Finite Group Databasebeta

← C137×#7780 #46857166

Subgroups — C137×#7780 #46857166

Order 35072.

478 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 98 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#2727×C137C211yesmaximal, characteristic
2#2058×C137C211yesmaximal, characteristic, complemented
3#2057×C137C211yesmaximal, characteristic, complemented
4#2013×C137C211yesmaximal, characteristic
5#1271×C137C211yesmaximal, characteristic, complemented
6#1271×C137C211yesmaximal, characteristic, complemented
7#1270×C137C211yesmaximal, characteristic
8C22×SD16×C137—21no—
9C2×C4⋊Q8×C137C2211yescharacteristic
10C2×#126×C137—21no—
11C2×#107×C137—21no—
12C2×#106×C137—21no—
13C2×#103×C137—21no—
14#399×C137—21no—
15#396×C137—21no—
16#394×C137—21no—
17#390×C137C2211yescharacteristic
18#389×C137—21no—
19#389×C137C2211yescharacteristic
20#389×C137C2211yescharacteristic
21#388×C137—21no—
22#340×C137—21no—
23#340×C137—21no—
24#336×C137C2211yescharacteristic
25#336×C137C2211yescharacteristic
26#336×C137C2211yescharacteristic
27C22×Q8×C137—21no—
28C22×D4×C137D411yescharacteristic
29C2×SD16×C137—42no—
30C2×SD16×C137—42no—
31C2×SD16×C137—42no—
32C22×C8×C137—21no—
33C22×C8×C137—21no—
34C22×C8×C137—21no—
35C4⋊Q8×C137—42no—
36#126×C137—42no—
37#126×C137—42no—
38C2×C4⋊C4×C137—21no—
39C2×C4⋊C4×C137—21no—
40C2×C4⋊C4×C137—21no—
41C2×C4⋊C4×C137—21no—
42C2×C4⋊C4×C137—21no—
43C2×C4⋊C4×C137—21no—
44C2×C4⋊C4×C137D411yescharacteristic
45C2×C4⋊C4×C137D411yescharacteristic
46C2×C4⋊C4×C137D411yescharacteristic
47C2×C4⋊C4×C137D411yescharacteristic
48C2×C4⋊C4×C137D411yescharacteristic
49C2×C22⋊C4×C137—21no—
50C2×C22⋊C4×C137—21no—
51C2×C22⋊C4×C137—21no—
52C2×C22⋊C4×C137—21no—
53C2×C22⋊C4×C137—21no—
54C2×C22⋊C4×C137—21no—
55C2×C42×C137—21no—
56C2×C42×C137C2311yescharacteristic
57#107×C137—42no—
58#106×C137—42no—
59#103×C137—42no—
60#95×C137—21no—
61#95×C137—21no—
62#95×C137—21no—
63C24×C137—21no—
64C2×Q8×C137—22no—
65C2×Q8×C137—41no—
66C2×Q8×C137—41no—
67C2×Q8×C137—41no—
68C2×Q8×C137—41no—
69C2×Q8×C137—41no—
70C2×D4×C137—21no—
71C2×D4×C137—21no—
72C2×D4×C137—41no—
73C2×D4×C137—41no—
74C22×C4×C137—21no—
75C22×C4×C137—21no—
76C22×C4×C137—21no—
77C22×C4×C137—21no—
78C22×C4×C137—21no—
79C22×C4×C137—41no—
80C22×C4×C137C2×D411yescharacteristic
81C22×C4×C137C2×D411yescharacteristic
82C22×C4×C137C2×D411yescharacteristic
83SD16×C137—48no—
84C2×C8×C137—22no—
85C2×C8×C137—22no—
86C2×C8×C137—22no—
87C2×C8×C137—41no—
88C2×C8×C137—41no—
89C2×C8×C137—41no—
90C2×C8×C137—41no—
91C2×C8×C137—41no—
92C2×C8×C137—41no—
93C4⋊C4×C137—21no—
94C4⋊C4×C137—22no—
95C4⋊C4×C137—22no—
96C4⋊C4×C137—21no—
97C4⋊C4×C137—21no—
98C4⋊C4×C137—41no—
99C4⋊C4×C137—41no—
100C4⋊C4×C137—41no—
101C4⋊C4×C137—42no—
102C4⋊C4×C137—41no—
103C4⋊C4×C137—41no—
104C4⋊C4×C137—41no—
105C4⋊C4×C137—41no—
106C4⋊C4×C137—41no—
107C4⋊C4×C137—41no—
108C4⋊C4×C137—81no—
109C4⋊C4×C137SD1612yescomplemented
110C22⋊C4×C137—42no—
