Finite Group Databasebeta

← C137×#32901 #46882287

Subgroups — C137×#32901 #46882287

Order 35072.

3706 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 3706 conjugacy classes of subgroups — too many to draw. Only its 366 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
1C2×#585×C137C211yesmaximal, characteristic
2C2×#458×C137C214yesmaximal, complemented
3#2406×C137C213yesmaximal, complemented
4#1790×C137C214yesmaximal, complemented
5#1782×C137C213yesmaximal, complemented
6#585×C137—216no—
7C2×2−1+4×C137—23no—
8C2×H(5,𝔽2)×C137—23no—
9C2×H(5,𝔽2)×C137C2214yes—
10C22×#41×C137—26no—
11C22×#41×C137C2213yes—
12C2×#123×C137C2216yescomplemented
13C22×C4⋊C4×C137C2211yescharacteristic, complemented
14#458×C137—216no—
15C2×#99×C137C22112yescomplemented
16#395×C137—23no—
17#394×C137—23no—
18#386×C137C2216yescomplemented
19#385×C137—26no—
20#381×C137C2213yescomplemented
212−1+4×C137—248no—
22H(5,𝔽2)×C137—248no—
23H(5,𝔽2)×C137—224no—
24H(5,𝔽2)×C137D4116yes—
25C2×#41×C137—24no—
26C2×#41×C137—212no—
27C2×#41×C137—248no—
28C2×#41×C137—248no—
29C2×#41×C137—224no—
30C2×#41×C137—26no—
31C2×#41×C137—212no—
32C2×#41×C137—26no—
33C22×Q8×C137—26no—
34C22×Q8×C137D413yes—
35C22×D4×C137—26no—
36C22×D4×C137—212no—
37C22×D4×C137D413yes—
38C22×D4×C137C2316yes—
39C23×C4×C137—24no—
40C23×C4×C137D416yes—
41C23×C4×C137C2311yescharacteristic
42C23×C4×C137C2311yescharacteristic
43C23×C4×C137C2311yescharacteristic
44#123×C137—224no—
45C2×C4⋊C4×C137—23no—
46C2×C4⋊C4×C137—26no—
47C2×C4⋊C4×C137—23no—
48C2×C22⋊C4×C137—26no—
49C2×C22⋊C4×C137—26no—
50C2×C22⋊C4×C137C2316yes—
51#99×C137—224no—
52#95×C137—23no—
53#95×C137—23no—
54#95×C137—26no—
55C24×C137—24no—
56C24×C137C2×D416yes—
57C24×C137C2411yescharacteristic
58#41×C137—232no—
59#41×C137—248no—
60#41×C137—296no—
61#41×C137—248no—
62#41×C137—248no—
63#41×C137—248no—
64C2×Q8×C137—24no—
65C2×Q8×C137—248no—
66C2×Q8×C137—26no—
67C2×Q8×C137—212no—
68C2×D4×C137—212no—
69C2×D4×C137—248no—
70C2×D4×C137—248no—
71C2×D4×C137—248no—
72C2×D4×C137—212no—
73C2×D4×C137—212no—
74C2×D4×C137—212no—
75C2×D4×C137—26no—
76C2×D4×C137C2×D4124yes—
77C22×C4×C137—212no—
78C22×C4×C137—232no—
79C22×C4×C137—212no—
80C22×C4×C137—24no—
81C22×C4×C137—26no—
82C22×C4×C137—26no—
83C22×C4×C137—212no—
84C22×C4×C137—23no—
85C22×C4×C137—26no—
86C22×C4×C137—23no—
87C22×C4×C137—24no—
88C22×C4×C137—42no—
89C22×C4×C137C2×D413yes—
90C22×C4×C137C2×D416yes—
91C22×C4×C137C2×D413yes—
92C4⋊C4×C137—41no—
93C4⋊C4×C137—43no—
94C4⋊C4×C137—412no—
95C22⋊C4×C137—212no—
96C22⋊C4×C137—412no—
97C22⋊C4×C137—412no—
