Finite Group Databasebeta

← #24918205

Subgroups — #24918205

Order 10624.

358 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 119 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#19128255C212yesmaximal
2#19128098C211yesmaximal, characteristic, complemented
3#19128079C211yesmaximal, characteristic, complemented
4#19128050C212yesmaximal, complemented
5#19127960C211yesmaximal, characteristic
6#19127958C212yesmaximal
7#19127956C211yesmaximal, characteristic
8#19127809C211yesmaximal, characteristic, complemented
9#19127806C212yesmaximal, complemented
10#19127653C211yesmaximal, characteristic
11#441×C83C211yesmaximal, characteristic, complemented
12C2×Dic332C2211yescharacteristic
13C2×#13050580C2212yes—
14C2×#13050573C2211yescharacteristic
15C2×#13050573C2211yescharacteristic
16#14581791C2212yes—
17#14581790C2212yes—
18#14581782—22no—
19#14581772C2212yes—
20#14581770—22no—
21#14581768C2212yes—
22C4×Dic166—22no—
23C4×Dic166C2212yes—
24#14581749—22no—
25C42×D83C2211yescharacteristic
26#14581724C2211yescharacteristic
27#14581723C2212yes—
28#14581717C2212yes—
29#14581716C2212yes—
30#14581715C2212yes—
31#14581703C2211yescharacteristic
32C8×Dic83C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×Q16×C83C2211yescharacteristic
35C4×Q8×C83C2212yes—
36#106×C83C2211yescharacteristic
37#103×C83C2212yes—
38#97×C83C2211yescharacteristic
39C2×Dic166—22no—
40C2×Dic166C2312yes—
41C2×C4×D83D412yes—
42C2×C4×D83C2311yescharacteristic
43Dic332—22no—
44#13050580—24no—
45#13050573—22no—
46#13050573—22no—
47C2×C83⋊C8C2311yescharacteristic
48C2×C83⋊C8C2311yescharacteristic
49C4⋊Dic83—22no—
50C4⋊Dic83C2312yes—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4—22no—
55Dic83⋊C4—22no—
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4—22no—
58C4×Dic83—22no—
59C4×Dic83D412yes—
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×D411yescharacteristic
79C2×Dic83C2×D412yes—
80C4×D83—21no—
81C4×D83—21no—
82C4×D83—24no—
83Dic166—22no—
84Dic166—22no—
85Dic166—44no—
86C8×C83—21no—
87C8×C83#4112yes—
88C83⋊C8—21no—
89C83⋊C8#4112yes—
90C22×C83C22×D411yescharacteristic
91C2×D83—21no—
92C2×D83—22no—
93C4×C83—22no—
94C4×C83—42no—
95C4×C83—42no—
96C4×C83C2×#4111yescharacteristic
97C4×C83C2×#4111yescharacteristic
98Dic83—21no—
99Dic83—42no—
100Dic83—42no—
101Dic83#13714yes—
102C2×C83#53211yescharacteristic
103C2×C83C2×#13712yes—
104D83—41no—
105#3185—831nomaximal, Sylow
106C83#318511yescharacteristic, complemented, Sylow
107#533—832no—
108#498—831no—
109#494—831no—
110#487—832no—
111#470—831no—
112#470—831no—
113#470—832no—
114#441—832no—
115#441—831no—
116#441D8311yescharacteristic, complemented
117C4×M16—831no—
118C2×Q16—832no—
119C2×Q16—831no—
120C2×Q16C2×D8311yescharacteristic
121C2×M16—831no—
122C2×M16—831no—
123C4⋊Q8—832no—
124C4⋊Q8—832no—
125#125—1662no—
126C22⋊Q8—832no—
127C22⋊Q8—832no—
128C22⋊Q8—1662no—
129C4×Q8—832no—
130C4×Q8—1662no—
131C4×Q8C2×D8312yes—
132#117—1662no—
133C2×C42—831no—
134#106—831no—
135#106—832no—
136#106C2×D8311yescharacteristic
137#103—832no—
138#103—832no—
139#103—832no—
140#103C2×D8312yes—
141#97—831no—
142#97C2×D8311yescharacteristic
143C4×C8—831no—
144C4×C8—831no—
145C2×Q8—832no—
146C2×Q8—1662no—
147C2×Q8C22×D8312yes—
148C22×C4—832no—
149C22×C4—831no—
150Q16—22no—
151Q16—1664no—
152Q16—1662no—
153M16—1662no—
154M16—1662no—
155C2×C8—831no—
156C2×C8—831no—
157C2×C8C22×D8311yescharacteristic
158C2×C8C22×D8311yescharacteristic
159C4⋊C4—22no—
160C4⋊C4—832no—
161C4⋊C4—1662no—
162C4⋊C4—1662no—
163C4⋊C4—1662no—
164C4⋊C4—1662no—
165C4⋊C4—1662no—
166C4⋊C4—1662no—
167C4⋊C4C22×D8312yes—
168C22⋊C4—1662no—
169C22⋊C4—1662no—
170C42—22no—
171C42—832no—
172C42—831no—
173C42—1662no—
174C42C22×D8311yescharacteristic
175C23—831no—
176Q8—22no—
177Q8—22no—
178Q8—3324no—
179Q8—1662no—
180Q8—1662no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—832no—
184C2×C4—831no—
185C2×C4—1664no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1661no—
189C2×C4—1661no—
190C2×C4D4×D8312yes—
191C2×C4C23×D8311yescharacteristic
192C8—21no—
193C8—832no—
194C8—1661no—
195C8#1305059012yes—
196C22—1662no—
197C22—1661no—
198C22C2×D4×D8311yescharacteristic
199C4—22no—
200C4—42no—
201C4—42no—
202C4—3322no—
203C4—3322no—
204C4—834no—
205C4—1661no—
206C4#41×D8311yescharacteristic
207C4C2×#1305059011yescharacteristic
208C2—3321no—
209C2#1912824811yescharacteristic
210C2#137×D8312yes—
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#3185
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. #19128098C2semidirect
3. #19128079C2semidirect
4. #19128050C2semidirect
8. #19127809C2semidirect
9. #19127806C2semidirect
11. #441×C83C2semidirect
106. C83#3185semidirect
116. #441D83semidirect

Maximal subgroups

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

subgroupindex
#191282552
#191280982
#191280792
#191280502
#191279602
#191279582
#191279562
#191278092
#191278062
#191276532
#441×C832
#318583