Finite Group Databasebeta

← #35299593

Subgroups — #35299593

Order 21248.

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

normal subgroups only

This group has 424 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#24917502C211yesmaximal, characteristic
2#24916403C211yesmaximal, characteristic
3#24906121C211yesmaximal, characteristic
4#24905892C211yesmaximal, characteristic
5#24905360C211yesmaximal, characteristic
6#24905308C211yesmaximal, characteristic
7C83×#1985C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9#19128036C412yes—
10#19127996—21no—
11#19127911—21no—
12#19127857C412yes—
13#19127829—22no—
14#19127829—22no—
15C4×#13050580—22no—
16#19127754C2211yescharacteristic
17#19127754C2211yescharacteristic
18#19127727—21no—
19#19127725C2211yescharacteristic
20#19127656—21no—
21#19127449—21no—
22#19127446—21no—
23Q8⋊(C83⋊C8)—22no—
24C2×C4×Q8×C83C2211yescharacteristic
25C2×C4⋊C8×C83C2211yescharacteristic
26C2×#97×C83C2211yescharacteristic
27#336×C83—21no—
28#336×C83—21no—
29C2×#13050580—22no—
30C2×#13050574—21no—
31#14581792D411yescharacteristic
32#14581792D411yescharacteristic
33#14581784—21no—
34#14581771—21no—
35#14581771C2×C412yes—
36C4×Dic166—21no—
37C4×Dic166D412yes—
38#14581751—21no—
39#14581725—22no—
40C4⋊(C83⋊C8)—21no—
41C4⋊(C83⋊C8)—21no—
42C4⋊(C83⋊C8)—21no—
43C4⋊(C83⋊C8)D412yes—
44Dic83⋊C8—24no—
45#14581716—22no—
46#14581715—22no—
47C22⋊(C83⋊C8)—21no—
48C22⋊(C83⋊C8)C2×C412yes—
49#14581702—22no—
50C4×C83⋊C8—21no—
51C4×C83⋊C8—21no—
52C8×Dic83—22no—
53C22×Q8×C83C2×C411yescharacteristic
54C22×C8×C83—21no—
55C22×C8×C83D411yescharacteristic
56C22×C8×C83D411yescharacteristic
57C4×Q8×C83—21no—
58C4×Q8×C83—21no—
59C4×Q8×C83—21no—
60C4×Q8×C83—21no—
61C2×C4⋊C4×C83—21no—
62C2×C4⋊C4×C83C2×C411yescharacteristic
63C2×C42×C83—21no—
64C2×C42×C83C2311yescharacteristic
65C4⋊C8×C83—21no—
66C4⋊C8×C83—21no—
67#97×C83—21no—
68#97×C83—21no—
69C2×Dic166—21no—
70C2×Dic166C22⋊C412yes—
71#13050580—44no—
72#13050574—41no—
73#13050574—41no—
74C2×C83⋊C8—22no—
75C2×C83⋊C8—22no—
76C2×C83⋊C8—41no—
77C4⋊Dic83—21no—
78C4⋊Dic83—21no—
79C4⋊Dic83—21no—
80C4⋊Dic83C22⋊C412yes—
81Dic83⋊C4—22no—
82Dic83⋊C4—22no—
83C22⋊Dic83—22no—
84C4×Dic83—22no—
85C2×Q8×C83—21no—
86C2×Q8×C83—21no—
87C2×Q8×C83—42no—
88C22×C4×C83—21no—
89C22×C4×C83—21no—
90C22×C4×C83C22×C411yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83#4111yescharacteristic
93C2×C8×C83—21no—
94C2×C8×C83—21no—
95C2×C8×C83—21no—
96C2×C8×C83—21no—
97C2×C8×C83—21no—
98C2×C8×C83—41no—
99C2×C8×C83—41no—
100C2×C8×C83—41no—
101C2×C8×C83Q1612yescomplemented
102C4⋊C4×C83—21no—
103C4⋊C4×C83—21no—
104C4⋊C4×C83—41no—
105C4⋊C4×C83—41no—
106C42×C83—41no—
107C42×C83—41no—
108C42×C83C2×D411yescharacteristic
109C42×C83C2×D411yescharacteristic
110C42×C83C2×D411yescharacteristic
111C42×C83#4111yescharacteristic
112C23×C83C4×D411yescharacteristic
113Q8×C83—22no—
114Q8×C83—22no—
115Q8×C83—41no—
116Q8×C83—41no—
117C2×C4×C83—22no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83—41no—
121C2×C4×C83—41no—
122C2×C4×C83—42no—
123C2×C4×C83—41no—
124C2×C4×C83C2×C22⋊C411yescharacteristic
125C2×C4×C83C4×D411yescharacteristic
126C2×C4×C83#12011yescharacteristic
127C2×C4×C83#12111yescharacteristic
128C2×C4×C83#12111yescharacteristic
129C2×C4×C83#12311yescharacteristic
130C2×C4×C83C2×Q1611yescharacteristic
131C2×C4×C83#13511yescharacteristic
132C2×C4×C83#13711yescharacteristic
133C2×C4×C83#13711yescharacteristic
134C2×Dic83—22no—
135C2×Dic83—22no—
136Dic166—42no—
137Dic166—81no—
138C8×C83—22no—
139C8×C83—41no—
140C8×C83—81no—
141C83⋊C8—42no—
142C83⋊C8—42no—
143C83⋊C8—81no—
144C22×C83—21no—
145C22×C83—21no—
146C22×C83#38611yescharacteristic
147C22×C83C4×Q1611yescharacteristic
148C22×C83#44111yescharacteristic
149C4×C83—21no—
150C4×C83—21no—
151C4×C83—21no—
152C4×C83—21no—
153C4×C83—42no—
154C4×C83—81no—
155C4×C83#45111yescharacteristic
156C4×C83#45211yescharacteristic
157C4×C83#46211yescharacteristic
158C4×C83#46411yescharacteristic
159Dic83—42no—
160Dic83—42no—
161#9048—831nomaximal, Sylow
162C2×C83—21no—
