Finite Group Databasebeta

← #24907982

Subgroups — #24907982

Order 10624.

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

normal subgroups only

This group has 312 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#19128467C211yesmaximal, characteristic
2#19128275C211yesmaximal, characteristic
3#19127710C211yesmaximal, characteristic, complemented
4#19127674C211yesmaximal, characteristic
5#19127458C211yesmaximal, characteristic, complemented
6#19127457C211yesmaximal, characteristic, complemented
7#419×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9#14581800—22no—
10#14581798—22no—
11C2×#13050574C2211yescharacteristic
12M16×D83—22no—
13#14581790C412yescomplemented
14#14581781C412yescomplemented
15#14581775—22no—
16#14581768—22no—
17#14581751C2211yescharacteristic
18#14581751C2211yescharacteristic
19#14581716—22no—
20#14581706—22no—
21C22×Q8×C83C2211yescharacteristic
22C2×M16×C83C2211yescharacteristic
23#117×C83C2211yescharacteristic
24#103×C83—22no—
25#13050590—21no—
26#13050590—21no—
27#13050590—21no—
28#13050590—21no—
29C2×Dic166C2×C411yescharacteristic
30C2×D332C2×C411yescharacteristic
31C2×C22⋊D83D412yes—
32C2×C4×D83D412yes—
33#13050574—21no—
34#13050574—21no—
35#13050573—24no—
36C2×C83⋊C8D412yes—
37C8×D83—24no—
38C4⋊Dic83—21no—
39C4⋊Dic83—21no—
40Dic83⋊C4—24no—
41C22⋊Dic83—21no—
42C22⋊Dic83—21no—
43(C2×D83)⋊C4—24no—
44C4×Dic83—21no—
45C4×Dic83C2×C412yes—
46C2×Q8×C83—21no—
47C2×Q8×C83—21no—
48C2×Q8×C83—21no—
49C2×Q8×C83—21no—
50C2×Q8×C83—21no—
51C2×Q8×C83C2×C412yes—
52C22×C4×C83—21no—
53C22×C4×C83C2311yescharacteristic
54M16×C83—21no—
55M16×C83—21no—
56C2×C8×C83D412yes—
57C4⋊C4×C83—21no—
58C22⋊C4×C83—21no—
59C42×C83—21no—
60C23×C83C2×D411yescharacteristic
61C22×D83C22⋊C412yes—
62Q8×C83—22no—
63Q8×C83—22no—
64Q8×C83—41no—
65Q8×C83—41no—
66C2×C4×C83—22no—
67C2×C4×C83—41no—
68C2×C4×C83—41no—
69C2×C4×C83—41no—
70C2×C4×C83C22×C411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83#4111yescharacteristic
75C2×C4×C83#4111yescharacteristic
76C22⋊D83—22no—
77C22⋊D83—22no—
78C2×Dic83—22no—
79C2×Dic83—41no—
80C2×Dic83C22⋊C412yes—
81D332—21no—
82D332—21no—
83C4×D83—22no—
84C4×D83—22no—
85Dic166—21no—
86Dic166—21no—
87C8×C83—22no—
88C83⋊C8—22no—
89C22×C83—41no—
90C22×C83C2×C22⋊C411yescharacteristic
91C22×C83#12111yescharacteristic
92C22×C83#12111yescharacteristic
93C2×D83—22no—
94C2×D83—42no—
95C4×C83—42no—
96C4×C83—81no—
97C4×C83C4×D411yescharacteristic
98C4×C83C4×D411yescharacteristic
99C4×C83#12011yescharacteristic
100C4×C83#12311yescharacteristic
101Dic83—22no—
102Dic83—42no—
103Dic83—81no—
104C2×C83—21no—
105C2×C83—21no—
106C2×C83—21no—
107C2×C83#38611yescharacteristic
108D83—42no—
109#1909—831nomaximal, Sylow
110C83#190911yescharacteristic, complemented, Sylow
111#568—831no—
112#536—831no—
113#419—831no—
114#419D8311yescharacteristic, complemented
115#410—831no—
116H(3,ℤ/4)—831no—
117H(3,ℤ/4)—831no—
118C2×#41—831no—
119C22×Q8C2×D8311yescharacteristic
120#131—1662no—
121#131—1662no—
122C2×M16—831no—
123C2×M16—1662no—
124C2×M16C2×D8311yescharacteristic
125C4⋊Q8—832no—
126#124—832no—
127#123—1662no—
128C22⋊Q8—1662no—
129#117—831no—
130#117—831no—
131#117C2×D8311yescharacteristic
132#103—22no—
133#103—1662no—
134#99—1662no—
135#41—1661no—
136#41—1661no—
137#41—1661no—
138#41—1661no—
139C2×Q8—21no—
140C2×Q8—21no—
141C2×Q8—21no—
142C2×Q8—21no—
143C2×Q8—21no—
144C2×Q8—831no—
145C2×Q8C4×D8312yes—
146C2×D4—832no—
147C2×D4—831no—
148C22×C4—21no—
149C22×C4—832no—
150C22×C4C22×D8311yescharacteristic
151M16—21no—
152M16—21no—
153M16—1664no—
154M16—1661no—
155M16—1661no—
156C2×C8—832no—
157C2×C8—1664no—
158C2×C8C22⋊D8312yes—
159C4⋊C4—21no—
160C4⋊C4—1664no—
161C4⋊C4—1661no—
162C4⋊C4—1661no—
163C22⋊C4—21no—
164C22⋊C4—1664no—
165C22⋊C4—1661no—
166C22⋊C4—1661no—
167C42—21no—
168C42—832no—
169C42—1661no—
170C23—832no—
171C23D4×D8311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—41no—
175Q8—41no—
176Q8—1661no—
177Q8—1661no—
178D4—1662no—
179D4—1662no—
180D4—1661no—
181D4—1661no—
182C2×C4—22no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—3321no—
187C2×C4—832no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4C2×C4×D8311yescharacteristic
192C2×C4D4×D8311yescharacteristic
193C2×C4D4×D8311yescharacteristic
194C2×C4C2×C22⋊D8311yescharacteristic
195C2×C4#1305058911yescharacteristic
196C2×C4#1305059011yescharacteristic
197C8—22no—
198C8—1662no—
199C22—41no—
200C22—3322no—
201C22—1662no—
202C22C22⋊C4×D8311yescharacteristic
203C22#1458176311yescharacteristic
204C22#1458176511yescharacteristic
205C4—42no—
206C4—81no—
207C4—3322no—
208C4—6641no—
209C4—1662no—
210C4(C22⋊D83)⋊C411yescharacteristic
211C4C4×C22⋊D8311yescharacteristic
212C4#1458176011yescharacteristic
213C4#1458177411yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—3322no—
218C2#1912757711yescharacteristic
subgroupG/H
centerC2#19127577
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#419×C83C2
socleC2×C83#386
prime radicalC2×#13050590C22
cosocleC22×C4×C83C23
hypercenter#419D83
hypocenterC83#1909
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
3. #19127710C2semidirect
5. #19127458C2semidirect
6. #19127457C2semidirect
7. #419×C83C2semidirect
13. #14581790C4semidirect
14. #14581781C4semidirect
110. C83#1909semidirect
114. #419D83semidirect

Maximal subgroups

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

subgroupindex
#191284672
#191282752
#191277102
#191276742
#191274582
#191274572
#419×C832
#190983