Finite Group Databasebeta

← C4×#354×C137 #46862012

Subgroups — C4×#354×C137 #46862012

Order 35072.

584 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 166 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
1C4×#124×C137C211yesmaximal, characteristic
2C2×#354×C137C212yesmaximal
3C4×#99×C137C212yesmaximal
4#1393×C137C212yesmaximal
5C2×#124×C137C2211yescharacteristic
6C2×C4×Q8×C137C411yescharacteristic, complemented
7C2×C4×D4×C137C2211yescharacteristic
8C2×#99×C137C2214yes—
9#391×C137C411yescharacteristic, complemented
10#390×C137C2211yescharacteristic
11#388×C137—22no—
12#382×C137C411yescharacteristic, complemented
13C4×C22⋊C4×C137—22no—
14C4×C22⋊C4×C137—22no—
15C43×C137C411yescharacteristic, complemented
16#354×C137C418yesdirect product factor
17#342×C137—22no—
18C22×Q8×C137C2×C411yescharacteristic
19C22×D4×C137C2311yescharacteristic
20C23×C4×C137D412yes—
21#124×C137—22no—
22#124×C137C2×C414yes—
23C4×Q8×C137—41no—
24C4×Q8×C137—41no—
25C4×D4×C137—41no—
26C4×D4×C137—41no—
27C2×C4⋊C4×C137—21no—
28C2×C4⋊C4×C137C2×C411yescharacteristic
29C2×C4⋊C4×C137C2×C411yescharacteristic
30C2×C22⋊C4×C137—22no—
31C2×C22⋊C4×C137—24no—
32C2×C22⋊C4×C137—22no—
33C2×C22⋊C4×C137D412yes—
34C2×C42×C137—21no—
35C2×C42×C137—21no—
36C2×C42×C137—21no—
37C2×C42×C137—42no—
38C2×C42×C137C2×C411yescharacteristic
39C2×C42×C137C2×C411yescharacteristic
40C2×C42×C137C2×C411yescharacteristic
41#99×C137—24no—
42#99×C137C2×C418yes—
43#95×C137—22no—
44#95×C137—22no—
45#95×C137—42no—
46C24×C137C2×D412yes—
47C2×Q8×C137—21no—
48C2×Q8×C137—41no—
49C2×Q8×C137—41no—
50C2×Q8×C137C4212yescomplemented
51C2×D4×C137—41no—
52C2×D4×C137—41no—
53C2×D4×C137C22×C412yes—
54C2×D4×C137#4112yes—
55C22×C4×C137—21no—
56C22×C4×C137—21no—
57C22×C4×C137—21no—
58C22×C4×C137—22no—
59C22×C4×C137—22no—
60C22×C4×C137—21no—
61C22×C4×C137—44no—
62C22×C4×C137—44no—
63C22×C4×C137C22⋊C412yes—
64C22×C4×C137C22⋊C411yescharacteristic
65C22×C4×C137C22⋊C411yescharacteristic
66C22×C4×C137C22×C411yescharacteristic
67C4⋊C4×C137—41no—
68C4⋊C4×C137—41no—
69C4⋊C4×C137—42no—
70C22⋊C4×C137—28no—
71C22⋊C4×C137—28no—
72C22⋊C4×C137—42no—
73C22⋊C4×C137—44no—
74C22⋊C4×C137—44no—
75C42×C137—21no—
76C42×C137—21no—
77C42×C137—21no—
78C42×C137—41no—
79C42×C137—42no—
80C42×C137—42no—
81C42×C137—41no—
82C42×C137—41no—
83C42×C137—84no—
84C42×C137C4212yescomplemented
85C23×C137—22no—
86C23×C137—22no—
87C23×C137—42no—
88C23×C137—42no—
89C23×C137C2×C22⋊C411yescharacteristic
90C23×C137C4×D414yes—
91Q8×C137—42no—
92Q8×C137—42no—
93D4×C137—42no—
94D4×C137—42no—
95C2×C4×C137—21no—
96C2×C4×C137—21no—
97C2×C4×C137—22no—
98C2×C4×C137—22no—
99C2×C4×C137—22no—
100C2×C4×C137—22no—
101C2×C4×C137—21no—
102C2×C4×C137—22no—
103C2×C4×C137—41no—
104C2×C4×C137—41no—
105C2×C4×C137—41no—
106C2×C4×C137—41no—
107C2×C4×C137—41no—
108C2×C4×C137—41no—
109C2×C4×C137—44no—
110C2×C4×C137—41no—
111C2×C4×C137—42no—
112C2×C4×C137—48no—
113C2×C4×C137—42no—
114C2×C4×C137—41no—
115C2×C4×C137—82no—
116C2×C4×C137—84no—
117C2×C4×C137—84no—
118C2×C4×C137#9911yescharacteristic
119C2×C4×C137#9911yescharacteristic
120C2×C4×C137C2×C4211yescharacteristic
121C2×C4×C137#11711yescharacteristic
122C22×C137—21no—
123C22×C137—21no—
124C22×C137—24no—
125C22×C137—44no—
126C22×C137—42no—
127C22×C137—42no—
128C22×C137—82no—
129C22×C137C4×C22⋊C411yescharacteristic
130C22×C137C2×#9911yescharacteristic
131C22×C137#41011yescharacteristic
132C4×C137—21no—
133C4×C137—21no—
134C4×C137—21no—
135C4×C137—42no—
136C4×C137—42no—
137C4×C137—42no—
138C4×C137—41no—
139C4×C137—42no—
140C4×C137—88no—
141C4×C137—81no—
142C4×C137#35412yesdirect product factor
143C2×C137—21no—
144C2×C137—21no—
145C2×C137—44no—
146C2×C137C4×#9911yescharacteristic
147C2×C137C2×#35411yescharacteristic
148C2×C137#212611yescharacteristic
149C4×#354C13711yesmaximal, characteristic, direct product factor, Sylow
150C137C4×#35411yescharacteristic, direct product factor, Sylow
