Finite Group Databasebeta

← #35315440

Subgroups — #35315440

Order 21248.

472 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 99 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#24912455C211yesmaximal, characteristic
2#24905781C211yesmaximal, characteristic
3#24905462C214yesmaximal, complemented
4C83×#1799C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic, complemented
6C2×#14581779—21no—
7C2×C22⋊(C83⋊C8)—21no—
8#19127626C412yescomplemented
9#19127626C2212yes—
10#19127606—21no—
11#19127561—21no—
12#19127561—21no—
13C4×C4⋊Dic83C411yescharacteristic, complemented
14#19127427—21no—
15#19127425C2212yes—
16C2×C4×D4×C83C2211yescharacteristic
17C2×#99×C83—21no—
18#380×C83—21no—
19#342×C83C2212yes—
20C22×Dic166—21no—
21C22×C83⋊C8—41no—
22#14581792—41no—
23#14581792—41no—
24#14581779—42no—
25#14581779—42no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83C2×C412yes—
29C2×C22⋊Dic83—22no—
30C2×C22⋊Dic83—22no—
31C2×C4×Dic83—21no—
32C22⋊(C83⋊C8)—22no—
33C22⋊(C83⋊C8)—41no—
34C83⋊#95—22no—
35C83⋊#95—22no—
36C83⋊#95C2×C412yes—
37C22×D4×C83D411yescharacteristic
38C23×C4×C83D411yescharacteristic
39C23×C4×C83D411yescharacteristic
40C4×D4×C83—41no—
41C4×D4×C83—41no—
42C2×C4⋊C4×C83—41no—
43C2×C4⋊C4×C83D411yescharacteristic
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83C2311yescharacteristic
47C2×C42×C83C2×C411yescharacteristic
48#99×C83—42no—
49#95×C83—41no—
50C2×Dic166—42no—
51C2×Dic166—42no—
52C2×Dic166—42no—
53C22×Dic83—22no—
54C22×Dic83—22no—
55C2×C83⋊C8—42no—
56C2×C83⋊C8—82no—
57C4⋊Dic83—42no—
58C4⋊Dic83—41no—
59C4⋊Dic83—41no—
60C4⋊Dic83—42no—
61C22⋊Dic83—44no—
62C22⋊Dic83—42no—
63C22⋊Dic83—42no—
64C4×Dic83—42no—
65C24×C83C22⋊C411yescharacteristic
66C24×C83C2×D411yescharacteristic
67C2×D4×C83—41no—
68C2×D4×C83—41no—
69C2×D4×C83D812yes—
70C2×D4×C83SD1612yes—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—42no—
76C22×C4×C83—42no—
77C22×C4×C83—42no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22×C411yescharacteristic
82C22×C4×C83C2×D411yescharacteristic
83C4⋊C4×C83—41no—
84C4⋊C4×C83—82no—
85C22⋊C4×C83—41no—
86C22⋊C4×C83—41no—
87C22⋊C4×C83—42no—
88C22⋊C4×C83—42no—
89C42×C83—21no—
90C42×C83—21no—
91C23×C83—21no—
92C23×C83—21no—
93C23×C83—21no—
94C23×C83—21no—
95C23×C83—41no—
96C23×C83—41no—
97C23×C83—41no—
98C23×C83—41no—
99C23×C83#10212yes—
100C23×C83C2×C22⋊C411yescharacteristic
101C23×C83C2×D811yescharacteristic
102C23×C83C2×SD1611yescharacteristic
103D4×C83—42no—
104D4×C83—42no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—42no—
112C2×C4×C83—42no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—81no—
117C2×C4×C83—81no—
118C2×C4×C83—82no—
119C2×C4×C83—82no—
120C2×C4×C83—82no—
121C2×C4×C83#9911yescharacteristic
122C2×C4×C83#9911yescharacteristic
123C2×C4×C83#10212yes—
124C2×Dic83—42no—
125C2×Dic83—42no—
126C2×Dic83—42no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129C2×Dic83—42no—
130Dic166—84no—
131C83⋊C8—82no—
132C22×C83—21no—
133C22×C83—21no—
134C22×C83—21no—
135C22×C83—22no—
136C22×C83—21no—
137C22×C83—42no—
138C22×C83—41no—
139C22×C83—41no—
140C22×C83—41no—
141C22×C83—41no—
142C22×C83—41no—
143C22×C83—41no—
144C22×C83—81no—
145C22×C83—81no—
146C22×C83C2×#9911yescharacteristic
147C22×C83C2×#10211yescharacteristic
148C22×C83#42111yescharacteristic
149C4×C83—21no—
150C4×C83—22no—
151C4×C83—41no—
152C4×C83—41no—
153C4×C83—41no—
154C4×C83—162no—
155C4×C83#35412yes—
156Dic83—82no—
157Dic83—82no—
158#11878—831nomaximal, Sylow
159C2×C83—21no—
160C2×C83—21no—
161C2×C83—41no—
162C2×C83—42no—
163C2×C83—41no—
