Finite Group Databasebeta

← #24913124

Subgroups — #24913124

Order 10624.

444 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 145 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#19127637C212yesmaximal
2#19127613C212yesmaximal
3#19127607C212yesmaximal, complemented
4#19127605C212yesmaximal, complemented
5#19127585C212yesmaximal, complemented
6#19127584C212yesmaximal, complemented
7#19127564C211yesmaximal, characteristic
8C4×C22⋊Dic83C211yesmaximal, characteristic
9C4×C22⋊C4×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C2×C4⋊Dic83C2212yes—
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×C22⋊Dic83C2212yes—
15C2×C22⋊Dic83C2211yescharacteristic
16C2×C22⋊Dic83C2211yescharacteristic
17C2×C4×Dic83C2211yescharacteristic
18C2×C4×Dic83C2211yescharacteristic
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C23×C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2212yes—
28C2×C42×C83C2211yescharacteristic
29C2×C42×C83C2211yescharacteristic
30#95×C83C2212yes—
31C2×Dic166—24no—
32C2×Dic166—24no—
33C2×Dic166—42no—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C4⋊Dic83—24no—
39Dic83⋊C4—24no—
40Dic83⋊C4—42no—
41C22⋊Dic83—24no—
42C22⋊Dic83—41no—
43C22⋊Dic83—41no—
44C4×Dic83—41no—
45C4×Dic83—41no—
46C24×C83C2311yescharacteristic
47C22×C4×C83—41no—
48C22×C4×C83—41no—
49C22×C4×C83Q814yescomplemented
50C22×C4×C83C2312yes—
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C22×C4×C83C2312yes—
54C22⋊C4×C83—24no—
55C42×C83—41no—
56C42×C83—41no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83—21no—
60C23×C83—21no—
61C23×C83C2×Q812yes—
62C23×C83C2411yescharacteristic
63C2×C4×C83—22no—
64C2×C4×C83—21no—
65C2×C4×C83—21no—
66C2×C4×C83—22no—
67C2×C4×C83—21no—
68C2×C4×C83—21no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83—44no—
73C2×C4×C83C2×Q812yes—
74C2×C4×C83C2×Q812yes—
75C2×C4×C83#4114yes—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83C2×Dic83—22no—
84C2×Dic83—22no—
85C2×Dic83—22no—
86C2×Dic83—22no—
87C2×Dic83#4114yes—
88Dic166—48no—
89C22×C83—22no—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83—42no—
93C22×C83—42no—
94C22×C83C22×Q811yescharacteristic
95C22×C83C2×#4112yes—
96C22×C83C2×#4112yes—
97C22×C83H(5,𝔽2)11yescharacteristic
98C22×C83H(5,𝔽2)11yescharacteristic
99C4×C83—22no—
100C4×C83—22no—
101C4×C83—24no—
102C4×C83—42no—
103Dic83—24no—
104Dic83—42no—
105Dic83—42no—
106Dic83—42no—
107C2×C83—42no—
108C2×C83#53311yescharacteristic
109C2×C83#53311yescharacteristic
110C2×C83#54311yescharacteristic
111C2×C83#55112yes—
112C2×C83#55912yes—
113#2607—831nomaximal, Sylow
114C83#260711yescharacteristic, complemented, Sylow
115#399—832no—
116#393—832no—
117#391—832no—
118#391—832no—
119#387—832no—
120#387—832no—
121#381—831no—
122C4×C22⋊C4—831no—
123C4×C22⋊C4D8311yescharacteristic
124C22×Q8—832no—
125C23×C4C2×D8311yescharacteristic
126C2×C4⋊C4—832no—
127C2×C4⋊C4—832no—
128C2×C4⋊C4—832no—
129C2×C22⋊C4—831no—
130C2×C22⋊C4—831no—
131C2×C22⋊C4—832no—
132C2×C22⋊C4C2×D8312yes—
133C2×C42—831no—
134C2×C42—831no—
135C2×C42C2×D8311yescharacteristic
136C2×C42C2×D8311yescharacteristic
137#95—832no—
138#95—832no—
139#95—832no—
140#95—832no—
141#95—832no—
142#95—832no—
143#95—832no—
144#95C2×D8312yes—
145C24C22×D8311yescharacteristic
146C2×Q8—3322no—
147C2×Q8—1664no—
148C2×Q8—1664no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—832no—
152C22×C4—832no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4Dic16614yescomplemented
156C22×C4C22×D8312yes—
157C22×C4C22×D8311yescharacteristic
158C22×C4C22×D8311yescharacteristic
159C22×C4C22×D8312yes—
160C4⋊C4—3322no—
161C4⋊C4—1664no—
162C4⋊C4—1664no—
163C22⋊C4—24no—
164C22⋊C4—3321no—
165C22⋊C4—3321no—
166C22⋊C4—1664no—
167C42—41no—
168C42—41no—
169C42—3321no—
170C42—3321no—
171C23—22no—
172C23—22no—
173C23—21no—
174C23—21no—
175C23C2×Dic16612yes—
176C23C23×D8311yescharacteristic
177Q8—3328no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—834no—
189C2×C4—1662no—
190C2×C4—1662no—
191C2×C4—1662no—
192C2×C4—1662no—
193C2×C4—1662no—
194C2×C4—1662no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1662no—
198C2×C4—1662no—
199C2×C4—1662no—
200C2×C4C2×Dic16612yes—
201C2×C4C2×Dic16612yes—
202C2×C4#1305059014yes—
203C22—22no—
204C22—22no—
205C22—22no—
206C22—42no—
207C22—42no—
208C22C22×Dic16611yescharacteristic
209C22#41×D8312yes—
210C22C2×#1305059012yes—
211C22#1458183711yescharacteristic
212C22#1458183711yescharacteristic
213C4—22no—
214C4—22no—
215C4—24no—
216C4—42no—
217C4—3322no—
218C4—3322no—
219C4—3322no—
220C4—1664no—
221C2—42no—
222C2#1912825411yescharacteristic
223C2#1912825811yescharacteristic
224C2#1912832711yescharacteristic
225C2#1912838012yes—
226C2#1912843412yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC4×C22⋊C4×C83C2
prime radicalC24×C83C23
hypercenterC4×C22⋊C4D83
hypocenterC83#2607
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. #19127607C2semidirect
4. #19127605C2semidirect
5. #19127585C2semidirect
6. #19127584C2semidirect
49. C22×C4×C83Q8semidirect
114. C83#2607semidirect
155. C22×C4Dic166semidirect

Maximal subgroups

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

subgroupindex
#191276372
#191276132
#191276072
#191276052
#191275852
#191275842
#191275642
C4×C22⋊Dic832
C4×C22⋊C4×C832
#260783