Finite Group Databasebeta

← C137×#22691 #46872077

Subgroups — C137×#22691 #46872077

Order 35072.

846 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#443×C137C211yesmaximal, characteristic
2#2952×C137C212yesmaximal
3#2913×C137C212yesmaximal, complemented
4#1951×C137C212yesmaximal, complemented
5#1860×C137C214yesmaximal, complemented
6#1846×C137C211yesmaximal, characteristic, complemented
7#1816×C137C212yesmaximal, complemented
8#1772×C137C211yesmaximal, characteristic, complemented
9C2×#135×C137C2211yescharacteristic
10C2×#131×C137C2212yes—
11#520×C137C2214yes—
12#443×C137C418yescomplemented
13#442×C137—22no—
14#441×C137—22no—
15C4×SD16×C137—24no—
16C2×#108×C137C2211yescharacteristic
17C2×#108×C137C2211yescharacteristic
18#428×C137—24no—
19#428×C137C2214yes—
20#427×C137—22no—
21#427×C137C2211yescharacteristic
22C2×#107×C137—22no—
23C2×#107×C137C2212yes—
24C2×#106×C137C2212yes—
25C2×#104×C137C2212yes—
26#417×C137C2214yes—
27#405×C137C2212yes—
28C2×C4×C8×C137C2211yescharacteristic
29#382×C137C2212yes—
30#343×C137C2214yes—
31#339×C137C2212yes—
32C2×#41×C137C2312yes—
33#135×C137C2×C414yes—
34#135×C137Q814yes—
35C2×Q16×C137D411yescharacteristic
36C2×SD16×C137D412yes—
37C2×D8×C137D411yescharacteristic
38#131×C137—24no—
39#131×C137C2×C418yescomplemented
40C2×M16×C137—22no—
41C2×M16×C137C2312yes—
42C2×M16×C137C2312yes—
43C22×C8×C137—22no—
44C22×C8×C137C2311yescharacteristic
45C22×C8×C137C2311yescharacteristic
46C4×Q8×C137—24no—
47C4×D4×C137—24no—
48C4×D4×C137—44no—
49#117×C137—24no—
50#117×C137C2312yes—
51C2×C4⋊C4×C137—24no—
52C2×C4⋊C4×C137C2314yes—
53C2×C42×C137C2311yescharacteristic
54#108×C137—22no—
55#108×C137C2×C414yescomplemented
56#107×C137—22no—
57#107×C137—22no—
58#107×C137—24no—
59#107×C137D412yes—
60#106×C137—22no—
61#106×C137—22no—
62#106×C137—24no—
63#106×C137D412yes—
64#104×C137C2×C418yescomplemented
65#103×C137—24no—
66#102×C137—24no—
67#97×C137—22no—
68C4×C8×C137—22no—
69C4×C8×C137C2×C414yescomplemented
70#95×C137—42no—
71#41×C137—22no—
72#41×C137—22no—
73#41×C137C22×C414yes—
74#41×C137C2×Q814yes—
75C2×Q8×C137C2×D412yes—
76C2×D4×C137—22no—
77C2×D4×C137C2×D412yes—
78C22×C4×C137—22no—
79C22×C4×C137—21no—
80C22×C4×C137—24no—
81C22×C4×C137—44no—
82C22×C4×C137C2411yescharacteristic
83Q16×C137C4⋊C414yes—
84SD16×C137C4⋊C418yes—
85D8×C137C4⋊C414yes—
86M16×C137—22no—
87M16×C137—28no—
88M16×C137—22no—
89M16×C137—22no—
90M16×C137C22×C414yes—
91C2×C8×C137—22no—
92C2×C8×C137—28no—
93C2×C8×C137—22no—
94C2×C8×C137C22×C412yes—
95C2×C8×C137C22×C412yes—
96C2×C8×C137C2×D411yescharacteristic
97C2×C8×C137C2×D411yescharacteristic
98C2×C8×C137C2×D411yescharacteristic
99C2×C8×C137C2×D411yescharacteristic
100C2×C8×C137C2×Q812yes—
101C2×C8×C137#4112yes—
102C4⋊C4×C137—24no—
103C4⋊C4×C137—24no—
104C4⋊C4×C137—44no—
105C4⋊C4×C137C2×D414yes—
106C22⋊C4×C137—22no—
107C22⋊C4×C137—44no—
108C42×C137—22no—
109C42×C137—21no—
110C42×C137—44no—
111C42×C137C22×C412yes—
112C23×C137—22no—
113C23×C137H(5,𝔽2)11yescharacteristic
