Finite Group Databasebeta

← #35322607

Subgroups — #35322607

Order 21248.

438 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 111 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#24916159C211yesmaximal, characteristic
2#24909092C212yesmaximal, complemented
3#24909073C212yesmaximal, complemented
4#24909068C211yesmaximal, characteristic, complemented
5C83×#2153C211yesmaximal, characteristic, complemented
6C2×#14581798C2211yescharacteristic
7C22×C8×D83C2211yescharacteristic
8#19128213C411yescharacteristic
9C2×#14581732C2212yes—
10#19127774—22no—
11C8×D332—22no—
12#19127728C411yescharacteristic
13#19127669C411yescharacteristic
14#19127664C411yescharacteristic
15#19127486—24no—
16(C2×D83)⋊C16—24no—
17#19127479—22no—
18C2×M32×C83C2212yes—
19#405×C83C2211yescharacteristic
20#364×C83—22no—
21C2×#13050590C2×C411yescharacteristic
22C22×C4×D83C2×C411yescharacteristic
23#14581798—22no—
24#14581798—22no—
25C2×#13050574C2×C411yescharacteristic
26C2×#13050573—21no—
27C22×C83⋊C8C2×C411yescharacteristic
28C2×C8×D83—22no—
29C2×C8×D83—21no—
30C2×C8×D83—21no—
31C2×C8×D83—21no—
32C2×C8×D83—22no—
33#14581776—21no—
34#14581772—21no—
35#14581767—21no—
36#14581762—21no—
37C4×D332—22no—
38#14581732—22no—
39#14581732—22no—
40C2×C83⋊C16D414yescomplemented
41C4⋊(C83⋊C8)—22no—
42(C2×D83)⋊C8—22no—
43(C2×D83)⋊C8—22no—
44C2×M16×C83C2×C411yescharacteristic
45C22×C8×C83C2311yescharacteristic
46#117×C83C2×C411yescharacteristic
47M32×C83—22no—
48M32×C83—22no—
49C2×C16×C83—22no—
50#97×C83D412yes—
51C4×C8×C83D412yes—
52C23×D83M1611yescharacteristic
53#13050590—22no—
54#13050590—22no—
55C2×Dic166M1611yescharacteristic
56C2×D332—22no—
57C2×D332M1611yescharacteristic
58C2×C22⋊D83—21no—
59C22×Dic83M1611yescharacteristic
60C2×C4×D83—21no—
61C2×C4×D83—21no—
62C2×C4×D83—22no—
63C2×C4×D83—21no—
64C2×C4×D83—22no—
65#13050574—22no—
66#13050573—42no—
67C2×C83⋊C8—21no—
68C2×C83⋊C8—21no—
69C2×C83⋊C8—22no—
70C8×D83—42no—
71C8×D83—41no—
72C8×D83—41no—
73C8×D83—41no—
74C8×D83—41no—
75C4⋊Dic83—22no—
76(C2×D83)⋊C4—22no—
77(C2×D83)⋊C4—22no—
78C83⋊C16—24no—
79C22×C4×C83C22×C411yescharacteristic
80M16×C83—22no—
81C2×C8×C83C22⋊C412yes—
82C2×C8×C83C2×D412yes—
83C2×C8×C83C2×D411yescharacteristic
84C2×C8×C83C2×D411yescharacteristic
85C2×C8×C83#4112yes—
86C4⋊C4×C83—21no—
87C22⋊C4×C83—21no—
88C42×C83C22⋊C412yes—
89C16×C83—42no—
90C23×C83C2×M1611yescharacteristic
91C22×D83—21no—
92C22×D83—22no—
93C22×D83—21no—
94C22×D83—42no—
95C2×C4×C83—22no—
96C2×C4×C83C2×C22⋊C411yescharacteristic
97C2×C4×C83C4×D412yes—
98C2×C4×C83C2×M1611yescharacteristic
99C2×C4×C83#13112yes—
100C22⋊D83—42no—
101C2×Dic83—21no—
102C2×Dic83—21no—
103C2×Dic83—22no—
104D332—22no—
105D332—44no—
106C4×D83—41no—
107C4×D83—41no—
108C4×D83—41no—
109C4×D83—41no—
110C4×D83—42no—
111Dic166—22no—
112C8×C83—22no—
113C8×C83#12112yes—
114C8×C83#12411yescharacteristic
115C8×C83#12711yescharacteristic
116C83⋊C8—41no—
117C83⋊C8—41no—
118C22×C83—41no—
119C22×C83#40711yescharacteristic
120C22×C83#43512yes—
121C2×D83—22no—
122C2×D83—24no—
123C2×D83—41no—
124C2×D83—42no—
125C2×D83—41no—
126C2×D83—42no—
127C2×D83—41no—
128C4×C83—42no—
129C4×C83#39011yescharacteristic
130C4×C83#40811yescharacteristic
131C4×C83#43512yes—
132Dic83—41no—
133Dic83—41no—
134#13200—831nomaximal, Sylow
135C2×C83—21no—
136C2×C83—22no—
137C2×C83#195511yescharacteristic
138D83—42no—
139D83—81no—
