Finite Group Databasebeta

← C137×#13292 #46862678

Subgroups — C137×#13292 #46862678

Order 35072.

484 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 126 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#2487×C137C211yesmaximal, characteristic
2#1799×C137C211yesmaximal, characteristic
3C4×#100×C137C211yesmaximal, characteristic
4#1390×C137C214yesmaximal
5C2×C4×D4×C137C2211yescharacteristic
6C22×C4⋊C4×C137C411yescharacteristic, complemented
7C2×#100×C137C2212yes—
8C2×#99×C137C2212yes—
9C4×M16×C137—21no—
10#400×C137—21no—
11#399×C137—21no—
12#389×C137—21no—
13#386×C137C2212yes—
14#385×C137C412yescomplemented
15#380×C137—21no—
16#380×C137—21no—
17#380×C137C411yescharacteristic, complemented
18#343×C137—21no—
19#342×C137—21no—
20C22×D4×C137C2311yescharacteristic
21C23×C4×C137C2×C411yescharacteristic
22C23×C4×C137C2×C412yes—
23C23×C4×C137D411yescharacteristic
24C2×M16×C137—22no—
25C4×D4×C137—41no—
26C4×D4×C137—41no—
27C2×C4⋊C4×C137—21no—
28C2×C4⋊C4×C137—21no—
29C2×C4⋊C4×C137—21no—
30C2×C4⋊C4×C137—22no—
31C2×C4⋊C4×C137—41no—
32C2×C4⋊C4×C137—41no—
33C2×C4⋊C4×C137—41no—
34C2×C4⋊C4×C137D411yescharacteristic
35C2×C22⋊C4×C137—22no—
36C2×C22⋊C4×C137—22no—
37C2×C22⋊C4×C137C2×C412yes—
38C2×C22⋊C4×C137C2×C411yescharacteristic
39C2×C22⋊C4×C137D411yescharacteristic
40C2×C42×C137D411yescharacteristic
41#100×C137—22no—
42#100×C137Q814yes—
43#99×C137—24no—
44#97×C137—41no—
45C4×C8×C137—41no—
46#95×C137—22no—
47#95×C137—21no—
48#95×C137—22no—
49#95×C137—41no—
50C24×C137C22×C411yescharacteristic
51C24×C137C2×D411yescharacteristic
52C2×D4×C137—41no—
53C2×D4×C137—41no—
54C2×D4×C137C2×Q812yes—
55C2×D4×C137#4112yes—
56C22×C4×C137—21no—
57C22×C4×C137—21no—
58C22×C4×C137—21no—
59C22×C4×C137—21no—
60C22×C4×C137—22no—
61C22×C4×C137—22no—
62C22×C4×C137—42no—
63C22×C4×C137—42no—
64C22×C4×C137—42no—
65C22×C4×C137—42no—
66C22×C4×C137—82no—
67C22×C4×C137C22⋊C411yescharacteristic
68C22×C4×C137C22⋊C411yescharacteristic
69C22×C4×C137C22⋊C411yescharacteristic
70C22×C4×C137C22⋊C411yescharacteristic
71C22×C4×C137C2×D411yescharacteristic
72M16×C137—24no—
73M16×C137—42no—
74C2×C8×C137—42no—
75C4⋊C4×C137—41no—
76C4⋊C4×C137—42no—
77C4⋊C4×C137—42no—
78C4⋊C4×C137—41no—
79C4⋊C4×C137—41no—
80C4⋊C4×C137—82no—
81C4⋊C4×C137—82no—
82C22⋊C4×C137—41no—
83C22⋊C4×C137—41no—
84C22⋊C4×C137—42no—
85C22⋊C4×C137—42no—
86C22⋊C4×C137—42no—
87C22⋊C4×C137—82no—
88C42×C137—21no—
89C42×C137—21no—
90C23×C137—21no—
91C23×C137—21no—
92C23×C137—21no—
93C23×C137—21no—
94C23×C137—41no—
95C23×C137—41no—
96C23×C137—41no—
97C23×C137—41no—
98C23×C137C2×C22⋊C411yescharacteristic
99C23×C137C4×Q812yes—
100C23×C137#12411yescharacteristic
101C23×C137C4⋊Q811yescharacteristic
102D4×C137—42no—
103D4×C137—42no—
104C2×C4×C137—21no—
105C2×C4×C137—21no—
106C2×C4×C137—22no—
107C2×C4×C137—21no—
108C2×C4×C137—21no—
109C2×C4×C137—21no—
110C2×C4×C137—42no—
111C2×C4×C137—42no—
112C2×C4×C137—42no—
113C2×C4×C137—41no—
114C2×C4×C137—41no—
115C2×C4×C137—41no—
116C2×C4×C137—42no—
117C2×C4×C137—42no—
118C2×C4×C137—42no—
119C2×C4×C137—42no—
120C2×C4×C137—42no—
121C2×C4×C137—42no—
122C2×C4×C137—81no—
123C2×C4×C137—84no—
124C2×C4×C137—82no—
125C2×C4×C137—82no—
126C2×C4×C137—82no—
127C2×C4×C137#9911yescharacteristic
128C2×C4×C137#9911yescharacteristic
129C2×C4×C137C22⋊Q812yes—
130C8×C137—44no—
131C22×C137—21no—
132C22×C137—21no—
133C22×C137—21no—
134C22×C137—22no—
135C22×C137—21no—
136C22×C137—42no—
137C22×C137—41no—
138C22×C137—41no—
139C22×C137—42no—
140C22×C137—41no—
141C22×C137—41no—
142C22×C137—41no—
143C22×C137—41no—
144C22×C137—81no—
145C22×C137#39111yescharacteristic