114Q8×C137—22no—
115Q8×C137C2×C4⋊C414yes—
116D4×C137—24no—
117D4×C137—24no—
118D4×C137—22no—
119D4×C137C2×C4⋊C414yes—
120C2×C4×C137—22no—
121C2×C4×C137—24no—
122C2×C4×C137—24no—
123C2×C4×C137—22no—
124C2×C4×C137—24no—
125C2×C4×C137—42no—
126C2×C4×C137—44no—
127C2×C4×C137—84no—
128C2×C4×C137—164no—
129C2×C4×C137C23×C411yescharacteristic
130C2×C4×C137C22×D411yescharacteristic
131C2×C4×C137C22×D411yescharacteristic
132C2×C4×C137C22×Q811yescharacteristic
133C2×C4×C137C2×#4111yescharacteristic
134C2×C4×C1372−1+411yescharacteristic
135C8×C137—24no—
136C8×C137—42no—
137C8×C137C2×C4⋊C412yes—
138C8×C137C2×C4⋊C412yes—
139C8×C137C4×D412yes—
140C8×C137C4×D412yes—
141C22×C137—22no—
142C22×C137—24no—
143C22×C137—42no—
144C22×C137—42no—
145C22×C137#52011yescharacteristic
146C22×C137D4211yescharacteristic
147C22×C137#55411yescharacteristic
148C4×C137—24no—
149C4×C137—42no—
150C4×C137—44no—
151C4×C137—84no—
152C4×C137C22×C4⋊C411yescharacteristic
153C4×C137C2×C4×D411yescharacteristic
154C4×C137#54711yescharacteristic
155C4×C137D4×Q811yescharacteristic
156C2×C137—21no—
157C2×C137—21no—
158C2×C137—44no—
159C2×C137D4×C4⋊C411yescharacteristic
160C2×C137#329012yes—
161#22691C13711yesmaximal, characteristic, direct product factor, Sylow
162C137#2269111yescharacteristic, direct product factor, Sylow
163C2×#443C2×C13711yescharacteristic
164#2952C2×C13712yes—
165#2913C2×C13712yescomplemented
166#1951C2×C13712yescomplemented
167#1860C2×C13714yescomplemented
168#1846C2×C13711yescharacteristic, complemented
169#1816C2×C13712yescomplemented
170#1772C2×C13711yescharacteristic, complemented
171C2×#135C22×C13711yescharacteristic
172C2×#131C22×C13712yes—
173#520C22×C13714yes—
174#443C4×C13718yescomplemented
175#442—22no—
176#441—22no—
177C4×SD16—24no—
178C2×#108C22×C13711yescharacteristic
179C2×#108C22×C13711yescharacteristic
180#428—24no—
181#428C22×C13714yes—
182#427—22no—
183#427C22×C13711yescharacteristic
184C2×#107—22no—
185C2×#107C22×C13712yes—
186C2×#106C22×C13712yes—
187C2×#104C22×C13712yes—
188#417C22×C13714yes—
189#405C22×C13712yes—
190C2×C4×C8C22×C13711yescharacteristic
191#382C22×C13712yes—
192#343C22×C13714yes—
193#339C22×C13712yes—
194C2×#41C23×C13712yes—
195#135C2×C4×C13714yes—
196#135Q8×C13714yes—
197C2×Q16D4×C13711yescharacteristic
198C2×SD16D4×C13712yes—
199C2×D8D4×C13711yescharacteristic
200#131—24no—
201#131C2×C4×C13718yescomplemented
202C2×M16—22no—
203C2×M16C23×C13712yes—
204C2×M16C23×C13712yes—
205C22×C8—22no—
206C22×C8C23×C13711yescharacteristic
207C22×C8C23×C13711yescharacteristic
208C4×Q8—24no—
209C4×D4—24no—
210C4×D4—44no—
211#117—24no—
212#117C23×C13712yes—
213C2×C4⋊C4—24no—
214C2×C4⋊C4C23×C13714yes—
215C2×C42C23×C13711yescharacteristic
216#108—22no—
217#108C2×C4×C13714yescomplemented
218#107—22no—
219#107—22no—
220#107—24no—
221#107D4×C13712yes—
222#106—22no—
223#106—22no—
224#106—24no—
225#106D4×C13712yes—
226#104C2×C4×C13718yescomplemented
227#103—24no—
228#102—24no—
229#97—22no—
230C4×C8—22no—
