Finite Group Databasebeta

← C137×#11790 #46861176

Subgroups — C137×#11790 #46861176

Order 35072.

444 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 106 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
2C4×#100×C137C211yesmaximal, characteristic
3#1465×C137C211yesmaximal, characteristic
4#1353×C137C214yesmaximal, complemented
5C2×C4×D4×C137C2211yescharacteristic
6C22×C4⋊C4×C137C411yescharacteristic
7C2×#100×C137—21no—
8#407×C137—21no—
9C4×M16×C137—21no—
10#400×C137C2212yes—
11#399×C137C412yes—
12#389×C137—21no—
13#386×C137—21no—
14#385×C137—21no—
15#380×C137—21no—
16#380×C137C411yescharacteristic
17#343×C137C2212yes—
18#324×C137C2212yes—
19#323×C137—21no—
20C22×D4×C137D411yescharacteristic
21C23×C4×C137—21no—
22C23×C4×C137C2×C411yescharacteristic
23C23×C4×C137D411yescharacteristic
24C2×M16×C137—22no—
25C2×M16×C137—41no—
26C4×D4×C137—41no—
27C4×D4×C137—41no—
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×C137C2×C412yes—
34C2×C4⋊C4×C137C2311yescharacteristic
35C2×C22⋊C4×C137—21no—
36C2×C22⋊C4×C137—22no—
37C2×C22⋊C4×C137C2×C411yescharacteristic
38C2×C22⋊C4×C137D411yescharacteristic
39C2×C42×C137D411yescharacteristic
40#100×C137—24no—
41C22⋊C8×C137—22no—
42C22⋊C8×C137—41no—
43#97×C137—41no—
44C4×C8×C137—41no—
45#95×C137—22no—
46#95×C137—22no—
47#95×C137C2×C412yes—
48C24×C137C22⋊C411yescharacteristic
49C24×C137C22⋊C411yescharacteristic
50C2×D4×C137—22no—
51C2×D4×C137—22no—
52C2×D4×C137—41no—
53C22×C4×C137—21no—
54C22×C4×C137—21no—
55C22×C4×C137—21no—
56C22×C4×C137—21no—
57C22×C4×C137—22no—
58C22×C4×C137—22no—
59C22×C4×C137—42no—
60C22×C4×C137—42no—
61C22×C4×C137—42no—
62C22×C4×C137—82no—
63C22×C4×C137C22⋊C411yescharacteristic
64C22×C4×C137C22⋊C411yescharacteristic
65C22×C4×C137C22×C411yescharacteristic
66C22×C4×C137C2×D411yescharacteristic
67C22×C4×C137C2×D411yescharacteristic
68M16×C137—24no—
69M16×C137—42no—
70M16×C137—82no—
71C2×C8×C137—42no—
72C2×C8×C137—42no—
73C4⋊C4×C137—41no—
74C4⋊C4×C137—42no—
75C4⋊C4×C137—42no—
76C4⋊C4×C137—41no—
77C4⋊C4×C137—41no—
78C4⋊C4×C137—82no—
79C22⋊C4×C137—41no—
80C22⋊C4×C137—41no—
81C22⋊C4×C137—42no—
82C22⋊C4×C137—82no—
83C42×C137—21no—
84C42×C137—21no—
85C23×C137—21no—
86C23×C137—21no—
87C23×C137—21no—
88C23×C137—21no—
89C23×C137—21no—
90C23×C137—41no—
91C23×C137—41no—
92C23×C137—41no—
93C23×C137—41no—
94C23×C137#10412yes—
95C23×C137C2×C22⋊C411yescharacteristic
96D4×C137—44no—
97C2×C4×C137—21no—
98C2×C4×C137—21no—
99C2×C4×C137—22no—
100C2×C4×C137—21no—
101C2×C4×C137—21no—
102C2×C4×C137—21no—
103C2×C4×C137—42no—
104C2×C4×C137—42no—
105C2×C4×C137—42no—
106C2×C4×C137—41no—
107C2×C4×C137—41no—
108C2×C4×C137—41no—
109C2×C4×C137—42no—
110C2×C4×C137—42no—
111C2×C4×C137—42no—
112C2×C4×C137—42no—
113C2×C4×C137—42no—
114C2×C4×C137—42no—
115C2×C4×C137—81no—
116C2×C4×C137—84no—
117C2×C4×C137#9911yescharacteristic
118C2×C4×C137#9911yescharacteristic
119C2×C4×C137#13712yes—
120C8×C137—44no—
121C8×C137—82no—
122C22×C137—21no—
123C22×C137—21no—
124C22×C137—22no—
125C22×C137—42no—
126C22×C137—42no—
127C22×C137—42no—
128C22×C137—41no—
129C22×C137—41no—
130C22×C137—41no—
131C22×C137—41no—
132C22×C137—41no—
133C22×C137—81no—
134C22×C137C2×#9911yescharacteristic
135C22×C137#41911yescharacteristic
136C22×C137C2×#10411yescharacteristic
137C4×C137—21no—
138C4×C137—22no—
139C4×C137—41no—
140C4×C137—41no—
141C4×C137—41no—
