Finite Group Databasebeta

← #35299713

Subgroups — #35299713

Order 21248.

444 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 103 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#24917785C211yesmaximal, characteristic
2#24916530C211yesmaximal, characteristic
3#24905885C212yesmaximal, complemented
4#24905301C212yesmaximal, complemented
5C83×#1525C211yesmaximal, characteristic
6#19128258C2211yescharacteristic
7#19128055—22no—
8#19128051C412yes—
9#19128047C412yes—
10#19127779—22no—
11D4×C83⋊C8—22no—
12#19127754C2212yes—
13#19127697—22no—
14#19127668—22no—
15C2×C4×C83⋊C8C2211yescharacteristic
16#19127433—22no—
17#19127433—22no—
18#532×C83C2211yescharacteristic
19C4×M16×C83C2212yes—
20#329×C83—22no—
21C2×#13050574—22no—
22C22×C83⋊C8—22no—
23C22×C83⋊C8—21no—
24#14581792C2×C412yes—
25#14581784—21no—
26#14581771—22no—
27C4×Dic166—22no—
28#14581751—21no—
29C4⋊(C83⋊C8)D412yes—
30C4⋊(C83⋊C8)D412yes—
31#14581716—24no—
32#14581713—24no—
33C22⋊(C83⋊C8)—24no—
34C22⋊(C83⋊C8)—22no—
35C4×C83⋊C8—21no—
36C4×C83⋊C8—22no—
37C4×C83⋊C8C2×C412yes—
38C2×#41×C83D412yes—
39C2×M16×C83—22no—
40C4⋊Q8×C83C2×C411yescharacteristic
41#127×C83C2×C411yescharacteristic
42#124×C83D412yes—
43#121×C83—22no—
44C4×D4×C83—22no—
45C2×C42×C83C2311yescharacteristic
46#97×C83—22no—
47C4×C8×C83—22no—
48C2×Dic166—22no—
49#13050574—82no—
50C2×C83⋊C8—22no—
51C2×C83⋊C8—22no—
52C2×C83⋊C8—22no—
53C2×C83⋊C8—41no—
54C2×C83⋊C8—41no—
55C2×C83⋊C8—82no—
56C4⋊Dic83—22no—
57C4⋊Dic83—22no—
58Dic83⋊C4—42no—
59C22⋊Dic83—41no—
60C4×Dic83—41no—
61#41×C83—42no—
62#41×C83—42no—
63C2×Q8×C83C22⋊C412yes—
64C2×D4×C83—22no—
65C2×D4×C83—22no—
66C2×D4×C83C22⋊C412yes—
67C22×C4×C83—22no—
68C22×C4×C83C2×D411yescharacteristic
69C22×C4×C83#4112yes—
70M16×C83—24no—
71M16×C83—42no—
72C2×C8×C83—42no—
73C4⋊C4×C83—22no—
74C22⋊C4×C83—24no—
75C42×C83C22×C411yescharacteristic
76C42×C83C2×D412yes—
77C42×C83C2×D411yescharacteristic
78C23×C83—22no—
79C23×C83#12311yescharacteristic
80Q8×C83—42no—
81D4×C83—42no—
82D4×C83—42no—
83D4×C83—42no—
84D4×C83—82no—
85C2×C4×C83—22no—
86C2×C4×C83—22no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—22no—
90C2×C4×C83—42no—
91C2×C4×C83—42no—
92C2×C4×C83C2×C22⋊C411yescharacteristic
93C2×C4×C83C4×D412yes—
94C2×C4×C83#12011yescharacteristic
95C2×C4×C83#12112yes—
96C2×Dic83—42no—
97Dic166—42no—
98Dic166—42no—
99C8×C83—44no—
100C83⋊C8—42no—
101C83⋊C8—42no—
102C83⋊C8—42no—
103C22×C83—22no—
104C22×C83—21no—
105C22×C83—44no—
106C22×C83—42no—
107C22×C83#38611yescharacteristic
108C4×C83—22no—
109C4×C83—21no—
110C4×C83—42no—
111C4×C83#44312yes—
112C4×C83#44412yes—
113C4×C83#45511yescharacteristic
114C4×C83#45511yescharacteristic
115Dic83—82no—
116#9067—831nomaximal, Sylow
117C2×C83—41no—
118C2×C83—82no—
119C2×C83#187612yes—
120C2×C83#188511yescharacteristic
121#3131—831no—
122#2979—831no—
