Finite Group Databasebeta

← C151×#12226 #50317151

Subgroups — C151×#12226 #50317151

Order 38656.

510 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 70 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#2455×C151C211yesmaximal, characteristic
2#2062×C151C211yesmaximal, characteristic, complemented
3#1614×C151C211yesmaximal, characteristic, complemented
4#1596×C151C211yesmaximal, characteristic
5#1352×C151C211yesmaximal, characteristic, complemented
6#1343×C151C211yesmaximal, characteristic
7#1326×C151C211yesmaximal, characteristic, complemented
8#543×C151C2211yescharacteristic
9C2×#123×C151—21no—
10C2×C4×D4×C151—21no—
11#484×C151—21no—
12#450×C151—21no—
13#399×C151C2211yescharacteristic
14#394×C151—21no—
15#392×C151C2211yescharacteristic
16#389×C151—21no—
17#388×C151—21no—
18#385×C151—21no—
19#385×C151—21no—
20#385×C151C2211yescharacteristic
21#342×C151—21no—
22#342×C151C2211yescharacteristic
23#328×C151—21no—
24#328×C151—21no—
25#323×C151C2211yescharacteristic
26#323×C151C2211yescharacteristic
27C22×D4×C151—21no—
28C22×D4×C151D411yescharacteristic
29C23×C4×C151—21no—
30C23×C4×C151D411yescharacteristic
31C2×SD16×C151—41no—
32#123×C151—44no—
33C22⋊Q8×C151—21no—
34C22⋊Q8×C151—21no—
35C22⋊Q8×C151—41no—
36C22⋊Q8×C151—41no—
37C4×D4×C151—44no—
38C2×C4⋊C4×C151—21no—
39C2×C4⋊C4×C151—21no—
40C2×C4⋊C4×C151—21no—
41C2×C4⋊C4×C151—21no—
42C2×C4⋊C4×C151—41no—
43C2×C4⋊C4×C151D411yescharacteristic
44C2×C22⋊C4×C151—21no—
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×C22⋊C4×C151—21no—
51C2×C22⋊C4×C151D411yescharacteristic
52C2×C22⋊C4×C151C2311yescharacteristic
53C2×C42×C151—21no—
54#107×C151—41no—
55#103×C151—41no—
56#102×C151—41no—
57C22⋊C8×C151—21no—
58C22⋊C8×C151—21no—
59#95×C151—21no—
60#95×C151—21no—
61#95×C151—41no—
62#95×C151D411yescharacteristic
63#95×C151D411yescharacteristic
64C24×C151—21no—
65C24×C151C2×D411yescharacteristic
66C2×Q8×C151—21no—
67C2×Q8×C151—21no—
68C2×D4×C151—42no—
69C2×D4×C151—41no—
70C2×D4×C151—42no—
71C2×D4×C151—41no—
72C2×D4×C151—41no—
73C2×D4×C151—42no—
74C22×C4×C151—21no—
75C22×C4×C151—21no—
76C22×C4×C151—21no—
77C22×C4×C151—21no—
78C22×C4×C151—21no—
79C22×C4×C151—21no—
80C22×C4×C151—21no—
81C22×C4×C151—42no—
82C22×C4×C151—42no—
83C22×C4×C151—41no—
84C22×C4×C151—41no—
85C22×C4×C151—42no—
86C22×C4×C151—81no—
87C22×C4×C151C2×D411yescharacteristic
88C22×C4×C151C2×D411yescharacteristic
89SD16×C151—82no—
90C2×C8×C151—41no—
91C2×C8×C151—41no—
92C4⋊C4×C151—21no—
93C4⋊C4×C151—21no—
94C4⋊C4×C151—41no—
95C4⋊C4×C151—42no—
96C4⋊C4×C151—42no—
97C4⋊C4×C151—42no—
98C4⋊C4×C151—42no—
99C4⋊C4×C151—41no—
100C4⋊C4×C151—41no—
101C4⋊C4×C151—82no—
102C22⋊C4×C151—24no—
103C22⋊C4×C151—22no—
104C22⋊C4×C151—21no—
105C22⋊C4×C151—21no—
106C22⋊C4×C151—41no—
107C22⋊C4×C151—42no—
108C22⋊C4×C151—42no—
109C22⋊C4×C151—42no—
110C22⋊C4×C151—41no—
111C22⋊C4×C151—42no—
112C22⋊C4×C151—41no—
113C22⋊C4×C151—41no—
114C42×C151—42no—
115C23×C151—21no—
116C23×C151—21no—
117C23×C151—22no—
118C23×C151—41no—
119C23×C151—41no—
120C23×C151—41no—
121C23×C151—41no—
122C23×C151—41no—
123C23×C151—41no—
124C23×C151—41no—
125C23×C151—41no—
126C23×C151#12011yescharacteristic
127C23×C151#13511yescharacteristic
128C23×C151#13711yescharacteristic
129Q8×C151—81no—
130Q8×C151—81no—
131D4×C151—42no—
132D4×C151—44no—
133D4×C151—44no—
134D4×C151—81no—
135C2×C4×C151—22no—
136C2×C4×C151—22no—
137C2×C4×C151—22no—
138C2×C4×C151—21no—
139C2×C4×C151—21no—
140C2×C4×C151—41no—
141C2×C4×C151—41no—
142C2×C4×C151—41no—
143C2×C4×C151—41no—
144C2×C4×C151—41no—
145C2×C4×C151—41no—
146C2×C4×C151—41no—
147C2×C4×C151—41no—
148C2×C4×C151—42no—
149C2×C4×C151—42no—
150C2×C4×C151—41no—
151C2×C4×C151—41no—
152C2×C4×C151—41no—
153C2×C4×C151—41no—
154C2×C4×C151—41no—
155C2×C4×C151—41no—
156C2×C4×C151—41no—
157C2×C4×C151—41no—
158C2×C4×C151—81no—
159C2×C4×C151—84no—
160C2×C4×C151—81no—
161C2×C4×C151—81no—
162C2×C4×C151—81no—
163C2×C4×C151—81no—
164C8×C151—81no—
165C8×C151—81no—
166C22×C151—21no—
167C22×C151—21no—
168C22×C151—21no—
169C22×C151—21no—
170C22×C151—41no—
171C22×C151—41no—
172C22×C151—42no—
173C22×C151—42no—
174C22×C151—41no—
175C22×C151—41no—
176C22×C151—42no—
177C22×C151—81no—
178C22×C151—81no—
179C22×C151—81no—
180C22×C151—81no—
181C22×C151—81no—
182C22×C151#45211yescharacteristic
