Finite Group Databasebeta

← #24907992

Subgroups — #24907992

Order 10624.

362 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 79 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#19128265C211yesmaximal, characteristic
2#19128175C211yesmaximal, characteristic, complemented
3#19127718C211yesmaximal, characteristic, complemented
4C2×#14581708C211yesmaximal, characteristic, complemented
5#19127458C211yesmaximal, characteristic, complemented
6#19127457C211yesmaximal, characteristic, complemented
7#421×C83C211yesmaximal, characteristic, complemented
8C22×D332C2211yescharacteristic
9C2×#13050574C2211yescharacteristic
10#14581788C412yescomplemented
11#14581781C412yescomplemented
12#14581766—22no—
13#14581760—22no—
14C4×D332—22no—
15#14581754—22no—
16#14581751C2211yescharacteristic
17#14581751C2211yescharacteristic
18C22⋊C4×D83—22no—
19#14581719—22no—
20#14581708—22no—
21C2×#41×C83C2211yescharacteristic
22C2×M16×C83C2211yescharacteristic
23#117×C83C2211yescharacteristic
24#104×C83—22no—
25C23×D83D412yes—
26C2×D332—21no—
27C2×D332—21no—
28C2×D332—21no—
29C2×D332—21no—
30C2×D332C2×C412yes—
31C2×D332D412yes—
32C2×C22⋊D83—24no—
33C2×C4×D83—24no—
34#13050574—21no—
35#13050574—21no—
36C2×C83⋊C8—21no—
37C4⋊Dic83—21no—
38C4⋊Dic83—21no—
39C22⋊Dic83—21no—
40C22⋊Dic83—21no—
41(C2×D83)⋊C4—24no—
42(C2×D83)⋊C4—24no—
43C4×Dic83C2×C412yescomplemented
44C4×Dic83D412yescomplemented
45#41×C83—21no—
46#41×C83—21no—
47#41×C83—21no—
48#41×C83—21no—
49C2×Q8×C83C2×C411yescharacteristic, complemented
50C2×D4×C83—21no—
51C2×D4×C83C2×C411yescharacteristic, complemented
52C22×C4×C83—21no—
53C22×C4×C83C2311yescharacteristic
54M16×C83—21no—
55M16×C83—21no—
56C2×C8×C83—21no—
57C4⋊C4×C83—21no—
58C22⋊C4×C83—21no—
59C42×C83D412yescomplemented
60C23×C83—21no—
61C23×C83C2×D411yescharacteristic
62C22×D83—22no—
63C22×D83—22no—
64C22×D83—82no—
65C22×D83C22⋊C414yes—
66Q8×C83—21no—
67Q8×C83—21no—
68D4×C83—21no—
69D4×C83—21no—
70D4×C83—41no—
71D4×C83—41no—
72C2×C4×C83—21no—
73C2×C4×C83—22no—
74C2×C4×C83—41no—
75C2×C4×C83—41no—
76C2×C4×C83C22×C411yescharacteristic
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2×D411yescharacteristic
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83#4111yescharacteristic
81C2×C4×C83#4111yescharacteristic
82C22⋊D83—84no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85D332—22no—
86D332—22no—
87D332—22no—
88D332—22no—
89C4×D83—84no—
90C8×C83—41no—
91C83⋊C8—41no—
92C22×C83—41no—
93C22×C83—42no—
94C22×C83—41no—
95C22×C83C2×C22⋊C411yescharacteristic
96C22×C83#12011yescharacteristic
97C22×C83#12311yescharacteristic
98C2×D83—24no—
99C2×D83—44no—
100C2×D83—44no—
101C2×D83—82no—
102C4×C83—42no—
103C4×C83—41no—
104C4×C83C4×D411yescharacteristic
105C4×C83C4×D411yescharacteristic
106C4×C83#12111yescharacteristic
107C4×C83#12111yescharacteristic
108Dic83—42no—
109Dic83—42no—
110C2×C83—21no—
111C2×C83—21no—
112C2×C83—21no—
113C2×C83—81no—
114C2×C83#38611yescharacteristic
115D83—44no—
116#1911—831nomaximal, Sylow
117C83#191111yescharacteristic, complemented, Sylow
118#535—831no—
