Finite Group Databasebeta

← C137×#11553 #46860939

Subgroups — C137×#11553 #46860939

Order 35072.

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

normal subgroups only

This group has 440 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#2504×C137C211yesmaximal, characteristic
2#1463×C137C211yesmaximal, characteristic, complemented
3C2×#324×C137C211yesmaximal, characteristic, complemented
4#1324×C137C212yesmaximal, complemented
5#1318×C137C212yesmaximal, complemented
6C2×#121×C137C411yescharacteristic
7C22×C4⋊C4×C137C2211yescharacteristic
8#407×C137—21no—
9C2×C22⋊C8×C137—21no—
10#397×C137—21no—
11#397×C137—21no—
12#390×C137C411yescharacteristic
13#390×C137C2211yescharacteristic
14#386×C137—21no—
15#386×C137—21no—
16#382×C137C411yescharacteristic
17#382×C137C2211yescharacteristic
18C4×C22⋊C4×C137C411yescharacteristic
19#324×C137—21no—
20#324×C137—21no—
21#324×C137C2212yes—
22#324×C137C2212yes—
23C22×D4×C137—21no—
24C22×D4×C137C2×C411yescharacteristic
25C23×C4×C137C2×C411yescharacteristic
26C23×C4×C137D411yescharacteristic
27C23×C4×C137D411yescharacteristic
28C2×M16×C137—41no—
29C22×C8×C137—41no—
30#121×C137—41no—
31#121×C137—41no—
32C2×C4⋊C4×C137—41no—
33C2×C4⋊C4×C137—41no—
34C2×C4⋊C4×C137C2×C411yescharacteristic
35C2×C4⋊C4×C137C2×C411yescharacteristic
36C2×C4⋊C4×C137D411yescharacteristic
37C2×C4⋊C4×C137D411yescharacteristic
38C2×C4⋊C4×C137C2311yescharacteristic
39C2×C22⋊C4×C137—21no—
40C2×C22⋊C4×C137—21no—
41C2×C22⋊C4×C137—21no—
42C2×C22⋊C4×C137—21no—
43C2×C22⋊C4×C137—21no—
44C2×C22⋊C4×C137—21no—
45C2×C42×C137C2×C411yescharacteristic
46C2×C42×C137C2×C411yescharacteristic
47C22⋊C8×C137—22no—
48C22⋊C8×C137—22no—
49C22⋊C8×C137—41no—
50C22⋊C8×C137—41no—
51#95×C137—21no—
52#95×C137—21no—
53#95×C137—21no—
54#95×C137—21no—
55#95×C137—21no—
56C24×C137—21no—
57C24×C137C22⋊C411yescharacteristic
58C2×D4×C137—22no—
59C2×D4×C137—41no—
60C2×D4×C137—41no—
61C2×D4×C137—42no—
62C2×D4×C137—42no—
63C2×D4×C137—41no—
64C2×D4×C137—41no—
65C22×C4×C137—21no—
66C22×C4×C137—21no—
67C22×C4×C137—21no—
68C22×C4×C137—21no—
69C22×C4×C137—21no—
70C22×C4×C137—21no—
71C22×C4×C137—22no—
72C22×C4×C137—21no—
73C22×C4×C137—21no—
74C22×C4×C137—21no—
75C22×C4×C137—41no—
76C22×C4×C137—81no—
77C22×C4×C137—81no—
78C22×C4×C137C22⋊C411yescharacteristic
79C22×C4×C137C22⋊C411yescharacteristic
80C22×C4×C137C22⋊C411yescharacteristic
81C22×C4×C137C22×C411yescharacteristic
82C22×C4×C137C2×D411yescharacteristic
83C22×C4×C137C2×D411yescharacteristic
84M16×C137—82no—
85C2×C8×C137—42no—
86C2×C8×C137—42no—
87C2×C8×C137—82no—
88C4⋊C4×C137—41no—
89C4⋊C4×C137—41no—
90C4⋊C4×C137—41no—
91C4⋊C4×C137—41no—
92C4⋊C4×C137—41no—
93C22⋊C4×C137—42no—
94C22⋊C4×C137—42no—
95C22⋊C4×C137—81no—
96C22⋊C4×C137—81no—
97C22⋊C4×C137—81no—
98C22⋊C4×C137—81no—
99C42×C137—22no—
100C42×C137—41no—
101C23×C137—22no—
102C23×C137—21no—
103C23×C137—21no—
104C23×C137—41no—
105C23×C137—41no—
106C23×C137—41no—
107C23×C137—41no—
108C23×C137—41no—
109C23×C137—41no—
110C23×C137—41no—
111C23×C137—41no—
112C23×C137#9911yescharacteristic
113C23×C137#9911yescharacteristic
114C23×C137C2×C22⋊C411yescharacteristic
115D4×C137—82no—
116D4×C137—82no—
117D4×C137—82no—
118C2×C4×C137—21no—
119C2×C4×C137—21no—
120C2×C4×C137—21no—
121C2×C4×C137—21no—
122C2×C4×C137—21no—
123C2×C4×C137—21no—
124C2×C4×C137—21no—
125C2×C4×C137—21no—
126C2×C4×C137—21no—
127C2×C4×C137—42no—
128C2×C4×C137—42no—
129C2×C4×C137—42no—
130C2×C4×C137—42no—
131C2×C4×C137—41no—
132C2×C4×C137—41no—
133C2×C4×C137—41no—
134C2×C4×C137—41no—
135C2×C4×C137—41no—
136C2×C4×C137—41no—
137C2×C4×C137—41no—
138C2×C4×C137—41no—
139C2×C4×C137—41no—
140C2×C4×C137—41no—
141C2×C4×C137—41no—
142C2×C4×C137—41no—
143C2×C4×C137—41no—
144C2×C4×C137—81no—
145C2×C4×C137—81no—
146C2×C4×C137#13112yes—
147C8×C137—82no—
148C8×C137—82no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—42no—
157C22×C137—42no—
158C22×C137—81no—
159C22×C137—81no—
160C22×C137—81no—
161C22×C137—81no—
162C22×C137—81no—
163C22×C137—81no—
164C22×C137—81no—
165C22×C137#35512yes—
166C22×C137#40811yescharacteristic
