—
162C2×C4×C137—22no—
163C2×C4×C137—22no—
164C2×C4×C137—22no—
165C2×C4×C137—24no—
166C2×C4×C137—24no—
167C2×C4×C137—28no—
168C2×C4×C137—24no—
169C2×C4×C137—42no—
170C2×C4×C137—42no—
171C2×C4×C137—44no—
172C2×C4×C137—42no—
173C2×C4×C137—44no—
174C2×C4×C137—42no—
175C2×C4×C137—44no—
176C2×C4×C137—48no—
177C2×C4×C137—42no—
178C2×C4×C137—44no—
179C2×C4×C137—84no—
180C2×C4×C137#12012yes—
181C2×C4×C137#12012yes—
182C22×C137—22no—
183C22×C137—24no—
184C22×C137—42no—
185C22×C137—42no—
186C22×C137—42no—
187C22×C137—44no—
188C22×C137—84no—
189C22×C137C2×#12011yescharacteristic
190C22×C137#52511yescharacteristic
191C22×C137#53511yescharacteristic
192C22×C137#54614yes—
193C4×C137—22no—
194C4×C137—22no—
195C4×C137—44no—
196C4×C137—42no—
197C4×C137—42no—
198C4×C137—44no—
199C4×C137—42no—
200C4×C137—82no—
201C4×C137—84no—
202C2×C137—21no—
203C2×C137—24no—
204C2×C137—21no—
205C2×C137—82no—
206C2×C137#240411yescharacteristic
207C2×C137#302812yes—
208#23989C13711yesmaximal, characteristic, direct product factor, Sylow
209C137#2398911yescharacteristic, direct product factor, Sylow
210#3491C2×C13711yescharacteristic
211#2614C2×C13712yescomplemented
212#2310C2×C13711yescharacteristic, complemented
213#2030C2×C13712yescomplemented
214#2022C2×C13712yes—
215#1899C2×C13712yescomplemented
216#1890C2×C13714yescomplemented
217#1820C2×C13711yescharacteristic, complemented
218C23×Q8C22×C13711yescharacteristic
219#552—24no—
220#551—24no—
221#543C22×C13711yescharacteristic
222#539—22no—
223#534C22×C13711yescharacteristic
224#529—22no—
225C2×#123C22×C13714yes—
226C2×#123C22×C13712yes—
227#518C22×C13712yes—
228C2×#117C22×C13711yescharacteristic
229C2×#117C22×C13711yescharacteristic
230C2×#99C22×C13714yes—
231#395—24no—
232#393—24no—
233#391—22no—
234#391—22no—
235#381C22×C13712yes—
236#381C22×C13712yes—
237#381C22×C13712yes—
238#381C22×C13714yes—
239#342C22×C13712yes—
240#342C22×C13712yes—
241#342C22×C13714yes—
242C22×Q8—21no—
243C22×Q8—21no—
244C22×Q8—24no—
245C22×Q8—24no—
246C22×Q8—42no—
247C22×D4C23×C13712yes—
248C23×C4C23×C13712yes—
249C23×C4C23×C13711yescharacteristic
250#126—24no—
251#126—24no—
252#125—22no—
253#125—24no—
254#124—22no—
255#123—24no—
256#123—22no—
257#123—22no—
258#123—24no—
259#123—22no—
260#123—28no—
261#123—44no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—24no—
265#121—22no—
266#120—22no—
267C4×Q8—22no—
268C4×D4—22no—
269C4×D4—24no—
270C4×D4—24no—
271#117—24no—
272#117—24no—
273#117—22no—
274#117—42no—
275C2×C4⋊C4—21no—
276C2×C4⋊C4—24no—
277C2×C4⋊C4D4×C13712yes—
278C2×C4⋊C4D4×C13714yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—24no—
281C2×C22⋊C4D4×C13714yes—
282C2×C22⋊C4C23×C13712yes—
283C2×C22⋊C4C23×C13714yes—
284C2×C22⋊C4C23×C13711yescharacteristic
285C2×C22⋊C4C23×C13711yescharacteristic
286C2×C22⋊C4C23×C13712yes—
287C2×C42—21no—
288C2×C42D4×C13712yes—
289#99—44no—
290#95—24no—
291#95—24no—
292#95—28no—
293#95—24no—
294C24C2×D4×C13712yes—
295C24C24×C13711yescharacteristic
296C2×Q8—24no—
297C2×Q8—24no—
298C2×Q8—22no—
299C2×Q8—416no—
300C2×Q8—42no—
301C2×Q8—42no—
302C2×Q8—42no—
303C2×Q8—42no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4C2×D4×C13712yes—
319C22×C4C2×D4×C13714yes—
320C22×C4C2×D4×C13711yescharacteristic
321C22×C4C2×D4×C13711yescharacteristic
322C22×C4C2×D4×C13712yes—
323C22×C4C2×D4×C13712yes—
324C22×C4C2×D4×C13714yes—
325C22×C4#41×C13714yes—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—48no—
334C4⋊C4—42no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—48no—
347C42—24no—
348C42—22no—
349C42—22no—
350C42—42no—
351C23—22no—
352C23—24no—
353C23—42no—
354C23—44no—
355C23C22×D4×C13711yescharacteristic
356C23C22×D4×C13712yes—
357C23C2×#41×C13712yes—
358C23H(5,𝔽2)×C13712yes—
359Q8—48no—
360Q8—48no—
361Q8—82no—
362D4—42no—
363D4—42no—
364D4—44no—
365D4—82no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—28no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—28no—
377C2×C4—24no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—42no—
384C2×C4—44no—
385C2×C4—48no—
386C2×C4—42no—
387C2×C4—44no—
388C2×C4—84no—
389C2×C4#120×C13712yes—
390C2×C4#120×C13712yes—
391C22—22no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—44no—
397C22—84no—
398C22C2×#120×C13711yescharacteristic
399C22#525×C13711yescharacteristic
400C22#535×C13711yescharacteristic
401C22#546×C13714yes—
402C4—22no—
403C4—22no—
404C4—44no—
405C4—42no—
406C4—42no—
407C4—44no—
408C4—42no—
409C4—82no—
410C4—84no—
411C2—21no—
412C2—24no—
413C2—21no—
414C2—82no—
415C2#2404×C13711yescharacteristic
416C2#3028×C13712yes—
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC24C24×C137
prime radical#534×C137C22
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2614×C137C2semidirect
3. #2310×C137C2semidirect
4. #2030×C137C2semidirect
6. #1899×C137C2semidirect
7. #1890×C137C2semidirect
8. #1820×C137C2semidirect
208. #23989C137direct (N × H)
209. C137#23989direct (N × H)
211. #2614C2×C137semidirect
212. #2310C2×C137semidirect
213. #2030C2×C137semidirect
215. #1899C2×C137semidirect
216. #1890C2×C137semidirect
217. #1820C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3491×C1372
#2614×C1372
#2310×C1372
#2030×C1372
#2022×C1372
#1899×C1372
#1890×C1372
#1820×C1372
#23989137