Finite Group Databasebeta

← #24918120

Subgroups — #24918120

Order 10624.

338 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 115 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#19128309C212yesmaximal
2#19128101C211yesmaximal, characteristic, complemented
3#19128079C211yesmaximal, characteristic, complemented
4#19128068C212yesmaximal, complemented
5#19127960C211yesmaximal, characteristic
6#19127956C211yesmaximal, characteristic
7#19127941C212yesmaximal
8#19127809C211yesmaximal, characteristic, complemented
9C4×#13050580C212yesmaximal, complemented
10#19127658C211yesmaximal, characteristic
11#441×C83C211yesmaximal, characteristic, complemented
12C2×Dic332C2211yescharacteristic
13C2×#13050580C2212yes—
14C2×#13050573C2211yescharacteristic
15C2×C8×D83C2211yescharacteristic
16#14581790C2212yes—
17#14581787—22no—
18#14581786—22no—
19#14581784—22no—
20#14581783C2212yes—
21#14581782—22no—
22#14581772C2212yes—
23#14581768C2212yes—
24C4×Dic166C2212yes—
25#14581749C2211yescharacteristic
26#14581725C2212yes—
27#14581724C2211yescharacteristic
28#14581717C2212yes—
29#14581716C2212yes—
30#14581715C2212yes—
31#14581702C2211yescharacteristic
32C4×C83⋊C8C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×Q16×C83C2211yescharacteristic
35C4×Q8×C83C2212yes—
36#106×C83C2211yescharacteristic
37#103×C83C2212yes—
38#97×C83C2211yescharacteristic
39C2×Dic166C2312yes—
40C2×C4×D83C2311yescharacteristic
41Dic332—22no—
42#13050580—24no—
43#13050573—22no—
44C2×C83⋊C8C2311yescharacteristic
45C2×C83⋊C8C2311yescharacteristic
46C8×D83—22no—
47C4⋊Dic83—22no—
48C4⋊Dic83—22no—
49C4⋊Dic83C2312yes—
50Dic83⋊C4—22no—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55Dic83⋊C4D412yes—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58(C2×D83)⋊C4D412yes—
59C4×Dic83—22no—
60C4×Dic83C2311yescharacteristic
61C2×Q8×C83C2312yes—
62Q16×C83—22no—
63C2×C8×C83C2311yescharacteristic
64C2×C8×C83C2311yescharacteristic
65C4⋊C4×C83—22no—
66C4⋊C4×C83C2312yes—
67C42×C83—22no—
68C42×C83C2311yescharacteristic
69C22×D83C2×D411yescharacteristic
70Q8×C83—22no—
71Q8×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83—22no—
74C2×C4×C83C2×D412yes—
75C2×C4×C83C2411yescharacteristic
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83C2×D412yes—
79C2×Dic83C2×D411yescharacteristic
80C4×D83—21no—
81C4×D83—21no—
82Dic166—22no—
83Dic166—22no—
84C8×C83—21no—
85C8×C83#4112yes—
86C83⋊C8—21no—
87C83⋊C8#4112yes—
88C22×C83C22×D411yescharacteristic
89C2×D83—21no—
90C2×D83—22no—
91C4×C83—22no—
92C4×C83—42no—
93C4×C83—42no—
94C4×C83C2×#4111yescharacteristic
95C4×C83C2×#4111yescharacteristic
96Dic83—21no—
97Dic83—22no—
98Dic83—42no—
99Dic83—42no—
100C2×C83#53211yescharacteristic
101C2×C83#57812yes—
102D83—41no—
103#3177—831nomaximal, Sylow
104C83#317711yescharacteristic, complemented, Sylow
105#541—832no—
106#499—831no—
107#494—831no—
108#491—832no—
109#470—831no—
110#470—831no—
111#467—832no—
112#441—831no—
113#441D8311yescharacteristic, complemented
114C4×Q16—832no—
115#405—831no—
116C2×Q16—832no—
117C2×Q16—831no—
118C2×Q16C2×D8311yescharacteristic
119C2×M16—831no—
120C22×C8—831no—
121C4⋊Q8—832no—
122#126—1662no—
123#126—1662no—
124#125—832no—
125#125—1662no—
126#125—1662no—
127C22⋊Q8—832no—
128C22⋊Q8—832no—
129C4×Q8—832no—
130C4×Q8C2×D8312yes—
131#117—831no—
132#107—832no—
133#106—831no—
134#106C2×D8311yescharacteristic
135#103—832no—
136#103—832no—
137#103—832no—
138#103C2×D8312yes—
139#97—831no—
140#97C2×D8311yescharacteristic
141C4×C8—831no—
142C4×C8—831no—
143C2×Q8—832no—
144C2×Q8C22×D8312yes—
145C22×C4—831no—
146Q16—22no—
147Q16—1664no—
148Q16—1662no—
149M16—1662no—
150C2×C8—831no—
151C2×C8—831no—
152C2×C8—1662no—
153C2×C8C22×D8311yescharacteristic
154C2×C8C22×D8311yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—832no—
157C4⋊C4—832no—
158C4⋊C4—1662no—
159C4⋊C4—1662no—
160C4⋊C4—1662no—
161C4⋊C4—1662no—
162C4⋊C4—1662no—
163C4⋊C4—1662no—
164C4⋊C4—1662no—
165C4⋊C4C22×D8312yes—
166C22⋊C4—832no—
167C22⋊C4—1662no—
168C22⋊C4—1662no—
169C42—22no—
170C42—831no—
171C42—1662no—
172C42C22×D8311yescharacteristic
173C23—831no—
174Q8—22no—
175Q8—22no—
176Q8—1662no—
177Q8—1662no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—832no—
181C2×C4—831no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4—1661no—
185C2×C4—1661no—
186C2×C4D4×D8312yes—
187C2×C4C23×D8311yescharacteristic
188C8—21no—
189C8—832no—
190C8—1661no—
191C8#1305059012yes—
192C22—1661no—
193C22—1662no—
194C22C2×D4×D8311yescharacteristic
195C4—22no—
196C4—42no—
197C4—42no—
198C4—3322no—
199C4—3322no—
200C4—1661no—
201C4—1662no—
202C4#41×D8311yescharacteristic
203C4C2×#1305059011yescharacteristic
204C2—3321no—
205C2#1912824811yescharacteristic
206C2#1912851712yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#441×C83C2
socleC22×C83C22×D4
prime radicalC22×D83C2×D4
hypercenter#441D83
hypocenterC83#3177
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #441

Complements

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

NH ≅ G/Ntype
2. #19128101C2semidirect
3. #19128079C2semidirect
4. #19128068C2semidirect
8. #19127809C2semidirect
9. C4×#13050580C2semidirect
11. #441×C83C2semidirect
104. C83#3177semidirect
113. #441D83semidirect

Maximal subgroups

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

subgroupindex
#191283092
#191281012
#191280792
#191280682
#191279602
#191279562
#191279412
#191278092
C4×#130505802
#191276582
#441×C832
#317783