Finite Group Databasebeta

← C2×#1402×C151 #50318484

Subgroups — C2×#1402×C151 #50318484

Order 38656.

564 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C2×#453×C151C211yesmaximal, characteristic
2C2×#353×C151C211yesmaximal, characteristic, complemented
3C2×#331×C151C211yesmaximal, characteristic, complemented
4#1402×C151C214yesmaximal, direct product factor
5C2×H(5,𝔽2)×C151C411yescharacteristic, complemented
6C2×AGL(1,ℤ/8)×C151—21no—
7C2×#127×C151C2211yescharacteristic
8#453×C151—21no—
9#453×C151—21no—
10#453×C151C412yescomplemented
11#453×C151C2212yes—
12C2×C4⋊C8×C151—21no—
13C2×#104×C151—21no—
14C2×#100×C151C411yescharacteristic, complemented
15C2×#99×C151—21no—
16#353×C151—21no—
17#353×C151C2212yescomplemented
18#331×C151—21no—
19#331×C151C2212yescomplemented
20H(5,𝔽2)×C151—21no—
21H(5,𝔽2)×C151—42no—
22H(5,𝔽2)×C151—41no—
23H(5,𝔽2)×C151C2×C412yescomplemented
24C2×#41×C151—21no—
25C2×#41×C151—41no—
26C22×D4×C151—41no—
27C22×D4×C151—41no—
28C22×D4×C151—41no—
29C22×D4×C151C2×C411yescharacteristic
30C22×D4×C151D411yescharacteristic
31AGL(1,ℤ/8)×C151—22no—
32AGL(1,ℤ/8)×C151—22no—
33AGL(1,ℤ/8)×C151—42no—
34C2×SD16×C151—41no—
35C2×D8×C151—41no—
36C2×M16×C151—21no—
37C22×C8×C151—41no—
38#127×C151—41no—
39#127×C151C2×C411yescharacteristic
40#127×C151D411yescharacteristic
41#127×C151D411yescharacteristic
42#127×C151C2311yescharacteristic
43C2×C22⋊C4×C151—41no—
44C2×C42×C151D411yescharacteristic
45C4⋊C8×C151—22no—
46C4⋊C8×C151—41no—
47#104×C151—22no—
48#104×C151—22no—
49#100×C151—21no—
50#100×C151C2×C412yescomplemented
51#99×C151—22no—
52#99×C151—41no—
53C24×C151—21no—
54C24×C151—21no—
55C24×C151—41no—
56#41×C151—22no—
57#41×C151—22no—
58#41×C151—41no—
59#41×C151—42no—
60#41×C151—44no—
61#41×C151—41no—
62#41×C151—42no—
63C2×Q8×C151—21no—
64C2×D4×C151—21no—
65C2×D4×C151—21no—
66C2×D4×C151—21no—
67C2×D4×C151—41no—
68C2×D4×C151—41no—
69C2×D4×C151—41no—
70C2×D4×C151—41no—
71C2×D4×C151—41no—
72C2×D4×C151—41no—
73C2×D4×C151—41no—
74C2×D4×C151—41no—
75C2×D4×C151—42no—
76C2×D4×C151—42no—
77C2×D4×C151—42no—
78C2×D4×C151—42no—
79C2×D4×C151—44no—
80C2×D4×C151—42no—
81C2×D4×C151—41no—
82C2×D4×C151—42no—
83C2×D4×C151—42no—
84C2×D4×C151—82no—
85C2×D4×C151—82no—
86C2×D4×C151C22⋊C411yescharacteristic
87C2×D4×C151C22⋊C411yescharacteristic
88C2×D4×C151C22×C411yescharacteristic
89C2×D4×C151C2×D411yescharacteristic
90C22×C4×C151—21no—
91C22×C4×C151—41no—
92C22×C4×C151—41no—
93C22×C4×C151—81no—
94C22×C4×C151C22⋊C411yescharacteristic
95SD16×C151—42no—
96SD16×C151—42no—
97D8×C151—42no—
98D8×C151—42no—
99M16×C151—22no—
100M16×C151—41no—
101C2×C8×C151—41no—
102C2×C8×C151—42no—
103C2×C8×C151—82no—
104C22⋊C4×C151—42no—
105C22⋊C4×C151—81no—
106C42×C151—21no—
107C42×C151C22⋊C411yescharacteristic
108C42×C151C2×D411yescharacteristic
109C23×C151—21no—
110C23×C151—21no—
111C23×C151—21no—
112C23×C151—21no—
113C23×C151—42no—
114C23×C151—41no—
115C23×C151—41no—
116C23×C151—41no—
117C23×C151—42no—
118C23×C151—41no—
119C23×C151—41no—
120C23×C151—82no—
121C23×C151—82no—
122C23×C151—162no—
