Finite Group Databasebeta

← C83×#9505 #35237329

Subgroups — C83×#9505 #35237329

Order 21248.

684 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 126 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
1C83×#2973C211yesmaximal, characteristic
2C83×#2409C211yesmaximal, characteristic
3C83×#1784C212yesmaximal, complemented
4C83×#1465C211yesmaximal, characteristic
5C83×#1305C212yesmaximal, complemented
6C23×Q8×C83C2211yescharacteristic
7C2×C22⋊Q8×C83—21no—
8C2×C4×D4×C83C2211yescharacteristic
9#435×C83—24no—
10#432×C83—21no—
11C2×#101×C83—24no—
12#407×C83C2212yes—
13#407×C83C2212yes—
14C2×C22⋊C8×C83—21no—
15#395×C83—21no—
16#393×C83—21no—
17#391×C83—21no—
18#386×C83C412yes—
19#385×C83C412yes—
20#380×C83C2211yescharacteristic
21#324×C83—21no—
22#323×C83—21no—
23C22×Q8×C83—21no—
24C22×Q8×C83—22no—
25C22×Q8×C83—21no—
26C22×Q8×C83—22no—
27C22×Q8×C83—44no—
28C22×D4×C83C2×C411yescharacteristic
29C23×C4×C83C2×C412yes—
30C23×C4×C83D411yescharacteristic
31C23×C4×C83C2311yescharacteristic
32C2×M16×C83—22no—
33C2×M16×C83D414yes—
34C22×C8×C83—22no—
35C22⋊Q8×C83—44no—
36C4×D4×C83—22no—
37C4×D4×C83—22no—
38C2×C4⋊C4×C83—22no—
39C2×C4⋊C4×C83—21no—
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C22⋊C4×C83—22no—
42C2×C22⋊C4×C83C2×C411yescharacteristic
43C2×C22⋊C4×C83C2×C412yes—
44C2×C22⋊C4×C83D411yescharacteristic
45C2×C42×C83D411yescharacteristic
46C4⋊C8×C83—22no—
47#101×C83—48no—
48C22⋊C8×C83—22no—
49C22⋊C8×C83—22no—
50C22⋊C8×C83—22no—
51C22⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53#97×C83—22no—
54#95×C83—22no—
55#95×C83—22no—
56#95×C83—21no—
57C24×C83C22⋊C411yescharacteristic
58C24×C83C22×C411yescharacteristic
59C2×Q8×C83—28no—
60C2×Q8×C83—24no—
61C2×Q8×C83—42no—
62C2×Q8×C83—44no—
63C2×Q8×C83—44no—
64C2×Q8×C83—48no—
65C2×Q8×C83—41no—
66C2×Q8×C83—42no—
67C2×Q8×C83—41no—
68C2×D4×C83—22no—
69C2×D4×C83—22no—
70C2×D4×C83—41no—
71C22×C4×C83—22no—
72C22×C4×C83—24no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—22no—
76C22×C4×C83—44no—
77C22×C4×C83—42no—
78C22×C4×C83—42no—
79C22×C4×C83C22⋊C411yescharacteristic
80C22×C4×C83C22⋊C411yescharacteristic
81C22×C4×C83C22⋊C411yescharacteristic
82C22×C4×C83C2×D412yes—
83C22×C4×C83C2×D411yescharacteristic
84C22×C4×C83C2×D411yescharacteristic
85C22×C4×C83#4112yes—
86M16×C83—44no—
87M16×C83—44no—
88C2×C8×C83—24no—
89C2×C8×C83—42no—
90C2×C8×C83—82no—
91C4⋊C4×C83—22no—
92C4⋊C4×C83—42no—
93C4⋊C4×C83—44no—
94C22⋊C4×C83—22no—
95C22⋊C4×C83—22no—
96C22⋊C4×C83—44no—
97C22⋊C4×C83—42no—
98C42×C83—21no—
99C42×C83—21no—
100C23×C83—21no—
101C23×C83—21no—
102C23×C83—21no—
103C23×C83—21no—
104C23×C83—41no—
105C23×C83—41no—
106C23×C83—41no—
107C23×C83—41no—
108C23×C83#10112yescomplemented
109C23×C83C2×C22⋊C411yescharacteristic
110C23×C83C4×D412yes—
111Q8×C83—48no—
112Q8×C83—44no—
113Q8×C83—44no—
114D4×C83—44no—
115C2×C4×C83—22no—
116C2×C4×C83—24no—
117C2×C4×C83—24no—
118C2×C4×C83—28no—
119C2×C4×C83—21no—
120C2×C4×C83—21no—
121C2×C4×C83—21no—
122C2×C4×C83—21no—
123C2×C4×C83—21no—
124C2×C4×C83—41no—
125C2×C4×C83—42no—
126C2×C4×C83—42no—
127C2×C4×C83—44no—
128C2×C4×C83—42no—
129C2×C4×C83—42no—
130C2×C4×C83—41no—
131C2×C4×C83—41no—
132C2×C4×C83—42no—
133C2×C4×C83—42no—
134C2×C4×C83—42no—
135C2×C4×C83—81no—
136C2×C4×C83#12011yescharacteristic
137C2×C4×C83#12112yes—
138C2×C4×C83#12311yescharacteristic
139C8×C83—44no—
140C8×C83—82no—
141C22×C83—21no—
142C22×C83—21no—
143C22×C83—22no—
144C22×C83—22no—
145C22×C83—22no—
146C22×C83—22no—
147C22×C83—42no—
148C22×C83—42no—
149C22×C83—42no—
150C22×C83—41no—
151C22×C83—41no—
152C22×C83—81no—
153C22×C83#38611yescharacteristic
154C22×C83C2×#10111yescharacteristic
155C22×C83#41311yescharacteristic
156C4×C83—22no—
157C4×C83—21no—
158C4×C83—44no—
159C4×C83—41no—
160C4×C83—41no—
161C4×C83—41no—
162C4×C83—82no—
163C4×C83B(4,𝔽2)12yes—
