Finite Group Databasebeta

← C137×#12135 #46861521

Subgroups — C137×#12135 #46861521

Order 35072.

414 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 70 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#2517×C137C211yesmaximal, characteristic, complemented
2#2044×C137C211yesmaximal, characteristic
3#1613×C137C211yesmaximal, characteristic
4#1607×C137C211yesmaximal, characteristic
5#1352×C137C211yesmaximal, characteristic, complemented
6#1344×C137C211yesmaximal, characteristic, complemented
7#1326×C137C211yesmaximal, characteristic, complemented
8#534×C137C2211yescharacteristic
9#481×C137—21no—
10#480×C137—21no—
11C2×#99×C137—21no—
12#399×C137C2211yescharacteristic
13#397×C137—21no—
14#397×C137C2211yescharacteristic
15#387×C137—21no—
16#387×C137—21no—
17#387×C137—21no—
18#386×C137—21no—
19#385×C137C2211yescharacteristic
20#382×C137—21no—
21C4×C22⋊C4×C137—21no—
22#342×C137C2211yescharacteristic
23#327×C137—21no—
24#327×C137—21no—
25#323×C137C2211yescharacteristic
26#323×C137C2211yescharacteristic
27C22×D4×C137—21no—
28C23×C4×C137D411yescharacteristic
29#123×C137—41no—
30#121×C137—21no—
31#121×C137—21no—
32#120×C137—41no—
33C2×C4⋊C4×C137—21no—
34C2×C4⋊C4×C137—21no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137D411yescharacteristic
37C2×C4⋊C4×C137D411yescharacteristic
38C2×C22⋊C4×C137—21no—
39C2×C22⋊C4×C137—21no—
40C2×C22⋊C4×C137—21no—
41C2×C22⋊C4×C137—41no—
42C2×C22⋊C4×C137D411yescharacteristic
43C2×C22⋊C4×C137C2311yescharacteristic
44C2×C42×C137—21no—
45C2×C42×C137—21no—
46#107×C137—41no—
47#106×C137—41no—
48#102×C137—41no—
49#102×C137—41no—
50#99×C137—42no—
51C22⋊C8×C137—21no—
52C22⋊C8×C137—21no—
53#95×C137—21no—
54#95×C137—21no—
55#95×C137—21no—
56#95×C137—21no—
57#95×C137—21no—
58#95×C137—21no—
59#95×C137—21no—
60#95×C137—41no—
61#95×C137D411yescharacteristic
62#95×C137D411yescharacteristic
63C24×C137—21no—
64C24×C137C2×D411yescharacteristic
65C2×D4×C137—21no—
66C2×D4×C137—21no—
67C2×D4×C137—41no—
68C2×D4×C137—41no—
69C2×D4×C137—41no—
70C2×D4×C137—41no—
71C2×D4×C137—41no—
72C2×D4×C137—41no—
73C22×C4×C137—21no—
74C22×C4×C137—21no—
75C22×C4×C137—21no—
76C22×C4×C137—21no—
77C22×C4×C137—21no—
78C22×C4×C137—21no—
79C22×C4×C137—21no—
80C22×C4×C137—42no—
81C22×C4×C137—41no—
82C22×C4×C137—41no—
83C22×C4×C137C2×D411yescharacteristic
84C22×C4×C137C2×D411yescharacteristic
85C2×C8×C137—41no—
86C2×C8×C137—41no—
87C4⋊C4×C137—21no—
88C4⋊C4×C137—21no—
89C4⋊C4×C137—41no—
90C4⋊C4×C137—41no—
91C4⋊C4×C137—42no—
92C4⋊C4×C137—42no—
93C4⋊C4×C137—42no—
94C22⋊C4×C137—21no—
95C22⋊C4×C137—21no—
96C22⋊C4×C137—41no—
97C22⋊C4×C137—42no—
98C22⋊C4×C137—41no—
99C22⋊C4×C137—41no—
100C22⋊C4×C137—41no—
101C22⋊C4×C137—41no—
102C22⋊C4×C137—41no—
103C22⋊C4×C137—41no—
104C22⋊C4×C137—82no—
105C42×C137—42no—
106C42×C137—81no—
107C23×C137—21no—
108C23×C137—21no—
109C23×C137—41no—
110C23×C137—41no—
111C23×C137—41no—
112C23×C137—41no—
113C23×C137—41no—
114C23×C137—81no—
115C23×C137—81no—
116C23×C137#12011yescharacteristic
117C23×C137#13511yescharacteristic
118C23×C137AGL(1,ℤ/8)11yescharacteristic
119D4×C137—81no—
120D4×C137—81no—
121D4×C137—81no—
122D4×C137—81no—
123D4×C137—81no—
124D4×C137—81no—
125C2×C4×C137—22no—
126C2×C4×C137—22no—
127C2×C4×C137—21no—
128C2×C4×C137—21no—
129C2×C4×C137—41no—
130C2×C4×C137—41no—
131C2×C4×C137—41no—
132C2×C4×C137—41no—
133C2×C4×C137—41no—
134C2×C4×C137—41no—
135C2×C4×C137—42no—
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—41no—
145C2×C4×C137—41no—
146C2×C4×C137—41no—
147C2×C4×C137—41no—
148C2×C4×C137—41no—
149C2×C4×C137—41no—
150C2×C4×C137—41no—
151C2×C4×C137—81no—
152C2×C4×C137—81no—
153C2×C4×C137—81no—
154C2×C4×C137—81no—
155C8×C137—81no—
156C8×C137—81no—
157C22×C137—21no—
158C22×C137—21no—
159C22×C137—21no—
160C22×C137—21no—
161C22×C137—41no—
162C22×C137—41no—
163C22×C137—41no—
164C22×C137—41no—
165C22×C137—81no—
166C22×C137—82no—
167C22×C137—81no—
168C22×C137—81no—
169C22×C137—81no—
