Finite Group Databasebeta

← #24918929

Subgroups — #24918929

Order 10624.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#19128324C212yesmaximal, complemented
2#19127990C212yesmaximal, complemented
3#19127984C212yesmaximal, complemented
4#19127954C211yesmaximal, characteristic
5#19127932C212yesmaximal, complemented
6#19127925C212yesmaximal, complemented
7#19127919C212yesmaximal, complemented
8#19127729C211yesmaximal, characteristic, complemented
9#470×C83C211yesmaximal, characteristic, complemented
10C2×#13050577C2212yes—
11C2×D664C2211yescharacteristic
12C22×C83⋊C8C2211yescharacteristic
13#14581791C2212yes—
14#14581787—22no—
15#14581781C2212yes—
16#14581778—22no—
17#14581774—22no—
18#14581772—22no—
19#14581766C2212yes—
20Q8×Dic83C2212yes—
21#14581754C2212yes—
22#14581751C2211yescharacteristic
23#14581724C2211yescharacteristic
24#14581723C2212yes—
25Dic83⋊C8C2212yes—
26Dic83⋊C8C2212yes—
27#14581716C2212yes—
28#14581714C2212yes—
29#14581712C2212yes—
30C2×Q16×C83C2211yescharacteristic
31C2×M16×C83C2211yescharacteristic
32C22⋊Q8×C83C2212yes—
33#106×C83C2211yescharacteristic
34#103×C83C2212yes—
35C2×Dic166—22no—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39#13050577—42no—
40D664—41no—
41C2×C83⋊C8—41no—
42C2×C83⋊C8C2312yes—
43C4⋊Dic83—22no—
44C4⋊Dic83D412yes—
45C4⋊Dic83C2312yes—
46Dic83⋊C4—22no—
47C22⋊Dic83D411yescharacteristic
48C22⋊Dic83D411yescharacteristic
49(C2×D83)⋊C4—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C4×Dic83C2312yes—
55C2×Q8×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57Q16×C83—41no—
58M16×C83—41no—
59C2×C8×C83C2311yescharacteristic
60C2×C8×C83C2311yescharacteristic
61C4⋊C4×C83—22no—
62C4⋊C4×C83C2312yes—
63C22⋊C4×C83—22no—
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66Q8×C83—22no—
67Q8×C83—22no—
68C2×C4×C83—21no—
69C2×C4×C83—21no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2411yescharacteristic
74C22⋊D83—82no—
75C2×Dic83—22no—
76C2×Dic83C2×D412yes—
77C2×Dic83C2×D412yes—
78D332—22no—
79D332—22no—
80C4×D83—82no—
81Dic166—82no—
82C8×C83—21no—
83C8×C83—21no—
84C83⋊C8—22no—
85C22×C83—21no—
86C22×C83—22no—
87C22×C83C22×D411yescharacteristic
88C2×D83—42no—
89C2×D83—42no—
90C2×D83—42no—
91C4×C83—21no—
92C4×C83—42no—
93C4×C83—42no—
94C4×C83H(5,𝔽2)11yescharacteristic
95C4×C832−1+411yescharacteristic
96Dic83—22no—
97Dic83—22no—
98Dic83—42no—
99C2×C83—41no—
100C2×C83#53711yescharacteristic
101C2×C83#57712yes—
102D83—82no—
103#3263—831nomaximal, Sylow
104C83#326311yescharacteristic, complemented, Sylow
105#542—832no—
106#476—832no—
107#475—832no—
108#470D8311yescharacteristic, complemented
109#469—831no—
110#465—832no—
111#464—832no—
112#463—832no—
113#424—831no—
114C2×Q16C2×D8311yescharacteristic
115C2×SD16—832no—
116C2×D8—831no—
117C2×M16C2×D8311yescharacteristic
118C22×C8—831no—
119C4⋊Q8—832no—
120#126—1662no—
121#124—832no—
122#124—1662no—
123#123—1662no—
124C22⋊Q8—1662no—
125C22⋊Q8C2×D8312yes—
126#121—832no—
127C4×Q8—832no—
128C4×D4—832no—
129#117—831no—
130#106—831no—
131#106—832no—
132#106C2×D8311yescharacteristic
133C4⋊C8—832no—
134C4⋊C8—832no—
135#103—832no—
136#103C2×D8312yes—
137#102—832no—
138#102—832no—
139C2×Q8—1662no—
140C2×Q8C22×D8312yes—
141C2×D4—832no—
142C2×D4—1662no—
143C22×C4—1662no—
144C22×C4C22×D8311yescharacteristic
145Q16—41no—
146SD16—3322no—
147D8—3321no—
148M16—41no—
149C2×C8—3321no—
150C2×C8—832no—
151C2×C8C22×D8311yescharacteristic
152C2×C8C22×D8311yescharacteristic
153C4⋊C4—22no—
154C4⋊C4—832no—
155C4⋊C4—832no—
156C4⋊C4—1662no—
157C4⋊C4—1662no—
158C4⋊C4C22×D8312yes—
159C22⋊C4—22no—
160C22⋊C4—831no—
161C22⋊C4—831no—
162C22⋊C4—1662no—
163C22⋊C4—1662no—
164C22⋊C4—1662no—
165C22⋊C4—1662no—
166C42—832no—
167C42—1662no—
168C23—1662no—
169C23D4×D8311yescharacteristic
170Q8—22no—
171Q8—22no—
172Q8—6642no—
173D4—6642no—
174D4—1662no—
175D4—1662no—
176C2×C4—21no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—832no—
181C2×C4—832no—
182C2×C4—6642no—
183C2×C4—1662no—
184C2×C4D4×D8311yescharacteristic
185C2×C4C23×D8311yescharacteristic
186C8—21no—
187C8—21no—
188C8—1662no—
189C22—22no—
190C22—21no—
191C22—3322no—
192C22—3322no—
193C22—3322no—
194C22C2×D4×D8311yescharacteristic
195C4—21no—
196C4—42no—
197C4—42no—
198C4—3322no—
199C4—1662no—
200C4—1662no—
201C4#1458183711yescharacteristic
202C4#1458183911yescharacteristic
203C2—41no—
204C2—6642no—
205C2#1912827811yescharacteristic
206C2#1912851012yes—
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#470×C83C2
socleC22×C83C22×D4
prime radical#19127954C2
hypercenter#470D83
hypocenterC83#3263
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. #19128324C2semidirect
2. #19127990C2semidirect
3. #19127984C2semidirect
5. #19127932C2semidirect
6. #19127925C2semidirect
7. #19127919C2semidirect
8. #19127729C2semidirect
9. #470×C83C2semidirect
104. C83#3263semidirect
108. #470D83semidirect

Maximal subgroups

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

subgroupindex
#191283242
#191279902
#191279842
#191279542
#191279322
#191279252
#191279192
#191277292
#470×C832
#326383