Finite Group Databasebeta

← C137×#8153 #46857539

Subgroups — C137×#8153 #46857539

Order 35072.

698 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C4⋊C4×C137C211yesmaximal, characteristic
2#1749×C137C212yesmaximal
3C2×#324×C137C212yesmaximal
4#1280×C137C212yesmaximal
5C22×C4⋊C4×C137C411yescharacteristic
6C22×C4⋊C4×C137C2211yescharacteristic
7C22×C42×C137C411yescharacteristic
8C22×C42×C137C411yescharacteristic
9C22×C42×C137C2211yescharacteristic
10C2×C22⋊C8×C137C2214yes—
11C2×#97×C137—22no—
12C4×C4⋊C4×C137—22no—
13C4×C4⋊C4×C137—22no—
14C2×#95×C137C411yescharacteristic
15C2×#95×C137C2211yescharacteristic
16#336×C137—22no—
17#324×C137C418yescomplemented
18C23×C4×C137C2×C411yescharacteristic
19C23×C4×C137C2×C411yescharacteristic
20C23×C4×C137C2×C411yescharacteristic
21C23×C4×C137C2×C411yescharacteristic
22C23×C4×C137C2×C411yescharacteristic
23C23×C4×C137C2×C411yescharacteristic
24C23×C4×C137C2311yescharacteristic
25C22×C8×C137—24no—
26C2×C4⋊C4×C137—22no—
27C2×C4⋊C4×C137—22no—
28C2×C4⋊C4×C137—22no—
29C2×C4⋊C4×C137—22no—
30C2×C4⋊C4×C137—22no—
31C2×C4⋊C4×C137C2×C414yes—
32C2×C42×C137—21no—
33C2×C42×C137—21no—
34C2×C42×C137—22no—
35C2×C42×C137—22no—
36C2×C42×C137—21no—
37C2×C42×C137—21no—
38C2×C42×C137—42no—
39C2×C42×C137—42no—
40C2×C42×C137D412yes—
41C2×C42×C137D412yes—
42C22⋊C8×C137—24no—
43C22⋊C8×C137C2×C418yes—
44#97×C137—44no—
45#95×C137—22no—
46#95×C137—22no—
47C24×C137C22×C411yescharacteristic
48C22×C4×C137—21no—
49C22×C4×C137—21no—
50C22×C4×C137—21no—
51C22×C4×C137—21no—
52C22×C4×C137—21no—
53C22×C4×C137—21no—
54C22×C4×C137—22no—
55C22×C4×C137—22no—
56C22×C4×C137—21no—
57C22×C4×C137—21no—
58C22×C4×C137—21no—
59C22×C4×C137—21no—
60C22×C4×C137—21no—
61C22×C4×C137—21no—
62C22×C4×C137—21no—
63C22×C4×C137—21no—
64C22×C4×C137—41no—
65C22×C4×C137—41no—
66C22×C4×C137—41no—
67C22×C4×C137—41no—
68C22×C4×C137—41no—
69C22×C4×C137—41no—
70C22×C4×C137—41no—
71C22×C4×C137—41no—
72C22×C4×C137—41no—
73C22×C4×C137—41no—
74C22×C4×C137C4212yes—
75C22×C4×C137C4212yes—
76C22×C4×C137C22⋊C412yes—
77C22×C4×C137C22⋊C411yescharacteristic
78C22×C4×C137C22⋊C411yescharacteristic
79C22×C4×C137M1612yes—
80C22×C4×C137M1612yes—
81C22×C4×C137M1612yes—
82C22×C4×C137M1612yes—
83C22×C4×C137C22×C412yes—
84C22×C4×C137C2×D412yes—
85C22×C4×C137#4112yes—
86C2×C8×C137—28no—
87C2×C8×C137—44no—
88C2×C8×C137—44no—
89C4⋊C4×C137—28no—
90C4⋊C4×C137—28no—
91C42×C137—24no—
92C42×C137—24no—
93C42×C137—24no—
94C42×C137—42no—
95C42×C137—42no—
96C42×C137—42no—
97C23×C137—21no—
98C23×C137—21no—
99C23×C137—21no—
100C23×C137—21no—
101C23×C137C2×C4211yescharacteristic
102C23×C137C2×C22⋊C411yescharacteristic
103C23×C137#11711yescharacteristic
104C23×C137C2×M1611yescharacteristic
105C23×C137C2×M1611yescharacteristic
106C23×C137C2×M1611yescharacteristic
107C23×C137C2×M1611yescharacteristic
108C2×C4×C137—24no—
109C2×C4×C137—24no—
110C2×C4×C137—22no—
111C2×C4×C137—22no—
112C2×C4×C137—21no—
113C2×C4×C137—21no—
114C2×C4×C137—22no—
115C2×C4×C137—22no—
116C2×C4×C137—22no—
117C2×C4×C137—21no—
118C2×C4×C137—22no—
119C2×C4×C137—21no—
120C2×C4×C137—22no—
121C2×C4×C137—22no—
122C2×C4×C137—22no—
123C2×C4×C137—22no—
124C2×C4×C137—22no—
125C2×C4×C137—22no—
126C2×C4×C137—22no—
127C2×C4×C137—22no—
128C2×C4×C137—42no—
129C2×C4×C137—42no—
130C2×C4×C137—42no—
131C2×C4×C137—42no—
132C2×C4×C137—42no—
133C2×C4×C137—42no—
134C2×C4×C137—42no—
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—42no—
142C2×C4×C137—42no—
143C2×C4×C137—42no—
144C2×C4×C137#9714yescomplemented
145C2×C4×C137#9912yes—
146C2×C4×C137#10112yes—
147C2×C4×C137C4×D414yes—
148C8×C137—48no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—21no—
153C22×C137—21no—
154C22×C137—21no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137C4×C22⋊C411yescharacteristic
162C22×C137C2×#9711yescharacteristic
163C22×C137C4×M1611yescharacteristic
164C22×C137#40711yescharacteristic
165C22×C137#40711yescharacteristic
166C22×C137C2×#9911yescharacteristic
167C22×C137#41011yescharacteristic
168C22×C137C2×#10111yescharacteristic
169C22×C137#41311yescharacteristic
170C22×C137#43211yescharacteristic
