Finite Group Databasebeta

← C83×#9192 #35237016

Subgroups — C83×#9192 #35237016

Order 21248.

458 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 134 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×#2921C211yesmaximal, characteristic
2C83×#2628C211yesmaximal, characteristic
3C83×#1548C212yesmaximal, complemented
4C83×#1458C211yesmaximal, characteristic
5C83×#1286C212yesmaximal, complemented
6C2×C4⋊Q8×C83C2211yescharacteristic
7C2×#117×C83C2211yescharacteristic
8#433×C83—24no—
9#431×C83C2212yes—
10#423×C83C2212yes—
11#407×C83—21no—
12C4×M16×C83—21no—
13#400×C83—21no—
14#399×C83—21no—
15#395×C83C412yes—
16#393×C83C412yes—
17#389×C83—21no—
18#382×C83C2211yescharacteristic
19#380×C83—21no—
20#333×C83—24no—
21#324×C83—21no—
22#323×C83—21no—
23C22×Q8×C83C2×C412yes—
24C23×C4×C83D411yescharacteristic
25C2×M16×C83—22no—
26C4⋊Q8×C83—22no—
27C4⋊Q8×C83—22no—
28#117×C83—22no—
29#117×C83—22no—
30C2×C4⋊C4×C83—21no—
31C2×C4⋊C4×C83—21no—
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83—22no—
34C2×C4⋊C4×C83C2×C411yescharacteristic
35C2×C4⋊C4×C83D411yescharacteristic
36C2×C22⋊C4×C83—22no—
37C2×C22⋊C4×C83C2×C411yescharacteristic
38C2×C22⋊C4×C83D411yescharacteristic
39C2×C42×C83D411yescharacteristic
40C2×C42×C83C2311yescharacteristic
41C4⋊C8×C83—22no—
42C4⋊C8×C83—22no—
43C4⋊C8×C83D414yes—
44C22⋊C8×C83—24no—
45C22⋊C8×C83—22no—
46#97×C83—22no—
47C4×C8×C83—22no—
48C4×C8×C83—22no—
49#95×C83—22no—
50#95×C83—22no—
51#95×C83—21no—
52#95×C83C2×C412yes—
53C24×C83C22⋊C411yescharacteristic
54C2×Q8×C83—24no—
55C2×Q8×C83—82no—
56C22×C4×C83—21no—
57C22×C4×C83—21no—
58C22×C4×C83—22no—
59C22×C4×C83—22no—
60C22×C4×C83—22no—
61C22×C4×C83—22no—
62C22×C4×C83C22⋊C411yescharacteristic
63C22×C4×C83C22⋊C411yescharacteristic
64C22×C4×C83C22⋊C411yescharacteristic
65C22×C4×C83C22×C411yescharacteristic
66C22×C4×C83C2×D411yescharacteristic
67C22×C4×C83C2×D411yescharacteristic
68M16×C83—42no—
69M16×C83—42no—
70C2×C8×C83—24no—
71C2×C8×C83—42no—
72C4⋊C4×C83—22no—
73C4⋊C4×C83—22no—
74C4⋊C4×C83—41no—
75C4⋊C4×C83—42no—
76C4⋊C4×C83—41no—
77C4⋊C4×C83—42no—
78C4⋊C4×C83—82no—
79C22⋊C4×C83—22no—
80C22⋊C4×C83—22no—
81C22⋊C4×C83—82no—
82C42×C83—21no—
83C42×C83—21no—
84C42×C83C2×D412yes—
85C42×C83#4112yes—
86C23×C83—21no—
87C23×C83—21no—
88C23×C83—21no—
89C23×C83—41no—
90C23×C83—41no—
91C23×C83C2×C22⋊C411yescharacteristic
92Q8×C83—84no—
93C2×C4×C83—21no—
94C2×C4×C83—21no—
95C2×C4×C83—21no—
96C2×C4×C83—21no—
97C2×C4×C83—21no—
98C2×C4×C83—41no—
99C2×C4×C83—41no—
100C2×C4×C83—42no—
101C2×C4×C83—42no—
102C2×C4×C83—41no—
103C2×C4×C83—41no—
104C2×C4×C83—42no—
105C2×C4×C83—42no—
106C2×C4×C83—42no—
107C2×C4×C83—42no—
108C2×C4×C83—42no—
109C2×C4×C83—42no—
110C2×C4×C83C4×D412yes—
111C2×C4×C83#12011yescharacteristic
112C2×C4×C83#12112yes—
113C2×C4×C83#12311yescharacteristic
114C2×C4×C83#13512yes—
115C2×C4×C83#13712yes—
116C8×C83—44no—
117C8×C83—82no—
118C22×C83—21no—
119C22×C83—21no—
120C22×C83—42no—
121C22×C83—41no—
122C22×C83—41no—
123C22×C83—41no—
124C22×C83—81no—
125C22×C83#38611yescharacteristic
126C22×C83#41611yescharacteristic
127C22×C83#41911yescharacteristic
