Finite Group Databasebeta

← C137×#9170 #46858556

Subgroups — C137×#9170 #46858556

Order 35072.

432 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3079×C137C211yesmaximal, characteristic
2#2925×C137C211yesmaximal, characteristic
3#1549×C137C211yesmaximal, characteristic, complemented
4C2×#333×C137C211yesmaximal, characteristic, complemented
5#1477×C137C211yesmaximal, characteristic
6#1301×C137C211yesmaximal, characteristic, complemented
7#1292×C137C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C137C2211yescharacteristic
9C4×#41×C137C2211yescharacteristic
10#477×C137—21no—
11#476×C137—21no—
12#475×C137C412yes—
13#474×C137C412yes—
14#433×C137—22no—
15#432×C137—21no—
16#431×C137—21no—
17#431×C137C2211yescharacteristic
18#428×C137—21no—
19#423×C137C2211yescharacteristic
20#423×C137C2211yescharacteristic
21C2×C4⋊C8×C137C2211yescharacteristic
22C2×C4⋊C8×C137C2211yescharacteristic
23#417×C137—21no—
24#408×C137—21no—
25#405×C137—21no—
26#333×C137—21no—
27#333×C137—21no—
28#332×C137—22no—
29Q8⋊C8×C137—21no—
30D4⋊C8×C137—21no—
31C2×#41×C137D411yescharacteristic
32C22×Q8×C137—21no—
33C2×M16×C137—21no—
34C2×M16×C137—21no—
35C22×C8×C137—21no—
36C22×C8×C137—21no—
37C4⋊Q8×C137—21no—
38C4⋊Q8×C137—21no—
39C4⋊Q8×C137—21no—
40C4⋊Q8×C137C2×C412yes—
41C4×Q8×C137C2×C411yescharacteristic
42C4×Q8×C137D411yescharacteristic
43C4×D4×C137—21no—
44C4×D4×C137—21no—
45C4×D4×C137C2×C411yescharacteristic
46C4×D4×C137D411yescharacteristic
47#117×C137—21no—
48#117×C137D411yescharacteristic
49C2×C4⋊C4×C137—21no—
50C2×C4⋊C4×C137—21no—
51C2×C42×C137—21no—
52C2×C42×C137C2311yescharacteristic
53#107×C137—22no—
54#106×C137—22no—
55C4⋊C8×C137—21no—
56C4⋊C8×C137—21no—
57C4⋊C8×C137—21no—
58C4⋊C8×C137—21no—
59C4⋊C8×C137—21no—
60C4⋊C8×C137—21no—
61C4⋊C8×C137C2×C412yes—
62C4⋊C8×C137D411yescharacteristic
63C4⋊C8×C137D411yescharacteristic
64C4⋊C8×C137D412yes—
65#103×C137—22no—
66#102×C137—22no—
67C22⋊C8×C137—22no—
68C22⋊C8×C137—22no—
69C22⋊C8×C137—21no—
70#97×C137—22no—
71C4×C8×C137—22no—
72C4×C8×C137—21no—
73#41×C137—21no—
74#41×C137—21no—
75#41×C137—21no—
76#41×C137—21no—
77C2×Q8×C137—22no—
78C2×Q8×C137—41no—
79C2×Q8×C137—82no—
80C2×Q8×C137C22⋊C411yescharacteristic
81C2×D4×C137—21no—
82C2×D4×C137C22⋊C411yescharacteristic
83C22×C4×C137—21no—
84C22×C4×C137—21no—
85C22×C4×C137—21no—
86C22×C4×C137—41no—
87C22×C4×C137C2×D411yescharacteristic
88C22×C4×C137#4111yescharacteristic
89C22×C4×C137#4111yescharacteristic
90M16×C137—41no—
91M16×C137—41no—
92M16×C137—41no—
93M16×C137—41no—
94C2×C8×C137—22no—
95C2×C8×C137—21no—
96C2×C8×C137—22no—
97C2×C8×C137—21no—
98C2×C8×C137—41no—
99C2×C8×C137—41no—
100C2×C8×C137—41no—
101C2×C8×C137—41no—
102C2×C8×C137—81no—
103C4⋊C4×C137—21no—
104C4⋊C4×C137—21no—
105C4⋊C4×C137—22no—
106C4⋊C4×C137—22no—
107C4⋊C4×C137—41no—
108C4⋊C4×C137—41no—
109C4⋊C4×C137C22⋊C411yescharacteristic
110C4⋊C4×C137C22⋊C411yescharacteristic
111C22⋊C4×C137—21no—
112C22⋊C4×C137—21no—
113C22⋊C4×C137—21no—
114C42×C137—21no—
115C42×C137—21no—
116C42×C137—21no—
117C42×C137C22×C411yescharacteristic
