Finite Group Databasebeta

← #35413697

Subgroups — #35413697

Order 21248.

1386 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1386 conjugacy classes of subgroups — too many to draw. Only its 235 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×#19128265C211yesmaximal, characteristic
2#24916098C212yesmaximal, complemented
3#24908311C214yesmaximal, complemented
4#24908301C214yesmaximal, complemented
5#24907391C211yesmaximal, characteristic, complemented
6#24907383C212yesmaximal, complemented
7C83×#1785C211yesmaximal, characteristic, complemented
8C23×D332C2211yescharacteristic
9#19128265—28no—
10C2×#14581788C2212yes—
11C2×#14581781C2212yes—
12C2×#14581766C414yes—
13C2×#14581760C414yes—
14C2×#14581751C2211yescharacteristic
15#19127759—28no—
16#19127724C2214yescomplemented
17C2×#14581711C2212yescomplemented
18C2×#14581709C2212yescomplemented
19C2×#14581708C2214yes—
20C2×#14581708C2214yes—
21#19127665C2212yes—
22M16×Dic83C2214yescomplemented
23C22×#41×C83C2211yescharacteristic
24C2×#101×C83C2212yescomplemented
25C2×#100×C83C2212yescomplemented
26#407×C83C2212yes—
27C24×D83C2×C412yes—
28C22×D332—24no—
29C22×D332—42no—
30C22×D332—42no—
31C22×D332C2×C412yes—
32C22×D332C2312yes—
33C22×C22⋊D83C2×C418yes—
34C2×#13050574C2314yes—
35#14581788—24no—
36#14581788—24no—
37#14581781—24no—
38#14581781—24no—
39#14581766—216no—
40#14581760—216no—
41#14581751D418yes—
42C2×C4⋊Dic83C2×C412yes—
43C2×C22⋊Dic83C2×C412yes—
44C2×(C2×D83)⋊C4C2×C418yes—
45C2×C4×Dic83C2312yes—
46Dic83⋊C8—28no—
47#14581711—42no—
48#14581709—42no—
49#14581708—44no—
50#14581708—44no—
51C22⋊(C83⋊C8)—24no—
52#14581702—24no—
53C8×Dic83—24no—
54C2×#41×C83—24no—
55C2×#41×C83—42no—
56C2×#41×C83—42no—
57C22×Q8×C83C2311yescharacteristic
58C22×D4×C83C2×C412yes—
59C22×D4×C83C2311yescharacteristic
60C23×C4×C83C2×C412yes—
61C23×C4×C83C2311yescharacteristic
62C2×M16×C83C2314yes—
63#101×C83—42no—
64#100×C83—42no—
65C22⋊C8×C83—24no—
66C23×D83—24no—
67C23×D83—42no—
68C23×D83—42no—
69C23×D83—84no—
70C23×D83C22×C414yes—
71C2×D332—24no—
72C2×D332—22no—
73C2×D332—22no—
74C2×D332—24no—
75C2×D332—24no—
76C2×D332—42no—
77C2×D332—42no—
78C2×D332—416no—
79C2×C22⋊D83—216no—
80C2×C22⋊D83—88no—
81C22×Dic83C22×C414yes—
82#13050574—24no—
83#13050574—24no—
84C2×C83⋊C8—24no—
85C4⋊Dic83C22⋊C418yes—
86C22⋊Dic83C22⋊C418yes—
87(C2×D83)⋊C4—216no—
88C4×Dic83C2×D414yes—
89C4×Dic83C2×D414yes—
90C24×C83C22×C412yes—
91C24×C83C22×C411yescharacteristic
92#41×C83—416no—
93C2×Q8×C83—22no—
94C2×Q8×C83—22no—
95C2×Q8×C83—22no—
96C2×D4×C83—24no—
97C2×D4×C83—22no—
98C2×D4×C83—22no—
99C2×D4×C83—22no—
100C2×D4×C83—42no—
101C2×D4×C83—42no—
102C22×C4×C83—22no—
103C22×C4×C83—22no—
104C22×C4×C83—24no—
105C22×C4×C83—42no—
106C22×C4×C83—42no—
107C22×C4×C83C22×C412yes—
