Finite Group Databasebeta

← #35559230

Subgroups — #35559230

Order 21248.

632 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127937C211yesmaximal, characteristic
2#24909654C214yesmaximal, complemented
3#24909648C212yesmaximal, complemented
4#24909643C212yesmaximal, complemented
5#24909252C212yesmaximal, complemented
6#24909178C212yesmaximal, complemented
7#24909074C211yesmaximal, characteristic, complemented
8C2×#427×C83C211yesmaximal, characteristic, complemented
9C22×D664C2211yescharacteristic
10C2×#14581766—21no—
11C2×#14581732C2212yes—
12#19127937—21no—
13#19127937—21no—
14#19127937C2212yes—
15#19127937C2212yes—
16C2×#14581726C2211yescharacteristic
17C2×#14581714—21no—
18#19127528C2214yescomplemented
19#19127526C2214yescomplemented
20#19127501C2214yescomplemented
21#19127501C2214yescomplemented
22C22⋊(C83⋊C16)C2212yescomplemented
23C22⋊(C83⋊C16)C2212yescomplemented
24C23×C8×C83C2211yescharacteristic
25C2×#117×C83—21no—
26#427×C83—21no—
27#427×C83—21no—
28#427×C83C2212yes—
29#427×C83C2212yescomplemented
30C2×#107×C83C2211yescharacteristic, complemented
31C2×#107×C83C2211yescharacteristic, complemented
32C2×#106×C83—21no—
33C22×D332—21no—
34C22×C22⋊D83—41no—
35C2×D664—21no—
36C2×D664—24no—
37C2×D664C2312yes—
38#14581766—22no—
39#14581766—22no—
40#14581766—41no—
41#14581766—41no—
42C2×C4⋊Dic83—21no—
43C2×(C2×D83)⋊C4—41no—
44#14581732—24no—
45C2×C83⋊C16C2314yes—
46#14581726—21no—
47#14581726C2312yes—
48#14581714—22no—
49#14581714—22no—
50C23×C4×C83D411yescharacteristic
51C22×C8×C83—24no—
52C22×C8×C83D411yescharacteristic
53C22×C8×C83D411yescharacteristic
54C22×C8×C83D411yescharacteristic
55C22×C8×C83C2311yescharacteristic
56C22×C8×C83C2311yescharacteristic
57C22×C8×C83C2311yescharacteristic
58#117×C83—22no—
59#117×C83—22no—
60#117×C83—41no—
61#117×C83—41no—
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C22⋊C4×C83—41no—
65C2×C42×C83—41no—
66#107×C83—21no—
67#107×C83—21no—
68#107×C83C2312yes—
69#107×C83C2312yes—
70#106×C83—22no—
71#106×C83—22no—
72C23×D83—41no—
73C2×D332—22no—
74C2×D332—42no—
75C2×D332—42no—
76C2×D332—41no—
77C2×C22⋊D83—42no—
78C2×C22⋊D83—42no—
79C2×C22⋊D83—84no—
80C22×Dic83—41no—
81D664—24no—
82D664—42no—
83C4⋊Dic83—22no—
84C4⋊Dic83—41no—
85(C2×D83)⋊C4—42no—
86(C2×D83)⋊C4—42no—
87C83⋊C16—24no—
88C24×C83D811yescharacteristic
89C22×C4×C83—22no—
90C22×C4×C83—22no—
91C22×C4×C83—41no—
92C22×C4×C83—41no—
93C22×C4×C83D811yescharacteristic
94C22×C4×C83D811yescharacteristic
95C22×C4×C83D811yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83C2×D411yescharacteristic
98C22×C4×C83C2×D411yescharacteristic
99C2×C8×C83—22no—
100C2×C8×C83—22no—
101C2×C8×C83—22no—
102C2×C8×C83—22no—
103C2×C8×C83—22no—
104C2×C8×C83C2×D411yescharacteristic
105C2×C8×C83C2×D411yescharacteristic
106C2×C8×C83C2×D411yescharacteristic
107C2×C8×C83#4114yes—
108C2×C8×C83C2411yescharacteristic
109C4⋊C4×C83—22no—
110C4⋊C4×C83—22no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C22⋊C4×C83—42no—
114C22⋊C4×C83—42no—
115C42×C83—42no—
116C42×C83—42no—
