Finite Group Databasebeta

← #35315563

Subgroups — #35315563

Order 21248.

440 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 103 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#24907455C211yesmaximal, characteristic
2#24905781C211yesmaximal, characteristic
3#24905493C212yesmaximal, complemented
4#24905490C212yesmaximal, complemented
5C83×#2564C211yesmaximal, characteristic
6#19127725—21no—
7C2×#14581709—21no—
8C2×C22⋊(C83⋊C8)—21no—
9#19127470C2212yes—
10#19127427C2212yes—
11#19127425—21no—
12C2×C22⋊Q8×C83C411yescharacteristic
13C2×C4×D4×C83C2211yescharacteristic
14#400×C83C2211yescharacteristic
15#399×C83C411yescharacteristic
16#395×C83C2211yescharacteristic
17#393×C83C411yescharacteristic
18#387×C83—21no—
19#386×C83—21no—
20#385×C83—21no—
21#380×C83—21no—
22C2×#95×C83C411yescharacteristic
23C2×#13050574—22no—
24C22×C83⋊C8—41no—
25C4⋊(C83⋊C8)—41no—
26C4⋊(C83⋊C8)—41no—
27#14581709—42no—
28C22⋊(C83⋊C8)—22no—
29C22⋊(C83⋊C8)—41no—
30C22×Q8×C83C2×C411yescharacteristic
31C22×D4×C83D411yescharacteristic
32C23×C4×C83—21no—
33C23×C4×C83C2×C411yescharacteristic
34C23×C4×C83D411yescharacteristic
35C22⋊Q8×C83—41no—
36C22⋊Q8×C83—41no—
37C4×D4×C83—41no—
38C4×D4×C83—41no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83—21no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83C2×C411yescharacteristic
43C2×C4⋊C4×C83C2311yescharacteristic
44C2×C22⋊C4×C83—21no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83—21no—
47C2×C22⋊C4×C83C2×C411yescharacteristic
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C42×C83D411yescharacteristic
50#95×C83—21no—
51#95×C83—21no—
52#95×C83—21no—
53#95×C83—21no—
54#95×C83—21no—
55#95×C83C2×C411yescharacteristic
56#95×C83C2×C411yescharacteristic
57#13050574—42no—
58#13050574—42no—
59C2×C83⋊C8—42no—
60C2×C83⋊C8—42no—
61C2×C83⋊C8—82no—
62C24×C83C22⋊C411yescharacteristic
63C24×C83C22⋊C411yescharacteristic
64C2×Q8×C83—41no—
65C2×Q8×C83—81no—
66C2×D4×C83—41no—
67C2×D4×C83—41no—
68C2×D4×C83SD1612yes—
69C2×D4×C83Q1612yes—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—42no—
79C22×C4×C83—42no—
80C22×C4×C83—81no—
81C22×C4×C83—81no—
82C22×C4×C83—81no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22×C411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83C2×D411yescharacteristic
88C4⋊C4×C83—41no—
89C4⋊C4×C83—41no—
90C4⋊C4×C83—42no—
91C4⋊C4×C83—41no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—81no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C22⋊C4×C83—42no—
97C22⋊C4×C83—42no—
98C22⋊C4×C83—81no—
99C42×C83—21no—
100C42×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—21no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83—41no—
109C23×C83#10312yes—
110C23×C83#10312yes—
111C23×C83C2×C22⋊C411yescharacteristic
112Q8×C83—82no—
113D4×C83—42no—
114D4×C83—42no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—22no—
118C2×C4×C83—21no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—41no—
122C2×C4×C83—42no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—41no—
128C2×C4×C83—41no—
129C2×C4×C83—41no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—81no—
138C2×C4×C83—81no—
139C2×C4×C83—81no—
140C2×C4×C83—84no—
141C2×C4×C83#9911yescharacteristic
142C2×C4×C83#9911yescharacteristic
143C2×C4×C83C2×SD1611yescharacteristic
144C2×C4×C83C2×Q1611yescharacteristic
145C83⋊C8—82no—
146C83⋊C8—82no—
147C22×C83—21no—
148C22×C83—21no—
149C22×C83—22no—
150C22×C83—22no—
151C22×C83—42no—
152C22×C83—42no—
153C22×C83—42no—
154C22×C83—41no—
155C22×C83—41no—
156C22×C83—41no—
157C22×C83—41no—
158C22×C83—81no—
159C22×C83C2×#9911yescharacteristic
160C22×C83C2×#10311yescharacteristic
161C22×C83#42111yescharacteristic
162C4×C83—21no—
163C4×C83—21no—
164C4×C83—21no—
165C4×C83—21no—
166C4×C83—42no—