111C22⋊C4×C137—42no—
112C22⋊C4×C137—41no—
113C22⋊C4×C137—41no—
114C22⋊C4×C137—41no—
115C22⋊C4×C137—41no—
116C22⋊C4×C137—81no—
117C22⋊C4×C137—81no—
118C42×C137—41no—
119C42×C137—42no—
120C23×C137—41no—
121C23×C137—41no—
122C23×C137—41no—
123C23×C137—41no—
124C23×C137—41no—
125C23×C137—41no—
126C23×C137—41no—
127C23×C137#12011yescharacteristic
128Q8×C137—42no—
129Q8×C137—42no—
130Q8×C137—81no—
131Q8×C137—81no—
132D4×C137—42no—
133D4×C137—81no—
134C2×C4×C137—21no—
135C2×C4×C137—21no—
136C2×C4×C137—21no—
137C2×C4×C137—21no—
138C2×C4×C137—21no—
139C2×C4×C137—21no—
140C2×C4×C137—22no—
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—41no—
152C2×C4×C137—41no—
153C2×C4×C137—41no—
154C2×C4×C137—41no—
155C2×C4×C137—42no—
156C2×C4×C137—81no—
157C2×C4×C137—81no—
158C2×C4×C137C2×SD1611yescharacteristic
159C2×C4×C137#13511yescharacteristic
160C2×C4×C137#13511yescharacteristic
161C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
162C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
163C2×C4×C137#13711yescharacteristic
164C8×C137—42no—
165C8×C137—42no—
166C8×C137—42no—
167C22×C137—81no—
168C22×C137—81no—
169C22×C137—81no—
170C22×C137—81no—
171C22×C137—81no—
172C22×C137—81no—
173C22×C137—81no—
174C22×C137#44911yescharacteristic
175C22×C137#45011yescharacteristic
176C22×C137#45211yescharacteristic
177C22×C137#45311yescharacteristic
178C22×C137#45611yescharacteristic
179C22×C137#45611yescharacteristic
180C22×C137B(4,𝔽2)11yescharacteristic
181C4×C137—21no—
182C4×C137—21no—
183C4×C137—21no—
184C4×C137—21no—
185C4×C137—21no—
186C4×C137—21no—
187C4×C137—42no—
188C4×C137—81no—
189C4×C137—82no—
190C4×C137—81no—
191C4×C137—81no—
192C4×C137—81no—
193C2×C137—161no—
194C2×C137#160011yescharacteristic
195C2×C137#160311yescharacteristic
196C2×C137#160411yescharacteristic
197C2×C137#163111yescharacteristic
198C2×C137#165011yescharacteristic
199C2×C137#165411yescharacteristic
200C2×C137#166011yescharacteristic
201#7780C13711yesmaximal, characteristic, direct product factor, Sylow
202C137#778011yescharacteristic, direct product factor, Sylow
203#2727C2×C13711yescharacteristic
204#2058C2×C13711yescharacteristic, complemented
205#2057C2×C13711yescharacteristic, complemented
206#2013C2×C13711yescharacteristic
207#1271C2×C13711yescharacteristic, complemented
208#1271C2×C13711yescharacteristic, complemented
209#1270C2×C13711yescharacteristic
210C22×SD16—21no—
211C2×C4⋊Q8C22×C13711yescharacteristic
212C2×#126—21no—
213C2×#107—21no—
214C2×#106—21no—
215C2×#103—21no—
216#399—21no—
217#396—21no—
218#394—21no—
219#390C22×C13711yescharacteristic
220#389—21no—
221#389C22×C13711yescharacteristic
222#389C22×C13711yescharacteristic
223#388—21no—
224#340—21no—
225#340—21no—
226#336C22×C13711yescharacteristic
227#336C22×C13711yescharacteristic
228#336C22×C13711yescharacteristic
229C22×Q8—21no—
230C22×D4D4×C13711yescharacteristic
231C2×SD16—42no—
