Finite Group Databasebeta

← #35299587

Subgroups — #35299587

Order 21248.

424 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 107 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#24916400C211yesmaximal, characteristic
3#24908404C211yesmaximal, characteristic
4#24906118C211yesmaximal, characteristic
5#24905358C211yesmaximal, characteristic
6#24905307C211yesmaximal, characteristic
7C83×#1487C211yesmaximal, characteristic
8#19128258C2211yescharacteristic
9#19128036C412yes—
10#19127996—21no—
11#19127911—21no—
12#19127857C412yes—
13#19127830—22no—
14#19127830—22no—
15C4×#13050580—22no—
16#19127754C2211yescharacteristic
17#19127729—21no—
18C2×C4⋊(C83⋊C8)C2211yescharacteristic
19#19127659—21no—
20C2×#14581703C2211yescharacteristic
21#19127456—21no—
22#19127456—21no—
23#19127449—21no—
24#19127448—21no—
25C2×C4×Q8×C83C2211yescharacteristic
26#431×C83C2211yescharacteristic
27#423×C83C2211yescharacteristic
28Q8⋊C8×C83—22no—
29C2×#13050580—22no—
30C22×C83⋊C8—21no—
31C22×C83⋊C8D411yescharacteristic
32C22×C83⋊C8D411yescharacteristic
33#14581792D411yescharacteristic
34#14581792D411yescharacteristic
35#14581784—21no—
36#14581771—21no—
37#14581771C2×C412yes—
38C4×Dic166—21no—
39C4×Dic166D412yes—
40#14581751—21no—
41#14581725—22no—
42C4⋊(C83⋊C8)—21no—
43C4⋊(C83⋊C8)—21no—
44C4⋊(C83⋊C8)—21no—
45Dic83⋊C8—24no—
46#14581716—22no—
47#14581715—22no—
48C22⋊(C83⋊C8)C2×C412yes—
49#14581703—21no—
50#14581703—21no—
51#14581702—22no—
52C4×C83⋊C8—21no—
53C8×Dic83—22no—
54C22×Q8×C83C2×C411yescharacteristic
55C2×M16×C83—21no—
56C4×Q8×C83—21no—
57C4×Q8×C83—21no—
58C4×Q8×C83—21no—
59C4×Q8×C83—21no—
60C2×C4⋊C4×C83—21no—
61C2×C4⋊C4×C83C2×C411yescharacteristic
62C2×C42×C83—21no—
63C2×C42×C83C2311yescharacteristic
64C4⋊C8×C83—21no—
65C4⋊C8×C83—21no—
66C4⋊C8×C83D412yes—
67C22⋊C8×C83—21no—
68C4×C8×C83—21no—
69C2×Dic166—21no—
70C2×Dic166C22⋊C412yes—
71#13050580—44no—
72C2×C83⋊C8—21no—
73C2×C83⋊C8—21no—
74C2×C83⋊C8—22no—
75C2×C83⋊C8—21no—
76C2×C83⋊C8—21no—
77C2×C83⋊C8—21no—
78C2×C83⋊C8—41no—
79C2×C83⋊C8—41no—
80C2×C83⋊C8—41no—
81C2×C83⋊C8Q1612yescomplemented
82C4⋊Dic83—21no—
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83C22⋊C412yes—
86Dic83⋊C4—22no—
87Dic83⋊C4—22no—
88C22⋊Dic83—22no—
89C4×Dic83—22no—
90C2×Q8×C83—21no—
91C2×Q8×C83—21no—
92C2×Q8×C83—42no—
93C22×C4×C83—21no—
94C22×C4×C83—21no—
95C22×C4×C83C22×C411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83#4111yescharacteristic
98M16×C83—41no—
99M16×C83—41no—
100C2×C8×C83—22no—
101C2×C8×C83—41no—
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—42no—
139C8×C83—81no—
140C83⋊C8—22no—
141C83⋊C8—42no—
142C83⋊C8—41no—
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#1985—831no—
170#1542—831no—
171#1487D8311yescharacteristic
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#431C2×D8311yescharacteristic
186#424—1661no—
187#423C2×D8311yescharacteristic
188C2×C4⋊C8—831no—
189#405—1661no—
190C2×#97—831no—
191#336—1661no—
192#336—1661no—
193#333—1661no—
194#332—1661no—
195Q8⋊C8—22no—
196C22×Q8C4×D8311yescharacteristic
197C2×Q16—1662no—
198C2×M16—21no—
199C22×C8—831no—
200C22×C8—831no—
201C22×C8—1661no—
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—1664no—
222C4⋊C8—1661no—
223C4⋊C8—1661no—
224C4⋊C8—1661no—
225C4⋊C8C22⋊D8312yes—
226#103—1662no—
227#103—1662no—
228C22⋊C8—21no—
229C22⋊C8—832no—
230#97—1662no—
231#97—1661no—
232#97—1661no—
233C4×C8—21no—
234C4×C8—1662no—
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×C4D4×D8311yescharacteristic
245C22×C4#1305058911yescharacteristic
246Q16—3324no—
247M16—41no—
248M16—41no—
249C2×C8—22no—
250C2×C8—41no—
251C2×C8—3321no—
252C2×C8—3321no—
253C2×C8—3321no—
254C2×C8—832no—
255C2×C8—1661no—
256C2×C8—1661no—
257C2×C8—1661no—
258C2×C8—1662no—
259C2×C8—1661no—
260C2×C8—1661no—
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
277C42C2×C22⋊D8311yescharacteristic
278C42#1305059011yescharacteristic
279C23(C22⋊D83)⋊C411yescharacteristic
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×C4C4×C22⋊D8311yescharacteristic
297C2×C4#1458176011yescharacteristic
298C2×C4#1458176311yescharacteristic
299C2×C4#1458176511yescharacteristic
300C2×C4#1458177411yescharacteristic
301C2×C4Q16×D8311yescharacteristic
302C2×C4#1458181311yescharacteristic
303C2×C4#1458182111yescharacteristic
304C2×C4#1458182411yescharacteristic
305C8—42no—
306C8—81no—
307C8—3322no—
308C8—3321no—
309C8—6641no—
310C8—1662no—
311C22—21no—
312C22—21no—
313C22#1912757711yescharacteristic
314C22#1912779311yescharacteristic
315C22#1912780511yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—42no—
321C4—81no—
322C4—3322no—
323C4—3322no—
324C4#1912785511yescharacteristic
325C4#1912786011yescharacteristic
326C4#1912791011yescharacteristic
327C4#1912792611yescharacteristic
328C2—21no—
329C2—21no—
330C2#2490780111yescharacteristic
331C2#2490782211yescharacteristic
332C2#2490787811yescharacteristic
subgroupG/H
centerC2×C4#14581760
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1487C2
socleC22×C83#386
prime radicalC23×C83C4×D4
cosocleC2×C42×C83C23
hypercenter#1487D83
hypocenterC83#9048
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
81. C2×C83⋊C8Q16semidirect
174. C83#9048semidirect

Maximal subgroups

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

subgroupindex
#249175022
#249164002
#249084042
#249061182
#249053582
#249053072
C83×#14872
#904883