167C4×C83—41no—
168C4×C83—41no—
169C4×C83—41no—
170C4×C83—81no—
171C4×C83—81no—
172C4×C83—81no—
173#11906—831nomaximal, Sylow
174C2×C83—21no—
175C2×C83—21no—
176C2×C83—42no—
177C2×C83—42no—
178C2×C83#152011yescharacteristic
179C2×C83#212111yescharacteristic
180C2×C83#213211yescharacteristic
181#2564D8311yescharacteristic
182#1800—831no—
183#1459—831no—
184#1354—832no—
185#1353—832no—
186C83#1190611yescharacteristic, complemented, Sylow
187C2×C22⋊Q8Dic8311yescharacteristic
188C2×C4×D4C2×D8311yescharacteristic
189#423—1661no—
190C2×#100—1661no—
191C2×C22⋊C8—1661no—
192#400C2×D8311yescharacteristic
193#399Dic8311yescharacteristic
194#395C2×D8311yescharacteristic
195#393Dic8311yescharacteristic
196#387—21no—
197#386—21no—
198#385—21no—
199#380—21no—
200C2×#95Dic8311yescharacteristic
201#343—832no—
202#324—832no—
203#323—1661no—
204C22×Q8C2×Dic8311yescharacteristic
205C22×D4C22⋊D8311yescharacteristic
206C23×C4—21no—
207C23×C4C2×Dic8311yescharacteristic
208C23×C4C22⋊D8311yescharacteristic
209C2×M16—1662no—
210C22×C8—3321no—
211C22⋊Q8—41no—
212C22⋊Q8—41no—
213C4×D4—41no—
214C4×D4—41no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C2×Dic8311yescharacteristic
219C2×C4⋊C4C22×D8311yescharacteristic
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4C2×Dic8311yescharacteristic
224C2×C22⋊C4C22⋊D8311yescharacteristic
225C2×C42C22⋊D8311yescharacteristic
226C4⋊C8—3321no—
227C4⋊C8—3321no—
228#100—3322no—
229C22⋊C8—3321no—
230C22⋊C8—1662no—
231#95—21no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95C2×Dic8311yescharacteristic
237#95C2×Dic8311yescharacteristic
238C24C22⋊Dic8311yescharacteristic
239C24C22⋊Dic8311yescharacteristic
240C2×Q8—41no—
241C2×Q8—81no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4#1305057712yes—
245C2×D4#1305058012yes—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—81no—
257C22×C4—81no—
258C22×C4—81no—
259C22×C4C22⋊Dic8311yescharacteristic
260C22×C4C22⋊Dic8311yescharacteristic
261C22×C4C22×Dic8311yescharacteristic
262C22×C4C2×C22⋊D8311yescharacteristic
263C22×C4C2×C22⋊D8311yescharacteristic
264M16—3322no—
265M16—3322no—
266C2×C8—3322no—
267C2×C8—3322no—
268C2×C8—6642no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—81no—
275C22⋊C4—41no—
276C22⋊C4—41no—
277C22⋊C4—42no—
278C22⋊C4—42no—
279C22⋊C4—81no—
280C42—21no—
281C42—21no—
282C23—21no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—41no—
287C23—41no—
288C23—41no—
289C23—41no—
290C23#1458171612yes—
291C23#1458171612yes—
292C23C2×C22⋊Dic8311yescharacteristic
293Q8—82no—
294D4—42no—
295D4—42no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—81no—
319C2×C4—81no—
320C2×C4—81no—
321C2×C4—84no—
322C2×C4#1458170711yescharacteristic
323C2×C4#1458170711yescharacteristic
324C2×C4C2×#1305057711yescharacteristic
325C2×C4C2×#1305058011yescharacteristic
326C8—6642no—
327C8—6642no—
328C22—21no—
329C22—21no—
330C22—22no—
331C22—22no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—41no—
339C22—81no—
340C22C2×#1458170711yescharacteristic
341C22C2×#1458171611yescharacteristic
342C22#1912771811yescharacteristic
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—41no—
351C4—81no—
352C4—81no—
353C4—81no—
354C2—21no—
355C2—21no—
356C2—42no—
357C2—42no—
358C2#2490603511yescharacteristic
359C2#2490910911yescharacteristic
360C2#2490914811yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#2564C2
socleC22×C83C2×#99
prime radicalC22×D4×C83D4
cosocleC2×C4⋊C4×C83C23
hypercenter#2564D83
hypocenterC83#11906
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #24905493C2semidirect
4. #24905490C2semidirect
186. C83#11906semidirect

Maximal subgroups

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

subgroupindex
#249074552
#249057812
#249054932
#249054902
C83×#25642
#1190683