Finite Group Databasebeta

← C151×#12199 #50317124

Subgroups — C151×#12199 #50317124

Order 38656.

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

normal subgroups only

This group has 434 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#2832×C151C211yesmaximal, characteristic, complemented
2#2007×C151C211yesmaximal, characteristic, complemented
3#1604×C151C211yesmaximal, characteristic, complemented
4#1598×C151C211yesmaximal, characteristic
5#1354×C151C211yesmaximal, characteristic, complemented
6#1345×C151C211yesmaximal, characteristic, complemented
7#1319×C151C211yesmaximal, characteristic, complemented
8C2×#137×C151—21no—
9#537×C151C2211yescharacteristic
10#448×C151—21no—
11#447×C151—21no—
12C2×#104×C151—21no—
13#400×C151—21no—
14#397×C151—21no—
15#395×C151C2211yescharacteristic
16#394×C151—21no—
17#394×C151C2211yescharacteristic
18#393×C151—21no—
19#393×C151—21no—
20#391×C151C2211yescharacteristic
21#388×C151—21no—
22#343×C151C2211yescharacteristic
23#327×C151—21no—
24#327×C151—21no—
25#324×C151C2211yescharacteristic
26#324×C151C2211yescharacteristic
27C2×#41×C151—21no—
28C22×Q8×C151D411yescharacteristic
29C22×Q8×C151D411yescharacteristic
30C22×D4×C151D411yescharacteristic
31#137×C151—42no—
32#137×C151—42no—
33C2×Q16×C151—41no—
34C2×SD16×C151—41no—
35C2×SD16×C151—41no—
36C2×D8×C151—41no—
37C2×M16×C151—21no—
38C22⋊Q8×C151—41no—
39#121×C151—21no—
40#121×C151—21no—
41#121×C151—41no—
42C2×C4⋊C4×C151—21no—
43C2×C4⋊C4×C151—21no—
44C2×C4⋊C4×C151C2311yescharacteristic
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×C42×C151D411yescharacteristic
52#104×C151—42no—
53#103×C151—41no—
54#102×C151—41no—
55C22⋊C8×C151—21no—
56C22⋊C8×C151—21no—
57#95×C151—21no—
58#95×C151—21no—
59#95×C151—21no—
60#95×C151—21no—
61#95×C151—21no—
62#95×C151—21no—
63#95×C151D411yescharacteristic
64#95×C151D411yescharacteristic
65C24×C151—21no—
66#41×C151—41no—
67#41×C151—41no—
68#41×C151—41no—
69#41×C151—41no—
70C2×Q8×C151—21no—
71C2×Q8×C151—21no—
72C2×Q8×C151—21no—
73C2×Q8×C151—41no—
74C2×Q8×C151—41no—
75C2×Q8×C151—41no—
76C2×Q8×C151—81no—
77C2×D4×C151—21no—
78C2×D4×C151—21no—
79C2×D4×C151—21no—
80C2×D4×C151—41no—
81C2×D4×C151—41no—
82C2×D4×C151—81no—
83C22×C4×C151—21no—
84C22×C4×C151—21no—
85C22×C4×C151—21no—
86C22×C4×C151—21no—
87C22×C4×C151—21no—
88C22×C4×C151—41no—
89C22×C4×C151C2×D411yescharacteristic
90C22×C4×C151C2×D411yescharacteristic
91C22×C4×C151C2×D411yescharacteristic
92Q16×C151—44no—
93SD16×C151—44no—
94SD16×C151—82no—
95D8×C151—82no—
96M16×C151—22no—
97M16×C151—41no—
98C2×C8×C151—41no—
99C2×C8×C151—41no—
100C2×C8×C151—41no—
101C4⋊C4×C151—21no—
102C4⋊C4×C151—21no—
103C4⋊C4×C151—41no—
104C4⋊C4×C151—81no—
105C4⋊C4×C151—81no—
106C22⋊C4×C151—41no—
107C22⋊C4×C151—41no—
108C22⋊C4×C151—41no—
109C22⋊C4×C151—41no—
110C22⋊C4×C151—81no—
111C22⋊C4×C151—81no—
112C22⋊C4×C151—81no—
113C22⋊C4×C151—81no—
114C22⋊C4×C151—81no—
115C42×C151—41no—
116C23×C151—41no—
117C23×C151—41no—
118C23×C151—41no—
119C23×C151—41no—
120C23×C151—41no—
121C23×C151—41no—
122C23×C151—41no—
123C23×C151—41no—
124C23×C151#12011yescharacteristic
125Q8×C151—42no—
126Q8×C151—41no—
127Q8×C151—41no—
128Q8×C151—42no—
129Q8×C151—81no—
130Q8×C151—81no—
131D4×C151—41no—
132D4×C151—41no—
133D4×C151—42no—
134D4×C151—81no—
135D4×C151—81no—
136D4×C151—81no—
137D4×C151—81no—
138D4×C151—81no—
139C2×C4×C151—21no—
140C2×C4×C151—21no—
141C2×C4×C151—21no—
142C2×C4×C151—21no—
143C2×C4×C151—41no—
144C2×C4×C151—41no—
145C2×C4×C151—41no—
146C2×C4×C151—41no—
147C2×C4×C151—41no—
148C2×C4×C151—41no—
149C2×C4×C151—41no—
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—41no—
159C2×C4×C151—41no—
160C2×C4×C151—41no—
161C2×C4×C151—81no—
162C2×C4×C151—81no—
163C2×C4×C151#13511yescharacteristic
164C2×C4×C151AGL(1,ℤ/8)11yescharacteristic
165C8×C151—42no—
166C8×C151—81no—
167C8×C151—81no—
168C22×C151—21no—
169C22×C151—21no—
170C22×C151—81no—
171C22×C151—81no—
172C22×C151—81no—
173C22×C151—81no—
174C22×C151—81no—
175C22×C151—81no—
176C22×C151—81no—
177C22×C151—81no—
178C22×C151—81no—
179C22×C151—81no—
180C22×C151#44911yescharacteristic
181C22×C151#45311yescharacteristic
182C22×C151B(4,𝔽2)11yescharacteristic
183C4×C151—21no—
184C4×C151—21no—
185C4×C151—41no—
186C4×C151—41no—
187C4×C151—81no—
188C4×C151—81no—
189C4×C151—81no—
190C4×C151—81no—
191C4×C151—81no—
