Finite Group Databasebeta

← #35322686

Subgroups — #35322686

Order 21248.

536 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 103 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#24911771C211yesmaximal, characteristic
2C2×#19127490C214yesmaximal
3C4×#14581709C211yesmaximal, characteristic
4C83×#1799C211yesmaximal, characteristic
5C2×#14581773C2212yes—
6C22×C4⋊Dic83C411yescharacteristic, complemented
7C2×#14581709C2212yes—
8C4×#13050574—21no—
9#19127631—21no—
10#19127626—21no—
11#19127599—21no—
12#19127562C412yescomplemented
13#19127561—21no—
14#19127561C411yescharacteristic, complemented
15#19127490—28no—
16#19127470—21no—
17C2×C4×D4×C83C2211yescharacteristic
18C2×#99×C83C2212yes—
19#380×C83—21no—
20#342×C83—21no—
21C23×Dic83C2×C412yes—
22C2×#13050574—22no—
23#14581773—24no—
24#14581773—24no—
25C2×C4⋊Dic83—21no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—41no—
29C2×C4⋊Dic83—41no—
30C2×Dic83⋊C4—22no—
31C2×C22⋊Dic83—22no—
32C2×C22⋊Dic83C2×C412yes—
33#14581709—22no—
34#14581709D414yes—
35#14581703—41no—
36C4×C83⋊C8—41no—
37C83⋊#95—22no—
38C83⋊#95—21no—
39C83⋊#95—22no—
40C22×D4×C83C2311yescharacteristic
41C23×C4×C83C2×C411yescharacteristic
42C23×C4×C83D411yescharacteristic
43C4×D4×C83—41no—
44C4×D4×C83—41no—
45C2×C4⋊C4×C83—41no—
46C2×C4⋊C4×C83D411yescharacteristic
47C2×C22⋊C4×C83—22no—
48C2×C22⋊C4×C83C2×C411yescharacteristic
49C2×C22⋊C4×C83D411yescharacteristic
50C2×C42×C83D411yescharacteristic
51#99×C83—24no—
52#95×C83—41no—
53C22×Dic83—22no—
54C22×Dic83—22no—
55C22×Dic83—24no—
56C22×Dic83—42no—
57C22×Dic83—42no—
58#13050574—24no—
59#13050574—42no—
60C2×C83⋊C8—42no—
61C4⋊Dic83—42no—
62C4⋊Dic83—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—41no—
65Dic83⋊C4—44no—
66C22⋊Dic83—24no—
67C22⋊Dic83—42no—
68C22⋊Dic83—42no—
69C24×C83C22×C411yescharacteristic
70C24×C83C2×D411yescharacteristic
71C2×D4×C83—41no—
72C2×D4×C83—41no—
73C2×D4×C83C2×D412yes—
74C2×D4×C83#4112yes—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—42no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83C22⋊C411yescharacteristic
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87C4⋊C4×C83—41no—
88C4⋊C4×C83—82no—
89C22⋊C4×C83—41no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C42×C83—21no—
94C42×C83—21no—
95C23×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×C83C4×D412yes—
105C23×C83#12411yescharacteristic
106C23×C83#12711yescharacteristic
107D4×C83—42no—
108D4×C83—42no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—22no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—42no—
116C2×C4×C83—42no—
117C2×C4×C83—42no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—81no—
122C2×C4×C83—82no—
123C2×C4×C83—82no—
124C2×C4×C83—82no—
125C2×C4×C83#9911yescharacteristic
126C2×C4×C83#9911yescharacteristic
127C2×C4×C83#12112yes—
128C2×Dic83—42no—
129C2×Dic83—42no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132C2×Dic83—42no—
133C2×Dic83—42no—
134C2×Dic83—84no—
135C83⋊C8—44no—
136C22×C83—21no—
137C22×C83—21no—
138C22×C83—22no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—42no—
142C22×C83—41no—
143C22×C83—41no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—81no—
150C22×C83#39011yescharacteristic
151C22×C83C2×#9911yescharacteristic
152C22×C83#41011yescharacteristic
153C4×C83—21no—
154C4×C83—22no—
155C4×C83—41no—
156C4×C83—41no—
157C4×C83—41no—
158C4×C83—162no—
159C4×C83#35212yescomplemented
160Dic83—82no—
161Dic83—82no—
162#13217—831nomaximal, Sylow
163C2×C83—21no—
164C2×C83—21no—
165C2×C83—41no—
166C2×C83—42no—
167C2×C83—41no—
168C2×C83#196411yescharacteristic
169C2×C83C2×#35211yescharacteristic
170C2×C83#212311yescharacteristic
171#2486—831no—
172C2×#352—834no—
