Finite Group Databasebeta

← #35313596

Subgroups — #35313596

Order 21248.

364 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 89 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#24916024C211yesmaximal, characteristic
2#349×D83C212yesmaximal
3#24905416C211yesmaximal, characteristic, complemented
4#24905415C212yesmaximal, complemented
5C83×#1328C211yesmaximal, characteristic, complemented
6C22×#13050573C411yescharacteristic
7C2×M16×D83C2211yescharacteristic
8#117×D83C411yescharacteristic
9M32×D83—22no—
10#19127729C2211yescharacteristic
11#19127728C411yescharacteristic
12#19127727C411yescharacteristic
13#19127726—22no—
14#19127723—22no—
15#19127483C2212yes—
16#19127482—22no—
17#424×C83C2211yescharacteristic
18#349×C83C2212yes—
19C22×C4×D83C2×C411yescharacteristic
20C2×#13050574C2×C411yescharacteristic
21C2×#13050573—21no—
22C2×#13050573—21no—
23C2×#13050573—42no—
24C2×#13050573D412yes—
25M16×D83—42no—
26C22×C83⋊C8C2×C411yescharacteristic
27C2×C8×D83D412yes—
28#14581751C2×C411yescharacteristic
29#14581750—21no—
30#14581749—21no—
31#14581748C812yes—
32C4⋊C4×D83—21no—
33C22⋊C4×D83C812yes—
34C42×D83—21no—
35#14581732—22no—
36#14581731—42no—
37C16×D83—42no—
38C4⋊(C83⋊C8)—22no—
39Dic83⋊C8—22no—
40Dic83⋊C8—22no—
41C2×M16×C83C2311yescharacteristic
42C22×C8×C83C2×C411yescharacteristic
43#117×C83C2×C411yescharacteristic
44M32×C83—22no—
45C4⋊C8×C83—22no—
46C23×D83C2×C811yescharacteristic
47C22×Dic83C2×C811yescharacteristic
48C2×C4×D83—22no—
49C2×C4×D83—22no—
50C2×C4×D83—22no—
51C2×C4×D83C22⋊C411yescharacteristic
52C2×C4×D83C22⋊C411yescharacteristic
53C2×C4×D83M1611yescharacteristic
54C2×C4×D83M1611yescharacteristic
55#13050574—41no—
56#13050573—42no—
57#13050573—42no—
58#13050573—42no—
59C2×C83⋊C8—21no—
60C2×C83⋊C8—41no—
61C2×C83⋊C8C22⋊C412yes—
62C8×D83—22no—
63C8×D83—22no—
64C4⋊Dic83—21no—
65Dic83⋊C4—22no—
66C22⋊Dic83C2×C812yes—
67(C2×D83)⋊C4—22no—
68C4×Dic83—21no—
69C4×Dic83—22no—
70C83⋊C16—42no—
71C22×C4×C83C22×C411yescharacteristic
72M16×C83—41no—
73C2×C8×C83—21no—
74C2×C8×C83—41no—
75C2×C8×C83C2×D412yes—
76C4⋊C4×C83—21no—
77C22⋊C4×C83C2×C812yes—
78C42×C83—21no—
79C16×C83—42no—
80C23×C83C22×C811yescharacteristic
81C22×D83—22no—
82C22×D83—81no—
83C22×D83C22⋊C811yescharacteristic
84C22×D83C22⋊C811yescharacteristic
85C2×C4×C83—21no—
86C2×C4×C83—21no—
87C2×C4×C83—22no—
88C2×C4×C83C2×C22⋊C411yescharacteristic
89C2×C4×C83C2×M1611yescharacteristic
90C2×Dic83—22no—
91C2×Dic83—22no—
92C2×Dic83C22⋊C811yescharacteristic
93C2×Dic83C22⋊C811yescharacteristic
94C4×D83—22no—
95C4×D83—21no—
96C4×D83—21no—
97C4×D83—22no—
98C4×D83—82no—
99C4×D83#9912yes—
100C4×D83#10112yes—
101C8×C83—22no—
102C8×C83—41no—
103C83⋊C8—22no—
104C83⋊C8—41no—
105C22×C83—21no—
106C22×C83—41no—
107C22×C83C2×C22⋊C811yescharacteristic
108C2×D83—21no—
109C2×D83—41no—
110C2×D83—41no—
111C2×D83—82no—
112C2×D83#32412yes—
113C4×C83—21no—
114C4×C83—42no—
115C4×C83C2×#9911yescharacteristic
116C4×C83C2×#10111yescharacteristic
117Dic83—21no—
118Dic83—42no—
119Dic83#32412yes—
120#11458—831nomaximal, Sylow
