Finite Group Databasebeta

← #35306651

Subgroups — #35306651

Order 21248.

542 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 105 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#24918401C212yesmaximal
2#24908287C211yesmaximal, characteristic
3#24906027C212yesmaximal, complemented
4#24905308C211yesmaximal, characteristic, complemented
5C83×#3162C211yesmaximal, characteristic, complemented
6C2×#14581789C2211yescharacteristic
7#19128075—22no—
8#19128049—22no—
9#19127901—24no—
10#19127851—24no—
11#19127754C2212yes—
12C2×#14581714—22no—
13#19127567C2211yescharacteristic
14#19127438—24no—
15#533×C83C2212yes—
16#501×C83—22no—
17#501×C83—22no—
18#427×C83—22no—
19C2×#97×C83C2211yescharacteristic
20C22×D332—21no—
21C22×D332—22no—
22C2×#13050577—28no—
23#14581789—22no—
24#14581789—21no—
25#14581789D412yes—
26C2×C4⋊Dic83—21no—
27C4⋊(C83⋊C8)—22no—
28#14581714—44no—
29#14581712—28no—
30C22⋊(C83⋊C8)D414yes—
31C4×C83⋊C8—22no—
32C83⋊#95—22no—
33C22×C8×C83D412yes—
34C4⋊Q8×C83D412yes—
35C4⋊Q8×C83D412yes—
36#125×C83—22no—
37C22⋊Q8×C83—22no—
38C22⋊Q8×C83—22no—
39C4×Q8×C83—22no—
40C4×Q8×C83—22no—
41#117×C83—22no—
42C2×C42×C83C2311yescharacteristic
43#107×C83—24no—
44#106×C83—22no—
45#106×C83—22no—
46#97×C83—22no—
47C23×D83—41no—
48C2×D332—24no—
49C2×D332—22no—
50C2×D332—42no—
51C2×D332—42no—
52C2×D332—41no—
53C2×D332—42no—
54C2×D332—84no—
55C22×Dic83—41no—
56#13050577—88no—
57C2×C83⋊C8—24no—
58C4⋊Dic83—42no—
59C2×Q8×C83—22no—
60C2×Q8×C83—22no—
61C22×C4×C83C2×D412yes—
62C22×C4×C83C2×D411yescharacteristic
63C2×C8×C83—22no—
64C2×C8×C83—22no—
65C2×C8×C83SD1614yescomplemented
66C4⋊C4×C83—22no—
67C4⋊C4×C83—22no—
68C4⋊C4×C83—22no—
69C4⋊C4×C83—22no—
70C4⋊C4×C83—42no—
71C4⋊C4×C83—42no—
72C22⋊C4×C83—42no—
73C42×C83—42no—
74C42×C83C2×D411yescharacteristic
75C42×C83C2×D412yes—
76C42×C83#4111yescharacteristic
77C23×C83#12711yescharacteristic
78C22×D83—42no—
79C22×D83—84no—
80C22×D83—81no—
81C22×D83—81no—
82Q8×C83—44no—
83Q8×C83—82no—
84C2×C4×C83—42no—
85C2×C4×C83—42no—
86C2×C4×C83—42no—
87C2×C4×C83#12012yes—
88C2×C4×C83#12011yescharacteristic
89C2×C4×C83#12111yescharacteristic
90C2×C4×C83#12112yes—
91C2×C4×C83C2×SD1612yes—
92C2×C4×C83#13712yes—
93C2×Dic83—82no—
94C2×Dic83—81no—
95D332—48no—
96D332—44no—
97D332—84no—
98D332—82no—
99C8×C83—24no—
100C83⋊C8—44no—
101C22×C83—22no—
102C22×C83#39211yescharacteristic
103C22×C83#49211yescharacteristic
104C22×C83#49711yescharacteristic
105C2×D83—84no—
106C2×D83—82no—
107C2×D83—162no—
108C2×D83—162no—
109C4×C83—22no—
110C4×C83—22no—
111C4×C83—82no—
112C4×C83—82no—
113C4×C83#44812yes—
114C4×C83#46112yes—
115Dic83—161no—
116#10220—831nomaximal, Sylow
117C2×C83—21no—
118C2×C83—21no—
119C2×C83#200812yes—
120C2×C83#201311yescharacteristic
121D83—162no—
122#3208—832no—
123#3162D8311yescharacteristic, complemented
124#1961—831no—
125#1518—832no—
126#1281—831no—
