Finite Group Databasebeta

← #1319×D83 #35313504

Subgroups — #1319×D83 #35313504

Order 21248.

618 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 101 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#391×D83C211yesmaximal, characteristic
2#324×D83C212yesmaximal
3#24905397C211yesmaximal, characteristic, complemented
4#24905396C212yesmaximal, complemented
5C83×#1319C211yesmaximal, characteristic, complemented
6C22×Q8×D83C411yescharacteristic
7C2×C4⋊C4×D83C2211yescharacteristic
8C2×C42×D83C411yescharacteristic
9C22⋊C8×D83—22no—
10#19127608C411yescharacteristic
11#19127607C411yescharacteristic
12#19127606C2211yescharacteristic
13#19127605—22no—
14#95×D83—22no—
15#19127427C2212yes—
16#19127426—22no—
17#391×C83C2211yescharacteristic
18#324×C83C2212yes—
19C22×Dic166—21no—
20C22×Dic166C2×C411yescharacteristic
21C2×Q8×D83—22no—
22C2×Q8×D83—41no—
23C2×Q8×D83—41no—
24C2×Q8×D83—42no—
25C2×Q8×D83C814yescomplemented
26C22×C4×D83—21no—
27C22×C4×D83—21no—
28C22×C4×D83C2×C411yescharacteristic
29C22×C4×D83D412yes—
30C2×C8×D83—42no—
31C2×C4⋊Dic83C2×C411yescharacteristic
32C2×Dic83⋊C4D412yes—
33C4⋊C4×D83—41no—
34C4⋊C4×D83—41no—
35C2×C4×Dic83—21no—
36C2×C4×Dic83C2×C411yescharacteristic
37C42×D83—42no—
38C22⋊(C83⋊C8)—22no—
39(C2×D83)⋊C8—42no—
40C83⋊#95—22no—
41C83⋊#95—22no—
42C83⋊#95—22no—
43C22×Q8×C83C2×C411yescharacteristic
44C2×C4⋊C4×C83C2311yescharacteristic
45C2×C42×C83C2×C411yescharacteristic
46C22⋊C8×C83—22no—
47#95×C83—22no—
48C23×D83C22⋊C411yescharacteristic
49C2×Dic166—21no—
50C2×Dic166—24no—
51C2×Dic166—42no—
52C2×Dic166—42no—
53C2×Dic166—41no—
54C2×Dic166—41no—
55C2×Dic166C2×C812yescomplemented
56Q8×D83—44no—
57Q8×D83—44no—
58Q8×D83—44no—
59Q8×D83—44no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83C22⋊C411yescharacteristic
63C22×Dic83C22⋊C412yes—
64C2×C4×D83—21no—
65C2×C4×D83—21no—
66C2×C4×D83—22no—
67C2×C4×D83—24no—
68C2×C4×D83—24no—
69C2×C4×D83—22no—
70C2×C4×D83—22no—
71C2×C4×D83—42no—
72C2×C4×D83—41no—
73C2×C4×D83—41no—
74C2×C4×D83—42no—
75C2×C4×D83—42no—
76C2×C4×D83—44no—
77C2×C4×D83C2×C812yes—
78C2×C4×D83M1612yes—
79C2×C83⋊C8—42no—
80C8×D83—84no—
81C4⋊Dic83—41no—
82Dic83⋊C4—42no—
83C4×Dic83—42no—
84C4×Dic83—41no—
85C2×Q8×C83—21no—
86C2×Q8×C83—41no—
87C2×Q8×C83—41no—
88C2×Q8×C83C2×C812yescomplemented
89C22×C4×C83—21no—
90C22×C4×C83—21no—
91C22×C4×C83C22×C411yescharacteristic
92C22×C4×C83C2×D412yes—
93C2×C8×C83—42no—
94C4⋊C4×C83—41no—
95C42×C83—41no—
96C23×C83C2×C22⋊C411yescharacteristic
97C22×D83—22no—
98C22×D83—22no—
99C22×D83C22⋊C812yes—
100C22×D83#9911yescharacteristic
101C22×D83#9911yescharacteristic
102C22×D83#10012yes—
103Q8×C83—42no—
104Q8×C83—42no—
105C2×C4×C83—21no—
106C2×C4×C83—21no—
107C2×C4×C83—22no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—41no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—42no—
115C2×C4×C83C22×C811yescharacteristic
116C2×C4×C83C2×M1611yescharacteristic
117C2×Dic83—21no—
118C2×Dic83—21no—
119C2×Dic83—22no—
120C2×Dic83—22no—
121C2×Dic83—22no—
122C2×Dic83—41no—
123C2×Dic83—41no—
124C2×Dic83—41no—
125C2×Dic83—41no—
126C2×Dic83—42no—
127C2×Dic83C22⋊C812yes—
128C4×D83—22no—
129C4×D83—22no—
130C4×D83—22no—
131C4×D83—22no—
132C4×D83—48no—
133C4×D83—44no—
134C4×D83—44no—
135C4×D83—44no—
136C4×D83—82no—
137Dic166—48no—
138Dic166—42no—
139Dic166—42no—
140C8×C83—82no—
141C83⋊C8—82no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83C2×C22⋊C811yescharacteristic
145C22×C83C2×#9911yescharacteristic
146C22×C83C2×#10011yescharacteristic
147C2×D83—21no—
148C2×D83—21no—
149C2×D83—24no—
150C2×D83—24no—
151C2×D83#32312yes—
152C2×D83#32312yes—
153C2×D83#35412yes—
154C2×D83#35612yes—
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—42no—
158C4×C83—42no—
159C4×C83—42no—
160Dic83—22no—
161Dic83—42no—
162Dic83—42no—
163Dic83—42no—
164C2×#1319—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83C2×#32311yescharacteristic
168C2×C83C2×#35411yescharacteristic
169C2×C83C2×#35611yescharacteristic
