Finite Group Databasebeta

← #35320439

Subgroups — #35320439

Order 21248.

412 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 97 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#24916181C211yesmaximal, characteristic
2#24909148C211yesmaximal, characteristic
3#24907511C212yesmaximal, complemented
4#24905585C212yesmaximal, complemented
5C83×#2132C211yesmaximal, characteristic
6C2×#14581799C411yescharacteristic
7C2×#14581799C2211yescharacteristic
8#19127773—22no—
9#19127773—22no—
10C2×#14581727—22no—
11#19127725C411yescharacteristic
12#19127685C2212yes—
13#19127668C411yescharacteristic
14#19127668C2211yescharacteristic
15C2×#14581703C411yescharacteristic
16#19127500—21no—
17#19127498—21no—
18C83⋊#338—22no—
19#532×C83C2211yescharacteristic
20#413×C83C2212yes—
21#356×C83—21no—
22#355×C83—21no—
23#14581799—42no—
24#14581799—42no—
25C2×#13050574—21no—
26C2×#13050574—22no—
27C2×#13050574C2×C411yescharacteristic
28C2×#13050574C2×C411yescharacteristic
29C2×#13050574D412yes—
30C22×C83⋊C8—21no—
31C22×C83⋊C8C2×C411yescharacteristic
32C22×C83⋊C8C2×C411yescharacteristic
33C22×C83⋊C8D412yes—
34#14581727—44no—
35C4⋊(C83⋊C8)—22no—
36#14581711—22no—
37#14581709—22no—
38C22⋊(C83⋊C8)—22no—
39C22⋊(C83⋊C8)—22no—
40#14581703—21no—
41#14581703—21no—
42C2×#41×C83C2×C411yescharacteristic
43C2×#41×C83C2311yescharacteristic
44C2×M16×C83—22no—
45C4⋊Q8×C83D411yescharacteristic
46#127×C83Q811yescharacteristic
47#124×C83D411yescharacteristic
48#124×C83Q811yescharacteristic
49#121×C83—22no—
50C4×D4×C83—22no—
51C2×C42×C83C2×C411yescharacteristic
52#101×C83—22no—
53#100×C83—22no—
54#13050574—41no—
55#13050574—42no—
56#13050574—42no—
57#13050574—41no—
58#13050574—42no—
59#13050574—42no—
60C2×C83⋊C8—22no—
61C2×C83⋊C8—24no—
62C2×C83⋊C8—21no—
63C2×C83⋊C8—21no—
64C2×C83⋊C8—42no—
65C2×C83⋊C8—41no—
66C2×C83⋊C8—42no—
67C2×C83⋊C8C22⋊C411yescharacteristic
68C2×C83⋊C8C22⋊C412yes—
69C2×C83⋊C8C22⋊C411yescharacteristic
70#41×C83—41no—
71#41×C83—41no—
72#41×C83—41no—
73#41×C83—41no—
74C2×Q8×C83C4⋊C411yescharacteristic
75C2×Q8×C83C2×D411yescharacteristic
76C2×D4×C83—21no—
77C2×D4×C83—22no—
78C2×D4×C83C4⋊C411yescharacteristic
79C2×D4×C83C2×Q811yescharacteristic
80C2×D4×C83#4112yes—
81C22×C4×C83—21no—
82C22×C4×C83—21no—
83C22×C4×C83C22×C411yescharacteristic
84C22×C4×C83C2×D412yes—
85M16×C83—42no—
86M16×C83—42no—
87C2×C8×C83—42no—
88C4⋊C4×C83—22no—
89C22⋊C4×C83—22no—
90C22⋊C4×C83—22no—
91C42×C83—21no—
92C42×C83C4⋊C411yescharacteristic
93C42×C83C4⋊C411yescharacteristic
94C23×C83—21no—
95C23×C83#11711yescharacteristic
96C23×C83C22⋊Q812yes—
97Q8×C83—41no—
98Q8×C83—41no—
99D4×C83—41no—
100D4×C83—42no—
101D4×C83—41no—
102D4×C83—42no—
103D4×C83—82no—
104C2×C4×C83—22no—
105C2×C4×C83—21no—
106C2×C4×C83—22no—
107C2×C4×C83—21no—
108C2×C4×C83—21no—
109C2×C4×C83—21no—
110C2×C4×C83—21no—
111C2×C4×C83—41no—
112C2×C4×C83—41no—
113C2×C4×C83—81no—
114C2×C4×C83C2×C22⋊C411yescharacteristic
115C2×C4×C83C2×C4⋊C411yescharacteristic
116C2×C4×C83#12112yes—
117C8×C83—82no—
118C83⋊C8—21no—
119C83⋊C8—22no—
120C83⋊C8—41no—
121C83⋊C8—41no—
122C83⋊C8—82no—
123C83⋊C8#9912yes—
124C22×C83—21no—
125C22×C83—22no—
126C22×C83—41no—
127C22×C83—42no—
128C22×C83—42no—
129C22×C83—41no—
130C22×C83#38011yescharacteristic
131C4×C83—22no—
132C4×C83—21no—
133C4×C83—41no—
134C4×C83—41no—
135C4×C83—41no—
136C4×C83C2×#9911yescharacteristic
137C4×C83#41011yescharacteristic
