Finite Group Databasebeta

← #35307476

Subgroups — #35307476

Order 21248.

400 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 93 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#24908435C212yesmaximal
2#24908282C211yesmaximal, characteristic
3#24905344C211yesmaximal, characteristic
4#24905344C212yesmaximal
5C83×#2844C211yesmaximal, characteristic
6C2×#14581792C2211yescharacteristic
7C2×#14581715—21no—
8C2×#14581715—21no—
9C2×#14581703C2212yes—
10C2×#14581703C2211yescharacteristic
11#19127463—22no—
12#19127463—22no—
13#400×C83—21no—
14#400×C83—22no—
15#395×C83—21no—
16#395×C83—22no—
17#382×C83C2212yes—
18#382×C83C2211yescharacteristic
19C22×Dic166—21no—
20C22×C83⋊C8—21no—
21C22×C83⋊C8—22no—
22#14581792—22no—
23#14581792—21no—
24#14581792Q812yescomplemented
25C2×C4⋊Dic83—22no—
26#14581715—42no—
27#14581715—42no—
28#14581703—22no—
29#14581703—22no—
30#14581703—22no—
31C22×Q8×C83—21no—
32C2×C4⋊C4×C83—21no—
33C2×C4⋊C4×C83—22no—
34C2×C4⋊C4×C83D411yescharacteristic
35C2×C4⋊C4×C83D411yescharacteristic
36C2×C4⋊C4×C83D412yes—
37C2×C4⋊C4×C83D412yes—
38C2×C42×C83C2311yescharacteristic
39#95×C83—22no—
40#95×C83—21no—
41#95×C83—21no—
42#95×C83—22no—
43#95×C83—22no—
44#95×C83—22no—
45C2×Dic166—22no—
46C2×Dic166—41no—
47C2×Dic166—41no—
48C2×Dic166—41no—
49C2×Dic166—41no—
50C2×Dic166—41no—
51C22×Dic83—41no—
52C2×C83⋊C8—24no—
53C2×C83⋊C8—22no—
54C2×C83⋊C8—41no—
55C2×C83⋊C8—42no—
56C2×C83⋊C8—42no—
57C2×C83⋊C8—41no—
58C4⋊Dic83—24no—
59C4⋊Dic83—42no—
60C2×Q8×C83—81no—
61C2×Q8×C83—82no—
62C22×C4×C83—21no—
63C22×C4×C83—22no—
64C22×C4×C83—21no—
65C22×C4×C83—22no—
66C22×C4×C83C2×D411yescharacteristic
67C22×C4×C83C2×D412yes—
68C4⋊C4×C83—42no—
69C4⋊C4×C83—42no—
70C4⋊C4×C83—41no—
71C4⋊C4×C83—41no—
72C4⋊C4×C83—81no—
73C4⋊C4×C83—82no—
74C42×C83C2×Q811yescharacteristic
75C42×C83#4112yes—
76C42×C83#4111yescharacteristic
77C23×C83#12011yescharacteristic
78Q8×C83—84no—
79C2×C4×C83—21no—
80C2×C4×C83—21no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83—41no—
84C2×C4×C83—41no—
85C2×C4×C83—42no—
86C2×C4×C83—42no—
87C2×C4×C83—42no—
88C2×C4×C83—42no—
89C2×C4×C83—42no—
90C2×C4×C83—42no—
91C2×C4×C83—42no—
92C2×C4×C83—42no—
93C2×C4×C83C22⋊Q812yes—
94C2×C4×C83C22⋊Q811yescharacteristic
95C2×C4×C83#12412yes—
96C2×C4×C83#12411yescharacteristic
97C2×Dic83—42no—
98C2×Dic83—81no—
99C2×Dic83—81no—
100Dic166—42no—
101Dic166—42no—
102Dic166—81no—
103Dic166—81no—
104C83⋊C8—42no—
105C83⋊C8—44no—
106C22×C83#39311yescharacteristic
107C22×C83#45311yescharacteristic
108C22×C83#45411yescharacteristic
109C22×C83#45512yes—
110C22×C83#45612yes—
111C4×C83—21no—
112C4×C83—21no—
113C4×C83—22no—
114C4×C83—22no—
115C4×C83—81no—
116C4×C83—82no—
117C4×C83—81no—
118C4×C83—82no—
119Dic83—82no—
120#10359—831nomaximal, Sylow
121C2×C83#165911yescharacteristic
122C2×C83#166011yescharacteristic
123C2×C83#166412yes—
124C2×C83#202711yescharacteristic
