Finite Group Databasebeta

← #35314708

Subgroups — #35314708

Order 21248.

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

normal subgroups only

This group has 452 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#24910731C211yesmaximal, characteristic
2#24905777C211yesmaximal, characteristic
3#24905492C212yesmaximal, complemented
4#24905489C212yesmaximal, complemented
5C83×#1742C211yesmaximal, characteristic
6C2×C4×Dic166C411yescharacteristic
7#19127754—21no—
8#19127640C2211yescharacteristic
9#19127637C411yescharacteristic
10#19127634—21no—
11#19127624—21no—
12#19127621C2211yescharacteristic
13#19127613C411yescharacteristic
14C4×Dic83⋊C4—21no—
15C4×C4⋊Dic83C411yescharacteristic
16C4×C22⋊Dic83—21no—
17#19127427C2212yes—
18#19127425—21no—
19C22×M16×C83—21no—
20C2×#117×C83C2211yescharacteristic
21#343×C83—21no—
22#343×C83C2212yes—
23C22×Dic166C2×C411yescharacteristic
24C4×Dic166—41no—
25C4×Dic166—41no—
26C2×C4⋊Dic83—21no—
27C2×C4⋊Dic83C2×C411yescharacteristic
28C2×Dic83⋊C4—21no—
29C2×Dic83⋊C4—21no—
30C2×Dic83⋊C4—21no—
31C2×C22⋊Dic83—21no—
32C2×C22⋊Dic83—21no—
33C2×C4×Dic83—21no—
34C2×C4×Dic83—21no—
35C4⋊(C83⋊C8)—41no—
36C22⋊(C83⋊C8)—22no—
37C4×C83⋊C8—41no—
38C83⋊#95—21no—
39C83⋊#95—21no—
40C83⋊#95—21no—
41C83⋊#95—21no—
42C83⋊#95C2×C411yescharacteristic
43C83⋊#95C2×C411yescharacteristic
44C23×C4×C83D411yescharacteristic
45C2×M16×C83—24no—
46C2×M16×C83—22no—
47C2×M16×C83—41no—
48C2×M16×C83—42no—
49C22×C8×C83—41no—
50#117×C83—41no—
51#117×C83—41no—
52C2×C4⋊C4×C83D411yescharacteristic
53C2×C4⋊C4×C83C2311yescharacteristic
54C2×C22⋊C4×C83C2×C411yescharacteristic
55C2×C22⋊C4×C83D411yescharacteristic
56C2×C42×C83C2×C411yescharacteristic
57C2×C42×C83D411yescharacteristic
58C2×Dic166—41no—
59C2×Dic166—42no—
60C22×Dic83—21no—
61C22×Dic83—21no—
62C22×Dic83—21no—
63C22×Dic83—21no—
64C2×C83⋊C8—42no—
65C4⋊Dic83—42no—
66C4⋊Dic83—41no—
67Dic83⋊C4—42no—
68Dic83⋊C4—42no—
69Dic83⋊C4—42no—
70C22⋊Dic83—42no—
71C22⋊Dic83—42no—
72C4×Dic83—41no—
73C4×Dic83—41no—
74C4×Dic83—42no—
75C24×C83C22⋊C411yescharacteristic
76C22×C4×C83—21no—
77C22×C4×C83—21no—
78C22×C4×C83—22no—
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83SD1612yes—
83C22×C4×C83Q1612yes—
84C22×C4×C83C22×C411yescharacteristic
85C22×C4×C83C2×D411yescharacteristic
86C22×C4×C83C2×D411yescharacteristic
87M16×C83—24no—
88M16×C83—44no—
89M16×C83—42no—
90C2×C8×C83—42no—
91C2×C8×C83—44no—
92C4⋊C4×C83—41no—
93C4⋊C4×C83—41no—
94C22⋊C4×C83—41no—
95C22⋊C4×C83—41no—
96C42×C83—21no—
97C42×C83—21no—
98C42×C83—21no—
99C42×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—41no—
103C23×C83—41no—
104C23×C83#10312yes—
105C23×C83C2×C22⋊C411yescharacteristic
106C2×C4×C83—21no—
107C2×C4×C83—21no—
108C2×C4×C83—22no—
109C2×C4×C83—22no—
110C2×C4×C83—22no—
111C2×C4×C83—22no—
112C2×C4×C83—21no—
113C2×C4×C83—21no—
114C2×C4×C83—21no—
115C2×C4×C83—21no—
116C2×C4×C83—41no—
117C2×C4×C83—41no—
118C2×C4×C83—41no—
119C2×C4×C83—41no—
120C2×C4×C83#9911yescharacteristic
121C2×C4×C83#9911yescharacteristic
122C2×C4×C83#10312yes—
123C2×C4×C83#10412yes—
124C2×C4×C83C2×SD1611yescharacteristic
125C2×C4×C83C2×Q1611yescharacteristic
126C2×Dic83—22no—
127C2×Dic83—22no—
128C2×Dic83—41no—
129C2×Dic83—41no—
130C2×Dic83—41no—
131C2×Dic83—41no—
132C2×Dic83—41no—
133C2×Dic83—41no—
134C2×Dic83—41no—
135C2×Dic83—41no—
136C2×Dic83—41no—
137C2×Dic83—41no—
138Dic166—42no—
139Dic166—42no—
140C8×C83—44no—
141C83⋊C8—82no—
142C22×C83—21no—
143C22×C83—21no—
144C22×C83—22no—
145C22×C83—42no—
146C22×C83—41no—
147C22×C83—41no—
148C22×C83—81no—
149C22×C83C2×#9911yescharacteristic
150C22×C83C2×#10311yescharacteristic
151C22×C83C2×#10411yescharacteristic
152C4×C83—21no—
153C4×C83—21no—
154C4×C83—41no—
155C4×C83—41no—
156C4×C83—41no—
157C4×C83—41no—
158C4×C83#32812yes—
159C4×C83#32812yes—
160Dic83—42no—
161Dic83—42no—
162Dic83—81no—
163Dic83—81no—
