Finite Group Databasebeta

← C151×#7788 #50312713

Subgroups — C151×#7788 #50312713

Order 38656.

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

normal subgroups only

This group has 418 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#2724×C151C211yesmaximal, characteristic
2#2099×C151C211yesmaximal, characteristic, complemented
3#2083×C151C211yesmaximal, characteristic, complemented
4#2071×C151C211yesmaximal, characteristic
5#1271×C151C211yesmaximal, characteristic, complemented
6#1271×C151C211yesmaximal, characteristic, complemented
7#1270×C151C211yesmaximal, characteristic
8C2×#123×C151—21no—
9C2×#107×C151—21no—
10C2×#102×C151—21no—
11#398×C151—21no—
12#397×C151—21no—
13#397×C151—21no—
14#390×C151C2211yescharacteristic
15#389×C151C2211yescharacteristic
16#389×C151C2211yescharacteristic
17#385×C151—21no—
18#385×C151—21no—
19#384×C151C2211yescharacteristic
20#340×C151—21no—
21#340×C151—21no—
22#340×C151—21no—
23#340×C151—21no—
24#336×C151C2211yescharacteristic
25#336×C151C2211yescharacteristic
26#336×C151C2211yescharacteristic
27C22×D4×C151D411yescharacteristic
28C23×C4×C151—21no—
29C22×C8×C151—21no—
30C22×C8×C151—21no—
31C22×C8×C151—21no—
32#123×C151—42no—
33#123×C151—42no—
34C2×C4⋊C4×C151—21no—
35C2×C4⋊C4×C151—21no—
36C2×C4⋊C4×C151—21no—
37C2×C4⋊C4×C151—21no—
38C2×C4⋊C4×C151—21no—
39C2×C4⋊C4×C151—21no—
40C2×C4⋊C4×C151D411yescharacteristic
41C2×C4⋊C4×C151D411yescharacteristic
42C2×C4⋊C4×C151D411yescharacteristic
43C2×C4⋊C4×C151D411yescharacteristic
44C2×C4⋊C4×C151D411yescharacteristic
45C2×C22⋊C4×C151—21no—
46C2×C22⋊C4×C151—21no—
47C2×C22⋊C4×C151—21no—
48C2×C22⋊C4×C151—21no—
49C2×C22⋊C4×C151—21no—
50C2×C42×C151C2311yescharacteristic
51#107×C151—42no—
52#102×C151—42no—
53#95×C151—21no—
54#95×C151—21no—
55#95×C151—21no—
56#95×C151—21no—
57#95×C151—21no—
58C24×C151—21no—
59C2×D4×C151—21no—
60C2×D4×C151—41no—
61C2×D4×C151—41no—
62C2×D4×C151D812yes—
63C22×C4×C151—21no—
64C22×C4×C151—21no—
65C22×C4×C151—21no—
66C22×C4×C151—21no—
67C22×C4×C151—21no—
68C22×C4×C151—42no—
69C22×C4×C151—41no—
70C22×C4×C151—81no—
71C22×C4×C151—81no—
72C22×C4×C151C2×D411yescharacteristic
73C22×C4×C151C2×D411yescharacteristic
74C22×C4×C151C2×D411yescharacteristic
75C2×C8×C151—22no—
76C2×C8×C151—41no—
77C2×C8×C151—41no—
78C2×C8×C151—41no—
79C2×C8×C151—41no—
80C2×C8×C151—41no—
81C2×C8×C151—41no—
82C2×C8×C151—41no—
83C2×C8×C151—41no—
84C4⋊C4×C151—21no—
85C4⋊C4×C151—21no—
86C4⋊C4×C151—21no—
87C4⋊C4×C151—21no—
88C4⋊C4×C151—41no—
89C4⋊C4×C151—41no—
90C4⋊C4×C151—41no—
91C4⋊C4×C151—42no—
92C4⋊C4×C151—41no—
93C4⋊C4×C151—41no—
94C4⋊C4×C151—41no—
95C4⋊C4×C151—41no—
96C4⋊C4×C151—41no—
97C4⋊C4×C151—41no—
98C4⋊C4×C151—41no—
99C4⋊C4×C151—41no—
100C4⋊C4×C151—81no—
101C22⋊C4×C151—42no—
102C22⋊C4×C151—42no—
103C22⋊C4×C151—42no—
104C22⋊C4×C151—41no—
105C22⋊C4×C151—41no—
106C22⋊C4×C151—41no—
107C22⋊C4×C151—41no—
108C42×C151—41no—
109C23×C151—22no—
110C23×C151—41no—
111C23×C151—41no—
112C23×C151—41no—
113C23×C151—41no—
114C23×C151—41no—
115C23×C151—41no—
116C23×C151#12011yescharacteristic
117D4×C151—42no—
118D4×C151—81no—
119C2×C4×C151—21no—
120C2×C4×C151—21no—
121C2×C4×C151—21no—
122C2×C4×C151—21no—
123C2×C4×C151—21no—
124C2×C4×C151—21no—
125C2×C4×C151—41no—
126C2×C4×C151—41no—
127C2×C4×C151—41no—
128C2×C4×C151—41no—
129C2×C4×C151—41no—
130C2×C4×C151—41no—
131C2×C4×C151—41no—
132C2×C4×C151—41no—
133C2×C4×C151—41no—
134C2×C4×C151—41no—
135C2×C4×C151—41no—
136C2×C4×C151—41no—
137C2×C4×C151—41no—
138C2×C4×C151—41no—
139C2×C4×C151—41no—
140C2×C4×C151—84no—
141C2×C4×C151—81no—
142C2×C4×C151—81no—
143C2×C4×C151—81no—
144C2×C4×C151C2×D811yescharacteristic
145C2×C4×C151#13511yescharacteristic
146C2×C4×C151#13511yescharacteristic
147C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C151#13711yescharacteristic
150C8×C151—42no—
151C8×C151—81no—
152C8×C151—81no—
153C22×C151—42no—
154C22×C151—42no—
155C22×C151—42no—
156C22×C151—81no—
157C22×C151—81no—
158C22×C151—81no—
159C22×C151—81no—
160C22×C151#44711yescharacteristic
161C22×C151#44911yescharacteristic
162C22×C151#45211yescharacteristic
163C22×C151#45411yescharacteristic
164C22×C151#45511yescharacteristic
165C22×C151#45611yescharacteristic
166C22×C151#45811yescharacteristic
167C4×C151—21no—
168C4×C151—21no—
169C4×C151—21no—
170C4×C151—21no—
171C4×C151—21no—
172C4×C151—21no—
173C4×C151—81no—