128C4×C83—22no—
129C4×C83—22no—
130C4×C83—41no—
131C4×C83—41no—
132C4×C83—41no—
133C4×C83—82no—
134C4×C83—82no—
135C4×C83#45212yes—
136C4×C83#48512yes—
137#9192C8311yesmaximal, characteristic, direct product factor, Sylow
138C2×C83—21no—
139C2×C83—21no—
140C2×C83—81no—
141C2×C83#187311yescharacteristic
142C2×C83#190911yescharacteristic
143C2×C83#191411yescharacteristic
144#2921C2×C8311yescharacteristic
145#2628C2×C8311yescharacteristic
146#1548C2×C8312yescomplemented
147#1458C2×C8311yescharacteristic
148#1286C2×C8312yescomplemented
149C83#919211yescharacteristic, direct product factor, Sylow
150C2×C4⋊Q8C22×C8311yescharacteristic
151C2×#117C22×C8311yescharacteristic
152#433—24no—
153#431C22×C8312yes—
154#423C22×C8312yes—
155#407—21no—
156C4×M16—21no—
157#400—21no—
158#399—21no—
159#395C4×C8312yes—
160#393C4×C8312yes—
161#389—21no—
162#382C22×C8311yescharacteristic
163#380—21no—
164#333—24no—
165#324—21no—
166#323—21no—
167C22×Q8C2×C4×C8312yes—
168C23×C4D4×C8311yescharacteristic
169C2×M16—22no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4C2×C4×C8311yescharacteristic
179C2×C4⋊C4D4×C8311yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×C4×C8311yescharacteristic
182C2×C22⋊C4D4×C8311yescharacteristic
183C2×C42D4×C8311yescharacteristic
184C2×C42C23×C8311yescharacteristic
185C4⋊C8—22no—
186C4⋊C8—22no—
187C4⋊C8D4×C8314yes—
188C22⋊C8—24no—
189C22⋊C8—22no—
190#97—22no—
191C4×C8—22no—
192C4×C8—22no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95C2×C4×C8312yes—
197C24C22⋊C4×C8311yescharacteristic
198C2×Q8—24no—
199C2×Q8—82no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C22⋊C4×C8311yescharacteristic
207C22×C4C22⋊C4×C8311yescharacteristic
208C22×C4C22⋊C4×C8311yescharacteristic
209C22×C4C22×C4×C8311yescharacteristic
210C22×C4C2×D4×C8311yescharacteristic
211C22×C4C2×D4×C8311yescharacteristic
212M16—42no—
213M16—42no—
214C2×C8—24no—
215C2×C8—42no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—41no—
219C4⋊C4—42no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42C2×D4×C8312yes—
229C42#41×C8312yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊C4×C8311yescharacteristic
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4C4×D4×C8312yes—
255C2×C4#120×C8311yescharacteristic
256C2×C4#121×C8312yes—
257C2×C4#123×C8311yescharacteristic
258C2×C4#135×C8312yes—
259C2×C4#137×C8312yes—
260C8—44no—
261C8—82no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#386×C8311yescharacteristic
270C22#416×C8311yescharacteristic
271C22#419×C8311yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#452×C8312yes—
280C4#485×C8312yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2C83×#187311yescharacteristic
285C2C83×#190911yescharacteristic
286C2C83×#191411yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC24×C83C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1548C2semidirect
5. C83×#1286C2semidirect
137. #9192C83direct (N × H)
146. #1548C2×C83semidirect
148. #1286C2×C83semidirect
149. C83#9192direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#29212
C83×#26282
C83×#15482
C83×#14582
C83×#12862
#919283