Finite Group Databasebeta

← #35318113

Subgroups — #35318113

Order 21248.

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

normal subgroups only

This group has 282 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#24920355C211yesmaximal, characteristic
2#24909690C211yesmaximal, characteristic
3#24909510C211yesmaximal, characteristic
4#24905519C211yesmaximal, characteristic
5Q8⋊(C83⋊C16)C211yesmaximal, characteristic
6#24905426C211yesmaximal, characteristic
7C83×#1670C211yesmaximal, characteristic
8#19128429—21no—
9C2×#14581738—21no—
10#19128106C2211yescharacteristic
11#19128103—21no—
12#19128009—22no—
13#19127987—21no—
14C4×Dic332—21no—
15C4×Dic332C2211yescharacteristic
16C4⋊(C83⋊C16)C2211yescharacteristic
17#19127505—22no—
18#19127505—21no—
19C4×C83⋊C16C2211yescharacteristic
20#494×C83C2211yescharacteristic
21#475×C83—21no—
22C8×Q8×C83C2211yescharacteristic
23#335×C83C2211yescharacteristic
24#332×C83—21no—
25C2×Dic332D411yescharacteristic
26#14581792—21no—
27#14581790—41no—
28#14581784—22no—
29#14581782—42no—
30C4×Dic166—22no—
31C4×Dic166—21no—
32#14581757—41no—
33#14581738—42no—
34C2×C83⋊C16—21no—
35C2×C83⋊C16—21no—
36#14581726—21no—
37#14581726D411yescharacteristic
38#14581726D412yes—
39#14581724—22no—
40#14581717—22no—
41#14581717—21no—
42C2×Q16×C83—21no—
43C2×Q16×C83—21no—
44C4⋊Q8×C83—21no—
45C4×Q8×C83D411yescharacteristic
46#106×C83—41no—
47C4⋊C8×C83—21no—
48C4⋊C8×C83—21no—
49C4⋊C8×C83D411yescharacteristic
50#103×C83—41no—
51C4×C8×C83—21no—
52C4×C8×C83C2311yescharacteristic
53C2×Dic166—21no—
54C2×Dic166—41no—
55Dic332—21no—
56Dic332SD1612yes—
57C4⋊Dic83—21no—
58C4⋊Dic83—22no—
59C4⋊Dic83—41no—
60Dic83⋊C4—42no—
61Dic83⋊C4—42no—
62Dic83⋊C4—42no—
63Dic83⋊C4—41no—
64C4×Dic83—42no—
65C4×Dic83—41no—
66C83⋊C16—41no—
67C83⋊C16—41no—
68C2×Q8×C83—41no—
69C2×Q8×C83D811yescharacteristic
70Q16×C83—42no—
71Q16×C83—41no—
72Q16×C83—41no—
73C2×C8×C83—21no—
74C2×C8×C83—41no—
75C2×C8×C83C2×D411yescharacteristic
76C2×C8×C83C2×D411yescharacteristic
77C4⋊C4×C83—21no—
78C4⋊C4×C83—41no—
79C4⋊C4×C83—41no—
80C4⋊C4×C83D811yescharacteristic
81C42×C83—21no—
82C42×C83C2×D411yescharacteristic
83Q8×C83—21no—
84Q8×C83—21no—
85Q8×C83—81no—
86Q8×C83—81no—
87C2×C4×C83—21no—
88C2×C4×C83—21no—
89C2×C4×C83—81no—
90C2×C4×C83#12011yescharacteristic
91C2×C4×C83C2×D811yescharacteristic
92C2×C4×C83AGL(1,ℤ/8)11yescharacteristic
93C2×Dic83—41no—
94C2×Dic83—41no—
95C2×Dic83—42no—
96Dic166—22no—
97Dic166—41no—
98Dic166—81no—
99Dic166—81no—
100C8×C83—21no—
101C8×C83—41no—
102C8×C83—81no—
103C8×C83C2×SD1611yescharacteristic
104C8×C83#13711yescharacteristic
105C22×C83#44711yescharacteristic
106C4×C83—21no—
107C4×C83—41no—
108C4×C83—41no—
109C4×C83—161no—
110C4×C83#44811yescharacteristic
111C4×C83#45411yescharacteristic
112C4×C83#50811yescharacteristic
113C4×C83#51011yescharacteristic
114Dic83—42no—
115Dic83—82no—
116Dic83—81no—
117#12357—831nomaximal, Sylow
118C2×C83#162711yescharacteristic
119C2×C83#218911yescharacteristic