174C4×C151—81no—
175C4×C151—81no—
176C4×C151—81no—
177C4×C151—81no—
178C4×C151—161no—
179C2×C151—82no—
180C2×C151#160711yescharacteristic
181C2×C151#160911yescharacteristic
182C2×C151#161511yescharacteristic
183C2×C151#162511yescharacteristic
184C2×C151#164011yescharacteristic
185C2×C151#164711yescharacteristic
186C2×C151#166411yescharacteristic
187#7788C15111yesmaximal, characteristic, direct product factor, Sylow
188C151#778811yescharacteristic, direct product factor, Sylow
189#2724C2×C15111yescharacteristic
190#2099C2×C15111yescharacteristic, complemented
191#2083C2×C15111yescharacteristic, complemented
192#2071C2×C15111yescharacteristic
193#1271C2×C15111yescharacteristic, complemented
194#1271C2×C15111yescharacteristic, complemented
195#1270C2×C15111yescharacteristic
196C2×#123—21no—
197C2×#107—21no—
198C2×#102—21no—
199#398—21no—
200#397—21no—
201#397—21no—
202#390C22×C15111yescharacteristic
203#389C22×C15111yescharacteristic
204#389C22×C15111yescharacteristic
205#385—21no—
206#385—21no—
207#384C22×C15111yescharacteristic
208#340—21no—
209#340—21no—
210#340—21no—
211#340—21no—
212#336C22×C15111yescharacteristic
213#336C22×C15111yescharacteristic
214#336C22×C15111yescharacteristic
215C22×D4D4×C15111yescharacteristic
216C23×C4—21no—
217C22×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220#123—42no—
221#123—42no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4D4×C15111yescharacteristic
229C2×C4⋊C4D4×C15111yescharacteristic
230C2×C4⋊C4D4×C15111yescharacteristic
231C2×C4⋊C4D4×C15111yescharacteristic
232C2×C4⋊C4D4×C15111yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C42C23×C15111yescharacteristic
239#107—42no—
240#102—42no—
241#95—21no—
242#95—21no—
243#95—21no—
244#95—21no—
245#95—21no—
246C24—21no—
247C2×D4—21no—
248C2×D4—41no—
249C2×D4—41no—
250C2×D4D8×C15112yes—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—81no—
259C22×C4—81no—
260C22×C4C2×D4×C15111yescharacteristic
261C22×C4C2×D4×C15111yescharacteristic
262C22×C4C2×D4×C15111yescharacteristic
263C2×C8—22no—
264C2×C8—41no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—41no—
268C2×C8—41no—
269C2×C8—41no—
270C2×C8—41no—
271C2×C8—41no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—42no—
280C4⋊C4—41no—
281C4⋊C4—41no—
282C4⋊C4—41no—
283C4⋊C4—41no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—81no—
289C22⋊C4—42no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—41no—
293C22⋊C4—41no—
294C22⋊C4—41no—
295C22⋊C4—41no—
296C42—41no—
297C23—22no—
298C23—41no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—41no—
304C23#120×C15111yescharacteristic
305D4—42no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—84no—
329C2×C4—81no—
330C2×C4—81no—
331C2×C4—81no—
332C2×C4C2×D8×C15111yescharacteristic
333C2×C4#135×C15111yescharacteristic
334C2×C4#135×C15111yescharacteristic
335C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
336C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
337C2×C4#137×C15111yescharacteristic
338C8—42no—
339C8—81no—
340C8—81no—
341C22—42no—
342C22—42no—
343C22—42no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22#447×C15111yescharacteristic
349C22#449×C15111yescharacteristic
350C22#452×C15111yescharacteristic
351C22#454×C15111yescharacteristic
352C22#455×C15111yescharacteristic
353C22#456×C15111yescharacteristic
354C22#458×C15111yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—81no—
362C4—81no—
363C4—81no—
364C4—81no—
365C4—81no—
366C4—161no—
367C2—82no—
368C2#1607×C15111yescharacteristic
369C2#1609×C15111yescharacteristic
370C2#1615×C15111yescharacteristic
371C2#1625×C15111yescharacteristic
372C2#1640×C15111yescharacteristic
373C2#1647×C15111yescharacteristic
374C2#1664×C15111yescharacteristic
subgroupG/H
center, socleC23×C151#120
commutator, Frattini, cosocleC2×C42C23×C151
prime radicalC22×D4×C151D4
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2099×C151C2semidirect
3. #2083×C151C2semidirect
5. #1271×C151C2semidirect
6. #1271×C151C2semidirect
187. #7788C151direct (N × H)
188. C151#7788direct (N × H)
190. #2099C2×C151semidirect
191. #2083C2×C151semidirect
193. #1271C2×C151semidirect
194. #1271C2×C151semidirect

Maximal subgroups

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

subgroupindex
#2724×C1512
#2099×C1512
#2083×C1512
#2071×C1512
#1271×C1512
#1270×C1512
#7788151