Finite Group Databasebeta

← C137×#9287 #46858673

Subgroups — C137×#9287 #46858673

Order 35072.

478 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3227×C137C211yesmaximal, characteristic
2#2932×C137C211yesmaximal, characteristic
3#1993×C137C211yesmaximal, characteristic, complemented
4#1548×C137C211yesmaximal, characteristic, complemented
5#1517×C137C211yesmaximal, characteristic
6#1286×C137C211yesmaximal, characteristic, complemented
7#1275×C137C211yesmaximal, characteristic, complemented
8C2×#131×C137—21no—
9#532×C137C2211yescharacteristic
10C2×C4⋊Q8×C137C2211yescharacteristic
11#498×C137C412yescomplemented
12#486×C137C412yescomplemented
13#481×C137—22no—
14#476×C137—22no—
15#435×C137—22no—
16#435×C137—22no—
17#431×C137C2211yescharacteristic
18#431×C137C2211yescharacteristic
19#423×C137C2211yescharacteristic
20#408×C137—21no—
21C2×#97×C137C2211yescharacteristic
22#382×C137C2211yescharacteristic
23#340×C137—21no—
24#340×C137—21no—
25#333×C137—22no—
26#327×C137—22no—
27C2×#41×C137—21no—
28C22×Q8×C137—21no—
29#131×C137—42no—
30#131×C137—42no—
31C2×M16×C137—22no—
32C2×M16×C137—21no—
33C22×C8×C137—22no—
34C22×C8×C137D411yescharacteristic
35C22×C8×C137D411yescharacteristic
36C4⋊Q8×C137—21no—
37C4⋊Q8×C137—21no—
38C4⋊Q8×C137—21no—
39C4⋊Q8×C137C2×C412yes—
40C4⋊Q8×C137C2×C411yescharacteristic
41#127×C137C2×C411yescharacteristic
42#124×C137—21no—
43#121×C137—21no—
44#121×C137D412yes—
45C4×D4×C137—21no—
46C4×D4×C137D412yes—
47C2×C4⋊C4×C137—21no—
48C2×C4⋊C4×C137—21no—
49C2×C4⋊C4×C137—21no—
50C2×C42×C137C2311yescharacteristic
51#106×C137—24no—
52C4⋊C8×C137—21no—
53C4⋊C8×C137—21no—
54C4⋊C8×C137—21no—
55C4⋊C8×C137D412yes—
56#102×C137—24no—
57C22⋊C8×C137—24no—
58C22⋊C8×C137—21no—
59C22⋊C8×C137—21no—
60#97×C137—21no—
61#97×C137—21no—
62C4×C8×C137—21no—
63C4×C8×C137C2×C412yes—
64#95×C137—21no—
65#95×C137—21no—
66#41×C137—41no—
67#41×C137—41no—
68#41×C137—41no—
69#41×C137—41no—
70C2×Q8×C137—21no—
71C2×Q8×C137—22no—
72C2×Q8×C137—41no—
73C2×Q8×C137—82no—
74C2×D4×C137—22no—
75C2×D4×C137—21no—
76C2×D4×C137—21no—
77C2×D4×C137C22⋊C412yes—
78C22×C4×C137—22no—
79C22×C4×C137—41no—
80C22×C4×C137—41no—
81C22×C4×C137C2×D411yescharacteristic
82C22×C4×C137C2×D411yescharacteristic
83C22×C4×C137#4111yescharacteristic
84M16×C137—44no—
85M16×C137—41no—
86M16×C137—41no—
87C2×C8×C137—21no—
88C2×C8×C137—21no—
89C2×C8×C137—22no—
90C2×C8×C137—21no—
91C2×C8×C137—22no—
92C2×C8×C137—21no—
93C2×C8×C137—21no—
94C2×C8×C137—44no—
95C2×C8×C137—41no—
96C2×C8×C137SD1612yes—
97C4⋊C4×C137—21no—
98C4⋊C4×C137—22no—
99C4⋊C4×C137—22no—
100C4⋊C4×C137—41no—
101C4⋊C4×C137—41no—
102C4⋊C4×C137—41no—
103C4⋊C4×C137—41no—
104C4⋊C4×C137C22⋊C412yes—
105C22⋊C4×C137—22no—
106C22⋊C4×C137—22no—
107C42×C137C22×C411yescharacteristic
108C42×C137C2×D411yescharacteristic
109C42×C137C2×D411yescharacteristic
110C42×C137#4111yescharacteristic
111C23×C137—22no—
112C23×C137#12111yescharacteristic
113Q8×C137—41no—
114Q8×C137—41no—
115Q8×C137—82no—