108C22×C4×C83C22×C411yescharacteristic
109C22×C4×C83C2×D414yes—
110C22×C4×C83C2411yescharacteristic
111M16×C83—24no—
112M16×C83—24no—
113C2×C8×C83—24no—
114C23×C83—21no—
115C23×C83—21no—
116C23×C83—22no—
117C23×C83—22no—
118C23×C83—22no—
119C23×C83—42no—
120C23×C83—44no—
121C23×C83C2×C22⋊C412yes—
122C23×C83C23×C411yescharacteristic
123C22×D83—24no—
124C22×D83—24no—
125C22×D83—24no—
126C22×D83—48no—
127C22×D83—48no—
128C22×D83—88no—
129C22×D83—88no—
130C22×D83—88no—
131Q8×C83—44no—
132D4×C83—48no—
133D4×C83—44no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—28no—
137C2×C4×C83—22no—
138C2×C4×C83—22no—
139C2×C4×C83—22no—
140C2×C4×C83—48no—
141C2×C4×C83C2×C22⋊C412yes—
142C2×C4×C83C22×D412yes—
143C2×C4×C83H(5,𝔽2)12yes—
144C2×C4×C83H(5,𝔽2)12yes—
145C22⋊D83—816no—
146C2×Dic83—44no—
147C2×Dic83C2×C22⋊C418yes—
148D332—48no—
149D332—44no—
150D332—44no—
151C8×C83—44no—
152C83⋊C8—44no—
153C22×C83—22no—
154C22×C83—22no—
155C22×C83—24no—
156C22×C83—42no—
157C22×C83—42no—
158C22×C83—42no—
159C22×C83—44no—
160C22×C83—82no—
161C22×C83—82no—
162C22×C83C22×C22⋊C411yescharacteristic
163C22×C83#51811yescharacteristic
164C22×C83#51811yescharacteristic
165C2×D83—44no—
166C2×D83—48no—
167C2×D83—816no—
168C2×D83—84no—
169C2×D83—84no—
170C4×C83—22no—
171C4×C83—42no—
172C4×C83#53514yes—
173Dic83—28no—
174#21801—831nomaximal, Sylow
175C2×C83—21no—
176C2×C83—21no—
177C2×C83—42no—
178C2×C83—82no—
179C2×C83#230911yescharacteristic
180C2×C83#288812yes—
181D83—84no—
182C2×#535—831no—
183#2921—832no—
184#1967—834no—
185#1965—834no—
186#1785—831no—
187#1785D8311yescharacteristic, complemented
188#1783—832no—
189C83#2180111yescharacteristic, complemented, Sylow
190C22×#41C2×D8311yescharacteristic
191C23×D4—831no—
192#535—1668no—
193C2×#127—832no—
194C2×#124—832no—
195C2×#121—834no—
196C2×#120—834no—
197C2×#117—831no—
198#433—1668no—
199#423—834no—
200C2×#101—832no—
201C2×#101C2×D8312yescomplemented
202C2×#100—832no—
203C2×#100—834no—
204C2×#100—834no—
205C2×#100C2×D8312yescomplemented
206#407—832no—
207#407C2×D8312yes—
208C4×M16—834no—
209C25—832no—
210C2×#41—24no—
211C2×#41—42no—
212C2×#41—42no—
213C22×Q8C22×D8311yescharacteristic
214C22×D4—3322no—
215C22×D4—3322no—
216C22×D4—838no—
217C22×D4—832no—
218C22×D4—832no—
219C22×D4—1664no—
220C22×D4C4×D8312yes—
221C22×D4C22×D8311yescharacteristic
222C23×C4C4×D8312yes—
223C23×C4C22×D8311yescharacteristic
224C2×M16—834no—
225C2×M16C22×D8314yes—
226#127—1664no—
227#127—1664no—
228#124—1664no—
229#124—1664no—
230#121—16616no—
231#120—16616no—
232#117—838no—
233C2×C4⋊C4—832no—
234C2×C22⋊C4—832no—
235C2×C22⋊C4—838no—
236C2×C42—832no—
237C4⋊C8—1668no—
238#101—42no—
239#101—3322no—
240#100—42no—
241#100—3324no—
242#100—3324no—
243#100—3322no—
244C22⋊C8—24no—
245C22⋊C8—1664no—
