Finite Group Databasebeta

← C151×#10020 #50314945

Subgroups — C151×#10020 #50314945

Order 38656.

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

normal subgroups only

This group has 598 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#3153×C151C211yesmaximal, characteristic
2#3112×C151C211yesmaximal, characteristic, complemented
3#2632×C151C211yesmaximal, characteristic, complemented
4#1533×C151C211yesmaximal, characteristic, complemented
5#1514×C151C211yesmaximal, characteristic, complemented
6#1491×C151C211yesmaximal, characteristic, complemented
7#1288×C151C211yesmaximal, characteristic, complemented
8C2×#137×C151—21no—
9C2×AGL(1,ℤ/8)×C151—21no—
10#532×C151C2211yescharacteristic
11C2×#124×C151C2211yescharacteristic
12C2×C4×Q8×C151C2211yescharacteristic
13#497×C151—21no—
14#496×C151—21no—
15#495×C151—22no—
16#480×C151—21no—
17#477×C151—21no—
18#460×C151—21no—
19#448×C151—21no—
20#440×C151—22no—
21#432×C151C2211yescharacteristic
22#432×C151C2211yescharacteristic
23C4×M16×C151C2211yescharacteristic
24#399×C151—21no—
25#395×C151—21no—
26#394×C151—21no—
27#393×C151—21no—
28#388×C151—22no—
29#383×C151C2211yescharacteristic
30#329×C151—22no—
31#327×C151—22no—
32Q8⋊C8×C151—22no—
33C2×#41×C151—21no—
34C22×Q8×C151D411yescharacteristic
35C22×Q8×C151D411yescharacteristic
36C22×D4×C151D411yescharacteristic
37#137×C151—44no—
38AGL(1,ℤ/8)×C151—44no—
39C2×Q16×C151—22no—
40C2×SD16×C151—22no—
41C2×SD16×C151—22no—
42C2×SD16×C151—22no—
43C2×D8×C151—22no—
44C2×M16×C151—21no—
45C4⋊Q8×C151D411yescharacteristic
46#127×C151D411yescharacteristic
47#124×C151—21no—
48#124×C151—21no—
49#124×C151—21no—
50#124×C151—21no—
51#124×C151—21no—
52#121×C151—21no—
53#121×C151—21no—
54C4×Q8×C151—21no—
55C4×Q8×C151—21no—
56C4×Q8×C151—21no—
57C4×Q8×C151—21no—
58C4×D4×C151—21no—
59C4×D4×C151—21no—
60C2×C4⋊C4×C151—21no—
61C2×C4⋊C4×C151—22no—
62C2×C4⋊C4×C151D411yescharacteristic
63C2×C22⋊C4×C151—22no—
64C2×C22⋊C4×C151—21no—
65C2×C22⋊C4×C151—21no—
66C2×C22⋊C4×C151—22no—
67C2×C42×C151—21no—
68C2×C42×C151C2311yescharacteristic
69#107×C151—22no—
70C4⋊C8×C151—21no—
71C4⋊C8×C151—21no—
72#103×C151—22no—
73#102×C151—22no—
74C22⋊C8×C151—21no—
75C22⋊C8×C151—21no—
76#97×C151—21no—
77#97×C151—21no—
78#97×C151D412yes—
79C4×C8×C151D412yes—
80#95×C151—22no—
81#95×C151—21no—
82#95×C151—21no—
83#95×C151D412yes—
84C24×C151—21no—
85#41×C151—42no—
86#41×C151—42no—
87C2×Q8×C151—21no—
88C2×Q8×C151—21no—
89C2×Q8×C151—21no—
90C2×Q8×C151—21no—
91C2×Q8×C151—21no—
92C2×Q8×C151—42no—
93C2×Q8×C151—42no—
94C2×D4×C151—21no—
95C2×D4×C151—21no—
96C2×D4×C151—21no—
97C2×D4×C151—21no—
98C2×D4×C151—21no—
99C2×D4×C151—42no—
100C2×D4×C151—41no—
101C22×C4×C151—21no—
102C22×C4×C151—22no—
103C22×C4×C151—21no—
104C22×C4×C151—21no—
105C22×C4×C151—41no—
106C22×C4×C151C2×D411yescharacteristic
107C22×C4×C151C2×D411yescharacteristic
108C22×C4×C151C2×D411yescharacteristic
109Q16×C151—44no—
110SD16×C151—44no—
111SD16×C151—44no—
112SD16×C151—44no—
113D8×C151—44no—
114M16×C151—24no—
115C2×C8×C151—22no—
116C2×C8×C151—22no—
117C2×C8×C151—41no—
118C4⋊C4×C151—21no—
119C4⋊C4×C151—21no—
120C4⋊C4×C151—21no—
121C4⋊C4×C151—21no—
122C4⋊C4×C151—41no—
123C4⋊C4×C151—41no—
124C4⋊C4×C151—82no—
125C22⋊C4×C151—44no—
126C22⋊C4×C151—41no—
