Finite Group Databasebeta

← #35313646

Subgroups — #35313646

Order 21248.

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

normal subgroups only

This group has 392 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#24916160C211yesmaximal, characteristic
2#24909072C211yesmaximal, characteristic, complemented
3C22⋊C16×D83C211yesmaximal, characteristic, complemented
4#24905404C211yesmaximal, characteristic, complemented
5#24905404C211yesmaximal, characteristic, complemented
6#24905389C211yesmaximal, characteristic, complemented
7C83×#1315C211yesmaximal, characteristic, complemented
8C2×#14581799C411yescharacteristic
9C22×C8×D83C2211yescharacteristic
10#19128208C411yescharacteristic
11C2×#14581731—21no—
12C2×C16×D83—21no—
13#19127775—21no—
14#19127775—21no—
15(C22⋊D83)⋊C8—21no—
16(C22⋊D83)⋊C8—21no—
17#19127727C2211yescharacteristic
18#19127664C411yescharacteristic
19#19127659C411yescharacteristic
20C22⋊(C83⋊C16)C2211yescharacteristic
21C22⋊(C83⋊C16)C2211yescharacteristic
22(C2×D83)⋊C16—21no—
23(C2×D83)⋊C16—21no—
24#408×C83C2211yescharacteristic
25C22⋊C16×C83C2211yescharacteristic
26C22⋊C16×C83C2211yescharacteristic
27C22×C4×D83C2×C411yescharacteristic
28#14581799—42no—
29C2×#13050574—21no—
30C2×#13050574C2×C411yescharacteristic
31C22×C83⋊C8—21no—
32C22×C83⋊C8C2×C411yescharacteristic
33C2×C8×D83—42no—
34C2×C8×D83D411yescharacteristic
35C2×C8×D83D411yescharacteristic
36C2×C8×D83D411yescharacteristic
37C2×C8×D83D411yescharacteristic
38#14581773—21no—
39#14581768C811yescharacteristic
40#14581768C811yescharacteristic
41#14581763C811yescharacteristic
42#14581763C811yescharacteristic
43#14581760—21no—
44C4×C22⋊D83—21no—
45C4×C22⋊D83—21no—
46#14581751C2×C411yescharacteristic
47#14581731—42no—
48C2×C83⋊C16—21no—
49C2×C83⋊C16—21no—
50C16×D83—42no—
51Dic83⋊C8—21no—
52Dic83⋊C8—21no—
53(C2×D83)⋊C8—21no—
54(C2×D83)⋊C8—21no—
55#14581702—21no—
56C8×Dic83—21no—
57C2×#41×C83C2×C411yescharacteristic
58C2×M16×C83C2×C411yescharacteristic
59C22×C8×C83C2311yescharacteristic
60C2×C16×C83—21no—
61C2×C16×C83—21no—
62C22⋊C8×C83—21no—
63C22⋊C8×C83—21no—
64C23×D83M1611yescharacteristic
65C2×C22⋊D83—21no—
66C2×C22⋊D83—21no—
67C22×Dic83M1611yescharacteristic
68C2×C4×D83—41no—
69C2×C4×D83—41no—
70C2×C4×D83C22⋊C411yescharacteristic
71C2×C4×D83C22⋊C411yescharacteristic
72C2×C4×D83C2×C811yescharacteristic
73C2×C4×D83C2×C811yescharacteristic
74#13050574—42no—
75#13050574—41no—
76C2×C83⋊C8—22no—
77C2×C83⋊C8—21no—
78C2×C83⋊C8—42no—
79C2×C83⋊C8C22⋊C411yescharacteristic
80C2×C83⋊C8C22⋊C411yescharacteristic
81C8×D83—22no—
82C8×D83—22no—
83C8×D83—41no—
84C8×D83—41no—
85C4⋊Dic83C2×C811yescharacteristic
86C4⋊Dic83C2×C811yescharacteristic
87Dic83⋊C4—21no—
88Dic83⋊C4—21no—
89C22⋊Dic83—21no—
90C22⋊Dic83—21no—
91C22⋊Dic83—21no—
92(C2×D83)⋊C4—21no—
93(C2×D83)⋊C4—21no—
