Finite Group Databasebeta

← #35301801

Subgroups — #35301801

Order 21248.

466 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 109 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#24916546C211yesmaximal, characteristic
2#24912672C211yesmaximal, characteristic
3#24907456C211yesmaximal, characteristic
4#24906144C211yesmaximal, characteristic
5#24905365C211yesmaximal, characteristic
6#24905364C211yesmaximal, characteristic
7C83×#1548C211yesmaximal, characteristic
8C2×C4×Dic166C2211yescharacteristic
9#19127831—22no—
10#19127828—22no—
11#19127755—22no—
12#19127754C2211yescharacteristic
13#19127725C2211yescharacteristic
14C2×#14581710—21no—
15#19127620C414yes—
16#19127606—21no—
17#19127597—22no—
18#19127570—21no—
19C4×C4⋊Dic83C2211yescharacteristic
20#19127469—21no—
21#19127450—22no—
22C2×C4⋊Q8×C83C2211yescharacteristic
23#431×C83C2211yescharacteristic
24#423×C83C2211yescharacteristic
25#333×C83—22no—
26C22×Dic166D411yescharacteristic
27C2×#13050574—21no—
28C4×Dic166—21no—
29C4×Dic166—21no—
30C4×Dic166—21no—
31C4×Dic166—21no—
32C2×C4⋊Dic83—21no—
33C2×C4⋊Dic83D411yescharacteristic
34C2×Dic83⋊C4—24no—
35C2×Dic83⋊C4C2×C412yes—
36C2×C4×Dic83—21no—
37C2×C4×Dic83D412yes—
38C4⋊(C83⋊C8)—21no—
39C4⋊(C83⋊C8)—21no—
40C4⋊(C83⋊C8)D412yes—
41Dic83⋊C8—24no—
42#14581710—42no—
43C22⋊(C83⋊C8)—21no—
44#14581702—24no—
45C4×C83⋊C8—21no—
46C83⋊#95—24no—
47C83⋊#95C2×C412yes—
48C22×Q8×C83C2×C412yes—
49C2×M16×C83—21no—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83—21no—
52C4⋊Q8×C83—21no—
53C4⋊Q8×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C4⋊C4×C83—21no—
56C2×C42×C83C2311yescharacteristic
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83D412yes—
60C22⋊C8×C83—21no—
61C4×C8×C83—21no—
62C2×Dic166—21no—
63C2×Dic166—21no—
64C2×Dic166—21no—
65C2×Dic166—21no—
66C2×Dic166—21no—
67C2×Dic166—21no—
68C22×Dic83—22no—
69C22×Dic83C22⋊C412yes—
70C22×Dic83C22⋊C412yes—
71#13050574—41no—
72#13050574—41no—
73C2×C83⋊C8—22no—
74C2×C83⋊C8—41no—
75C4⋊Dic83—24no—
76C4⋊Dic83—21no—
77C4⋊Dic83—21no—
78Dic83⋊C4—22no—
79Dic83⋊C4—22no—
80Dic83⋊C4—84no—
81C4×Dic83—22no—
82C4×Dic83—22no—
83C4×Dic83—41no—
84C4×Dic83—41no—
85C2×Q8×C83—22no—
86C2×Q8×C83—22no—
87C2×Q8×C83—82no—
88C22×C4×C83—22no—
89C22×C4×C83C22×C411yescharacteristic
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92M16×C83—41no—
93M16×C83—41no—
94C2×C8×C83—22no—
95C2×C8×C83—41no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—41no—
100C42×C83C2×D411yescharacteristic
101C42×C83C2×D411yescharacteristic
102C42×C83#4111yescharacteristic
103C42×C83#4111yescharacteristic
104C23×C83C2×C22⋊C411yescharacteristic
105Q8×C83—82no—
106Q8×C83—82no—
107C2×C4×C83—22no—
108C2×C4×C83—41no—
109C2×C4×C83—44no—
110C2×C4×C83—42no—
111C2×C4×C83C4×D411yescharacteristic
112C2×C4×C83C4×D411yescharacteristic
113C2×C4×C83#12011yescharacteristic
114C2×C4×C83#12111yescharacteristic
115C2×C4×C83#12111yescharacteristic
116C2×C4×C83#12311yescharacteristic
117C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
118C2×C4×C83#13711yescharacteristic
119C2×C4×C83#13711yescharacteristic
120C2×C4×C83#13711yescharacteristic
121C2×Dic83—24no—
122C2×Dic83—22no—
123C2×Dic83—22no—
124C2×Dic83—22no—
125C2×Dic83—22no—
126C2×Dic83—42no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129Dic166—41no—
130Dic166—41no—
131Dic166—41no—
132Dic166—41no—
133C8×C83—42no—
134C8×C83—81no—
135C83⋊C8—42no—
136C83⋊C8—81no—
137C22×C83—21no—
138C22×C83—21no—
139C22×C83#38611yescharacteristic
140C22×C83#41711yescharacteristic
141C22×C83#41911yescharacteristic
142C4×C83—21no—
143C4×C83—21no—
144C4×C83—21no—
145C4×C83—82no—
146C4×C83#45612yes—
147C4×C83#46811yescharacteristic
148C4×C83#46811yescharacteristic
149C4×C83#47011yescharacteristic
150C4×C83#47011yescharacteristic
151Dic83—44no—
152Dic83—42no—
153Dic83—42no—
154#9400—831nomaximal, Sylow
155C2×C83—21no—