116Q8×C137—82no—
117D4×C137—42no—
118D4×C137—42no—
119D4×C137—42no—
120D4×C137—41no—
121D4×C137—41no—
122D4×C137—81no—
123C2×C4×C137—22no—
124C2×C4×C137—41no—
125C2×C4×C137—41no—
126C2×C4×C137—42no—
127C2×C4×C137—42no—
128C2×C4×C137—42no—
129C2×C4×C137—82no—
130C2×C4×C137—81no—
131C2×C4×C137—81no—
132C2×C4×C137—81no—
133C2×C4×C137C2×C22⋊C411yescharacteristic
134C2×C4×C137C4×D411yescharacteristic
135C2×C4×C137C4×D411yescharacteristic
136C2×C4×C137#12011yescharacteristic
137C2×C4×C137#12111yescharacteristic
138C2×C4×C137#12311yescharacteristic
139C2×C4×C137C2×SD1611yescharacteristic
140C2×C4×C137#13511yescharacteristic
141C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
142C2×C4×C137#13711yescharacteristic
143C8×C137—22no—
144C8×C137—42no—
145C8×C137—42no—
146C8×C137—41no—
147C8×C137—81no—
148C22×C137—21no—
149C22×C137—21no—
150C22×C137—42no—
151C22×C137—42no—
152C22×C137—42no—
153C22×C137#38611yescharacteristic
154C22×C137#46111yescharacteristic
155C22×C137#46311yescharacteristic
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—21no—
159C4×C137—21no—
160C4×C137—42no—
161C4×C137—82no—
162C4×C137—161no—
163C4×C137C4×SD1611yescharacteristic
164C4×C137#44011yescharacteristic
165C4×C137#45011yescharacteristic
166C4×C137#45211yescharacteristic
167C2×C137—21no—
168C2×C137—21no—
169C2×C137—82no—
170C2×C137#187711yescharacteristic
171C2×C137#190311yescharacteristic
172C2×C137#190911yescharacteristic
173#9287C13711yesmaximal, characteristic, direct product factor, Sylow
174C137#928711yescharacteristic, direct product factor, Sylow
175#3227C2×C13711yescharacteristic
176#2932C2×C13711yescharacteristic
177#1993C2×C13711yescharacteristic, complemented
178#1548C2×C13711yescharacteristic, complemented
179#1517C2×C13711yescharacteristic
180#1286C2×C13711yescharacteristic, complemented
181#1275C2×C13711yescharacteristic, complemented
182C2×#131—21no—
183#532C22×C13711yescharacteristic
184C2×C4⋊Q8C22×C13711yescharacteristic
185#498C4×C13712yescomplemented
186#486C4×C13712yescomplemented
187#481—22no—
188#476—22no—
189#435—22no—
190#435—22no—
191#431C22×C13711yescharacteristic
192#431C22×C13711yescharacteristic
193#423C22×C13711yescharacteristic
194#408—21no—
195C2×#97C22×C13711yescharacteristic
196#382C22×C13711yescharacteristic
197#340—21no—
198#340—21no—
199#333—22no—
200#327—22no—
201C2×#41—21no—
202C22×Q8—21no—
203#131—42no—
204#131—42no—
205C2×M16—22no—
206C2×M16—21no—
207C22×C8—22no—
208C22×C8D4×C13711yescharacteristic
209C22×C8D4×C13711yescharacteristic
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8C2×C4×C13712yes—
214C4⋊Q8C2×C4×C13711yescharacteristic
215#127C2×C4×C13711yescharacteristic
216#124—21no—
217#121—21no—
218#121D4×C13712yes—
219C4×D4—21no—
220C4×D4D4×C13712yes—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C42C23×C13711yescharacteristic
225#106—24no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8D4×C13712yes—
230#102—24no—
231C22⋊C8—24no—
232C22⋊C8—21no—
