Finite Group Databasebeta

← #35302191

Subgroups — #35302191

Order 21248.

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

normal subgroups only

This group has 556 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#24916492C211yesmaximal, characteristic
2#24911780C211yesmaximal, characteristic
3#24907551C212yesmaximal, complemented
4#24905330C212yesmaximal, complemented
5C83×#1799C211yesmaximal, characteristic
6C22×C4⋊Dic83C2211yescharacteristic
7#19127773—24no—
8#19127758—21no—
9#19127665—21no—
10C2×C22⋊(C83⋊C8)C2212yes—
11#19127640C412yescomplemented
12#19127637C412yescomplemented
13#19127599—21no—
14#19127581—21no—
15#19127573—21no—
16#19127561—21no—
17#19127561C2211yescharacteristic
18#19127463—24no—
19C2×C4×D4×C83C2211yescharacteristic
20C2×#99×C83—21no—
21#380×C83C2212yes—
22#342×C83—21no—
23C23×Dic83—21no—
24C2×#13050574—22no—
25C22×C83⋊C8D414yes—
26C2×C4⋊Dic83—22no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—21no—
29C2×C4⋊Dic83—22no—
30C2×C4⋊Dic83C2×C412yes—
31C2×Dic83⋊C4—22no—
32C2×C22⋊Dic83—22no—
33C2×C22⋊Dic83—21no—
34C4⋊(C83⋊C8)—22no—
35C22⋊(C83⋊C8)—22no—
36C22⋊(C83⋊C8)—22no—
37C22⋊(C83⋊C8)—22no—
38#14581703—22no—
39C83⋊#95—22no—
40C83⋊#95—22no—
41C83⋊#95C2×C412yes—
42C22×D4×C83D411yescharacteristic
43C23×C4×C83D411yescharacteristic
44C23×C4×C83C2311yescharacteristic
45C4×D4×C83—22no—
46C4×D4×C83—22no—
47C2×C4⋊C4×C83—22no—
48C2×C4⋊C4×C83C2×C411yescharacteristic
49C2×C22⋊C4×C83—22no—
50C2×C22⋊C4×C83C2×C411yescharacteristic
51C2×C22⋊C4×C83D411yescharacteristic
52C2×C42×C83D411yescharacteristic
53#99×C83—42no—
54#95×C83—22no—
55C22×Dic83—22no—
56C22×Dic83—22no—
57C22×Dic83—42no—
58C22×Dic83—82no—
59#13050574—82no—
60C2×C83⋊C8—24no—
61C2×C83⋊C8—44no—
62C4⋊Dic83—24no—
63C4⋊Dic83—42no—
64C4⋊Dic83—41no—
65C4⋊Dic83—41no—
66Dic83⋊C4—82no—
67C22⋊Dic83—41no—
68C22⋊Dic83—41no—
69C22⋊Dic83—82no—
70C24×C83C22⋊C411yescharacteristic
71C24×C83C2×D411yescharacteristic
72C2×D4×C83—22no—
73C2×D4×C83—21no—
74C2×D4×C83—22no—
75C2×D4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—42no—
79C22×C4×C83—42no—
80C22×C4×C83—42no—
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D412yes—
86C22×C4×C83C2×D411yescharacteristic
87C22×C4×C83#4112yes—
88C4⋊C4×C83—22no—
89C4⋊C4×C83—44no—
90C22⋊C4×C83—22no—
91C22⋊C4×C83—22no—
92C22⋊C4×C83—42no—
93C22⋊C4×C83—42no—
94C42×C83—21no—
95C42×C83—21no—
96C23×C83—21no—
97C23×C83—21no—
98C23×C83—21no—
99C23×C83—41no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83C2×C22⋊C411yescharacteristic
104C23×C83#12112yes—
105C23×C83AGL(1,ℤ/8)11yescharacteristic
106C23×C83#13712yes—
107C23×C83#13711yescharacteristic
108D4×C83—42no—
109D4×C83—42no—
110C2×C4×C83—24no—
111C2×C4×C83—22no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—21no—
117C2×C4×C83—41no—
118C2×C4×C83—42no—
119C2×C4×C83—42no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—81no—
123C2×C4×C83—82no—
124C2×C4×C83—82no—
125C2×C4×C83—82no—
126C2×C4×C83C4×D412yes—
127C2×C4×C83#12011yescharacteristic
128C2×C4×C83#12311yescharacteristic
129C2×Dic83—42no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132C2×Dic83—42no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—84no—
136C83⋊C8—44no—
137C22×C83—21no—
138C22×C83—22no—
139C22×C83—21no—
140C22×C83—22no—
141C22×C83—41no—
142C22×C83—42no—
143C22×C83—41no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—81no—
149C22×C83#38611yescharacteristic
150C22×C83#41711yescharacteristic
151C22×C83#41911yescharacteristic
152C22×C83#46812yes—
153C22×C83#47012yes—
154C4×C83—21no—
155C4×C83—22no—
156C4×C83—41no—
157C4×C83—41no—
158C4×C83—41no—
159C4×C83—162no—
160C4×C83#45812yes—
161Dic83—82no—
162Dic83—82no—
163#9473—831nomaximal, Sylow
164C2×C83—21no—
165C2×C83—22no—
166C2×C83—21no—
167C2×C83—22no—
168C2×C83—81no—
169C2×C83#189011yescharacteristic
