Finite Group Databasebeta

← C137×#8779 #46858165

Subgroups — C137×#8779 #46858165

Order 35072.

466 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 114 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#2357×C137C211yesmaximal, characteristic
2C4×#99×C137C211yesmaximal, characteristic
3#1465×C137C211yesmaximal, characteristic
4#1297×C137C212yesmaximal, complemented
5#1284×C137C212yesmaximal, complemented
6C2×C4×D4×C137C2211yescharacteristic
7C2×#99×C137—21no—
8#407×C137C2212yes—
9#390×C137—21no—
10#388×C137—21no—
11#388×C137—21no—
12#382×C137C411yescharacteristic, complemented
13#381×C137C411yescharacteristic, complemented
14#381×C137C2211yescharacteristic
15C4×C4⋊C4×C137—21no—
16C4×C4⋊C4×C137C411yescharacteristic, complemented
17C4×C22⋊C4×C137C411yescharacteristic, complemented
18C4×C22⋊C4×C137C2212yes—
19C4×C22⋊C4×C137C2211yescharacteristic
20#342×C137—21no—
21#324×C137—21no—
22#323×C137—21no—
23C22×D4×C137D411yescharacteristic
24C23×C4×C137C2×C411yescharacteristic
25C23×C4×C137C2311yescharacteristic
26C2×M16×C137—22no—
27C4×D4×C137—22no—
28C4×D4×C137—22no—
29C2×C4⋊C4×C137—21no—
30C2×C4⋊C4×C137—21no—
31C2×C4⋊C4×C137D411yescharacteristic
32C2×C22⋊C4×C137—22no—
33C2×C22⋊C4×C137—21no—
34C2×C22⋊C4×C137—21no—
35C2×C22⋊C4×C137—21no—
36C2×C22⋊C4×C137—21no—
37C2×C22⋊C4×C137D411yescharacteristic
38C2×C22⋊C4×C137D411yescharacteristic
39C2×C42×C137—21no—
40C2×C42×C137—21no—
41C2×C42×C137—21no—
42C2×C42×C137—22no—
43C2×C42×C137C2×C411yescharacteristic
44#99×C137—24no—
45C22⋊C8×C137—22no—
46C22⋊C8×C137—22no—
47#95×C137—21no—
48#95×C137—21no—
49#95×C137—21no—
50#95×C137—22no—
51#95×C137C2×C411yescharacteristic
52#95×C137C2×C411yescharacteristic
53#95×C137C2×C411yescharacteristic
54#95×C137C2×C411yescharacteristic
55C24×C137C22⋊C411yescharacteristic
56C24×C137C2×D411yescharacteristic
57C2×D4×C137—22no—
58C2×D4×C137—22no—
59C2×D4×C137—41no—
60C22×C4×C137—21no—
61C22×C4×C137—21no—
62C22×C4×C137—21no—
63C22×C4×C137—21no—
64C22×C4×C137—21no—
65C22×C4×C137—21no—
66C22×C4×C137—42no—
67C22×C4×C137—81no—
68C22×C4×C137—81no—
69C22×C4×C137C22⋊C411yescharacteristic
70C22×C4×C137C22⋊C411yescharacteristic
71C22×C4×C137C22⋊C411yescharacteristic
72C22×C4×C137C22×C411yescharacteristic
73C22×C4×C137C2×D411yescharacteristic
74C22×C4×C137#4112yes—
75C22×C4×C137#4112yes—
76M16×C137—82no—
77C2×C8×C137—42no—
78C4⋊C4×C137—22no—
79C4⋊C4×C137—42no—
80C4⋊C4×C137—42no—
81C22⋊C4×C137—22no—
82C22⋊C4×C137—22no—
83C22⋊C4×C137—24no—
84C22⋊C4×C137—42no—
85C22⋊C4×C137—42no—
86C22⋊C4×C137—42no—
87C22⋊C4×C137—81no—
88C22⋊C4×C137—81no—
89C42×C137—21no—
90C42×C137—21no—
91C42×C137—42no—
92C42×C137—81no—
93C42×C137—81no—
94C42×C137—82no—
95C23×C137—21no—
96C23×C137—21no—
97C23×C137—21no—
98C23×C137—21no—
99C23×C137—21no—
100C23×C137—41no—
101C23×C137—41no—
102C23×C137—41no—
103C23×C137—41no—
104C23×C137C2×C22⋊C411yescharacteristic
105C23×C137#12312yes—
106D4×C137—44no—
107C2×C4×C137—21no—
108C2×C4×C137—21no—
109C2×C4×C137—22no—
110C2×C4×C137—21no—
111C2×C4×C137—21no—
112C2×C4×C137—41no—
113C2×C4×C137—42no—
114C2×C4×C137—42no—
115C2×C4×C137—41no—
116C2×C4×C137—41no—
117C2×C4×C137—41no—
118C2×C4×C137—41no—
119C2×C4×C137—41no—
120C2×C4×C137—41no—
121C2×C4×C137—41no—
122C2×C4×C137—44no—
123C2×C4×C137—41no—
124C2×C4×C137—41no—
125C2×C4×C137—41no—
126C2×C4×C137—41no—
127C2×C4×C137—41no—
128C2×C4×C137—41no—
129C2×C4×C137—41no—
130C2×C4×C137—81no—
131C2×C4×C137—81no—
132C2×C4×C137—82no—
133C2×C4×C137—82no—
134C2×C4×C137#10412yes—
135C2×C4×C137#11711yescharacteristic
136C2×C4×C137#11711yescharacteristic
137C2×C4×C137#12312yes—
138C8×C137—82no—
139C22×C137—21no—
140C22×C137—21no—
141C22×C137—22no—
142C22×C137—22no—
143C22×C137—22no—
144C22×C137—41no—
145C22×C137—41no—
146C22×C137—41no—
147C22×C137—81no—
148C22×C137—81no—
149C22×C137—81no—
150C22×C137—81no—
151C22×C137#38111yescharacteristic
152C22×C137C2×#10411yescharacteristic
153C22×C137#42111yescharacteristic
154C4×C137—22no—
155C4×C137—22no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—42no—
159C4×C137—41no—
160C4×C137—41no—
161C4×C137—81no—
162C4×C137—81no—
163C4×C137—81no—