140#2929—831no—
141#2153D8311yescharacteristic, complemented
142#2117—832no—
143#2112—832no—
144#2110—831no—
145C83#1320011yescharacteristic, complemented, Sylow
146C2×#131—831no—
147C23×C8—831no—
148#525—831no—
149C2×M32—832no—
150C2×M32C2×D8312yes—
151#435—1662no—
152C8×D4—1662no—
153#424—831no—
154#408—831no—
155#407—831no—
156#405C2×D8311yescharacteristic
157#364—22no—
158#350—1664no—
159C22⋊C16—1664no—
160#347—1662no—
161C2×#41—831no—
162C23×C4—831no—
163#131—1662no—
164#131—1662no—
165C2×M16—831no—
166C2×M16—1661no—
167C2×M16C4×D8311yescharacteristic
168C22×C8—831no—
169C22×C8—1662no—
170C22×C8—1661no—
171C22×C8—1661no—
172C22×C8—1661no—
173C22×C8—1662no—
174C22×C8C22×D8311yescharacteristic
175#123—1661no—
176C22⋊Q8—1661no—
177#121—1661no—
178#120—1661no—
179C4×D4—1662no—
180#117C4×D8311yescharacteristic
181M32—22no—
182M32—22no—
183M32—1662no—
184M32—1662no—
185C2×C16—22no—
186C2×C16—834no—
187C4⋊C8—1662no—
188C22⋊C8—1662no—
189C22⋊C8—1662no—
190#97D33212yes—
191C4×C8C22⋊D8312yes—
192C24—831no—
193#41—1662no—
194#41—1662no—
195C2×Q8—831no—
196C2×D4—831no—
197C2×D4—1662no—
198C2×D4—1661no—
199C22×C4—831no—
200C22×C4—1661no—
201C22×C4—1662no—
202C22×C4—1661no—
203C22×C4—1661no—
204C22×C4—1662no—
205C22×C4C2×C4×D8311yescharacteristic
206M16—22no—
207M16—3322no—
208M16—1662no—
209C2×C8—3322no—
210C2×C8—3321no—
211C2×C8—3321no—
212C2×C8—3321no—
213C2×C8—3321no—
214C2×C8—1661no—
215C2×C8—1661no—
216C2×C8—1662no—
217C2×C8(C2×D83)⋊C412yes—
218C2×C8D4×D8312yes—
219C2×C8C2×C22⋊D8311yescharacteristic
220C2×C8C2×D33211yescharacteristic
221C2×C8#1305059112yes—
222C4⋊C4—21no—
223C4⋊C4—1662no—
224C22⋊C4—21no—
225C22⋊C4—1662no—
226C22⋊C4—1662no—
227C42(C2×D83)⋊C412yes—
228C16—42no—
229C16—1664no—
230C23—3322no—
231C23—1661no—
232C23—1661no—
233C23—1662no—
234C23M16×D8311yescharacteristic
235Q8—1662no—
236D4—3324no—
237D4—3322no—
238D4—1662no—
239C2×C4—22no—
240C2×C4—3321no—
241C2×C4—3321no—
242C2×C4—3321no—
243C2×C4—3321no—
244C2×C4—3322no—
245C2×C4—1661no—
246C2×C4—1661no—
247C2×C4—1662no—
248C2×C4C2×(C2×D83)⋊C411yescharacteristic
249C2×C4#1458175412yes—
250C2×C4M16×D8311yescharacteristic
251C2×C4#1458179812yes—
252C8—22no—
253C8—3321no—
254C8—3321no—
255C8#1458176712yes—
256C8#1458178111yescharacteristic
257C8#1458178811yescharacteristic
258C22—41no—
259C22—3321no—
260C22—3322no—
261C22—3322no—
262C22—3321no—
263C22—3321no—
264C22—1662no—
265C22—1664no—
266C22#1912766411yescharacteristic
267C22#1912777112yes—
268C4—42no—
269C4—3321no—
270C4—3321no—
271C4#1912760211yescharacteristic
272C4#1912766711yescharacteristic
273C4#1912777112yes—
274C2—21no—
275C2—22no—
276C2—3322no—
277C2—6641no—
278C2#2490826011yescharacteristic
subgroupG/H
centerC8#14581788
commutatorC22×C4×C83C22×C4
FrattiniC22×C8C22×D83
FittingC83×#2153C2
socleC2×C83#1955
prime radical#19128213C4
cosocleC22×C8×C83C23
hypercenter#2153D83
hypocenterC83#13200
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2153

Complements

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

NH ≅ G/Ntype
2. #24909092C2semidirect
3. #24909073C2semidirect
4. #24909068C2semidirect
5. C83×#2153C2semidirect
40. C2×C83⋊C16D4semidirect
141. #2153D83semidirect
145. C83#13200semidirect

Maximal subgroups

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

subgroupindex
#249161592
#249090922
#249090732
#249090682
C83×#21532
#1320083