Finite Group Databasebeta

← C137×#10466 #46859852

Subgroups — C137×#10466 #46859852

Order 35072.

420 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#3037×C137C211yesmaximal, characteristic
2#1805×C137C211yesmaximal, characteristic, complemented
3#1571×C137C211yesmaximal, characteristic, complemented
4#1540×C137C211yesmaximal, characteristic
5#1517×C137C211yesmaximal, characteristic
6#1461×C137C211yesmaximal, characteristic, complemented
7#1286×C137C211yesmaximal, characteristic, complemented
8C2×#137×C137—21no—
9#532×C137C2211yescharacteristic
10C2×C4×Q8×C137C2211yescharacteristic
11#464×C137—22no—
12#462×C137—21no—
13#460×C137—21no—
14#431×C137C2211yescharacteristic
15#431×C137C2211yescharacteristic
16#423×C137C2211yescharacteristic
17C2×C4⋊C8×C137C2211yescharacteristic
18#417×C137—21no—
19C2×#103×C137—21no—
20#382×C137C2211yescharacteristic
21#340×C137—21no—
22#335×C137—22no—
23#332×C137—21no—
24#331×C137—21no—
25#327×C137—22no—
26#324×C137—22no—
27C2×#41×C137—21no—
28C22×Q8×C137D411yescharacteristic
29#137×C137—42no—
30#137×C137—42no—
31C2×Q16×C137—22no—
32C2×SD16×C137—22no—
33C2×M16×C137—21no—
34C22×C8×C137—21no—
35C4⋊Q8×C137D411yescharacteristic
36#127×C137D411yescharacteristic
37#124×C137—21no—
38#121×C137—21no—
39#121×C137—21no—
40C4×Q8×C137—21no—
41C4×Q8×C137—21no—
42C4×Q8×C137—21no—
43C4×Q8×C137—21no—
44C4×D4×C137—21no—
45C4×D4×C137—21no—
46C2×C4⋊C4×C137—21no—
47C2×C4⋊C4×C137—21no—
48C2×C4⋊C4×C137D411yescharacteristic
49C2×C42×C137—21no—
50C2×C42×C137C2311yescharacteristic
51C4⋊C8×C137—21no—
52C4⋊C8×C137—21no—
53C4⋊C8×C137—21no—
54C4⋊C8×C137—21no—
55C4⋊C8×C137—21no—
56C4⋊C8×C137D412yes—
57#103×C137—22no—
58#103×C137—42no—
59#102×C137—22no—
60C22⋊C8×C137—21no—
61C22⋊C8×C137D412yes—
62C4×C8×C137—21no—
63C4×C8×C137—21no—
64#95×C137—21no—
65#95×C137—21no—
66#41×C137—41no—
67#41×C137—41no—
68#41×C137—41no—
69#41×C137—41no—
70C2×Q8×C137—21no—
71C2×Q8×C137—21no—
72C2×Q8×C137—21no—
73C2×Q8×C137—21no—
74C2×Q8×C137—21no—
75C2×Q8×C137—21no—
76C2×Q8×C137D812yes—
77C2×D4×C137—21no—
78C2×D4×C137—21no—
79C2×D4×C137—21no—
80C2×D4×C137—41no—
81C22×C4×C137—21no—
82C22×C4×C137—21no—
83C22×C4×C137—41no—
84C22×C4×C137—41no—
85C22×C4×C137C2×D411yescharacteristic
86C22×C4×C137C2×D411yescharacteristic
87C22×C4×C137#4111yescharacteristic
88Q16×C137—44no—
89SD16×C137—44no—
90M16×C137—41no—
91M16×C137—41no—
92C2×C8×C137—22no—
93C2×C8×C137—22no—
94C2×C8×C137—41no—
95C2×C8×C137—41no—
96C2×C8×C137—41no—
97C2×C8×C137—41no—
98C4⋊C4×C137—21no—
99C4⋊C4×C137—21no—
100C4⋊C4×C137—21no—
101C4⋊C4×C137—21no—
102C4⋊C4×C137—41no—
103C4⋊C4×C137—41no—
104C4⋊C4×C137—41no—
105C4⋊C4×C137—41no—
106C22⋊C4×C137—41no—
107C22⋊C4×C137—41no—
108C42×C137—41no—
109C42×C137—41no—
110C42×C137C2×D411yescharacteristic
111C42×C137C2×D411yescharacteristic
112C42×C137#4111yescharacteristic
113C42×C137#4111yescharacteristic
114C23×C137—41no—
115C23×C137#12111yescharacteristic
