Finite Group Databasebeta

← #35314799

Subgroups — #35314799

Order 21248.

420 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 83 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#24913380C211yesmaximal, characteristic
2C4×#14581709C211yesmaximal, characteristic
3#24905494C212yesmaximal, complemented
4#24905470C212yesmaximal, complemented
5C83×#1459C211yesmaximal, characteristic
6C2×#14581771C411yescharacteristic
7C2×#14581709—21no—
8C4×#13050574—21no—
9#19127631C2211yescharacteristic
10#19127629—21no—
11#19127626C411yescharacteristic
12#19127621C2211yescharacteristic
13#19127615—21no—
14#19127613C411yescharacteristic
15#19127585—21no—
16#19127574—21no—
17#19127564C411yescharacteristic
18#19127470C2212yes—
19C2×C4×D4×C83C2211yescharacteristic
20C2×C22⋊C8×C83—21no—
21#324×C83C2212yes—
22#323×C83—21no—
23C22×Dic166C2×C411yescharacteristic
24C2×#13050574—22no—
25#14581771—41no—
26#14581771—41no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C22⋊Dic83—21no—
34C2×C22⋊Dic83—21no—
35#14581709—24no—
36#14581703—41no—
37C4×C83⋊C8—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95—21no—
43C83⋊#95C2×C411yescharacteristic
44C83⋊#95C2×C411yescharacteristic
45C22×D4×C83D411yescharacteristic
46C23×C4×C83C2×C411yescharacteristic
47C23×C4×C83D411yescharacteristic
48C22×C8×C83—41no—
49C4×D4×C83—41no—
50C4×D4×C83—41no—
51C2×C4⋊C4×C83C2311yescharacteristic
52C2×C22⋊C4×C83C2×C411yescharacteristic
53C2×C22⋊C4×C83D411yescharacteristic
54C2×C42×C83D411yescharacteristic
55C22⋊C8×C83—22no—
56C22⋊C8×C83—41no—
57C2×Dic166—41no—
58C2×Dic166—81no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63#13050574—24no—
64#13050574—42no—
65C2×C83⋊C8—42no—
66C4⋊Dic83—41no—
67Dic83⋊C4—41no—
68Dic83⋊C4—41no—
69Dic83⋊C4—42no—
70Dic83⋊C4—81no—
71C22⋊Dic83—42no—
72C22⋊Dic83—42no—
73C22⋊Dic83—81no—
74C22⋊Dic83—81no—
75C24×C83C22⋊C411yescharacteristic
76C24×C83C22⋊C411yescharacteristic
77C2×D4×C83—22no—
78C2×D4×C83—22no—
79C2×D4×C83—41no—
80C22×C4×C83—21no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—81no—
85C22×C4×C83—81no—
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22×C411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C2×C8×C83—42no—
92C2×C8×C83—82no—
93C4⋊C4×C83—41no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C42×C83—21no—
97C42×C83—21no—
98C23×C83—21no—
99C23×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—41no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83#10412yes—
108C23×C83C2×C22⋊C411yescharacteristic
109D4×C83—44no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—41no—
116C2×C4×C83—42no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—81no—
121C2×C4×C83—81no—
122C2×C4×C83—81no—
123C2×C4×C83#9911yescharacteristic
124C2×C4×C83#9911yescharacteristic
125C2×C4×C83#13512yes—
126C2×Dic83—41no—
127C2×Dic83—41no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138Dic166—82no—
139C8×C83—82no—
140C83⋊C8—44no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—22no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—81no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83C2×#9911yescharacteristic
154C22×C83#41611yescharacteristic
155C22×C83C2×#10411yescharacteristic
156C4×C83—21no—
157C4×C83—21no—
158C4×C83—22no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162Dic83—81no—
163Dic83—81no—
164Dic83—81no—
165Dic83—81no—
166#11739—831nomaximal, Sylow
167C2×C83—21no—
168C2×C83—21no—
169C2×C83—42no—
170C2×C83—81no—
171C2×C83#150111yescharacteristic
172C2×C83#212111yescharacteristic
173C2×C83#213511yescharacteristic
174#2638—831no—