127C83#1022011yescharacteristic, complemented, Sylow
128#533C2×D8312yes—
129C2×#127—831no—
130#501—22no—
131#501—22no—
132#492—1662no—
133#486—1662no—
134#460—1664no—
135#450—1664no—
136#431—832no—
137#427—22no—
138C2×#102—1662no—
139C2×#97C2×D8311yescharacteristic
140#382—831no—
141#328—1664no—
142C22×D4—1662no—
143C22×D4—1661no—
144C2×SD16—1668no—
145C22×C8C22⋊D8312yes—
146C4⋊Q8C22⋊D8312yes—
147C4⋊Q8C22⋊D8312yes—
148#127—832no—
149#127—1662no—
150#127—1661no—
151#125—22no—
152C22⋊Q8—22no—
153C22⋊Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156#117—22no—
157C2×C4⋊C4—1661no—
158C2×C42C22×D8311yescharacteristic
159#107—24no—
160#106—22no—
161#106—22no—
162C4⋊C8—1662no—
163#102—3324no—
164#102—1668no—
165C22⋊C8—834no—
166#97—22no—
167C4×C8—1662no—
168#95—1662no—
169C24—3321no—
170C2×Q8—22no—
171C2×Q8—22no—
172C2×D4—3321no—
173C2×D4—3322no—
174C2×D4—3322no—
175C2×D4—3322no—
176C2×D4—6644no—
177C2×D4—1664no—
178C2×D4—1662no—
179C22×C4—3321no—
180C22×C4D4×D8312yes—
181C22×C4C2×C22⋊D8311yescharacteristic
182SD16—6648no—
183C2×C8—22no—
184C2×C8—22no—
185C2×C8—1664no—
186C2×C8#1305057714yescomplemented
187C4⋊C4—22no—
188C4⋊C4—22no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—3322no—
194C22⋊C4—42no—
195C42—42no—
196C42D4×D8311yescharacteristic
197C42C2×C22⋊D8312yes—
198C42#1305058911yescharacteristic
199C23—3322no—
200C23—6644no—
201C23—6641no—
202C23—6641no—
203C23#1458178811yescharacteristic
204Q8—44no—
205Q8—82no—
206D4—3328no—
207D4—3324no—
208D4—6644no—
209D4—6642no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—42no—
213C2×C4—6642no—
214C2×C4—6641no—
215C2×C4#1458176012yes—
216C2×C4#1458176111yescharacteristic
217C2×C4#1458176311yescharacteristic
218C2×C4#1458176412yes—
219C2×C4C2×#1305057712yes—
220C2×C4#1458182312yes—
221C8—24no—
222C8—3324no—
223C22—22no—
224C22—13282no—
225C22—13282no—
226C22—6642no—
227C22—6644no—
228C22#1912760911yescharacteristic
229C22#1912807511yescharacteristic
230C22#1912809311yescharacteristic
231C4—22no—
232C4—22no—
233C4—82no—
234C4—82no—
235C4—13281no—
236C4#1912784012yes—
237C4#1912790312yes—
238C2—21no—
239C2—21no—
240C2—13282no—
241C2#2490851312yes—
242C2#2490854611yescharacteristic
subgroupG/H
centerC22#19127609
commutator, cosocleC2×C42×C83C23
FrattiniC2×C42C22×D83
FittingC83×#3162C2
socleC22×C83#392
prime radicalC2×#14581789C22
hypercenter#3162D83
hypocenterC83#10220
LCSG ▷ C2×C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3162

Complements

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

NH ≅ G/Ntype
3. #24906027C2semidirect
4. #24905308C2semidirect
5. C83×#3162C2semidirect
65. C2×C8×C83SD16semidirect
123. #3162D83semidirect
127. C83#10220semidirect
186. C2×C8#13050577semidirect

Maximal subgroups

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

subgroupindex
#249184012
#249082872
#249060272
#249053082
C83×#31622
#1022083