94C4×Dic83—21no—
95C83⋊C16—41no—
96C83⋊C16—41no—
97#41×C83—41no—
98#41×C83—41no—
99C2×Q8×C83C2×C811yescharacteristic
100C2×D4×C83—21no—
101C2×D4×C83C2×C811yescharacteristic
102C22×C4×C83—21no—
103C22×C4×C83C22×C411yescharacteristic
104M16×C83—41no—
105C2×C8×C83—21no—
106C2×C8×C83—41no—
107C2×C8×C83C2×D411yescharacteristic
108C2×C8×C83C2×D411yescharacteristic
109C16×C83—41no—
110C16×C83—41no—
111C23×C83—21no—
112C23×C83C2×M1611yescharacteristic
113C22×D83—22no—
114C22×D83—41no—
115C22×D83—41no—
116C22×D83C22⋊C811yescharacteristic
117C22×D83C22⋊C811yescharacteristic
118Q8×C83—41no—
119D4×C83—41no—
120D4×C83—42no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83C2×C22⋊C411yescharacteristic
127C2×C4×C83C22×C811yescharacteristic
128C22⋊D83—42no—
129C22⋊D83—42no—
130C2×Dic83—21no—
131C2×Dic83—21no—
132C2×Dic83—41no—
133C2×Dic83C22⋊C811yescharacteristic
134C2×Dic83C22⋊C811yescharacteristic
135C4×D83—21no—
136C4×D83—21no—
137C4×D83—21no—
138C4×D83—21no—
139C4×D83—41no—
140C4×D83—41no—
141C8×C83—21no—
142C8×C83—21no—
143C8×C83—41no—
144C83⋊C8—21no—
145C83⋊C8—41no—
146C83⋊C8#9912yes—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—21no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83C2×C22⋊C811yescharacteristic
154C2×D83—21no—
155C2×D83—21no—
156C2×D83—21no—
157C2×D83—21no—
158C2×D83—21no—
159C2×D83—44no—
160C2×D83#35012yes—
161C4×C83—21no—
162C4×C83—41no—
163C4×C83C2×#9911yescharacteristic
164C4×C83#41311yescharacteristic
165Dic83—21no—
166Dic83—21no—
167Dic83—41no—
168Dic83—41no—
169#11470—831nomaximal, Sylow
170C2×C83—41no—
171C2×C83—81no—
172C2×C83#145911yescharacteristic
173C2×C83C2×#35011yescharacteristic
174C2×C83#211711yescharacteristic
175D83—22no—
176D83—41no—
177#2929—831no—
178#2112—831no—
179C2×C22⋊C16—831no—
180#1323—831no—
181#1323—831no—
182#1315—831no—
183#1315D8311yescharacteristic, complemented
184C83#1147011yescharacteristic, complemented, Sylow
185C2×#131—831no—
186C23×C8—831no—
187#525—831no—
188C2×M32—1661no—
189C22×C16—1661no—
190#435—1661no—
191#435—1661no—
192C8×D4—1661no—
193C8×D4—1661no—
194#424—831no—
195#408C2×D8311yescharacteristic
196#407—831no—
197#405—831no—
198C22⋊C16—831no—
199C22⋊C16—831no—
200C22⋊C16—1661no—
201C22⋊C16—1661no—
202C22⋊C16C2×D8311yescharacteristic
203C22⋊C16C2×D8311yescharacteristic
204C2×#41C4×D8311yescharacteristic
205C23×C4—831no—
206#131—3322no—
207C2×M16—831no—
208C2×M16—1661no—
209C2×M16C4×D8311yescharacteristic
210C22×C8—3322no—
211C22×C8—831no—
212C22×C8—831no—
213C22×C8—831no—
214C22×C8—831no—
215C22×C8—831no—
216C22×C8—1661no—
217C22×C8C22×D8311yescharacteristic
218#123—1661no—
219C22⋊Q8—831no—
220C22⋊Q8—831no—
221#121—831no—
222#121—831no—
223#120—1661no—
224C4×D4—1661no—
225C4×D4—1661no—
226#117—831no—
227M32—3322no—
228C2×C16—21no—
229C2×C16—21no—
230C2×C16—3322no—