164C4×C137—84no—
165C2×C137—21no—
166C2×C137—21no—
167C2×C137—42no—
168C2×C137—81no—
169C2×C137#182611yescharacteristic
170C2×C137#182711yescharacteristic
171C2×C137#182811yescharacteristic
172#8779C13711yesmaximal, characteristic, direct product factor, Sylow
173C137#877911yescharacteristic, direct product factor, Sylow
174#2357C2×C13711yescharacteristic
175C4×#99C2×C13711yescharacteristic
176#1465C2×C13711yescharacteristic
177#1297C2×C13712yescomplemented
178#1284C2×C13712yescomplemented
179C2×C4×D4C22×C13711yescharacteristic
180C2×#99—21no—
181#407C22×C13712yes—
182#390—21no—
183#388—21no—
184#388—21no—
185#382C4×C13711yescharacteristic, complemented
186#381C4×C13711yescharacteristic, complemented
187#381C22×C13711yescharacteristic
188C4×C4⋊C4—21no—
189C4×C4⋊C4C4×C13711yescharacteristic, complemented
190C4×C22⋊C4C4×C13711yescharacteristic, complemented
191C4×C22⋊C4C22×C13712yes—
192C4×C22⋊C4C22×C13711yescharacteristic
193#342—21no—
194#324—21no—
195#323—21no—
196C22×D4D4×C13711yescharacteristic
197C23×C4C2×C4×C13711yescharacteristic
198C23×C4C23×C13711yescharacteristic
199C2×M16—22no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4D4×C13711yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4D4×C13711yescharacteristic
211C2×C22⋊C4D4×C13711yescharacteristic
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42C2×C4×C13711yescharacteristic
217#99—24no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220#95—21no—
221#95—21no—
222#95—21no—
223#95—22no—
224#95C2×C4×C13711yescharacteristic
225#95C2×C4×C13711yescharacteristic
226#95C2×C4×C13711yescharacteristic
227#95C2×C4×C13711yescharacteristic
228C24C22⋊C4×C13711yescharacteristic
229C24C2×D4×C13711yescharacteristic
230C2×D4—22no—
231C2×D4—22no—
232C2×D4—41no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—42no—
240C22×C4—81no—
241C22×C4—81no—
242C22×C4C22⋊C4×C13711yescharacteristic
243C22×C4C22⋊C4×C13711yescharacteristic
244C22×C4C22⋊C4×C13711yescharacteristic
245C22×C4C22×C4×C13711yescharacteristic
246C22×C4C2×D4×C13711yescharacteristic
247C22×C4#41×C13712yes—
248C22×C4#41×C13712yes—
249M16—82no—
250C2×C8—42no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C22⋊C4—22no—
255C22⋊C4—22no—
256C22⋊C4—24no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—81no—
261C22⋊C4—81no—
262C42—21no—
263C42—21no—
264C42—42no—
265C42—81no—
266C42—81no—
267C42—82no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23—41no—
276C23—41no—
277C23C2×C22⋊C4×C13711yescharacteristic
278C23#123×C13712yes—
279D4—44no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—41no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—44no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4#104×C13712yes—
308C2×C4#117×C13711yescharacteristic
309C2×C4#117×C13711yescharacteristic
310C2×C4#123×C13712yes—
311C8—82no—
312C22—21no—
313C22—21no—
314C22—22no—
315C22—22no—
316C22—22no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22—81no—
322C22—81no—
323C22—81no—
324C22#381×C13711yescharacteristic
325C22C2×#104×C13711yescharacteristic
326C22#421×C13711yescharacteristic
327C4—22no—
328C4—22no—
329C4—21no—
330C4—21no—
331C4—42no—
332C4—41no—
333C4—41no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—84no—
338C2—21no—
339C2—21no—
340C2—42no—
341C2—81no—
342C2#1826×C13711yescharacteristic
343C2#1827×C13711yescharacteristic
344C2#1828×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#381
commutatorC22×C4C22×C4×C137
Frattini, cosocleC23×C4C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1297×C137C2semidirect
5. #1284×C137C2semidirect
12. #382×C137C4semidirect
13. #381×C137C4semidirect
16. C4×C4⋊C4×C137C4semidirect
17. C4×C22⋊C4×C137C4semidirect
172. #8779C137direct (N × H)
173. C137#8779direct (N × H)
177. #1297C2×C137semidirect
178. #1284C2×C137semidirect
185. #382C4×C137semidirect
186. #381C4×C137semidirect
189. C4×C4⋊C4C4×C137semidirect
190. C4×C22⋊C4C4×C137semidirect

Maximal subgroups

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

subgroupindex
#2357×C1372
C4×#99×C1372
#1465×C1372
#1297×C1372
#1284×C1372
#8779137