Finite Group Databasebeta

← #35305392

Subgroups — #35305392

Order 21248.

528 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 105 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#24917467C212yesmaximal
2#24908261C211yesmaximal, characteristic
3#24905866C212yesmaximal, complemented
4#24905294C211yesmaximal, characteristic, complemented
5C83×#3162C211yesmaximal, characteristic, complemented
6C2×C4×D332C2211yescharacteristic
7#19128029—22no—
8#19127988—22no—
9#19127903—22no—
10#19127851—22no—
11C4×#13050577—24no—
12#19127754C2212yes—
13C2×C4⋊(C83⋊C8)C2211yescharacteristic
14C2×(C2×D83)⋊C8—22no—
15Q8⋊(C83⋊C8)—24no—
16#533×C83C2212yes—
17#501×C83—22no—
18#501×C83—22no—
19#427×C83—22no—
20C2×#97×C83C2211yescharacteristic
21C22×D332D411yescharacteristic
22C22×C4×D83—21no—
23C2×#13050577—24no—
24C22×C83⋊C8—21no—
25C4×D332—22no—
26C4×D332—22no—
27C2×C4⋊Dic83D411yescharacteristic
28C2×(C2×D83)⋊C4—21no—
29#14581723—24no—
30C4⋊(C83⋊C8)—22no—
31C4⋊(C83⋊C8)—22no—
32#14581716—24no—
33#14581712—24no—
34C22⋊(C83⋊C8)—22no—
35(C2×D83)⋊C8—44no—
36C4×C83⋊C8D414yes—
37C22×C8×C83D412yes—
38C4⋊Q8×C83D412yes—
39C4⋊Q8×C83D412yes—
40#125×C83—22no—
41C22⋊Q8×C83—22no—
42C22⋊Q8×C83—22no—
43C4×Q8×C83—22no—
44C4×Q8×C83—22no—
45#117×C83—22no—
46C2×C42×C83C2311yescharacteristic
47#107×C83—22no—
48#107×C83—22no—
49#106×C83—24no—
50#97×C83—22no—
51C23×D83—21no—
52C2×D332—22no—
53C2×D332—42no—
54C22×Dic83—21no—
55C2×C4×D83—41no—
56C2×C4×D83—41no—
57C2×C4×D83—42no—
58C2×C4×D83—42no—
59#13050577—48no—
60C2×C83⋊C8—24no—
61C2×C83⋊C8—41no—
62C2×C83⋊C8—42no—
63C4⋊Dic83—22no—
64(C2×D83)⋊C4—42no—
65C2×Q8×C83—22no—
66C2×Q8×C83—22no—
67C22×C4×C83C2×D411yescharacteristic
68C22×C4×C83C2×D411yescharacteristic
69C22×C4×C83#4111yescharacteristic
70C2×C8×C83—22no—
71C2×C8×C83—22no—
72C2×C8×C83SD1614yescomplemented
73C4⋊C4×C83—22no—
74C4⋊C4×C83—22no—
75C4⋊C4×C83—22no—
76C4⋊C4×C83—22no—
77C4⋊C4×C83—42no—
78C4⋊C4×C83—42no—
79C22⋊C4×C83—42no—
80C42×C83—42no—
81C42×C83C2×D412yes—
82C42×C83C2×D412yes—
83C23×C83#12111yescharacteristic
84C22×D83—22no—
85C22×D83—41no—
86C22×D83—42no—
87C22×D83—41no—
88C22×D83—42no—
89Q8×C83—44no—
90Q8×C83—44no—
91C2×C4×C83—41no—
92C2×C4×C83—41no—
93C2×C4×C83—42no—
94C2×C4×C83—42no—
95C2×C4×C83#12012yes—
96C2×C4×C83#12011yescharacteristic
97C2×C4×C83#12112yes—
98C2×C4×C83#12711yescharacteristic
99C2×C4×C83C2×SD1612yes—
100C2×C4×C83#13712yes—
101C2×Dic83—41no—
102C2×Dic83—42no—
103D332—24no—
104D332—42no—
105C4×D83—82no—
106C4×D83—81no—
107C4×D83—81no—
108C8×C83—24no—
109C83⋊C8—44no—
110C83⋊C8—81no—
111C22×C83—21no—
112C22×C83—21no—
113C22×C83#39211yescharacteristic
114C22×C83#46112yes—
115C2×D83—44no—
116C2×D83—42no—
117C2×D83—82no—
118C2×D83—81no—
119C2×D83—81no—
120C4×C83—22no—
121C4×C83—22no—
122C4×C83—82no—
123C4×C83—82no—
124C4×C83#44812yes—
125C4×C83#49211yescharacteristic
126C4×C83#49711yescharacteristic
127Dic83—81no—
128#10036—831nomaximal, Sylow
129C2×C83—22no—
130C2×C83#201011yescharacteristic
131C2×C83#201311yescharacteristic
132C2×C83#201711yescharacteristic
133D83—82no—
134#3162D8311yescharacteristic, complemented