120C2×C83#219511yescharacteristic
121#3383—831no—
122#2241—831no—
123#2209—831no—
124#1670D8311yescharacteristic
125#1364—831no—
126Q8⋊C16—831no—
127#1332—831no—
128C83#1235711yescharacteristic, complemented, Sylow
129#557—1661no—
130C2×Q32—1661no—
131#500—831no—
132#499—1661no—
133#494C2×D8311yescharacteristic
134#479—1662no—
135#475—21no—
136#475—1661no—
137C8×Q8C2×D8311yescharacteristic
138C4×Q16—831no—
139C4×Q16—1661no—
140C4⋊C16—831no—
141#358—1661no—
142#358—1662no—
143C4×C16—831no—
144#335C2×D8311yescharacteristic
145#332—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—831no—
149C4⋊Q8—21no—
150C4⋊Q8—3321no—
151C4⋊Q8—1661no—
152#125—3322no—
153#125—1662no—
154C4×Q8—3321no—
155C4×Q8—1662no—
156C4×Q8—1661no—
157C4×Q8C22⋊D8311yescharacteristic
158Q32—3322no—
159C2×C16—1661no—
160C2×C16—1661no—
161#107—831no—
162#107—832no—
163#107—1661no—
164#106—41no—
165#106—1662no—
166C4⋊C8—21no—
167C4⋊C8—21no—
168C4⋊C8C22⋊D8311yescharacteristic
169#103—41no—
170#103—1662no—
171#103—1661no—
172C4×C8—21no—
173C4×C8C22×D8311yescharacteristic
174C2×Q8—41no—
175C2×Q8—3321no—
176C2×Q8—1661no—
177C2×Q8#1305057511yescharacteristic
178Q16—42no—
179Q16—41no—
180Q16—41no—
181Q16—832no—
182Q16—1661no—
183C2×C8—21no—
184C2×C8—41no—
185C2×C8D4×D8311yescharacteristic
186C2×C8D4×D8311yescharacteristic
187C4⋊C4—21no—
188C4⋊C4—41no—
189C4⋊C4—41no—
190C4⋊C4—3322no—
191C4⋊C4—3322no—
192C4⋊C4—3322no—
193C4⋊C4—3321no—
194C4⋊C4—3321no—
195C4⋊C4—1661no—
196C4⋊C4—1662no—
197C4⋊C4#1305057511yescharacteristic
198C42—21no—
199C42—3322no—
200C42—3321no—
201C42C2×C22⋊D8311yescharacteristic
202C16—3321no—
203C16—3321no—
204Q8—21no—
205Q8—21no—
206Q8—81no—
207Q8—81no—
208Q8—3321no—
209Q8—6641no—
210Q8—6641no—
211Q8—1662no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—81no—
215C2×C4—3321no—
216C2×C4—3321no—
217C2×C4—3322no—
218C2×C4#1458176011yescharacteristic
219C2×C4C2×#1305057511yescharacteristic
220C2×C4#1458181711yescharacteristic
221C8—21no—
222C8—41no—
223C8—81no—
224C8SD16×D8311yescharacteristic
225C8#1458182111yescharacteristic
226C22#1912783411yescharacteristic
227C4—21no—
228C4—41no—
229C4—41no—
230C4—161no—
231C4—3322no—
232C4—6642no—
233C4—6641no—
234C4#1912783811yescharacteristic
235C4#1912787211yescharacteristic
236C4#1912813111yescharacteristic
237C4#1912814211yescharacteristic
238C2#2490656411yescharacteristic
239C2#2490940111yescharacteristic
240C2#2490943911yescharacteristic
subgroupG/H
centerC22#19127834
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1670C2
socle, prime radicalC22×C83#447
hypercenter#1670D83
hypocenterC83#12357
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C8×Q8 ◁ #1670

Complements

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

NH ≅ G/Ntype
128. C83#12357semidirect

Maximal subgroups

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

subgroupindex
#249203552
#249096902
#249095102
#249055192
Q8⋊(C83⋊C16)2
#249054262
C83×#16702
#1235783