173#1799D8311yescharacteristic
174C4×#100—831no—
175C83#1321711yescharacteristic, complemented, Sylow
176C2×#123—832no—
177C2×C4×D4C2×D8311yescharacteristic
178C22×C4⋊C4—831no—
179C2×#100—832no—
180C2×#99C2×D8312yes—
181C4×M16—1661no—
182#398—1661no—
183#397—1661no—
184#389—1661no—
185#381—832no—
186#380—21no—
187#380—831no—
188#380—1661no—
189#352—1668no—
190#343—1661no—
191#342—21no—
192C22×D4C22×D8311yescharacteristic
193C23×C4—832no—
194C23×C4C4×D8311yescharacteristic
195C23×C4C22⋊D8311yescharacteristic
196C2×M16—1662no—
197#123—1664no—
198#123—1664no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—3321no—
203C2×C4⋊C4—3321no—
204C2×C4⋊C4—1661no—
205C2×C4⋊C4—1661no—
206C2×C4⋊C4—1661no—
207C2×C4⋊C4—1662no—
208C2×C4⋊C4D33211yescharacteristic
209C2×C22⋊C4—22no—
210C2×C22⋊C4—832no—
211C2×C22⋊C4—1662no—
212C2×C22⋊C4C4×D8311yescharacteristic
213C2×C22⋊C4C22⋊D8311yescharacteristic
214C2×C42D33211yescharacteristic
215#100—834no—
216#100—1662no—
217#99—24no—
218#97—3321no—
219C4×C8—3321no—
220#95—41no—
221#95—1661no—
222#95—1662no—
223#95—1662no—
224C24C2×C4×D8311yescharacteristic
225C24C2×C22⋊D8311yescharacteristic
226C2×D4—41no—
227C2×D4—41no—
228C2×D4D4×D8312yes—
229C2×D4#1305059112yes—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—42no—
237C22×C4—3322no—
238C22×C4—3322no—
239C22×C4—1662no—
240C22×C4—1662no—
241C22×C4—1664no—
242C22×C4(C2×D83)⋊C411yescharacteristic
243C22×C4(C2×D83)⋊C411yescharacteristic
244C22×C4(C2×D83)⋊C411yescharacteristic
245C22×C4(C2×D83)⋊C411yescharacteristic
246C22×C4C2×D33211yescharacteristic
247M16—3322no—
248M16—1664no—
249C2×C8—3322no—
250C4⋊C4—41no—
251C4⋊C4—82no—
252C4⋊C4—3322no—
253C4⋊C4—3322no—
254C4⋊C4—3324no—
255C4⋊C4—3321no—
256C4⋊C4—3321no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—3322no—
262C22⋊C4—3322no—
263C22⋊C4—1664no—
264C42—21no—
265C42—21no—
266C23—21no—
267C23—21no—
268C23—21no—
269C23—21no—
270C23—41no—
271C23—41no—
272C23—41no—
273C23—41no—
274C23C2×(C2×D83)⋊C411yescharacteristic
275C23#1458175412yes—
276C23#1458178111yescharacteristic
277C23#1458178811yescharacteristic
278D4—42no—
279D4—42no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—81no—
293C2×C4—82no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—3322no—
297C2×C4—3322no—
298C2×C4—3322no—
299C2×C4—3322no—
300C2×C4—3322no—
301C2×C4—3322no—
302C2×C4—6644no—
303C2×C4#1458170611yescharacteristic
304C2×C4#1458170611yescharacteristic
305C2×C4#1458176712yes—
306C8—3324no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—21no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—42no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1912760211yescharacteristic
322C22C2×#1458170611yescharacteristic
323C22#1912767411yescharacteristic
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—162no—
330C4—6642no—
331C4—6642no—
332C4#1912749012yescomplemented
333C2—21no—
334C2—21no—
335C2—41no—
336C2—42no—
337C2—41no—
338C2#2490829811yescharacteristic
339C2C2×#1912749011yescharacteristic
340C2#2490911611yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC24×C83C22×C4
FrattiniC22×D4C22×D83
FittingC83×#1799C2
socleC22×C83C2×#99
prime radical, cosocleC22×D4×C83C23
hypercenter#1799D83
hypocenterC83#13217
LCSG ▷ C24×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1799

Complements

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

NH ≅ G/Ntype
6. C22×C4⋊Dic83C4semidirect
12. #19127562C4semidirect
14. #19127561C4semidirect
159. C4×C83#352semidirect
175. C83#13217semidirect
332. C4#19127490semidirect

Maximal subgroups

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

subgroupindex
#249117712
C2×#191274902
C4×#145817092
C83×#17992
#1321783