117C23×C83—22no—
118C23×C83—22no—
119C23×C83—41no—
120C23×C83C2×D811yescharacteristic
121C23×C83C2×D811yescharacteristic
122C23×C83C2×D811yescharacteristic
123C22×D83—42no—
124C22×D83—82no—
125C22×D83—82no—
126C22×D83—82no—
127C2×C4×C83—22no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—21no—
132C2×C4×C83—22no—
133C2×C4×C83—22no—
134C2×C4×C83—42no—
135C2×C4×C83—42no—
136C2×C4×C83—82no—
137C2×C4×C83—82no—
138C2×C4×C83#12312yes—
139C2×C4×C83C2×D811yescharacteristic
140C2×C4×C83C2×D811yescharacteristic
141C2×C4×C83C2×D811yescharacteristic
142C2×C4×C83#13712yes—
143C2×C4×C83C22×D411yescharacteristic
144C22⋊D83—84no—
145C22⋊D83—84no—
146C2×Dic83—42no—
147C2×Dic83—82no—
148D332—42no—
149D332—42no—
150D332—82no—
151C8×C83—21no—
152C8×C83—21no—
153C8×C83#12312yes—
154C8×C83C2×#4112yes—
155C22×C83—21no—
156C22×C83—21no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—21no—
160C22×C83—21no—
161C22×C83—21no—
162C22×C83—44no—
163C22×C83#48012yes—
164C22×C83#50912yes—
165C22×C83C22×D811yescharacteristic
166C2×D83—82no—
167C2×D83—82no—
168C2×D83—82no—
169C2×D83—164no—
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—82no—
173C4×C83—82no—
174C4×C83#48012yes—
175C4×C83C2×#12311yescharacteristic
176C4×C83C2×#13711yescharacteristic
177Dic83—82no—
178#33648—831nomaximal, Sylow
179C2×C83—22no—
180C2×C83—22no—
181C2×C83—41no—
182C2×C83C2×#48011yescharacteristic
183C2×C83C2×#50911yescharacteristic
184C2×C83#341411yescharacteristic
185D83—162no—
186C2×#466—831no—
187C2×#427D8311yescharacteristic, complemented
188#2234—834no—
189#2233—832no—
190#2232—832no—
191#2159—832no—
192#2140—832no—
193#2112—831no—
194C83#3364811yescharacteristic, complemented, Sylow
195C22×D8—831no—
196C23×C8C2×D8311yescharacteristic
197C2×#121—1661no—
198C2×#117—21no—
199C2×M32—832no—
200#466—832no—
201#466—832no—
202#466—1661no—
203#466—1661no—
204#427—21no—
205#427—21no—
206#427C2×D8312yes—
207#427C2×D8312yescomplemented
208C2×#107—831no—
209C2×#107C2×D8311yescharacteristic, complemented
210C2×#107C2×D8311yescharacteristic, complemented
211C2×#106—21no—
212C2×#102—1661no—
213#367—834no—
214#366—834no—
215#357—834no—
216#357—834no—
217C22⋊C16—832no—
218C22⋊C16—832no—
219C22×D4—3321no—
220C22×D4—1661no—
221C23×C4C22⋊D8311yescharacteristic
222C2×D8—832no—
223C2×D8—1661no—
224C2×D8—1664no—
225C22×C8—24no—
226C22×C8C22⋊D8311yescharacteristic
227C22×C8C22⋊D8311yescharacteristic
228C22×C8C22⋊D8311yescharacteristic
229C22×C8C22×D8311yescharacteristic
230C22×C8C22×D8311yescharacteristic
231C22×C8C22×D8311yescharacteristic
232#121—3321no—
233#121—3321no—
234#121—1662no—
235#121—1662no—
236#117—22no—
237#117—22no—
238#117—41no—
239#117—41no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—1661no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—3321no—
245C2×C42—41no—
246M32—1664no—
247C2×C16—834no—
248#107—21no—
249#107—21no—
250#107—832no—
251#107—1661no—
252#107C22×D8312yes—
253#107C22×D8312yes—
254#106—22no—
255#106—22no—
256#102—1662no—
257#102—1662no—
258C24—3321no—
259C24#1305057511yescharacteristic
