Finite Group Databasebeta

← #35315741

Subgroups — #35315741

Order 21248.

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

normal subgroups only

This group has 300 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
2#24909261C211yesmaximal, characteristic
3#24906210C211yesmaximal, characteristic
4#24905519C211yesmaximal, characteristic, complemented
5#24905517C211yesmaximal, characteristic, complemented
6#24905512C211yesmaximal, characteristic, complemented
7C83×#1362C211yesmaximal, characteristic, complemented
8#19128256—21no—
9#19128106C2211yescharacteristic
10#19128051—21no—
11#19127939—21no—
12#19127939C412yes—
13Q16×Dic83—21no—
14Q16×Dic83C412yes—
15#19127753—21no—
16C4×C8×D83C2211yescharacteristic
17C4⋊(C83⋊C16)C2211yescharacteristic
18(C2×D83)⋊C16—21no—
19C16×Dic83—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—
36#14581726—21no—
37#14581726C2×C412yes—
38C4⋊(C83⋊C8)—21no—
39#14581716—22no—
40#14581716—22no—
41(C2×D83)⋊C8—41no—
42C4×C83⋊C8D411yescharacteristic
43C8×Dic83—41no—
44C8×Dic83C2×C411yescharacteristic
45C8×Dic83D411yescharacteristic
46C2×Q16×C83—21no—
47C2×Q16×C83C2×C412yes—
48C4⋊Q8×C83—21no—
49C2×C16×C83—21no—
50C4⋊C8×C83—21no—
51C4×C8×C83C2311yescharacteristic
52C2×Dic166—41no—
53C2×C4×D83C22⋊C411yescharacteristic
54C2×C4×D83D811yescharacteristic
55C2×C4×D83SD1611yescharacteristic
56C2×C83⋊C8—41no—
57C2×C83⋊C8C22⋊C411yescharacteristic
58C2×C83⋊C8C22⋊C411yescharacteristic
59C8×D83—21no—
60C8×D83—21no—
61C8×D83—41no—
62C4⋊Dic83—22no—
63C4⋊Dic83—41no—
64C4⋊Dic83—42no—
65Dic83⋊C4—42no—
66(C2×D83)⋊C4—42no—
67C4×Dic83—42no—
68C4×Dic83C22⋊C411yescharacteristic
69C4×Dic83D811yescharacteristic
70C4×Dic83SD1611yescharacteristic
71C83⋊C16—41no—
72C2×Q8×C83—22no—
73Q16×C83—42no—
74Q16×C83—41no—
75Q16×C83—41no—
76C2×C8×C83—41no—
77C2×C8×C83C22×C411yescharacteristic
78C2×C8×C83C2×D411yescharacteristic
79C4⋊C4×C83—41no—
80C4⋊C4×C83—41no—
81C42×C83C2×D411yescharacteristic
82C16×C83—41no—
83C22×D83#10211yescharacteristic
84Q8×C83—42no—
85Q8×C83—42no—
86C2×C4×C83—42no—
87C2×C4×C83C2×C22⋊C411yescharacteristic
88C2×C4×C83C2×D811yescharacteristic
89C2×C4×C83C2×SD1611yescharacteristic
90C2×Dic83—42no—
91C2×Dic83#10211yescharacteristic
92C2×Dic83#10211yescharacteristic
93C2×Dic83#10211yescharacteristic
94C4×D83—21no—
95C4×D83—21no—
96C4×D83—21no—
97C4×D83—21no—
98C4×D83—22no—
99Dic166—81no—
100Dic166—81no—
101C8×C83—21no—
102C8×C83—81no—
103C8×C83#13511yescharacteristic
104C8×C83#13511yescharacteristic
105C83⋊C8—21no—
106C83⋊C8—21no—
107C83⋊C8—81no—
108C22×C83C2×#10211yescharacteristic
109C2×D83—21no—
110C2×D83—21no—
111C2×D83—21no—
112C4×C83—21no—
113C4×C83—82no—
114C4×C83#41311yescharacteristic
115C4×C83#41611yescharacteristic
116C4×C83#50811yescharacteristic
117C4×C83#50811yescharacteristic
118Dic83—21no—
119Dic83—21no—
120Dic83—82no—
121Dic83#36212yes—
122#11950—831nomaximal, Sylow
