Finite Group Databasebeta

← C137×#13073 #46862459

Subgroups — C137×#13073 #46862459

Order 35072.

478 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#504×C137C211yesmaximal, characteristic, complemented
2C2×#472×C137C211yesmaximal, characteristic
3C2×#361×C137C211yesmaximal, characteristic, complemented
4C2×#357×C137C211yesmaximal, characteristic, complemented
5#1816×C137C211yesmaximal, characteristic, complemented
6#1387×C137C211yesmaximal, characteristic, complemented
7#1381×C137C211yesmaximal, characteristic, complemented
8C2×AGL(1,ℤ/8)×C137—21no—
9C22×D8×C137C2211yescharacteristic
10C2×#131×C137C2211yescharacteristic
11#504×C137—22no—
12#504×C137—22no—
13C2×M32×C137—21no—
14C2×M32×C137C2211yescharacteristic
15C22×C16×C137—21no—
16C22×C16×C137C2211yescharacteristic
17#472×C137—22no—
18#472×C137C414yescomplemented
19C2×#108×C137C2211yescharacteristic
20C2×#108×C137C2211yescharacteristic
21C2×#107×C137C2211yescharacteristic
22#417×C137—21no—
23C2×#100×C137—21no—
24#361×C137—22no—
25#357×C137—21no—
26#357×C137—21no—
27#343×C137—21no—
28C2×#41×C137D411yescharacteristic
29C22×D4×C137—21no—
30AGL(1,ℤ/8)×C137—24no—
31AGL(1,ℤ/8)×C137—42no—
32C2×SD16×C137—41no—
33C2×D8×C137—21no—
34C2×D8×C137—21no—
35C2×D8×C137—21no—
36C2×D8×C137—21no—
37C2×D8×C137—21no—
38C2×D8×C137—41no—
39C2×D8×C137C2×C412yes—
40#131×C137—21no—
41#131×C137—21no—
42#131×C137C2×C412yes—
43#131×C137D412yes—
44C2×M16×C137—21no—
45C2×M16×C137—21no—
46C2×M16×C137—21no—
47C2×M16×C137D411yescharacteristic
48C22×C8×C137—21no—
49C22×C8×C137C2311yescharacteristic
50C2×C4⋊C4×C137—21no—
51M32×C137—24no—
52M32×C137—21no—
53M32×C137D412yes—
54C2×C16×C137—24no—
55C2×C16×C137—21no—
56C2×C16×C137—21no—
57C2×C16×C137—21no—
58C2×C16×C137D411yescharacteristic
59C2×C16×C137D411yescharacteristic
60#108×C137—21no—
61#108×C137—21no—
62#108×C137—21no—
63#108×C137C2×C412yes—
64#107×C137—21no—
65#107×C137—21no—
66#103×C137—41no—
67#102×C137—41no—
68#100×C137—24no—
69C24×C137—41no—
70#41×C137—21no—
71#41×C137—21no—
72#41×C137C22⋊C412yes—
73#41×C137SD1612yes—
74C2×Q8×C137D811yescharacteristic
75C2×D4×C137—21no—
76C2×D4×C137—22no—
77C2×D4×C137—41no—
78C2×D4×C137—41no—
79C2×D4×C137—41no—
80C2×D4×C137—41no—
81C2×D4×C137—41no—
82C2×D4×C137D811yescharacteristic
83C22×C4×C137—21no—
84C22×C4×C137—41no—
85C22×C4×C137C2×D411yescharacteristic
86SD16×C137—44no—
87D8×C137—22no—
88D8×C137—22no—
89D8×C137—44no—
90D8×C137—41no—
91D8×C137—41no—
92M16×C137—21no—
93M16×C137—22no—
94M16×C137—22no—
95M16×C137—22no—
96M16×C137—41no—
97M16×C137—41no—
98M16×C137—41no—
99M16×C137C22⋊C412yes—
100C2×C8×C137—21no—
101C2×C8×C137—22no—
102C2×C8×C137—22no—
103C2×C8×C137—41no—
104C2×C8×C137—41no—
105C2×C8×C137C22×C411yescharacteristic
106C2×C8×C137C2×D411yescharacteristic
107C2×C8×C137C2×D411yescharacteristic
108C2×C8×C137C2×D411yescharacteristic
109C2×C8×C137#4111yescharacteristic
110C2×C8×C137#4111yescharacteristic
111C4⋊C4×C137—41no—
112C4⋊C4×C137—41no—
113C16×C137—21no—
114C16×C137—22no—
115C16×C137—21no—
116C23×C137—21no—
117C23×C137—42no—
118C23×C137—81no—
119C23×C137—81no—
