Finite Group Databasebeta

← #35314616

Subgroups — #35314616

Order 21248.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 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#24913213C211yesmaximal, characteristic
2#24907202C211yesmaximal, characteristic, complemented
3#24905467C214yesmaximal, complemented
4C83×#1458C211yesmaximal, characteristic, complemented
5C22×#13050574—21no—
6C2×#14581789C411yescharacteristic
7C2×#14581766—21no—
8#19127618—21no—
9#19127618C2212yes—
10#19127610—21no—
11#19127610C412yes—
12#19127603—21no—
13#19127567C411yescharacteristic
14#19127470—21no—
15#19127470C2212yes—
16C2×#117×C83C2211yescharacteristic
17#431×C83—21no—
18#324×C83C2212yes—
19#323×C83—21no—
20C22×D332—21no—
21C22×D332—21no—
22C22×D332C2×C412yes—
23C22×C22⋊D83—22no—
24C2×#13050574—24no—
25C2×#13050574—22no—
26C2×#13050574—41no—
27C2×#13050574—42no—
28C22×C83⋊C8—41no—
29#14581789—41no—
30#14581789—41no—
31#14581766—42no—
32#14581766—42no—
33C2×C4⋊Dic83—21no—
34C2×(C2×D83)⋊C4—22no—
35C2×(C2×D83)⋊C4—22no—
36C2×(C2×D83)⋊C4—22no—
37C83⋊#95—21no—
38C83⋊#95C2×C412yes—
39C23×C4×C83D411yescharacteristic
40#117×C83—41no—
41#117×C83—41no—
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C4⋊C4×C83C2311yescharacteristic
44C2×C22⋊C4×C83C2×C411yescharacteristic
45C2×C22⋊C4×C83D411yescharacteristic
46C2×C42×C83C2×C411yescharacteristic
47C2×C42×C83D411yescharacteristic
48C4⋊C8×C83—41no—
49C22⋊C8×C83—22no—
50C4×C8×C83—41no—
51C23×D83—22no—
52C2×D332—42no—
53C2×D332—42no—
54C2×D332—41no—
55C2×D332—41no—
56C2×D332—42no—
57C2×D332—42no—
58C2×D332—82no—
59C2×D332—82no—
60C2×C22⋊D83—44no—
61C2×C22⋊D83—82no—
62C2×C22⋊D83—82no—
63C22×Dic83—22no—
64#13050574—24no—
65#13050574—44no—
66#13050574—42no—
67C2×C83⋊C8—42no—
68C2×C83⋊C8—44no—
69C4⋊Dic83—42no—
70(C2×D83)⋊C4—42no—
71(C2×D83)⋊C4—42no—
72(C2×D83)⋊C4—82no—
73(C2×D83)⋊C4—82no—
74C24×C83C22⋊C411yescharacteristic
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—22no—
78C22×C4×C83C22⋊C411yescharacteristic
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83D812yes—
82C22×C4×C83SD1612yes—
83C22×C4×C83C22×C411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C2×C8×C83—42no—
87C4⋊C4×C83—41no—
88C4⋊C4×C83—41no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C42×C83—21no—
92C42×C83—21no—
93C42×C83—21no—
94C42×C83—21no—
95C23×C83—21no—
96C23×C83—21no—
97C23×C83—41no—
98C23×C83—41no—
99C23×C83#10212yes—
100C23×C83C2×C22⋊C411yescharacteristic
101C22×D83—42no—
102C22×D83—42no—
103C22×D83—42no—
104C22×D83—42no—
105C22×D83—42no—
106C22×D83—42no—
107C22×D83—42no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—22no—
113C2×C4×C83—22no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—21no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83#9911yescharacteristic
123C2×C4×C83#9911yescharacteristic
124C2×C4×C83#10212yes—
125C2×C4×C83#10412yes—
126C2×C4×C83C2×D811yescharacteristic
127C2×C4×C83C2×SD1611yescharacteristic
128C22⋊D83—88no—
129C2×Dic83—42no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132D332—84no—
133D332—84no—
134D332—82no—
135D332—82no—
136C8×C83—82no—
137C83⋊C8—44no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—22no—
141C22×C83—42no—
142C22×C83—41no—
143C22×C83—41no—
144C22×C83—81no—
145C22×C83C2×#9911yescharacteristic
146C22×C83C2×#10211yescharacteristic
147C22×C83C2×#10411yescharacteristic
148C2×D83—82no—
149C2×D83—82no—
150C2×D83—82no—
151C2×D83—82no—
152C2×D83—82no—
153C2×D83—82no—
154C2×D83—82no—
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—41no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83#32712yes—
162C4×C83#32712yes—
163Dic83—82no—
164#11699—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83—42no—