142C4×C137—82no—
143C4×C137—82no—
144C4×C137#35212yes—
145C2×C137—21no—
146C2×C137—21no—
147C2×C137—42no—
148C2×C137—81no—
149C2×C137#150811yescharacteristic
150C2×C137C2×#35211yescharacteristic
151C2×C137#213411yescharacteristic
152#11790C13711yesmaximal, characteristic, direct product factor, Sylow
153C137#1179011yescharacteristic, direct product factor, Sylow
154#2487C2×C13711yescharacteristic
155C4×#100C2×C13711yescharacteristic
156#1465C2×C13711yescharacteristic
157#1353C2×C13714yescomplemented
158C2×C4×D4C22×C13711yescharacteristic
159C22×C4⋊C4C4×C13711yescharacteristic
160C2×#100—21no—
161#407—21no—
162C4×M16—21no—
163#400C22×C13712yes—
164#399C4×C13712yes—
165#389—21no—
166#386—21no—
167#385—21no—
168#380—21no—
169#380C4×C13711yescharacteristic
170#343C22×C13712yes—
171#324C22×C13712yes—
172#323—21no—
173C22×D4D4×C13711yescharacteristic
174C23×C4—21no—
175C23×C4C2×C4×C13711yescharacteristic
176C23×C4D4×C13711yescharacteristic
177C2×M16—22no—
178C2×M16—41no—
179C4×D4—41no—
180C4×D4—41no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—41no—
185C2×C4⋊C4—41no—
186C2×C4⋊C4C2×C4×C13712yes—
187C2×C4⋊C4C23×C13711yescharacteristic
188C2×C22⋊C4—21no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4C2×C4×C13711yescharacteristic
191C2×C22⋊C4D4×C13711yescharacteristic
192C2×C42D4×C13711yescharacteristic
193#100—24no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196#97—41no—
197C4×C8—41no—
198#95—22no—
199#95—22no—
200#95C2×C4×C13712yes—
201C24C22⋊C4×C13711yescharacteristic
202C24C22⋊C4×C13711yescharacteristic
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—41no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—82no—
216C22×C4C22⋊C4×C13711yescharacteristic
217C22×C4C22⋊C4×C13711yescharacteristic
218C22×C4C22×C4×C13711yescharacteristic
219C22×C4C2×D4×C13711yescharacteristic
220C22×C4C2×D4×C13711yescharacteristic
221M16—24no—
222M16—42no—
223M16—82no—
224C2×C8—42no—
225C2×C8—42no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4—82no—
232C22⋊C4—41no—
233C22⋊C4—41no—
234C22⋊C4—42no—
235C22⋊C4—82no—
236C42—21no—
237C42—21no—
238C23—21no—
239C23—21no—
240C23—21no—
241C23—21no—
242C23—21no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#104×C13712yes—
248C23C2×C22⋊C4×C13711yescharacteristic
249D4—44no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—84no—
270C2×C4#99×C13711yescharacteristic
271C2×C4#99×C13711yescharacteristic
272C2×C4#137×C13712yes—
273C8—44no—
274C8—82no—
275C22—21no—
276C22—21no—
277C22—22no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—81no—
287C22C2×#99×C13711yescharacteristic
288C22#419×C13711yescharacteristic
289C22C2×#104×C13711yescharacteristic
290C4—22no—
291C4—21no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4#352×C13712yes—
298C2—21no—
299C2—21no—
300C2—42no—
301C2—81no—
302C2#1508×C13711yescharacteristic
303C2C2×#352×C13711yescharacteristic
304C2#2134×C13711yescharacteristic
subgroupG/H
center, socleC22×C137C2×#99
commutatorC22×C4C22×C4×C137
Frattini, cosocleC2×C4⋊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. #1353×C137C2semidirect
152. #11790C137direct (N × H)
153. C137#11790direct (N × H)
157. #1353C2×C137semidirect

Maximal subgroups

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

subgroupindex
#2487×C1372
C4×#100×C1372
#1465×C1372
#1353×C1372
#11790137