125C2×C83#202812yes—
126#2844D8311yescharacteristic
127#1993—832no—
128#1960—831no—
129#1295—831no—
130#1295—832no—
131C83#1035911yescharacteristic, complemented, Sylow
132C2×C4⋊Q8—831no—
133C2×#103—1661no—
134C2×#103—1661no—
135C2×#97—831no—
136C2×#97—832no—
137#400—22no—
138#400—21no—
139#395—22no—
140#395—21no—
141#382C2×D8311yescharacteristic
142#382C2×D8312yes—
143#340—1662no—
144#340—1662no—
145C22×Q8—21no—
146C22×Q8—1661no—
147C22×C8—1662no—
148C22×C8—1661no—
149C4⋊Q8—832no—
150C4⋊Q8—1661no—
151C4⋊Q8—1662no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—1662no—
155C2×C4⋊C4C22⋊D8312yes—
156C2×C4⋊C4C22⋊D8312yes—
157C2×C4⋊C4C22⋊D8311yescharacteristic
158C2×C4⋊C4C22⋊D8311yescharacteristic
159C2×C42C22×D8311yescharacteristic
160#103—3322no—
161#103—3322no—
162#97—1662no—
163#97—1662no—
164#97—1662no—
165#95—22no—
166#95—21no—
167#95—21no—
168#95—22no—
169#95—22no—
170#95—22no—
171C2×Q8—81no—
172C2×Q8—82no—
173C2×Q8—3321no—
174C2×Q8—3321no—
175C2×Q8—3321no—
176C2×Q8—3321no—
177C2×Q8—3321no—
178C2×Q8—1662no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—3321no—
184C22×C4C2×C22⋊D8312yes—
185C22×C4C2×C22⋊D8311yescharacteristic
186C2×C8—3321no—
187C2×C8—3322no—
188C2×C8—3322no—
189C2×C8—3321no—
190C2×C8—1662no—
191C2×C8—1664no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—81no—
197C4⋊C4—82no—
198C4⋊C4—3322no—
199C4⋊C4—1664no—
200C42Q8×D8311yescharacteristic
201C42#1305059112yes—
202C42#1305059111yescharacteristic
203C23#1458176111yescharacteristic
204Q8—84no—
205Q8—3322no—
206Q8—3322no—
207Q8—6641no—
208Q8—6641no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—42no—
216C2×C4—42no—
217C2×C4—42no—
218C2×C4—42no—
219C2×C4—42no—
220C2×C4—42no—
221C2×C4—42no—
222C2×C4—42no—
223C2×C4—3322no—
224C2×C4—6641no—
225C2×C4—6641no—
226C2×C4#1458176812yes—
227C2×C4#1458176811yescharacteristic
228C2×C4#1458178112yes—
229C2×C4#1458178111yescharacteristic
230C8—3322no—
231C8—3324no—
232C22#1912761211yescharacteristic
233C22#1912786911yescharacteristic
234C22#1912787411yescharacteristic
235C22#1912787912yes—
236C22#1912788412yes—
237C4—21no—
238C4—21no—
239C4—22no—
240C4—22no—
241C4—81no—
242C4—82no—
243C4—81no—
244C4—82no—
245C4—6642no—
246C2#2490676911yescharacteristic
247C2#2490677411yescharacteristic
248C2#2490679712yes—
249C2#2490861611yescharacteristic
250C2#2490862112yes—
subgroupG/H
centerC23#14581761
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#2844C2
socle, prime radicalC23×C83#120
hypercenter#2844D83
hypocenterC83#10359
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2844

Complements

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

NH ≅ G/Ntype
24. #14581792Q8semidirect
131. C83#10359semidirect

Maximal subgroups

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

subgroupindex
#249084352
#249082822
#249053442
C83×#28442
#1035983