120C23×C137—161no—
121C23×C137—161no—
122C23×C137#13511yescharacteristic
123Q8×C137—21no—
124Q8×C137#10212yes—
125D4×C137—22no—
126D4×C137—22no—
127D4×C137—21no—
128D4×C137—42no—
129D4×C137—42no—
130D4×C137—81no—
131D4×C137—81no—
132D4×C137#10212yes—
133C2×C4×C137—21no—
134C2×C4×C137—22no—
135C2×C4×C137—22no—
136C2×C4×C137—81no—
137C2×C4×C137—161no—
138C2×C4×C137C2×C22⋊C411yescharacteristic
139C2×C4×C137#12111yescharacteristic
140C2×C4×C137#12111yescharacteristic
141C2×C4×C137C2×D811yescharacteristic
142C2×C4×C137C2×SD1611yescharacteristic
143C2×C4×C137#13511yescharacteristic
144C8×C137—22no—
145C8×C137—42no—
146C8×C137—81no—
147C8×C137C4×D411yescharacteristic
148C8×C137C4×D411yescharacteristic
149C8×C137#12011yescharacteristic
150C8×C137#12311yescharacteristic
151C22×C137—21no—
152C22×C137—21no—
153C22×C137—22no—
154C22×C137—41no—
155C22×C137—41no—
156C22×C137—82no—
157C22×C137—162no—
158C22×C137—162no—
159C22×C137—161no—
160C22×C137—161no—
161C22×C137#41611yescharacteristic
162C22×C137#46511yescharacteristic
163C22×C137#46611yescharacteristic
164C4×C137—22no—
165C4×C137—161no—
166C4×C137#38611yescharacteristic
167C4×C137C2×#10211yescharacteristic
168C4×C137#46511yescharacteristic
169C4×C137#46611yescharacteristic
170C2×C137—21no—
171C2×C137—21no—
172C2×C137—42no—
173C2×C137—162no—
174C2×C137#189311yescharacteristic
175C2×C137#222111yescharacteristic
176C2×C137#222311yescharacteristic
177#13073C13711yesmaximal, characteristic, direct product factor, Sylow
178C137#1307311yescharacteristic, direct product factor, Sylow
179C2×#504C2×C13711yescharacteristic, complemented
180C2×#472C2×C13711yescharacteristic
181C2×#361C2×C13711yescharacteristic, complemented
182C2×#357C2×C13711yescharacteristic, complemented
183#1816C2×C13711yescharacteristic, complemented
184#1387C2×C13711yescharacteristic, complemented
185#1381C2×C13711yescharacteristic, complemented
186C2×AGL(1,ℤ/8)—21no—
187C22×D8C22×C13711yescharacteristic
188C2×#131C22×C13711yescharacteristic
189#504—22no—
190#504—22no—
191C2×M32—21no—
192C2×M32C22×C13711yescharacteristic
193C22×C16—21no—
194C22×C16C22×C13711yescharacteristic
195#472—22no—
196#472C4×C13714yescomplemented
197C2×#108C22×C13711yescharacteristic
198C2×#108C22×C13711yescharacteristic
199C2×#107C22×C13711yescharacteristic
200#417—21no—
201C2×#100—21no—
202#361—22no—
203#357—21no—
204#357—21no—
205#343—21no—
206C2×#41D4×C13711yescharacteristic
207C22×D4—21no—
208AGL(1,ℤ/8)—24no—
209AGL(1,ℤ/8)—42no—
210C2×SD16—41no—
211C2×D8—21no—
212C2×D8—21no—
213C2×D8—21no—
214C2×D8—21no—
215C2×D8—21no—
216C2×D8—41no—
217C2×D8C2×C4×C13712yes—
218#131—21no—
219#131—21no—
220#131C2×C4×C13712yes—
221#131D4×C13712yes—
222C2×M16—21no—
223C2×M16—21no—
224C2×M16—21no—
225C2×M16D4×C13711yescharacteristic
226C22×C8—21no—
227C22×C8C23×C13711yescharacteristic
228C2×C4⋊C4—21no—
229M32—24no—
230M32—21no—
231M32D4×C13712yes—
232C2×C16—24no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16D4×C13711yescharacteristic
237C2×C16D4×C13711yescharacteristic
238#108—21no—
239#108—21no—
240#108—21no—
