Finite Group Databasebeta

← C137×#8535 #46857921

Subgroups — C137×#8535 #46857921

Order 35072.

1062 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 150 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#3490×C137C211yesmaximal, characteristic
2#1898×C137C214yesmaximal, complemented
3#1819×C137C211yesmaximal, characteristic, complemented
4#1440×C137C211yesmaximal, characteristic, complemented
5#551×C137—24no—
6#546×C137—24no—
7#543×C137C411yescharacteristic, complemented
8#538×C137—22no—
9#534×C137C411yescharacteristic, complemented
10#525×C137—22no—
11C2×#120×C137C2212yes—
12#518×C137C412yescomplemented
13C22×C22⋊C4×C137C2211yescharacteristic
14C22×C22⋊C4×C137C2212yes—
15C2×#99×C137—24no—
16#379×C137C2212yes—
17C2×#95×C137—22no—
18#342×C137—22no—
19#342×C137—24no—
20C25×C137C2311yescharacteristic
21C2×#41×C137—22no—
22C22×D4×C137C2×C412yes—
23C22×D4×C137D414yes—
24C23×C4×C137—22no—
25C23×C4×C137D412yes—
26#126×C137—24no—
27#124×C137—24no—
28#123×C137—24no—
29#123×C137—24no—
30#123×C137—22no—
31C22⋊Q8×C137—22no—
32C22⋊Q8×C137—22no—
33C22⋊Q8×C137—24no—
34#121×C137—22no—
35#121×C137—24no—
36#121×C137—22no—
37#120×C137—22no—
38#120×C137—22no—
39#120×C137—22no—
40#120×C137—24no—
41C4×D4×C137—24no—
42C4×D4×C137—24no—
43C4×D4×C137—24no—
44#117×C137—22no—
45#117×C137—22no—
46C2×C22⋊C4×C137—22no—
47C2×C22⋊C4×C137—24no—
48C2×C22⋊C4×C137—24no—
49C2×C22⋊C4×C137—24no—
50C2×C22⋊C4×C137—22no—
51C2×C22⋊C4×C137—44no—
52C2×C22⋊C4×C137C2×C411yescharacteristic
53C2×C22⋊C4×C137C2×C411yescharacteristic
54C2×C22⋊C4×C137C2×C412yes—
55C2×C22⋊C4×C137D412yes—
56C2×C22⋊C4×C137D414yes—
57#99×C137—48no—
58#95×C137—44no—
59C24×C137—42no—
60C24×C137—42no—
61C24×C137—42no—
62C24×C137C22⋊C412yes—
63C24×C137C22×C411yescharacteristic
64C24×C137C2×D411yescharacteristic
65C24×C137C2×D414yes—
66C24×C137C2×D411yescharacteristic
67#41×C137—82no—
68#41×C137—82no—
69C2×Q8×C137—22no—
70C2×D4×C137—22no—
71C2×D4×C137—22no—
72C2×D4×C137—24no—
73C2×D4×C137—22no—
74C2×D4×C137—24no—
75C2×D4×C137—44no—
76C2×D4×C137—44no—
77C2×D4×C137—44no—
78C22×C4×C137—22no—
79C22×C4×C137—24no—
80C22×C4×C137—22no—
81C22×C4×C137—24no—
82C22×C4×C137—42no—
83C22×C4×C137—42no—
84C22×C4×C137—42no—
85C22×C4×C137—42no—
86C22×C4×C137—42no—
87C22×C4×C137—42no—
88C22×C4×C137—42no—
89C22×C4×C137—42no—
90C22×C4×C137—42no—
91C22×C4×C137C22⋊C412yes—
92C22×C4×C137C22⋊C414yes—
93C22×C4×C137C22⋊C412yes—
94C22×C4×C137C22⋊C412yes—
95C4⋊C4×C137—24no—
96C4⋊C4×C137—22no—
97C4⋊C4×C137—22no—
98C4⋊C4×C137—24no—
99C22⋊C4×C137—24no—
100C22⋊C4×C137—24no—
101C22⋊C4×C137—22no—
102C22⋊C4×C137—22no—
103C22⋊C4×C137—22no—
104C22⋊C4×C137—22no—
105C22⋊C4×C137—24no—
106C22⋊C4×C137—24no—
107C22⋊C4×C137—24no—
108C22⋊C4×C137—28no—
109C22⋊C4×C137—44no—
110C22⋊C4×C137—44no—
111C22⋊C4×C137—44no—
112C22⋊C4×C137—44no—
113C42×C137—24no—
