Finite Group Databasebeta

← #35299463

Subgroups — #35299463

Order 21248.

660 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×#19127631C211yesmaximal, characteristic
2#24907547C212yesmaximal
3C2×#19127427C211yesmaximal, characteristic
4#24905341C212yesmaximal
5C83×#2316C211yesmaximal, characteristic
6C22×C4⋊Dic83C2211yescharacteristic
7C22×Dic83⋊C4—21no—
8C2×C22⋊(C83⋊C8)C2212yes—
9#19127631—22no—
10#19127631C414yescomplemented
11C2×C83⋊#95—21no—
12C2×C83⋊#95C2211yescharacteristic
13#19127463—24no—
14#19127427—22no—
15C2×#117×C83—21no—
16C22×C4⋊C4×C83C2211yescharacteristic
17#389×C83—24no—
18#380×C83—21no—
19#380×C83C2212yes—
20C23×Dic83—21no—
21C23×Dic83—21no—
22C22×C83⋊C8—22no—
23C2×C4⋊Dic83—22no—
24C2×C4⋊Dic83—21no—
25C2×C4⋊Dic83—22no—
26C2×C4⋊Dic83C2×C412yes—
27C2×Dic83⋊C4—24no—
28C2×Dic83⋊C4—42no—
29C2×Dic83⋊C4—42no—
30C22⋊(C83⋊C8)—22no—
31C22⋊(C83⋊C8)—22no—
32C83⋊#95—24no—
33C83⋊#95—21no—
34C83⋊#95C2×C412yes—
35C23×C4×C83D411yescharacteristic
36C23×C4×C83D411yescharacteristic
37C23×C4×C83C2311yescharacteristic
38#117×C83—42no—
39#117×C83—42no—
40C2×C4⋊C4×C83—22no—
41C2×C4⋊C4×C83—22no—
42C2×C4⋊C4×C83—22no—
43C2×C4⋊C4×C83—22no—
44C2×C4⋊C4×C83C2×C412yes—
45C2×C4⋊C4×C83D412yes—
46C2×C4⋊C4×C83D414yes—
47C2×C22⋊C4×C83—22no—
48C2×C42×C83—22no—
49#95×C83—22no—
50#95×C83—22no—
51C22×Dic83—24no—
52C22×Dic83—22no—
53C22×Dic83—22no—
54C22×Dic83—42no—
55C22×Dic83—41no—
56C22×Dic83—41no—
57C22×Dic83—41no—
58C22×Dic83—41no—
59C22×Dic83—41no—
60C22×Dic83—41no—
61C22×Dic83—41no—
62C22×Dic83—41no—
63C2×C83⋊C8—44no—
64C2×C83⋊C8—42no—
65C4⋊Dic83—24no—
66C4⋊Dic83—42no—
67Dic83⋊C4—48no—
68C24×C83C2×D411yescharacteristic
69C22×C4×C83—21no—
70C22×C4×C83—21no—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—24no—
76C22×C4×C83—42no—
77C22×C4×C83—42no—
78C22×C4×C83C22⋊C412yes—
79C22×C4×C83C22⋊C412yes—
80C22×C4×C83D812yes—
81C22×C4×C83SD1612yes—
82C22×C4×C83C22×C411yescharacteristic
83C22×C4×C83C2×D412yes—
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83#4112yes—
86C4⋊C4×C83—24no—
87C4⋊C4×C83—24no—
88C4⋊C4×C83—44no—
89C4⋊C4×C83—44no—
90C4⋊C4×C83—82no—
91C22⋊C4×C83—42no—
92C22⋊C4×C83—42no—
93C42×C83—44no—
94C23×C83—22no—
95C23×C83—22no—
96C23×C83C2×C22⋊C411yescharacteristic
97C23×C83#12011yescharacteristic
98C23×C83#12311yescharacteristic
99C23×C83C2×D811yescharacteristic
100C23×C83C2×SD1611yescharacteristic
101C23×C83AGL(1,ℤ/8)11yescharacteristic
102C23×C83#13711yescharacteristic
103C2×C4×C83—22no—
104C2×C4×C83—22no—
105C2×C4×C83—22no—
106C2×C4×C83—22no—
107C2×C4×C83—24no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83—42no—
114C2×C4×C83—42no—
115C2×C4×C83—42no—
116C2×C4×C83—44no—
117C2×C4×C83—44no—
118C2×C4×C83—44no—
119C2×C4×C83—81no—
120C2×C4×C83#10214yes—
121C2×C4×C83C4×D412yes—
122C2×C4×C83#12112yes—
123C2×Dic83—24no—
124C2×Dic83—44no—
125C2×Dic83—44no—
126C2×Dic83—42no—
127C2×Dic83—42no—
128C2×Dic83—42no—
129C2×Dic83—42no—
130C2×Dic83—42no—
131C2×Dic83—81no—
132C2×Dic83—81no—
133C2×Dic83—81no—
134C2×Dic83—81no—
135C83⋊C8—82no—
136C22×C83—22no—
137C22×C83—22no—
138C22×C83—22no—
139C22×C83—22no—
140C22×C83—21no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83#38611yescharacteristic
145C22×C83C2×#10211yescharacteristic
146C22×C83#41711yescharacteristic
147C22×C83#44711yescharacteristic
148C22×C83#44811yescharacteristic
149C22×C83#45612yes—
150C22×C83#45812yes—
151C22×C83#48011yescharacteristic
152C22×C83#48111yescharacteristic
153C4×C83—24no—
154C4×C83—22no—
155C4×C83—22no—
156C4×C83—42no—
157C4×C83—84no—
158Dic83—44no—
159Dic83—82no—
160#9027—831nomaximal, Sylow
161C2×C83—21no—
162C2×C83—21no—
163C2×C83—21no—
164C2×C83—21no—
165C2×C83#161012yes—
166C2×C83#161112yes—