233C22⋊C8—21no—
234#97—21no—
235#97—21no—
236C4×C8—21no—
237C4×C8C2×C4×C13712yes—
238#95—21no—
239#95—21no—
240#41—41no—
241#41—41no—
242#41—41no—
243#41—41no—
244C2×Q8—21no—
245C2×Q8—22no—
246C2×Q8—41no—
247C2×Q8—82no—
248C2×D4—22no—
249C2×D4—21no—
250C2×D4—21no—
251C2×D4C22⋊C4×C13712yes—
252C22×C4—22no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×D4×C13711yescharacteristic
256C22×C4C2×D4×C13711yescharacteristic
257C22×C4#41×C13711yescharacteristic
258M16—44no—
259M16—41no—
260M16—41no—
261C2×C8—21no—
262C2×C8—21no—
263C2×C8—22no—
264C2×C8—21no—
265C2×C8—22no—
266C2×C8—21no—
267C2×C8—21no—
268C2×C8—44no—
269C2×C8—41no—
270C2×C8SD16×C13712yes—
271C4⋊C4—21no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4C22⋊C4×C13712yes—
279C22⋊C4—22no—
280C22⋊C4—22no—
281C42C22×C4×C13711yescharacteristic
282C42C2×D4×C13711yescharacteristic
283C42C2×D4×C13711yescharacteristic
284C42#41×C13711yescharacteristic
285C23—22no—
286C23#121×C13711yescharacteristic
287Q8—41no—
288Q8—41no—
289Q8—82no—
290Q8—82no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—41no—
295D4—41no—
296D4—81no—
297C2×C4—22no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—82no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4C2×C22⋊C4×C13711yescharacteristic
308C2×C4C4×D4×C13711yescharacteristic
309C2×C4C4×D4×C13711yescharacteristic
310C2×C4#120×C13711yescharacteristic
311C2×C4#121×C13711yescharacteristic
312C2×C4#123×C13711yescharacteristic
313C2×C4C2×SD16×C13711yescharacteristic
314C2×C4#135×C13711yescharacteristic
315C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
316C2×C4#137×C13711yescharacteristic
317C8—22no—
318C8—42no—
319C8—42no—
320C8—41no—
321C8—81no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—42no—
326C22—42no—
327C22#386×C13711yescharacteristic
328C22#461×C13711yescharacteristic
329C22#463×C13711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—42no—
335C4—82no—
336C4—161no—
337C4C4×SD16×C13711yescharacteristic
338C4#440×C13711yescharacteristic
339C4#450×C13711yescharacteristic
340C4#452×C13711yescharacteristic
341C2—21no—
342C2—21no—
343C2—82no—
344C2#1877×C13711yescharacteristic
345C2#1903×C13711yescharacteristic
346C2#1909×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#386
commutatorC42C22×C4×C137
Frattini, cosocleC2×C42C23×C137
prime radical#532×C137C22
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. #1993×C137C2semidirect
4. #1548×C137C2semidirect
6. #1286×C137C2semidirect
7. #1275×C137C2semidirect
11. #498×C137C4semidirect
12. #486×C137C4semidirect
173. #9287C137direct (N × H)
174. C137#9287direct (N × H)
177. #1993C2×C137semidirect
178. #1548C2×C137semidirect
180. #1286C2×C137semidirect
181. #1275C2×C137semidirect
185. #498C4×C137semidirect
186. #486C4×C137semidirect

Maximal subgroups

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

subgroupindex
#3227×C1372
#2932×C1372
#1993×C1372
#1548×C1372
#1517×C1372
#1286×C1372
#1275×C1372
#9287137