114C23×C137—22no—
115C23×C137—22no—
116C23×C137—22no—
117C23×C137—22no—
118C23×C137—22no—
119C23×C137—24no—
120C23×C137—24no—
121C23×C137—22no—
122C23×C137—42no—
123C23×C137—44no—
124C23×C137—42no—
125C23×C137—44no—
126C23×C137—42no—
127C23×C137—44no—
128C23×C137—82no—
129C23×C137—82no—
130C23×C137—82no—
131C23×C137—82no—
132C23×C137C2×C22⋊C412yes—
133C23×C137C2×C22⋊C411yescharacteristic
134C23×C137#12012yes—
135C23×C137#12012yes—
136Q8×C137—82no—
137D4×C137—44no—
138D4×C137—44no—
139D4×C137—44no—
140D4×C137—44no—
141D4×C137—84no—
142D4×C137—82no—
143C2×C4×C137—22no—
144C2×C4×C137—22no—
145C2×C4×C137—24no—
146C2×C4×C137—22no—
147C2×C4×C137—42no—
148C2×C4×C137—42no—
149C2×C4×C137—42no—
150C2×C4×C137—42no—
151C2×C4×C137—42no—
152C2×C4×C137—44no—
153C2×C4×C137—44no—
154C2×C4×C137—44no—
155C2×C4×C137—44no—
156C2×C4×C137—82no—
157C2×C4×C137—84no—
158C2×C4×C137—82no—
159C2×C4×C137—82no—
160C2×C4×C137—82no—
161C2×C4×C137—82no—
162C2×C4×C137—82no—
163C2×C4×C137—84no—
164C22×C137—22no—
165C22×C137—22no—
166C22×C137—22no—
167C22×C137—22no—
168C22×C137—22no—
169C22×C137—22no—
170C22×C137—22no—
171C22×C137—22no—
172C22×C137—22no—
173C22×C137—24no—
174C22×C137—24no—
175C22×C137—42no—
176C22×C137—44no—
177C22×C137—84no—
178C22×C137—82no—
179C22×C137—82no—
180C22×C137—84no—
181C22×C137—162no—
182C22×C137#37911yescharacteristic
183C22×C137#41012yes—
184C22×C137B(4,𝔽2)12yes—
185C22×C137#45812yes—
186C4×C137—42no—
187C4×C137—42no—
188C4×C137—42no—
189C4×C137—44no—
190C4×C137—84no—
191C4×C137—162no—
192C2×C137—22no—
193C2×C137—22no—
194C2×C137—22no—
195C2×C137—42no—
196C2×C137—42no—
197C2×C137—82no—
198C2×C137#178211yescharacteristic
199C2×C137#179012yes—
200#8535C13711yesmaximal, characteristic, direct product factor, Sylow
201C137#853511yescharacteristic, direct product factor, Sylow
202#3490C2×C13711yescharacteristic
203#1898C2×C13714yescomplemented
204#1819C2×C13711yescharacteristic, complemented
205#1440C2×C13711yescharacteristic, complemented
206#551—24no—
207#546—24no—
208#543C4×C13711yescharacteristic, complemented
209#538—22no—
210#534C4×C13711yescharacteristic, complemented
211#525—22no—
212C2×#120C22×C13712yes—
213#518C4×C13712yescomplemented
214C22×C22⋊C4C22×C13711yescharacteristic
215C22×C22⋊C4C22×C13712yes—
216C2×#99—24no—
217#379C22×C13712yes—
218C2×#95—22no—
219#342—22no—
220#342—24no—
221C25C23×C13711yescharacteristic
222C2×#41—22no—
223C22×D4C2×C4×C13712yes—
224C22×D4D4×C13714yes—
225C23×C4—22no—
226C23×C4D4×C13712yes—
227#126—24no—
228#124—24no—
229#123—24no—
230#123—24no—
231#123—22no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234C22⋊Q8—24no—
235#121—22no—
236#121—24no—
237#121—22no—
238#120—22no—
239#120—22no—
240#120—22no—
241#120—24no—
242C4×D4—24no—
243C4×D4—24no—
244C4×D4—24no—
245#117—22no—
246#117—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—44no—
