Finite Group Databasebeta

← #24915944

Subgroups — #24915944

Order 10624.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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
1C2×#14581839C211yesmaximal, characteristic
2#19128467C212yesmaximal, complemented
3#19127685C211yesmaximal, characteristic, complemented
4#19127682C214yesmaximal, complemented
5C2×#14581710C212yesmaximal, complemented
6#101×D83C214yesmaximal, complemented
7#413×C83C211yesmaximal, characteristic, complemented
8#14581839—24no—
9#14581839—24no—
10C2×#13050590C412yes—
11C2×#13050590C2211yescharacteristic
12C2×#13050589C414yes—
13C2×#13050589C2212yes—
14C22×Dic166C412yes—
15C22×Dic166C2211yescharacteristic
16C2×Q8×D83C2212yes—
17#14581800—24no—
18#14581798—24no—
19C2×#13050574C2212yes—
20M16×D83C2218yes—
21#14581711C2212yes—
22#14581710C2218yes—
23#14581709C2212yes—
24C2×#41×C83C2211yescharacteristic
25C2×M16×C83C2212yes—
26#101×C83C2212yes—
27#100×C83C2212yes—
28#13050590—24no—
29#13050590—24no—
30#13050590D414yes—
31#13050590D414yes—
32#13050589—216no—
33#13050589—28no—
34C2×Dic166—22no—
35C2×Dic166—22no—
36C2×Dic166—24no—
37C2×Dic166—24no—
38C2×Dic166C2×C414yes—
39C2×Dic166C2×C412yes—
40C2×Dic166C2×C411yescharacteristic
41C2×Dic166C2312yes—
42Q8×D83—28no—
43C2×D332C2×C411yescharacteristic
44C2×C22⋊D83C2×C414yes—
45C2×C22⋊D83C2×C412yes—
46C22×Dic83C2×C414yes—
47C22×Dic83C2×C412yes—
48C2×C4×D83C2×C414yes—
49C2×C4×D83C2312yes—
50#13050574—22no—
51#13050574C2314yes—
52#13050573—28no—
53C2×C83⋊C8—22no—
54C8×D83—28no—
55#41×C83—22no—
56#41×C83—22no—
57C2×Q8×C83C2311yescharacteristic
58C2×D4×C83C2×C412yes—
59C2×D4×C83C2311yescharacteristic
60C22×C4×C83C2×C412yes—
61C22×C4×C83C2311yescharacteristic
62M16×C83—22no—
63M16×C83C2314yes—
64C2×C8×C83—22no—
65C23×C83C22×C412yes—
66C23×C83C22×C411yescharacteristic
67C22×D83C22×C412yes—
68Q8×C83—22no—
69D4×C83—24no—
70D4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—22no—
73C2×C4×C83C22×C412yes—
74C2×C4×C83C22×C411yescharacteristic
75C2×C4×C83C2×D412yes—
76C2×C4×C83C2×D412yes—
77C2×C4×C83C2411yescharacteristic
78C22⋊D83—28no—
79C22⋊D83C22⋊C418yes—
80C2×Dic83—24no—
81C2×Dic83—28no—
82C2×Dic83C22×C414yes—
83C2×Dic83C22×C412yes—
84D332C22⋊C412yes—
85D332C22⋊C412yes—
86C4×D83—28no—
87C4×D83C2×D418yes—
88Dic166—22no—
89Dic166—22no—
90Dic166—24no—
91Dic166—28no—
92Dic166C22⋊C412yes—
93Dic166C22⋊C412yes—
94C8×C83—24no—
95C83⋊C8—24no—
96C22×C83—22no—
97C22×C83—41no—
98C22×C83—42no—
99C22×C83C2×C22⋊C412yes—
100C22×C83C23×C411yescharacteristic
101C2×D83—42no—
102C2×D83C2×C22⋊C414yes—
103C4×C83—22no—
104C4×C83C2×C22⋊C411yescharacteristic
105C4×C83C2×C22⋊C411yescharacteristic
106C4×C83C22×D412yes—
107Dic83—24no—
108Dic83C2×C22⋊C414yes—
109C2×C83—21no—
110C2×C83—22no—
111C2×C83—42no—
112C2×C83C22×C22⋊C411yescharacteristic
