Finite Group Databasebeta

← #35315213

Subgroups — #35315213

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#24910722C211yesmaximal, characteristic
2#24905801C211yesmaximal, characteristic
3#24905489C212yesmaximal, complemented
4#24905468C212yesmaximal, complemented
5C83×#1800C211yesmaximal, characteristic
6C2×#14581751C411yescharacteristic
7#19127665—21no—
8#19127640C2211yescharacteristic
9#19127637C411yescharacteristic
10#19127631C2211yescharacteristic
11#19127629—21no—
12#19127626C411yescharacteristic
13#19127615—21no—
14C4×Dic83⋊C4—21no—
15C4×C22⋊Dic83—21no—
16C4×C22⋊Dic83C411yescharacteristic
17#19127427C2212yes—
18#19127425—21no—
19C2×C4×D4×C83C2211yescharacteristic
20#423×C83—21no—
21C2×#100×C83—21no—
22#343×C83C2212yes—
23C2×#13050574—41no—
24#14581751—41no—
25#14581751—41no—
26C2×C4⋊Dic83C2×C411yescharacteristic
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×C22⋊Dic83—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C22⋊Dic83—21no—
34C2×C4×Dic83—21no—
35C2×C4×Dic83—21no—
36C22⋊(C83⋊C8)—22no—
37C22⋊(C83⋊C8)—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C22×D4×C83D411yescharacteristic
45C23×C4×C83C2×C411yescharacteristic
46C23×C4×C83D411yescharacteristic
47C2×M16×C83—22no—
48C4×D4×C83—41no—
49C4×D4×C83—41no—
50C2×C4⋊C4×C83C2311yescharacteristic
51C2×C22⋊C4×C83C2×C411yescharacteristic
52C2×C22⋊C4×C83D411yescharacteristic
53C2×C42×C83D411yescharacteristic
54C4⋊C8×C83—41no—
55C4⋊C8×C83—41no—
56#100×C83—42no—
57C22×Dic83—21no—
58C22×Dic83—21no—
59C22×Dic83—21no—
60C22×Dic83—21no—
61#13050574—82no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—41no—
65Dic83⋊C4—42no—
66Dic83⋊C4—41no—
67Dic83⋊C4—41no—
68C22⋊Dic83—42no—
69C22⋊Dic83—42no—
70C22⋊Dic83—42no—
71C22⋊Dic83—42no—
72C4×Dic83—42no—
73C4×Dic83—41no—
74C4×Dic83—41no—
75C24×C83C22⋊C411yescharacteristic
76C24×C83C22⋊C411yescharacteristic
77C2×D4×C83—21no—
78C2×D4×C83—21no—
79C2×D4×C83—41no—
80C2×D4×C83—41no—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83—21no—
84C22×C4×C83—21no—
85C22×C4×C83—42no—
86C22×C4×C83—42no—
87C22×C4×C83C22⋊C411yescharacteristic
88C22×C4×C83C22⋊C411yescharacteristic
89C22×C4×C83C22×C411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92M16×C83—42no—
93M16×C83—42no—
94C2×C8×C83—42no—
95C4⋊C4×C83—41no—
96C22⋊C4×C83—41no—
97C22⋊C4×C83—41no—
98C42×C83—21no—
99C42×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—21no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83—41no—
110C23×C83C2×C22⋊C411yescharacteristic
111D4×C83—42no—
112D4×C83—42no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—22no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—21no—
119C2×C4×C83—42no—
120C2×C4×C83—42no—
121C2×C4×C83—42no—
122C2×C4×C83—41no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—81no—
126C2×C4×C83#9911yescharacteristic
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
129C2×C4×C83#13711yescharacteristic
130C2×Dic83—22no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138C2×Dic83—41no—
139C2×Dic83—41no—
140C2×Dic83—41no—
141C2×Dic83—41no—
142C8×C83—82no—
143C83⋊C8—82no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83—42no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—41no—
153C22×C83—81no—
154C22×C83—81no—
155C22×C83—81no—
156C22×C83C2×#9911yescharacteristic
157C22×C83#41711yescharacteristic
158C22×C83#42111yescharacteristic
159C4×C83—21no—
160C4×C83—21no—
161C4×C83—21no—
162C4×C83—41no—
163C4×C83—41no—
164C4×C83—41no—
165C4×C83#35212yes—
166Dic83—42no—
167Dic83—81no—
168Dic83—81no—
169Dic83—81no—
170#11830—831nomaximal, Sylow
171C2×C83—21no—
172C2×C83—21no—
173C2×C83—81no—
174C2×C83—81no—