156C2×C83—21no—
157C2×C83#188611yescharacteristic
158C2×C83#189411yescharacteristic
159C2×C83#191411yescharacteristic
160#2983—831no—
161#2567—831no—
162#1801—831no—
163#1548—831no—
164#1548D8311yescharacteristic
165#1303—831no—
166#1303—831no—
167C83#940011yescharacteristic, complemented, Sylow
168C2×C4⋊Q8C2×D8311yescharacteristic
169C2×C4×Q8—831no—
170#446—1662no—
171#446—1662no—
172#432—1662no—
173#431—831no—
174#431C2×D8311yescharacteristic
175#423—831no—
176#423C2×D8311yescharacteristic
177C2×#101—1661no—
178#395—834no—
179#391—1661no—
180#389—1662no—
181#384—1661no—
182C4×C4⋊C4—831no—
183#343—1661no—
184#333—22no—
185#333—1662no—
186C22×Q8—831no—
187C22×Q8C4×D8312yes—
188C2×M16—21no—
189C2×M16—1661no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4×Q8—1661no—
195C4×Q8—1661no—
196C4×Q8—1661no—
197C4×Q8—1661no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—832no—
201C2×C4⋊C4—831no—
202C2×C4⋊C4—1664no—
203C2×C4⋊C4—1661no—
204C2×C42—832no—
205C2×C42—1661no—
206C2×C42C22×D8311yescharacteristic
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—832no—
210C4⋊C8—1664no—
211C4⋊C8—1661no—
212C4⋊C8—1661no—
213C4⋊C8C22⋊D8312yes—
214#101—3322no—
215C22⋊C8—21no—
216C22⋊C8—1661no—
217#97—1664no—
218C4×C8—21no—
219C4×C8—1661no—
220#95—832no—
221#95—1664no—
222C2×Q8—22no—
223C2×Q8—22no—
224C2×Q8—82no—
225C2×Q8—1661no—
226C2×Q8—1661no—
227C2×Q8—1661no—
228C2×Q8—1661no—
229C2×Q8—1661no—
230C2×Q8—1661no—
231C22×C4—22no—
232C22×C4—832no—
233C22×C4—832no—
234C22×C4—1662no—
235C22×C4C2×C4×D8311yescharacteristic
236C22×C4D4×D8311yescharacteristic
237C22×C4D4×D8311yescharacteristic
238M16—41no—
239M16—41no—
240M16—3321no—
241M16—3321no—
242C2×C8—22no—
243C2×C8—41no—
244C2×C8—3321no—
245C2×C8—1662no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—6644no—
251C4⋊C4—1664no—
252C4⋊C4—1661no—
253C4⋊C4—1661no—
254C4⋊C4—1662no—
255C4⋊C4—1662no—
256C42—3321no—
257C42—3321no—
258C42—1662no—
259C42—1662no—
260C42D4×D8311yescharacteristic
261C42C2×C22⋊D8311yescharacteristic
262C42#1305058911yescharacteristic
263C42#1305059011yescharacteristic
264C23C22⋊C4×D8311yescharacteristic
265Q8—82no—
266Q8—82no—
267Q8—3321no—
268Q8—3321no—
269Q8—3321no—
270Q8—3321no—
271C2×C4—22no—
272C2×C4—41no—
273C2×C4—44no—
274C2×C4—42no—
275C2×C4—3322no—
276C2×C4—3322no—
277C2×C4—3322no—
278C2×C4—1664no—
279C2×C4—1662no—
280C2×C4—1662no—
281C2×C4—1662no—
282C2×C4—1662no—
283C2×C4(C22⋊D83)⋊C411yescharacteristic
284C2×C4C4×C22⋊D8311yescharacteristic
285C2×C4#1458176011yescharacteristic
286C2×C4#1458176311yescharacteristic
287C2×C4#1458176511yescharacteristic
288C2×C4#1458177411yescharacteristic
289C2×C4#1458181911yescharacteristic
290C2×C4#1458182111yescharacteristic
291C2×C4#1458182411yescharacteristic
292C2×C4#1458182411yescharacteristic
293C8—42no—
294C8—81no—
295C8—3322no—
296C8—6641no—
297C22—21no—
298C22—21no—
299C22#1912757711yescharacteristic
300C22#1912770111yescharacteristic
301C22#1912770811yescharacteristic
302C4—21no—
303C4—21no—
304C4—21no—
305C4—82no—
306C4—3324no—
307C4—3322no—
308C4—3322no—
309C4#1912788212yes—
310C4#1912794411yescharacteristic
311C4#1912795011yescharacteristic
312C4#1912795611yescharacteristic
313C4#1912795811yescharacteristic
314C2—21no—
315C2—21no—
316C2#2490785311yescharacteristic
317C2#2490789311yescharacteristic
318C2#2490800711yescharacteristic
subgroupG/H
centerC22#19127577
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1548C2
socleC22×C83#386
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1548D83
hypocenterC83#9400
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1548

Complements

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

NH ≅ G/Ntype
167. C83#9400semidirect

Maximal subgroups

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

subgroupindex
#249165462
#249126722
#249074562
#249061442
#249053652
#249053642
C83×#15482
#940083