Finite Group Databasebeta

← C137×#15171 #46864557

Subgroups — C137×#15171 #46864557

Order 35072.

1302 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#401×C137C211yesmaximal, characteristic, complemented
2C2×#396×C137C211yesmaximal, characteristic
3#1442×C137C213yesmaximal, complemented
4#1440×C137C213yesmaximal
5#1437×C137C213yesmaximal
6#1436×C137C213yesmaximal, complemented
7C4×#95×C137C211yesmaximal, characteristic, complemented
8C22×C4⋊C4×C137C2213yes—
9C22×C22⋊C4×C137C2211yescharacteristic
10C22×C22⋊C4×C137C2213yes—
11C22×C22⋊C4×C137C2213yes—
12C22×C42×C137C2213yes—
13#401×C137—24no—
14#396×C137—24no—
15C2×#95×C137C2213yes—
16C2×#95×C137C2213yes—
17C2×#95×C137C2213yes—
18C2×#95×C137C2213yes—
19C2×#95×C137C2213yes—
20C2×#95×C137C2213yes—
21C2×#95×C137C2213yes—
22C2×#95×C137C2211yescharacteristic
23C25×C137C2311yescharacteristic
24C23×C4×C137C2311yescharacteristic
25C23×C4×C137C2313yes—
26C23×C4×C137C2313yes—
27C23×C4×C137C2313yes—
28C23×C4×C137C2313yes—
29C23×C4×C137C2311yescharacteristic
30C2×C4⋊C4×C137—26no—
31C2×C4⋊C4×C137—26no—
32C2×C4⋊C4×C137—43no—
33C2×C22⋊C4×C137—26no—
34C2×C22⋊C4×C137—26no—
35C2×C22⋊C4×C137—26no—
36C2×C22⋊C4×C137—26no—
37C2×C22⋊C4×C137—26no—
38C2×C22⋊C4×C137—43no—
39C2×C22⋊C4×C137—43no—
40C2×C42×C137—26no—
41C2×C42×C137—43no—
42C2×C42×C137—43no—
43#95×C137—26no—
44#95×C137—26no—
45#95×C137—26no—
46#95×C137—26no—
47#95×C137—26no—
48#95×C137—26no—
49#95×C137—43no—
50#95×C137D414yes—
51C24×C137—21no—
52C24×C137—23no—
53C24×C137—23no—
54C24×C137—23no—
55C24×C137—23no—
56C24×C137—21no—
57C24×C137#4112yes—
58C24×C137C2411yescharacteristic
59C22×C4×C137—23no—
60C22×C4×C137—21no—
61C22×C4×C137—23no—
62C22×C4×C137—23no—
63C22×C4×C137—23no—
64C22×C4×C137—23no—
65C22×C4×C137—23no—
66C22×C4×C137—23no—
67C22×C4×C137—23no—
68C22×C4×C137—23no—
69C22×C4×C137—23no—
70C22×C4×C137—23no—
71C22×C4×C137—23no—
72C22×C4×C137—23no—
73C22×C4×C137—23no—
74C22×C4×C137—23no—
75C22×C4×C137—23no—
76C22×C4×C137—23no—
77C22×C4×C137—23no—
78C22×C4×C137—23no—
79C22×C4×C137—21no—
80C22×C4×C137—23no—
81C22×C4×C137—23no—
82C22×C4×C137—23no—
83C22×C4×C137—23no—
84C22×C4×C137—23no—
85C22×C4×C137—23no—
86C22×C4×C137—23no—
87C22×C4×C137—23no—
88C22×C4×C137—23no—
89C22×C4×C137C2×D416yes—
90C22×C4×C137#4116yes—
91C22×C4×C137#4116yes—
92C22×C4×C137#4116yes—
93C4⋊C4×C137—412no—
94C22⋊C4×C137—412no—
95C22⋊C4×C137—412no—
96C22⋊C4×C137—44no—
97C42×C137—412no—
98C23×C137—26no—
99C23×C137—22no—
100C23×C137—26no—
101C23×C137—41no—
102C23×C137—43no—
103C23×C137—43no—
104C23×C137—43no—
105C23×C137—43no—
106C23×C137—43no—
107C23×C137—43no—
108C23×C137—43no—
109C23×C137—43no—
110C23×C137—43no—
111C23×C137C22×D411yescharacteristic
112C23×C137C2×#4113yes—
113C23×C137C2×#4113yes—
114C23×C137C2×#4111yescharacteristic
115C23×C137C2×#4113yes—
116C23×C137H(5,𝔽2)13yes—
117C23×C1372−1+411yescharacteristic
118C2×C4×C137—26no—
119C2×C4×C137—26no—
