Finite Group Databasebeta

← C151×#11816 #50316741

Subgroups — C151×#11816 #50316741

Order 38656.

388 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 98 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#2396×C151C211yesmaximal, characteristic
2#1822×C151C211yesmaximal, characteristic
3#1464×C151C211yesmaximal, characteristic
4#1354×C151C212yesmaximal, complemented
5#1353×C151C212yesmaximal, complemented
6C2×C4×Q8×C151C411yescharacteristic
7C2×#117×C151C2211yescharacteristic
8#432×C151—21no—
9#407×C151—21no—
10#400×C151C2211yescharacteristic
11#399×C151C411yescharacteristic
12#398×C151—21no—
13#396×C151—21no—
14#395×C151C2211yescharacteristic
15#393×C151C411yescharacteristic
16C4×C4⋊C4×C151—21no—
17C4×C4⋊C4×C151C411yescharacteristic
18C4×C22⋊C4×C151—21no—
19#343×C151—21no—
20#343×C151C2212yes—
21#324×C151C2212yes—
22#323×C151—21no—
23C22×Q8×C151C2×C411yescharacteristic
24C23×C4×C151D411yescharacteristic
25C2×M16×C151—22no—
26C4×Q8×C151—41no—
27C4×Q8×C151—41no—
28#117×C151—41no—
29#117×C151—41no—
30C2×C4⋊C4×C151—21no—
31C2×C4⋊C4×C151—21no—
32C2×C4⋊C4×C151—21no—
33C2×C4⋊C4×C151—21no—
34C2×C4⋊C4×C151C2×C411yescharacteristic
35C2×C4⋊C4×C151D411yescharacteristic
36C2×C4⋊C4×C151C2311yescharacteristic
37C2×C22⋊C4×C151—21no—
38C2×C22⋊C4×C151—21no—
39C2×C22⋊C4×C151C2×C411yescharacteristic
40C2×C22⋊C4×C151D411yescharacteristic
41C2×C42×C151—21no—
42C2×C42×C151—21no—
43C2×C42×C151C2×C411yescharacteristic
44C2×C42×C151D411yescharacteristic
45C4⋊C8×C151—41no—
46C22⋊C8×C151—22no—
47C22⋊C8×C151—41no—
48C22⋊C8×C151—41no—
49#97×C151—41no—
50#95×C151—21no—
51#95×C151—21no—
52#95×C151—21no—
53#95×C151—21no—
54#95×C151C2×C411yescharacteristic
55#95×C151C2×C411yescharacteristic
56C24×C151C22⋊C411yescharacteristic
57C2×Q8×C151—42no—
58C2×Q8×C151—41no—
59C22×C4×C151—21no—
60C22×C4×C151—21no—
61C22×C4×C151—21no—
62C22×C4×C151—21no—
63C22×C4×C151—22no—
64C22×C4×C151—21no—
65C22×C4×C151—21no—
66C22×C4×C151—41no—
67C22×C4×C151C22⋊C411yescharacteristic
68C22×C4×C151C22⋊C411yescharacteristic
69C22×C4×C151C22⋊C411yescharacteristic
70C22×C4×C151C22×C411yescharacteristic
71C22×C4×C151C2×D411yescharacteristic
72C22×C4×C151C2×D411yescharacteristic
73M16×C151—42no—
74M16×C151—42no—
75C2×C8×C151—42no—
76C2×C8×C151—42no—
77C4⋊C4×C151—41no—
78C4⋊C4×C151—41no—
79C4⋊C4×C151—41no—
80C4⋊C4×C151—42no—
81C4⋊C4×C151—42no—
82C4⋊C4×C151—42no—
83C4⋊C4×C151—42no—
84C22⋊C4×C151—41no—
85C22⋊C4×C151—41no—
86C22⋊C4×C151—42no—
87C22⋊C4×C151—42no—
88C42×C151—21no—
89C42×C151—21no—
90C42×C151—21no—
91C42×C151—21no—
92C42×C151—42no—
93C42×C151—41no—
94C42×C151—41no—
95C23×C151—21no—
96C23×C151—21no—
97C23×C151—21no—
98C23×C151—41no—
99C23×C151—41no—
100C23×C151C2×C22⋊C411yescharacteristic
101Q8×C151—42no—
102Q8×C151—42no—
103C2×C4×C151—21no—
104C2×C4×C151—21no—
105C2×C4×C151—21no—
106C2×C4×C151—21no—
107C2×C4×C151—22no—
108C2×C4×C151—22no—
109C2×C4×C151—21no—
110C2×C4×C151—21no—
111C2×C4×C151—21no—
112C2×C4×C151—21no—
113C2×C4×C151—42no—
114C2×C4×C151—42no—
115C2×C4×C151—41no—
116C2×C4×C151—41no—
117C2×C4×C151—41no—
118C2×C4×C151—41no—
119C2×C4×C151—41no—
120C2×C4×C151—41no—
121C2×C4×C151—41no—
122C2×C4×C151—41no—
123C2×C4×C151—41no—
124C2×C4×C151—41no—
125C2×C4×C151—41no—
126C2×C4×C151—41no—
127C2×C4×C151—41no—
128C2×C4×C151—41no—
129C2×C4×C151#9911yescharacteristic
130C2×C4×C151#9911yescharacteristic
131C2×C4×C151#13712yes—
132C8×C151—82no—
133C8×C151—82no—
134C22×C151—21no—
135C22×C151—21no—
136C22×C151—42no—
137C22×C151—41no—
138C22×C151—41no—
139C22×C151—41no—
140C22×C151—81no—
141C22×C151C2×#9911yescharacteristic
142C22×C151#41911yescharacteristic
143C22×C151#42111yescharacteristic
144C4×C151—22no—
145C4×C151—21no—
146C4×C151—21no—