175C4×#100—831no—
176#1459D8311yescharacteristic
177#1354—832no—
178#1346—832no—
179C83#1173911yescharacteristic, complemented, Sylow
180C2×C22⋊Q8—831no—
181C2×C4×D4C2×D8311yescharacteristic
182C2×#100—1661no—
183C2×C22⋊C8—21no—
184C4×M16—1661no—
185#398—831no—
186#397—831no—
187#397—1661no—
188#395—831no—
189#394—1661no—
190#393—831no—
191#387—1661no—
192#385—1661no—
193#381—831no—
194#343—832no—
195#324C2×D8312yes—
196#323—21no—
197C22×Q8—831no—
198C22×D4D33211yescharacteristic
199C23×C4C4×D8311yescharacteristic
200C23×C4C22⋊D8311yescharacteristic
201C2×M16—1662no—
202C22×C8—41no—
203C22⋊Q8—3321no—
204C22⋊Q8—3321no—
205C4×D4—41no—
206C4×D4—41no—
207C2×C4⋊C4—831no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4—1661no—
210C2×C4⋊C4—1661no—
211C2×C4⋊C4C22×D8311yescharacteristic
212C2×C22⋊C4—1661no—
213C2×C22⋊C4—1661no—
214C2×C22⋊C4—1661no—
215C2×C22⋊C4—1661no—
216C2×C22⋊C4C4×D8311yescharacteristic
217C2×C22⋊C4C22⋊D8311yescharacteristic
218C2×C42D33211yescharacteristic
219#100—1664no—
220C22⋊C8—22no—
221C22⋊C8—41no—
222#97—3321no—
223C4×C8—3321no—
224#95—831no—
225#95—831no—
226#95—1661no—
227#95—1661no—
228#95—1661no—
229#95—1661no—
230#95—1661no—
231C24(C2×D83)⋊C411yescharacteristic
232C24(C2×D83)⋊C411yescharacteristic
233C2×Q8—3321no—
234C2×Q8—6641no—
235C2×D4—22no—
236C2×D4—22no—
237C2×D4—41no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—81no—
243C22×C4—81no—
244C22×C4—1661no—
245C22×C4—1661no—
246C22×C4—1661no—
247C22×C4—1661no—
248C22×C4(C2×D83)⋊C411yescharacteristic
249C22×C4(C2×D83)⋊C411yescharacteristic
250C22×C4C2×C4×D8311yescharacteristic
251C22×C4C2×C22⋊D8311yescharacteristic
252C22×C4C2×D33211yescharacteristic
253M16—3322no—
254M16—1664no—
255C2×C8—42no—
256C2×C8—82no—
257C2×C8—3322no—
258C4⋊C4—41no—
259C4⋊C4—3321no—
260C4⋊C4—3322no—
261C4⋊C4—3321no—
262C4⋊C4—3321no—
263C4⋊C4—6641no—
264C22⋊C4—41no—
265C22⋊C4—41no—
266C22⋊C4—3322no—
267C22⋊C4—3322no—
268C22⋊C4—6641no—
269C22⋊C4—6641no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23#1458171812yes—
282C23C2×(C2×D83)⋊C411yescharacteristic
283Q8—6642no—
284D4—44no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—81no—
296C2×C4—81no—
297C2×C4—81no—
298C2×C4—3321no—
299C2×C4—3321no—
300C2×C4—3321no—
301C2×C4—3321no—
302C2×C4—3321no—
303C2×C4—3321no—
304C2×C4—3321no—
305C2×C4—3321no—
306C2×C4—3321no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—3321no—
310C2×C4#1458170611yescharacteristic
311C2×C4#1458170611yescharacteristic
312C2×C4#1458181412yes—
313C8—82no—
314C8—3324no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—42no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22C2×#1458170611yescharacteristic
328C22#1912769811yescharacteristic
329C22C2×#1458171811yescharacteristic
330C4—21no—
331C4—21no—
332C4—22no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—6641no—
337C4—6641no—
338C4—6641no—
339C4—6641no—
340C2—21no—
341C2—21no—
342C2—42no—
343C2—81no—
344C2#2490595511yescharacteristic
345C2#2490910711yescharacteristic
346C2#2490915711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1459C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C4⋊C4×C83C23
hypercenter#1459D83
hypocenterC83#11739
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1459

Complements

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

NH ≅ G/Ntype
3. #24905494C2semidirect
4. #24905470C2semidirect
179. C83#11739semidirect

Maximal subgroups

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

subgroupindex
#249133802
C4×#145817092
#249054942
#249054702
C83×#14592
#1173983