120C2×C4×C137—26no—
121C2×C4×C137—26no—
122C2×C4×C137—26no—
123C2×C4×C137—26no—
124C2×C4×C137—26no—
125C2×C4×C137—26no—
126C2×C4×C137—26no—
127C2×C4×C137—26no—
128C2×C4×C137—26no—
129C2×C4×C137—43no—
130C2×C4×C137—43no—
131C2×C4×C137—43no—
132C2×C4×C137—43no—
133C2×C4×C137—43no—
134C2×C4×C137—43no—
135C2×C4×C137—43no—
136C2×C4×C137—43no—
137C2×C4×C137—43no—
138C2×C4×C137—43no—
139C2×C4×C137—43no—
140C2×C4×C137—43no—
141C2×C4×C137—41no—
142C2×C4×C137—43no—
143C2×C4×C137—43no—
144C2×C4×C137—43no—
145C2×C4×C137—43no—
146C2×C4×C137—43no—
147C2×C4×C137—43no—
148C2×C4×C137—43no—
149C2×C4×C137—43no—
150C2×C4×C137—43no—
151C2×C4×C137#12614yes—
152C22×C137—46no—
153C22×C137—46no—
154C22×C137—42no—
155C22×C137—83no—
156C22×C137—81no—
157C22×C137—81no—
158C22×C137—83no—
159C22×C137C2×#12611yescharacteristic
160C22×C137#52913yes—
161C22×C137#52913yes—
162C22×C137#52913yes—
163C22×C137#53213yes—
164C22×C137#53813yes—
165C22×C137#53913yes—
166C22×C137#54613yes—
167C22×C137#54811yescharacteristic
168C22×C137#55113yes—
169C22×C137#55313yes—
170C22×C137#55313yes—
171C22×C137#55613yes—
172C4×C137—24no—
173C4×C137—46no—
174C4×C137—46no—
175C4×C137—46no—
176C4×C137—83no—
177C4×C137—81no—
178C2×C137—82no—
179C2×C137#240111yescharacteristic
180C2×C137#255913yes—
181C2×C137#256013yes—
182C2×C137#257613yes—
183C2×C137#259413yes—
184C2×C137#264311yescharacteristic
185C2×C137#286011yescharacteristic
186#15171C13711yesmaximal, characteristic, direct product factor, Sylow
187C137#1517111yescharacteristic, direct product factor, Sylow
188C2×#401C2×C13711yescharacteristic, complemented
189C2×#396C2×C13711yescharacteristic
190#1442C2×C13713yescomplemented
191#1440C2×C13713yes—
192#1437C2×C13713yes—
193#1436C2×C13713yescomplemented
194C4×#95C2×C13711yescharacteristic, complemented
195C22×C4⋊C4C22×C13713yes—
196C22×C22⋊C4C22×C13711yescharacteristic
197C22×C22⋊C4C22×C13713yes—
198C22×C22⋊C4C22×C13713yes—
199C22×C42C22×C13713yes—
200#401—24no—
201#396—24no—
202C2×#95C22×C13713yes—
203C2×#95C22×C13713yes—
204C2×#95C22×C13713yes—
205C2×#95C22×C13713yes—
206C2×#95C22×C13713yes—
207C2×#95C22×C13713yes—
208C2×#95C22×C13713yes—
209C2×#95C22×C13711yescharacteristic
210C25C23×C13711yescharacteristic
211C23×C4C23×C13711yescharacteristic
212C23×C4C23×C13713yes—
213C23×C4C23×C13713yes—
214C23×C4C23×C13713yes—
215C23×C4C23×C13713yes—
216C23×C4C23×C13711yescharacteristic
217C2×C4⋊C4—26no—
218C2×C4⋊C4—26no—
219C2×C4⋊C4—43no—
220C2×C22⋊C4—26no—
221C2×C22⋊C4—26no—
222C2×C22⋊C4—26no—
223C2×C22⋊C4—26no—
224C2×C22⋊C4—26no—
225C2×C22⋊C4—43no—
226C2×C22⋊C4—43no—
227C2×C42—26no—
228C2×C42—43no—
229C2×C42—43no—
230#95—26no—
231#95—26no—
232#95—26no—
233#95—26no—
234#95—26no—
235#95—26no—
236#95—43no—
237#95D4×C13714yes—
238C24—21no—
239C24—23no—
240C24—23no—
241C24—23no—
242C24—23no—
243C24—21no—
244C24#41×C13712yes—
245C24C24×C13711yescharacteristic
246C22×C4—23no—
247C22×C4—21no—