232C2×SD16—42no—
233C2×SD16—42no—
234C22×C8—21no—
235C22×C8—21no—
236C22×C8—21no—
237C4⋊Q8—42no—
238#126—42no—
239#126—42no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4D4×C13711yescharacteristic
247C2×C4⋊C4D4×C13711yescharacteristic
248C2×C4⋊C4D4×C13711yescharacteristic
249C2×C4⋊C4D4×C13711yescharacteristic
250C2×C4⋊C4D4×C13711yescharacteristic
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C42—21no—
258C2×C42C23×C13711yescharacteristic
259#107—42no—
260#106—42no—
261#103—42no—
262#95—21no—
263#95—21no—
264#95—21no—
265C24—21no—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—41no—
275C2×D4—41no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4C2×D4×C13711yescharacteristic
283C22×C4C2×D4×C13711yescharacteristic
284C22×C4C2×D4×C13711yescharacteristic
285SD16—48no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—41no—
294C2×C8—41no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4—81no—
311C4⋊C4SD16×C13712yescomplemented
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C22⋊C4—81no—
319C22⋊C4—81no—
320C42—41no—
321C42—42no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23#120×C13711yescharacteristic
330Q8—42no—
331Q8—42no—
332Q8—81no—
333Q8—81no—
334D4—42no—
335D4—81no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4C2×SD16×C13711yescharacteristic
361C2×C4#135×C13711yescharacteristic
362C2×C4#135×C13711yescharacteristic
363C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
364C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
365C2×C4#137×C13711yescharacteristic
366C8—42no—
367C8—42no—
368C8—42no—
369C22—81no—
370C22—81no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22#449×C13711yescharacteristic
377C22#450×C13711yescharacteristic
378C22#452×C13711yescharacteristic
379C22#453×C13711yescharacteristic
380C22#456×C13711yescharacteristic
381C22#456×C13711yescharacteristic
382C22B(4,𝔽2)×C13711yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—42no—
390C4—81no—
391C4—82no—
392C4—81no—
393C4—81no—
394C4—81no—
395C2—161no—
396C2#1600×C13711yescharacteristic
397C2#1603×C13711yescharacteristic
398C2#1604×C13711yescharacteristic
399C2#1631×C13711yescharacteristic
400C2#1650×C13711yescharacteristic
401C2#1654×C13711yescharacteristic
402C2#1660×C13711yescharacteristic
subgroupG/H
center, socleC23×C137#120
commutator, Frattini, cosocleC2×C42C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2058×C137C2semidirect
3. #2057×C137C2semidirect
5. #1271×C137C2semidirect
6. #1271×C137C2semidirect
109. C4⋊C4×C137SD16semidirect
201. #7780C137direct (N × H)
202. C137#7780direct (N × H)
204. #2058C2×C137semidirect
205. #2057C2×C137semidirect
207. #1271C2×C137semidirect
208. #1271C2×C137semidirect
311. C4⋊C4SD16×C137semidirect

Maximal subgroups

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

subgroupindex
#2727×C1372
#2058×C1372
#2057×C1372
#2013×C1372
#1271×C1372
#1270×C1372
#7780137