116Q8×C137—22no—
117Q8×C137—22no—
118Q8×C137—41no—
119Q8×C137—41no—
120Q8×C137—41no—
121Q8×C137—41no—
122D4×C137—41no—
123D4×C137—41no—
124D4×C137—81no—
125D4×C137—81no—
126D4×C137—81no—
127D4×C137—81no—
128C2×C4×C137—22no—
129C2×C4×C137—41no—
130C2×C4×C137—41no—
131C2×C4×C137—41no—
132C2×C4×C137—41no—
133C2×C4×C137—41no—
134C2×C4×C137—41no—
135C2×C4×C137—81no—
136C2×C4×C137—81no—
137C2×C4×C137—81no—
138C2×C4×C137—81no—
139C2×C4×C137—81no—
140C2×C4×C137—81no—
141C2×C4×C137#12011yescharacteristic
142C2×C4×C137#12111yescharacteristic
143C2×C4×C137#12111yescharacteristic
144C2×C4×C137#12311yescharacteristic
145C2×C4×C137#12311yescharacteristic
146C2×C4×C137#12411yescharacteristic
147C2×C4×C137C2×D811yescharacteristic
148C2×C4×C137#13511yescharacteristic
149C2×C4×C137AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C137#13711yescharacteristic
151C8×C137—42no—
152C8×C137—42no—
153C8×C137—81no—
154C8×C137—81no—
155C22×C137—21no—
156C22×C137—21no—
157C22×C137—81no—
158C22×C137—81no—
159C22×C137—81no—
160C22×C137#39411yescharacteristic
161C22×C137#46611yescharacteristic
162C22×C137#46811yescharacteristic
163C4×C137—21no—
164C4×C137—21no—
165C4×C137—21no—
166C4×C137—21no—
167C4×C137—42no—
168C4×C137—81no—
169C4×C137—81no—
170C4×C137—161no—
171C4×C137#44711yescharacteristic
172C4×C137#45211yescharacteristic
173C4×C137#48011yescharacteristic
174C4×C137#48211yescharacteristic
175C2×C137—21no—
176C2×C137—21no—
177C2×C137—161no—
178C2×C137#203311yescharacteristic
179C2×C137#204911yescharacteristic
180C2×C137#205311yescharacteristic
181#10466C13711yesmaximal, characteristic, direct product factor, Sylow
182C137#1046611yescharacteristic, direct product factor, Sylow
183#3037C2×C13711yescharacteristic
184#1805C2×C13711yescharacteristic, complemented
185#1571C2×C13711yescharacteristic, complemented
186#1540C2×C13711yescharacteristic
187#1517C2×C13711yescharacteristic
188#1461C2×C13711yescharacteristic, complemented
189#1286C2×C13711yescharacteristic, complemented
190C2×#137—21no—
191#532C22×C13711yescharacteristic
192C2×C4×Q8C22×C13711yescharacteristic
193#464—22no—
194#462—21no—
195#460—21no—
196#431C22×C13711yescharacteristic
197#431C22×C13711yescharacteristic
198#423C22×C13711yescharacteristic
199C2×C4⋊C8C22×C13711yescharacteristic
200#417—21no—
201C2×#103—21no—
202#382C22×C13711yescharacteristic
203#340—21no—
204#335—22no—
205#332—21no—
206#331—21no—
207#327—22no—
208#324—22no—
209C2×#41—21no—
210C22×Q8D4×C13711yescharacteristic
211#137—42no—
212#137—42no—
213C2×Q16—22no—
214C2×SD16—22no—
215C2×M16—21no—
216C22×C8—21no—
217C4⋊Q8D4×C13711yescharacteristic
218#127D4×C13711yescharacteristic
219#124—21no—
220#121—21no—
221#121—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8—21no—
225C4×Q8—21no—
226C4×D4—21no—
227C4×D4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4D4×C13711yescharacteristic
231C2×C42—21no—
