Finite Group Databasebeta

← #35301813

Subgroups — #35301813

Order 21248.

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

normal subgroups only

This group has 486 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#24916546C211yesmaximal, characteristic
2#24912451C211yesmaximal, characteristic
3#24907456C211yesmaximal, characteristic
4#24906136C211yesmaximal, characteristic
5#24906134C211yesmaximal, characteristic
6#24905366C211yesmaximal, characteristic
7C83×#1303C211yesmaximal, characteristic
8C2×#14581792—21no—
9C2×#14581792C2211yescharacteristic
10C2×C4×Dic166C2211yescharacteristic
11#19127831—22no—
12#19127828—22no—
13#19127755—22no—
14#19127754C2211yescharacteristic
15#19127725C2211yescharacteristic
16C2×#14581710—21no—
17#19127633C414yes—
18#19127607—21no—
19#19127600—22no—
20#19127469—21no—
21#19127448—22no—
22#19127446—22no—
23#431×C83C2211yescharacteristic
24#423×C83C2211yescharacteristic
25C4×C4⋊C4×C83C2211yescharacteristic
26C22×Dic166—21no—
27C22×Dic166D411yescharacteristic
28C2×#13050574—21no—
29#14581792—21no—
30#14581792—21no—
31#14581792—21no—
32#14581792—21no—
33#14581792—42no—
34#14581792—42no—
35C4×Dic166—21no—
36C4×Dic166—21no—
37C4×Dic166—21no—
38C4×Dic166—21no—
39C2×C4⋊Dic83—24no—
40C2×C4⋊Dic83—21no—
41C2×C4⋊Dic83C2×C412yes—
42C2×C4⋊Dic83D411yescharacteristic
43C2×Dic83⋊C4C2×C412yes—
44C2×C4×Dic83D412yes—
45C4⋊(C83⋊C8)—21no—
46C4⋊(C83⋊C8)—21no—
47C4⋊(C83⋊C8)D412yes—
48Dic83⋊C8—24no—
49#14581710—42no—
50C22⋊(C83⋊C8)—21no—
51#14581702—24no—
52C4×C83⋊C8—21no—
53C83⋊#95—24no—
54C2×M16×C83—21no—
55C2×C4⋊C4×C83—21no—
56C2×C42×C83—21no—
57C2×C42×C83C2311yescharacteristic
58C4⋊C8×C83—21no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83D412yes—
61C22⋊C8×C83—21no—
62C4×C8×C83—21no—
63#95×C83C2×C412yes—
64C2×Dic166—21no—
65C2×Dic166—21no—
66C2×Dic166—21no—
67C2×Dic166—21no—
68C2×Dic166—21no—
69C2×Dic166—21no—
70C2×Dic166—41no—
71C2×Dic166—41no—
72C2×Dic166—41no—
73C2×Dic166—41no—
74C2×Dic166—41no—
75C2×Dic166—41no—
76C22×Dic83—22no—
77C22×Dic83C22⋊C412yes—
78C22×Dic83C22⋊C412yes—
79#13050574—41no—
80#13050574—41no—
81C2×C83⋊C8—22no—
82C2×C83⋊C8—41no—
83C4⋊Dic83—21no—
84C4⋊Dic83—21no—
85C4⋊Dic83—22no—
86C4⋊Dic83—22no—
87C4⋊Dic83—48no—
88C4⋊Dic83—41no—
89C4⋊Dic83—41no—
90Dic83⋊C4—22no—
91Dic83⋊C4—22no—
92C4×Dic83—22no—
93C4×Dic83—22no—
94C22×C4×C83—22no—
95C22×C4×C83C22×C411yescharacteristic
96C22×C4×C83C2×D411yescharacteristic
97C22×C4×C83C2×D411yescharacteristic
98M16×C83—41no—
99M16×C83—41no—
100C2×C8×C83—22no—
101C2×C8×C83—41no—
102C4⋊C4×C83—24no—
103C42×C83—41no—
104C42×C83—41no—
105C42×C83C2×D411yescharacteristic
106C42×C83C2×D411yescharacteristic
107C42×C83#4111yescharacteristic
108C42×C83#4111yescharacteristic
109C23×C83C2×C22⋊C411yescharacteristic
110C2×C4×C83—22no—
111C2×C4×C83—24no—
112C2×C4×C83—41no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83C4×D411yescharacteristic
116C2×C4×C83C4×D411yescharacteristic
117C2×C4×C83#12011yescharacteristic
118C2×C4×C83#12111yescharacteristic
119C2×C4×C83#12111yescharacteristic
120C2×C4×C83#12311yescharacteristic
121C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
122C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
123C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
124C2×C4×C83#13711yescharacteristic
125C2×Dic83—22no—
126C2×Dic83—22no—
127C2×Dic83—22no—
128C2×Dic83—22no—
129C2×Dic83—42no—
130C2×Dic83—42no—
131C2×Dic83—42no—
132C2×Dic83—42no—
133Dic166—41no—
134Dic166—41no—
135Dic166—41no—
136Dic166—41no—
137Dic166—81no—
138Dic166—81no—
139Dic166—81no—
140Dic166—81no—
141C8×C83—42no—
142C8×C83—81no—
143C83⋊C8—42no—
144C83⋊C8—81no—
145C22×C83—21no—
146C22×C83—21no—
147C22×C83#38611yescharacteristic
148C22×C83#41711yescharacteristic
149C22×C83#41811yescharacteristic
150C4×C83—21no—
151C4×C83—21no—
152C4×C83—21no—
153C4×C83—44no—
154C4×C83#45612yes—
155C4×C83#46811yescharacteristic
156C4×C83#46811yescharacteristic
157C4×C83#46911yescharacteristic
158C4×C83#46911yescharacteristic
159Dic83—42no—
160Dic83—42no—
