Finite Group Databasebeta

← #35313585

Subgroups — #35313585

Order 21248.

434 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 131 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#24905777C211yesmaximal, characteristic
2#24905777C211yesmaximal, characteristic
3#24905411C214yesmaximal
4C83×#2317C211yesmaximal, characteristic
5#19127754—21no—
6#19127754—21no—
7#19127427—21no—
8#19127427—21no—
9#19127425C2212yes—
10#19127425C2212yes—
11C2×#117×C83C411yescharacteristic
12C2×#117×C83C2211yescharacteristic
13C22×C4⋊C4×C83C411yescharacteristic
14#389×C83—21no—
15#389×C83—21no—
16#385×C83C412yes—
17#385×C83C2212yes—
18#384×C83—21no—
19#382×C83—21no—
20C4⋊(C83⋊C8)—41no—
21C4⋊(C83⋊C8)—41no—
22C22⋊(C83⋊C8)—22no—
23C22⋊(C83⋊C8)—22no—
24C4×C83⋊C8—41no—
25C4×C83⋊C8—41no—
26C23×C4×C83C2×C411yescharacteristic
27C23×C4×C83C2×C412yes—
28#117×C83—22no—
29#117×C83—41no—
30#117×C83—41no—
31#117×C83C814yescomplemented
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83—21no—
34C2×C4⋊C4×C83—21no—
35C2×C4⋊C4×C83—22no—
36C2×C4⋊C4×C83—22no—
37C2×C4⋊C4×C83—41no—
38C2×C4⋊C4×C83—41no—
39C2×C4⋊C4×C83D411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C22⋊C4×C83C2×C412yes—
42C2×C22⋊C4×C83C2×C411yescharacteristic
43C2×C22⋊C4×C83C2311yescharacteristic
44C2×C42×C83—21no—
45C2×C42×C83D411yescharacteristic
46C2×C42×C83D411yescharacteristic
47#95×C83—22no—
48#95×C83—22no—
49C2×C83⋊C8—42no—
50C2×C83⋊C8—42no—
51C24×C83C22×C411yescharacteristic
52C22×C4×C83—21no—
53C22×C4×C83—21no—
54C22×C4×C83—22no—
55C22×C4×C83—22no—
56C22×C4×C83—22no—
57C22×C4×C83—42no—
58C22×C4×C83—82no—
59C22×C4×C83C22⋊C411yescharacteristic
60C22×C4×C83C22⋊C411yescharacteristic
61C22×C4×C83C22⋊C411yescharacteristic
62C22×C4×C83C22⋊C411yescharacteristic
63C22×C4×C83C2×C812yes—
64C22×C4×C83M1612yes—
65C22×C4×C83C2×D411yescharacteristic
66C22×C4×C83C2×D411yescharacteristic
67C4⋊C4×C83—24no—
68C4⋊C4×C83—41no—
69C4⋊C4×C83—41no—
70C4⋊C4×C83—42no—
71C4⋊C4×C83—41no—
72C4⋊C4×C83—41no—
73C4⋊C4×C83—82no—
74C4⋊C4×C83—82no—
75C22⋊C4×C83—22no—
76C22⋊C4×C83—41no—
77C22⋊C4×C83—41no—
78C22⋊C4×C83C2×C814yes—
79C42×C83—24no—
80C42×C83—21no—
81C42×C83—21no—
82C42×C83—21no—
83C42×C83—21no—
84C23×C83—21no—
85C23×C83—21no—
86C23×C83—41no—
87C23×C83—41no—
88C23×C83C2×C22⋊C411yescharacteristic
89C23×C83C22×C811yescharacteristic
90C23×C83C2×M1611yescharacteristic
91C2×C4×C83—21no—
92C2×C4×C83—21no—
93C2×C4×C83—22no—
94C2×C4×C83—22no—
95C2×C4×C83—22no—
96C2×C4×C83—22no—
97C2×C4×C83—24no—
98C2×C4×C83—21no—
99C2×C4×C83—21no—
100C2×C4×C83—21no—
101C2×C4×C83—21no—
102C2×C4×C83—42no—
103C2×C4×C83—42no—
104C2×C4×C83—41no—
105C2×C4×C83—41no—
106C2×C4×C83—41no—
107C2×C4×C83—41no—
108C2×C4×C83—42no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83—84no—
112C2×C4×C83C22⋊C812yes—
113C2×C4×C83C22⋊C812yes—
114C2×C4×C83#9911yescharacteristic
115C2×C4×C83#9911yescharacteristic
116C2×C4×C83#10112yes—
117C83⋊C8—82no—
118C83⋊C8—82no—
119C22×C83—21no—
120C22×C83—21no—
121C22×C83—21no—
122C22×C83—21no—
123C22×C83—41no—
124C22×C83—41no—
125C22×C83—42no—
126C22×C83—41no—
127C22×C83—41no—
128C22×C83C2×C22⋊C811yescharacteristic
129C22×C83C2×#9911yescharacteristic
130C22×C83C2×#10111yescharacteristic
131C4×C83—22no—
132C4×C83—44no—
133C4×C83—41no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83—41no—
137C4×C83—82no—
138C4×C83#32412yes—
139C4×C83#32412yes—
140#11454—831nomaximal, Sylow
141C2×C83—21no—
142C2×C83—21no—
