Finite Group Databasebeta

← #35308019

Subgroups — #35308019

Order 21248.

408 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×#19127911C211yesmaximal, characteristic
2#24906235C211yesmaximal, characteristic
3#24906136C211yesmaximal, characteristic
4#24906136C211yesmaximal, characteristic
5#24905858C211yesmaximal, characteristic
6#24905358C211yesmaximal, characteristic
7C83×#1801C211yesmaximal, characteristic
8C22×#13050580—21no—
9C2×#14581792C2211yescharacteristic
10#19127911—22no—
11#19127911—22no—
12#19127754C2211yescharacteristic
13#19127754C2211yescharacteristic
14C2×C4⋊(C83⋊C8)C2211yescharacteristic
15C2×#14581715—21no—
16#19127451—22no—
17#19127448—22no—
18#19127448—22no—
19Q8⋊(C83⋊C8)—22no—
20C2×C4×Q8×C83C2211yescharacteristic
21#423×C83C2211yescharacteristic
22#423×C83C2211yescharacteristic
23C2×#101×C83—21no—
24#343×C83—21no—
25C22×Dic166—21no—
26C2×#13050580—24no—
27C2×#13050580—42no—
28C2×#13050580—42no—
29C22×C83⋊C8—21no—
30#14581792—21no—
31#14581792—21no—
32#14581792—21no—
33#14581792D412yes—
34C2×C4⋊Dic83—21no—
35C2×C4⋊Dic83—21no—
36C4⋊(C83⋊C8)—21no—
37C4⋊(C83⋊C8)—21no—
38C4⋊(C83⋊C8)D412yes—
39C4⋊(C83⋊C8)D412yes—
40#14581715—24no—
41C22⋊(C83⋊C8)—21no—
42C22⋊(C83⋊C8)—21no—
43C4×C83⋊C8—21no—
44C4×C83⋊C8—21no—
45C22×Q8×C83D411yescharacteristic
46C2×M16×C83—21no—
47C2×M16×C83—21no—
48C4×Q8×C83—21no—
49C4×Q8×C83—21no—
50C4×Q8×C83—21no—
51C4×Q8×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83D411yescharacteristic
54C2×C42×C83—21no—
55C2×C42×C83C2311yescharacteristic
56C4⋊C8×C83—21no—
57C4⋊C8×C83—21no—
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60#101×C83—42no—
61C2×Dic166—22no—
62C2×Dic166—41no—
63C2×Dic166—41no—
64C2×Dic166—41no—
65C2×Dic166—41no—
66C2×Dic166—41no—
67C22×Dic83—41no—
68#13050580—48no—
69C2×C83⋊C8—22no—
70C2×C83⋊C8—22no—
71C2×C83⋊C8—41no—
72C2×C83⋊C8—41no—
73C2×C83⋊C8—41no—
74C4⋊Dic83—22no—
75C4⋊Dic83—22no—
76C4⋊Dic83—41no—
77C4⋊Dic83—41no—
78C2×Q8×C83—21no—
79C2×Q8×C83—21no—
80C2×Q8×C83—21no—
81C2×Q8×C83—21no—
82C2×Q8×C83—21no—
83C2×Q8×C83SD1612yes—
84C22×C4×C83—21no—
85C22×C4×C83—21no—
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83#4111yescharacteristic
88C22×C4×C83#4111yescharacteristic
89M16×C83—41no—
90M16×C83—41no—
91M16×C83—41no—
92M16×C83—41no—
93C2×C8×C83—41no—
94C2×C8×C83—41no—
95C4⋊C4×C83—21no—
96C4⋊C4×C83—21no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C42×C83—41no—
100C42×C83—41no—
101C42×C83C2×D411yescharacteristic
102C42×C83C2×D411yescharacteristic
103C42×C83C2×D411yescharacteristic
104C42×C83#4111yescharacteristic
105C23×C83#12311yescharacteristic
106Q8×C83—22no—
107Q8×C83—22no—
108Q8×C83—41no—
109Q8×C83—41no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—41no—
113C2×C4×C83—41no—
114C2×C4×C83—41no—
115C2×C4×C83—41no—
116C2×C4×C83—81no—
117C2×C4×C83#12011yescharacteristic
118C2×C4×C83#12111yescharacteristic
119C2×C4×C83#12111yescharacteristic
120C2×C4×C83#12111yescharacteristic
121C2×C4×C83#12311yescharacteristic
122C2×C4×C83#12411yescharacteristic
123C2×C4×C83C2×SD1611yescharacteristic
124C2×C4×C83#13511yescharacteristic
125C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
127C2×Dic83—42no—
128C2×Dic83—81no—
129C2×Dic83—81no—
130Dic166—42no—
131Dic166—42no—
132Dic166—81no—
133Dic166—81no—
134C8×C83—81no—
135C8×C83—81no—
136C83⋊C8—42no—
137C83⋊C8—42no—
138C83⋊C8—81no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83#39411yescharacteristic
142C22×C83#48111yescharacteristic
143C22×C83#48211yescharacteristic
144C4×C83—21no—
145C4×C83—21no—
146C4×C83—21no—
147C4×C83—42no—
148C4×C83—81no—
149C4×C83#44911yescharacteristic
150C4×C83#45011yescharacteristic
151C4×C83#46511yescharacteristic
152C4×C83#46911yescharacteristic
153C4×C83#47312yes—
154Dic83—82no—
155#10447—831nomaximal, Sylow