147C4×C151—42no—
148C4×C151—41no—
149C4×C151—42no—
150C4×C151—41no—
151C4×C151—41no—
152C4×C151—41no—
153C4×C151—81no—
154C4×C151—81no—
155C2×C151—21no—
156C2×C151—21no—
157C2×C151—81no—
158C2×C151#151111yescharacteristic
159C2×C151#212011yescharacteristic
160C2×C151#213211yescharacteristic
161#11816C15111yesmaximal, characteristic, direct product factor, Sylow
162C151#1181611yescharacteristic, direct product factor, Sylow
163#2396C2×C15111yescharacteristic
164#1822C2×C15111yescharacteristic
165#1464C2×C15111yescharacteristic
166#1354C2×C15112yescomplemented
167#1353C2×C15112yescomplemented
168C2×C4×Q8C4×C15111yescharacteristic
169C2×#117C22×C15111yescharacteristic
170#432—21no—
171#407—21no—
172#400C22×C15111yescharacteristic
173#399C4×C15111yescharacteristic
174#398—21no—
175#396—21no—
176#395C22×C15111yescharacteristic
177#393C4×C15111yescharacteristic
178C4×C4⋊C4—21no—
179C4×C4⋊C4C4×C15111yescharacteristic
180C4×C22⋊C4—21no—
181#343—21no—
182#343C22×C15112yes—
183#324C22×C15112yes—
184#323—21no—
185C22×Q8C2×C4×C15111yescharacteristic
186C23×C4D4×C15111yescharacteristic
187C2×M16—22no—
188C4×Q8—41no—
189C4×Q8—41no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4C2×C4×C15111yescharacteristic
197C2×C4⋊C4D4×C15111yescharacteristic
198C2×C4⋊C4C23×C15111yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4C2×C4×C15111yescharacteristic
202C2×C22⋊C4D4×C15111yescharacteristic
203C2×C42—21no—
204C2×C42—21no—
205C2×C42C2×C4×C15111yescharacteristic
206C2×C42D4×C15111yescharacteristic
207C4⋊C8—41no—
208C22⋊C8—22no—
209C22⋊C8—41no—
210C22⋊C8—41no—
211#97—41no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95C2×C4×C15111yescharacteristic
217#95C2×C4×C15111yescharacteristic
218C24C22⋊C4×C15111yescharacteristic
219C2×Q8—42no—
220C2×Q8—41no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—41no—
229C22×C4C22⋊C4×C15111yescharacteristic
230C22×C4C22⋊C4×C15111yescharacteristic
231C22×C4C22⋊C4×C15111yescharacteristic
232C22×C4C22×C4×C15111yescharacteristic
233C22×C4C2×D4×C15111yescharacteristic
234C22×C4C2×D4×C15111yescharacteristic
235M16—42no—
236M16—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C22⋊C4—41no—
247C22⋊C4—41no—
248C22⋊C4—42no—
249C22⋊C4—42no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—42no—
255C42—41no—
256C42—41no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23C2×C22⋊C4×C15111yescharacteristic
263Q8—42no—
264Q8—42no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4#99×C15111yescharacteristic
292C2×C4#99×C15111yescharacteristic
293C2×C4#137×C15112yes—
294C8—82no—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—42no—
299C22—41no—
300C22—41no—
301C22—41no—
302C22—81no—
303C22C2×#99×C15111yescharacteristic
304C22#419×C15111yescharacteristic
305C22#421×C15111yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—42no—
310C4—41no—
311C4—42no—
312C4—41no—
313C4—41no—
314C4—41no—
315C4—81no—
316C4—81no—
317C2—21no—
318C2—21no—
319C2—81no—
320C2#1511×C15111yescharacteristic
321C2#2120×C15111yescharacteristic
322C2#2132×C15111yescharacteristic
subgroupG/H
center, socleC22×C151C2×#99
commutatorC22×C4C22×C4×C151
Frattini, cosocleC2×C4⋊C4C23×C151
prime radicalC24×C151C22⋊C4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C22×C4×C151 ◁ C2×#117×C151 ◁ G

Complements

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

NH ≅ G/Ntype
4. #1354×C151C2semidirect
5. #1353×C151C2semidirect
161. #11816C151direct (N × H)
162. C151#11816direct (N × H)
166. #1354C2×C151semidirect
167. #1353C2×C151semidirect

Maximal subgroups

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

subgroupindex
#2396×C1512
#1822×C1512
#1464×C1512
#1354×C1512
#1353×C1512
#11816151