143C2×C83—41no—
144C2×C83—41no—
145C2×C83C2×#32411yescharacteristic
146C2×C83#212211yescharacteristic
147C2×C83#212311yescharacteristic
148#2317D8311yescharacteristic
149#1458—831no—
150#1458—831no—
151#1326—834no—
152C83#1145411yescharacteristic, complemented, Sylow
153C2×#117Dic8311yescharacteristic
154C2×#117C2×D8311yescharacteristic
155C22×C4⋊C4Dic8311yescharacteristic
156#431—1661no—
157#431—1661no—
158#389—21no—
159#389—21no—
160#385Dic8312yes—
161#385C2×D8312yes—
162#384—21no—
163#382—21no—
164#324—1661no—
165#324—1661no—
166#323—832no—
167#323—832no—
168C23×C4C2×Dic8311yescharacteristic
169C23×C4C2×Dic8312yes—
170#117—22no—
171#117—41no—
172#117—41no—
173#117C83⋊C814yescomplemented
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—41no—
180C2×C4⋊C4—41no—
181C2×C4⋊C4C22⋊D8311yescharacteristic
182C2×C4⋊C4C22⋊D8311yescharacteristic
183C2×C22⋊C4C2×Dic8312yes—
184C2×C22⋊C4C2×Dic8311yescharacteristic
185C2×C22⋊C4C22×D8311yescharacteristic
186C2×C42—21no—
187C2×C42C22⋊D8311yescharacteristic
188C2×C42C22⋊D8311yescharacteristic
189C4⋊C8—3321no—
190C4⋊C8—3321no—
191C22⋊C8—1662no—
192C22⋊C8—1662no—
193C4×C8—3321no—
194C4×C8—3321no—
195#95—22no—
196#95—22no—
197C24C22×Dic8311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—42no—
204C22×C4—82no—
205C22×C4C22⋊Dic8311yescharacteristic
206C22×C4C22⋊Dic8311yescharacteristic
207C22×C4C22⋊Dic8311yescharacteristic
208C22×C4C22⋊Dic8311yescharacteristic
209C22×C4C2×C83⋊C812yes—
210C22×C4#1305057412yes—
211C22×C4C2×C22⋊D8311yescharacteristic
212C22×C4C2×C22⋊D8311yescharacteristic
213C2×C8—3322no—
214C2×C8—3322no—
215C4⋊C4—24no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—42no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—82no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—41no—
225C22⋊C4—41no—
226C22⋊C4C2×C83⋊C814yes—
227C42—24no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C23—21no—
233C23—21no—
234C23—41no—
235C23—41no—
236C23C2×C22⋊Dic8311yescharacteristic
237C23C22×C83⋊C811yescharacteristic
238C23C2×#1305057411yescharacteristic
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—24no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—21no—
249C2×C4—21no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—84no—
260C2×C4C22⋊(C83⋊C8)12yes—
261C2×C4C22⋊(C83⋊C8)12yes—
262C2×C4#1458170711yescharacteristic
263C2×C4#1458170711yescharacteristic
264C2×C4#1458171112yes—
265C8—6642no—
266C8—6642no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—41no—
272C22—41no—
273C22—42no—
274C22—41no—
275C22—41no—
276C22C2×C22⋊(C83⋊C8)11yescharacteristic
277C22C2×#1458170711yescharacteristic
278C22C2×#1458171111yescharacteristic
279C4—22no—
280C4—44no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—41no—
285C4—82no—
286C4#1912742712yes—
287C4#1912742712yes—
288C2—21no—
289C2—21no—
290C2—41no—
291C2—41no—
292C2C2×#1912742711yescharacteristic
293C2#2490911311yescharacteristic
294C2#2490911711yescharacteristic
subgroupG/H
centerC22C2×#14581707
commutatorC23×C83C22×C8
FrattiniC2×C22⋊C4C22×D83
FittingC83×#2317C2
socleC22×C83C2×#99
prime radicalC24×C83C22×C4
cosocleC2×C22⋊C4×C83C23
hypercenter#2317D83
hypocenterC83#11454
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
31. #117×C83C8semidirect
152. C83#11454semidirect
173. #117C83⋊C8semidirect

Maximal subgroups

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

subgroupindex
#249057772
#249054112
C83×#23172
#1145483