Finite Group Databasebeta

← #24918864

Subgroups — #24918864

Order 10624.

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

normal subgroups only

This group has 354 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#19128235C212yesmaximal, complemented
2#19128009C212yesmaximal, complemented
3#19128002C212yesmaximal, complemented
4#19127959C211yesmaximal, characteristic, complemented
5#19127954C211yesmaximal, characteristic
6#19127947C212yesmaximal, complemented
7#19127924C212yesmaximal, complemented
8#19127917C212yesmaximal, complemented
9#423×C83C211yesmaximal, characteristic, complemented
10C2×Dic332C2211yescharacteristic
11C2×#13050579C2212yes—
12C2×D664C2211yescharacteristic
13#14581786—22no—
14#14581784C2212yes—
15#14581780C2212yes—
16#14581775—22no—
17#14581771C2212yes—
18#14581766C2212yes—
19C4×Dic166C2212yes—
20C4×D332C2212yes—
21C4×C22⋊D83—22no—
22#14581749—22no—
23#14581726C2212yes—
24#14581724C2211yescharacteristic
25#14581724C2211yescharacteristic
26#14581717C2212yes—
27#14581717C2212yes—
28#14581714C2212yes—
29#14581714C2212yes—
30C2×M16×C83C2211yescharacteristic
31C2×M16×C83C2211yescharacteristic
32C2×C42×C83C2211yescharacteristic
33C4⋊C8×C83C2212yes—
34C4⋊C8×C83C2212yes—
35C2×Dic166C2312yes—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39Dic332—41no—
40#13050579—42no—
41D664—41no—
42C4⋊Dic83C2312yes—
43C4⋊Dic83C2312yes—
44Dic83⋊C4—22no—
45Dic83⋊C4—22no—
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×C83D412yes—
55C22×C4×C83C2311yescharacteristic
56M16×C83—41no—
57M16×C83—41no—
58C2×C8×C83C2311yescharacteristic
59C2×C8×C83C2312yes—
60C2×C8×C83C2311yescharacteristic
61C42×C83D411yescharacteristic
62C42×C83D411yescharacteristic
63C42×C83C2312yes—
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—42no—
69C2×C4×C83C2×D411yescharacteristic
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2411yescharacteristic
73C22⋊D83—44no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77D332—22no—
78D332—22no—
79C4×D83—44no—
80Dic166—22no—
81Dic166—22no—
82C8×C83—21no—
83C8×C83—22no—
84C8×C83—21no—
85C22×C83—21no—
86C22×C83—22no—
87C22×C83C22×D411yescharacteristic
88C2×D83—42no—
89C2×D83—42no—
90C2×D83—42no—
91C4×C83—21no—
92C4×C83—22no—
93C4×C83—22no—
94C4×C83H(5,𝔽2)11yescharacteristic
95C4×C832−1+411yescharacteristic
96Dic83—42no—
97Dic83—42no—
98Dic83—42no—
99C2×C83—41no—
100C2×C83#53711yescharacteristic
101C2×C83#57512yes—
102D83—82no—
103#3256—831nomaximal, Sylow
104C83#325611yescharacteristic, complemented, Sylow
105#529—832no—
106#479—832no—
107#478—832no—
108#470—831no—
109#469—831no—
110#468—832no—
111#464—832no—
112#463—832no—
113#423D8311yescharacteristic, complemented
114C2×Q16—831no—
115C2×SD16—832no—
116C2×D8—831no—
117C2×M16C2×D8311yescharacteristic
118C2×M16C2×D8311yescharacteristic
119#126—1662no—
120#125—832no—
121#124—832no—
122#123—1662no—
123C22⋊Q8—832no—
124#121—832no—
125C4×Q8—832no—
126C4×D4—832no—
127C4×D4—1662no—
128#117—1662no—
129C2×C42C2×D8311yescharacteristic
130#107—832no—
131#106—831no—
132#106—831no—
133C4⋊C8C2×D8312yes—
134C4⋊C8C2×D8312yes—
135#103—832no—
136#103—832no—
137#102—832no—
138#102—832no—
139C2×Q8—832no—
140C2×D4—832no—
141C2×D4—1662no—
142C22×C4—1662no—
143C22×C4D33212yes—
144C22×C4C22×D8311yescharacteristic
145Q16—3321no—
146SD16—3322no—
147D8—3321no—
148M16—41no—
149M16—41no—
150C2×C8C22×D8311yescharacteristic
151C2×C8C22×D8312yes—
152C2×C8C22×D8311yescharacteristic
153C4⋊C4—832no—
154C4⋊C4—832no—
155C4⋊C4—1662no—
156C4⋊C4—1662no—
157C4⋊C4—1662no—
158C4⋊C4—1662no—
159C22⋊C4—1662no—
160C22⋊C4—1662no—
161C22⋊C4—1662no—
162C22⋊C4—1662no—
163C22⋊C4—1662no—
164C42—1662no—
165C42D33211yescharacteristic
166C42D33211yescharacteristic
167C42C22×D8312yes—
168C23—1662no—
169C23C2×D33211yescharacteristic
170Q8—1662no—
171Q8—1662no—
172D4—3324no—
173D4—1662no—
174D4—1662no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—42no—
178C2×C4—3324no—
179C2×C4—1662no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4C2×D33212yes—
183C2×C4C2×D33211yescharacteristic
184C2×C4C2×D33212yes—
185C2×C4C23×D8311yescharacteristic
186C8—21no—
187C8—22no—
188C8—21no—
189C22—22no—
190C22—21no—
191C22—3322no—
192C22—3322no—
193C22—3322no—
194C22C22×D33211yescharacteristic
195C4—22no—
196C4—22no—
197C4—21no—
198C4—3322no—
199C4—3322no—
200C4—3322no—
201C4#1458183711yescharacteristic
202C4#1458184011yescharacteristic
203C2—41no—
204C2—6642no—
205C2#1912828211yescharacteristic
206C2#1912850112yes—
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#423×C83C2
socleC22×C83C22×D4
prime radical#19127954C2
hypercenter#423D83
hypocenterC83#3256
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #423

Complements

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

NH ≅ G/Ntype
1. #19128235C2semidirect
2. #19128009C2semidirect
3. #19128002C2semidirect
4. #19127959C2semidirect
6. #19127947C2semidirect
7. #19127924C2semidirect
8. #19127917C2semidirect
9. #423×C83C2semidirect
104. C83#3256semidirect
113. #423D83semidirect

Maximal subgroups

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

subgroupindex
#191282352
#191280092
#191280022
#191279592
#191279542
#191279472
#191279242
#191279172
#423×C832
#325683