113D83—24no—
114#2889—831nomaximal, Sylow
115C83#288911yescharacteristic, complemented, Sylow
116C2×2−1+4—831no—
117#568—832no—
118#413—831no—
119#413—834no—
120#413D8311yescharacteristic, complemented
121C2×#101—832no—
122C2×#101—834no—
1232−1+4—1664no—
1242−1+4—1664no—
125C2×#41—834no—
126C2×#41—832no—
127C2×#41—832no—
128C2×#41—831no—
129C2×#41C2×D8311yescharacteristic
130C22×Q8—832no—
131C22×Q8—832no—
132C22×Q8—831no—
133#131—1664no—
134#131—1664no—
135C2×M16—832no—
136C2×M16—838no—
137C2×M16C2×D8312yes—
138#101—832no—
139#101—838no—
140#101C2×D8312yes—
141#100—832no—
142#100C2×D8312yes—
143#41—22no—
144#41—22no—
145#41—834no—
146#41—834no—
147#41—1664no—
148#41—1664no—
149#41—1668no—
150#41—16616no—
151C2×Q8—831no—
152C2×Q8—834no—
153C2×Q8—832no—
154C2×Q8—832no—
155C2×Q8—1664no—
156C2×Q8—1664no—
157C2×Q8—1668no—
158C2×Q8—1662no—
159C2×Q8—1662no—
160C2×Q8C22×D8311yescharacteristic
161C2×D4—832no—
162C2×D4—834no—
163C2×D4—831no—
164C2×D4C4×D8312yes—
165C2×D4C22×D8311yescharacteristic
166C22×C4—832no—
167C22×C4—834no—
168C22×C4—832no—
169C22×C4—834no—
170C22×C4C4×D8312yes—
171C22×C4C22×D8311yescharacteristic
172M16—22no—
173M16—834no—
174M16—1668no—
175M16—1662no—
176M16C22×D8314yes—
177C2×C8—22no—
178C2×C8—1668no—
179C2×C8—1662no—
180C23—832no—
181C23C2×C4×D8312yes—
182C23C2×C4×D8311yescharacteristic
183Q8—22no—
184Q8—832no—
185Q8—832no—
186Q8—1668no—
187Q8—1664no—
188Q8—1662no—
189Q8—1662no—
190D4—24no—
191D4—22no—
192D4—838no—
193D4—832no—
194D4—832no—
195D4—1668no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—832no—
199C2×C4—838no—
200C2×C4—834no—
201C2×C4—1668no—
202C2×C4—1668no—
203C2×C4—1664no—
204C2×C4C2×C4×D8312yes—
205C2×C4C2×C4×D8311yescharacteristic
206C2×C4D4×D8312yes—
207C2×C4D4×D8312yes—
208C2×C4C23×D8311yescharacteristic
209C8—24no—
210C8—1664no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—3322no—
215C22—834no—
216C22C22⋊C4×D8312yes—
217C22C22×C4×D8311yescharacteristic
218C4—22no—
219C4—834no—
220C4—1664no—
221C4C22⋊C4×D8311yescharacteristic
222C4C22⋊C4×D8311yescharacteristic
223C4C2×D4×D8312yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—1664no—
228C2C2×C22⋊C4×D8311yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D83
commutatorC22×C83C23×C4
FrattiniC2×C4C23×D83
Fitting#413×C83C2
socleC2×C83C22×C22⋊C4
prime radicalC2×#14581839C2
cosocleC2×C4×C83C24
hypercenter#413D83
hypocenterC83#2889
LCSG ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #19128467C2semidirect
3. #19127685C2semidirect
4. #19127682C2semidirect
5. C2×#14581710C2semidirect
6. #101×D83C2semidirect
7. #413×C83C2semidirect
115. C83#2889semidirect
120. #413D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818392
#191284672
#191276852
#191276822
C2×#145817102
#101×D832
#413×C832
#288983