121C2×C83—21no—
122C2×C83—41no—
123C2×C83C2×#32411yescharacteristic
124D83—22no—
125D83—41no—
126#2905—831no—
127C2×#349—832no—
128#1328—831no—
129#1328—832no—
130#1328D8311yescharacteristic, complemented
131C83#1145811yescharacteristic, complemented, Sylow
132C22×M16—831no—
133C22×M16—831no—
134C2×#117—831no—
135C2×M32—1662no—
136#424—831no—
137#424—831no—
138#424—831no—
139#424—1662no—
140#424C2×D8311yescharacteristic
141#423—1662no—
142#349—832no—
143#349—1662no—
144#349C2×D8312yes—
145C23×C4—831no—
146C2×M16—3322no—
147C2×M16—3322no—
148C2×M16—831no—
149C2×M16—832no—
150C2×M16—1661no—
151C2×M16—1661no—
152C2×M16C22×D8311yescharacteristic
153C22×C8—831no—
154C22×C8—832no—
155C22×C8C4×D8311yescharacteristic
156#117—831no—
157#117—832no—
158#117—1661no—
159#117—1661no—
160#117C4×D8311yescharacteristic
161C2×C4⋊C4—1661no—
162C2×C22⋊C4—832no—
163C2×C42—1661no—
164M32—22no—
165M32—3322no—
166M32—1662no—
167C2×C16—3322no—
168C4⋊C8—22no—
169C4⋊C8—1662no—
170C4⋊C8—1662no—
171C4⋊C8—1662no—
172C24—831no—
173C22×C4—831no—
174C22×C4—831no—
175C22×C4—831no—
176C22×C4—831no—
177C22×C4—831no—
178C22×C4—1662no—
179C22×C4—1662no—
180C22×C4—1662no—
181C22×C4C2×C4×D8311yescharacteristic
182M16—41no—
183M16—3322no—
184M16—3322no—
185M16—3322no—
186M16—3321no—
187C2×C8—21no—
188C2×C8—41no—
189C2×C8—3321no—
190C2×C8—832no—
191C2×C8—1662no—
192C2×C8—1662no—
193C2×C8—1661no—
194C2×C8D4×D8312yes—
195C4⋊C4—21no—
196C4⋊C4—1662no—
197C4⋊C4—1661no—
198C22⋊C4—832no—
199C22⋊C4—1662no—
200C22⋊C4C8×D8312yes—
201C42—21no—
202C42—1662no—
203C42—1661no—
204C16—42no—
205C16—3322no—
206C23—831no—
207C23—831no—
208C23—6641no—
209C23—1662no—
210C23C2×C8×D8311yescharacteristic
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—22no—
214C2×C4—831no—
215C2×C4—832no—
216C2×C4—832no—
217C2×C4—831no—
218C2×C4—6642no—
219C2×C4—1662no—
220C2×C4—1662no—
221C2×C4—1662no—
222C2×C4—1662no—
223C2×C4—1661no—
224C2×C4—1661no—
225C2×C4C22⋊C4×D8311yescharacteristic
226C2×C4M16×D8311yescharacteristic
227C8—22no—
228C8—41no—
229C8—3321no—
230C8—1662no—
231C22—21no—
232C22—41no—
233C22—3321no—
234C22—3321no—
235C22—832no—
236C22—6642no—
237C22—1661no—
238C22C22⋊C8×D8311yescharacteristic
239C4—21no—
240C4—42no—
241C4—3322no—
242C4—832no—
243C4—1661no—
244C4#99×D8311yescharacteristic
245C4#101×D8311yescharacteristic
246C2—21no—
247C2—41no—
248C2—3321no—
249C2—1662no—
250C2#324×D8311yescharacteristic
subgroupG/H
centerC4#99×D83
commutatorC23×C83C22×C8
FrattiniC2×M16C22×D83
FittingC83×#1328C2
socleC2×C83C2×#324
prime radicalC23×D83C2×C8
cosocleC2×M16×C83C23
hypercenter#1328D83
hypocenterC83#11458
LCSG ▷ C23×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×M16 ◁ #1328

Complements

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

NH ≅ G/Ntype
3. #24905416C2semidirect
4. #24905415C2semidirect
5. C83×#1328C2semidirect
130. #1328D83semidirect
131. C83#11458semidirect

Maximal subgroups

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

subgroupindex
#249160242
#349×D832
#249054162
#249054152
C83×#13282
#1145883