Finite Group Databasebeta

← #35314305

Subgroups — #35314305

Order 21248.

306 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 89 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#24916437C211yesmaximal, characteristic
2#24909264C211yesmaximal, characteristic, complemented
3#24909217C211yesmaximal, characteristic, complemented
4#24905445C211yesmaximal, characteristic, complemented
5#24905422C211yesmaximal, characteristic, complemented
6#24905420C211yesmaximal, characteristic, complemented
7C83×#1338C211yesmaximal, characteristic, complemented
8#19128238C411yescharacteristic
9C2×#14581731—21no—
10Q8×C83⋊C8C411yescharacteristic
11#19127780C2211yescharacteristic
12#19127774—21no—
13#19127766—21no—
14#19127766C411yescharacteristic
15#19127761—21no—
16#19127757C411yescharacteristic
17#19127658—21no—
18C4×C8×D83C2211yescharacteristic
19C4⋊(C83⋊C16)C2211yescharacteristic
20Dic83⋊C16—21no—
21Dic83⋊C16—21no—
22(C2×D83)⋊C16—21no—
23#19127477C2211yescharacteristic
24#446×C83C2211yescharacteristic
25C4⋊C16×C83C2211yescharacteristic
26#346×C83C2211yescharacteristic
27C2×#13050573—21no—
28C2×C8×D83—21no—
29C2×C8×D83D411yescharacteristic
30C2×C8×D83D411yescharacteristic
31#14581787—21no—
32#14581783C811yescharacteristic
33#14581781C811yescharacteristic
34#14581777—21no—
35#14581768C811yescharacteristic
36#14581767C811yescharacteristic
37C4×D332—21no—
38C4×D332C2×C411yescharacteristic
39#14581749—21no—
40C42×D83C2×C411yescharacteristic
41#14581731—42no—
42C2×C83⋊C16—21no—
43C2×C83⋊C16—21no—
44C4⋊(C83⋊C8)—21no—
45C4⋊(C83⋊C8)C2×C411yescharacteristic
46(C2×D83)⋊C8—21no—
47(C2×D83)⋊C8—21no—
48(C2×D83)⋊C8—21no—
49#14581702—21no—
50C4×C83⋊C8—21no—
51C4×C83⋊C8C2×C411yescharacteristic
52C8×Dic83D411yescharacteristic
53C8×Dic83D411yescharacteristic
54C4×Q8×C83C2×C411yescharacteristic
55C2×C16×C83—21no—
56C2×C16×C83—21no—
57C4⋊C8×C83—21no—
58C4⋊C8×C83C2×C411yescharacteristic
59#97×C83—21no—
60C4×C8×C83C2311yescharacteristic
61C2×D332—21no—
62C2×D332C2×C811yescharacteristic
63C2×C4×D83—21no—
64C2×C4×D83C22⋊C411yescharacteristic
65C2×C4×D83C2×C811yescharacteristic
66C2×C4×D83M1611yescharacteristic
67#13050573—41no—
68#13050573—41no—
69C2×C83⋊C8—21no—
70C2×C83⋊C8C22⋊C411yescharacteristic
71C2×C83⋊C8C22⋊C411yescharacteristic
72C8×D83—21no—
73C8×D83—21no—
74C8×D83—41no—
75C8×D83—42no—
76C4⋊Dic83—21no—
77C4⋊Dic83C2×C811yescharacteristic
78Dic83⋊C4—21no—
79Dic83⋊C4—21no—
80(C2×D83)⋊C4—21no—
81(C2×D83)⋊C4—21no—
82(C2×D83)⋊C4—21no—
83(C2×D83)⋊C4—21no—
84(C2×D83)⋊C4—21no—
85C4×Dic83C22⋊C411yescharacteristic
86C4×Dic83C2×C811yescharacteristic
87C4×Dic83M1611yescharacteristic
88C83⋊C16—41no—
89C83⋊C16—41no—
90C2×Q8×C83C2×C811yescharacteristic
91C2×C8×C83—21no—
92C2×C8×C83C2×D411yescharacteristic
93C2×C8×C83C2×D411yescharacteristic
94C4⋊C4×C83—21no—
95C4⋊C4×C83C2×C811yescharacteristic
96C42×C83—21no—
97C42×C83C22×C411yescharacteristic
98C16×C83—41no—
99C16×C83—41no—
100C22×D83—21no—
101C22×D83C22⋊C811yescharacteristic
102Q8×C83—41no—
103C2×C4×C83—21no—
104C2×C4×C83—21no—
105C2×C4×C83C2×C22⋊C411yescharacteristic
106C2×C4×C83C22×C811yescharacteristic
107C2×C4×C83C2×M1611yescharacteristic
108C2×Dic83—21no—
109C2×Dic83C22⋊C811yescharacteristic
110C2×Dic83C22⋊C811yescharacteristic
111C2×Dic83C22⋊C811yescharacteristic
112D332—41no—
113D332—42no—
114C4×D83—21no—
115C4×D83—21no—
116C4×D83—41no—
117C4×D83—41no—
118C4×D83—41no—
119C4×D83—41no—
120C8×C83—21no—
121C8×C83—41no—
122C8×C83AGL(1,ℤ/8)11yescharacteristic
123C8×C83#13711yescharacteristic
124C83⋊C8—21no—
125C83⋊C8—21no—
126C83⋊C8—41no—
127C22×C83C2×C22⋊C811yescharacteristic
128C2×D83—21no—
129C2×D83—21no—
130C2×D83—21no—
131C2×D83—41no—
132C2×D83—41no—
133C2×D83—41no—
134C4×C83—21no—
135C4×C83—21no—
136C4×C83—41no—
137C4×C83—41no—
138C4×C83#41711yescharacteristic
139C4×C83#42111yescharacteristic