135#3101—832no—
136#1955—831no—
137#1481—832no—
138#1275—831no—
139C83#1003611yescharacteristic, complemented, Sylow
140#533C2×D8312yes—
141C2×C4×D4—831no—
142#501—22no—
143#501—22no—
144#483—1662no—
145#476—1662no—
146#461—1662no—
147#450—1662no—
148C4×SD16—1664no—
149#431—832no—
150#427—22no—
151C2×C4⋊C8—831no—
152C2×C22⋊C8—1662no—
153C2×#97C2×D8311yescharacteristic
154Q8⋊C8—1664no—
155C22×D4—831no—
156C23×C4—1661no—
157C2×SD16—1664no—
158C22×C8—1661no—
159C22×C8C22⋊D8312yes—
160C4⋊Q8C22⋊D8312yes—
161C4⋊Q8C22⋊D8312yes—
162#125—22no—
163C22⋊Q8—22no—
164C22⋊Q8—22no—
165C4×Q8—22no—
166C4×Q8—22no—
167C4×D4—1662no—
168C4×D4—1662no—
169#117—22no—
170C2×C4⋊C4—831no—
171C2×C22⋊C4—1661no—
172C2×C42C22×D8311yescharacteristic
173#107—22no—
174#107—22no—
175#106—24no—
176#106—1664no—
177C4⋊C8—1662no—
178C4⋊C8—1662no—
179#103—1664no—
180#102—1664no—
181C22⋊C8—3324no—
182C22⋊C8—1662no—
183#97—22no—
184C4×C8—834no—
185C24—1661no—
186C2×Q8—22no—
187C2×Q8—22no—
188C2×D4—3322no—
189C2×D4—1662no—
190C22×C4—3321no—
191C22×C4—3321no—
192C22×C4—3322no—
193C22×C4—3322no—
194C22×C4—1661no—
195C22×C4D4×D8311yescharacteristic
196C22×C4C2×C22⋊D8311yescharacteristic
197C22×C4#1305058911yescharacteristic
198SD16—3328no—
199C2×C8—22no—
200C2×C8—22no—
201C2×C8—3321no—
202C2×C8—3322no—
203C2×C8—1664no—
204C2×C8#1305057714yescomplemented
205C4⋊C4—22no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—42no—
210C4⋊C4—42no—
211C4⋊C4—1662no—
212C22⋊C4—42no—
213C22⋊C4—3322no—
214C42—42no—
215C42D4×D8312yes—
216C42C2×C22⋊D8312yes—
217C23—3321no—
218C23—3322no—
219C23—3321no—
220C23—3322no—
221C23—1662no—
222C23#1458176311yescharacteristic
223Q8—44no—
224Q8—44no—
225D4—3322no—
226D4—1664no—
227C2×C4—41no—
228C2×C4—41no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—3321no—
232C2×C4—3322no—
233C2×C4—6641no—
234C2×C4—6641no—
235C2×C4—6642no—
236C2×C4#1458176012yes—
237C2×C4#1458176111yescharacteristic
238C2×C4#1458176412yes—
239C2×C4#1458178811yescharacteristic
240C2×C4C2×#1305057712yes—
241C2×C4#1458182312yes—
242C8—24no—
243C8—3324no—
244C8—6641no—
245C22—21no—
246C22—21no—
247C22—3324no—
248C22—3322no—
249C22—6642no—
250C22—6641no—
251C22—6641no—
252C22#1912760911yescharacteristic
253C22#1912790312yes—
254C4—22no—
255C4—22no—
256C4—82no—
257C4—82no—
258C4—6641no—
259C4#1912784012yes—
260C4#1912807511yescharacteristic
261C4#1912809311yescharacteristic
262C2—22no—
263C2—6642no—
264C2#2490852711yescharacteristic
265C2#2490854611yescharacteristic
266C2#2490856611yescharacteristic
subgroupG/H
centerC2×C4#14581788
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#3162C2
socleC22×C83#392
prime radicalC22×D332D4
hypercenter#3162D83
hypocenterC83#10036
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #24905866C2semidirect
4. #24905294C2semidirect
5. C83×#3162C2semidirect
72. C2×C8×C83SD16semidirect
134. #3162D83semidirect
139. C83#10036semidirect
204. C2×C8#13050577semidirect

Maximal subgroups

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

subgroupindex
#249174672
#249082612
#249058662
#249052942
C83×#31622
#1003683