Finite Group Databasebeta

← #24912469

Subgroups — #24912469

Order 10624.

544 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 185 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×#14581769C212yesmaximal
2#19127633C212yesmaximal, complemented
3#19127622C212yesmaximal, complemented
4#19127607C212yesmaximal, complemented
5#19127599C212yesmaximal, complemented
6#19127576C212yesmaximal
7#19127561C211yesmaximal, characteristic
8C2×C83⋊#95C211yesmaximal, characteristic
9C4×C22⋊C4×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C23×Dic83C2212yes—
12#14581769—28no—
13C2×C4⋊Dic83C2212yes—
14C2×C4⋊Dic83C2212yes—
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C22⋊Dic83C2212yes—
20C83⋊#95C2211yescharacteristic
21C83⋊#95C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2211yescharacteristic
26C83⋊#95C2211yescharacteristic
27C23×C4×C83C2211yescharacteristic
28C2×C22⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2211yescharacteristic
30C2×C42×C83C2212yes—
31#95×C83C2212yes—
32C2×Dic166—24no—
33C2×Dic166—24no—
34C2×Dic166—42no—
35C22×Dic83—24no—
36C22×Dic83—24no—
37C22×Dic83—42no—
38C22×Dic83C2312yes—
39C22×Dic83C2312yes—
40C22×Dic83C2312yes—
41C22×Dic83C2312yes—
42C4⋊Dic83—24no—
43C4⋊Dic83—24no—
44C4⋊Dic83—42no—
45C4⋊Dic83—42no—
46Dic83⋊C4—24no—
47Dic83⋊C4—24no—
48C22⋊Dic83—24no—
49C24×C83C2311yescharacteristic
50C22×C4×C83—42no—
51C22×C4×C83Q814yescomplemented
52C22×C4×C83C2312yes—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2312yes—
55C22×C4×C83C2311yescharacteristic
56C22⋊C4×C83D414yes—
57C22⋊C4×C83Q814yescomplemented
58C42×C83—42no—
59C23×C83—22no—
60C23×C83—22no—
61C23×C83C2×D412yes—
62C23×C83C2×Q812yes—
63C23×C83C2×Q812yes—
64C23×C83C2411yescharacteristic
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—22no—
68C2×C4×C83—22no—
69C2×C4×C83—22no—
70C2×C4×C83—22no—
71C2×C4×C83—44no—
72C2×C4×C83C2×D414yes—
73C2×C4×C83C2×Q814yes—
74C2×C4×C83C2×Q812yes—
75C2×C4×C83C2×Q812yes—
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—48no—
88C2×Dic83#4114yes—
89Dic166—48no—
90C22×C83—22no—
91C22×C83—22no—
92C22×C83—24no—
93C22×C83—24no—
94C22×C83C4⋊Q814yes—
95C22×C83C22×D411yescharacteristic
96C22×C83C22×Q811yescharacteristic
97C22×C83C22×Q811yescharacteristic
98C22×C83C2×#4112yes—
99C22×C83H(5,𝔽2)12yes—
100C4×C83—22no—
101C4×C83—22no—
102C4×C83—24no—
103C4×C83—24no—
104Dic83—24no—
105Dic83—42no—
106Dic83—42no—
107Dic83—42no—
108C2×C83—24no—
109C2×C83C2×C4⋊Q811yescharacteristic
110C2×C83#53311yescharacteristic
111C2×C83#54311yescharacteristic
112C2×C83#54612yes—
113C2×C83#55412yes—
114#2549—831nomaximal, Sylow
115C83#254911yescharacteristic, complemented, Sylow
116C2×C22⋊Q8—832no—
117#398—832no—
118#395—832no—
119#391—832no—
120#389—832no—
121#385—832no—
122#380—831no—
123C4×C22⋊C4D8311yescharacteristic
124C2×#95—831no—
125C22×Q8—832no—
126C23×C4—832no—
127C23×C4C2×D8311yescharacteristic
128C22⋊Q8—1668no—
129C2×C4⋊C4—832no—
130C2×C4⋊C4—832no—
131C2×C4⋊C4—832no—
132C2×C4⋊C4—832no—
133C2×C4⋊C4—832no—
134C2×C4⋊C4—832no—
135C2×C22⋊C4—832no—
136C2×C22⋊C4C2×D8311yescharacteristic
137C2×C22⋊C4C2×D8311yescharacteristic
138C2×C42C2×D8312yes—
139#95—832no—
140#95—831no—
141#95—831no—
142#95—831no—
143#95—831no—
144#95—832no—
145#95—832no—
146#95C2×D8312yes—
147C24C22×D8311yescharacteristic
148C2×Q8—3322no—
149C2×Q8—1664no—
150C2×Q8—1664no—
151C22×C4—42no—
152C22×C4—3322no—
153C22×C4—832no—
154C22×C4—832no—
155C22×C4—832no—
156C22×C4—832no—
157C22×C4—1664no—
158C22×C4—1664no—
159C22×C4Dic16614yescomplemented
160C22×C4C22×D8312yes—
161C22×C4C22×D8311yescharacteristic
162C22×C4C22×D8311yescharacteristic
163C22×C4C22×D8312yes—
164C4⋊C4—3322no—
165C4⋊C4—3322no—
166C4⋊C4—1664no—
167C4⋊C4—1664no—
168C4⋊C4—1664no—
169C4⋊C4—1664no—
170C22⋊C4—1664no—
171C22⋊C4Dic16614yescomplemented
172C22⋊C4D33214yes—
173C42—42no—
174C23—22no—
175C23—22no—
176C23C2×D33212yes—
177C23C2×Dic16612yes—
178C23C2×Dic16612yes—
179C23C23×D8311yescharacteristic
180Q8—3328no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—44no—
188C2×C4—3328no—
189C2×C4—834no—
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×C4—1662no—
201C2×C4C2×D33214yes—
202C2×C4C2×Dic16614yes—
203C2×C4C2×Dic16612yes—
204C2×C4C2×Dic16612yes—
205C22—22no—
206C22—24no—
207C22—24no—
208C22—22no—
209C22#1458179214yes—
210C22C22×D33211yescharacteristic
211C22C22×Dic16611yescharacteristic
212C22C22×Dic16611yescharacteristic
213C22#41×D8312yes—
214C22#1458183712yes—
215C4—22no—
216C4—22no—
217C4—24no—
218C4—24no—
219C4—3322no—
220C4—3322no—
221C4—3322no—
222C4—1664no—
223C2—24no—
224C2C2×#1458179211yescharacteristic
225C2#1912825411yescharacteristic
226C2#1912832711yescharacteristic
227C2#1912834612yes—
228C2#1912841012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC4×C22⋊C4×C83C2
prime radicalC24×C83C23
hypercenterC4×C22⋊C4D83
hypocenterC83#2549
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #19127633C2semidirect
3. #19127622C2semidirect
4. #19127607C2semidirect
5. #19127599C2semidirect
51. C22×C4×C83Q8semidirect
57. C22⋊C4×C83Q8semidirect
115. C83#2549semidirect
159. C22×C4Dic166semidirect
171. C22⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#145817692
#191276332
#191276222
#191276072
#191275992
#191275762
#191275612
C2×C83⋊#952
C4×C22⋊C4×C832
#254983