98C23×C137—216no—
99C23×C137—26no—
100C23×C137—26no—
101C23×C137—41no—
102C23×C137—41no—
103C23×C137—46no—
104C23×C137—46no—
105C23×C137—84no—
106C23×C137—84no—
107C23×C137#120112yes—
108C23×C137C22×D413yes—
109C23×C137C22×D414yes—
110Q8×C137—216no—
111Q8×C137—224no—
112D4×C137—248no—
113D4×C137—224no—
114D4×C137—248no—
115C2×C4×C137—224no—
116C2×C4×C137—248no—
117C2×C4×C137—224no—
118C2×C4×C137—26no—
119C2×C4×C137—26no—
120C2×C4×C137—26no—
121C2×C4×C137—26no—
122C2×C4×C137—26no—
123C2×C4×C137—43no—
124C2×C4×C137—41no—
125C2×C4×C137—46no—
126C2×C4×C137—46no—
127C2×C4×C137—46no—
128C2×C4×C137—46no—
129C2×C4×C137—41no—
130C2×C4×C137#120112yes—
131C2×C4×C137#12011yescharacteristic
132C2×C4×C137#12013yes—
133C22×C137—23no—
134C22×C137—23no—
135C22×C137—212no—
136C22×C137—44no—
137C22×C137—412no—
138C22×C137—46no—
139C22×C137—46no—
140C22×C137—86no—
141C22×C137—816no—
142C22×C137C2×#12016yes—
143C22×C137C2×#12011yescharacteristic
144C4×C137—21no—
145C4×C137—212no—
146C4×C137—23no—
147C4×C137—42no—
148C4×C137—86no—
149C4×C137—81no—
150C2×C137—26no—
151C2×C137—412no—
152C2×C137#284611yescharacteristic
153C2×C137#302511yescharacteristic
154C2×C137#302811yescharacteristic
155#32901C13711yesmaximal, characteristic, direct product factor, Sylow
156C137#3290111yescharacteristic, direct product factor, Sylow
157C2×#585C2×C13711yescharacteristic
158C2×#458C2×C13714yescomplemented
159#2406C2×C13713yescomplemented
160#1790C2×C13714yescomplemented
161#1782C2×C13713yescomplemented
162#585—216no—
163C2×2−1+4—23no—
164C2×H(5,𝔽2)—23no—
165C2×H(5,𝔽2)C22×C13714yes—
166C22×#41—26no—
167C22×#41C22×C13713yes—
168C2×#123C22×C13716yescomplemented
169C22×C4⋊C4C22×C13711yescharacteristic, complemented
170#458—216no—
171C2×#99C22×C137112yescomplemented
172#395—23no—
173#394—23no—
174#386C22×C13716yescomplemented
175#385—26no—
176#381C22×C13713yescomplemented
1772−1+4—248no—
178H(5,𝔽2)—248no—
179H(5,𝔽2)—224no—
180H(5,𝔽2)D4×C137116yes—
181C2×#41—24no—
182C2×#41—212no—
183C2×#41—248no—
184C2×#41—248no—
185C2×#41—224no—
186C2×#41—26no—
187C2×#41—212no—
188C2×#41—26no—
189C22×Q8—26no—
190C22×Q8D4×C13713yes—
191C22×D4—26no—
192C22×D4—212no—
193C22×D4D4×C13713yes—
194C22×D4C23×C13716yes—
195C23×C4—24no—
196C23×C4D4×C13716yes—
197C23×C4C23×C13711yescharacteristic
198C23×C4C23×C13711yescharacteristic
199C23×C4C23×C13711yescharacteristic
200#123—224no—
201C2×C4⋊C4—23no—
202C2×C4⋊C4—23no—
203C2×C4⋊C4—26no—
204C2×C22⋊C4—26no—
205C2×C22⋊C4—26no—
206C2×C22⋊C4C23×C13716yes—
207#99—224no—
208#95—23no—
209#95—23no—
210#95—26no—
211C24—24no—