123#1525D8311yescharacteristic
124#1485—832no—
125#1278—832no—
126C83#906711yescharacteristic, complemented, Sylow
127#533—831no—
128#532C2×D8311yescharacteristic
129#488—1662no—
130#487—832no—
131#486—832no—
132#436—1662no—
133C8×D4—1662no—
134#431—832no—
135#416—1662no—
136#408—1662no—
137C4×M16C2×D8312yes—
138C2×C4×C8—831no—
139#329—22no—
140Q8⋊C8—1662no—
141Q8⋊C8—1662no—
142C2×#41D33212yes—
143C2×M16—22no—
144C2×M16—1662no—
145C22×C8—1661no—
146C22×C8—1662no—
147C4⋊Q8—832no—
148C4⋊Q8C4×D8311yescharacteristic
149#127C4×D8311yescharacteristic
150#125—1661no—
151#124C22⋊D8312yes—
152C22⋊Q8—1662no—
153#121—22no—
154C4×Q8—1662no—
155C4×D4—22no—
156#117—1661no—
157C2×C42C22×D8311yescharacteristic
158C4⋊C8—832no—
159C4⋊C8—832no—
160#103—1664no—
161#102—1664no—
162C22⋊C8—1664no—
163C22⋊C8—1662no—
164#97—22no—
165C4×C8—22no—
166C4×C8—832no—
167C4×C8—1662no—
168C4×C8—1661no—
169#41—42no—
170#41—42no—
171C2×Q8—1662no—
172C2×Q8(C2×D83)⋊C412yes—
173C2×D4—22no—
174C2×D4—22no—
175C2×D4(C2×D83)⋊C412yes—
176C22×C4—22no—
177C22×C4C2×D33211yescharacteristic
178C22×C4#1305058912yes—
179M16—24no—
180M16—42no—
181M16—6642no—
182C2×C8—42no—
183C2×C8—3321no—
184C2×C8—3321no—
185C2×C8—6642no—
186C2×C8—1662no—
187C2×C8—1662no—
188C2×C8—1662no—
189C4⋊C4—22no—
190C4⋊C4—3322no—
191C4⋊C4—1662no—
192C4⋊C4—1662no—
193C22⋊C4—24no—
194C22⋊C4—3321no—
195C42—3321no—
196C42C2×C4×D8311yescharacteristic
197C42D4×D8312yes—
198C42C2×C22⋊D8311yescharacteristic
199C23—22no—
200C23#1458177611yescharacteristic
201Q8—42no—
202Q8—3322no—
203Q8—3322no—
204D4—42no—
205D4—42no—
206D4—42no—
207D4—82no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—22no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—3322no—
216C2×C4C2×(C2×D83)⋊C411yescharacteristic
217C2×C4(C22⋊D83)⋊C412yes—
218C2×C4#1458176211yescharacteristic
219C2×C4#1458176412yes—
220C8—44no—
221C8—3322no—
222C8—3322no—
223C8—3322no—
224C22—22no—
225C22—21no—
226C22—44no—
227C22—42no—
228C22#1912757811yescharacteristic
229C4—22no—
230C4—21no—
231C4—42no—
232C4—6642no—
233C4#1912781512yes—
234C4#1912782012yes—
235C4#1912788111yescharacteristic
236C4#1912788111yescharacteristic
237C2—41no—
238C2—82no—
239C2#2490778812yes—
240C2#2490784911yescharacteristic
subgroupG/H
centerC2×C4#14581762
commutatorC42×C83C22×C4
FrattiniC2×C42C22×D83
FittingC83×#1525C2
socleC22×C83#386
prime radical#532×C83C22
cosocleC2×C42×C83C23
hypercenter#1525D83
hypocenterC83#9067
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1525

Complements

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

NH ≅ G/Ntype
3. #24905885C2semidirect
4. #24905301C2semidirect
126. C83#9067semidirect

Maximal subgroups

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

subgroupindex
#249177852
#249165302
#249058852
#249053012
C83×#15252
#906783