Finite Group Databasebeta

← #35314615

Subgroups — #35314615

Order 21248.

552 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 115 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#24907202C211yesmaximal, characteristic, complemented
2#24905777C211yesmaximal, characteristic, complemented
3#24905466C214yesmaximal, complemented
4C83×#2619C211yesmaximal, characteristic
5C22×#13050574—21no—
6#19127754—21no—
7#19127470—21no—
8#19127470C2212yes—
9#19127427C2212yes—
10#19127425—21no—
11C2×#127×C83C411yescharacteristic
12C2×#121×C83—21no—
13C2×#117×C83C2211yescharacteristic
14#394×C83—21no—
15#394×C83C2212yes—
16#392×C83—21no—
17#392×C83C412yes—
18#390×C83—21no—
19#382×C83C411yescharacteristic
20C2×#13050574—24no—
21C2×#13050574—22no—
22C2×#13050574—41no—
23C2×#13050574—42no—
24C22×C83⋊C8—41no—
25C4⋊(C83⋊C8)—41no—
26C22⋊(C83⋊C8)—22no—
27C4×C83⋊C8—41no—
28C22×D4×C83—22no—
29C22×D4×C83—21no—
30C22×D4×C83—21no—
31C22×D4×C83C2×C412yes—
32C23×C4×C83D411yescharacteristic
33#127×C83—41no—
34#127×C83—41no—
35#121×C83—42no—
36#121×C83—42no—
37#117×C83—41no—
38#117×C83—41no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C4⋊C4×C83C2311yescharacteristic
42C2×C22⋊C4×C83—22no—
43C2×C22⋊C4×C83—22no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83C2×C411yescharacteristic
46C2×C22⋊C4×C83D411yescharacteristic
47C2×C42×C83C2×C411yescharacteristic
48C2×C42×C83D411yescharacteristic
49#95×C83—21no—
50#95×C83C2×C412yes—
51#13050574—24no—
52#13050574—44no—
53#13050574—42no—
54C2×C83⋊C8—42no—
55C2×C83⋊C8—44no—
56C2×C83⋊C8—42no—
57C24×C83—22no—
58C24×C83C22⋊C411yescharacteristic
59C2×D4×C83—42no—
60C2×D4×C83—42no—
61C2×D4×C83—44no—
62C2×D4×C83—41no—
63C2×D4×C83—41no—
64C2×D4×C83—42no—
65C2×D4×C83—42no—
66C2×D4×C83—82no—
67C2×D4×C83—82no—
68C2×D4×C83—82no—
69C2×D4×C83—82no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—22no—
73C22×C4×C83—22no—
74C22×C4×C83C22⋊C411yescharacteristic
75C22×C4×C83C22⋊C411yescharacteristic
76C22×C4×C83C22⋊C411yescharacteristic
77C22×C4×C83D812yes—
78C22×C4×C83SD1612yes—
79C22×C4×C83C22×C411yescharacteristic
80C22×C4×C83C2×D411yescharacteristic
81C22×C4×C83C2×D411yescharacteristic
82C4⋊C4×C83—41no—
83C4⋊C4×C83—41no—
84C4⋊C4×C83—42no—
85C22⋊C4×C83—41no—
86C22⋊C4×C83—41no—
87C22⋊C4×C83—42no—
88C22⋊C4×C83—42no—
89C22⋊C4×C83—82no—
90C22⋊C4×C83—82no—
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—42no—
100C23×C83—42no—
101C23×C83—42no—
102C23×C83—42no—
103C23×C83—42no—
104C23×C83—42no—
105C23×C83—42no—
106C23×C83#10212yes—
107C23×C83C2×C22⋊C411yescharacteristic
108D4×C83—84no—
109D4×C83—84no—
110D4×C83—88no—
111D4×C83—82no—
112D4×C83—82no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—22no—
116C2×C4×C83—22no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—41no—
124C2×C4×C83—41no—
125C2×C4×C83—41no—
126C2×C4×C83—41no—
127C2×C4×C83—42no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83#9911yescharacteristic
131C2×C4×C83#9911yescharacteristic
132C2×C4×C83#10212yes—
133C2×C4×C83#10412yes—
134C2×C4×C83C2×D811yescharacteristic
135C2×C4×C83C2×SD1611yescharacteristic
136C83⋊C8—44no—
137C83⋊C8—82no—
138C22×C83—21no—
139C22×C83—21no—
140C22×C83—22no—
141C22×C83—42no—
142C22×C83—41no—
143C22×C83—41no—
144C22×C83—81no—
145C22×C83—82no—
146C22×C83—82no—
147C22×C83—82no—
148C22×C83—82no—
149C22×C83—82no—
150C22×C83—82no—
151C22×C83—82no—
152C22×C83C2×#9911yescharacteristic
153C22×C83C2×#10211yescharacteristic
154C22×C83C2×#10411yescharacteristic
155C4×C83—21no—
156C4×C83—21no—
157C4×C83—41no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—82no—
