Finite Group Databasebeta

← C137×#12105 #46861491

Subgroups — C137×#12105 #46861491

Order 35072.

476 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 82 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#3186×C137C211yesmaximal, characteristic
2#2185×C137C212yesmaximal, complemented
3#2001×C137C211yesmaximal, characteristic, complemented
4#1347×C137C212yesmaximal, complemented
5#1314×C137C211yesmaximal, characteristic, complemented
6C22×Q16×C137C2211yescharacteristic, complemented
7C2×C22⋊Q8×C137—21no—
8C2×#121×C137—21no—
9C2×SD32×C137—22no—
10#480×C137—22no—
11#466×C137—21no—
12#466×C137C2212yes—
13#447×C137—22no—
14C2×#103×C137—21no—
15C2×C22⋊C8×C137C2211yescharacteristic
16#358×C137—22no—
17C22⋊C16×C137C2212yescomplemented
18#340×C137C2211yescharacteristic, complemented
19C22×Q8×C137—21no—
20C22×D4×C137—21no—
21C22×D4×C137—41no—
22C23×C4×C137D411yescharacteristic
23C2×Q16×C137—22no—
24C2×Q16×C137—42no—
25C2×D8×C137—21no—
26C2×D8×C137D412yes—
27C22×C8×C137D411yescharacteristic
28C22×C8×C137C2311yescharacteristic
29C22⋊Q8×C137—42no—
30C22⋊Q8×C137—42no—
31#121×C137—22no—
32#121×C137—22no—
33#121×C137—41no—
34#121×C137—41no—
35C2×C4⋊C4×C137—21no—
36C2×C4⋊C4×C137—21no—
37C2×C4⋊C4×C137—41no—
38C2×C22⋊C4×C137—41no—
39C2×C22⋊C4×C137—41no—
40SD32×C137—44no—
41C2×C16×C137—22no—
42#107×C137—21no—
43#107×C137D412yes—
44#103×C137—42no—
45#102×C137—22no—
46#102×C137—22no—
47C22⋊C8×C137—22no—
48C24×C137—41no—
49C24×C137D811yescharacteristic
50C2×Q8×C137—22no—
51C2×Q8×C137—41no—
52C2×Q8×C137—41no—
53C2×Q8×C137—41no—
54C2×Q8×C137—41no—
55C2×Q8×C137—41no—
56C2×D4×C137—22no—
57C2×D4×C137—41no—
58C2×D4×C137—42no—
59C2×D4×C137—42no—
60C2×D4×C137—81no—
61C2×D4×C137—81no—
62C2×D4×C137—82no—
63C2×D4×C137—82no—
64C22×C4×C137—21no—
65C22×C4×C137—21no—
66C22×C4×C137—41no—
67C22×C4×C137—41no—
68C22×C4×C137—41no—
69C22×C4×C137—41no—
70C22×C4×C137—41no—
71C22×C4×C137D811yescharacteristic
72C22×C4×C137C2×D411yescharacteristic
73Q16×C137—24no—
74Q16×C137—42no—
75D8×C137—42no—
76D8×C137—81no—
77C2×C8×C137—21no—
78C2×C8×C137—41no—
79C2×C8×C137—41no—
80C2×C8×C137C2×D412yes—
81C4⋊C4×C137—22no—
82C4⋊C4×C137—22no—
83C4⋊C4×C137—41no—
84C4⋊C4×C137—41no—
85C4⋊C4×C137—44no—
86C22⋊C4×C137—42no—
87C22⋊C4×C137—44no—
88C22⋊C4×C137—42no—
89C16×C137—42no—
90C23×C137—21no—
91C23×C137—21no—
92C23×C137—41no—
93C23×C137—41no—
94C23×C137—42no—
95C23×C137—82no—
96C23×C137—82no—
97C23×C137—82no—
98C23×C137SD3212yescomplemented
99C23×C137C2×D811yescharacteristic
100Q8×C137—42no—
101Q8×C137—42no—
102Q8×C137—81no—
103Q8×C137—81no—
104D4×C137—42no—
105D4×C137—42no—
106D4×C137—82no—
107D4×C137—84no—
108D4×C137—82no—
109D4×C137—82no—
110C2×C4×C137—21no—
111C2×C4×C137—21no—
112C2×C4×C137—21no—
113C2×C4×C137—41no—
114C2×C4×C137—41no—
115C2×C4×C137—41no—
116C2×C4×C137—42no—
117C2×C4×C137—42no—
118C2×C4×C137—42no—
119C2×C4×C137—81no—
120C2×C4×C137—81no—
121C2×C4×C137—82no—
122C2×C4×C137—81no—
123C2×C4×C137—81no—
124C2×C4×C137—81no—
125C2×C4×C137#12011yescharacteristic
126C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
127C8×C137—22no—
128C8×C137—41no—
129C22×C137—21no—
130C22×C137—21no—
131C22×C137—22no—
132C22×C137—42no—
133C22×C137—41no—
134C22×C137—41no—
135C22×C137—81no—
136C22×C137—82no—
137C22×C137—82no—
138C22×C137—82no—
139C22×C137—164no—
140C22×C137#44711yescharacteristic
141C22×C137C2×SD3211yescharacteristic
142C22×C137#50911yescharacteristic
143C4×C137—21no—
144C4×C137—41no—
145C4×C137—82no—
146C4×C137—82no—
147C4×C137—82no—
148C4×C137#45311yescharacteristic
149C4×C137B(4,𝔽2)11yescharacteristic
150C2×C137—21no—
151C2×C137—21no—
152C2×C137—42no—
153C2×C137—162no—
154C2×C137#159511yescharacteristic
155C2×C137#218711yescharacteristic
156C2×C137#219111yescharacteristic