212C24C2×D4×C13716yes—
213C24C24×C13711yescharacteristic
214#41—232no—
215#41—248no—
216#41—296no—
217#41—248no—
218#41—248no—
219#41—248no—
220C2×Q8—24no—
221C2×Q8—248no—
222C2×Q8—26no—
223C2×Q8—212no—
224C2×D4—212no—
225C2×D4—248no—
226C2×D4—248no—
227C2×D4—248no—
228C2×D4—212no—
229C2×D4—212no—
230C2×D4—212no—
231C2×D4—26no—
232C2×D4C2×D4×C137124yes—
233C22×C4—212no—
234C22×C4—232no—
235C22×C4—212no—
236C22×C4—24no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—212no—
240C22×C4—23no—
241C22×C4—26no—
242C22×C4—23no—
243C22×C4—24no—
244C22×C4—42no—
245C22×C4C2×D4×C13713yes—
246C22×C4C2×D4×C13716yes—
247C22×C4C2×D4×C13713yes—
248C4⋊C4—41no—
249C4⋊C4—43no—
250C4⋊C4—412no—
251C22⋊C4—212no—
252C22⋊C4—412no—
253C22⋊C4—412no—
254C23—216no—
255C23—26no—
256C23—26no—
257C23—41no—
258C23—41no—
259C23—46no—
260C23—46no—
261C23—84no—
262C23—84no—
263C23#120×C137112yes—
264C23C22×D4×C13713yes—
265C23C22×D4×C13714yes—
266Q8—216no—
267Q8—224no—
268D4—248no—
269D4—224no—
270D4—248no—
271C2×C4—224no—
272C2×C4—248no—
273C2×C4—224no—
274C2×C4—26no—
275C2×C4—26no—
276C2×C4—26no—
277C2×C4—26no—
278C2×C4—26no—
279C2×C4—43no—
280C2×C4—41no—
281C2×C4—46no—
282C2×C4—46no—
283C2×C4—46no—
284C2×C4—46no—
285C2×C4—41no—
286C2×C4#120×C13711yescharacteristic
287C2×C4#120×C13713yes—
288C2×C4#120×C137112yes—
289C22—23no—
290C22—23no—
291C22—212no—
292C22—44no—
293C22—412no—
294C22—46no—
295C22—46no—
296C22—86no—
297C22—816no—
298C22C2×#120×C13716yes—
299C22C2×#120×C13711yescharacteristic
300C4—21no—
301C4—212no—
302C4—23no—
303C4—42no—
304C4—86no—
305C4—81no—
306C2—26no—
307C2—412no—
308C2#2846×C13711yescharacteristic
309C2#3025×C13711yescharacteristic
310C2#3028×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#120
commutator, Frattini, cosocleC24C24×C137
prime radicalC2×#585×C137C2
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#458×C137C2semidirect
3. #2406×C137C2semidirect
4. #1790×C137C2semidirect
5. #1782×C137C2semidirect
12. C2×#123×C137C22semidirect
13. C22×C4⋊C4×C137C22semidirect
15. C2×#99×C137C22semidirect
18. #386×C137C22semidirect
20. #381×C137C22semidirect
155. #32901C137direct (N × H)
156. C137#32901direct (N × H)
158. C2×#458C2×C137semidirect
159. #2406C2×C137semidirect
160. #1790C2×C137semidirect
161. #1782C2×C137semidirect
168. C2×#123C22×C137semidirect
169. C22×C4⋊C4C22×C137semidirect
171. C2×#99C22×C137semidirect
174. #386C22×C137semidirect
176. #381C22×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#585×C1372
C2×#458×C1372
#2406×C1372
#1790×C1372
#1782×C1372
#32901137