246#97—1664no—
247C4×C8—1664no—
248C24—3322no—
249C24—3322no—
250C24—834no—
251C24—6644no—
252C24—1664no—
253C24C2×C4×D8312yes—
254C24C2×C4×D8311yescharacteristic
255#41—416no—
256C2×Q8—22no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—42no—
264C2×D4—42no—
265C2×D4—33216no—
266C2×D4—3322no—
267C2×D4—3322no—
268C2×D4—6648no—
269C2×D4—16616no—
270C2×D4—1662no—
271C2×D4—1662no—
272C2×D4—1664no—
273C2×D4—1664no—
274C2×D4—1664no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—834no—
281C22×C4C2×C4×D8312yes—
282C22×C4C2×C4×D8311yescharacteristic
283C22×C4D4×D8314yes—
284C22×C4C23×D8311yescharacteristic
285M16—24no—
286M16—24no—
287M16—1664no—
288M16—1664no—
289C2×C8—24no—
290C2×C8—1664no—
291C4⋊C4—838no—
292C22⋊C4—838no—
293C22⋊C4—16616no—
294C42—834no—
295C42—834no—
296C23—21no—
297C23—21no—
298C23—22no—
299C23—22no—
300C23—22no—
301C23—42no—
302C23—44no—
303C23—3328no—
304C23—3328no—
305C23—6648no—
306C23—6648no—
307C23—6648no—
308C23—1664no—
309C23—1664no—
310C23—1664no—
311C23C22⋊C4×D8312yes—
312C23C22×C4×D8311yescharacteristic
313Q8—44no—
314D4—48no—
315D4—44no—
316D4—3328no—
317D4—3324no—
318D4—3324no—
319D4—66416no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—28no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—48no—
327C2×C4—3324no—
328C2×C4—838no—
329C2×C4C22⋊C4×D8312yes—
330C2×C4C2×D4×D8312yes—
331C2×C4#1458183712yes—
332C2×C4#1458183712yes—
333C8—44no—
334C8—3324no—
335C22—22no—
336C22—22no—
337C22—24no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22—44no—
342C22—82no—
343C22—82no—
344C22—3324no—
345C22—3328no—
346C22—6644no—
347C22—6644no—
348C22—66416no—
349C22C2×C22⋊C4×D8311yescharacteristic
350C22#1912817411yescharacteristic
351C22#1912817411yescharacteristic
352C4—22no—
353C4—42no—
354C4—1668no—
355C4#1912826614yes—
356C2—21no—
357C2—21no—
358C2—42no—
359C2—82no—
360C2—6644no—
361C2#2491004411yescharacteristic
362C2#2491594012yes—
subgroupG/H
centerC22C2×C22⋊C4×D83
commutatorC23×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1785C2
socleC22×C83C22×C22⋊C4
prime radicalC2×#19128265C2
cosocleC22×C4×C83C24
hypercenter#1785D83
hypocenterC83#21801
LCSG ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #24916098C2semidirect
3. #24908311C2semidirect
4. #24908301C2semidirect
5. #24907391C2semidirect
6. #24907383C2semidirect
7. C83×#1785C2semidirect
16. #19127724C22semidirect
17. C2×#14581711C22semidirect
18. C2×#14581709C22semidirect
22. M16×Dic83C22semidirect
24. C2×#101×C83C22semidirect
25. C2×#100×C83C22semidirect
187. #1785D83semidirect
189. C83#21801semidirect
201. C2×#101C2×D83semidirect
205. C2×#100C2×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282652
#249160982
#249083112
#249083012
#249073912
#249073832
C83×#17852
#2180183