167C2×C83#187411yescharacteristic
168C2×C83#188611yescharacteristic
169C2×C83#189011yescharacteristic
170C2×#398—831no—
171#2316D8311yescharacteristic
172#1823—832no—
173C2×#324—831no—
174#1294—832no—
175C83#902711yescharacteristic, complemented, Sylow
176C2×#117—21no—
177C22×C4⋊C4—831no—
178C22×C4⋊C4—1661no—
179C22×C4⋊C4C2×D8311yescharacteristic
180C2×C22⋊C8—832no—
181#398—834no—
182#398—1662no—
183#389—24no—
184#380—21no—
185#380C2×D8312yes—
186C2×#95—831no—
187C2×#95—1661no—
188#340—1664no—
189#324—1662no—
190C23×C4—1661no—
191C23×C4—1661no—
192C23×C4C22⋊D8311yescharacteristic
193C23×C4C22⋊D8311yescharacteristic
194C23×C4C22×D8311yescharacteristic
195C22×C8—1662no—
196#117—42no—
197#117—42no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—3322no—
203C2×C4⋊C4—3322no—
204C2×C4⋊C4—832no—
205C2×C4⋊C4—1664no—
206C2×C4⋊C4—1661no—
207C2×C4⋊C4—1662no—
208C2×C4⋊C4—1662no—
209C2×C4⋊C4C4×D8312yes—
210C2×C4⋊C4D33212yes—
211C2×C4⋊C4C22⋊D8314yes—
212C2×C22⋊C4—22no—
213C2×C42—22no—
214C22⋊C8—1662no—
215C22⋊C8—1662no—
216#95—22no—
217#95—22no—
218#95—832no—
219#95—1664no—
220#95—1661no—
221C24C2×C22⋊D8311yescharacteristic
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—24no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4—3322no—
232C22×C4—3321no—
233C22×C4—3321no—
234C22×C4—3321no—
235C22×C4—3321no—
236C22×C4—3321no—
237C22×C4—3321no—
238C22×C4—3321no—
239C22×C4—3321no—
240C22×C4—1662no—
241C22×C4—1664no—
242C22×C4—1662no—
243C22×C4(C2×D83)⋊C412yes—
244C22×C4(C2×D83)⋊C412yes—
245C22×C4#1305057512yes—
246C22×C4#1305057712yes—
247C22×C4C2×C4×D8311yescharacteristic
248C22×C4C2×C22⋊D8312yes—
249C22×C4C2×D33211yescharacteristic
250C22×C4#1305059012yes—
251C2×C8—3324no—
252C2×C8—3322no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—82no—
258C4⋊C4—3322no—
259C4⋊C4—3328no—
260C4⋊C4—1664no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—44no—
264C23—22no—
265C23—22no—
266C23C2×(C2×D83)⋊C411yescharacteristic
267C23#1458176111yescharacteristic
268C23#1458177511yescharacteristic
269C23C2×#1305057511yescharacteristic
270C23C2×#1305057711yescharacteristic
271C23#1458181811yescharacteristic
272C23#1458182311yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—44no—
287C2×C4—44no—
288C2×C4—44no—
289C2×C4—81no—
290C2×C4—3322no—
291C2×C4—3324no—
292C2×C4—3324no—
293C2×C4—3322no—
294C2×C4—3322no—
295C2×C4—3322no—
296C2×C4—3322no—
297C2×C4—6641no—
298C2×C4—6641no—
299C2×C4—6641no—
300C2×C4—6641no—
301C2×C4—1664no—
302C2×C4#1458171214yes—
303C2×C4C4×C22⋊D8312yes—
304C2×C4#1458176612yes—
305C8—6642no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—22no—
314C22#1912758011yescharacteristic
315C22C2×#1458171211yescharacteristic
316C22#1912770011yescharacteristic
317C22#1912783511yescharacteristic
318C22#1912784011yescharacteristic
319C22#1912788412yes—
320C22#1912789112yes—
321C22#1912801411yescharacteristic
322C22#1912801911yescharacteristic
323C4—24no—
324C4—22no—
325C4—22no—
326C4—42no—
327C4—84no—
328C4—3324no—
329C4—6642no—
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#2490646612yes—
335C2#2490647112yes—
336C2#2490778011yescharacteristic
337C2#2490785611yescharacteristic
338C2#2490787611yescharacteristic
subgroupG/H
centerC23#14581761
commutatorC22×C4×C83C22×C4
FrattiniC23×C4C22×D83
FittingC83×#2316C2
socleC23×C83#120
prime radicalC24×C83C2×D4
cosocleC23×C4×C83C23
hypercenter#2316D83
hypocenterC83#9027
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
10. #19127631C4semidirect
175. C83#9027semidirect

Maximal subgroups

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

subgroupindex
C2×#191276312
#249075472
C2×#191274272
#249053412
C83×#23162
#902783