118C42×C137C2×D411yescharacteristic
119C42×C137C2×D411yescharacteristic
120C42×C137C2×D411yescharacteristic
121C23×C137—21no—
122C23×C137#12311yescharacteristic
123Q8×C137—21no—
124Q8×C137—21no—
125Q8×C137—82no—
126Q8×C137—82no—
127D4×C137—21no—
128D4×C137—21no—
129D4×C137—41no—
130D4×C137—41no—
131C2×C4×C137—21no—
132C2×C4×C137—21no—
133C2×C4×C137—21no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
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—81no—
143C2×C4×C137—82no—
144C2×C4×C137C2×C22⋊C411yescharacteristic
145C2×C4×C137C4×D411yescharacteristic
146C2×C4×C137C4×D411yescharacteristic
147C2×C4×C137#12011yescharacteristic
148C2×C4×C137#12111yescharacteristic
149C2×C4×C137#12111yescharacteristic
150C2×C4×C137#13511yescharacteristic
151C2×C4×C137#13511yescharacteristic
152C2×C4×C137#13711yescharacteristic
153C2×C4×C137#13711yescharacteristic
154C8×C137—42no—
155C8×C137—41no—
156C8×C137—41no—
157C8×C137—42no—
158C8×C137—81no—
159C8×C137—81no—
160C22×C137—21no—
161C22×C137—21no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137#38611yescharacteristic
166C22×C137#48511yescharacteristic
167C22×C137#48511yescharacteristic
168C4×C137—21no—
169C4×C137—21no—
170C4×C137—21no—
171C4×C137—41no—
172C4×C137—41no—
173C4×C137—41no—
174C4×C137—82no—
175C4×C137#41611yescharacteristic
176C4×C137#41911yescharacteristic
177C4×C137#45211yescharacteristic
178C4×C137#45211yescharacteristic
179C4×C137#47111yescharacteristic
180C4×C137#47311yescharacteristic
181C2×C137—21no—
182C2×C137—21no—
183C2×C137—81no—
184C2×C137#187311yescharacteristic
185C2×C137#190311yescharacteristic
186C2×C137#190811yescharacteristic
187#9170C13711yesmaximal, characteristic, direct product factor, Sylow
188C137#917011yescharacteristic, direct product factor, Sylow
189#3079C2×C13711yescharacteristic
190#2925C2×C13711yescharacteristic
191#1549C2×C13711yescharacteristic, complemented
192C2×#333C2×C13711yescharacteristic, complemented
193#1477C2×C13711yescharacteristic
194#1301C2×C13711yescharacteristic, complemented
195#1292C2×C13711yescharacteristic, complemented
196C2×C4⋊Q8C22×C13711yescharacteristic
197C4×#41C22×C13711yescharacteristic
198#477—21no—
199#476—21no—
200#475C4×C13712yes—
201#474C4×C13712yes—
202#433—22no—
203#432—21no—
204#431—21no—
205#431C22×C13711yescharacteristic
206#428—21no—
207#423C22×C13711yescharacteristic
208#423C22×C13711yescharacteristic
209C2×C4⋊C8C22×C13711yescharacteristic
210C2×C4⋊C8C22×C13711yescharacteristic
211#417—21no—
212#408—21no—
213#405—21no—
214#333—21no—
215#333—21no—
216#332—22no—
217Q8⋊C8—21no—
218D4⋊C8—21no—
219C2×#41D4×C13711yescharacteristic
220C22×Q8—21no—
221C2×M16—21no—
222C2×M16—21no—
223C22×C8—21no—
224C22×C8—21no—
225C4⋊Q8—21no—
226C4⋊Q8—21no—
227C4⋊Q8—21no—
228C4⋊Q8C2×C4×C13712yes—
229C4×Q8C2×C4×C13711yescharacteristic
230C4×Q8D4×C13711yescharacteristic
231C4×D4—21no—
232C4×D4—21no—
233C4×D4C2×C4×C13711yescharacteristic
234C4×D4D4×C13711yescharacteristic
235#117—21no—
236#117D4×C13711yescharacteristic
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C42—21no—