170C22×C137—161no—
171C22×C137#44911yescharacteristic
172C22×C137#45411yescharacteristic
173C22×C137B(4,𝔽2)11yescharacteristic
174C4×C137—42no—
175C4×C137—41no—
176C4×C137—42no—
177C4×C137—41no—
178C4×C137—81no—
179C4×C137—81no—
180C4×C137—81no—
181C4×C137—81no—
182C4×C137—81no—
183C4×C137—161no—
184C2×C137—21no—
185C2×C137—21no—
186C2×C137—41no—
187C2×C137—41no—
188C2×C137—161no—
189C2×C137#159711yescharacteristic
190C2×C137#219711yescharacteristic
191C2×C137#219811yescharacteristic
192#12135C13711yesmaximal, characteristic, direct product factor, Sylow
193C137#1213511yescharacteristic, direct product factor, Sylow
194#2517C2×C13711yescharacteristic, complemented
195#2044C2×C13711yescharacteristic
196#1613C2×C13711yescharacteristic
197#1607C2×C13711yescharacteristic
198#1352C2×C13711yescharacteristic, complemented
199#1344C2×C13711yescharacteristic, complemented
200#1326C2×C13711yescharacteristic, complemented
201#534C22×C13711yescharacteristic
202#481—21no—
203#480—21no—
204C2×#99—21no—
205#399C22×C13711yescharacteristic
206#397—21no—
207#397C22×C13711yescharacteristic
208#387—21no—
209#387—21no—
210#387—21no—
211#386—21no—
212#385C22×C13711yescharacteristic
213#382—21no—
214C4×C22⋊C4—21no—
215#342C22×C13711yescharacteristic
216#327—21no—
217#327—21no—
218#323C22×C13711yescharacteristic
219#323C22×C13711yescharacteristic
220C22×D4—21no—
221C23×C4D4×C13711yescharacteristic
222#123—41no—
223#121—21no—
224#121—21no—
225#120—41no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4D4×C13711yescharacteristic
230C2×C4⋊C4D4×C13711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4D4×C13711yescharacteristic
236C2×C22⋊C4C23×C13711yescharacteristic
237C2×C42—21no—
238C2×C42—21no—
239#107—41no—
240#106—41no—
241#102—41no—
242#102—41no—
243#99—42no—
244C22⋊C8—21no—
245C22⋊C8—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—41no—
254#95D4×C13711yescharacteristic
255#95D4×C13711yescharacteristic
256C24—21no—
257C24C2×D4×C13711yescharacteristic
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4C2×D4×C13711yescharacteristic
277C22×C4C2×D4×C13711yescharacteristic
278C2×C8—41no—
279C2×C8—41no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C22⋊C4—21no—
288C22⋊C4—21no—
289C22⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C22⋊C4—82no—
298C42—42no—
299C42—81no—
300C23—21no—
301C23—21no—
302C23—41no—
303C23—41no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—81no—
308C23—81no—
309C23#120×C13711yescharacteristic
310C23#135×C13711yescharacteristic
311C23AGL(1,ℤ/8)×C13711yescharacteristic
312D4—81no—
313D4—81no—
314D4—81no—
315D4—81no—
316D4—81no—
317D4—81no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—81no—
347C2×C4—81no—
348C8—81no—
349C8—81no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—81no—
359C22—82no—
360C22—81no—
361C22—81no—
362C22—81no—
363C22—161no—
364C22#449×C13711yescharacteristic
365C22#454×C13711yescharacteristic
366C22B(4,𝔽2)×C13711yescharacteristic
367C4—42no—
368C4—41no—
369C4—42no—
370C4—41no—
371C4—81no—
372C4—81no—
373C4—81no—
374C4—81no—
375C4—81no—
376C4—161no—
377C2—21no—
378C2—21no—
379C2—41no—
380C2—41no—
381C2—161no—
382C2#1597×C13711yescharacteristic
383C2#2197×C13711yescharacteristic
384C2#2198×C13711yescharacteristic
subgroupG/H
center, socleC22×C137B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C137
prime radical#534×C137C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C137 ◁ C2×C22⋊C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2517×C137C2semidirect
5. #1352×C137C2semidirect
6. #1344×C137C2semidirect
7. #1326×C137C2semidirect
192. #12135C137direct (N × H)
193. C137#12135direct (N × H)
194. #2517C2×C137semidirect
198. #1352C2×C137semidirect
199. #1344C2×C137semidirect
200. #1326C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2517×C1372
#2044×C1372
#1613×C1372
#1607×C1372
#1352×C1372
#1344×C1372
#1326×C1372
#12135137