253C2×C22⋊C4C2×C4×C13711yescharacteristic
254C2×C22⋊C4C2×C4×C13711yescharacteristic
255C2×C22⋊C4C2×C4×C13712yes—
256C2×C22⋊C4D4×C13712yes—
257C2×C22⋊C4D4×C13714yes—
258#99—48no—
259#95—44no—
260C24—42no—
261C24—42no—
262C24—42no—
263C24C22⋊C4×C13712yes—
264C24C22×C4×C13711yescharacteristic
265C24C2×D4×C13711yescharacteristic
266C24C2×D4×C13714yes—
267C24C2×D4×C13711yescharacteristic
268#41—82no—
269#41—82no—
270C2×Q8—22no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—24no—
274C2×D4—22no—
275C2×D4—24no—
276C2×D4—44no—
277C2×D4—44no—
278C2×D4—44no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4C22⋊C4×C13712yes—
293C22×C4C22⋊C4×C13714yes—
294C22×C4C22⋊C4×C13712yes—
295C22×C4C22⋊C4×C13712yes—
296C4⋊C4—24no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—24no—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—22no—
305C22⋊C4—22no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—28no—
310C22⋊C4—44no—
311C22⋊C4—44no—
312C22⋊C4—44no—
313C22⋊C4—44no—
314C42—24no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—22no—
320C23—24no—
321C23—24no—
322C23—22no—
323C23—42no—
324C23—44no—
325C23—42no—
326C23—44no—
327C23—42no—
328C23—44no—
329C23—82no—
330C23—82no—
331C23—82no—
332C23—82no—
333C23C2×C22⋊C4×C13712yes—
334C23C2×C22⋊C4×C13711yescharacteristic
335C23#120×C13712yes—
336C23#120×C13712yes—
337Q8—82no—
338D4—44no—
339D4—44no—
340D4—44no—
341D4—44no—
342D4—84no—
343D4—82no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—82no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4—84no—
365C22—22no—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—24no—
375C22—24no—
376C22—42no—
377C22—44no—
378C22—84no—
379C22—82no—
380C22—82no—
381C22—84no—
382C22—162no—
383C22#379×C13711yescharacteristic
384C22#410×C13712yes—
385C22B(4,𝔽2)×C13712yes—
386C22#458×C13712yes—
387C4—42no—
388C4—42no—
389C4—42no—
390C4—44no—
391C4—84no—
392C4—162no—
393C2—22no—
394C2—22no—
395C2—22no—
396C2—42no—
397C2—42no—
398C2—82no—
399C2#1782×C13711yescharacteristic
400C2#1790×C13712yes—
subgroupG/H
center, socleC22×C137#379
commutatorC24C22×C4×C137
Frattini, cosocleC25C23×C137
prime radical#3490×C137C2
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C25×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1898×C137C2semidirect
3. #1819×C137C2semidirect
4. #1440×C137C2semidirect
7. #543×C137C4semidirect
9. #534×C137C4semidirect
12. #518×C137C4semidirect
200. #8535C137direct (N × H)
201. C137#8535direct (N × H)
203. #1898C2×C137semidirect
204. #1819C2×C137semidirect
205. #1440C2×C137semidirect
208. #543C4×C137semidirect
210. #534C4×C137semidirect
213. #518C4×C137semidirect

Maximal subgroups

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

subgroupindex
#3490×C1372
#1898×C1372
#1819×C1372
#1440×C1372
#8535137