183C22×C151#45611yescharacteristic
184C22×C151B(4,𝔽2)11yescharacteristic
185C4×C151—42no—
186C4×C151—42no—
187C4×C151—41no—
188C4×C151—42no—
189C4×C151—41no—
190C4×C151—81no—
191C4×C151—82no—
192C4×C151—81no—
193C4×C151—81no—
194C4×C151—81no—
195C2×C151—21no—
196C2×C151—21no—
197C2×C151—41no—
198C2×C151—41no—
199C2×C151—82no—
200C2×C151#160311yescharacteristic
201C2×C151#219611yescharacteristic
202C2×C151#219811yescharacteristic
203#12226C15111yesmaximal, characteristic, direct product factor, Sylow
204C151#1222611yescharacteristic, direct product factor, Sylow
205#2455C2×C15111yescharacteristic
206#2062C2×C15111yescharacteristic, complemented
207#1614C2×C15111yescharacteristic, complemented
208#1596C2×C15111yescharacteristic
209#1352C2×C15111yescharacteristic, complemented
210#1343C2×C15111yescharacteristic
211#1326C2×C15111yescharacteristic, complemented
212#543C22×C15111yescharacteristic
213C2×#123—21no—
214C2×C4×D4—21no—
215#484—21no—
216#450—21no—
217#399C22×C15111yescharacteristic
218#394—21no—
219#392C22×C15111yescharacteristic
220#389—21no—
221#388—21no—
222#385—21no—
223#385—21no—
224#385C22×C15111yescharacteristic
225#342—21no—
226#342C22×C15111yescharacteristic
227#328—21no—
228#328—21no—
229#323C22×C15111yescharacteristic
230#323C22×C15111yescharacteristic
231C22×D4—21no—
232C22×D4D4×C15111yescharacteristic
233C23×C4—21no—
234C23×C4D4×C15111yescharacteristic
235C2×SD16—41no—
236#123—44no—
237C22⋊Q8—21no—
238C22⋊Q8—21no—
239C22⋊Q8—41no—
240C22⋊Q8—41no—
241C4×D4—44no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4D4×C15111yescharacteristic
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4D4×C15111yescharacteristic
256C2×C22⋊C4C23×C15111yescharacteristic
257C2×C42—21no—
258#107—41no—
259#103—41no—
260#102—41no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95D4×C15111yescharacteristic
267#95D4×C15111yescharacteristic
268C24—21no—
269C24C2×D4×C15111yescharacteristic
270C2×Q8—21no—
271C2×Q8—21no—
272C2×D4—42no—
273C2×D4—41no—
274C2×D4—42no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—42no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—81no—
291C22×C4C2×D4×C15111yescharacteristic
292C22×C4C2×D4×C15111yescharacteristic
293SD16—82no—
294C2×C8—41no—
295C2×C8—41no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—41no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—82no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—41no—
311C22⋊C4—42no—
312C22⋊C4—42no—
313C22⋊C4—42no—
314C22⋊C4—41no—
315C22⋊C4—42no—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—42no—
319C23—21no—
320C23—21no—
321C23—22no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23—41no—
326C23—41no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23#120×C15111yescharacteristic
331C23#135×C15111yescharacteristic
332C23#137×C15111yescharacteristic
333Q8—81no—
334Q8—81no—
335D4—42no—
336D4—44no—
337D4—44no—
338D4—81no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—81no—
363C2×C4—84no—
364C2×C4—81no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—81no—
368C8—81no—
369C8—81no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22—81no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#452×C15111yescharacteristic
387C22#456×C15111yescharacteristic
388C22B(4,𝔽2)×C15111yescharacteristic
389C4—42no—
390C4—42no—
391C4—41no—
392C4—42no—
393C4—41no—
394C4—81no—
395C4—82no—
396C4—81no—
397C4—81no—
398C4—81no—
399C2—21no—
400C2—21no—
401C2—41no—
402C2—41no—
403C2—82no—
404C2#1603×C15111yescharacteristic
405C2#2196×C15111yescharacteristic
406C2#2198×C15111yescharacteristic
subgroupG/H
center, socleC22×C151B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C151
prime radical#392×C151C22
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C151 ◁ C2×C22⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #2062×C151C2semidirect
3. #1614×C151C2semidirect
5. #1352×C151C2semidirect
7. #1326×C151C2semidirect
203. #12226C151direct (N × H)
204. C151#12226direct (N × H)
206. #2062C2×C151semidirect
207. #1614C2×C151semidirect
209. #1352C2×C151semidirect
211. #1326C2×C151semidirect

Maximal subgroups

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

subgroupindex
#2455×C1512
#2062×C1512
#1614×C1512
#1596×C1512
#1352×C1512
#1343×C1512
#1326×C1512
#12226151