127C22⋊C4×C151—41no—
128C22⋊C4×C151—41no—
129C22⋊C4×C151—41no—
130C22⋊C4×C151—41no—
131C22⋊C4×C151—41no—
132C22⋊C4×C151—82no—
133C42×C151—41no—
134C42×C151—41no—
135C42×C151C2×D411yescharacteristic
136C42×C151C2×D411yescharacteristic
137C42×C151C2×D411yescharacteristic
138C42×C151#4111yescharacteristic
139C23×C151—42no—
140C23×C151—42no—
141C23×C151—41no—
142C23×C151—41no—
143C23×C151—42no—
144C23×C151#12011yescharacteristic
145Q8×C151—22no—
146Q8×C151—22no—
147Q8×C151—41no—
148Q8×C151—41no—
149Q8×C151—42no—
150Q8×C151—41no—
151Q8×C151—41no—
152Q8×C151—81no—
153D4×C151—41no—
154D4×C151—41no—
155D4×C151—42no—
156D4×C151—42no—
157D4×C151—81no—
158D4×C151—81no—
159D4×C151—82no—
160C2×C4×C151—22no—
161C2×C4×C151—22no—
162C2×C4×C151—41no—
163C2×C4×C151—41no—
164C2×C4×C151—41no—
165C2×C4×C151—41no—
166C2×C4×C151—42no—
167C2×C4×C151—42no—
168C2×C4×C151—41no—
169C2×C4×C151—41no—
170C2×C4×C151—41no—
171C2×C4×C151—42no—
172C2×C4×C151—42no—
173C2×C4×C151—42no—
174C2×C4×C151—82no—
175C2×C4×C151#12011yescharacteristic
176C2×C4×C151#12011yescharacteristic
177C2×C4×C151#12111yescharacteristic
178C2×C4×C151#12111yescharacteristic
179C2×C4×C151#12111yescharacteristic
180C2×C4×C151#12711yescharacteristic
181C8×C151—24no—
182C8×C151—42no—
183C8×C151—81no—
184C22×C151—21no—
185C22×C151—21no—
186C22×C151—84no—
187C22×C151—81no—
188C22×C151—81no—
189C22×C151—81no—
190C22×C151—81no—
191C22×C151—81no—
192C22×C151—81no—
193C22×C151#39211yescharacteristic
194C22×C151#45312yes—
195C4×C151—21no—
196C4×C151—21no—
197C4×C151—21no—
198C4×C151—21no—
199C4×C151—21no—
200C4×C151—21no—
201C4×C151—42no—
202C4×C151—82no—
203C4×C151—81no—
204C4×C151—81no—
205C4×C151—81no—
206C2×C151—22no—
207C2×C151—161no—
208C2×C151—161no—
209C2×C151#200211yescharacteristic
210C2×C151#200911yescharacteristic
211C2×C151#201011yescharacteristic
212#10020C15111yesmaximal, characteristic, direct product factor, Sylow
213C151#1002011yescharacteristic, direct product factor, Sylow
214#3153C2×C15111yescharacteristic
215#3112C2×C15111yescharacteristic, complemented
216#2632C2×C15111yescharacteristic, complemented
217#1533C2×C15111yescharacteristic, complemented
218#1514C2×C15111yescharacteristic, complemented
219#1491C2×C15111yescharacteristic, complemented
220#1288C2×C15111yescharacteristic, complemented
221C2×#137—21no—
222C2×AGL(1,ℤ/8)—21no—
223#532C22×C15111yescharacteristic
224C2×#124C22×C15111yescharacteristic
225C2×C4×Q8C22×C15111yescharacteristic
226#497—21no—
227#496—21no—
228#495—22no—
229#480—21no—
230#477—21no—
231#460—21no—
232#448—21no—
233#440—22no—
234#432C22×C15111yescharacteristic
235#432C22×C15111yescharacteristic
236C4×M16C22×C15111yescharacteristic
237#399—21no—
238#395—21no—
239#394—21no—
240#393—21no—
241#388—22no—
242#383C22×C15111yescharacteristic
243#329—22no—
244#327—22no—
245Q8⋊C8—22no—
246C2×#41—21no—
247C22×Q8D4×C15111yescharacteristic
248C22×Q8D4×C15111yescharacteristic
249C22×D4D4×C15111yescharacteristic
250#137—44no—
251AGL(1,ℤ/8)—44no—
252C2×Q16—22no—
253C2×SD16—22no—
254C2×SD16—22no—
255C2×SD16—22no—
256C2×D8—22no—
257C2×M16—21no—
258C4⋊Q8D4×C15111yescharacteristic
259#127D4×C15111yescharacteristic
260#124—21no—
261#124—21no—
262#124—21no—
263#124—21no—
264#124—21no—
265#121—21no—