231C2×C16—1661no—
232C2×C16—1661no—
233C4⋊C8—1661no—
234C4⋊C8—1661no—
235C22⋊C8—21no—
236C22⋊C8—21no—
237C22⋊C8—1661no—
238C22⋊C8—1661no—
239#97—1661no—
240C4×C8—1661no—
241C24—831no—
242#41—41no—
243#41—41no—
244C2×Q8C8×D8311yescharacteristic
245C2×D4—21no—
246C2×D4—1661no—
247C2×D4—1661no—
248C2×D4C8×D8311yescharacteristic
249C22×C4—21no—
250C22×C4—3321no—
251C22×C4—3321no—
252C22×C4—831no—
253C22×C4—831no—
254C22×C4—831no—
255C22×C4—831no—
256C22×C4—831no—
257C22×C4C2×C4×D8311yescharacteristic
258M16—41no—
259M16—3322no—
260M16—3321no—
261C2×C8—21no—
262C2×C8—41no—
263C2×C8—3321no—
264C2×C8—3321no—
265C2×C8—3322no—
266C2×C8—831no—
267C2×C8—831no—
268C2×C8—1662no—
269C2×C8—1662no—
270C2×C8—1661no—
271C2×C8—1662no—
272C2×C8D4×D8311yescharacteristic
273C2×C8D4×D8311yescharacteristic
274C4⋊C4—831no—
275C4⋊C4—831no—
276C4⋊C4—1661no—
277C4⋊C4—1661no—
278C22⋊C4—1661no—
279C22⋊C4—1661no—
280C22⋊C4—1661no—
281C22⋊C4—1661no—
282C22⋊C4—1661no—
283C42—1661no—
284C16—41no—
285C16—41no—
286C16—3321no—
287C16—3321no—
288C23—21no—
289C23—3321no—
290C23—3321no—
291C23—831no—
292C23—831no—
293C23—1662no—
294C23M16×D8311yescharacteristic
295Q8—41no—
296D4—41no—
297D4—42no—
298D4—3322no—
299D4—3322no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—3321no—
306C2×C4—3321no—
307C2×C4—3321no—
308C2×C4—831no—
309C2×C4—831no—
310C2×C4—1661no—
311C2×C4—1661no—
312C2×C4—1661no—
313C2×C4—1661no—
314C2×C4—1661no—
315C2×C4—1661no—
316C2×C4C22⋊C4×D8311yescharacteristic
317C2×C4C2×C8×D8311yescharacteristic
318C8—21no—
319C8—21no—
320C8—41no—
321C8—3321no—
322C8—832no—
323C8—1661no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—3324no—
331C22—832no—
332C22—1661no—
333C22—1661no—
334C22—1661no—
335C22—1661no—
336C22—1661no—
337C22C22⋊C8×D8311yescharacteristic
338C4—21no—
339C4—41no—
340C4—3321no—
341C4—3321no—
342C4—1661no—
343C4—1661no—
344C4#99×D8311yescharacteristic
345C4#1912768211yescharacteristic
346C2—41no—
347C2—81no—
348C2—3321no—
349C2—1662no—
350C2#2490577811yescharacteristic
351C2#350×D8311yescharacteristic
352C2#2490908911yescharacteristic
subgroupG/H
centerC2×C4C22⋊C4×D83
commutatorC2×C4×C83C22×C8
FrattiniC22×C8C22×D83
FittingC83×#1315C2
socleC22×C83C2×C22⋊C8
prime radical#19128208C4
cosocleC22×C8×C83C23
hypercenter#1315D83
hypocenterC83#11470
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1315

Complements

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

NH ≅ G/Ntype
2. #24909072C2semidirect
3. C22⋊C16×D83C2semidirect
4. #24905404C2semidirect
5. #24905404C2semidirect
6. #24905389C2semidirect
7. C83×#1315C2semidirect
183. #1315D83semidirect
184. C83#11470semidirect

Maximal subgroups

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

subgroupindex
#249161602
#249090722
C22⋊C16×D832
#249054042
#249053892
C83×#13152
#1147083