Finite Group Databasebeta

← #35301830

Subgroups — #35301830

Order 21248.

706 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 123 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#24916504C211yesmaximal, characteristic, complemented
2#24907455C211yesmaximal, characteristic, complemented
3#24906131C212yesmaximal, complemented
4#24905366C212yesmaximal, complemented
5C83×#2541C211yesmaximal, characteristic
6#19127779—24no—
7#19127754C2212yes—
8#19127725C2212yes—
9C2×#14581709—21no—
10#19127665—22no—
11#19127470—21no—
12#19127445—24no—
13C2×#127×C83—21no—
14C2×#127×C83C2211yescharacteristic
15C2×#121×C83—22no—
16C2×C4×D4×C83C2211yescharacteristic
17#392×C83C414yes—
18#390×C83—21no—
19C4×C4⋊C4×C83C2211yescharacteristic
20C2×#13050574—24no—
21C2×#13050574—22no—
22C4⋊(C83⋊C8)—22no—
23C4⋊(C83⋊C8)—22no—
24C4⋊(C83⋊C8)D414yescomplemented
25#14581709—42no—
26C22⋊(C83⋊C8)—24no—
27C22⋊(C83⋊C8)—22no—
28C4×C83⋊C8—22no—
29C22×D4×C83—24no—
30C22×D4×C83—21no—
31C22×D4×C83—21no—
32C22×D4×C83C2×C412yes—
33C22×D4×C83D411yescharacteristic
34C23×C4×C83D412yes—
35#127×C83—22no—
36#127×C83—22no—
37#127×C83—44no—
38#121×C83—48no—
39C4×D4×C83—22no—
40C4×D4×C83—22no—
41C2×C4⋊C4×C83—21no—
42C2×C4⋊C4×C83D411yescharacteristic
43C2×C22⋊C4×C83—24no—
44C2×C22⋊C4×C83C2×C412yes—
45C2×C42×C83—21no—
46C2×C42×C83C2311yescharacteristic
47#95×C83C2×C412yes—
48#13050574—42no—
49#13050574—42no—
50#13050574—84no—
51C2×C83⋊C8—24no—
52C2×C83⋊C8—42no—
53C24×C83—22no—
54C24×C83C22⋊C412yes—
55C2×D4×C83—24no—
56C2×D4×C83—22no—
57C2×D4×C83—22no—
58C2×D4×C83—21no—
59C2×D4×C83—21no—
60C2×D4×C83—44no—
61C2×D4×C83—48no—
62C2×D4×C83—48no—
63C2×D4×C83—41no—
64C2×D4×C83—42no—
65C2×D4×C83—41no—
66C2×D4×C83—42no—
67C2×D4×C83—41no—
68C2×D4×C83—41no—
69C2×D4×C83—84no—
70C2×D4×C83—82no—
71C22×C4×C83—22no—
72C22×C4×C83—22no—
73C22×C4×C83—22no—
74C22×C4×C83—44no—
75C22×C4×C83C22⋊C412yes—
76C22×C4×C83C22×C411yescharacteristic
77C22×C4×C83C2×D411yescharacteristic
78C22×C4×C83C2×D411yescharacteristic
79C4⋊C4×C83—24no—
80C4⋊C4×C83—22no—
81C22⋊C4×C83—24no—
82C22⋊C4×C83—48no—
83C42×C83—42no—
84C42×C83C2×D412yes—
85C42×C83#4112yes—
86C23×C83—22no—
87C23×C83—22no—
88C23×C83—22no—
89C23×C83—42no—
90C23×C83—42no—
91C23×C83—44no—
92C23×C83—44no—
93C23×C83—42no—
94C23×C83—42no—
95C23×C83—42no—
96C23×C83C2×C22⋊C411yescharacteristic
97D4×C83—42no—
98D4×C83—42no—
99D4×C83—84no—
100D4×C83—82no—
101D4×C83—816no—
102D4×C83—82no—
103D4×C83—82no—
104D4×C83—82no—
105C2×C4×C83—22no—
106C2×C4×C83—24no—
107C2×C4×C83—22no—
108C2×C4×C83—41no—
109C2×C4×C83—42no—
110C2×C4×C83—42no—
111C2×C4×C83—44no—
112C2×C4×C83—42no—
113C2×C4×C83—82no—
114C2×C4×C83C4×D412yes—
115C2×C4×C83#12011yescharacteristic
116C2×C4×C83#12112yes—
117C2×C4×C83#12311yescharacteristic
118C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
119C2×C4×C83AGL(1,ℤ/8)12yes—
120C2×C4×C83#13711yescharacteristic
121C83⋊C8—44no—
122C83⋊C8—82no—
123C22×C83—21no—
124C22×C83—21no—
125C22×C83—44no—
126C22×C83—42no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—82no—
130C22×C83—84no—
131C22×C83—84no—
132C22×C83—82no—
133C22×C83—82no—
134C22×C83—82no—
135C22×C83#38611yescharacteristic
136C22×C83#41711yescharacteristic
137C22×C83#41811yescharacteristic
138C4×C83—21no—
139C4×C83—22no—
140C4×C83—44no—
141C4×C83—42no—
142C4×C83#45312yes—
143C4×C83#46812yes—
144C4×C83#46912yes—
145#9405—831nomaximal, Sylow
146C2×C83—21no—
147C2×C83—21no—
148C2×C83—82no—
149C2×C83—162no—
150C2×C83#188711yescharacteristic
