Finite Group Databasebeta

← #35315893

Subgroups — #35315893

Order 21248.

306 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 85 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#24918047C211yesmaximal, characteristic
2C4⋊C16×D83C211yesmaximal, characteristic
3#24906225C211yesmaximal, characteristic
4#24905519C211yesmaximal, characteristic, complemented
5#24905517C211yesmaximal, characteristic, complemented
6#24905512C211yesmaximal, characteristic, complemented
7C83×#1362C211yesmaximal, characteristic, complemented
8#19128256—21no—
9C2×C16×D83—21no—
10#19128106C2211yescharacteristic
11#19128051—21no—
12#19127939—21no—
13#19127939C412yes—
14Q16×Dic83—21no—
15Q16×Dic83C412yes—
16#19127757—21no—
17C4×C8×D83C2211yescharacteristic
18C4⋊(C83⋊C16)C2211yescharacteristic
19Dic83⋊C16—21no—
20#19127454C2211yescharacteristic
21#19127451—21no—
22#494×C83C2211yescharacteristic
23C4⋊C16×C83C2211yescharacteristic
24#335×C83C2211yescharacteristic
25C2×C8×D83C2×C411yescharacteristic
26C2×C8×D83D411yescharacteristic
27#14581792—21no—
28#14581783—41no—
29#14581772—41no—
30#14581768—22no—
31Q8×Dic83—22no—
32#14581757—41no—
33#14581750—41no—
34C42×D83D411yescharacteristic
35C2×C83⋊C16—21no—
36C16×D83—42no—
37#14581726—21no—
38#14581726C2×C412yes—
39C4⋊(C83⋊C8)—21no—
40#14581716—22no—
41#14581716—22no—
42(C2×D83)⋊C8—41no—
43#14581702—41no—
44C4×C83⋊C8D411yescharacteristic
45C8×Dic83C2×C411yescharacteristic
46C8×Dic83D411yescharacteristic
47C2×Q16×C83—21no—
48C2×Q16×C83C2×C412yes—
49C4⋊Q8×C83—21no—
50C2×C16×C83—21no—
51C4⋊C8×C83—21no—
52C4×C8×C83C2311yescharacteristic
53C2×Dic166—41no—
54C2×C4×D83C22⋊C411yescharacteristic
55C2×C4×D83D811yescharacteristic
56C2×C4×D83SD1611yescharacteristic
57C2×C83⋊C8—41no—
58C2×C83⋊C8C22⋊C411yescharacteristic
59C2×C83⋊C8C22⋊C411yescharacteristic
60C8×D83—21no—
61C8×D83—21no—
62C8×D83—41no—
63C4⋊Dic83—22no—
64C4⋊Dic83—41no—
65C4⋊Dic83—42no—
66Dic83⋊C4—42no—
67(C2×D83)⋊C4—42no—
68C4×Dic83—42no—
69C4×Dic83C22⋊C411yescharacteristic
70C4×Dic83D811yescharacteristic
71C4×Dic83SD1611yescharacteristic
72C83⋊C16—41no—
73C2×Q8×C83—22no—
74Q16×C83—42no—
75Q16×C83—41no—
76Q16×C83—41no—
77C2×C8×C83—41no—
78C2×C8×C83C22×C411yescharacteristic
79C2×C8×C83C2×D411yescharacteristic
80C4⋊C4×C83—41no—
81C4⋊C4×C83—41no—
82C42×C83C2×D411yescharacteristic
83C16×C83—41no—
84C22×D83#10211yescharacteristic
85Q8×C83—42no—
86Q8×C83—42no—
87C2×C4×C83—42no—
88C2×C4×C83C2×C22⋊C411yescharacteristic
89C2×C4×C83C2×D811yescharacteristic
90C2×C4×C83C2×SD1611yescharacteristic
91C2×Dic83—42no—
92C2×Dic83#10211yescharacteristic
93C2×Dic83#10211yescharacteristic
94C2×Dic83#10211yescharacteristic
95C4×D83—22no—
96C4×D83—21no—
97C4×D83—21no—
98C4×D83—21no—
99C4×D83—21no—
100Dic166—81no—
101Dic166—81no—
102C8×C83—21no—
103C8×C83—81no—
104C8×C83AGL(1,ℤ/8)11yescharacteristic
105C8×C83#13711yescharacteristic
106C83⋊C8—21no—
107C83⋊C8—21no—
108C83⋊C8—81no—
109C22×C83C2×#10211yescharacteristic
110C2×D83—21no—
111C2×D83—21no—
112C2×D83#36212yes—
113C4×C83—21no—
114C4×C83—82no—
115C4×C83#41311yescharacteristic
116C4×C83#41711yescharacteristic
117C4×C83#50811yescharacteristic
118C4×C83#50811yescharacteristic
119Dic83—21no—
120Dic83—21no—
121Dic83—21no—
