Finite Group Databasebeta

← #35318132

Subgroups — #35318132

Order 21248.

296 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 69 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#24920328C211yesmaximal, characteristic
2#24909691C211yesmaximal, characteristic, complemented
3#24909508C211yesmaximal, characteristic
4#24906834C211yesmaximal, characteristic, complemented
5#24905522C211yesmaximal, characteristic, complemented
6#24905443C211yesmaximal, characteristic, complemented
7C83×#1331C211yesmaximal, characteristic
8#19128408—21no—
9C2×#14581736—21no—
10#19128106C2211yescharacteristic
11#19128102—21no—
12#19128081C2211yescharacteristic
13#19127999—22no—
14#19127991—21no—
15#19127987—21no—
16C8×Dic166C2211yescharacteristic
17D8×Dic83—21no—
18C4⋊(C83⋊C16)C2211yescharacteristic
19#19127506—22no—
20#19127504—21no—
21#19127446—21no—
22C4×D8×C83C2211yescharacteristic
23C4×C16×C83C2211yescharacteristic
24#335×C83C2211yescharacteristic
25C2×Dic332—21no—
26C2×Dic332—21no—
27#14581792—21no—
28#14581792—21no—
29#14581783—22no—
30#14581771—41no—
31#14581769—42no—
32C4×Dic166D411yescharacteristic
33D4×Dic83—22no—
34C2×C4⋊Dic83—41no—
35#14581736—42no—
36C2×C83⋊C16—21no—
37#14581726—21no—
38#14581726D412yes—
39#14581724—41no—
40#14581723—22no—
41C4⋊(C83⋊C8)D411yescharacteristic
42Dic83⋊C8—21no—
43#14581717—41no—
44#14581713—22no—
45C8×Dic83—21no—
46C2×D8×C83D411yescharacteristic
47C4×D4×C83—21no—
48C2×C16×C83—21no—
49#107×C83D411yescharacteristic
50C4⋊C8×C83—21no—
51#102×C83—21no—
52C4×C8×C83C2311yescharacteristic
53C2×Dic166—41no—
54C2×Dic166—41no—
55C2×Dic166D811yescharacteristic
56C22×Dic83—42no—
57Dic332—42no—
58Dic332—41no—
59Dic332—41no—
60C2×C83⋊C8—21no—
61C4⋊Dic83—22no—
62C4⋊Dic83—42no—
63C4⋊Dic83—41no—
64C4⋊Dic83—41no—
65C4⋊Dic83—41no—
66C4⋊Dic83D811yescharacteristic
67Dic83⋊C4—21no—
68Dic83⋊C4—42no—
69C22⋊Dic83—42no—
70C4×Dic83—21no—
71C83⋊C16—41no—
72C2×D4×C83—21no—
73C22×C4×C83—41no—
74D8×C83—21no—
75D8×C83SD1612yes—
76C2×C8×C83—41no—
77C2×C8×C83C2×D411yescharacteristic
78C2×C8×C83C2×D411yescharacteristic
79C4⋊C4×C83—21no—
80C22⋊C4×C83—41no—
81C42×C83C2×D411yescharacteristic
82C16×C83—41no—
83C23×C83—41no—
84D4×C83—22no—
85D4×C83—41no—
86C2×C4×C83—41no—
87C2×C4×C83—81no—
88C2×C4×C83—81no—
89C2×C4×C83#12011yescharacteristic
90C2×C4×C83C2×D811yescharacteristic
91C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
92C2×Dic83—21no—
93C2×Dic83—21no—
94C2×Dic83—42no—
95C2×Dic83—81no—
96C2×Dic83—162no—
97Dic166—21no—
98Dic166—21no—
99Dic166—81no—
100Dic166—81no—
101Dic166—161no—
102C8×C83—21no—
103C8×C83—81no—
104C8×C83C2×SD1611yescharacteristic
105C8×C83#13711yescharacteristic
106C83⋊C8—41no—
107C22×C83—42no—
108C22×C83—81no—
109C22×C83—81no—
110C22×C83#44711yescharacteristic
111C4×C83—21no—
112C4×C83—81no—
113C4×C83#44811yescharacteristic
114C4×C83#45411yescharacteristic
115C4×C83#50811yescharacteristic
116C4×C83#51011yescharacteristic
117Dic83—41no—
118Dic83—41no—
119Dic83—82no—
120Dic83—161no—
121#12359—831nomaximal, Sylow
122C2×C83—82no—
123C2×C83#162711yescharacteristic
124C2×C83#218911yescharacteristic