151C2×C83#189511yescharacteristic
152C2×C83#191411yescharacteristic
153#2975—831no—
154#2541D8311yescharacteristic
155#1800—831no—
156#1545—832no—
157#1303—832no—
158C83#940511yescharacteristic, complemented, Sylow
159C2×#127—21no—
160C2×#127C2×D8311yescharacteristic
161C2×#121—22no—
162C2×C4×D4C2×D8311yescharacteristic
163#436—1664no—
164#431—832no—
165#423—832no—
166C2×#100—1661no—
167#407—1662no—
168#392Dic8314yes—
169#390—21no—
170C4×C4⋊C4C2×D8311yescharacteristic
171#343—1661no—
172#331—1664no—
173C22×D4—24no—
174C22×D4—21no—
175C22×D4—21no—
176C22×D4C2×Dic8312yes—
177C22×D4C22⋊D8311yescharacteristic
178C23×C4C22⋊D8312yes—
179C2×M16—1662no—
180C2×M16—1664no—
181#127—22no—
182#127—22no—
183#127—44no—
184#121—48no—
185C4×D4—22no—
186C4×D4—22no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4C22⋊D8311yescharacteristic
189C2×C22⋊C4—24no—
190C2×C22⋊C4C2×Dic8312yes—
191C2×C42—21no—
192C2×C42C22×D8311yescharacteristic
193C4⋊C8—834no—
194C4⋊C8—1662no—
195C4⋊C8—1662no—
196#100—3322no—
197C22⋊C8—1664no—
198C22⋊C8—1662no—
199C4×C8—1662no—
200#95C2×Dic8312yes—
201C24—22no—
202C24C22⋊Dic8312yes—
203C2×D4—22no—
204C2×D4—21no—
205C2×D4—22no—
206C2×D4—21no—
207C2×D4—24no—
208C2×D4—44no—
209C2×D4—48no—
210C2×D4—48no—
211C2×D4—41no—
212C2×D4—42no—
213C2×D4—41no—
214C2×D4—41no—
215C2×D4—42no—
216C2×D4—41no—
217C2×D4—84no—
218C2×D4—82no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—44no—
223C22×C4C22⋊Dic8312yes—
224C22×C4C22×Dic8311yescharacteristic
225C22×C4C2×C22⋊D8311yescharacteristic
226C22×C4C2×C22⋊D8311yescharacteristic
227M16—3322no—
228M16—3322no—
229M16—6644no—
230C2×C8—3322no—
231C2×C8—1664no—
232C4⋊C4—24no—
233C4⋊C4—22no—
234C22⋊C4—24no—
235C22⋊C4—48no—
236C42—42no—
237C42D4×D8312yes—
238C42#1305058912yes—
239C23—22no—
240C23—22no—
241C23—22no—
242C23—42no—
243C23—42no—
244C23—44no—
245C23—44no—
246C23—42no—
247C23—42no—
248C23—42no—
249C23C2×C22⋊Dic8311yescharacteristic
250D4—42no—
251D4—42no—
252D4—84no—
253D4—82no—
254D4—816no—
255D4—82no—
256D4—82no—
257D4—82no—
258C2×C4—22no—
259C2×C4—24no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—44no—
265C2×C4—42no—
266C2×C4—82no—
267C2×C4D4×Dic8312yes—
268C2×C4#1458176011yescharacteristic
269C2×C4#1458176412yes—
270C2×C4#1458177311yescharacteristic
271C2×C4#1458181711yescharacteristic
272C2×C4#1458181812yes—
273C2×C4#1458182211yescharacteristic
274C8—3324no—
275C8—6642no—
276C22—21no—
277C22—21no—
278C22—44no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—82no—
283C22—84no—
284C22—84no—
285C22—82no—
286C22—82no—
287C22—82no—
288C22#1912758111yescharacteristic
289C22#1912770211yescharacteristic
290C22#1912770611yescharacteristic
291C4—21no—
292C4—22no—
293C4—44no—
294C4—42no—
295C4#1912786812yes—
296C4#1912794512yes—
297C4#1912795212yes—
298C2—21no—
299C2—21no—
300C2—82no—
301C2—162no—
302C2#2490786211yescharacteristic
303C2#2490790211yescharacteristic
304C2#2490801111yescharacteristic
subgroupG/H
centerC22#19127581
commutatorC22×C4×C83C22×C4
FrattiniC2×C42C22×D83
Fitting, prime radicalC83×#2541C2
socleC22×C83#386
cosocleC2×C42×C83C23
hypercenter#2541D83
hypocenterC83#9405
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
1. #24916504C2semidirect
2. #24907455C2semidirect
3. #24906131C2semidirect
4. #24905366C2semidirect
24. C4⋊(C83⋊C8)D4semidirect
158. C83#9405semidirect

Maximal subgroups

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

subgroupindex
#249165042
#249074552
#249061312
#249053662
C83×#25412
#940583