Finite Group Databasebeta

← #35306339

Subgroups — #35306339

Order 21248.

476 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 105 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#24918611C211yesmaximal, characteristic
2C2×#19128097C211yesmaximal, characteristic
3#24908452C211yesmaximal, characteristic
4#24906135C211yesmaximal, characteristic
5#24905918C211yesmaximal, characteristic
6#24905301C211yesmaximal, characteristic
7C83×#3139C211yesmaximal, characteristic
8C2×#14581792C2211yescharacteristic
9#19128104—21no—
10#19128097—22no—
11#19128097—22no—
12#19128051—21no—
13#19128034—22no—
14#19127996—22no—
15#19127754C2211yescharacteristic
16C2×#14581723—22no—
17#19127725C2211yescharacteristic
18C2×C4×C83⋊C8C2211yescharacteristic
19#19127469—22no—
20#19127447—22no—
21Q8⋊(C83⋊C8)—22no—
22#533×C83C2211yescharacteristic
23C2×C4⋊Q8×C83C2211yescharacteristic
24#491×C83—22no—
25#487×C83—21no—
26#487×C83—21no—
27#419×C83—22no—
28C4×M16×C83C2211yescharacteristic
29C22×Dic166—21no—
30C2×#13050574—21no—
31C22×C83⋊C8—21no—
32#14581792—21no—
33#14581792—21no—
34#14581792—21no—
35#14581792—21no—
36C2×C4⋊Dic83—21no—
37C2×C4⋊Dic83D412yes—
38#14581725—24no—
39#14581723—28no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42C4⋊(C83⋊C8)D412yes—
43#14581716—24no—
44C22⋊(C83⋊C8)—21no—
45C4×C83⋊C8—21no—
46C4×C83⋊C8—21no—
47C4×C83⋊C8D412yes—
48C22×Q8×C83—21no—
49C2×M16×C83D412yes—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83—21no—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83D411yescharacteristic
54C4⋊Q8×C83D411yescharacteristic
55C4⋊Q8×C83D412yes—
56#125×C83—21no—
57C22⋊Q8×C83—21no—
58C22⋊Q8×C83—21no—
59C4×Q8×C83—21no—
60C4×Q8×C83—21no—
61#117×C83—21no—
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C42×C83C2311yescharacteristic
65#103×C83—24no—
66#103×C83—24no—
67#97×C83—21no—
68C4×C8×C83—21no—
69C2×Dic166—41no—
70C2×Dic166—41no—
71C2×Dic166—82no—
72C22×Dic83—22no—
73#13050574—42no—
74C2×C83⋊C8—24no—
75C2×C83⋊C8—22no—
76C2×C83⋊C8—22no—
77C2×C83⋊C8—41no—
78C4⋊Dic83—22no—
79C4⋊Dic83—22no—
80C4⋊Dic83—41no—
81C4⋊Dic83—41no—
82C2×Q8×C83—22no—
83C2×Q8×C83—21no—
84C2×Q8×C83—21no—
85C2×Q8×C83—41no—
86C2×Q8×C83—42no—
87C2×Q8×C83—42no—
88C22×C4×C83—41no—
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83#4111yescharacteristic
92M16×C83—22no—
93M16×C83—22no—
94C2×C8×C83—22no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—22no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83—22no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83—41no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83—41no—
105C22⋊C4×C83—41no—
106C42×C83—41no—
107C42×C83C2×D411yescharacteristic
108C42×C83C2×D411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83C2×D411yescharacteristic
111C23×C83#12111yescharacteristic
112Q8×C83—44no—
113Q8×C83—42no—
114Q8×C83—42no—
115Q8×C83—81no—
116Q8×C83—81no—
117C2×C4×C83—22no—
118C2×C4×C83—22no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—42no—
122C2×C4×C83—82no—
123C2×C4×C83#12011yescharacteristic
124C2×C4×C83#12011yescharacteristic
125C2×C4×C83#12011yescharacteristic
126C2×C4×C83#12111yescharacteristic
127C2×C4×C83#12111yescharacteristic
128C2×C4×C83#12711yescharacteristic
129C2×C4×C83#13512yes—
130C2×C4×C83#13712yes—
131C2×Dic83—42no—
132C2×Dic83—82no—
133Dic166—82no—
134Dic166—82no—
135C8×C83—42no—
136C83⋊C8—24no—
137C83⋊C8—42no—
138C83⋊C8—81no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83#39211yescharacteristic
142C22×C83#46412yes—
143C4×C83—21no—
144C4×C83—21no—
145C4×C83—81no—
146C4×C83—82no—
147C4×C83—81no—
148C4×C83#45212yes—
149C4×C83#45612yes—
150C4×C83#47112yes—
151C4×C83#49511yescharacteristic
152C4×C83#49711yescharacteristic
153Dic83—82no—