125C2×C83#219411yescharacteristic
126#3380—831no—
127#2241—831no—
128#2208—831no—
129#1670—831no—
130#1365—831no—
131Q8⋊C16—831no—
132#1331D8311yescharacteristic
133C83#1235911yescharacteristic, complemented, Sylow
134#554—1661no—
135C2×SD32—1661no—
136#500—831no—
137#499—1661no—
138#494—831no—
139#478—1662no—
140#476—1661no—
141#475—1661no—
142C8×Q8—831no—
143C4×D8—1661no—
144C4×D8C2×D8311yescharacteristic
145C4⋊C16—831no—
146#358—1662no—
147#358—1661no—
148C4×C16C2×D8311yescharacteristic
149#335C2×D8311yescharacteristic
150#332—1661no—
151C2×Q16—1661no—
152C2×Q16—1661no—
153C2×D8D33211yescharacteristic
154C4⋊Q8—1661no—
155C4⋊Q8—1661no—
156#125—1662no—
157C22⋊Q8—3322no—
158C22⋊Q8—3321no—
159C4×Q8—831no—
160C4×D4—21no—
161C4×D4—1662no—
162C2×C4⋊C4—3321no—
163SD32—3322no—
164C2×C16—21no—
165C2×C16—1661no—
166#107—832no—
167#107—1661no—
168#107D33211yescharacteristic
169#106—3321no—
170#106—1662no—
171C4⋊C8—21no—
172C4⋊C8—831no—
173C4⋊C8—1661no—
174#103—3321no—
175#102—21no—
176#102—1662no—
177C4×C8—1661no—
178C4×C8C22×D8311yescharacteristic
179C2×Q8—3321no—
180C2×Q8—3321no—
181C2×Q8—831no—
182C2×D4—21no—
183C22×C4—41no—
184C22×C4—3322no—
185Q16—3322no—
186Q16—3321no—
187Q16—3321no—
188D8—21no—
189D8#1305057912yes—
190C2×C8—41no—
191C2×C8—1661no—
192C2×C8D4×D8311yescharacteristic
193C2×C8C2×D33211yescharacteristic
194C4⋊C4—21no—
195C4⋊C4—3322no—
196C4⋊C4—3322no—
197C4⋊C4—3321no—
198C4⋊C4—3321no—
199C4⋊C4—3321no—
200C4⋊C4—831no—
201C4⋊C4—1662no—
202C4⋊C4—1661no—
203C22⋊C4—41no—
204C22⋊C4—3322no—
205C42—1661no—
206C42D4×D8311yescharacteristic
207C16—41no—
208C16—3321no—
209C23—41no—
210Q8—13281no—
211Q8—6641no—
212Q8—6641no—
213Q8—1661no—
214Q8—1661no—
215D4—22no—
216D4—41no—
217C2×C4—41no—
218C2×C4—81no—
219C2×C4—81no—
220C2×C4—13282no—
221C2×C4—3322no—
222C2×C4—6641no—
223C2×C4—1661no—
224C2×C4—1661no—
225C2×C4#1458176211yescharacteristic
226C2×C4D8×D8311yescharacteristic
227C2×C4#1458181911yescharacteristic
228C8—21no—
229C8—81no—
230C8—3321no—
231C8C2×#1305057911yescharacteristic
232C8#1458182511yescharacteristic
233C22—42no—
234C22—81no—
235C22—81no—
236C22#1912783611yescharacteristic
237C4—21no—
238C4—81no—
239C4—13281no—
240C4—3321no—
241C4—3321no—
242C4—6642no—
243C4#1912784211yescharacteristic
244C4#1912787511yescharacteristic
245C4#1912813011yescharacteristic
246C4#1912814011yescharacteristic
247C2—82no—
248C2#2490656911yescharacteristic
249C2#2490940411yescharacteristic
250C2#2490943511yescharacteristic
subgroupG/H
centerC22#19127836
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1331C2
socleC22×C83#447
prime radicalC2×D8×C83D4
hypercenter#1331D83
hypocenterC83#12359
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1331

Complements

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

NH ≅ G/Ntype
2. #24909691C2semidirect
4. #24906834C2semidirect
5. #24905522C2semidirect
6. #24905443C2semidirect
133. C83#12359semidirect

Maximal subgroups

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

subgroupindex
#249203282
#249096912
#249095082
#249068342
#249055222
#249054432
C83×#13312
#1235983