164C2×C83#151911yescharacteristic
165C2×C83#212311yescharacteristic
166C2×C83C2×#35411yescharacteristic
167#2547—831no—
168#1799D8311yescharacteristic
169#1459—831no—
170#1344—834no—
171C83#1187811yescharacteristic, complemented, Sylow
172C2×C4⋊Q8—831no—
173C2×#124—1661no—
174C2×C4×D4C2×D8311yescharacteristic
175C2×#99—21no—
176C2×C22⋊C8—1661no—
177#397—832no—
178#397—832no—
179#391—1661no—
180#380—21no—
181#380—1661no—
182#380—1661no—
183C4×C4⋊C4—831no—
184#342C2×D8312yes—
185#324—1661no—
186#323—832no—
187C22×Q8—1661no—
188C22×D4C22⋊D8311yescharacteristic
189C23×C4D33211yescharacteristic
190C23×C4C22⋊D8311yescharacteristic
191C22×C8—3321no—
192C4⋊Q8—3321no—
193C4⋊Q8—3321no—
194#124—3322no—
195#124—3322no—
196C4×D4—41no—
197C4×D4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—832no—
200C2×C4⋊C4—1661no—
201C2×C4⋊C4—1661no—
202C2×C4⋊C4D33211yescharacteristic
203C2×C22⋊C4—22no—
204C2×C22⋊C4—1662no—
205C2×C22⋊C4—1662no—
206C2×C22⋊C4C4×D8311yescharacteristic
207C2×C22⋊C4C22×D8311yescharacteristic
208C2×C42—1661no—
209C2×C42C4×D8311yescharacteristic
210#99—42no—
211C22⋊C8—3321no—
212C22⋊C8—1662no—
213#95—41no—
214#95—832no—
215#95—1662no—
216#95—1662no—
217C24(C2×D83)⋊C411yescharacteristic
218C24C2×C22⋊D8311yescharacteristic
219C2×Q8—3322no—
220C2×Q8—3322no—
221C2×Q8—3322no—
222C2×D4—41no—
223C2×D4—41no—
224C2×D4#1305057512yes—
225C2×D4#1305057712yes—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—42no—
231C22×C4—42no—
232C22×C4—42no—
233C22×C4—1662no—
234C22×C4—1662no—
235C22×C4(C2×D83)⋊C411yescharacteristic
236C22×C4(C2×D83)⋊C411yescharacteristic
237C22×C4(C2×D83)⋊C411yescharacteristic
238C22×C4C2×C4×D8311yescharacteristic
239C22×C4C2×D33211yescharacteristic
240C2×C8—3322no—
241C2×C8—6642no—
242C4⋊C4—41no—
243C4⋊C4—82no—
244C4⋊C4—3322no—
245C4⋊C4—3322no—
246C4⋊C4—3321no—
247C4⋊C4—3321no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—3322no—
253C22⋊C4—3322no—
254C22⋊C4—3324no—
255C42—21no—
256C42—21no—
257C42—3322no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—41no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23#1458171212yes—
267C23C2×(C2×D83)⋊C411yescharacteristic
268C23C2×#1305057511yescharacteristic
269C23C2×#1305057711yescharacteristic
270Q8—6644no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—3322no—
290C2×C4—3322no—
291C2×C4—3322no—
292C2×C4—3322no—
293C2×C4—3322no—
294C2×C4—3322no—
295C2×C4#1458170611yescharacteristic
296C2×C4#1458170611yescharacteristic
297C2×C4#1458171212yes—
298C8—6642no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—41no—
305C22—41no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—81no—
312C22—81no—
313C22C2×#1458170611yescharacteristic
314C22C2×#1458171211yescharacteristic
315C22#1912771911yescharacteristic
316C4—22no—
317C4—21no—
318C4—41no—
319C4—41no—
320C4—41no—
321C4—162no—
322C4—6642no—
323C4—6642no—
324C4#1912749512yes—
325C2—21no—
326C2—21no—
327C2—41no—
328C2—41no—
329C2—42no—
330C2#2490603011yescharacteristic
331C2#2490911611yescharacteristic
332C2C2×#1912749511yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C22⋊C4C22×D83
FittingC83×#1799C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C22⋊C4×C83C23
hypercenter#1799D83
hypocenterC83#11878
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #24905462C2semidirect
5. C2×#14581792C4semidirect
8. #19127626C4semidirect
13. C4×C4⋊Dic83C4semidirect
171. C83#11878semidirect

Maximal subgroups

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

subgroupindex
#249124552
#249057812
#249054622
C83×#17992
#1187883