163C2×C83—21no—
164C2×C83#187811yescharacteristic
165C2×C83#188111yescharacteristic
166C2×C83#189111yescharacteristic
167#3104—831no—
168#2961—831no—
169#1985D8311yescharacteristic
170#1542—831no—
171#1487—831no—
172#1301—831no—
173#1281—831no—
174C83#904811yescharacteristic, complemented, Sylow
175#533—831no—
176C2×C4×Q8C2×D8311yescharacteristic
177#484—832no—
178#477—1661no—
179#462—1661no—
180#451—832no—
181#446—1662no—
182#446—1662no—
183C4×Q16—1662no—
184#431—831no—
185#431—831no—
186#424—1661no—
187#423—831no—
188C2×C4⋊C8C2×D8311yescharacteristic
189#405—1661no—
190C2×#97C2×D8311yescharacteristic
191#336—21no—
192#336—21no—
193#333—1661no—
194#332—1661no—
195Q8⋊C8—1662no—
196C22×Q8C4×D8311yescharacteristic
197C2×Q16—1662no—
198C2×M16—1661no—
199C22×C8—21no—
200C22×C8C22⋊D8311yescharacteristic
201C22×C8C22⋊D8311yescharacteristic
202C4⋊Q8—831no—
203C4⋊Q8—831no—
204#125—1661no—
205C22⋊Q8—832no—
206C22⋊Q8—1661no—
207C4×Q8—21no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—832no—
212C4×Q8—1661no—
213#117—1661no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C4×D8311yescharacteristic
216C2×C42—21no—
217C2×C42C22×D8311yescharacteristic
218#107—1662no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—832no—
222C4⋊C8—1664no—
223C4⋊C8—1661no—
224C4⋊C8—1661no—
225C4⋊C8—1661no—
226#103—1662no—
227#103—1662no—
228C22⋊C8—832no—
229C22⋊C8—1661no—
230#97—21no—
231#97—21no—
232#97—1662no—
233C4×C8—1662no—
234C4×C8—1661no—
235C4×C8—1661no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—42no—
239C2×Q8—832no—
240C2×Q8—1661no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4C2×C4×D8311yescharacteristic
244C22×C4C2×C22⋊D8311yescharacteristic
245C22×C4#1305059011yescharacteristic
246Q16—3324no—
247M16—3321no—
248M16—3321no—
249C2×C8—21no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—3321no—
258C2×C8—1662no—
259C2×C8—1662no—
260C2×C8#1305058012yescomplemented
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—832no—
266C4⋊C4—1661no—
267C4⋊C4—1661no—
268C4⋊C4—1661no—
269C4⋊C4—1662no—
270C4⋊C4—1662no—
271C22⋊C4—1662no—
272C42—41no—
273C42—41no—
274C42—1662no—
275C42D4×D8311yescharacteristic
276C42D4×D8311yescharacteristic
277C42D4×D8311yescharacteristic
278C42#1305058911yescharacteristic
279C23C4×C22⋊D8311yescharacteristic
280Q8—22no—
281Q8—22no—
282Q8—41no—
283Q8—41no—
284Q8—3322no—
285Q8—6641no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—42no—
292C2×C4—41no—
293C2×C4—1662no—
294C2×C4—1662no—
295C2×C4C22⋊C4×D8311yescharacteristic
296C2×C4(C22⋊D83)⋊C411yescharacteristic
297C2×C4#1458176011yescharacteristic
298C2×C4#1458176311yescharacteristic
299C2×C4#1458176511yescharacteristic
300C2×C4#1458177411yescharacteristic
301C2×C4C2×#1305058011yescharacteristic
302C2×C4#1458181211yescharacteristic
303C2×C4#1458182211yescharacteristic
304C2×C4#1458182311yescharacteristic
305C8—22no—
306C8—41no—
307C8—81no—
308C8—3322no—
309C8—3322no—
310C8—6641no—
311C22—21no—
312C22—21no—
313C22#1912757711yescharacteristic
314C22C4×#1305058011yescharacteristic
315C22#1912780611yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—42no—
321C4—81no—
322C4—3322no—
323C4—3322no—
324C4#1912785611yescharacteristic
325C4#1912786111yescharacteristic
326C4#1912790811yescharacteristic
327C4#1912792011yescharacteristic
328C2—21no—
329C2—21no—
330C2#2490780711yescharacteristic
331C2#2490782811yescharacteristic
332C2#2490787811yescharacteristic
subgroupG/H
centerC2×C4#14581760
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1985C2
socleC22×C83#386
prime radicalC23×C83C4×D4
cosocleC2×C42×C83C23
hypercenter#1985D83
hypocenterC83#9048
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1985

Complements

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

NH ≅ G/Ntype
101. C2×C8×C83Q16semidirect
174. C83#9048semidirect
260. C2×C8#13050580semidirect

Maximal subgroups

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

subgroupindex
#249175022
#249164032
#249061212
#249058922
#249053602
#249053082
C83×#19852
#904883