168C2×C83C2×#32711yescharacteristic
169C2×C83#212211yescharacteristic
170C2×C83#213411yescharacteristic
171D83—162no—
172#2619—831no—
173#1742—831no—
174#1458D8311yescharacteristic, complemented
175#1345—834no—
176C83#1169911yescharacteristic, complemented, Sylow
177C22×M16—1661no—
178C2×#127—831no—
179C2×#121—1661no—
180C2×#117C2×D8311yescharacteristic
181#431—21no—
182#394—832no—
183#394—1661no—
184#392—832no—
185#392—1661no—
186#390—1661no—
187#382—831no—
188#343—832no—
189#343—1661no—
190#324C2×D8312yes—
191#323—21no—
192C22×D4—832no—
193C22×D4—1662no—
194C22×D4—1661no—
195C22×D4—1661no—
196C23×C4D33211yescharacteristic
197C2×M16—3321no—
198C2×M16—3322no—
199C2×M16—1664no—
200C2×M16—1662no—
201C22×C8—3321no—
202#127—3321no—
203#127—3321no—
204#121—3322no—
205#121—3322no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4D33211yescharacteristic
210C2×C4⋊C4C22×D8311yescharacteristic
211C2×C22⋊C4—1662no—
212C2×C22⋊C4—1662no—
213C2×C22⋊C4—1662no—
214C2×C22⋊C4C4×D8311yescharacteristic
215C2×C22⋊C4C22⋊D8311yescharacteristic
216C2×C42C4×D8311yescharacteristic
217C2×C42C22⋊D8311yescharacteristic
218C4⋊C8—41no—
219C22⋊C8—22no—
220C4×C8—41no—
221#95—832no—
222#95—1661no—
223C24—1662no—
224C24(C2×D83)⋊C411yescharacteristic
225C2×D4—3322no—
226C2×D4—3322no—
227C2×D4—3324no—
228C2×D4—3321no—
229C2×D4—3321no—
230C2×D4—3322no—
231C2×D4—3322no—
232C2×D4—6642no—
233C2×D4—6642no—
234C2×D4—6642no—
235C2×D4—6642no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—1662no—
240C22×C4(C2×D83)⋊C411yescharacteristic
241C22×C4(C2×D83)⋊C411yescharacteristic
242C22×C4(C2×D83)⋊C411yescharacteristic
243C22×C4D66412yes—
244C22×C4#1305057912yes—
245C22×C4C2×C4×D8311yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247C22×C4C2×D33211yescharacteristic
248M16—3324no—
249M16—3322no—
250M16—1664no—
251C2×C8—42no—
252C2×C8—3322no—
253C2×C8—3324no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—3322no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—3322no—
260C22⋊C4—3322no—
261C22⋊C4—6642no—
262C22⋊C4—6642no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—3322no—
272C23—3322no—
273C23—3322no—
274C23—3322no—
275C23—3322no—
276C23—3322no—
277C23—3322no—
278C23#1458171412yes—
279C23C2×(C2×D83)⋊C411yescharacteristic
280D4—6644no—
281D4—6644no—
282D4—6648no—
283D4—6642no—
284D4—6642no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—3322no—
300C2×C4—3322no—
301C2×C4—3322no—
302C2×C4#1458170611yescharacteristic
303C2×C4#1458170611yescharacteristic
304C2×C4#1458171412yes—
305C2×C4#1458171812yes—
306C2×C4C2×D66411yescharacteristic
307C2×C4C2×#1305057911yescharacteristic
308C8—82no—
309C8—3324no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—6642no—
318C22—6642no—
319C22—6642no—
320C22—6642no—
321C22—6642no—
322C22—6642no—
323C22—6642no—
324C22C2×#1458170611yescharacteristic
325C22C2×#1458171411yescharacteristic
326C22C2×#1458171811yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—6642no—
334C4#1912743612yes—
335C4#1912743612yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—13282no—
340C2C2×#1912743611yescharacteristic
341C2#2490911211yescharacteristic
342C2#2490915411yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1458C2
socleC22×C83C2×#99
prime radical#24913213C2
cosocleC2×C4⋊C4×C83C23
hypercenter#1458D83
hypocenterC83#11699
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #24907202C2semidirect
3. #24905467C2semidirect
4. C83×#1458C2semidirect
174. #1458D83semidirect
176. C83#11699semidirect

Maximal subgroups

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

subgroupindex
#249132132
#249072022
#249054672
C83×#14582
#1169983