175C2×C83#151311yescharacteristic
176C2×C83C2×#35211yescharacteristic
177C2×C83#213211yescharacteristic
178#2395—831no—
179#1800D8311yescharacteristic
180#1465—831no—
181#1353—832no—
182#1346—832no—
183C83#1183011yescharacteristic, complemented, Sylow
184C2×C4×D4C2×D8311yescharacteristic
185C2×#117—831no—
186#423—21no—
187C2×#100—21no—
188#407—1661no—
189#400—831no—
190#399—831no—
191#398—831no—
192#397—831no—
193#397—1661no—
194#394—1661no—
195C4×C4⋊C4—1661no—
196C4×C22⋊C4—831no—
197C4×C22⋊C4—1661no—
198#343C2×D8312yes—
199#324—832no—
200#323—1661no—
201C22×D4C22⋊D8311yescharacteristic
202C23×C4C4×D8311yescharacteristic
203C23×C4D33211yescharacteristic
204C2×M16—22no—
205C2×M16—3321no—
206C4×D4—41no—
207C4×D4—41no—
208#117—3321no—
209#117—3321no—
210C2×C4⋊C4—831no—
211C2×C4⋊C4—831no—
212C2×C4⋊C4—1661no—
213C2×C4⋊C4—1661no—
214C2×C4⋊C4C22×D8311yescharacteristic
215C2×C22⋊C4—1661no—
216C2×C22⋊C4—1661no—
217C2×C22⋊C4—1661no—
218C2×C22⋊C4—1661no—
219C2×C22⋊C4C4×D8311yescharacteristic
220C2×C22⋊C4D33211yescharacteristic
221C2×C42—1661no—
222C2×C42—1661no—
223C2×C42C22⋊D8311yescharacteristic
224C4⋊C8—41no—
225C4⋊C8—41no—
226#100—42no—
227C22⋊C8—3321no—
228C22⋊C8—1662no—
229#95—831no—
230#95—831no—
231#95—1661no—
232#95—1661no—
233#95—1661no—
234#95—1661no—
235C24(C2×D83)⋊C411yescharacteristic
236C24(C2×D83)⋊C411yescharacteristic
237C2×D4—21no—
238C2×D4—21no—
239C2×D4—41no—
240C2×D4—41no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—1661no—
248C22×C4—1661no—
249C22×C4—1661no—
250C22×C4—1661no—
251C22×C4(C2×D83)⋊C411yescharacteristic
252C22×C4(C2×D83)⋊C411yescharacteristic
253C22×C4C2×C4×D8311yescharacteristic
254C22×C4C2×C22⋊D8311yescharacteristic
255C22×C4C2×D33211yescharacteristic
256M16—42no—
257M16—42no—
258M16—6642no—
259C2×C8—42no—
260C2×C8—3322no—
261C4⋊C4—41no—
262C4⋊C4—3321no—
263C4⋊C4—3321no—
264C4⋊C4—3322no—
265C4⋊C4—3321no—
266C4⋊C4—3321no—
267C22⋊C4—41no—
268C22⋊C4—41no—
269C22⋊C4—3322no—
270C22⋊C4—3322no—
271C22⋊C4—3322no—
272C22⋊C4—3322no—
273C42—21no—
274C42—21no—
275C42—3322no—
276C42—3321no—
277C42—3321no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23—41no—
285C23—41no—
286C23—41no—
287C23—41no—
288C23C2×(C2×D83)⋊C411yescharacteristic
289D4—42no—
290D4—42no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—81no—
304C2×C4—3321no—
305C2×C4—3321no—
306C2×C4—3321no—
307C2×C4—3321no—
308C2×C4—3321no—
309C2×C4—3321no—
310C2×C4—3321no—
311C2×C4—3321no—
312C2×C4—3321no—
313C2×C4—3321no—
314C2×C4—3321no—
315C2×C4—1662no—
316C2×C4#1458170611yescharacteristic
317C2×C4#1458170611yescharacteristic
318C2×C4#1458181811yescharacteristic
319C2×C4#1458182311yescharacteristic
320C8—82no—
321C8—6642no—
322C22—21no—
323C22—21no—
324C22—42no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22C2×#1458170611yescharacteristic
335C22#1912770011yescharacteristic
336C22#1912771911yescharacteristic
337C4—21no—
338C4—21no—
339C4—21no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—3322no—
344C4—6641no—
345C4—6641no—
346C4—6641no—
347C4#1912749212yes—
348C2—21no—
349C2—21no—
350C2—81no—
351C2—81no—
352C2#2490600411yescharacteristic
353C2C2×#1912749211yescharacteristic
354C2#2490914611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1800C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C4⋊C4×C83C23
hypercenter#1800D83
hypocenterC83#11830
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #24905489C2semidirect
4. #24905468C2semidirect
183. C83#11830semidirect

Maximal subgroups

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

subgroupindex
#249107222
#249058012
#249054892
#249054682
C83×#18002
#1183083