162C4×C83#32712yes—
163C4×C83#32712yes—
164#11699—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83—42no—
168C2×C83—162no—
169C2×C83C2×#32711yescharacteristic
170C2×C83#212211yescharacteristic
171C2×C83#213411yescharacteristic
172#2619D8311yescharacteristic
173#1742—831no—
174#1458—831no—
175#1345—834no—
176C83#1169911yescharacteristic, complemented, Sylow
177C22×M16—1661no—
178C2×#127Dic8311yescharacteristic
179C2×#121—21no—
180C2×#117C2×D8311yescharacteristic
181#431—1661no—
182#394—21no—
183#394C2×D8312yes—
184#392—21no—
185#392Dic8312yes—
186#390—21no—
187#382Dic8311yescharacteristic
188#343—832no—
189#343—1661no—
190#324—832no—
191#323—1661no—
192C22×D4—22no—
193C22×D4—21no—
194C22×D4—21no—
195C22×D4C2×Dic8312yes—
196C23×C4C22⋊D8311yescharacteristic
197C2×M16—3321no—
198C2×M16—3322no—
199C2×M16—1664no—
200C2×M16—1662no—
201C22×C8—3321no—
202#127—41no—
203#127—41no—
204#121—42no—
205#121—42no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4C22⋊D8311yescharacteristic
210C2×C4⋊C4C22×D8311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×Dic8311yescharacteristic
215C2×C22⋊C4C22⋊D8311yescharacteristic
216C2×C42C2×Dic8311yescharacteristic
217C2×C42C22⋊D8311yescharacteristic
218C4⋊C8—3321no—
219C22⋊C8—1662no—
220C4×C8—3321no—
221#95—21no—
222#95C2×Dic8312yes—
223C24—22no—
224C24C22⋊Dic8311yescharacteristic
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—44no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—82no—
233C2×D4—82no—
234C2×D4—82no—
235C2×D4—82no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4C22⋊Dic8311yescharacteristic
241C22×C4C22⋊Dic8311yescharacteristic
242C22×C4C22⋊Dic8311yescharacteristic
243C22×C4#1305057512yes—
244C22×C4#1305057812yes—
245C22×C4C22×Dic8311yescharacteristic
246C22×C4C2×C22⋊D8311yescharacteristic
247C22×C4C2×C22⋊D8311yescharacteristic
248M16—3324no—
249M16—3322no—
250M16—1664no—
251C2×C8—3322no—
252C2×C8—3324no—
253C2×C8—3322no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—82no—
262C22⋊C4—82no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—42no—
277C23—42no—
278C23#1458171312yes—
279C23C2×C22⋊Dic8311yescharacteristic
280D4—84no—
281D4—84no—
282D4—88no—
283D4—82no—
284D4—82no—
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—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4#1458170711yescharacteristic
303C2×C4#1458170711yescharacteristic
304C2×C4#1458171312yes—
305C2×C4#1458171912yes—
306C2×C4C2×#1305057511yescharacteristic
307C2×C4C2×#1305057811yescharacteristic
308C8—3324no—
309C8—6642no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—82no—
322C22—82no—
323C22—82no—
324C22C2×#1458170711yescharacteristic
325C22C2×#1458171311yescharacteristic
326C22C2×#1458171911yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—82no—
334C4#1912743512yes—
335C4#1912743512yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—162no—
340C2C2×#1912743511yescharacteristic
341C2#2490911311yescharacteristic
342C2#2490915511yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
Fitting, prime radicalC83×#2619C2
socleC22×C83C2×#99
cosocleC2×C4⋊C4×C83C23
hypercenter#2619D83
hypocenterC83#11699
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2619

Complements

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

NH ≅ G/Ntype
1. #24907202C2semidirect
2. #24905777C2semidirect
3. #24905466C2semidirect
176. C83#11699semidirect

Maximal subgroups

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

subgroupindex
#249072022
#249057772
#249054662
C83×#26192
#1169983