154#10173—831nomaximal, Sylow
155C2×C83—22no—
156C2×C83#200511yescharacteristic
157C2×C83#201511yescharacteristic
158C2×C83#201911yescharacteristic
159#3230—831no—
160C2×#498—831no—
161#3139D8311yescharacteristic
162#1997—831no—
163#1546—831no—
164#1493—831no—
165#1278—831no—
166C83#1017311yescharacteristic, complemented, Sylow
167#533C2×D8311yescharacteristic
168C2×C4⋊Q8—831no—
169C2×C4⋊Q8C2×D8311yescharacteristic
170#500—1661no—
171#498—1662no—
172#498—1662no—
173#491—22no—
174#487—21no—
175#487—21no—
176#487—1661no—
177#484—1662no—
178#477—1662no—
179#431—831no—
180C2×#106—1662no—
181#423—831no—
182#419—22no—
183C4×M16C2×D8311yescharacteristic
184C2×C4×C8—831no—
185#343—1662no—
186#332—1662no—
187Q8⋊C8—1662no—
188C22×Q8—21no—
189C22×Q8—1661no—
190C2×M16—1661no—
191C2×M16C22⋊D8312yes—
192C22×C8—1661no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—1661no—
197C4⋊Q8—1661no—
198C4⋊Q8—1661no—
199C4⋊Q8—1661no—
200C4⋊Q8C22⋊D8312yes—
201C4⋊Q8C22⋊D8311yescharacteristic
202C4⋊Q8C22⋊D8311yescharacteristic
203#125—21no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C4×Q8—21no—
207C4×Q8—21no—
208#117—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—832no—
212C2×C4⋊C4—1661no—
213C2×C42C22×D8311yescharacteristic
214#107—1664no—
215#106—1668no—
216C4⋊C8—832no—
217C4⋊C8—1661no—
218C4⋊C8—1661no—
219#103—24no—
220#103—24no—
221#103—1664no—
222C22⋊C8—1661no—
223#97—21no—
224C4×C8—21no—
225C4×C8—832no—
226C4×C8—1661no—
227C4×C8—1661no—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—21no—
231C2×Q8—41no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—3321no—
235C2×Q8—3321no—
236C2×Q8—6642no—
237C22×C4—41no—
238C22×C4—1662no—
239C22×C4D4×D8311yescharacteristic
240C22×C4C2×C22⋊D8311yescharacteristic
241C22×C4#1305058911yescharacteristic
242M16—22no—
243M16—22no—
244M16—3322no—
245C2×C8—22no—
246C2×C8—3321no—
247C2×C8—1662no—
248C2×C8—1664no—
249C2×C8—1662no—
250C4⋊C4—22no—
251C4⋊C4—21no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—21no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—3321no—
261C4⋊C4—3321no—
262C4⋊C4—1662no—
263C4⋊C4—1662no—
264C22⋊C4—41no—
265C42—41no—
266C42D4×D8311yescharacteristic
267C42D4×D8311yescharacteristic
268C42C2×C22⋊D8311yescharacteristic
269C42C2×C22⋊D8311yescharacteristic
270C23#1458176311yescharacteristic
271Q8—44no—
272Q8—42no—
273Q8—42no—
274Q8—81no—
275Q8—81no—
276Q8—6642no—
277Q8—6642no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—3322no—
285C2×C4—6642no—
286C2×C4#1458176011yescharacteristic
287C2×C4#1458176011yescharacteristic
288C2×C4#1458176111yescharacteristic
289C2×C4#1458176411yescharacteristic
290C2×C4#1458176411yescharacteristic
291C2×C4#1458178811yescharacteristic
292C2×C4#1458181212yes—
293C2×C4#1458182212yes—
294C8—42no—
295C8—3322no—
296C8—6641no—
297C8—1664no—
298C22—21no—
299C22—21no—
300C22#1912760911yescharacteristic
301C22#1912792012yes—
302C4—21no—
303C4—21no—
304C4—82no—
305C4—81no—
306C4—81no—
307C4—6642no—
308C4#1912786312yes—
309C4#1912788212yes—
310C4#1912796612yes—
311C4#1912808311yescharacteristic
312C4#1912809611yescharacteristic
313C2—22no—
314C2#2490849611yescharacteristic
315C2#2490855711yescharacteristic
316C2#2490857411yescharacteristic
subgroupG/H
centerC22#19127609
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#3139C2
socleC22×C83#392
prime radicalC23×C83#121
hypercenter#3139D83
hypocenterC83#10173
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3139

Complements

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

NH ≅ G/Ntype
166. C83#10173semidirect

Maximal subgroups

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

subgroupindex
#249186112
C2×#191280972
#249084522
#249061352
#249059182
#249053012
C83×#31392
#1017383