156C2×C83—21no—
157C2×C83—21no—
158C2×C83#203411yescharacteristic
159C2×C83#204711yescharacteristic
160C2×C83#204911yescharacteristic
161C2×#462—831no—
162#1801D8311yescharacteristic
163#1570—831no—
164#1546—831no—
165#1546—831no—
166#1479—831no—
167#1301—831no—
168C83#1044711yescharacteristic, complemented, Sylow
169C22×Q16—1661no—
170C2×C4⋊Q8—831no—
171C2×C4×Q8C2×D8311yescharacteristic
172#462—1662no—
173#462—1662no—
174#431—831no—
175#431—831no—
176#423C2×D8311yescharacteristic
177#423C2×D8311yescharacteristic
178C2×C4⋊C8—831no—
179C2×#103—1661no—
180C2×#101—21no—
181#343—21no—
182#334—1662no—
183#332—1662no—
184#332—1662no—
185Q8⋊C8—1662no—
186C22×Q8—1661no—
187C22×Q8C22⋊D8311yescharacteristic
188C2×Q16—3322no—
189C2×Q16—3322no—
190C2×Q16—1664no—
191C2×M16—21no—
192C2×M16—21no—
193C22×C8—1661no—
194C4⋊Q8—832no—
195C4⋊Q8—1661no—
196C4⋊Q8—1661no—
197C4⋊Q8—1661no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1661no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4C22⋊D8311yescharacteristic
206C2×C42—21no—
207C2×C42C22×D8311yescharacteristic
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—832no—
213C4⋊C8—832no—
214C4⋊C8—1661no—
215C4⋊C8—1661no—
216#103—1664no—
217#101—42no—
218C22⋊C8—1661no—
219C22⋊C8—1661no—
220C4×C8—1661no—
221C4×C8—1661no—
222C2×Q8—21no—
223C2×Q8—21no—
224C2×Q8—21no—
225C2×Q8—21no—
226C2×Q8—21no—
227C2×Q8—3321no—
228C2×Q8—3321no—
229C2×Q8—3321no—
230C2×Q8—3321no—
231C2×Q8—3321no—
232C2×Q8—1662no—
233C2×Q8#1305057712yes—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—3321no—
237C22×C4C2×C22⋊D8311yescharacteristic
238C22×C4#1305059011yescharacteristic
239C22×C4#1305059011yescharacteristic
240Q16—3328no—
241M16—41no—
242M16—41no—
243M16—41no—
244M16—41no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8—3321no—
248C2×C8—3321no—
249C2×C8—3321no—
250C2×C8—1662no—
251C2×C8—1662no—
252C4⋊C4—21no—
253C4⋊C4—21no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—3321no—
257C4⋊C4—3321no—
258C4⋊C4—1662no—
259C4⋊C4—1662no—
260C42—41no—
261C42—41no—
262C42D4×D8311yescharacteristic
263C42D4×D8311yescharacteristic
264C42D4×D8311yescharacteristic
265C42#1305058911yescharacteristic
266C23#1458177511yescharacteristic
267Q8—22no—
268Q8—22no—
269Q8—41no—
270Q8—41no—
271Q8—3322no—
272Q8—3322no—
273Q8—6641no—
274Q8—6641no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—3322no—
283C2×C4—6641no—
284C2×C4—6641no—
285C2×C4#1458176011yescharacteristic
286C2×C4#1458176411yescharacteristic
287C2×C4#1458176511yescharacteristic
288C2×C4#1458176511yescharacteristic
289C2×C4#1458177411yescharacteristic
290C2×C4#1458177811yescharacteristic
291C2×C4C2×#1305057711yescharacteristic
292C2×C4#1458181211yescharacteristic
293C2×C4#1458181711yescharacteristic
294C2×C4#1458181811yescharacteristic
295C8—81no—
296C8—81no—
297C8—3322no—
298C8—3322no—
299C8—6641no—
300C22—21no—
301C22—21no—
302C22#1912761611yescharacteristic
303C22#1912801911yescharacteristic
304C22#1912802511yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—42no—
309C4—81no—
310C4—6642no—
311C4#1912784411yescharacteristic
312C4#1912785111yescharacteristic
313C4#1912793211yescharacteristic
314C4#1912795211yescharacteristic
315C4#1912797512yes—
316C2—21no—
317C2—21no—
318C2#2490865211yescharacteristic
319C2#2490872911yescharacteristic
320C2#2490874311yescharacteristic
subgroupG/H
centerC22#19127616
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1801C2
socleC22×C83#394
prime radicalC23×C83#123
hypercenter#1801D83
hypocenterC83#10447
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
168. C83#10447semidirect

Maximal subgroups

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

subgroupindex
C2×#191279112
#249062352
#249061362
#249058582
#249053582
C83×#18012
#1044783