241#108C2×C4×C13712yes—
242#107—21no—
243#107—21no—
244#103—41no—
245#102—41no—
246#100—24no—
247C24—41no—
248#41—21no—
249#41—21no—
250#41C22⋊C4×C13712yes—
251#41SD16×C13712yes—
252C2×Q8D8×C13711yescharacteristic
253C2×D4—21no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4D8×C13711yescharacteristic
261C22×C4—21no—
262C22×C4—41no—
263C22×C4C2×D4×C13711yescharacteristic
264SD16—44no—
265D8—22no—
266D8—22no—
267D8—44no—
268D8—41no—
269D8—41no—
270M16—21no—
271M16—22no—
272M16—22no—
273M16—22no—
274M16—41no—
275M16—41no—
276M16—41no—
277M16C22⋊C4×C13712yes—
278C2×C8—21no—
279C2×C8—22no—
280C2×C8—22no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8C22×C4×C13711yescharacteristic
284C2×C8C2×D4×C13711yescharacteristic
285C2×C8C2×D4×C13711yescharacteristic
286C2×C8C2×D4×C13711yescharacteristic
287C2×C8#41×C13711yescharacteristic
288C2×C8#41×C13711yescharacteristic
289C4⋊C4—41no—
290C4⋊C4—41no—
291C16—21no—
292C16—22no—
293C16—21no—
294C23—21no—
295C23—42no—
296C23—81no—
297C23—81no—
298C23—161no—
299C23—161no—
300C23#135×C13711yescharacteristic
301Q8—21no—
302Q8#102×C13712yes—
303D4—22no—
304D4—22no—
305D4—21no—
306D4—42no—
307D4—42no—
308D4—81no—
309D4—81no—
310D4#102×C13712yes—
311C2×C4—21no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—81no—
315C2×C4—161no—
316C2×C4C2×C22⋊C4×C13711yescharacteristic
317C2×C4#121×C13711yescharacteristic
318C2×C4#121×C13711yescharacteristic
319C2×C4C2×D8×C13711yescharacteristic
320C2×C4C2×SD16×C13711yescharacteristic
321C2×C4#135×C13711yescharacteristic
322C8—22no—
323C8—42no—
324C8—81no—
325C8C4×D4×C13711yescharacteristic
326C8C4×D4×C13711yescharacteristic
327C8#120×C13711yescharacteristic
328C8#123×C13711yescharacteristic
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—82no—
335C22—162no—
336C22—162no—
337C22—161no—
338C22—161no—
339C22#416×C13711yescharacteristic
340C22#465×C13711yescharacteristic
341C22#466×C13711yescharacteristic
342C4—22no—
343C4—161no—
344C4#386×C13711yescharacteristic
345C4C2×#102×C13711yescharacteristic
346C4#465×C13711yescharacteristic
347C4#466×C13711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—162no—
352C2#1893×C13711yescharacteristic
353C2#2221×C13711yescharacteristic
354C2#2223×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#466
commutatorC2×C8C22×C4×C137
Frattini, cosocleC22×C8C23×C137
prime radicalC2×#472×C137C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×#131×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#504×C137C2semidirect
3. C2×#361×C137C2semidirect
4. C2×#357×C137C2semidirect
5. #1816×C137C2semidirect
6. #1387×C137C2semidirect
7. #1381×C137C2semidirect
18. #472×C137C4semidirect
177. #13073C137direct (N × H)
178. C137#13073direct (N × H)
179. C2×#504C2×C137semidirect
181. C2×#361C2×C137semidirect
182. C2×#357C2×C137semidirect
183. #1816C2×C137semidirect
184. #1387C2×C137semidirect
185. #1381C2×C137semidirect
196. #472C4×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#504×C1372
C2×#472×C1372
C2×#361×C1372
C2×#357×C1372
#1816×C1372
#1387×C1372
#1381×C1372
#13073137