Finite Group Databasebeta

← #24918924

Subgroups — #24918924

Order 10624.

346 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128322C212yesmaximal, complemented
2#19127997C212yesmaximal, complemented
3#19127982C212yesmaximal, complemented
4#19127947C212yesmaximal, complemented
5#19127942C211yesmaximal, characteristic, complemented
6#19127937C211yesmaximal, characteristic
7#19127924C212yesmaximal, complemented
8#19127917C212yesmaximal, complemented
9#424×C83C211yesmaximal, characteristic, complemented
10C2×Dic332C2211yescharacteristic
11C2×#13050579C2212yes—
12C2×D664C2211yescharacteristic
13#14581792C2212yes—
14#14581787—22no—
15#14581780C2212yes—
16#14581778—22no—
17#14581774—22no—
18#14581771C2212yes—
19#14581770—22no—
20#14581766C2212yes—
21C4×Dic166C2212yes—
22C4×D332C2212yes—
23#14581726C2211yescharacteristic
24#14581726C2211yescharacteristic
25#14581726C2212yes—
26#14581717C2212yes—
27#14581717C2212yes—
28#14581714C2212yes—
29#14581714C2212yes—
30C2×M16×C83C2211yescharacteristic
31C22×C8×C83C2211yescharacteristic
32#117×C83C2211yescharacteristic
33C4⋊C8×C83C2212yes—
34C4⋊C8×C83C2212yes—
35C2×Dic166—22no—
36C2×Dic166C2312yes—
37C2×D332C2312yes—
38C2×C22⋊D83—22no—
39C2×C4×D83—22no—
40Dic332—41no—
41#13050579—42no—
42D664—41no—
43C4⋊Dic83—22no—
44C4⋊Dic83C2312yes—
45C4⋊Dic83C2312yes—
46Dic83⋊C4—22no—
47Dic83⋊C4—22no—
48C22⋊Dic83—22no—
49(C2×D83)⋊C4—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C22×C4×C83C2311yescharacteristic
55M16×C83—41no—
56C2×C8×C83—41no—
57C2×C8×C83C2311yescharacteristic
58C2×C8×C83C2312yes—
59C2×C8×C83C2311yescharacteristic
60C4⋊C4×C83D411yescharacteristic
61C4⋊C4×C83D411yescharacteristic
62C22⋊C4×C83D412yes—
63C42×C83C2312yes—
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83C2×D412yes—
69C2×C4×C83C2×D412yes—
70C2×C4×C83C2×D411yescharacteristic
71C2×C4×C83C2411yescharacteristic
72C22⋊D83—82no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76D332—22no—
77D332—22no—
78C4×D83—82no—
79Dic166—22no—
80Dic166—22no—
81Dic166—82no—
82C8×C83—21no—
83C8×C83—21no—
84C8×C83—22no—
85C22×C83—21no—
86C22×C83—21no—
87C22×C83—21no—
88C22×C83C22×D411yescharacteristic
89C2×D83—42no—
90C2×D83—42no—
91C2×D83—42no—
92C4×C83—21no—
93C4×C83—22no—
94C4×C83—22no—
95C4×C83H(5,𝔽2)11yescharacteristic
96C4×C832−1+411yescharacteristic
97Dic83—42no—
98Dic83—42no—
99Dic83—42no—
100C2×C83—41no—
101C2×C83#53711yescharacteristic
102C2×C83#57611yescharacteristic
103C2×C83#57811yescharacteristic
104D83—82no—
105#3262—831nomaximal, Sylow
106C83#326211yescharacteristic, complemented, Sylow
107#542—832no—
108#477—832no—
109#474—832no—
110#468—832no—
111#467—831no—
112#466—831no—
113#464—832no—
114#463—832no—
115#424D8311yescharacteristic, complemented
116C2×Q16—831no—
117C2×SD16—832no—
118C2×D8—831no—
119C2×M16C2×D8311yescharacteristic
120C22×C8C2×D8311yescharacteristic
121C4⋊Q8—832no—
122#126—1662no—
123#124—832no—
124#124—1662no—
125#123—1662no—
126C22⋊Q8—832no—
127C22⋊Q8—1662no—
128#121—832no—
129C4×Q8—832no—
130C4×D4—832no—
131#117C2×D8311yescharacteristic
132#107—831no—
133#107—831no—
134#107—832no—
135C4⋊C8C2×D8312yes—
136C4⋊C8C2×D8312yes—
137#103—832no—
138#103—832no—
139#102—832no—
140#102—832no—
141C2×Q8—832no—
142C2×Q8—1662no—
143C2×D4—832no—
144C2×D4—1662no—
145C22×C4—1662no—
146C22×C4C22×D8311yescharacteristic
147Q16—3321no—
148SD16—3322no—
149D8—3321no—
150M16—41no—
151C2×C8—41no—
152C2×C8C22×D8311yescharacteristic
153C2×C8C22×D8312yes—
154C2×C8C22×D8311yescharacteristic
155C4⋊C4—832no—
156C4⋊C4—832no—
157C4⋊C4—1662no—
158C4⋊C4—1662no—
159C4⋊C4—1662no—
160C4⋊C4D33211yescharacteristic
161C4⋊C4D33211yescharacteristic
162C22⋊C4—1662no—
163C22⋊C4—1662no—
164C22⋊C4—1662no—
165C22⋊C4—1662no—
166C22⋊C4—1662no—
167C22⋊C4D33212yes—
168C42—1662no—
169C42C22×D8312yes—
170C23—1662no—
171C23C2×D33211yescharacteristic
172Q8—6642no—
173Q8—1662no—
174Q8—1662no—
175D4—6642no—
176D4—1662no—
177D4—1662no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—6642no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4C2×D33212yes—
185C2×C4C2×D33212yes—
186C2×C4C2×D33211yescharacteristic
187C2×C4C23×D8311yescharacteristic
188C8—21no—
189C8—21no—
190C8—22no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—3322no—
195C22—3322no—
196C22—3322no—
197C22C22×D33211yescharacteristic
198C4—22no—
199C4—22no—
200C4—21no—
201C4—3322no—
202C4—3322no—
203C4—3322no—
204C4#1458183711yescharacteristic
205C4#1458184011yescharacteristic
206C2—41no—
207C2—6642no—
208C2#1912828211yescharacteristic
209C2#1912850611yescharacteristic
210C2#1912851811yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#424×C83C2
socleC22×C83C22×D4
prime radical#19127937C2
hypercenter#424D83
hypocenterC83#3262
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #19128322C2semidirect
2. #19127997C2semidirect
3. #19127982C2semidirect
4. #19127947C2semidirect
5. #19127942C2semidirect
7. #19127924C2semidirect
8. #19127917C2semidirect
9. #424×C83C2semidirect
106. C83#3262semidirect
115. #424D83semidirect

Maximal subgroups

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

subgroupindex
#191283222
#191279972
#191279822
#191279472
#191279422
#191279372
#191279242
#191279172
#424×C832
#326283