231C4×C8C2×C4×C13714yescomplemented
232#95—42no—
233#41—22no—
234#41—22no—
235#41C22×C4×C13714yes—
236#41C2×Q8×C13714yes—
237C2×Q8C2×D4×C13712yes—
238C2×D4—22no—
239C2×D4C2×D4×C13712yes—
240C22×C4—22no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—44no—
244C22×C4C24×C13711yescharacteristic
245Q16C4⋊C4×C13714yes—
246SD16C4⋊C4×C13718yes—
247D8C4⋊C4×C13714yes—
248M16—22no—
249M16—28no—
250M16—22no—
251M16—22no—
252M16C22×C4×C13714yes—
253C2×C8—22no—
254C2×C8—28no—
255C2×C8—22no—
256C2×C8C22×C4×C13712yes—
257C2×C8C22×C4×C13712yes—
258C2×C8C2×D4×C13711yescharacteristic
259C2×C8C2×D4×C13711yescharacteristic
260C2×C8C2×D4×C13711yescharacteristic
261C2×C8C2×D4×C13711yescharacteristic
262C2×C8C2×Q8×C13712yes—
263C2×C8#41×C13712yes—
264C4⋊C4—24no—
265C4⋊C4—24no—
266C4⋊C4—44no—
267C4⋊C4C2×D4×C13714yes—
268C22⋊C4—22no—
269C22⋊C4—44no—
270C42—22no—
271C42—21no—
272C42—44no—
273C42C22×C4×C13712yes—
274C23—22no—
275C23H(5,𝔽2)×C13711yescharacteristic
276Q8—22no—
277Q8C2×C4⋊C4×C13714yes—
278D4—24no—
279D4—24no—
280D4—22no—
281D4C2×C4⋊C4×C13714yes—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—24no—
287C2×C4—42no—
288C2×C4—44no—
289C2×C4—84no—
290C2×C4—164no—
291C2×C4C23×C4×C13711yescharacteristic
292C2×C4C22×D4×C13711yescharacteristic
293C2×C4C22×D4×C13711yescharacteristic
294C2×C4C22×Q8×C13711yescharacteristic
295C2×C4C2×#41×C13711yescharacteristic
296C2×C42−1+4×C13711yescharacteristic
297C8—24no—
298C8—42no—
299C8C2×C4⋊C4×C13712yes—
300C8C2×C4⋊C4×C13712yes—
301C8C4×D4×C13712yes—
302C8C4×D4×C13712yes—
303C22—22no—
304C22—24no—
305C22—42no—
306C22—42no—
307C22#520×C13711yescharacteristic
308C22D42×C13711yescharacteristic
309C22#554×C13711yescharacteristic
310C4—24no—
311C4—42no—
312C4—44no—
313C4—84no—
314C4C22×C4⋊C4×C13711yescharacteristic
315C4C2×C4×D4×C13711yescharacteristic
316C4#547×C13711yescharacteristic
317C4D4×Q8×C13711yescharacteristic
318C2—21no—
319C2—21no—
320C2—44no—
321C2D4×C4⋊C4×C13711yescharacteristic
322C2#3290×C13712yes—
subgroupG/H
center, socleC22×C137D42
commutatorC2×C4C23×C4×C137
Frattini, cosocleC22×C4C24×C137
prime radicalC2×#135×C137C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
3. #2913×C137C2semidirect
4. #1951×C137C2semidirect
5. #1860×C137C2semidirect
6. #1846×C137C2semidirect
7. #1816×C137C2semidirect
8. #1772×C137C2semidirect
12. #443×C137C4semidirect
39. #131×C137C2×C4semidirect
55. #108×C137C2×C4semidirect
64. #104×C137C2×C4semidirect
69. C4×C8×C137C2×C4semidirect
161. #22691C137direct (N × H)
162. C137#22691direct (N × H)
165. #2913C2×C137semidirect
166. #1951C2×C137semidirect
167. #1860C2×C137semidirect
168. #1846C2×C137semidirect
169. #1816C2×C137semidirect
170. #1772C2×C137semidirect
174. #443C4×C137semidirect
201. #131C2×C4×C137semidirect
217. #108C2×C4×C137semidirect
226. #104C2×C4×C137semidirect
231. C4×C8C2×C4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#443×C1372
#2952×C1372
#2913×C1372
#1951×C1372
#1860×C1372
#1846×C1372
#1816×C1372
#1772×C1372
#22691137