164#11721—831nomaximal, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83—42no—
168C2×C83C2×#32811yescharacteristic
169C2×C83#212011yescharacteristic
170C2×C83#213211yescharacteristic
171#2396—831no—
172#1742D8311yescharacteristic
173#1458—831no—
174#1354—832no—
175#1353—832no—
176C83#1172111yescharacteristic, complemented, Sylow
177C22×M16—21no—
178C2×C4×Q8—831no—
179C2×#117C2×D8311yescharacteristic
180#431—1661no—
181#400—831no—
182#399—831no—
183#398—1661no—
184#396—1661no—
185#395—831no—
186#393—831no—
187C4×C4⋊C4—831no—
188C4×C4⋊C4—1661no—
189C4×C22⋊C4—1661no—
190#343—21no—
191#343C2×D8312yes—
192#324—832no—
193#323—1661no—
194C22×Q8—831no—
195C23×C4C22⋊D8311yescharacteristic
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—41no—
199C2×M16—42no—
200C22×C8—41no—
201C4×Q8—3321no—
202C4×Q8—3321no—
203#117—41no—
204#117—41no—
205C2×C4⋊C4—831no—
206C2×C4⋊C4—1661no—
207C2×C4⋊C4—1661no—
208C2×C4⋊C4—1661no—
209C2×C4⋊C4—1661no—
210C2×C4⋊C4C22⋊D8311yescharacteristic
211C2×C4⋊C4C22×D8311yescharacteristic
212C2×C22⋊C4—1661no—
213C2×C22⋊C4—1661no—
214C2×C22⋊C4C4×D8311yescharacteristic
215C2×C22⋊C4D33211yescharacteristic
216C2×C42—1661no—
217C2×C42—1661no—
218C2×C42C4×D8311yescharacteristic
219C2×C42D33211yescharacteristic
220C4⋊C8—3321no—
221C22⋊C8—1662no—
222C4×C8—3321no—
223#95—831no—
224#95—831no—
225#95—1661no—
226#95—1661no—
227#95—1661no—
228#95—1661no—
229C24(C2×D83)⋊C411yescharacteristic
230C2×Q8—3321no—
231C2×Q8—3322no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—1661no—
236C22×C4—1661no—
237C22×C4—1661no—
238C22×C4—1661no—
239C22×C4(C2×D83)⋊C411yescharacteristic
240C22×C4(C2×D83)⋊C411yescharacteristic
241C22×C4(C2×D83)⋊C411yescharacteristic
242C22×C4#1305057812yes—
243C22×C4#1305058012yes—
244C22×C4C2×C4×D8311yescharacteristic
245C22×C4C2×C22⋊D8311yescharacteristic
246C22×C4C2×D33211yescharacteristic
247M16—24no—
248M16—44no—
249M16—42no—
250C2×C8—42no—
251C2×C8—44no—
252C2×C8—3322no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—3321no—
256C4⋊C4—3322no—
257C4⋊C4—3322no—
258C4⋊C4—3322no—
259C4⋊C4—3322no—
260C22⋊C4—41no—
261C22⋊C4—41no—
262C22⋊C4—3322no—
263C22⋊C4—3322no—
264C42—21no—
265C42—21no—
266C42—21no—
267C42—21no—
268C42—3321no—
269C42—3321no—
270C42—3322no—
271C23—21no—
272C23—21no—
273C23—41no—
274C23—41no—
275C23#1458171512yes—
276C23C2×(C2×D83)⋊C411yescharacteristic
277Q8—3322no—
278Q8—3322no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—3321no—
294C2×C4—3321no—
295C2×C4—3321no—
296C2×C4—3321no—
297C2×C4—3321no—
298C2×C4—3321no—
299C2×C4—3321no—
300C2×C4—3321no—
301C2×C4—3321no—
302C2×C4—3321no—
303C2×C4—1662no—
304C2×C4—1662no—
305C2×C4#1458170611yescharacteristic
306C2×C4#1458170611yescharacteristic
307C2×C4#1458171512yes—
308C2×C4#1458172012yes—
309C2×C4C2×#1305057811yescharacteristic
310C2×C4C2×#1305058011yescharacteristic
311C8—44no—
312C8—6642no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—42no—
317C22—41no—
318C22—41no—
319C22—81no—
320C22C2×#1458170611yescharacteristic
321C22C2×#1458171511yescharacteristic
322C22C2×#1458172011yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—3322no—
330C4—3322no—
331C4—6641no—
332C4—6641no—
333C4#1912743712yes—
334C4#1912743712yes—
335C2—21no—
336C2—21no—
337C2—42no—
338C2C2×#1912743711yescharacteristic
339C2#2490910511yescharacteristic
340C2#2490914611yescharacteristic
subgroupG/H
centerC2×C4#14581706
commutatorC22×C4×C83C22×C4
FrattiniC2×C4⋊C4C22×D83
FittingC83×#1742C2
socleC22×C83C2×#99
prime radicalC24×C83C22⋊C4
cosocleC2×C4⋊C4×C83C23
hypercenter#1742D83
hypocenterC83#11721
LCSG ▷ C22×C4×C83 ▷ C23×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
3. #24905492C2semidirect
4. #24905489C2semidirect
176. C83#11721semidirect

Maximal subgroups

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

subgroupindex
#249107312
#249057772
#249054922
#249054892
C83×#17422
#1172183