240C2×C42C23×C13711yescharacteristic
241#107—22no—
242#106—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8C2×C4×C13712yes—
250C4⋊C8D4×C13712yes—
251C4⋊C8D4×C13711yescharacteristic
252C4⋊C8D4×C13711yescharacteristic
253#103—22no—
254#102—22no—
255C22⋊C8—22no—
256C22⋊C8—22no—
257C22⋊C8—21no—
258#97—22no—
259C4×C8—22no—
260C4×C8—21no—
261#41—21no—
262#41—21no—
263#41—21no—
264#41—21no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—82no—
268C2×Q8C22⋊C4×C13711yescharacteristic
269C2×D4—21no—
270C2×D4C22⋊C4×C13711yescharacteristic
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—41no—
275C22×C4C2×D4×C13711yescharacteristic
276C22×C4#41×C13711yescharacteristic
277C22×C4#41×C13711yescharacteristic
278M16—41no—
279M16—41no—
280M16—41no—
281M16—41no—
282C2×C8—22no—
283C2×C8—21no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—81no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4C22⋊C4×C13711yescharacteristic
298C4⋊C4C22⋊C4×C13711yescharacteristic
299C22⋊C4—21no—
300C22⋊C4—21no—
301C22⋊C4—21no—
302C42—21no—
303C42—21no—
304C42—21no—
305C42C22×C4×C13711yescharacteristic
306C42C2×D4×C13711yescharacteristic
307C42C2×D4×C13711yescharacteristic
308C42C2×D4×C13711yescharacteristic
309C23—21no—
310C23#123×C13711yescharacteristic
311Q8—21no—
312Q8—21no—
313Q8—82no—
314Q8—82no—
315D4—21no—
316D4—21no—
317D4—41no—
318D4—41no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—81no—
331C2×C4—82no—
332C2×C4C2×C22⋊C4×C13711yescharacteristic
333C2×C4C4×D4×C13711yescharacteristic
334C2×C4C4×D4×C13711yescharacteristic
335C2×C4#120×C13711yescharacteristic
336C2×C4#121×C13711yescharacteristic
337C2×C4#121×C13711yescharacteristic
338C2×C4#135×C13711yescharacteristic
339C2×C4#135×C13711yescharacteristic
340C2×C4#137×C13711yescharacteristic
341C2×C4#137×C13711yescharacteristic
342C8—42no—
343C8—41no—
344C8—41no—
345C8—42no—
346C8—81no—
347C8—81no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22#386×C13711yescharacteristic
354C22#485×C13711yescharacteristic
355C22#485×C13711yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—41no—
360C4—41no—
361C4—41no—
362C4—82no—
363C4#416×C13711yescharacteristic
364C4#419×C13711yescharacteristic
365C4#452×C13711yescharacteristic
366C4#452×C13711yescharacteristic
367C4#471×C13711yescharacteristic
368C4#473×C13711yescharacteristic
369C2—21no—
370C2—21no—
371C2—81no—
372C2#1873×C13711yescharacteristic
373C2#1903×C13711yescharacteristic
374C2#1908×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#386
commutatorC42C22×C4×C137
Frattini, cosocleC2×C42C23×C137
prime radicalC2×#41×C137D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #1549×C137C2semidirect
4. C2×#333×C137C2semidirect
6. #1301×C137C2semidirect
7. #1292×C137C2semidirect
187. #9170C137direct (N × H)
188. C137#9170direct (N × H)
191. #1549C2×C137semidirect
192. C2×#333C2×C137semidirect
194. #1301C2×C137semidirect
195. #1292C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3079×C1372
#2925×C1372
#1549×C1372
C2×#333×C1372
#1477×C1372
#1301×C1372
#1292×C1372
#9170137