157#12105C13711yesmaximal, characteristic, direct product factor, Sylow
158C137#1210511yescharacteristic, direct product factor, Sylow
159#3186C2×C13711yescharacteristic
160#2185C2×C13712yescomplemented
161#2001C2×C13711yescharacteristic, complemented
162#1347C2×C13712yescomplemented
163#1314C2×C13711yescharacteristic, complemented
164C22×Q16C22×C13711yescharacteristic, complemented
165C2×C22⋊Q8—21no—
166C2×#121—21no—
167C2×SD32—22no—
168#480—22no—
169#466—21no—
170#466C22×C13712yes—
171#447—22no—
172C2×#103—21no—
173C2×C22⋊C8C22×C13711yescharacteristic
174#358—22no—
175C22⋊C16C22×C13712yescomplemented
176#340C22×C13711yescharacteristic, complemented
177C22×Q8—21no—
178C22×D4—21no—
179C22×D4—41no—
180C23×C4D4×C13711yescharacteristic
181C2×Q16—22no—
182C2×Q16—42no—
183C2×D8—21no—
184C2×D8D4×C13712yes—
185C22×C8D4×C13711yescharacteristic
186C22×C8C23×C13711yescharacteristic
187C22⋊Q8—42no—
188C22⋊Q8—42no—
189#121—22no—
190#121—22no—
191#121—41no—
192#121—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C22⋊C4—41no—
198SD32—44no—
199C2×C16—22no—
200#107—21no—
201#107D4×C13712yes—
202#103—42no—
203#102—22no—
204#102—22no—
205C22⋊C8—22no—
206C24—41no—
207C24D8×C13711yescharacteristic
208C2×Q8—22no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×D4—22no—
215C2×D4—41no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—81no—
219C2×D4—81no—
220C2×D4—82no—
221C2×D4—82no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—41no—
227C22×C4—41no—
228C22×C4—41no—
229C22×C4D8×C13711yescharacteristic
230C22×C4C2×D4×C13711yescharacteristic
231Q16—24no—
232Q16—42no—
233D8—42no—
234D8—81no—
235C2×C8—21no—
236C2×C8—41no—
237C2×C8—41no—
238C2×C8C2×D4×C13712yes—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—44no—
244C22⋊C4—42no—
245C22⋊C4—44no—
246C22⋊C4—42no—
247C16—42no—
248C23—21no—
249C23—21no—
250C23—41no—
251C23—41no—
252C23—42no—
253C23—82no—
254C23—82no—
255C23—82no—
256C23SD32×C13712yescomplemented
257C23C2×D8×C13711yescharacteristic
258Q8—42no—
259Q8—42no—
260Q8—81no—
261Q8—81no—
262D4—42no—
263D4—42no—
264D4—82no—
265D4—84no—
266D4—82no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—82no—
279C2×C4—81no—
280C2×C4—81no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4#120×C13711yescharacteristic
284C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
285C8—22no—
286C8—41no—
287C22—21no—
288C22—21no—
289C22—22no—
290C22—42no—
291C22—41no—
292C22—41no—
293C22—81no—
294C22—82no—
295C22—82no—
296C22—82no—
297C22—164no—
298C22#447×C13711yescharacteristic
299C22C2×SD32×C13711yescharacteristic
300C22#509×C13711yescharacteristic
301C4—21no—
302C4—41no—
303C4—82no—
304C4—82no—
305C4—82no—
306C4#453×C13711yescharacteristic
307C4B(4,𝔽2)×C13711yescharacteristic
308C2—21no—
309C2—21no—
310C2—42no—
311C2—162no—
312C2#1595×C13711yescharacteristic
313C2#2187×C13711yescharacteristic
314C2#2191×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#447
commutator, Frattini, cosocleC22×C8C23×C137
prime radical#3186×C137C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×C22⋊C8×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2185×C137C2semidirect
3. #2001×C137C2semidirect
4. #1347×C137C2semidirect
5. #1314×C137C2semidirect
6. C22×Q16×C137C22semidirect
17. C22⋊C16×C137C22semidirect
18. #340×C137C22semidirect
98. C23×C137SD32semidirect
157. #12105C137direct (N × H)
158. C137#12105direct (N × H)
160. #2185C2×C137semidirect
161. #2001C2×C137semidirect
162. #1347C2×C137semidirect
163. #1314C2×C137semidirect
164. C22×Q16C22×C137semidirect
175. C22⋊C16C22×C137semidirect
176. #340C22×C137semidirect
256. C23SD32×C137semidirect

Maximal subgroups

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

subgroupindex
#3186×C1372
#2185×C1372
#2001×C1372
#1347×C1372
#1314×C1372
#12105137