Finite Group Databasebeta

← C137×#11823 #46861209

Subgroups — C137×#11823 #46861209

Order 35072.

446 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#2398×C137C211yesmaximal, characteristic
2#1799×C137C211yesmaximal, characteristic
3#1465×C137C211yesmaximal, characteristic
4#1351×C137C212yesmaximal, complemented
5#1344×C137C212yesmaximal, complemented
6C2×C4×Q8×C137C411yescharacteristic, complemented
7C2×C4×D4×C137C2211yescharacteristic
8C2×#99×C137—21no—
9#407×C137—21no—
10#400×C137—21no—
11#397×C137C411yescharacteristic, complemented
12#397×C137C2211yescharacteristic
13#394×C137—21no—
14#393×C137C411yescharacteristic, complemented
15#393×C137C2211yescharacteristic
16#380×C137—21no—
17C4×C22⋊C4×C137—21no—
18C4×C22⋊C4×C137—21no—
19#376×C137C411yescharacteristic, complemented
20#342×C137C2212yes—
21#324×C137—21no—
22#323×C137C2212yes—
23C22×Q8×C137C2×C411yescharacteristic
24C22×D4×C137D411yescharacteristic
25C23×C4×C137D411yescharacteristic
26C23×C4×C137D411yescharacteristic
27C2×M16×C137—41no—
28C4×Q8×C137—41no—
29C4×Q8×C137—41no—
30C4×D4×C137—41no—
31C4×D4×C137—41no—
32C2×C4⋊C4×C137—21no—
33C2×C4⋊C4×C137—41no—
34C2×C4⋊C4×C137C2×C411yescharacteristic
35C2×C4⋊C4×C137D411yescharacteristic
36C2×C22⋊C4×C137—21no—
37C2×C22⋊C4×C137—21no—
38C2×C22⋊C4×C137—21no—
39C2×C22⋊C4×C137—21no—
40C2×C22⋊C4×C137—22no—
41C2×C22⋊C4×C137C2×C411yescharacteristic
42C2×C22⋊C4×C137C2311yescharacteristic
43C2×C42×C137—21no—
44C2×C42×C137—21no—
45C2×C42×C137C2×C411yescharacteristic
46#99×C137—42no—
47C22⋊C8×C137—22no—
48C22⋊C8×C137—41no—
49#95×C137—21no—
50#95×C137—21no—
51#95×C137—21no—
52#95×C137—21no—
53#95×C137—21no—
54#95×C137—41no—
55#95×C137C2×C411yescharacteristic
56#95×C137C2×C411yescharacteristic
57C24×C137C22⋊C411yescharacteristic
58C24×C137C2×D411yescharacteristic
59C2×Q8×C137—42no—
60C2×Q8×C137—41no—
61C2×D4×C137—21no—
62C2×D4×C137—21no—
63C2×D4×C137—41no—
64C2×D4×C137—41no—
65C22×C4×C137—21no—
66C22×C4×C137—21no—
67C22×C4×C137—21no—
68C22×C4×C137—21no—
69C22×C4×C137—21no—
70C22×C4×C137—21no—
71C22×C4×C137—21no—
72C22×C4×C137—21no—
73C22×C4×C137—42no—
74C22×C4×C137—42no—
75C22×C4×C137—42no—
76C22×C4×C137C22⋊C411yescharacteristic
77C22×C4×C137C22⋊C411yescharacteristic
78C22×C4×C137C22⋊C411yescharacteristic
79C22×C4×C137C22×C411yescharacteristic
80C22×C4×C137C2×D411yescharacteristic
81M16×C137—82no—
82C2×C8×C137—42no—
83C4⋊C4×C137—41no—
84C4⋊C4×C137—41no—
85C4⋊C4×C137—42no—
86C4⋊C4×C137—82no—
87C22⋊C4×C137—41no—
88C22⋊C4×C137—41no—
89C22⋊C4×C137—42no—
90C22⋊C4×C137—42no—
91C22⋊C4×C137—42no—
92C22⋊C4×C137—41no—
93C22⋊C4×C137—41no—
94C22⋊C4×C137—42no—
95C22⋊C4×C137—42no—
96C42×C137—21no—
97C42×C137—21no—
98C42×C137—42no—
99C42×C137—41no—
100C42×C137—41no—
101C23×C137—21no—
102C23×C137—21no—
103C23×C137—21no—
104C23×C137—21no—
105C23×C137—21no—
106C23×C137—41no—
107C23×C137—41no—
108C23×C137—41no—
109C23×C137—41no—
110C23×C137C2×C22⋊C411yescharacteristic
111C23×C137AGL(1,ℤ/8)11yescharacteristic
112C23×C137#13711yescharacteristic
113Q8×C137—42no—
114Q8×C137—42no—
115D4×C137—42no—
116D4×C137—42no—
117C2×C4×C137—22no—
118C2×C4×C137—22no—
119C2×C4×C137—21no—
120C2×C4×C137—21no—
121C2×C4×C137—21no—
122C2×C4×C137—21no—
123C2×C4×C137—21no—
124C2×C4×C137—21no—
125C2×C4×C137—41no—
126C2×C4×C137—41no—
127C2×C4×C137—42no—
128C2×C4×C137—42no—
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—41no—
136C2×C4×C137—41no—
137C2×C4×C137—41no—
138C2×C4×C137—41no—
139C2×C4×C137—41no—
140C2×C4×C137—41no—
141C2×C4×C137—81no—
142C2×C4×C137—81no—
143C2×C4×C137—82no—
144C2×C4×C137—82no—
145C2×C4×C137—82no—
146C2×C4×C137#9911yescharacteristic
147C2×C4×C137#9911yescharacteristic
148C8×C137—82no—
149C22×C137—21no—
150C22×C137—21no—
151C22×C137—21no—
152C22×C137—21no—
153C22×C137—41no—
154C22×C137—41no—
155C22×C137—42no—
156C22×C137—41no—
157C22×C137—41no—
158C22×C137—41no—
159C22×C137—41no—
160C22×C137—41no—
161C22×C137—81no—
162C22×C137—81no—
163C22×C137C2×#9911yescharacteristic
164C22×C137#41711yescharacteristic
165C22×C137#42111yescharacteristic
166C4×C137—21no—
167C4×C137—42no—
168C4×C137—41no—
169C4×C137—41no—
170C4×C137—41no—
171C4×C137—41no—
172C4×C137—81no—
173C4×C137—81no—
174C4×C137—81no—
175C4×C137—162no—
176C4×C137#35412yes—