119#518—831no—
120#421—831no—
121#421D8311yescharacteristic, complemented
122C2×#100—831no—
123H(3,ℤ/4)—831no—
124H(3,ℤ/4)—831no—
125C2×#41C2×D8311yescharacteristic
126C22×D4—831no—
127C2×M16—831no—
128C2×M16C2×D8311yescharacteristic
129#127—832no—
130#124—832no—
131#121—1662no—
132#120—1662no—
133C4×D4—1662no—
134C4×D4—1662no—
135#117—831no—
136#117—831no—
137#117C2×D8311yescharacteristic
138C2×C22⋊C4—1662no—
139#104—22no—
140#104—1662no—
141#100—1662no—
142C24—832no—
143#41—21no—
144#41—21no—
145#41—21no—
146#41—21no—
147C2×Q8C4×D8311yescharacteristic, complemented
148C2×D4—21no—
149C2×D4—832no—
150C2×D4—832no—
151C2×D4—1664no—
152C2×D4—1661no—
153C2×D4—1661no—
154C2×D4—1661no—
155C2×D4—1661no—
156C2×D4C4×D8311yescharacteristic, complemented
157C22×C4—21no—
158C22×C4—1664no—
159C22×C4C22×D8311yescharacteristic
160M16—21no—
161M16—21no—
162M16—1661no—
163M16—1661no—
164C2×C8—21no—
165C2×C8—1661no—
166C4⋊C4—21no—
167C4⋊C4—1661no—
168C4⋊C4—1661no—
169C22⋊C4—21no—
170C22⋊C4—1664no—
171C22⋊C4—1664no—
172C22⋊C4—1661no—
173C22⋊C4—1661no—
174C42—832no—
175C42—832no—
176C42C22⋊D8312yescomplemented
177C23—21no—
178C23—834no—
179C23—6642no—
180C23—1662no—
181C23—1662no—
182C23D4×D8311yescharacteristic
183Q8—21no—
184Q8—21no—
185D4—21no—
186D4—21no—
187D4—41no—
188D4—41no—
189D4—6644no—
190D4—1662no—
191D4—1662no—
192D4—1662no—
193D4—1662no—
194C2×C4—22no—
195C2×C4—21no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—6644no—
199C2×C4—1662no—
200C2×C4—1662no—
201C2×C4C2×C4×D8311yescharacteristic
202C2×C4D4×D8311yescharacteristic
203C2×C4D4×D8311yescharacteristic
204C2×C4C2×C22⋊D8311yescharacteristic
205C2×C4#1305058911yescharacteristic
206C2×C4#1305059011yescharacteristic
207C8—41no—
208C8—3321no—
209C22—41no—
210C22—42no—
211C22—41no—
212C22—3324no—
213C22—3324no—
214C22—6642no—
215C22—1664no—
216C22C22⋊C4×D8311yescharacteristic
217C22#1458176011yescharacteristic
218C22#1458177411yescharacteristic
219C4—42no—
220C4—41no—
221C4—3322no—
222C4—3322no—
223C4(C22⋊D83)⋊C411yescharacteristic
224C4C4×C22⋊D8311yescharacteristic
225C4#1458176311yescharacteristic
226C4#1458176511yescharacteristic
227C2—21no—
228C2—21no—
229C2—21no—
230C2—81no—
231C2—3324no—
232C2#1912757711yescharacteristic
subgroupG/H
centerC2#19127577
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#421×C83C2
socleC2×C83#386
prime radical#19128265C2
cosocleC22×C4×C83C23
hypercenter#421D83
hypocenterC83#1911
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. #19128175C2semidirect
3. #19127718C2semidirect
4. C2×#14581708C2semidirect
5. #19127458C2semidirect
6. #19127457C2semidirect
7. #421×C83C2semidirect
10. #14581788C4semidirect
11. #14581781C4semidirect
43. C4×Dic83C2×C4semidirect
44. C4×Dic83D4semidirect
49. C2×Q8×C83C2×C4semidirect
51. C2×D4×C83C2×C4semidirect
59. C42×C83D4semidirect
117. C83#1911semidirect
121. #421D83semidirect
147. C2×Q8C4×D83semidirect
156. C2×D4C4×D83semidirect
176. C42C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
#191281752
#191277182
C2×#145817082
#191274582
#191274572
#421×C832
#191183