171C22×C137#43211yescharacteristic
172C4×C137—22no—
173C4×C137—22no—
174C4×C137—24no—
175C4×C137—24no—
176C4×C137—42no—
177C4×C137—42no—
178C4×C137—42no—
179C4×C137—42no—
180C2×C137—21no—
181C2×C137—21no—
182C2×C137—21no—
183C2×C137—21no—
184C2×C137#146311yescharacteristic
185C2×C137#146411yescharacteristic
186C2×C137#146511yescharacteristic
187C2×C137#146711yescharacteristic
188C2×C137#174911yescharacteristic
189C2×C137C4×#9911yescharacteristic
190C2×C137C4×#10111yescharacteristic
191#8153C13711yesmaximal, characteristic, direct product factor, Sylow
192C137#815311yescharacteristic, direct product factor, Sylow
193C2×C4×C4⋊C4C2×C13711yescharacteristic
194#1749C2×C13712yes—
195C2×#324C2×C13712yes—
196#1280C2×C13712yes—
197C22×C4⋊C4C4×C13711yescharacteristic
198C22×C4⋊C4C22×C13711yescharacteristic
199C22×C42C4×C13711yescharacteristic
200C22×C42C4×C13711yescharacteristic
201C22×C42C22×C13711yescharacteristic
202C2×C22⋊C8C22×C13714yes—
203C2×#97—22no—
204C4×C4⋊C4—22no—
205C4×C4⋊C4—22no—
206C2×#95C4×C13711yescharacteristic
207C2×#95C22×C13711yescharacteristic
208#336—22no—
209#324C4×C13718yescomplemented
210C23×C4C2×C4×C13711yescharacteristic
211C23×C4C2×C4×C13711yescharacteristic
212C23×C4C2×C4×C13711yescharacteristic
213C23×C4C2×C4×C13711yescharacteristic
214C23×C4C2×C4×C13711yescharacteristic
215C23×C4C2×C4×C13711yescharacteristic
216C23×C4C23×C13711yescharacteristic
217C22×C8—24no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4C2×C4×C13714yes—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42D4×C13712yes—
233C2×C42D4×C13712yes—
234C22⋊C8—24no—
235C22⋊C8C2×C4×C13718yes—
236#97—44no—
237#95—22no—
238#95—22no—
239C24C22×C4×C13711yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C42×C13712yes—
267C22×C4C42×C13712yes—
268C22×C4C22⋊C4×C13712yes—
269C22×C4C22⋊C4×C13711yescharacteristic
270C22×C4C22⋊C4×C13711yescharacteristic
271C22×C4M16×C13712yes—
272C22×C4M16×C13712yes—
273C22×C4M16×C13712yes—
274C22×C4M16×C13712yes—
275C22×C4C22×C4×C13712yes—
276C22×C4C2×D4×C13712yes—
277C22×C4#41×C13712yes—
278C2×C8—28no—
279C2×C8—44no—
280C2×C8—44no—
281C4⋊C4—28no—
282C4⋊C4—28no—
283C42—24no—
284C42—24no—
285C42—24no—
286C42—42no—
287C42—42no—
288C42—42no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23C2×C42×C13711yescharacteristic
294C23C2×C22⋊C4×C13711yescharacteristic
295C23#117×C13711yescharacteristic
296C23C2×M16×C13711yescharacteristic
297C23C2×M16×C13711yescharacteristic
298C23C2×M16×C13711yescharacteristic
299C23C2×M16×C13711yescharacteristic
300C2×C4—24no—
301C2×C4—24no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4#97×C13714yescomplemented
337C2×C4#99×C13712yes—
338C2×C4#101×C13712yes—
339C2×C4C4×D4×C13714yes—
340C8—48no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22C4×C22⋊C4×C13711yescharacteristic
354C22C2×#97×C13711yescharacteristic
355C22C4×M16×C13711yescharacteristic
356C22#407×C13711yescharacteristic
357C22#407×C13711yescharacteristic
358C22C2×#99×C13711yescharacteristic
359C22#410×C13711yescharacteristic
360C22C2×#101×C13711yescharacteristic
361C22#413×C13711yescharacteristic
362C22#432×C13711yescharacteristic
363C22#432×C13711yescharacteristic
364C4—22no—
365C4—22no—
366C4—24no—
367C4—24no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C2—21no—
373C2—21no—
374C2—21no—
375C2—21no—
376C2#1463×C13711yescharacteristic
377C2#1464×C13711yescharacteristic
378C2#1465×C13711yescharacteristic
379C2#1467×C13711yescharacteristic
380C2#1749×C13711yescharacteristic
381C2C4×#99×C13711yescharacteristic
382C2C4×#101×C13711yescharacteristic
subgroupG/H
center, socleC23×C137C2×C22⋊C4
commutatorC23C2×C42×C137
Frattini, cosocleC23×C4C23×C137
prime radicalC24×C137C22×C4
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C23×C137 ◁ C22×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
17. #324×C137C4semidirect
144. C2×C4×C137#97semidirect
191. #8153C137direct (N × H)
192. C137#8153direct (N × H)
209. #324C4×C137semidirect
336. C2×C4#97×C137semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C4⋊C4×C1372
#1749×C1372
C2×#324×C1372
#1280×C1372
#8153137