177C2×C137—21no—
178C2×C137—21no—
179C2×C137—41no—
180C2×C137—41no—
181C2×C137—81no—
182C2×C137#151311yescharacteristic
183C2×C137#212011yescharacteristic
184C2×C137C2×#35411yescharacteristic
185#11823C13711yesmaximal, characteristic, direct product factor, Sylow
186C137#1182311yescharacteristic, direct product factor, Sylow
187#2398C2×C13711yescharacteristic
188#1799C2×C13711yescharacteristic
189#1465C2×C13711yescharacteristic
190#1351C2×C13712yescomplemented
191#1344C2×C13712yescomplemented
192C2×C4×Q8C4×C13711yescharacteristic, complemented
193C2×C4×D4C22×C13711yescharacteristic
194C2×#99—21no—
195#407—21no—
196#400—21no—
197#397C4×C13711yescharacteristic, complemented
198#397C22×C13711yescharacteristic
199#394—21no—
200#393C4×C13711yescharacteristic, complemented
201#393C22×C13711yescharacteristic
202#380—21no—
203C4×C22⋊C4—21no—
204C4×C22⋊C4—21no—
205#376C4×C13711yescharacteristic, complemented
206#342C22×C13712yes—
207#324—21no—
208#323C22×C13712yes—
209C22×Q8C2×C4×C13711yescharacteristic
210C22×D4D4×C13711yescharacteristic
211C23×C4D4×C13711yescharacteristic
212C23×C4D4×C13711yescharacteristic
213C2×M16—41no—
214C4×Q8—41no—
215C4×Q8—41no—
216C4×D4—41no—
217C4×D4—41no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4C2×C4×C13711yescharacteristic
221C2×C4⋊C4D4×C13711yescharacteristic
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4C2×C4×C13711yescharacteristic
228C2×C22⋊C4C23×C13711yescharacteristic
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C2×C4×C13711yescharacteristic
232#99—42no—
233C22⋊C8—22no—
234C22⋊C8—41no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—21no—
240#95—41no—
241#95C2×C4×C13711yescharacteristic
242#95C2×C4×C13711yescharacteristic
243C24C22⋊C4×C13711yescharacteristic
244C24C2×D4×C13711yescharacteristic
245C2×Q8—42no—
246C2×Q8—41no—
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—41no—
250C2×D4—41no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4C22⋊C4×C13711yescharacteristic
263C22×C4C22⋊C4×C13711yescharacteristic
264C22×C4C22⋊C4×C13711yescharacteristic
265C22×C4C22×C4×C13711yescharacteristic
266C22×C4C2×D4×C13711yescharacteristic
267M16—82no—
268C2×C8—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—82no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—42no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—41no—
279C22⋊C4—41no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C42—21no—
283C42—21no—
284C42—42no—
285C42—41no—
286C42—41no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23C2×C22⋊C4×C13711yescharacteristic
297C23AGL(1,ℤ/8)×C13711yescharacteristic
298C23#137×C13711yescharacteristic
299Q8—42no—
300Q8—42no—
301D4—42no—
302D4—42no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—81no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4#99×C13711yescharacteristic
333C2×C4#99×C13711yescharacteristic
334C8—82no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—41no—
340C22—41no—
341C22—42no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—81no—
348C22—81no—
349C22C2×#99×C13711yescharacteristic
350C22#417×C13711yescharacteristic
351C22#421×C13711yescharacteristic
352C4—21no—
353C4—41no—
354C4—42no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—81no—
359C4—81no—
360C4—81no—
361C4—162no—
362C4#354×C13712yes—
363C2—21no—
364C2—21no—
365C2—41no—
366C2—41no—
367C2—81no—
368C2#1513×C13711yescharacteristic
369C2#2120×C13711yescharacteristic
370C2C2×#354×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C22⋊C4C23×C137
prime radicalC22×D4×C137D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C137 ◁ C22×C4×C137 ◁ C2×C4×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1351×C137C2semidirect
5. #1344×C137C2semidirect
6. C2×C4×Q8×C137C4semidirect
11. #397×C137C4semidirect
14. #393×C137C4semidirect
19. #376×C137C4semidirect
185. #11823C137direct (N × H)
186. C137#11823direct (N × H)
190. #1351C2×C137semidirect
191. #1344C2×C137semidirect
192. C2×C4×Q8C4×C137semidirect
197. #397C4×C137semidirect
200. #393C4×C137semidirect
205. #376C4×C137semidirect

Maximal subgroups

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

subgroupindex
#2398×C1372
#1799×C1372
#1465×C1372
#1351×C1372
#1344×C1372
#11823137