167C22×C137C2×#9911yescharacteristic
168C22×C137#41311yescharacteristic
169C4×C137—22no—
170C4×C137—41no—
171C4×C137—41no—
172C4×C137—41no—
173C4×C137—41no—
174C4×C137—41no—
175C4×C137—81no—
176C4×C137—81no—
177C4×C137—81no—
178C2×C137—21no—
179C2×C137—21no—
180C2×C137—22no—
181C2×C137—41no—
182C2×C137—161no—
183C2×C137#146811yescharacteristic
184C2×C137#212711yescharacteristic
185C2×C137C2×#35511yescharacteristic
186#11553C13711yesmaximal, characteristic, direct product factor, Sylow
187C137#1155311yescharacteristic, direct product factor, Sylow
188#2504C2×C13711yescharacteristic
189#1463C2×C13711yescharacteristic, complemented
190C2×#324C2×C13711yescharacteristic, complemented
191#1324C2×C13712yescomplemented
192#1318C2×C13712yescomplemented
193C2×#121C4×C13711yescharacteristic
194C22×C4⋊C4C22×C13711yescharacteristic
195#407—21no—
196C2×C22⋊C8—21no—
197#397—21no—
198#397—21no—
199#390C4×C13711yescharacteristic
200#390C22×C13711yescharacteristic
201#386—21no—
202#386—21no—
203#382C4×C13711yescharacteristic
204#382C22×C13711yescharacteristic
205C4×C22⋊C4C4×C13711yescharacteristic
206#324—21no—
207#324—21no—
208#324C22×C13712yes—
209#324C22×C13712yes—
210C22×D4—21no—
211C22×D4C2×C4×C13711yescharacteristic
212C23×C4C2×C4×C13711yescharacteristic
213C23×C4D4×C13711yescharacteristic
214C23×C4D4×C13711yescharacteristic
215C2×M16—41no—
216C22×C8—41no—
217#121—41no—
218#121—41no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4C2×C4×C13711yescharacteristic
222C2×C4⋊C4C2×C4×C13711yescharacteristic
223C2×C4⋊C4D4×C13711yescharacteristic
224C2×C4⋊C4D4×C13711yescharacteristic
225C2×C4⋊C4C23×C13711yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C42C2×C4×C13711yescharacteristic
233C2×C42C2×C4×C13711yescharacteristic
234C22⋊C8—22no—
235C22⋊C8—22no—
236C22⋊C8—41no—
237C22⋊C8—41no—
238#95—21no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243C24—21no—
244C24C22⋊C4×C13711yescharacteristic
245C2×D4—22no—
246C2×D4—41no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—41no—
263C22×C4—81no—
264C22×C4—81no—
265C22×C4C22⋊C4×C13711yescharacteristic
266C22×C4C22⋊C4×C13711yescharacteristic
267C22×C4C22⋊C4×C13711yescharacteristic
268C22×C4C22×C4×C13711yescharacteristic
269C22×C4C2×D4×C13711yescharacteristic
270C22×C4C2×D4×C13711yescharacteristic
271M16—82no—
272C2×C8—42no—
273C2×C8—42no—
274C2×C8—82no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—81no—
283C22⋊C4—81no—
284C22⋊C4—81no—
285C22⋊C4—81no—
286C42—22no—
287C42—41no—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—41no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23#99×C13711yescharacteristic
300C23#99×C13711yescharacteristic
301C23C2×C22⋊C4×C13711yescharacteristic
302D4—82no—
303D4—82no—
304D4—82no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—81no—
332C2×C4—81no—
333C2×C4#131×C13712yes—
334C8—82no—
335C8—82no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—42no—
344C22—42no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22—81no—
349C22—81no—
350C22—81no—
351C22—81no—
352C22#355×C13712yes—
353C22#408×C13711yescharacteristic
354C22C2×#99×C13711yescharacteristic
355C22#413×C13711yescharacteristic
356C4—22no—
357C4—41no—
358C4—41no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—81no—
363C4—81no—
364C4—81no—
365C2—21no—
366C2—21no—
367C2—22no—
368C2—41no—
369C2—161no—
370C2#1468×C13711yescharacteristic
371C2#2127×C13711yescharacteristic
372C2C2×#355×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C4⋊C4C23×C137
prime radicalC2×#121×C137C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C24×C137 ◁ C22×C4⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1463×C137C2semidirect
3. C2×#324×C137C2semidirect
4. #1324×C137C2semidirect
5. #1318×C137C2semidirect
186. #11553C137direct (N × H)
187. C137#11553direct (N × H)
189. #1463C2×C137semidirect
190. C2×#324C2×C137semidirect
191. #1324C2×C137semidirect
192. #1318C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2504×C1372
#1463×C1372
C2×#324×C1372
#1324×C1372
#1318×C1372
#11553137