123C23×C151#9911yescharacteristic
124Q8×C151—22no—
125Q8×C151—41no—
126D4×C151—22no—
127D4×C151—41no—
128D4×C151—41no—
129D4×C151—41no—
130D4×C151—41no—
131D4×C151—41no—
132D4×C151—42no—
133D4×C151—44no—
134D4×C151—44no—
135D4×C151—42no—
136D4×C151—42no—
137D4×C151—81no—
138D4×C151—81no—
139D4×C151—81no—
140D4×C151—82no—
141D4×C151—82no—
142D4×C151—81no—
143D4×C151—81no—
144C2×C4×C151—22no—
145C2×C4×C151—22no—
146C2×C4×C151—21no—
147C2×C4×C151—21no—
148C2×C4×C151—21no—
149C2×C4×C151—21no—
150C2×C4×C151—42no—
151C2×C4×C151—42no—
152C2×C4×C151—42no—
153C2×C4×C151—41no—
154C2×C4×C151—42no—
155C2×C4×C151—82no—
156C2×C4×C151—162no—
157C2×C4×C151#9911yescharacteristic
158C2×C4×C151C2×C22⋊C411yescharacteristic
159C8×C151—42no—
160C8×C151—82no—
161C22×C151—22no—
162C22×C151—42no—
163C22×C151—41no—
164C22×C151—41no—
165C22×C151—41no—
166C22×C151—41no—
167C22×C151—81no—
168C22×C151—81no—
169C22×C151—81no—
170C22×C151—81no—
171C22×C151—82no—
172C22×C151—82no—
173C22×C151—82no—
174C22×C151—161no—
175C22×C151—161no—
176C22×C151—162no—
177C22×C151—162no—
178C22×C151AΣL(1,𝔽16)11yescharacteristic
179C22×C151AΣL(1,𝔽16)11yescharacteristic
180C22×C151C2×#9911yescharacteristic
181C4×C151—21no—
182C4×C151—22no—
183C4×C151—21no—
184C4×C151—21no—
185C4×C151—21no—
186C4×C151—42no—
187C4×C151—162no—
188C2×C151—21no—
189C2×C151—21no—
190C2×C151—81no—
191C2×C151—81no—
192C2×C151—81no—
193C2×C151—81no—
194C2×C151—82no—
195C2×C151#140212yesdirect product factor
196C2×C151C2×AΣL(1,𝔽16)11yescharacteristic
197C2×#1402C15111yesmaximal, characteristic, direct product factor, Sylow
198C151C2×#140211yescharacteristic, direct product factor, Sylow
199C2×#453C2×C15111yescharacteristic
200C2×#353C2×C15111yescharacteristic, complemented
201C2×#331C2×C15111yescharacteristic, complemented
202#1402C2×C15114yesdirect product factor
203C2×H(5,𝔽2)C4×C15111yescharacteristic, complemented
204C2×AGL(1,ℤ/8)—21no—
205C2×#127C22×C15111yescharacteristic
206#453—21no—
207#453—21no—
208#453C4×C15112yescomplemented
209#453C22×C15112yes—
210C2×C4⋊C8—21no—
211C2×#104—21no—
212C2×#100C4×C15111yescharacteristic, complemented
213C2×#99—21no—
214#353—21no—
215#353C22×C15112yescomplemented
216#331—21no—
217#331C22×C15112yescomplemented
218H(5,𝔽2)—21no—
219H(5,𝔽2)—41no—
220H(5,𝔽2)—42no—
221H(5,𝔽2)C2×C4×C15112yescomplemented
222C2×#41—21no—
223C2×#41—41no—
224C22×D4—41no—
225C22×D4—41no—
226C22×D4—41no—
227C22×D4C2×C4×C15111yescharacteristic
228C22×D4D4×C15111yescharacteristic
229AGL(1,ℤ/8)—22no—
230AGL(1,ℤ/8)—22no—
231AGL(1,ℤ/8)—42no—
232C2×SD16—41no—
233C2×D8—41no—
234C2×M16—21no—
235C22×C8—41no—
236#127—41no—
237#127C2×C4×C15111yescharacteristic
238#127D4×C15111yescharacteristic
239#127D4×C15111yescharacteristic
240#127C23×C15111yescharacteristic
241C2×C22⋊C4—41no—
242C2×C42D4×C15111yescharacteristic
243C4⋊C8—22no—
244C4⋊C8—41no—
245#104—22no—
246#104—22no—
247#100—21no—
248#100C2×C4×C15112yescomplemented
249#99—22no—
250#99—41no—
251C24—21no—
252C24—21no—
253C24—41no—
254#41—22no—
255#41—22no—
256#41—41no—
257#41—42no—
258#41—44no—