170D83—22no—
171D83#131914yesdirect product factor
172C2×#391—831no—
173C2×#324—832no—
174#1319—831no—
175#1319—832no—
176#1319D8311yescharacteristic, direct product factor
177C83C2×#131911yescharacteristic, complemented, Sylow
178C23×Q8—831no—
179C22×C4⋊C4—831no—
180C22×C42—831no—
181C2×C22⋊C8—1662no—
182#391—831no—
183#391—831no—
184#391—831no—
185#391—1662no—
186#391C2×D8311yescharacteristic
187C2×#95—1662no—
188#324—832no—
189#324—1662no—
190#324C2×D8312yes—
191C22×Q8—3322no—
192C22×Q8—3321no—
193C22×Q8—3321no—
194C22×Q8—834no—
195C22×Q8—831no—
196C22×Q8—1661no—
197C22×Q8—1662no—
198C22×Q8C4×D8311yescharacteristic
199C23×C4—832no—
200C23×C4—831no—
201C23×C4—1661no—
202C23×C4—1661no—
203C22×C8—3322no—
204C2×C4⋊C4—3321no—
205C2×C4⋊C4—3321no—
206C2×C4⋊C4—831no—
207C2×C4⋊C4—832no—
208C2×C4⋊C4C22×D8311yescharacteristic
209C2×C42—3322no—
210C2×C42—831no—
211C2×C42—1661no—
212C2×C42C4×D8311yescharacteristic
213C22⋊C8—22no—
214C22⋊C8—3322no—
215C22⋊C8—1662no—
216#95—22no—
217#95—1662no—
218#95—1662no—
219#95—1662no—
220C24—831no—
221C2×Q8—21no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—3321no—
225C2×Q8—3321no—
226C2×Q8—3322no—
227C2×Q8—3322no—
228C2×Q8—3324no—
229C2×Q8—3324no—
230C2×Q8—3324no—
231C2×Q8—3324no—
232C2×Q8—832no—
233C2×Q8—1664no—
234C2×Q8—1661no—
235C2×Q8C8×D8312yescomplemented
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—3322no—
239C22×C4—3321no—
240C22×C4—3321no—
241C22×C4—3322no—
242C22×C4—3322no—
243C22×C4—3324no—
244C22×C4—832no—
245C22×C4—831no—
246C22×C4—832no—
247C22×C4—832no—
248C22×C4—1662no—
249C22×C4—1664no—
250C22×C4—1664no—
251C22×C4—1662no—
252C22×C4—1662no—
253C22×C4—1661no—
254C22×C4—1661no—
255C22×C4—1661no—
256C22×C4—1661no—
257C22×C4C2×C4×D8311yescharacteristic
258C22×C4D4×D8312yes—
259C2×C8—42no—
260C2×C8—3322no—
261C2×C8—6644no—
262C4⋊C4—41no—
263C4⋊C4—3322no—
264C4⋊C4—3321no—
265C42—41no—
266C42—3321no—
267C42—3322no—
268C23—831no—
269C23—832no—
270C23—831no—
271C23—832no—
272C23—1662no—
273C23—1662no—
274C23C22⋊C4×D8311yescharacteristic
275Q8—42no—
276Q8—42no—
277Q8—3322no—
278Q8—3322no—
279Q8—3328no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—3324no—
291C2×C4—3324no—
292C2×C4—3324no—
293C2×C4—3328no—
294C2×C4—3321no—
295C2×C4—3321no—
296C2×C4—3321no—
297C2×C4—3321no—
298C2×C4—3322no—
299C2×C4—832no—
300C2×C4—6642no—
301C2×C4—1662no—
302C2×C4—1662no—
303C2×C4—1662no—
304C2×C4—1662no—
305C2×C4—1662no—
306C2×C4—1662no—
307C2×C4—1662no—
308C2×C4—1661no—
309C2×C4—1661no—
310C2×C4C2×C8×D8311yescharacteristic
311C2×C4M16×D8311yescharacteristic
312C8—82no—
313C8—6642no—
314C22—21no—
315C22—21no—
316C22—832no—
317C22—832no—
318C22—832no—
319C22—832no—
320C22—1664no—
321C22—1664no—
322C22—1661no—
323C22—1661no—
324C22C22⋊C8×D8311yescharacteristic
325C22#99×D8311yescharacteristic
326C22#100×D8311yescharacteristic
327C4—21no—
328C4—21no—
329C4—42no—
330C4—42no—
331C4—42no—
332C4—3322no—
333C4—3322no—
334C4—3322no—
335C4—1662no—
336C2—21no—
337C2—21no—
338C2—834no—
339C2—1662no—
340C2#323×D8311yescharacteristic
341C2#354×D8311yescharacteristic
342C2#356×D8311yescharacteristic
subgroupG/H
centerC22#99×D83
commutatorC2×C4×C83C22×C8
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1319C2
socleC22×C83C2×#99
prime radicalC23×D83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1319D83
hypocenterC83C2×#1319
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #1319

Complements

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

NH ≅ G/Ntype
3. #24905397C2semidirect
4. #24905396C2semidirect
5. C83×#1319C2semidirect
25. C2×Q8×D83C8semidirect
55. C2×Dic166C2×C8semidirect
88. C2×Q8×C83C2×C8semidirect
171. D83#1319direct (N × H)
176. #1319D83direct (N × H)
177. C83C2×#1319semidirect
235. C2×Q8C8×D83semidirect

Maximal subgroups

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

subgroupindex
#391×D832
#324×D832
#249053972
#249053962
C83×#13192
C2×#131983