140Dic83—21no—
141Dic83—21no—
142Dic83—21no—
143Dic83—41no—
144#11627—831nomaximal, Sylow
145C2×C83#147711yescharacteristic
146C2×C83#211511yescharacteristic
147C2×C83#211711yescharacteristic
148D83—41no—
149D83—81no—
150#2966—831no—
151#2163—831no—
152#2150—831no—
153#1338—831no—
154#1338D8311yescharacteristic, complemented
155#1330—831no—
156#1330—831no—
157C83#1162711yescharacteristic, complemented, Sylow
158#529—831no—
159C2×M32—1661no—
160#446C2×D8311yescharacteristic
161C8×Q8—831no—
162#436—831no—
163#435—1661no—
164C8×D4—831no—
165C8×D4—1661no—
166#433—1661no—
167#432—831no—
168#405—1661no—
169C2×C4×C8—831no—
170C4⋊C16—831no—
171C4⋊C16—1661no—
172C4⋊C16—1661no—
173C4⋊C16C2×D8311yescharacteristic
174C22⋊C16—1661no—
175#346—831no—
176#346C2×D8311yescharacteristic
177C2×M16—1661no—
178C22×C8—831no—
179C22×C8—831no—
180C22×C8—1661no—
181#126—1661no—
182#125—831no—
183#124—831no—
184#123—1661no—
185C22⋊Q8—831no—
186#121—831no—
187C4×Q8C4×D8311yescharacteristic
188C4×D4—831no—
189C4×D4—1661no—
190#117—1661no—
191C2×C42—831no—
192M32—3322no—
193C2×C16—21no—
194C2×C16—21no—
195C2×C16—1661no—
196C2×C16—1661no—
197C4⋊C8—21no—
198C4⋊C8—831no—
199C4⋊C8—1661no—
200C4⋊C8C4×D8311yescharacteristic
201C22⋊C8—1661no—
202C22⋊C8—1661no—
203C22⋊C8—1661no—
204#97—21no—
205#97—1661no—
206C4×C8—831no—
207C4×C8—831no—
208C4×C8—831no—
209C4×C8—1661no—
210C4×C8C22×D8311yescharacteristic
211C2×Q8C8×D8311yescharacteristic
212C2×D4—831no—
213C2×D4—1661no—
214C22×C4—831no—
215C22×C4—831no—
216C22×C4—831no—
217C22×C4—1661no—
218M16—3321no—
219M16—3321no—
220C2×C8—21no—
221C2×C8—3321no—
222C2×C8—3322no—
223C2×C8—831no—
224C2×C8—831no—
225C2×C8—1661no—
226C2×C8—1661no—
227C2×C8—1661no—
228C2×C8D4×D8311yescharacteristic
229C2×C8D4×D8311yescharacteristic
230C4⋊C4—21no—
231C4⋊C4—831no—
232C4⋊C4—1661no—
233C4⋊C4—1661no—
234C4⋊C4—1661no—
235C4⋊C4C8×D8311yescharacteristic
236C22⋊C4—1661no—
237C22⋊C4—1661no—
238C22⋊C4—1661no—
239C22⋊C4—1661no—
240C22⋊C4—1661no—
241C42—21no—
242C42—831no—
243C42—831no—
244C42—831no—
245C42C2×C4×D8311yescharacteristic
246C16—41no—
247C16—41no—
248C16—3321no—
249C16—3321no—
250C23—831no—
251C23—1661no—
252Q8—41no—
253D4—3321no—
254D4—3322no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—3321no—
258C2×C4—3321no—
259C2×C4—3321no—
260C2×C4—3321no—
261C2×C4—831no—
262C2×C4—831no—
263C2×C4—831no—
264C2×C4—1661no—
265C2×C4—1661no—
266C2×C4—1661no—
267C2×C4C22⋊C4×D8311yescharacteristic
268C2×C4C2×C8×D8311yescharacteristic
269C2×C4M16×D8311yescharacteristic
270C8—21no—
271C8—41no—
272C8—3321no—
273C8—1661no—
274C8—1661no—
275C8#1458181911yescharacteristic
276C8#1458182411yescharacteristic
277C22—3321no—
278C22—3321no—
279C22—3321no—
280C22—1661no—
281C22—1661no—
282C22—1661no—
283C22C22⋊C8×D8311yescharacteristic
284C4—21no—
285C4—21no—
286C4—41no—
287C4—41no—
288C4—3321no—
289C4—1661no—
290C4—1661no—
291C4—1661no—
292C4#1912770111yescharacteristic
293C4#1912771611yescharacteristic
294C2—3321no—
295C2—6641no—
296C2#2490584911yescharacteristic
297C2#2490908211yescharacteristic
298C2#2490908911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC2×C4×C83C22×C8
FrattiniC4×C8C22×D83
FittingC83×#1338C2
socleC22×C83C2×C22⋊C8
prime radical#14581767C8
cosocleC4×C8×C83C23
hypercenter#1338D83
hypocenterC83#11627
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1338

Complements

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

NH ≅ G/Ntype
2. #24909264C2semidirect
3. #24909217C2semidirect
4. #24905445C2semidirect
5. #24905422C2semidirect
6. #24905420C2semidirect
7. C83×#1338C2semidirect
154. #1338D83semidirect
157. C83#11627semidirect

Maximal subgroups

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

subgroupindex
#249164372
#249092642
#249092172
#249054452
#249054222
#249054202
C83×#13382
#1162783