Finite Group Databasebeta

← #24920554

Subgroups — #24920554

Order 10624.

380 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 113 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#19128433C212yesmaximal, complemented
2#19128087C212yesmaximal, complemented
3#19127925C212yesmaximal, complemented
4#19127913C212yesmaximal, complemented
5#19127894C212yesmaximal, complemented
6#19127831C211yesmaximal, characteristic, complemented
7#19127802C212yesmaximal, complemented
8C4×D664C211yesmaximal, characteristic, complemented
9#495×C83C211yesmaximal, characteristic, complemented
10C2×#13050577C2212yes—
11C2×D664C2211yescharacteristic
12C2×#13050575C2212yes—
13#14581788C2212yes—
14#14581781C2212yes—
15#14581778—22no—
16#14581776—22no—
17#14581774—22no—
18Q8×Dic83C2212yes—
19C4×Dic166C2211yescharacteristic
20C4×D332C2212yes—
21D4×Dic83C2212yes—
22(C22⋊D83)⋊C4—22no—
23#14581726C2211yescharacteristic
24C4⋊(C83⋊C8)C2211yescharacteristic
25Dic83⋊C8C2212yes—
26#14581716C2212yes—
27#14581714C2212yes—
28#14581713C2212yes—
29#14581702C2212yes—
30C2×Q16×C83C2211yescharacteristic
31C2×SD16×C83C2212yes—
32C2×D8×C83C2211yescharacteristic
33#124×C83C2212yes—
34C4×C8×C83C2211yescharacteristic
35C2×Dic166D411yescharacteristic
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C22⋊D83—22no—
39C22×Dic83—22no—
40C2×C4×D83—22no—
41#13050577—42no—
42D664—22no—
43#13050575—42no—
44C2×C83⋊C8C2312yes—
45C4⋊Dic83—22no—
46C4⋊Dic83D411yescharacteristic
47C4⋊Dic83C2312yes—
48Dic83⋊C4D412yes—
49C22⋊Dic83—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C4×Dic83C2312yes—
55C2×Q8×C83C2312yes—
56C2×D4×C83C2312yes—
57Q16×C83—41no—
58SD16×C83—24no—
59D8×C83—41no—
60C2×C8×C83C2311yescharacteristic
61C2×C8×C83C2311yescharacteristic
62C22⋊C4×C83—22no—
63C22⋊C4×C83—22no—
64C42×C83C2311yescharacteristic
65C23×C83—22no—
66C22×D83—22no—
67Q8×C83—22no—
68Q8×C83—22no—
69D4×C83—22no—
70D4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×D411yescharacteristic
74C2×C4×C83C2411yescharacteristic
75C22⋊D83—44no—
76C22⋊D83—82no—
77C2×Dic83—22no—
78C2×Dic83—82no—
79C2×Dic83C2×D412yes—
80C2×Dic83C2×D412yes—
81D332—22no—
82D332—22no—
83C4×D83—82no—
84Dic166—41no—
85C8×C83—21no—
86C8×C83#4112yes—
87C83⋊C8—22no—
88C22×C83—42no—
89C22×C83—42no—
90C22×C83—42no—
91C22×C83C22×D411yescharacteristic
92C2×D83—42no—
93C2×D83—42no—
94C2×D83—42no—
95C4×C83—21no—
96C4×C83—21no—
97C4×C83—42no—
98C4×C83C2×#4111yescharacteristic
99C4×C83H(5,𝔽2)11yescharacteristic
100Dic83—22no—
101Dic83—22no—
102Dic83—42no—
103C2×C83—82no—
104C2×C83#55011yescharacteristic
105C2×C83#57511yescharacteristic
106C2×C83#57611yescharacteristic
107D83—82no—
108#3403—831nomaximal, Sylow
109C83#340311yescharacteristic, complemented, Sylow
110#559—832no—
111#496—832no—
112#495D8311yescharacteristic, complemented
113#464—832no—
114#463—832no—
115#459—832no—
116#446—831no—
117#440—832no—
118C4×D8—831no—
119C2×Q16C2×D8311yescharacteristic
120C2×SD16—832no—
121C2×SD16C2×D8312yes—
122C2×D8—832no—
123C2×D8—831no—
124C2×D8C2×D8311yescharacteristic
125#127—832no—
126#124—832no—
127#124—1662no—
128#124C2×D8312yes—
129#123—1662no—
130#123—1662no—
131C4×Q8—831no—
132C4×Q8—832no—
133C4×D4—832no—
134C4×D4—832no—
135C4×D4—1662no—
136#107—831no—
137C4⋊C8—832no—
138C4⋊C8—831no—
139#103—832no—
140#102—832no—
141#102—832no—
142#97—832no—
143C4×C8C2×D8311yescharacteristic
144C2×Q8—831no—
145C2×Q8C22×D8312yes—
146C2×D4—832no—
147C2×D4—1662no—
148C2×D4—1662no—
149C2×D4C22×D8312yes—
150C22×C4—1662no—
151C22×C4—1662no—
152Q16—41no—
153SD16—24no—
154SD16—3322no—
155D8—41no—
156D8—3322no—
157D8—1662no—
158C2×C8—832no—
159C2×C8C22×D8311yescharacteristic
160C2×C8C22×D8311yescharacteristic
161C4⋊C4—832no—
162C4⋊C4—831no—
163C4⋊C4—832no—
164C4⋊C4—1662no—
165C22⋊C4—22no—
166C22⋊C4—22no—
167C22⋊C4—1662no—
168C22⋊C4—1662no—
169C22⋊C4—1662no—
170C22⋊C4—1662no—
171C42—832no—
172C42—1662no—
173C42C22×D8311yescharacteristic
174C23—22no—
175C23—1662no—
176Q8—22no—
177Q8—22no—
178Q8—3321no—
179D4—22no—
180D4—22no—
181D4—3324no—
182D4—6642no—
183D4—1662no—
184D4—1662no—
185C2×C4—22no—
186C2×C4—832no—
187C2×C4—832no—
188C2×C4—6642no—
189C2×C4—6642no—
190C2×C4—1662no—
191C2×C4D4×D8311yescharacteristic
192C2×C4D4×D8311yescharacteristic
193C2×C4C23×D8311yescharacteristic
194C8—21no—
195C8—1662no—
196C8#1305058912yes—
197C22—42no—
198C22—42no—
199C22—42no—
200C22—3322no—
201C22—3322no—
202C22—3322no—
203C22C2×D4×D8311yescharacteristic
204C4—21no—
205C4—21no—
206C4—42no—
207C4—3322no—
208C4—1662no—
209C4—1662no—
210C4C2×#1305058911yescharacteristic
211C4#1458183711yescharacteristic
212C2—82no—
213C2—6642no—
214C2#1912837111yescharacteristic
215C2#1912849811yescharacteristic
216C2#1912850311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#495×C83C2
socleC22×C83C22×D4
hypercenter#495D83
hypocenterC83#3403
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #495

Complements

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

NH ≅ G/Ntype
1. #19128433C2semidirect
2. #19128087C2semidirect
3. #19127925C2semidirect
4. #19127913C2semidirect
5. #19127894C2semidirect
6. #19127831C2semidirect
7. #19127802C2semidirect
8. C4×D664C2semidirect
9. #495×C83C2semidirect
109. C83#3403semidirect
112. #495D83semidirect

Maximal subgroups

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

subgroupindex
#191284332
#191280872
#191279252
#191279132
#191278942
#191278312
#191278022
C4×D6642
#495×C832
#340383