Finite Group Databasebeta

← #35308902

Subgroups — #35308902

Order 21248.

406 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 97 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#24915735C211yesmaximal, characteristic
2#24907632C211yesmaximal, characteristic
3#24907631C212yesmaximal
4#24905345C212yesmaximal
5C83×#1295C211yesmaximal, characteristic
6C2×#14581726—22no—
7C2×#14581725—21no—
8C2×#14581724—22no—
9C2×#14581723—21no—
10C2×#14581703C2211yescharacteristic
11#19127643—22no—
12#19127640—21no—
13#19127622—22no—
14#19127621—21no—
15#19127571C2211yescharacteristic
16#19127567C2212yes—
17C2×#97×C83C2212yes—
18#382×C83C2211yescharacteristic
19C22×Dic166—21no—
20C22×C83⋊C8—21no—
21C2×C4⋊Dic83—21no—
22C2×C4⋊Dic83—21no—
23C2×C4⋊Dic83—22no—
24C2×C4⋊Dic83D412yes—
25C2×C4⋊Dic83D412yes—
26C2×Dic83⋊C4—22no—
27#14581726—44no—
28#14581725—42no—
29#14581724—44no—
30#14581723—42no—
31#14581703—21no—
32#14581703Q812yescomplemented
33C83⋊#95—21no—
34C83⋊#95—22no—
35C83⋊#95—21no—
36C83⋊#95—22no—
37C83⋊#95—22no—
38C22×C8×C83—22no—
39C2×C4⋊C4×C83D411yescharacteristic
40C2×C4⋊C4×C83D411yescharacteristic
41C2×C42×C83C2311yescharacteristic
42#97×C83—22no—
43#97×C83Q814yescomplemented
44#95×C83—22no—
45C2×Dic166—82no—
46C2×Dic166—81no—
47C22×Dic83—22no—
48C22×Dic83—22no—
49C22×Dic83—41no—
50C2×C83⋊C8—22no—
51C2×C83⋊C8—22no—
52C2×C83⋊C8—22no—
53C4⋊Dic83—42no—
54C4⋊Dic83—42no—
55C4⋊Dic83—41no—
56C4⋊Dic83—42no—
57C4⋊Dic83—41no—
58C4⋊Dic83—42no—
59C4⋊Dic83—81no—
60Dic83⋊C4—82no—
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83C2×D412yes—
64C22×C4×C83C2×D411yescharacteristic
65C2×C8×C83—24no—
66C2×C8×C83—24no—
67C2×C8×C83—24no—
68C4⋊C4×C83—41no—
69C4⋊C4×C83—41no—
70C42×C83C2×Q811yescharacteristic
71C42×C83C2×Q812yes—
72C42×C83#4111yescharacteristic
73C23×C83#12011yescharacteristic
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83—21no—
78C2×C4×C83—41no—
79C2×C4×C83—41no—
80C2×C4×C83—42no—
81C2×C4×C83—42no—
82C2×C4×C83C22⋊Q812yes—
83C2×C4×C83C22⋊Q811yescharacteristic
84C2×C4×C83C4⋊Q812yes—
85C2×C4×C83C4⋊Q811yescharacteristic
86C2×Dic83—42no—
87C2×Dic83—42no—
88C2×Dic83—42no—
89C2×Dic83—42no—
90C2×Dic83—42no—
91C2×Dic83—42no—
92C2×Dic83—82no—
93C2×Dic83—81no—
94Dic166—84no—
95C8×C83—28no—
96C83⋊C8—24no—
97C22×C83#39511yescharacteristic
98C22×C83#45312yes—
99C22×C83#45311yescharacteristic
100C22×C83#45612yes—
101C22×C83#45611yescharacteristic
102C4×C83—21no—
103C4×C83—22no—
104C4×C83—22no—
105C4×C83—21no—
106C4×C83—81no—
107C4×C83—81no—
108Dic83—82no—
109Dic83—82no—
110Dic83—161no—
111#10583—831nomaximal, Sylow
112C2×C83#165511yescharacteristic
113C2×C83#166011yescharacteristic
114C2×C83#166012yes—
115C2×C83#206012yes—
116C2×C83#206011yescharacteristic
117#2844—831no—
118#1847—832no—
119#1847—831no—
120#1295—832no—