123C2×C83#154011yescharacteristic
124C2×C83#213811yescharacteristic
125C2×C83C2×#36211yescharacteristic
126D83—41no—
127#3170—831no—
128#2162—831no—
129#1564—831no—
130#1364—831no—
131#1364—831no—
132#1362—831no—
133#1362D8311yescharacteristic, complemented
134C83#1195011yescharacteristic, complemented, Sylow
135#533—1661no—
136#500—831no—
137#494C2×D8311yescharacteristic
138#487—1661no—
139#467—832no—
140#467—1661no—
141C4×Q16—832no—
142C4×Q16—1661no—
143#431—1661no—
144C2×C4×C8—831no—
145C4⋊C16—831no—
146C4⋊C16C2×D8311yescharacteristic
147C22⋊C16—1661no—
148C4×C16—1661no—
149#335—831no—
150#335C2×D8311yescharacteristic
151#334—1661no—
152C2×Q16—21no—
153C2×Q16C4×D8312yes—
154C22×C8—831no—
155C22×C8—831no—
156C4⋊Q8—21no—
157C4⋊Q8—1661no—
158#125—3321no—
159C22⋊Q8—3321no—
160C22⋊Q8—1662no—
161C4×Q8—3321no—
162C4×Q8—1662no—
163#117—3321no—
164C2×C42—831no—
165C2×C16—21no—
166C2×C16—1661no—
167#107—832no—
168#107—1661no—
169C4⋊C8—21no—
170C4⋊C8—1661no—
171#103—1662no—
172#103—1662no—
173C22⋊C8—3321no—
174C4×C8—3321no—
175C4×C8—831no—
176C4×C8—831no—
177C4×C8—831no—
178C4×C8C22×D8311yescharacteristic
179C2×Q8—22no—
180C2×Q8—3321no—
181C22×C4—831no—
182C22×C4—831no—
183C22×C4—831no—
184Q16—42no—
185Q16—41no—
186Q16—41no—
187C2×C8—41no—
188C2×C8—3321no—
189C2×C8—3321no—
190C2×C8—831no—
191C2×C8—831no—
192C2×C8—1661no—
193C2×C8—1661no—
194C2×C8C2×C4×D8311yescharacteristic
195C2×C8D4×D8311yescharacteristic
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—3322no—
199C4⋊C4—3322no—
200C4⋊C4—3321no—
201C4⋊C4—1662no—
202C22⋊C4—3322no—
203C42—3322no—
204C42—831no—
205C42—831no—
206C42—831no—
207C42D4×D8311yescharacteristic
208C16—41no—
209C16—3321no—
210C23—831no—
211Q8—42no—
212Q8—42no—
213Q8—6641no—
214Q8—6641no—
215C2×C4—42no—
216C2×C4—3322no—
217C2×C4—831no—
218C2×C4—831no—
219C2×C4—831no—
220C2×C4—1662no—
221C2×C4—1661no—
222C2×C4—1661no—
223C2×C4—1661no—
224C2×C4—1661no—
225C2×C4C22⋊C4×D8311yescharacteristic
226C2×C4D8×D8311yescharacteristic
227C2×C4SD16×D8311yescharacteristic
228C8—21no—
229C8—81no—
230C8—6641no—
231C8—1661no—
232C8—1661no—
233C8#1458181111yescharacteristic
234C8#1458181311yescharacteristic
235C22—1661no—
236C22—1661no—
237C22—1661no—
238C22#102×D8311yescharacteristic
239C4—21no—
240C4—82no—
241C4—832no—
242C4—6642no—
243C4—1661no—
244C4—1661no—
245C4#1912768211yescharacteristic
246C4#1912769411yescharacteristic
247C4#1912813211yescharacteristic
248C4#1912813411yescharacteristic
249C2—3321no—
250C2#2490610911yescharacteristic
251C2#2490916911yescharacteristic
252C2#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#11950
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
133. #1362D83semidirect
134. C83#11950semidirect

Maximal subgroups

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

subgroupindex
#249180472
#249092612
#249062102
#249055192
#249055172
#249055122
C83×#13622
#1195083