Finite Group Databasebeta

← C137×#10564 #46859950

Subgroups — C137×#10564 #46859950

Order 35072.

574 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 114 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#3186×C137C211yesmaximal, characteristic
2#1964×C137C212yesmaximal, complemented
3#1893×C137C212yesmaximal, complemented
4#1850×C137C211yesmaximal, characteristic, complemented
5#1289×C137C211yesmaximal, characteristic, complemented
6C23×C8×C137C2211yescharacteristic
7C2×#121×C137C2212yes—
8#480×C137—22no—
9#466×C137—21no—
10#466×C137—21no—
11#447×C137—22no—
12C2×C4⋊C8×C137—21no—
13C2×#102×C137—22no—
14C2×#99×C137—22no—
15C2×#99×C137—22no—
16C2×C22⋊C8×C137C2211yescharacteristic
17#380×C137C2212yes—
18C4×C22⋊C4×C137C2211yescharacteristic
19#340×C137—22no—
20#336×C137—21no—
21C22×D4×C137—22no—
22C22×D4×C137D412yes—
23C23×C4×C137C2311yescharacteristic
24C2×D8×C137—21no—
25C2×D8×C137—21no—
26C22×C8×C137—22no—
27C22×C8×C137—21no—
28C22×C8×C137—42no—
29C22×C8×C137D411yescharacteristic
30C22×C8×C137D411yescharacteristic
31#121×C137—22no—
32#121×C137—22no—
33#121×C137—22no—
34#121×C137—22no—
35C2×C4⋊C4×C137—22no—
36C2×C4⋊C4×C137D412yes—
37C2×C22⋊C4×C137—22no—
38C2×C22⋊C4×C137—22no—
39C2×C22⋊C4×C137D411yescharacteristic
40C2×C22⋊C4×C137D411yescharacteristic
41C2×C42×C137—21no—
42#107×C137—21no—
43#107×C137—21no—
44C4⋊C8×C137—42no—
45#102×C137—22no—
46#102×C137—22no—
47#102×C137—44no—
48#99×C137—44no—
49#99×C137—44no—
50C22⋊C8×C137—22no—
51#95×C137—21no—
52#95×C137—22no—
53C24×C137—22no—
54C24×C137C2×D411yescharacteristic
55C2×D4×C137—22no—
56C2×D4×C137—22no—
57C2×D4×C137—42no—
58C2×D4×C137—42no—
59C2×D4×C137—42no—
60C2×D4×C137—42no—
61C2×D4×C137—42no—
62C2×D4×C137—42no—
63C2×D4×C137—42no—
64C2×D4×C137—42no—
65C22×C4×C137—21no—
66C22×C4×C137—21no—
67C22×C4×C137—22no—
68C22×C4×C137—41no—
69C22×C4×C137—41no—
70C22×C4×C137—42no—
71C22×C4×C137C2×D412yes—
72C22×C4×C137C2×D411yescharacteristic
73C22×C4×C137#4111yescharacteristic
74C22×C4×C137#4112yes—
75D8×C137—81no—
76D8×C137—81no—
77C2×C8×C137—24no—
78C2×C8×C137—22no—
79C2×C8×C137—21no—
80C2×C8×C137—21no—
81C2×C8×C137—41no—
82C2×C8×C137—42no—
83C2×C8×C137—41no—
84C2×C8×C137—41no—
85C2×C8×C137—41no—
86C4⋊C4×C137—22no—
87C4⋊C4×C137—22no—
88C4⋊C4×C137—42no—
89C4⋊C4×C137—42no—
90C22⋊C4×C137—21no—
91C22⋊C4×C137—22no—
92C22⋊C4×C137—42no—
93C22⋊C4×C137—42no—
94C22⋊C4×C137—42no—
95C22⋊C4×C137—42no—
96C22⋊C4×C137D812yescomplemented
97C42×C137—41no—
98C42×C137—41no—
99C23×C137—41no—
100C23×C137—41no—
101C23×C137—42no—
102C23×C137—42no—
103C23×C137—42no—
104C23×C137—82no—
105C23×C137—82no—
106C23×C137#12112yes—
107C23×C137#12111yescharacteristic
108C23×C137C2×D811yescharacteristic
109C23×C137#13511yescharacteristic
110C23×C137AGL(1,ℤ/8)12yes—
111D4×C137—44no—
112D4×C137—42no—
113D4×C137—42no—
114D4×C137—82no—
115D4×C137—82no—
116D4×C137—82no—
117D4×C137—82no—
118C2×C4×C137—21no—
119C2×C4×C137—22no—
120C2×C4×C137—21no—
121C2×C4×C137—22no—
122C2×C4×C137—22no—
123C2×C4×C137—41no—
124C2×C4×C137—42no—
125C2×C4×C137—41no—
126C2×C4×C137—41no—
127C2×C4×C137—41no—
128C2×C4×C137—41no—
129C2×C4×C137—42no—
130C2×C4×C137—42no—
131C2×C4×C137—42no—
132C2×C4×C137—41no—
133C2×C4×C137—82no—
134C2×C4×C137—82no—
135C2×C4×C137—82no—
136C2×C4×C137#12011yescharacteristic
137C2×C4×C137#12312yes—
138C2×C4×C137#12411yescharacteristic
139C8×C137—42no—
140C8×C137—41no—
141C8×C137—41no—
142C22×C137—21no—
143C22×C137—22no—
144C22×C137—21no—
145C22×C137—22no—
146C22×C137—41no—
147C22×C137—41no—
148C22×C137—42no—
149C22×C137—82no—
150C22×C137—82no—
151C22×C137—82no—
152C22×C137—162no—
153C22×C137—162no—
154C22×C137#39411yescharacteristic
155C22×C137#45912yes—
156C22×C137#46312yes—
157C22×C137#46611yescharacteristic
158C22×C137#46911yescharacteristic
159C4×C137—22no—
160C4×C137—21no—
161C4×C137—42no—
162C4×C137—81no—
163C4×C137—82no—
164C4×C137—162no—
165C4×C137B(4,𝔽2)11yescharacteristic
166C4×C137#45811yescharacteristic
167C2×C137—21no—