192C4×C151—81no—
193C2×C151—21no—
194C2×C151—21no—
195C2×C151—161no—
196C2×C151—161no—
197C2×C151#160011yescharacteristic
198C2×C151#219911yescharacteristic
199C2×C151#220311yescharacteristic
200#12199C15111yesmaximal, characteristic, direct product factor, Sylow
201C151#1219911yescharacteristic, direct product factor, Sylow
202#2832C2×C15111yescharacteristic, complemented
203#2007C2×C15111yescharacteristic, complemented
204#1604C2×C15111yescharacteristic, complemented
205#1598C2×C15111yescharacteristic
206#1354C2×C15111yescharacteristic, complemented
207#1345C2×C15111yescharacteristic, complemented
208#1319C2×C15111yescharacteristic, complemented
209C2×#137—21no—
210#537C22×C15111yescharacteristic
211#448—21no—
212#447—21no—
213C2×#104—21no—
214#400—21no—
215#397—21no—
216#395C22×C15111yescharacteristic
217#394—21no—
218#394C22×C15111yescharacteristic
219#393—21no—
220#393—21no—
221#391C22×C15111yescharacteristic
222#388—21no—
223#343C22×C15111yescharacteristic
224#327—21no—
225#327—21no—
226#324C22×C15111yescharacteristic
227#324C22×C15111yescharacteristic
228C2×#41—21no—
229C22×Q8D4×C15111yescharacteristic
230C22×Q8D4×C15111yescharacteristic
231C22×D4D4×C15111yescharacteristic
232#137—42no—
233#137—42no—
234C2×Q16—41no—
235C2×SD16—41no—
236C2×SD16—41no—
237C2×D8—41no—
238C2×M16—21no—
239C22⋊Q8—41no—
240#121—21no—
241#121—21no—
242#121—41no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C23×C15111yescharacteristic
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C42D4×C15111yescharacteristic
253#104—42no—
254#103—41no—
255#102—41no—
256C22⋊C8—21no—
257C22⋊C8—21no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95D4×C15111yescharacteristic
265#95D4×C15111yescharacteristic
266C24—21no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—81no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—81no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—41no—
290C22×C4C2×D4×C15111yescharacteristic
291C22×C4C2×D4×C15111yescharacteristic
292C22×C4C2×D4×C15111yescharacteristic
293Q16—44no—
294SD16—44no—
295SD16—82no—
296D8—82no—
297M16—22no—
298M16—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—41no—
305C4⋊C4—81no—
306C4⋊C4—81no—
307C22⋊C4—41no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—81no—
312C22⋊C4—81no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C22⋊C4—81no—
316C42—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23—41no—
323C23—41no—
324C23—41no—
325C23#120×C15111yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—42no—
330Q8—81no—
331Q8—81no—
332D4—41no—
333D4—41no—
334D4—42no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—81no—
339D4—81no—
340C2×C4—21no—
341C2×C4—21no—
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—41no—
353C2×C4—41no—
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—81no—
364C2×C4#135×C15111yescharacteristic
365C2×C4AGL(1,ℤ/8)×C15111yescharacteristic
366C8—42no—
367C8—81no—
368C8—81no—
369C22—21no—
370C22—21no—
371C22—81no—
372C22—81no—
373C22—81no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22—81no—
380C22—81no—
381C22#449×C15111yescharacteristic
382C22#453×C15111yescharacteristic
383C22B(4,𝔽2)×C15111yescharacteristic
384C4—21no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—81no—
390C4—81no—
391C4—81no—
392C4—81no—
393C4—81no—
394C2—21no—
395C2—21no—
396C2—161no—
397C2—161no—
398C2#1600×C15111yescharacteristic
399C2#2199×C15111yescharacteristic
400C2#2203×C15111yescharacteristic
subgroupG/H
center, socleC22×C151B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C151
prime radical#1598×C151C2
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C151 ◁ C2×C4⋊C4×C151 ◁ G

Complements

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

NH ≅ G/Ntype
1. #2832×C151C2semidirect
2. #2007×C151C2semidirect
3. #1604×C151C2semidirect
5. #1354×C151C2semidirect
6. #1345×C151C2semidirect
7. #1319×C151C2semidirect
200. #12199C151direct (N × H)
201. C151#12199direct (N × H)
202. #2832C2×C151semidirect
203. #2007C2×C151semidirect
204. #1604C2×C151semidirect
206. #1354C2×C151semidirect
207. #1345C2×C151semidirect
208. #1319C2×C151semidirect

Maximal subgroups

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

subgroupindex
#2832×C1512
#2007×C1512
#1604×C1512
#1598×C1512
#1354×C1512
#1345×C1512
#1319×C1512
#12199151