122Dic83—82no—
123#11984—831nomaximal, Sylow
124C2×C83#154711yescharacteristic
125C2×C83#213811yescharacteristic
126C2×C83C2×#36211yescharacteristic
127D83—22no—
128#3170—831no—
129C2×C4⋊C16—831no—
130#1568—831no—
131#1364—831no—
132#1364—831no—
133#1362—831no—
134#1362D8311yescharacteristic, complemented
135C83#1198411yescharacteristic, complemented, Sylow
136#533—1661no—
137C22×C16—1661no—
138#500—831no—
139#494C2×D8311yescharacteristic
140#487—1661no—
141#467—832no—
142#467—1661no—
143C4×Q16—832no—
144C4×Q16—1661no—
145#432—1661no—
146C2×C4×C8—831no—
147C4⋊C16—831no—
148C4⋊C16—1661no—
149C4⋊C16C2×D8311yescharacteristic
150#335—831no—
151#335C2×D8311yescharacteristic
152#334—1661no—
153C2×Q16—21no—
154C2×Q16C4×D8312yes—
155C22×C8—831no—
156C22×C8—831no—
157C4⋊Q8—21no—
158C4⋊Q8—1661no—
159#125—3321no—
160C22⋊Q8—3321no—
161C22⋊Q8—1662no—
162C4×Q8—3321no—
163C4×Q8—1662no—
164#117—3321no—
165C2×C42—831no—
166C2×C16—21no—
167C2×C16—3322no—
168C2×C16—1661no—
169#107—832no—
170#107—1661no—
171C4⋊C8—21no—
172C4⋊C8—1661no—
173#103—1662no—
174#103—1662no—
175C22⋊C8—3321no—
176#97—3321no—
177C4×C8—831no—
178C4×C8—831no—
179C4×C8—831no—
180C4×C8C22×D8311yescharacteristic
181C2×Q8—22no—
182C2×Q8—3321no—
183C22×C4—831no—
184C22×C4—831no—
185C22×C4—831no—
186Q16—42no—
187Q16—41no—
188Q16—41no—
189C2×C8—41no—
190C2×C8—3321no—
191C2×C8—3321no—
192C2×C8—831no—
193C2×C8—831no—
194C2×C8—1661no—
195C2×C8—1661no—
196C2×C8C2×C4×D8311yescharacteristic
197C2×C8D4×D8311yescharacteristic
198C4⋊C4—41no—
199C4⋊C4—41no—
200C4⋊C4—3322no—
201C4⋊C4—3322no—
202C4⋊C4—3321no—
203C4⋊C4—1662no—
204C22⋊C4—3322no—
205C42—3322no—
206C42—831no—
207C42—831no—
208C42—831no—
209C42D4×D8311yescharacteristic
210C16—41no—
211C16—3321no—
212C23—831no—
213Q8—42no—
214Q8—42no—
215Q8—6641no—
216Q8—6641no—
217C2×C4—42no—
218C2×C4—3322no—
219C2×C4—831no—
220C2×C4—831no—
221C2×C4—831no—
222C2×C4—1662no—
223C2×C4—1661no—
224C2×C4—1661no—
225C2×C4—1661no—
226C2×C4—1661no—
227C2×C4C22⋊C4×D8311yescharacteristic
228C2×C4D8×D8311yescharacteristic
229C2×C4SD16×D8311yescharacteristic
230C8—21no—
231C8—81no—
232C8—6641no—
233C8—1661no—
234C8—1661no—
235C8#1458181611yescharacteristic
236C8#1458182111yescharacteristic
237C22—832no—
238C22—1661no—
239C22—1661no—
240C22#102×D8311yescharacteristic
241C4—21no—
242C4—82no—
243C4—6642no—
244C4—1661no—
245C4—1661no—
246C4—1661no—
247C4#1912768211yescharacteristic
248C4#1912769911yescharacteristic
249C4#1912813211yescharacteristic
250C4#1912813411yescharacteristic
251C2—1662no—
252C2#2490613711yescharacteristic
253C2#2490916911yescharacteristic
254C2#362×D8311yescharacteristic
subgroupG/H
centerC22#102×D83
commutatorC2×C8×C83C22×C4
FrattiniC4×C8C22×D83
FittingC83×#1362C2
socleC22×C83C2×#102
prime radicalC22×D83#102
cosocleC4×C8×C83C23
hypercenter#1362D83
hypocenterC83#11984
LCSG ▷ C2×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
4. #24905519C2semidirect
5. #24905517C2semidirect
6. #24905512C2semidirect
7. C83×#1362C2semidirect
134. #1362D83semidirect
135. C83#11984semidirect

Maximal subgroups

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

subgroupindex
#249180472
C4⋊C16×D832
#249062252
#249055192
#249055172
#249055122
C83×#13622
#1198483