259#41—41no—
260#41—42no—
261C2×Q8—21no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—44no—
278C2×D4—42no—
279C2×D4—41no—
280C2×D4—42no—
281C2×D4—42no—
282C2×D4—82no—
283C2×D4—82no—
284C2×D4C22⋊C4×C15111yescharacteristic
285C2×D4C22⋊C4×C15111yescharacteristic
286C2×D4C22×C4×C15111yescharacteristic
287C2×D4C2×D4×C15111yescharacteristic
288C22×C4—21no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—81no—
292C22×C4C22⋊C4×C15111yescharacteristic
293SD16—42no—
294SD16—42no—
295D8—42no—
296D8—42no—
297M16—22no—
298M16—41no—
299C2×C8—41no—
300C2×C8—42no—
301C2×C8—82no—
302C22⋊C4—42no—
303C22⋊C4—81no—
304C42—21no—
305C42C22⋊C4×C15111yescharacteristic
306C42C2×D4×C15111yescharacteristic
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—42no—
312C23—41no—
313C23—41no—
314C23—41no—
315C23—42no—
316C23—41no—
317C23—41no—
318C23—82no—
319C23—82no—
320C23—162no—
321C23#99×C15111yescharacteristic
322Q8—22no—
323Q8—41no—
324D4—22no—
325D4—41no—
326D4—41no—
327D4—41no—
328D4—41no—
329D4—41no—
330D4—42no—
331D4—44no—
332D4—44no—
333D4—42no—
334D4—42no—
335D4—81no—
336D4—81no—
337D4—81no—
338D4—82no—
339D4—82no—
340D4—81no—
341D4—81no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—42no—
353C2×C4—82no—
354C2×C4—162no—
355C2×C4#99×C15111yescharacteristic
356C2×C4C2×C22⋊C4×C15111yescharacteristic
357C8—42no—
358C8—82no—
359C22—22no—
360C22—42no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22—81no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22—161no—
373C22—161no—
374C22—162no—
375C22—162no—
376C22AΣL(1,𝔽16)×C15111yescharacteristic
377C22AΣL(1,𝔽16)×C15111yescharacteristic
378C22C2×#99×C15111yescharacteristic
379C4—21no—
380C4—22no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—42no—
385C4—162no—
386C2—21no—
387C2—21no—
388C2—81no—
389C2—81no—
390C2—81no—
391C2—81no—
392C2—82no—
393C2#1402×C15112yesdirect product factor
394C2C2×AΣL(1,𝔽16)×C15111yescharacteristic
subgroupG/H
center, socleC22×C151AΣL(1,𝔽16)
commutatorC2×D4C22×C4×C151
Frattini, cosocle#127C23×C151
prime radicalC2×#453×C151C2
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C151 ◁ C23×C151 ◁ C22×C4×C151 ◁ C2×#127×C151 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#353×C151C2semidirect
3. C2×#331×C151C2semidirect
4. #1402×C151C2direct (N × H)
5. C2×H(5,𝔽2)×C151C4semidirect
10. #453×C151C4semidirect
14. C2×#100×C151C4semidirect
17. #353×C151C22semidirect
19. #331×C151C22semidirect
23. H(5,𝔽2)×C151C2×C4semidirect
50. #100×C151C2×C4semidirect
195. C2×C151#1402direct (N × H)
197. C2×#1402C151direct (N × H)
198. C151C2×#1402direct (N × H)
200. C2×#353C2×C151semidirect
201. C2×#331C2×C151semidirect
202. #1402C2×C151direct (N × H)
203. C2×H(5,𝔽2)C4×C151semidirect
208. #453C4×C151semidirect
212. C2×#100C4×C151semidirect
215. #353C22×C151semidirect
217. #331C22×C151semidirect
221. H(5,𝔽2)C2×C4×C151semidirect
248. #100C2×C4×C151semidirect
393. C2#1402×C151direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#453×C1512
C2×#353×C1512
C2×#331×C1512
#1402×C1512
C2×#1402151