161Dic83—82no—
162#9402—831nomaximal, Sylow
163C2×C83—21no—
164C2×C83—21no—
165C2×C83#188611yescharacteristic
166C2×C83#189511yescharacteristic
167C2×C83#191511yescharacteristic
168#2983—831no—
169#2546—831no—
170#1801—831no—
171#1546—831no—
172#1546—831no—
173#1303—831no—
174#1303D8311yescharacteristic
175C83#940211yescharacteristic, complemented, Sylow
176C2×C4⋊Q8—831no—
177C2×C4⋊Q8—1661no—
178C2×C4×Q8—831no—
179#446—1662no—
180#446—1662no—
181#432—1662no—
182#431—831no—
183#431C2×D8311yescharacteristic
184#423—831no—
185#423C2×D8311yescharacteristic
186C2×#101—1661no—
187#398—834no—
188#391—1661no—
189#389—1662no—
190C4×C4⋊C4C2×D8311yescharacteristic
191#343—1661no—
192#332—1662no—
193#332—1662no—
194C22×Q8—831no—
195C22×Q8—1661no—
196C2×M16—21no—
197C2×M16—1661no—
198C4⋊Q8—3322no—
199C4⋊Q8—3322no—
200C4⋊Q8—1661no—
201C4⋊Q8—1661no—
202C4⋊Q8—1661no—
203C4⋊Q8—1661no—
204C4×Q8—1661no—
205C4×Q8—1661no—
206C4×Q8—1661no—
207C4×Q8—1661no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—832no—
210C2×C4⋊C4—831no—
211C2×C4⋊C4—832no—
212C2×C4⋊C4—1664no—
213C2×C4⋊C4—1661no—
214C2×C42—21no—
215C2×C42—832no—
216C2×C42C22×D8311yescharacteristic
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—832no—
220C4⋊C8—1664no—
221C4⋊C8—1661no—
222C4⋊C8—1661no—
223C4⋊C8D33212yes—
224#101—3322no—
225C22⋊C8—21no—
226C22⋊C8—1661no—
227#97—1664no—
228C4×C8—21no—
229C4×C8—1661no—
230#95—1664no—
231#95C4×D8312yes—
232C2×Q8—3321no—
233C2×Q8—3321no—
234C2×Q8—3321no—
235C2×Q8—3321no—
236C2×Q8—3321no—
237C2×Q8—3321no—
238C2×Q8—1661no—
239C2×Q8—1661no—
240C2×Q8—1661no—
241C2×Q8—1661no—
242C2×Q8—1661no—
243C2×Q8—1661no—
244C22×C4—22no—
245C22×C4—832no—
246C22×C4—832no—
247C22×C4—1662no—
248C22×C4C2×C4×D8311yescharacteristic
249C22×C4D4×D8311yescharacteristic
250C22×C4D4×D8311yescharacteristic
251M16—41no—
252M16—41no—
253M16—3321no—
254M16—3321no—
255C2×C8—22no—
256C2×C8—41no—
257C2×C8—3321no—
258C2×C8—1662no—
259C4⋊C4—24no—
260C4⋊C4—3328no—
261C4⋊C4—3321no—
262C4⋊C4—3321no—
263C4⋊C4—1662no—
264C4⋊C4—1662no—
265C4⋊C4—1661no—
266C4⋊C4—1661no—
267C4⋊C4—1662no—
268C4⋊C4—1662no—
269C42—41no—
270C42—41no—
271C42—1662no—
272C42—1662no—
273C42D4×D8311yescharacteristic
274C42C2×D33211yescharacteristic
275C42#1305059011yescharacteristic
276C42#1305059111yescharacteristic
277C23C22⋊C4×D8311yescharacteristic
278Q8—3321no—
279Q8—3321no—
280Q8—3321no—
281Q8—3321no—
282Q8—6641no—
283Q8—6641no—
284Q8—6641no—
285Q8—6641no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—42no—
291C2×C4—3322no—
292C2×C4—3322no—
293C2×C4—3322no—
294C2×C4—3322no—
295C2×C4—1662no—
296C2×C4—1662no—
297C2×C4—1662no—
298C2×C4—1662no—
299C2×C4#1458175411yescharacteristic
300C2×C4C4×D33211yescharacteristic
301C2×C4#1458176211yescharacteristic
302C2×C4#1458176511yescharacteristic
303C2×C4#1458176711yescharacteristic
304C2×C4#1458177711yescharacteristic
305C2×C4#1458181611yescharacteristic
306C2×C4#1458181911yescharacteristic
307C2×C4#1458181911yescharacteristic
308C2×C4#1458182411yescharacteristic
309C8—42no—
310C8—81no—
311C8—3322no—
312C8—6641no—
313C22—21no—
314C22—21no—
315C22#1912757911yescharacteristic
316C22#1912770111yescharacteristic
317C22#1912770411yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—44no—
322C4—3322no—
323C4—3322no—
324C4—6642no—
325C4#1912788512yes—
326C4#1912794811yescharacteristic
327C4#1912795011yescharacteristic
328C4#1912795311yescharacteristic
329C4#1912795511yescharacteristic
330C2—21no—
331C2—21no—
332C2#2490785511yescharacteristic
333C2#2490790011yescharacteristic
334C2#2490801411yescharacteristic
subgroupG/H
centerC22#19127579
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1303C2
socleC22×C83#386
prime radicalC23×C83C2×C22⋊C4
cosocleC2×C42×C83C23
hypercenter#1303D83
hypocenterC83#9402
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
175. C83#9402semidirect

Maximal subgroups

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

subgroupindex
#249165462
#249124512
#249074562
#249061362
#249061342
#249053662
C83×#13032
#940283