Finite Group Databasebeta

← C137×#8667 #46858053

Subgroups — C137×#8667 #46858053

Order 35072.

454 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
1C8×#41×C137C211yesmaximal, characteristic
2#1806×C137C212yesmaximal
3C4×#107×C137C211yesmaximal, characteristic, complemented
4#1567×C137C211yesmaximal, characteristic, complemented
5#1303×C137C212yesmaximal, complemented
6C4×#41×C137C2211yescharacteristic
7C8×Q8×C137C411yescharacteristic, complemented
8C8×Q8×C137C411yescharacteristic, complemented
9C8×D4×C137—21no—
10C8×D4×C137—21no—
11C8×D4×C137C411yescharacteristic, complemented
12C8×D4×C137C411yescharacteristic, complemented
13#431×C137—21no—
14#431×C137C2211yescharacteristic
15C2×#107×C137—21no—
16#423×C137C2212yes—
17C2×#104×C137—22no—
18C2×C4×C8×C137—21no—
19C2×C4×C8×C137C2211yescharacteristic
20C4×C4⋊C4×C137C2212yes—
21#340×C137—21no—
22#339×C137—22no—
23#335×C137—21no—
24#335×C137—21no—
25C2×#41×C137D411yescharacteristic
26C2×M16×C137—22no—
27C22×C8×C137—21no—
28C22×C8×C137—21no—
29C22×C8×C137D411yescharacteristic
30C22×C8×C137D411yescharacteristic
31C4×Q8×C137C2×C411yescharacteristic
32C4×Q8×C137Q811yescharacteristic
33C4×D4×C137—21no—
34C4×D4×C137—21no—
35C4×D4×C137C2×C411yescharacteristic
36C4×D4×C137Q811yescharacteristic
37#117×C137—21no—
38#117×C137D411yescharacteristic
39C2×C4⋊C4×C137—22no—
40C2×C42×C137—21no—
41C2×C42×C137—22no—
42C2×C42×C137C2311yescharacteristic
43#107×C137—24no—
44C4⋊C8×C137—21no—
45C4⋊C8×C137—21no—
46C4⋊C8×C137—22no—
47C4⋊C8×C137—22no—
48C4⋊C8×C137C2×C411yescharacteristic
49C4⋊C8×C137C2×C411yescharacteristic
50#104×C137—44no—
51C22⋊C8×C137—21no—
52C22⋊C8×C137—21no—
53C22⋊C8×C137—21no—
54C4×C8×C137—21no—
55C4×C8×C137—21no—
56C4×C8×C137—21no—
57C4×C8×C137C2×C411yescharacteristic
58C4×C8×C137C2×C411yescharacteristic
59C4×C8×C137D411yescharacteristic
60C4×C8×C137D411yescharacteristic
61#95×C137—22no—
62#41×C137—21no—
63#41×C137—21no—
64#41×C137D812yes—
65#41×C137Q1612yes—
66C2×Q8×C137C4⋊C411yescharacteristic
67C2×D4×C137—21no—
68C2×D4×C137C4⋊C411yescharacteristic
69C22×C4×C137—21no—
70C22×C4×C137—21no—
71C22×C4×C137—21no—
72C22×C4×C137—42no—
73C22×C4×C137C2×D411yescharacteristic
74C22×C4×C137C2×D411yescharacteristic
75C22×C4×C137#4111yescharacteristic
76M16×C137—42no—
77M16×C137—42no—
78C2×C8×C137—22no—
79C2×C8×C137—21no—
80C2×C8×C137—21no—
81C2×C8×C137—41no—
82C2×C8×C137—42no—
83C2×C8×C137—42no—
84C2×C8×C137—42no—
85C2×C8×C137C22⋊C411yescharacteristic
86C2×C8×C137C22⋊C411yescharacteristic
87C2×C8×C137C22⋊C411yescharacteristic
88C2×C8×C137C22⋊C411yescharacteristic
89C4⋊C4×C137—21no—
90C4⋊C4×C137—21no—
91C4⋊C4×C137—24no—
92C4⋊C4×C137—42no—
93C4⋊C4×C137C4⋊C411yescharacteristic
94C4⋊C4×C137C4⋊C411yescharacteristic
95C22⋊C4×C137—21no—
96C22⋊C4×C137—21no—
97C22⋊C4×C137—21no—
98C42×C137—21no—
99C42×C137—21no—
100C42×C137—21no—
101C42×C137—42no—
102C42×C137—42no—
103C42×C137C22×C411yescharacteristic
104C42×C137C2×D411yescharacteristic
105C42×C137C2×Q811yescharacteristic
106C42×C137#4111yescharacteristic
107C23×C137—21no—
108C23×C137#12111yescharacteristic
109Q8×C137—21no—
110Q8×C137#10712yescomplemented
111D4×C137—22no—
112D4×C137—22no—
113D4×C137—21no—
114D4×C137#10712yescomplemented
115C2×C4×C137—21no—
116C2×C4×C137—21no—