260C2×D4—3322no—
261C2×D4—3322no—
262C2×D4—3322no—
263C2×D4—3321no—
264C2×D4—3322no—
265C2×D4—6644no—
266C2×D4—1662no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—3321no—
272C22×C4#1305057511yescharacteristic
273C22×C4#1305057511yescharacteristic
274C22×C4#1305057511yescharacteristic
275C22×C4C2×C22⋊D8311yescharacteristic
276C22×C4C2×C22⋊D8311yescharacteristic
277C22×C4C2×C22⋊D8311yescharacteristic
278D8—3322no—
279D8—1664no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8C2×C22⋊D8311yescharacteristic
286C2×C8C2×C22⋊D8311yescharacteristic
287C2×C8C2×C22⋊D8311yescharacteristic
288C2×C8#1305059014yes—
289C2×C8C23×D8311yescharacteristic
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—41no—
293C4⋊C4—41no—
294C4⋊C4—3321no—
295C4⋊C4—1662no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—3322no—
299C22⋊C4—3322no—
300C42—42no—
301C42—42no—
302C16—1664no—
303C23—22no—
304C23—22no—
305C23—41no—
306C23—3322no—
307C23—6642no—
308C23—6642no—
309C23—6642no—
310C23C2×#1305057511yescharacteristic
311C23C2×#1305057511yescharacteristic
312C23C2×#1305057511yescharacteristic
313D4—3322no—
314D4—3322no—
315D4—6644no—
316D4—6644no—
317D4—6642no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—3322no—
330C2×C4—6642no—
331C2×C4#1458177512yes—
332C2×C4C2×#1305057511yescharacteristic
333C2×C4C2×#1305057511yescharacteristic
334C2×C4C2×#1305057511yescharacteristic
335C2×C4#1458182312yes—
336C2×C4C22×C22⋊D8311yescharacteristic
337C8—21no—
338C8—21no—
339C8#1458177512yes—
340C8C2×#1305059012yes—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—44no—
349C22—13284no—
350C22—6642no—
351C22—6642no—
352C22—6642no—
353C22#1912801412yes—
354C22#1912813612yes—
355C22C22×#1305057511yescharacteristic
356C4—21no—
357C4—21no—
358C4—82no—
359C4—82no—
360C4—6642no—
361C4#1912801412yes—
362C4C2×#1458177511yescharacteristic
363C4C2×#1458182311yescharacteristic
364C2—22no—
365C2—22no—
366C2—41no—
367C2—13282no—
368C2C2×#1912801411yescharacteristic
369C2C2×#1912813611yescharacteristic
370C2#2492059711yescharacteristic
subgroupG/H
centerC22C22×#13050575
commutator, cosocleC2×C8×C83C24
FrattiniC2×C8C23×D83
FittingC2×#427×C83C2
socleC22×C83C22×D8
prime radicalC2×#19127937C2
hypercenterC2×#427D83
hypocenterC83#33648
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
2. #24909654C2semidirect
3. #24909648C2semidirect
4. #24909643C2semidirect
5. #24909252C2semidirect
6. #24909178C2semidirect
7. #24909074C2semidirect
8. C2×#427×C83C2semidirect
18. #19127528C22semidirect
19. #19127526C22semidirect
20. #19127501C22semidirect
21. #19127501C22semidirect
22. C22⋊(C83⋊C16)C22semidirect
23. C22⋊(C83⋊C16)C22semidirect
29. #427×C83C22semidirect
30. C2×#107×C83C22semidirect
31. C2×#107×C83C22semidirect
187. C2×#427D83semidirect
194. C83#33648semidirect
207. #427C2×D83semidirect
209. C2×#107C2×D83semidirect
210. C2×#107C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191279372
#249096542
#249096482
#249096432
#249092522
#249091782
#249090742
C2×#427×C832
#3364883