121#1295D8311yescharacteristic
122C83#1058311yescharacteristic, complemented, Sylow
123C2×#107—1662no—
124C2×#107—1661no—
125C2×#106—1662no—
126C2×#106—1661no—
127C2×#97—831no—
128C2×#97C2×D8312yes—
129#400—1662no—
130#400—1661no—
131#395—1662no—
132#395—1661no—
133#384—831no—
134#382—832no—
135#382C2×D8311yescharacteristic
136C22×Q8—1661no—
137C22×C8—22no—
138C22×C8—1661no—
139C2×C4⋊C4—832no—
140C2×C4⋊C4—832no—
141C2×C4⋊C4—1662no—
142C2×C4⋊C4—1661no—
143C2×C4⋊C4—1662no—
144C2×C4⋊C4—1661no—
145C2×C4⋊C4D33211yescharacteristic
146C2×C4⋊C4D33211yescharacteristic
147C2×C42C22×D8311yescharacteristic
148#107—3322no—
149#107—3324no—
150#106—3322no—
151#106—3324no—
152#97—22no—
153#97—832no—
154#97—1661no—
155#97Dic16614yescomplemented
156#95—22no—
157#95—1662no—
158#95—1661no—
159#95—1661no—
160#95—1662no—
161#95—1662no—
162C2×Q8—6642no—
163C2×Q8—6641no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—3321no—
167C22×C4—1662no—
168C22×C4—1662no—
169C22×C4D4×D8312yes—
170C22×C4C2×D33211yescharacteristic
171C2×C8—24no—
172C2×C8—24no—
173C2×C8—24no—
174C2×C8—1662no—
175C2×C8—1662no—
176C2×C8—1662no—
177C4⋊C4—41no—
178C4⋊C4—41no—
179C4⋊C4—3322no—
180C4⋊C4—3322no—
181C4⋊C4—3322no—
182C4⋊C4—3321no—
183C4⋊C4—3322no—
184C4⋊C4—3321no—
185C4⋊C4—6642no—
186C4⋊C4—6641no—
187C42Q8×D8311yescharacteristic
188C42C2×Dic16612yes—
189C42#1305058911yescharacteristic
190C23#1458176211yescharacteristic
191Q8—6644no—
192C2×C4—22no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—42no—
197C2×C4—41no—
198C2×C4—41no—
199C2×C4—42no—
200C2×C4—3322no—
201C2×C4—3322no—
202C2×C4—3322no—
203C2×C4—3322no—
204C2×C4—3322no—
205C2×C4—3322no—
206C2×C4—6642no—
207C2×C4—6641no—
208C2×C4#1458176912yes—
209C2×C4#1458177211yescharacteristic
210C2×C4#1458179112yes—
211C2×C4#1458179211yescharacteristic
212C8—28no—
213C8—1664no—
214C22#1912762211yescharacteristic
215C22#1912787012yes—
216C22#1912787111yescharacteristic
217C22#1912788512yes—
218C22#1912788611yescharacteristic
219C4—22no—
220C4—22no—
221C4—21no—
222C4—21no—
223C4—81no—
224C4—81no—
225C4—13281no—
226C4—6642no—
227C4—6642no—
228C2#2490675311yescharacteristic
229C2#2490677511yescharacteristic
230C2#2490677612yes—
231C2#2490880712yes—
232C2#2490880911yescharacteristic
subgroupG/H
centerC23#14581762
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#1295C2
socle, prime radicalC23×C83#120
hypercenter#1295D83
hypocenterC83#10583
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1295

Complements

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

NH ≅ G/Ntype
32. #14581703Q8semidirect
43. #97×C83Q8semidirect
122. C83#10583semidirect
155. #97Dic166semidirect

Maximal subgroups

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

subgroupindex
#249157352
#249076322
#249076312
#249053452
C83×#12952
#1058383