Finite Group Databasebeta

← #24918909

Subgroups — #24918909

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#19128313C211yesmaximal, characteristic, complemented
2#19128005C212yesmaximal, complemented
3#19128001C212yesmaximal, complemented
4#19127945C212yesmaximal, complemented
5#19127928C212yesmaximal, complemented
6#19127906C212yesmaximal, complemented
7#19127898C212yesmaximal, complemented
8#19127728C211yesmaximal, characteristic, complemented
9#540×C83C211yesmaximal, characteristic
10C2×#13050578C2212yes—
11C2×#13050578C2212yes—
12C2×#13050574C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14#14581792C2211yescharacteristic
15#14581785—22no—
16#14581784C2211yescharacteristic
17#14581782—22no—
18#14581771C2212yes—
19C4×Dic166C2212yes—
20#14581725C2212yes—
21#14581723C2212yes—
22C4⋊(C83⋊C8)C2212yes—
23C4⋊(C83⋊C8)C2212yes—
24#14581715C2212yes—
25#14581715C2212yes—
26#14581713C2212yes—
27#14581713C2212yes—
28#127×C83C2211yescharacteristic
29#125×C83C2211yescharacteristic
30#123×C83—22no—
31#121×C83—22no—
32#121×C83C2212yes—
33C4×D4×C83C2212yes—
34#117×C83C2211yescharacteristic
35C2×Dic166C2312yes—
36#13050578—42no—
37#13050578—42no—
38#13050574—41no—
39C2×C83⋊C8—41no—
40C2×C83⋊C8C2312yes—
41C2×C83⋊C8C2312yes—
42C4⋊Dic83—22no—
43C4⋊Dic83C2312yes—
44Dic83⋊C4—22no—
45Dic83⋊C4—22no—
46Dic83⋊C4—22no—
47C22⋊Dic83—22no—
48C4×Dic83—22no—
49C2×D4×C83—22no—
50C2×D4×C83—22no—
51C2×D4×C83C2312yes—
52C22×C4×C83—22no—
53C22×C4×C83C2311yescharacteristic
54C4⋊C4×C83—22no—
55C4⋊C4×C83D412yes—
56C4⋊C4×C83C2312yes—
57C22⋊C4×C83—22no—
58C22⋊C4×C83—22no—
59C22⋊C4×C83D412yes—
60C42×C83C2311yescharacteristic
61C42×C83C2311yescharacteristic
62C23×C83—22no—
63C23×C83C2×D411yescharacteristic
64D4×C83—22no—
65D4×C83—22no—
66D4×C83—82no—
67D4×C83—82no—
68C2×C4×C83—21no—
69C2×C4×C83—21no—
70C2×C4×C83—22no—
71C2×C4×C83—82no—
72C2×C4×C83C2×D412yes—
73C2×C4×C83C2×D412yes—
74C2×C4×C83C2×D411yescharacteristic
75C2×C4×C83C2411yescharacteristic
76C2×Dic83—22no—
77C2×Dic83—22no—
78Dic166—22no—
79Dic166—22no—
80C83⋊C8—22no—
81C83⋊C8—22no—
82C22×C83—21no—
83C22×C83—22no—
84C22×C83—42no—
85C22×C83—42no—
86C22×C83—42no—
87C22×C83C22×D411yescharacteristic
88C4×C83—21no—
89C4×C83—22no—
90C4×C83—22no—
91C4×C83—42no—
92C4×C83H(5,𝔽2)11yescharacteristic
93C4×C832−1+411yescharacteristic
94Dic83—42no—
95Dic83—42no—
96C2×C83—41no—
97C2×C83—82no—
98C2×C83#53711yescharacteristic
99C2×C83#57712yes—
100#3260—831nomaximal, Sylow
101C83#326011yescharacteristic, complemented, Sylow
102#541—831no—
103#540D8311yescharacteristic
104#479—832no—
105#478—832no—
106#468—832no—
107#465—832no—
108#461—832no—
109#460—832no—
110#424—831no—
111C2×SD16—832no—
112C2×SD16—832no—
113C2×M16—831no—
114C22×C8—831no—
115C4⋊Q8—831no—
116#127C2×D8311yescharacteristic
117#126—1662no—
118#125—831no—
119#125—1662no—
120#125C2×D8311yescharacteristic
121#123—22no—
122C22⋊Q8—832no—
123#121—22no—
124#121C2×D8312yes—
125C4×Q8—832no—
126C4×D4C2×D8312yes—
127#117C2×D8311yescharacteristic
128#107—832no—
129#106—832no—
130C4⋊C8—832no—
131C4⋊C8—832no—
132#103—832no—
133#103—832no—
134#102—832no—
135#102—832no—
136C2×Q8—832no—
137C2×D4—22no—
138C2×D4—22no—
139C2×D4C22×D8312yes—
140C22×C4—22no—
141C22×C4C22×D8311yescharacteristic
142SD16—3322no—
143SD16—3322no—
144M16—3321no—
145C2×C8—3321no—
146C2×C8—832no—
147C2×C8—832no—
148C4⋊C4—22no—
149C4⋊C4—832no—
150C4⋊C4—1662no—
151C4⋊C4—1662no—
152C4⋊C4—1662no—
153C4⋊C4—1662no—
154C4⋊C4C22⋊D8312yes—
155C4⋊C4C22×D8312yes—
156C22⋊C4—22no—
157C22⋊C4—22no—
158C22⋊C4—1662no—
159C22⋊C4C22⋊D8312yes—
160C42—1662no—
161C42C22×D8311yescharacteristic
162C42C22×D8311yescharacteristic
163C23—22no—
164C23C2×C22⋊D8311yescharacteristic
165Q8—1662no—
166Q8—1662no—
167D4—22no—
168D4—22no—
169D4—82no—
170D4—82no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—82no—
175C2×C4—1662no—
176C2×C4—1662no—
177C2×C4C2×C22⋊D8312yes—
178C2×C4C2×C22⋊D8312yes—
179C2×C4C2×C22⋊D8311yescharacteristic
180C2×C4C23×D8311yescharacteristic
181C8—1662no—
182C8—1662no—
183C22—22no—
184C22—21no—
185C22—42no—
186C22—42no—
187C22—42no—
188C22C22×C22⋊D8311yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—3322no—
194C4—3322no—
195C4#1458183811yescharacteristic
196C4#1458183911yescharacteristic
197C2—41no—
198C2—82no—
199C2#1912827911yescharacteristic
200C2#1912851212yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting, prime radical#540×C83C2
socleC22×C83C22×D4
hypercenter#540D83
hypocenterC83#3260
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #540

Complements

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

NH ≅ G/Ntype
1. #19128313C2semidirect
2. #19128005C2semidirect
3. #19128001C2semidirect
4. #19127945C2semidirect
5. #19127928C2semidirect
6. #19127906C2semidirect
7. #19127898C2semidirect
8. #19127728C2semidirect
101. C83#3260semidirect

Maximal subgroups

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

subgroupindex
#191283132
#191280052
#191280012
#191279452
#191279282
#191279062
#191278982
#191277282
#540×C832
#326083