151C4×#124C2×C13711yescharacteristic
152C2×#354C2×C13712yes—
153C4×#99C2×C13712yes—
154#1393C2×C13712yes—
155C2×#124C22×C13711yescharacteristic
156C2×C4×Q8C4×C13711yescharacteristic, complemented
157C2×C4×D4C22×C13711yescharacteristic
158C2×#99C22×C13714yes—
159#391C4×C13711yescharacteristic, complemented
160#390C22×C13711yescharacteristic
161#388—22no—
162#382C4×C13711yescharacteristic, complemented
163C4×C22⋊C4—22no—
164C4×C22⋊C4—22no—
165C43C4×C13711yescharacteristic, complemented
166#354C4×C13718yesdirect product factor
167#342—22no—
168C22×Q8C2×C4×C13711yescharacteristic
169C22×D4C23×C13711yescharacteristic
170C23×C4D4×C13712yes—
171#124—22no—
172#124C2×C4×C13714yes—
173C4×Q8—41no—
174C4×Q8—41no—
175C4×D4—41no—
176C4×D4—41no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4C2×C4×C13711yescharacteristic
179C2×C4⋊C4C2×C4×C13711yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—22no—
183C2×C22⋊C4D4×C13712yes—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—42no—
188C2×C42C2×C4×C13711yescharacteristic
189C2×C42C2×C4×C13711yescharacteristic
190C2×C42C2×C4×C13711yescharacteristic
191#99—24no—
192#99C2×C4×C13718yes—
193#95—22no—
194#95—22no—
195#95—42no—
196C24C2×D4×C13712yes—
197C2×Q8—21no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8C42×C13712yescomplemented
201C2×D4—41no—
202C2×D4—41no—
203C2×D4C22×C4×C13712yes—
204C2×D4#41×C13712yes—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—44no—
212C22×C4—44no—
213C22×C4C22⋊C4×C13711yescharacteristic
214C22×C4C22⋊C4×C13712yes—
215C22×C4C22⋊C4×C13711yescharacteristic
216C22×C4C22×C4×C13711yescharacteristic
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C22⋊C4—28no—
221C22⋊C4—28no—
222C22⋊C4—42no—
223C22⋊C4—44no—
224C22⋊C4—44no—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—41no—
229C42—42no—
230C42—42no—
231C42—41no—
232C42—41no—
233C42—84no—
234C42C42×C13712yescomplemented
235C23—22no—
236C23—22no—
237C23—42no—
238C23—42no—
239C23C2×C22⋊C4×C13711yescharacteristic
240C23C4×D4×C13714yes—
241Q8—42no—
242Q8—42no—
243D4—42no—
244D4—42no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—44no—
260C2×C4—41no—
261C2×C4—42no—
262C2×C4—48no—
263C2×C4—42no—
264C2×C4—41no—
265C2×C4—82no—
266C2×C4—84no—
267C2×C4—84no—
268C2×C4#99×C13711yescharacteristic
269C2×C4#99×C13711yescharacteristic
270C2×C4C2×C42×C13711yescharacteristic
271C2×C4#117×C13711yescharacteristic
272C22—21no—
273C22—21no—
274C22—24no—
275C22—44no—
276C22—42no—
277C22—42no—
278C22—82no—
279C22C4×C22⋊C4×C13711yescharacteristic
280C22C2×#99×C13711yescharacteristic
281C22#410×C13711yescharacteristic
282C4—21no—
283C4—21no—
284C4—21no—
285C4—42no—
286C4—42no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—88no—
291C4—81no—
292C4#354×C13712yesdirect product factor
293C2—21no—
294C2—21no—
295C2—44no—
296C2C4×#99×C13711yescharacteristic
297C2C2×#354×C13711yescharacteristic
298C2#2126×C13711yescharacteristic
subgroupG/H
centerC2×C4×C137#99
commutatorC2×C4C2×C42×C137
Frattini, cosocleC22×D4C23×C137
socleC22×C137C2×#99
prime radicalC22×D4×C137C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C137 ◁ C22×C4×C137 ◁ C2×C4×D4×C137 ◁ G

Complements

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

NH ≅ G/Ntype
6. C2×C4×Q8×C137C4semidirect
9. #391×C137C4semidirect
12. #382×C137C4semidirect
15. C43×C137C4semidirect
16. #354×C137C4direct (N × H)
50. C2×Q8×C137C42semidirect
84. C42×C137C42semidirect
142. C4×C137#354direct (N × H)
149. C4×#354C137direct (N × H)
150. C137C4×#354direct (N × H)
156. C2×C4×Q8C4×C137semidirect
159. #391C4×C137semidirect
162. #382C4×C137semidirect
165. C43C4×C137semidirect
166. #354C4×C137direct (N × H)
200. C2×Q8C42×C137semidirect
234. C42C42×C137semidirect
292. C4#354×C137direct (N × H)

Maximal subgroups

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

subgroupindex
C4×#124×C1372
C2×#354×C1372
C4×#99×C1372
#1393×C1372
C4×#354137