Finite Group Databasebeta

← #24913633

Subgroups — #24913633

Order 10624.

424 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 141 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#19127643C212yesmaximal, complemented
2#19127624C212yesmaximal
3#19127622C212yesmaximal, complemented
4#19127613C212yesmaximal
5#19127607C212yesmaximal, complemented
6#19127585C212yesmaximal, complemented
7#19127564C211yesmaximal, characteristic
8#19127561C211yesmaximal, characteristic
9C4×C22⋊C4×C83C211yesmaximal, characteristic
10C22×Dic166C2212yes—
11C2×C4⋊Dic83C2212yes—
12C2×C4⋊Dic83C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×C22⋊Dic83C2211yescharacteristic
16C2×C22⋊Dic83C2211yescharacteristic
17C2×C22⋊Dic83C2212yes—
18C83⋊#95C2212yes—
19C83⋊#95C2211yescharacteristic
20C83⋊#95C2211yescharacteristic
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C23×C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2212yes—
28C2×C42×C83C2212yes—
29#95×C83C2211yescharacteristic
30#95×C83C2211yescharacteristic
31C2×Dic166—24no—
32C2×Dic166—42no—
33C2×Dic166—42no—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C4⋊Dic83—42no—
39C4⋊Dic83—42no—
40Dic83⋊C4—24no—
41Dic83⋊C4—42no—
42C22⋊Dic83—42no—
43C22⋊Dic83—41no—
44C22⋊Dic83—41no—
45C24×C83C2311yescharacteristic
46C22×C4×C83—42no—
47C22×C4×C83Q814yescomplemented
48C22×C4×C83C2312yes—
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2312yes—
52C22⋊C4×C83—24no—
53C42×C83—42no—
54C23×C83—22no—
55C23×C83—22no—
56C23×C83—21no—
57C23×C83—21no—
58C23×C83C2×Q812yes—
59C23×C83C2411yescharacteristic
60C2×C4×C83—22no—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—22no—
67C2×C4×C83—44no—
68C2×C4×C83C2×Q812yes—
69C2×C4×C83C2×Q812yes—
70C2×C4×C83#4114yes—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—22no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—22no—
83Dic166—48no—
84C22×C83—22no—
85C22×C83—22no—
86C22×C83—22no—
87C22×C83—42no—
88C22×C83—42no—
89C22×C83C22×Q811yescharacteristic
90C22×C83C2×#4112yes—
91C22×C83H(5,𝔽2)12yes—
92C22×C83H(5,𝔽2)11yescharacteristic
93C22×C832−1+411yescharacteristic
94C4×C83—24no—
95C4×C83—22no—
96C4×C83—22no—
97C4×C83—42no—
98Dic83—42no—
99Dic83—42no—
100Dic83—42no—
101Dic83—42no—
102C2×C83—42no—
103C2×C83#53311yescharacteristic
104C2×C83#53812yes—
105C2×C83#54212yes—
106C2×C83#54311yescharacteristic
107C2×C83#54411yescharacteristic
108#2665—831nomaximal, Sylow
109C83#266511yescharacteristic, complemented, Sylow
110#400—832no—
111#396—832no—
112#395—832no—
113#393—832no—
114#391—832no—
115#387—832no—
116#381—831no—
117#380—831no—
118C4×C22⋊C4D8311yescharacteristic
119C22×Q8—832no—
120C23×C4C2×D8311yescharacteristic
121C2×C4⋊C4—832no—
122C2×C4⋊C4—832no—
123C2×C4⋊C4—832no—
124C2×C4⋊C4—832no—
125C2×C22⋊C4—831no—
126C2×C22⋊C4—831no—
127C2×C22⋊C4—832no—
128C2×C22⋊C4C2×D8312yes—
129C2×C42C2×D8312yes—
130#95—832no—
131#95—831no—
132#95—831no—
133#95—832no—
134#95—832no—
135#95—832no—
136#95—832no—
137#95—832no—
138#95C2×D8311yescharacteristic
139#95C2×D8311yescharacteristic
140C24C22×D8311yescharacteristic
141C2×Q8—3322no—
142C2×Q8—3322no—
143C2×Q8—1664no—
144C22×C4—42no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4—832no—
148C22×C4—832no—
149C22×C4Dic16614yescomplemented
150C22×C4C22×D8312yes—
151C22×C4C22×D8311yescharacteristic
152C22×C4C22×D8311yescharacteristic
153C22×C4C22×D8312yes—
154C4⋊C4—3322no—
155C4⋊C4—3322no—
156C4⋊C4—3322no—
157C4⋊C4—1664no—
158C22⋊C4—24no—
159C22⋊C4—3322no—
160C22⋊C4—3321no—
161C22⋊C4—3321no—
162C42—42no—
163C23—22no—
164C23—22no—
165C23—21no—
166C23—21no—
167C23C2×Dic16612yes—
168C23C23×D8311yescharacteristic
169Q8—3328no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—44no—
178C2×C4—1662no—
179C2×C4—1662no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1662no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4—1662no—
187C2×C4—1662no—
188C2×C4—1662no—
189C2×C4—1662no—
190C2×C4C2×Dic16612yes—
191C2×C4C2×Dic16612yes—
192C2×C4#1305059014yes—
193C22—22no—
194C22—22no—
195C22—22no—
196C22—42no—
197C22—42no—
198C22C22×Dic16611yescharacteristic
199C22C2×#1305059012yes—
200C22#1458183712yes—
201C22#1458183811yescharacteristic
202C22#1458183911yescharacteristic
203C4—22no—
204C4—22no—
205C4—24no—
206C4—42no—
207C4—3322no—
208C4—3322no—
209C4—3322no—
210C4—3322no—
211C2—42no—
212C2#1912825811yescharacteristic
213C2#1912828412yes—
214C2#1912831612yes—
215C2#1912832711yescharacteristic
216C2#1912832911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC4×C22⋊C4×C83C2
prime radicalC24×C83C23
hypercenterC4×C22⋊C4D83
hypocenterC83#2665
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #19127643C2semidirect
3. #19127622C2semidirect
5. #19127607C2semidirect
6. #19127585C2semidirect
47. C22×C4×C83Q8semidirect
109. C83#2665semidirect
149. C22×C4Dic166semidirect

Maximal subgroups

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

subgroupindex
#191276432
#191276242
#191276222
#191276132
#191276072
#191275852
#191275642
#191275612
C4×C22⋊C4×C832
#266583