232C2×C42C23×C13711yescharacteristic
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8D4×C13712yes—
239#103—22no—
240#103—42no—
241#102—22no—
242C22⋊C8—21no—
243C22⋊C8D4×C13712yes—
244C4×C8—21no—
245C4×C8—21no—
246#95—21no—
247#95—21no—
248#41—41no—
249#41—41no—
250#41—41no—
251#41—41no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—21no—
255C2×Q8—21no—
256C2×Q8—21no—
257C2×Q8—21no—
258C2×Q8D8×C13712yes—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—41no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4C2×D4×C13711yescharacteristic
268C22×C4C2×D4×C13711yescharacteristic
269C22×C4#41×C13711yescharacteristic
270Q16—44no—
271SD16—44no—
272M16—41no—
273M16—41no—
274C2×C8—22no—
275C2×C8—22no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C22⋊C4—41no—
289C22⋊C4—41no—
290C42—41no—
291C42—41no—
292C42C2×D4×C13711yescharacteristic
293C42C2×D4×C13711yescharacteristic
294C42#41×C13711yescharacteristic
295C42#41×C13711yescharacteristic
296C23—41no—
297C23#121×C13711yescharacteristic
298Q8—22no—
299Q8—22no—
300Q8—41no—
301Q8—41no—
302Q8—41no—
303Q8—41no—
304D4—41no—
305D4—41no—
306D4—81no—
307D4—81no—
308D4—81no—
309D4—81no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—81no—
322C2×C4—81no—
323C2×C4#120×C13711yescharacteristic
324C2×C4#121×C13711yescharacteristic
325C2×C4#121×C13711yescharacteristic
326C2×C4#123×C13711yescharacteristic
327C2×C4#123×C13711yescharacteristic
328C2×C4#124×C13711yescharacteristic
329C2×C4C2×D8×C13711yescharacteristic
330C2×C4#135×C13711yescharacteristic
331C2×C4AGL(1,ℤ/8)×C13711yescharacteristic
332C2×C4#137×C13711yescharacteristic
333C8—42no—
334C8—42no—
335C8—81no—
336C8—81no—
337C22—21no—
338C22—21no—
339C22—81no—
340C22—81no—
341C22—81no—
342C22#394×C13711yescharacteristic
343C22#466×C13711yescharacteristic
344C22#468×C13711yescharacteristic
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—42no—
350C4—81no—
351C4—81no—
352C4—161no—
353C4#447×C13711yescharacteristic
354C4#452×C13711yescharacteristic
355C4#480×C13711yescharacteristic
356C4#482×C13711yescharacteristic
357C2—21no—
358C2—21no—
359C2—161no—
360C2#2033×C13711yescharacteristic
361C2#2049×C13711yescharacteristic
362C2#2053×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#394
commutator, Frattini, cosocleC2×C42C23×C137
prime radical#532×C137C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C2×C42×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1805×C137C2semidirect
3. #1571×C137C2semidirect
6. #1461×C137C2semidirect
7. #1286×C137C2semidirect
181. #10466C137direct (N × H)
182. C137#10466direct (N × H)
184. #1805C2×C137semidirect
185. #1571C2×C137semidirect
188. #1461C2×C137semidirect
189. #1286C2×C137semidirect

Maximal subgroups

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

subgroupindex
#3037×C1372
#1805×C1372
#1571×C1372
#1540×C1372
#1517×C1372
#1461×C1372
#1286×C1372
#10466137