146C22×C137C2×#9911yescharacteristic
147C22×C137#41011yescharacteristic
148C4×C137—21no—
149C4×C137—22no—
150C4×C137—41no—
151C4×C137—41no—
152C4×C137—41no—
153C4×C137—82no—
154C4×C137—82no—
155C4×C137—162no—
156C4×C137#35212yes—
157C2×C137—21no—
158C2×C137—21no—
159C2×C137—41no—
160C2×C137—42no—
161C2×C137—41no—
162C2×C137#199611yescharacteristic
163C2×C137C2×#35211yescharacteristic
164C2×C137#212211yescharacteristic
165#13292C13711yesmaximal, characteristic, direct product factor, Sylow
166C137#1329211yescharacteristic, direct product factor, Sylow
167#2487C2×C13711yescharacteristic
168#1799C2×C13711yescharacteristic
169C4×#100C2×C13711yescharacteristic
170#1390C2×C13714yes—
171C2×C4×D4C22×C13711yescharacteristic
172C22×C4⋊C4C4×C13711yescharacteristic, complemented
173C2×#100C22×C13712yes—
174C2×#99C22×C13712yes—
175C4×M16—21no—
176#400—21no—
177#399—21no—
178#389—21no—
179#386C22×C13712yes—
180#385C4×C13712yescomplemented
181#380—21no—
182#380—21no—
183#380C4×C13711yescharacteristic, complemented
184#343—21no—
185#342—21no—
186C22×D4C23×C13711yescharacteristic
187C23×C4C2×C4×C13711yescharacteristic
188C23×C4C2×C4×C13712yes—
189C23×C4D4×C13711yescharacteristic
190C2×M16—22no—
191C4×D4—41no—
192C4×D4—41no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4D4×C13711yescharacteristic
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4C2×C4×C13712yes—
204C2×C22⋊C4C2×C4×C13711yescharacteristic
205C2×C22⋊C4D4×C13711yescharacteristic
206C2×C42D4×C13711yescharacteristic
207#100—22no—
208#100Q8×C13714yes—
209#99—24no—
210#97—41no—
211C4×C8—41no—
212#95—22no—
213#95—21no—
214#95—22no—
215#95—41no—
216C24C22×C4×C13711yescharacteristic
217C24C2×D4×C13711yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4C2×Q8×C13712yes—
221C2×D4#41×C13712yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—22no—
227C22×C4—22no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—82no—
233C22×C4C22⋊C4×C13711yescharacteristic
234C22×C4C22⋊C4×C13711yescharacteristic
235C22×C4C22⋊C4×C13711yescharacteristic
236C22×C4C22⋊C4×C13711yescharacteristic
237C22×C4C2×D4×C13711yescharacteristic
238M16—24no—
239M16—42no—
240C2×C8—42no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C22⋊C4—82no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23C2×C22⋊C4×C13711yescharacteristic
265C23C4×Q8×C13712yes—
266C23#124×C13711yescharacteristic
267C23C4⋊Q8×C13711yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—81no—
289C2×C4—84no—
290C2×C4—82no—
291C2×C4—82no—
292C2×C4—82no—
293C2×C4#99×C13711yescharacteristic
294C2×C4#99×C13711yescharacteristic
295C2×C4C22⋊Q8×C13712yes—
296C8—44no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22#391×C13711yescharacteristic
312C22C2×#99×C13711yescharacteristic
313C22#410×C13711yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—82no—
320C4—82no—
321C4—162no—
322C4#352×C13712yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#1996×C13711yescharacteristic
329C2C2×#352×C13711yescharacteristic
330C2#2122×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC24C22×C4×C137
Frattini, cosocleC22×D4C23×C137
prime radicalC22×D4×C137C23
LCSG ▷ C24 ▷ C22 ▷ 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
6. C22×C4⋊C4×C137C4semidirect
14. #385×C137C4semidirect
17. #380×C137C4semidirect
165. #13292C137direct (N × H)
166. C137#13292direct (N × H)
172. C22×C4⋊C4C4×C137semidirect
180. #385C4×C137semidirect
183. #380C4×C137semidirect

Maximal subgroups

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

subgroupindex
#2487×C1372
#1799×C1372
C4×#100×C1372
#1390×C1372
#13292137