248C22×C4—23no—
249C22×C4—23no—
250C22×C4—23no—
251C22×C4—23no—
252C22×C4—23no—
253C22×C4—23no—
254C22×C4—23no—
255C22×C4—23no—
256C22×C4—23no—
257C22×C4—23no—
258C22×C4—23no—
259C22×C4—23no—
260C22×C4—23no—
261C22×C4—23no—
262C22×C4—23no—
263C22×C4—23no—
264C22×C4—23no—
265C22×C4—23no—
266C22×C4—21no—
267C22×C4—23no—
268C22×C4—23no—
269C22×C4—23no—
270C22×C4—23no—
271C22×C4—23no—
272C22×C4—23no—
273C22×C4—23no—
274C22×C4—23no—
275C22×C4—23no—
276C22×C4C2×D4×C13716yes—
277C22×C4#41×C13716yes—
278C22×C4#41×C13716yes—
279C22×C4#41×C13716yes—
280C4⋊C4—412no—
281C22⋊C4—412no—
282C22⋊C4—412no—
283C22⋊C4—44no—
284C42—412no—
285C23—26no—
286C23—22no—
287C23—26no—
288C23—41no—
289C23—43no—
290C23—43no—
291C23—43no—
292C23—43no—
293C23—43no—
294C23—43no—
295C23—43no—
296C23—43no—
297C23—43no—
298C23C22×D4×C13711yescharacteristic
299C23C2×#41×C13713yes—
300C23C2×#41×C13713yes—
301C23C2×#41×C13711yescharacteristic
302C23C2×#41×C13713yes—
303C23H(5,𝔽2)×C13713yes—
304C232−1+4×C13711yescharacteristic
305C2×C4—26no—
306C2×C4—26no—
307C2×C4—26no—
308C2×C4—26no—
309C2×C4—26no—
310C2×C4—26no—
311C2×C4—26no—
312C2×C4—26no—
313C2×C4—26no—
314C2×C4—26no—
315C2×C4—26no—
316C2×C4—43no—
317C2×C4—43no—
318C2×C4—43no—
319C2×C4—43no—
320C2×C4—43no—
321C2×C4—43no—
322C2×C4—43no—
323C2×C4—43no—
324C2×C4—43no—
325C2×C4—43no—
326C2×C4—43no—
327C2×C4—43no—
328C2×C4—41no—
329C2×C4—43no—
330C2×C4—43no—
331C2×C4—43no—
332C2×C4—43no—
333C2×C4—43no—
334C2×C4—43no—
335C2×C4—43no—
336C2×C4—43no—
337C2×C4—43no—
338C2×C4#126×C13714yes—
339C22—46no—
340C22—46no—
341C22—42no—
342C22—83no—
343C22—83no—
344C22—81no—
345C22—81no—
346C22C2×#126×C13711yescharacteristic
347C22#529×C13713yes—
348C22#529×C13713yes—
349C22#529×C13713yes—
350C22#532×C13713yes—
351C22#538×C13713yes—
352C22#539×C13713yes—
353C22#546×C13713yes—
354C22#548×C13711yescharacteristic
355C22#551×C13713yes—
356C22#553×C13713yes—
357C22#553×C13713yes—
358C22#556×C13713yes—
359C4—24no—
360C4—46no—
361C4—46no—
362C4—46no—
363C4—83no—
364C4—81no—
365C2—82no—
366C2#2401×C13711yescharacteristic
367C2#2559×C13713yes—
368C2#2560×C13713yes—
369C2#2576×C13713yes—
370C2#2594×C13713yes—
371C2#2643×C13711yescharacteristic
372C2#2860×C13711yescharacteristic
subgroupG/H
center, socleC24×C137C24
commutator, Frattini, cosocleC24C24×C137
prime radicalC25×C137C23
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C137 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#401×C137C2semidirect
3. #1442×C137C2semidirect
6. #1436×C137C2semidirect
7. C4×#95×C137C2semidirect
186. #15171C137direct (N × H)
187. C137#15171direct (N × H)
188. C2×#401C2×C137semidirect
190. #1442C2×C137semidirect
193. #1436C2×C137semidirect
194. C4×#95C2×C137semidirect

Maximal subgroups

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

subgroupindex
C2×#401×C1372
C2×#396×C1372
#1442×C1372
#1440×C1372
#1437×C1372
#1436×C1372
C4×#95×C1372
#15171137