138#12810—831nomaximal, Sylow
139C2×C83—21no—
140C2×C83—41no—
141C2×C83—42no—
142C2×C83—81no—
143C2×C83#179911yescharacteristic
144#2932—831no—
145#2132—831no—
146#2132D8311yescharacteristic
147#1813—832no—
148#1392—832no—
149C83#1281011yescharacteristic, complemented, Sylow
150C2×#131—831no—
151C2×#131—831no—
152#532C2×D8311yescharacteristic
153#435—1662no—
154#435—1662no—
155C2×#108—1662no—
156#423—831no—
157#413—832no—
158#413C2×D8312yes—
159#408—831no—
160#408—831no—
161C2×#97—831no—
162#356—21no—
163#356—1661no—
164#355—21no—
165#355—1661no—
166#338—1662no—
167C2×#41C4×D8311yescharacteristic
168C2×#41C22×D8311yescharacteristic
169#131—3322no—
170#131—3322no—
171C2×M16—22no—
172C2×M16—831no—
173C2×M16—832no—
174C2×M16—831no—
175C2×M16—1662no—
176C2×M16—1661no—
177C22×C8—831no—
178C22×C8—831no—
179C22×C8—832no—
180C22×C8—1661no—
181C4⋊Q8C22⋊D8311yescharacteristic
182#127Dic16611yescharacteristic
183#124Dic16611yescharacteristic
184#124C22⋊D8311yescharacteristic
185#121—22no—
186C4×D4—22no—
187C2×C42C4×D8311yescharacteristic
188#108—3324no—
189C4⋊C8—1662no—
190#101—22no—
191#101—1662no—
192#100—22no—
193#100—1662no—
194C22⋊C8—1662no—
195C22⋊C8—1662no—
196#97—1661no—
197#97—1661no—
198#41—41no—
199#41—41no—
200#41—41no—
201#41—41no—
202C2×Q8Dic83⋊C411yescharacteristic
203C2×Q8C2×C22⋊D8311yescharacteristic
204C2×D4—21no—
205C2×D4—22no—
206C2×D4Dic83⋊C411yescharacteristic
207C2×D4C2×Dic16611yescharacteristic
208C2×D4#1305058912yes—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4C2×C4×D8311yescharacteristic
212C22×C4D4×D8312yes—
213M16—42no—
214M16—42no—
215M16—3322no—
216M16—3321no—
217M16—3322no—
218M16—3322no—
219M16—3322no—
220M16—3321no—
221C2×C8—42no—
222C2×C8—3322no—
223C2×C8—3321no—
224C2×C8—3322no—
225C2×C8—831no—
226C2×C8—831no—
227C2×C8—832no—
228C2×C8—1661no—
229C2×C8—1661no—
230C2×C8—1664no—
231C2×C8—1662no—
232C4⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—22no—
235C42—21no—
236C42Dic83⋊C411yescharacteristic
237C42Dic83⋊C411yescharacteristic
238C23—21no—
239C23#1458174811yescharacteristic
240C23#1458176912yes—
241Q8—41no—
242Q8—41no—
243D4—41no—
244D4—42no—
245D4—41no—
246D4—42no—
247D4—82no—
248C2×C4—22no—
249C2×C4—21no—
250C2×C4—22no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—81no—
258C2×C4C22⋊C4×D8311yescharacteristic
259C2×C4C2×Dic83⋊C411yescharacteristic
260C2×C4#1458176412yes—
261C8—82no—
262C8—3321no—
263C8—3321no—
264C8—832no—
265C8—6642no—
266C8—1661no—
267C8—1662no—
268C22—21no—
269C22—22no—
270C22—41no—
271C22—42no—
272C22—42no—
273C22—41no—
274C22#1912755911yescharacteristic
275C4—22no—
276C4—21no—
277C4—41no—
278C4—41no—
279C4—41no—
280C4#99×D8311yescharacteristic
281C4#1912767311yescharacteristic
282C2—21no—
283C2—41no—
284C2—42no—
285C2—81no—
286C2#2490744911yescharacteristic
subgroupG/H
centerC4#19127673
commutatorC22×C4×C83C22×C4
FrattiniC2×#41C22×D83
FittingC83×#2132C2
socleC2×C83#1799
prime radical#532×C83C22
cosocleC2×#41×C83C23
hypercenter#2132D83
hypocenterC83#12810
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #2132

Complements

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

NH ≅ G/Ntype
3. #24907511C2semidirect
4. #24905585C2semidirect
149. C83#12810semidirect

Maximal subgroups

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

subgroupindex
#249161812
#249091482
#249075112
#249055852
C83×#21322
#1281083