Finite Group Databasebeta

← #24918904

Subgroups — #24918904

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#19128306C211yesmaximal, characteristic
2#19128008C212yesmaximal, complemented
3#19127998C212yesmaximal, complemented
4#19127949C212yesmaximal, complemented
5#19127932C212yesmaximal, complemented
6#19127903C212yesmaximal, complemented
7#19127901C212yesmaximal, complemented
8#19127728C211yesmaximal, characteristic, complemented
9#541×C83C211yesmaximal, characteristic, complemented
10C2×#13050577C2212yes—
11C2×#13050577C2212yes—
12C2×#13050574C2211yescharacteristic
13C22×C83⋊C8C2211yescharacteristic
14#14581789C2211yescharacteristic
15#14581784C2211yescharacteristic
16#14581777—22no—
17#14581766C2212yes—
18#14581765—22no—
19C4×D332C2212yes—
20#14581725C2212yes—
21#14581723C2212yes—
22C4⋊(C83⋊C8)C2212yes—
23C4⋊(C83⋊C8)C2212yes—
24#14581716C2212yes—
25#14581716C2212yes—
26#14581712C2212yes—
27#14581712C2212yes—
28C4⋊Q8×C83C2211yescharacteristic
29#126×C83—22no—
30#125×C83—22no—
31#125×C83C2211yescharacteristic
32C22⋊Q8×C83C2212yes—
33C4×Q8×C83C2212yes—
34#117×C83C2211yescharacteristic
35C2×D332—22no—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39#13050577—42no—
40#13050577—42no—
41#13050574—41no—
42C2×C83⋊C8—41no—
43C2×C83⋊C8C2312yes—
44C2×C83⋊C8C2312yes—
45C4⋊Dic83C2312yes—
46Dic83⋊C4—22no—
47(C2×D83)⋊C4—22no—
48(C2×D83)⋊C4—22no—
49C2×Q8×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—22no—
52C4⋊C4×C83—22no—
53C4⋊C4×C83—22no—
54C4⋊C4×C83—22no—
55C4⋊C4×C83D412yes—
56C4⋊C4×C83C2312yes—
57C22⋊C4×C83—22no—
58C22⋊C4×C83D412yes—
59C42×C83—22no—
60C42×C83C2311yescharacteristic
61C42×C83C2311yescharacteristic
62C23×C83C2×D411yescharacteristic
63C22×D83—22no—
64Q8×C83—22no—
65Q8×C83—22no—
66C2×C4×C83—21no—
67C2×C4×C83—21no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×D412yes—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2411yescharacteristic
74C22⋊D83—82no—
75C2×Dic83—22no—
76D332—22no—
77D332—22no—
78D332—82no—
79C4×D83—82no—
80C83⋊C8—22no—
81C83⋊C8—22no—
82C22×C83—21no—
83C22×C83—22no—
84C22×C83C22×D411yescharacteristic
85C2×D83—42no—
86C2×D83—42no—
87C2×D83—42no—
88C4×C83—21no—
89C4×C83—22no—
90C4×C83—22no—
91C4×C83—42no—
92C4×C83—42no—
93C4×C83H(5,𝔽2)11yescharacteristic
94C4×C832−1+411yescharacteristic
95Dic83—42no—
96C2×C83—41no—
97C2×C83#53711yescharacteristic
98C2×C83#57712yes—
99D83—82no—
100#3260—831nomaximal, Sylow
101C83#326011yescharacteristic, complemented, Sylow
102#541D8311yescharacteristic, complemented
103#540—831no—
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⋊Q8C2×D8311yescharacteristic
116#127—831no—
117#126—22no—
118#125—22no—
119#125—831no—
120#125C2×D8311yescharacteristic
121#123—1662no—
122C22⋊Q8C2×D8312yes—
123#121—832no—
124#121—1662no—
125C4×Q8C2×D8312yes—
126C4×D4—832no—
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×Q8C22×D8312yes—
137C2×D4—832no—
138C2×D4—1662no—
139C2×D4—1662no—
140C22×C4—1662no—
141C22×C4C22×D8311yescharacteristic
142SD16—3322no—
143SD16—3322no—
144M16—3321no—
145C2×C8—3321no—
146C2×C8—832no—
147C2×C8—832no—
148C4⋊C4—22no—
149C4⋊C4—22no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—832no—
153C4⋊C4—1662no—
154C4⋊C4C22⋊D8312yes—
155C4⋊C4C22×D8312yes—
156C22⋊C4—22no—
157C22⋊C4—1662no—
158C22⋊C4—1662no—
159C22⋊C4C22⋊D8312yes—
160C42—22no—
161C42C22×D8311yescharacteristic
162C42C22×D8311yescharacteristic
163C23—1662no—
164C23C2×C22⋊D8311yescharacteristic
165Q8—22no—
166Q8—22no—
167D4—6642no—
168D4—6642no—
169D4—1662no—
170D4—1662no—
171C2×C4—21no—
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—6642no—
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—3322no—
186C22—3322no—
187C22—3322no—
188C22C22×C22⋊D8311yescharacteristic
189C4—22no—
190C4—22no—
191C4—21no—
192C4—42no—
193C4—42no—
194C4—3322no—
195C4#1458183811yescharacteristic
196C4#1458183911yescharacteristic
197C2—41no—
198C2—6642no—
199C2#1912827911yescharacteristic
200C2#1912850912yes—
subgroupG/H
centerC22C22×C22⋊D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#541×C83C2
socleC22×C83C22×D4
prime radical#19128306C2
hypercenter#541D83
hypocenterC83#3260
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #117 ◁ #541

Complements

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

NH ≅ G/Ntype
2. #19128008C2semidirect
3. #19127998C2semidirect
4. #19127949C2semidirect
5. #19127932C2semidirect
6. #19127903C2semidirect
7. #19127901C2semidirect
8. #19127728C2semidirect
9. #541×C83C2semidirect
101. C83#3260semidirect
102. #541D83semidirect

Maximal subgroups

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

subgroupindex
#191283062
#191280082
#191279982
#191279492
#191279322
#191279032
#191279012
#191277282
#541×C832
#326083