164#9505C8311yesmaximal, characteristic, direct product factor, Sylow
165C2×C83—21no—
166C2×C83—21no—
167C2×C83—42no—
168C2×C83—42no—
169C2×C83#188911yescharacteristic
170C2×C83#190111yescharacteristic
171C2×C83#191511yescharacteristic
172#2973C2×C8311yescharacteristic
173#2409C2×C8311yescharacteristic
174#1784C2×C8312yescomplemented
175#1465C2×C8311yescharacteristic
176#1305C2×C8312yescomplemented
177C83#950511yescharacteristic, direct product factor, Sylow
178C23×Q8C22×C8311yescharacteristic
179C2×C22⋊Q8—21no—
180C2×C4×D4C22×C8311yescharacteristic
181#435—24no—
182#432—21no—
183C2×#101—24no—
184#407C22×C8312yes—
185#407C22×C8312yes—
186C2×C22⋊C8—21no—
187#395—21no—
188#393—21no—
189#391—21no—
190#386C4×C8312yes—
191#385C4×C8312yes—
192#380C22×C8311yescharacteristic
193#324—21no—
194#323—21no—
195C22×Q8—21no—
196C22×Q8—22no—
197C22×Q8—21no—
198C22×Q8—22no—
199C22×Q8—44no—
200C22×D4C2×C4×C8311yescharacteristic
201C23×C4C2×C4×C8312yes—
202C23×C4D4×C8311yescharacteristic
203C23×C4C23×C8311yescharacteristic
204C2×M16—22no—
205C2×M16D4×C8314yes—
206C22×C8—22no—
207C22⋊Q8—44no—
208C4×D4—22no—
209C4×D4—22no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4D4×C8311yescharacteristic
213C2×C22⋊C4—22no—
214C2×C22⋊C4C2×C4×C8311yescharacteristic
215C2×C22⋊C4C2×C4×C8312yes—
216C2×C22⋊C4D4×C8311yescharacteristic
217C2×C42D4×C8311yescharacteristic
218C4⋊C8—22no—
219#101—48no—
220C22⋊C8—22no—
221C22⋊C8—22no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224C22⋊C8—22no—
225#97—22no—
226#95—22no—
227#95—22no—
228#95—21no—
229C24C22⋊C4×C8311yescharacteristic
230C24C22×C4×C8311yescharacteristic
231C2×Q8—28no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—48no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×Q8—41no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—41no—
243C22×C4—22no—
244C22×C4—24no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—22no—
248C22×C4—44no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4C22⋊C4×C8311yescharacteristic
252C22×C4C22⋊C4×C8311yescharacteristic
253C22×C4C22⋊C4×C8311yescharacteristic
254C22×C4C2×D4×C8312yes—
255C22×C4C2×D4×C8311yescharacteristic
256C22×C4C2×D4×C8311yescharacteristic
257C22×C4#41×C8312yes—
258M16—44no—
259M16—44no—
260C2×C8—24no—
261C2×C8—42no—
262C2×C8—82no—
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—44no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C42—21no—
271C42—21no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23#101×C8312yescomplemented
281C23C2×C22⋊C4×C8311yescharacteristic
282C23C4×D4×C8312yes—
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—24no—
290C2×C4—28no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—42no—
299C2×C4—44no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—81no—
308C2×C4#120×C8311yescharacteristic
309C2×C4#121×C8312yes—
310C2×C4#123×C8311yescharacteristic
311C8—44no—
312C8—82no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—22no—
319C22—42no—
320C22—42no—
321C22—42no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#386×C8311yescharacteristic
326C22C2×#101×C8311yescharacteristic
327C22#413×C8311yescharacteristic
328C4—22no—
329C4—21no—
330C4—44no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4B(4,𝔽2)×C8312yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—42no—
340C2C83×#188911yescharacteristic
341C2C83×#190111yescharacteristic
342C2C83×#191511yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC24C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC22×D4×C83C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1784C2semidirect
5. C83×#1305C2semidirect
108. C23×C83#101semidirect
164. #9505C83direct (N × H)
174. #1784C2×C83semidirect
176. #1305C2×C83semidirect
177. C83#9505direct (N × H)
280. C23#101×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#29732
C83×#24092
C83×#17842
C83×#14652
C83×#13052
#950583