266#121—21no—
267C4×Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×D4—21no—
272C4×D4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4D4×C15111yescharacteristic
276C2×C22⋊C4—22no—
277C2×C22⋊C4—21no—
278C2×C22⋊C4—21no—
279C2×C22⋊C4—22no—
280C2×C42—21no—
281C2×C42C23×C15111yescharacteristic
282#107—22no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285#103—22no—
286#102—22no—
287C22⋊C8—21no—
288C22⋊C8—21no—
289#97—21no—
290#97—21no—
291#97D4×C15112yes—
292C4×C8D4×C15112yes—
293#95—22no—
294#95—21no—
295#95—21no—
296#95D4×C15112yes—
297C24—21no—
298#41—42no—
299#41—42no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—42no—
306C2×Q8—42no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4—41no—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—41no—
319C22×C4C2×D4×C15111yescharacteristic
320C22×C4C2×D4×C15111yescharacteristic
321C22×C4C2×D4×C15111yescharacteristic
322Q16—44no—
323SD16—44no—
324SD16—44no—
325SD16—44no—
326D8—44no—
327M16—24no—
328C2×C8—22no—
329C2×C8—22no—
330C2×C8—41no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—82no—
338C22⋊C4—44no—
339C22⋊C4—41no—
340C22⋊C4—41no—
341C22⋊C4—41no—
342C22⋊C4—41no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—82no—
346C42—41no—
347C42—41no—
348C42C2×D4×C15111yescharacteristic
349C42C2×D4×C15111yescharacteristic
350C42C2×D4×C15111yescharacteristic
351C42#41×C15111yescharacteristic
352C23—42no—
353C23—42no—
354C23—41no—
355C23—41no—
356C23—42no—
357C23#120×C15111yescharacteristic
358Q8—22no—
359Q8—22no—
360Q8—41no—
361Q8—41no—
362Q8—41no—
363Q8—42no—
364Q8—41no—
365Q8—81no—
366D4—41no—
367D4—42no—
368D4—41no—
369D4—42no—
370D4—81no—
371D4—81no—
372D4—82no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—82no—
388C2×C4#120×C15111yescharacteristic
389C2×C4#120×C15111yescharacteristic
390C2×C4#121×C15111yescharacteristic
391C2×C4#121×C15111yescharacteristic
392C2×C4#121×C15111yescharacteristic
393C2×C4#127×C15111yescharacteristic
394C8—24no—
395C8—42no—
396C8—81no—
397C22—21no—
398C22—21no—
399C22—84no—
400C22—81no—
401C22—81no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—81no—
406C22#392×C15111yescharacteristic
407C22#453×C15112yes—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—42no—
415C4—82no—
416C4—81no—
417C4—81no—
418C4—81no—
419C2—22no—
420C2—161no—
421C2—161no—
422C2#2002×C15111yescharacteristic
423C2#2009×C15111yescharacteristic
424C2#2010×C15111yescharacteristic
subgroupG/H
center, socleC22×C151#392
commutator, Frattini, cosocleC2×C42C23×C151
prime radical#3153×C151C2
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C151 ◁ C2×C42×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. #3112×C151C2semidirect
3. #2632×C151C2semidirect
4. #1533×C151C2semidirect
5. #1514×C151C2semidirect
6. #1491×C151C2semidirect
7. #1288×C151C2semidirect
212. #10020C151direct (N × H)
213. C151#10020direct (N × H)
215. #3112C2×C151semidirect
216. #2632C2×C151semidirect
217. #1533C2×C151semidirect
218. #1514C2×C151semidirect
219. #1491C2×C151semidirect
220. #1288C2×C151semidirect

Maximal subgroups

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

subgroupindex
#3153×C1512
#3112×C1512
#2632×C1512
#1533×C1512
#1514×C1512
#1491×C1512
#1288×C1512
#10020151