Finite Group Databasebeta

← #35322716

Subgroups — #35322716

Order 21248.

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

normal subgroups only

This group has 552 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#24912455C211yesmaximal, characteristic
2C2×#19127495C214yesmaximal
3C4×#14581707C211yesmaximal, characteristic
4C83×#1799C211yesmaximal, characteristic
5C2×#14581792C411yescharacteristic, complemented
6C2×#14581779C2212yes—
7C2×#14581707C2212yes—
8#19127626—21no—
9#19127626—21no—
10#19127606C412yescomplemented
11#19127561—21no—
12#19127561—21no—
13C4×C4⋊Dic83C411yescharacteristic, complemented
14C4×C22⋊Dic83—21no—
15#19127495—28no—
16#19127468—21no—
17C2×C4×D4×C83C2211yescharacteristic
18C2×#99×C83C2212yes—
19#380×C83—21no—
20#342×C83—21no—
21C22×Dic166C2×C412yes—
22#14581792—41no—
23#14581792—41no—
24#14581779—24no—
25#14581779—24no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83—21no—
28C2×C4⋊Dic83—21no—
29C2×C22⋊Dic83—22no—
30C2×C22⋊Dic83—22no—
31C2×C22⋊Dic83—22no—
32C2×C4×Dic83—41no—
33C2×C4×Dic83C2×C412yes—
34#14581707—22no—
35#14581707D414yes—
36C83⋊#95—21no—
37C83⋊#95—22no—
38C83⋊#95—22no—
39C83⋊#95—41no—
40C22×D4×C83C2311yescharacteristic
41C23×C4×C83D411yescharacteristic
42C23×C4×C83D411yescharacteristic
43C4×D4×C83—41no—
44C4×D4×C83—41no—
45C2×C4⋊C4×C83—41no—
46C2×C4⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83—22no—
48C2×C22⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83D411yescharacteristic
50C2×C42×C83C2×C411yescharacteristic
51#99×C83—24no—
52#95×C83—41no—
53C2×Dic166—24no—
54C2×Dic166—42no—
55C2×Dic166—42no—
56C22×Dic83—22no—
57C22×Dic83—22no—
58C22×Dic83—42no—
59C4⋊Dic83—42no—
60C4⋊Dic83—42no—
61C4⋊Dic83—41no—
62C4⋊Dic83—41no—
63C22⋊Dic83—24no—
64C22⋊Dic83—42no—
65C22⋊Dic83—44no—
66C22⋊Dic83—42no—
67C22⋊Dic83—42no—
68C4×Dic83—24no—
69C4×Dic83—82no—
70C24×C83C2×D411yescharacteristic
71C24×C83C2×D411yescharacteristic
72C2×D4×C83—41no—
73C2×D4×C83—41no—
74C2×D4×C83C2×D412yes—
75C2×D4×C83#4112yes—
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—21no—
80C22×C4×C83—42no—
81C22×C4×C83—42no—
82C22×C4×C83—42no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22×C411yescharacteristic
88C4⋊C4×C83—41no—
89C4⋊C4×C83—82no—
90C22⋊C4×C83—41no—
91C22⋊C4×C83—41no—
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—21no—
100C23×C83—41no—
101C23×C83—41no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83C2×C22⋊C411yescharacteristic
105C23×C83#12112yes—
106C23×C83#12411yescharacteristic
107C23×C83#12711yescharacteristic
108D4×C83—42no—
109D4×C83—42no—
110C2×C4×C83—21no—
111C2×C4×C83—21no—
112C2×C4×C83—22no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
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—81no—
123C2×C4×C83—82no—
124C2×C4×C83—82no—
125C2×C4×C83—82no—
126C2×C4×C83#9911yescharacteristic
127C2×C4×C83#9911yescharacteristic
128C2×C4×C83C4×D412yes—
129C2×Dic83—44no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132C2×Dic83—42no—
133C2×Dic83—42no—
134C2×Dic83—42no—
135C2×Dic83—42no—
136C2×Dic83—82no—
137C2×Dic83—82no—
138Dic166—84no—
139C22×C83—21no—
140C22×C83—21no—
141C22×C83—22no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—42no—
145C22×C83—41no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—41no—
149C22×C83—41no—
150C22×C83—41no—
151C22×C83—81no—
152C22×C83—81no—
153C22×C83#39011yescharacteristic
154C22×C83C2×#9911yescharacteristic
155C22×C83#41011yescharacteristic
156C4×C83—21no—
157C4×C83—22no—
158C4×C83—41no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—162no—
162C4×C83#35412yescomplemented
163Dic83—82no—
164Dic83—84no—
165Dic83—82no—
166#13224—831nomaximal, Sylow
167C2×C83—21no—
168C2×C83—21no—
169C2×C83—42no—
170C2×C83—41no—
171C2×C83—41no—
172C2×C83#196411yescharacteristic