117C2×C4×C137—21no—
118C2×C4×C137—22no—
119C2×C4×C137—22no—
120C2×C4×C137—21no—
121C2×C4×C137—21no—
122C2×C4×C137—41no—
123C2×C4×C137—42no—
124C2×C4×C137—41no—
125C2×C4×C137—41no—
126C2×C4×C137—44no—
127C2×C4×C137—82no—
128C2×C4×C137—82no—
129C2×C4×C137C2×C22⋊C411yescharacteristic
130C2×C4×C137C2×C4⋊C411yescharacteristic
131C2×C4×C137#11711yescharacteristic
132C2×C4×C137#12111yescharacteristic
133C2×C4×C137C22⋊Q811yescharacteristic
134C2×C4×C137C22⋊Q811yescharacteristic
135C2×C4×C137C2×D811yescharacteristic
136C2×C4×C137C2×Q1611yescharacteristic
137C2×C4×C137#13511yescharacteristic
138C2×C4×C137#13511yescharacteristic
139C8×C137—21no—
140C8×C137—21no—
141C8×C137—21no—
142C8×C137—41no—
143C8×C137—41no—
144C8×C137—82no—
145C8×C137#10412yescomplemented
146C22×C137—21no—
147C22×C137—21no—
148C22×C137—22no—
149C22×C137—41no—
150C22×C137—41no—
151C22×C137#38011yescharacteristic
152C22×C137#46611yescharacteristic
153C22×C137#46711yescharacteristic
154C4×C137—21no—
155C4×C137—22no—
156C4×C137—21no—
157C4×C137—21no—
158C4×C137—41no—
159C4×C137—41no—
160C4×C137—84no—
161C4×C137C2×#10411yescharacteristic
162C4×C137#42111yescharacteristic
163C4×C137C2×#10711yescharacteristic
164C4×C137#42711yescharacteristic
165C4×C137#46611yescharacteristic
166C4×C137#46711yescharacteristic
167C2×C137—21no—
168C2×C137—21no—
169C2×C137—42no—
170C2×C137#180611yescharacteristic
171C2×C137#180811yescharacteristic
172C2×C137#181411yescharacteristic
173#8667C13711yesmaximal, characteristic, direct product factor, Sylow
174C137#866711yescharacteristic, direct product factor, Sylow
175C8×#41C2×C13711yescharacteristic
176#1806C2×C13712yes—
177C4×#107C2×C13711yescharacteristic, complemented
178#1567C2×C13711yescharacteristic, complemented
179#1303C2×C13712yescomplemented
180C4×#41C22×C13711yescharacteristic
181C8×Q8C4×C13711yescharacteristic, complemented
182C8×Q8C4×C13711yescharacteristic, complemented
183C8×D4—21no—
184C8×D4—21no—
185C8×D4C4×C13711yescharacteristic, complemented
186C8×D4C4×C13711yescharacteristic, complemented
187#431—21no—
188#431C22×C13711yescharacteristic
189C2×#107—21no—
190#423C22×C13712yes—
191C2×#104—22no—
192C2×C4×C8—21no—
193C2×C4×C8C22×C13711yescharacteristic
194C4×C4⋊C4C22×C13712yes—
195#340—21no—
196#339—22no—
197#335—21no—
198#335—21no—
199C2×#41D4×C13711yescharacteristic
200C2×M16—22no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8D4×C13711yescharacteristic
204C22×C8D4×C13711yescharacteristic
205C4×Q8C2×C4×C13711yescharacteristic
206C4×Q8Q8×C13711yescharacteristic
207C4×D4—21no—
208C4×D4—21no—
209C4×D4C2×C4×C13711yescharacteristic
210C4×D4Q8×C13711yescharacteristic
211#117—21no—
212#117D4×C13711yescharacteristic
213C2×C4⋊C4—22no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42C23×C13711yescharacteristic
217#107—24no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8C2×C4×C13711yescharacteristic
223C4⋊C8C2×C4×C13711yescharacteristic
224#104—44no—
225C22⋊C8—21no—
226C22⋊C8—21no—
227C22⋊C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8C2×C4×C13711yescharacteristic
232C4×C8C2×C4×C13711yescharacteristic
233C4×C8D4×C13711yescharacteristic
234C4×C8D4×C13711yescharacteristic
235#95—22no—
236#41—21no—
237#41—21no—
238#41D8×C13712yes—
239#41Q16×C13712yes—