168C2×C137—21no—
169C2×C137—22no—
170C2×C137—22no—
171C2×C137—162no—
172C2×C137#204411yescharacteristic
173C2×C137#205311yescharacteristic
174C2×C137#205411yescharacteristic
175#10564C13711yesmaximal, characteristic, direct product factor, Sylow
176C137#1056411yescharacteristic, direct product factor, Sylow
177#3186C2×C13711yescharacteristic
178#1964C2×C13712yescomplemented
179#1893C2×C13712yescomplemented
180#1850C2×C13711yescharacteristic, complemented
181#1289C2×C13711yescharacteristic, complemented
182C23×C8C22×C13711yescharacteristic
183C2×#121C22×C13712yes—
184#480—22no—
185#466—21no—
186#466—21no—
187#447—22no—
188C2×C4⋊C8—21no—
189C2×#102—22no—
190C2×#99—22no—
191C2×#99—22no—
192C2×C22⋊C8C22×C13711yescharacteristic
193#380C22×C13712yes—
194C4×C22⋊C4C22×C13711yescharacteristic
195#340—22no—
196#336—21no—
197C22×D4—22no—
198C22×D4D4×C13712yes—
199C23×C4C23×C13711yescharacteristic
200C2×D8—21no—
201C2×D8—21no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—42no—
205C22×C8D4×C13711yescharacteristic
206C22×C8D4×C13711yescharacteristic
207#121—22no—
208#121—22no—
209#121—22no—
210#121—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4D4×C13712yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4D4×C13711yescharacteristic
216C2×C22⋊C4D4×C13711yescharacteristic
217C2×C42—21no—
218#107—21no—
219#107—21no—
220C4⋊C8—42no—
221#102—22no—
222#102—22no—
223#102—44no—
224#99—44no—
225#99—44no—
226C22⋊C8—22no—
227#95—21no—
228#95—22no—
229C24—22no—
230C24C2×D4×C13711yescharacteristic
231C2×D4—22no—
232C2×D4—22no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4C2×D4×C13712yes—
248C22×C4C2×D4×C13711yescharacteristic
249C22×C4#41×C13711yescharacteristic
250C22×C4#41×C13712yes—
251D8—81no—
252D8—81no—
253C2×C8—24no—
254C2×C8—22no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—41no—
258C2×C8—42no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—42no—
266C22⋊C4—21no—
267C22⋊C4—22no—
268C22⋊C4—42no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—42no—
272C22⋊C4D8×C13712yescomplemented
273C42—41no—
274C42—41no—
275C23—41no—
276C23—41no—
277C23—42no—
278C23—42no—
279C23—42no—
280C23—82no—
281C23—82no—
282C23#121×C13712yes—
283C23#121×C13711yescharacteristic
284C23C2×D8×C13711yescharacteristic
285C23#135×C13711yescharacteristic
286C23AGL(1,ℤ/8)×C13712yes—
287D4—44no—
288D4—42no—
289D4—42no—
290D4—82no—
291D4—82no—
292D4—82no—
293D4—82no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4#120×C13711yescharacteristic
313C2×C4#123×C13712yes—
314C2×C4#124×C13711yescharacteristic
315C8—42no—
316C8—41no—
317C8—41no—
318C22—21no—
319C22—22no—
320C22—21no—
321C22—22no—
322C22—41no—
323C22—41no—
324C22—42no—
325C22—82no—
326C22—82no—
327C22—82no—
328C22—162no—
329C22—162no—
330C22#394×C13711yescharacteristic
331C22#459×C13712yes—
332C22#463×C13712yes—
333C22#466×C13711yescharacteristic
334C22#469×C13711yescharacteristic
335C4—22no—
336C4—21no—
337C4—42no—
338C4—81no—
339C4—82no—
340C4—162no—
341C4B(4,𝔽2)×C13711yescharacteristic
342C4#458×C13711yescharacteristic
343C2—21no—
344C2—21no—
345C2—22no—
346C2—22no—
347C2—162no—
348C2#2044×C13711yescharacteristic
349C2#2053×C13711yescharacteristic
350C2#2054×C13711yescharacteristic
subgroupG/H
center, socleC22×C137#394
commutator, Frattini, cosocleC23×C4C23×C137
prime radical#3186×C137C2
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C137 ◁ C23×C4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
2. #1964×C137C2semidirect
3. #1893×C137C2semidirect
4. #1850×C137C2semidirect
5. #1289×C137C2semidirect
96. C22⋊C4×C137D8semidirect
175. #10564C137direct (N × H)
176. C137#10564direct (N × H)
178. #1964C2×C137semidirect
179. #1893C2×C137semidirect
180. #1850C2×C137semidirect
181. #1289C2×C137semidirect
272. C22⋊C4D8×C137semidirect

Maximal subgroups

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

subgroupindex
#3186×C1372
#1964×C1372
#1893×C1372
#1850×C1372
#1289×C1372
#10564137