173C2×C83C2×#35411yescharacteristic
174C2×C83#212911yescharacteristic
175#2547—831no—
176C2×#354—834no—
177#1799D8311yescharacteristic
178C4×#99—831no—
179C83#1322411yescharacteristic, complemented, Sylow
180C2×C4⋊Q8—831no—
181C2×#124—832no—
182C2×C4×D4C2×D8311yescharacteristic
183C2×#99—832no—
184C2×#99C2×D8312yes—
185#397—1661no—
186#397—1661no—
187#391—832no—
188#380—21no—
189#380—1661no—
190#380—1661no—
191C4×C4⋊C4—831no—
192C4×C22⋊C4—1661no—
193#354—1668no—
194#342—21no—
195#342—1661no—
196C22×Q8—832no—
197C22×D4C22×D8311yescharacteristic
198C23×C4D33211yescharacteristic
199C23×C4C22⋊D8311yescharacteristic
200C4⋊Q8—3321no—
201C4⋊Q8—3321no—
202#124—1664no—
203#124—1664no—
204C4×D4—41no—
205C4×D4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—1661no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4—1661no—
210C2×C4⋊C4C4×D8311yescharacteristic
211C2×C22⋊C4—22no—
212C2×C22⋊C4—1662no—
213C2×C22⋊C4—1662no—
214C2×C22⋊C4—1662no—
215C2×C22⋊C4D33211yescharacteristic
216C2×C22⋊C4C22⋊D8311yescharacteristic
217C2×C42—3321no—
218C2×C42—832no—
219C2×C42C4×D8311yescharacteristic
220#99—24no—
221#99—834no—
222#99—1662no—
223#95—41no—
224#95—3321no—
225#95—1661no—
226#95—1662no—
227#95—1662no—
228C24C2×C22⋊D8311yescharacteristic
229C24C2×D33211yescharacteristic
230C2×Q8—3322no—
231C2×Q8—3322no—
232C2×Q8—1664no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4D4×D8312yes—
236C2×D4#1305059112yes—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—3322no—
245C22×C4—1662no—
246C22×C4—1662no—
247C22×C4(C2×D83)⋊C411yescharacteristic
248C22×C4(C2×D83)⋊C411yescharacteristic
249C22×C4(C2×D83)⋊C411yescharacteristic
250C22×C4(C2×D83)⋊C411yescharacteristic
251C22×C4C2×C4×D8311yescharacteristic
252C4⋊C4—41no—
253C4⋊C4—82no—
254C4⋊C4—3322no—
255C4⋊C4—3322no—
256C4⋊C4—3321no—
257C4⋊C4—3321no—
258C22⋊C4—41no—
259C22⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—3322no—
263C22⋊C4—3324no—
264C22⋊C4—3322no—
265C22⋊C4—3322no—
266C22⋊C4—1664no—
267C42—21no—
268C42—21no—
269C42—6642no—
270C42—1664no—
271C23—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23C2×(C2×D83)⋊C411yescharacteristic
280C23#1458176712yes—
281C23#1458178111yescharacteristic
282C23#1458178811yescharacteristic
283Q8—6644no—
284D4—42no—
285D4—42no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—82no—
300C2×C4—82no—
301C2×C4—82no—
302C2×C4—3324no—
303C2×C4—3322no—
304C2×C4—3322no—
305C2×C4—3322no—
306C2×C4—3322no—
307C2×C4—3322no—
308C2×C4—3322no—
309C2×C4—6642no—
310C2×C4—6642no—
311C2×C4#1458170611yescharacteristic
312C2×C4#1458170611yescharacteristic
313C2×C4#1458175412yes—
314C22—21no—
315C22—21no—
316C22—21no—
317C22—22no—
318C22—21no—
319C22—42no—
320C22—41no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—81no—
327C22—81no—
328C22#1912760211yescharacteristic
329C22C2×#1458170611yescharacteristic
330C22#1912767411yescharacteristic
331C4—22no—
332C4—21no—
333C4—41no—
334C4—41no—
335C4—41no—
336C4—162no—
337C4—6642no—
338C4—6644no—
339C4—6642no—
340C4#1912749512yescomplemented
341C2—21no—
342C2—21no—
343C2—41no—
344C2—42no—
345C2—41no—
346C2#2490829811yescharacteristic
347C2C2×#1912749511yescharacteristic
348C2#2490913711yescharacteristic
subgroupG/H
centerC22C2×#14581706
commutatorC22×C4×C83C22×C4
FrattiniC22×D4C22×D83
FittingC83×#1799C2
socleC22×C83C2×#99
prime radical, cosocleC22×D4×C83C23
hypercenter#1799D83
hypocenterC83#13224
LCSG ▷ C22×C4×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
5. C2×#14581792C4semidirect
10. #19127606C4semidirect
13. C4×C4⋊Dic83C4semidirect
162. C4×C83#354semidirect
179. C83#13224semidirect
340. C4#19127495semidirect

Maximal subgroups

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

subgroupindex
#249124552
C2×#191274952
C4×#145817072
C83×#17992
#1322483