240C2×Q8C4⋊C4×C13711yescharacteristic
241C2×D4—21no—
242C2×D4C4⋊C4×C13711yescharacteristic
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—42no—
247C22×C4C2×D4×C13711yescharacteristic
248C22×C4C2×D4×C13711yescharacteristic
249C22×C4#41×C13711yescharacteristic
250M16—42no—
251M16—42no—
252C2×C8—22no—
253C2×C8—21no—
254C2×C8—21no—
255C2×C8—41no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8C22⋊C4×C13711yescharacteristic
260C2×C8C22⋊C4×C13711yescharacteristic
261C2×C8C22⋊C4×C13711yescharacteristic
262C2×C8C22⋊C4×C13711yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—24no—
266C4⋊C4—42no—
267C4⋊C4C4⋊C4×C13711yescharacteristic
268C4⋊C4C4⋊C4×C13711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—42no—
276C42—42no—
277C42C22×C4×C13711yescharacteristic
278C42C2×D4×C13711yescharacteristic
279C42C2×Q8×C13711yescharacteristic
280C42#41×C13711yescharacteristic
281C23—21no—
282C23#121×C13711yescharacteristic
283Q8—21no—
284Q8#107×C13712yescomplemented
285D4—22no—
286D4—22no—
287D4—21no—
288D4#107×C13712yescomplemented
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—44no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4C2×C22⋊C4×C13711yescharacteristic
304C2×C4C2×C4⋊C4×C13711yescharacteristic
305C2×C4#117×C13711yescharacteristic
306C2×C4#121×C13711yescharacteristic
307C2×C4C22⋊Q8×C13711yescharacteristic
308C2×C4C22⋊Q8×C13711yescharacteristic
309C2×C4C2×D8×C13711yescharacteristic
310C2×C4C2×Q16×C13711yescharacteristic
311C2×C4#135×C13711yescharacteristic
312C2×C4#135×C13711yescharacteristic
313C8—21no—
314C8—21no—
315C8—21no—
316C8—41no—
317C8—41no—
318C8—82no—
319C8#104×C13712yescomplemented
320C22—21no—
321C22—21no—
322C22—22no—
323C22—41no—
324C22—41no—
325C22#380×C13711yescharacteristic
326C22#466×C13711yescharacteristic
327C22#467×C13711yescharacteristic
328C4—21no—
329C4—22no—
330C4—21no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—84no—
335C4C2×#104×C13711yescharacteristic
336C4#421×C13711yescharacteristic
337C4C2×#107×C13711yescharacteristic
338C4#427×C13711yescharacteristic
339C4#466×C13711yescharacteristic
340C4#467×C13711yescharacteristic
341C2—21no—
342C2—21no—
343C2—42no—
344C2#1806×C13711yescharacteristic
345C2#1808×C13711yescharacteristic
346C2#1814×C13711yescharacteristic
subgroupG/H
centerC2×C4×C137#121
commutatorC42C22×C4×C137
Frattini, cosocleC2×C42C23×C137
socleC22×C137#380
prime radicalC2×#41×C137D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. C4×#107×C137C2semidirect
4. #1567×C137C2semidirect
5. #1303×C137C2semidirect
7. C8×Q8×C137C4semidirect
8. C8×Q8×C137C4semidirect
11. C8×D4×C137C4semidirect
12. C8×D4×C137C4semidirect
110. Q8×C137#107semidirect
114. D4×C137#107semidirect
145. C8×C137#104semidirect
173. #8667C137direct (N × H)
174. C137#8667direct (N × H)
177. C4×#107C2×C137semidirect
178. #1567C2×C137semidirect
179. #1303C2×C137semidirect
181. C8×Q8C4×C137semidirect
182. C8×Q8C4×C137semidirect
185. C8×D4C4×C137semidirect
186. C8×D4C4×C137semidirect
284. Q8#107×C137semidirect
288. D4#107×C137semidirect
319. C8#104×C137semidirect

Maximal subgroups

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

subgroupindex
C8×#41×C1372
#1806×C1372
C4×#107×C1372
#1567×C1372
#1303×C1372
#8667137