170C2×C83#189411yescharacteristic
171C2×C83#191011yescharacteristic
172#2973—831no—
173#2487—831no—
174#1824—832no—
175#1799D8311yescharacteristic
176#1289—832no—
177C83#947311yescharacteristic, complemented, Sylow
178C2×C4×D4C2×D8311yescharacteristic
179C22×C4⋊C4—831no—
180#435—1664no—
181#432—1661no—
182C2×#99—21no—
183#407—1661no—
184C2×C22⋊C8—832no—
185#400—832no—
186#399—832no—
187#389—1661no—
188#386—1661no—
189#385—1661no—
190#380—831no—
191#380—1661no—
192#380C2×D8312yes—
193#342—21no—
194#340—1664no—
195C22×D4C22⋊D8311yescharacteristic
196C23×C4—1661no—
197C23×C4D33211yescharacteristic
198C23×C4C22×D8311yescharacteristic
199C2×M16—1662no—
200C22×C8—834no—
201C4×D4—22no—
202C4×D4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—832no—
205C2×C4⋊C4—1661no—
206C2×C4⋊C4—1661no—
207C2×C4⋊C4—1662no—
208C2×C4⋊C4—1662no—
209C2×C4⋊C4—1662no—
210C2×C4⋊C4C4×D8311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1661no—
213C2×C22⋊C4—1662no—
214C2×C22⋊C4C4×D8311yescharacteristic
215C2×C22⋊C4C22⋊D8311yescharacteristic
216C2×C42D33211yescharacteristic
217C4⋊C8—1662no—
218#99—42no—
219C22⋊C8—1662no—
220C22⋊C8—1662no—
221C22⋊C8—1662no—
222#97—1662no—
223#95—22no—
224#95—832no—
225#95—1662no—
226#95—1662no—
227C24(C2×D83)⋊C411yescharacteristic
228C24C2×C22⋊D8311yescharacteristic
229C2×D4—22no—
230C2×D4—21no—
231C2×D4—22no—
232C2×D4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—3322no—
239C22×C4—6642no—
240C22×C4—1662no—
241C22×C4—1662no—
242C22×C4(C2×D83)⋊C411yescharacteristic
243C22×C4(C2×D83)⋊C411yescharacteristic
244C22×C4(C2×D83)⋊C411yescharacteristic
245C22×C4C2×C4×D8311yescharacteristic
246C22×C4D4×D8312yes—
247C22×C4C2×D33211yescharacteristic
248C22×C4#1305058912yes—
249M16—6642no—
250C2×C8—3324no—
251C2×C8—1664no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C4⋊C4—3322no—
255C4⋊C4—3321no—
256C4⋊C4—3321no—
257C4⋊C4—6642no—
258C4⋊C4—1664no—
259C22⋊C4—22no—
260C22⋊C4—22no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—3321no—
264C22⋊C4—3321no—
265C22⋊C4—6642no—
266C42—21no—
267C42—21no—
268C23—21no—
269C23—21no—
270C23—21no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23—41no—
275C23C2×(C2×D83)⋊C411yescharacteristic
276C23#1458176412yes—
277C23#1458181811yescharacteristic
278C23#1458182212yes—
279C23#1458182311yescharacteristic
280D4—42no—
281D4—42no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—81no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—3322no—
299C2×C4—3322no—
300C2×C4—3322no—
301C2×C4—3322no—
302C2×C4—3322no—
303C2×C4—3322no—
304C2×C4—6644no—
305C2×C4(C22⋊D83)⋊C412yes—
306C2×C4#1458176211yescharacteristic
307C2×C4#1458177611yescharacteristic
308C8—3324no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—41no—
314C22—42no—
315C22—41no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1912757811yescharacteristic
322C22#1912770011yescharacteristic
323C22#1912770911yescharacteristic
324C22#1912794512yes—
325C22#1912795712yes—
326C4—21no—
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—162no—
332C4—6642no—
333C4—6642no—
334C4#1912789212yes—
335C2—22no—
336C2—21no—
337C2—21no—
338C2—22no—
339C2—81no—
340C2#2490787411yescharacteristic
341C2#2490789411yescharacteristic
342C2#2490798811yescharacteristic
subgroupG/H
centerC22#19127578
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#1799C2
socleC22×C83#386
prime radicalC22×D4×C83D4
cosocleC23×C4×C83C23
hypercenter#1799D83
hypocenterC83#9473
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
3. #24907551C2semidirect
4. #24905330C2semidirect
11. #19127640C4semidirect
12. #19127637C4semidirect
177. C83#9473semidirect

Maximal subgroups

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

subgroupindex
#249164922
#249117802
#249075512
#249053302
C83×#17992
#947383