Finite Group Databasebeta

← #24913660

Subgroups — #24913660

Order 10624.

408 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 137 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#19127634C212yesmaximal, complemented
3#19127634C212yesmaximal, complemented
4#19127624C212yesmaximal
5#19127598C212yesmaximal, complemented
6#19127584C212yesmaximal, complemented
7#19127564C211yesmaximal, characteristic
8C4×C22⋊Dic83C211yesmaximal, characteristic
9#380×C83C211yesmaximal, characteristic
10C2×C4⋊Dic83C2212yes—
11C2×C4⋊Dic83C2212yes—
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2212yes—
18C2×C4×Dic83C2212yes—
19C83⋊#95C2211yescharacteristic
20C83⋊#95C2211yescharacteristic
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2211yescharacteristic
26C23×C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2212yes—
28C2×C22⋊C4×C83C2211yescharacteristic
29C2×C22⋊C4×C83C2211yescharacteristic
30#95×C83C2212yes—
31C22×Dic83C2312yes—
32C22×Dic83C2312yes—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C4⋊Dic83—24no—
36C4⋊Dic83—42no—
37Dic83⋊C4—24no—
38Dic83⋊C4—42no—
39Dic83⋊C4—42no—
40Dic83⋊C4—42no—
41C22⋊Dic83—24no—
42C22⋊Dic83—42no—
43C4×Dic83—42no—
44C24×C83C2311yescharacteristic
45C22×C4×C83—42no—
46C22×C4×C83Q814yes—
47C22×C4×C83C2312yes—
48C22×C4×C83C2312yes—
49C22×C4×C83C2311yescharacteristic
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—42no—
52C22⋊C4×C83—41no—
53C22⋊C4×C83—41no—
54C23×C83—22no—
55C23×C83—21no—
56C23×C83—22no—
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—22no—
68C2×C4×C83—44no—
69C2×C4×C83C2×Q812yes—
70C2×C4×C83C2×Q812yes—
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#4114yes—
83C22×C83—22no—
84C22×C83—22no—
85C22×C83—22no—
86C22×C83—42no—
87C22×C83—42no—
88C22×C83C22×Q811yescharacteristic
89C22×C83C2×#4112yes—
90C22×C83H(5,𝔽2)12yes—
91C22×C83H(5,𝔽2)11yescharacteristic
92C22×C832−1+411yescharacteristic
93C4×C83—22no—
94C4×C83—22no—
95C4×C83—42no—
96C4×C83—42no—
97Dic83—24no—
98Dic83—42no—
99Dic83—42no—
100Dic83—42no—
101C2×C83—42no—
102C2×C83#53311yescharacteristic
103C2×C83#53912yes—
104C2×C83#54012yes—
105C2×C83#54311yescharacteristic
106C2×C83#54411yescharacteristic
107#2667—831nomaximal, Sylow
108C83#266711yescharacteristic, complemented, Sylow
109#399—832no—
110#398—832no—
111#398—832no—
112#396—832no—
113#389—832no—
114#387—832no—
115#381—831no—
116#380D8311yescharacteristic
117C4×C22⋊C4—831no—
118C23×C4C2×D8311yescharacteristic
119C2×C4⋊C4—832no—
120C2×C4⋊C4—832no—
121C2×C4⋊C4—832no—
122C2×C4⋊C4—832no—
123C2×C4⋊C4—832no—
124C2×C4⋊C4—832no—
125C2×C4⋊C4C2×D8312yes—
126C2×C22⋊C4—832no—
127C2×C22⋊C4—832no—
128C2×C22⋊C4C2×D8311yescharacteristic
129C2×C22⋊C4C2×D8311yescharacteristic
130C2×C42—832no—
131#95—831no—
132#95—831no—
133#95—831no—
134#95—831no—
135#95—832no—
136#95—832no—
137#95—832no—
138#95C2×D8312yes—
139C24C22×D8311yescharacteristic
140C22×C4—42no—
141C22×C4—832no—
142C22×C4—832no—
143C22×C4—832no—
144C22×C4—832no—
145C22×C4Dic16614yes—
146C22×C4C22×D8312yes—
147C22×C4C22×D8312yes—
148C22×C4C22×D8311yescharacteristic
149C22×C4C22×D8311yescharacteristic
150C4⋊C4—42no—
151C4⋊C4—3322no—
152C4⋊C4—3322no—
153C4⋊C4—3322no—
154C4⋊C4—3322no—
155C4⋊C4—1664no—
156C4⋊C4—1664no—
157C22⋊C4—41no—
158C22⋊C4—41no—
159C22⋊C4—3322no—
160C22⋊C4—1664no—
161C42—3322no—
162C23—22no—
163C23—21no—
164C23—22no—
165C23—21no—
166C23C2×Dic16612yes—
167C23C23×D8311yescharacteristic
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—44no—
177C2×C4—834no—
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×C4C2×Dic16612yes—
190C2×C4C2×Dic16612yes—
191C22—22no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22C22×Dic16611yescharacteristic
197C22#41×D8312yes—
198C22#1458183712yes—
199C22#1458183711yescharacteristic
200C22#1458184011yescharacteristic
201C4—22no—
202C4—22no—
203C4—42no—
204C4—42no—
205C4—3322no—
206C4—3322no—
207C4—3322no—
208C4—1664no—
209C2—42no—
210C2#1912825411yescharacteristic
211C2#1912829712yes—
212C2#1912830112yes—
213C2#1912832711yescharacteristic
214C2#1912833211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
Fitting#380×C83C2
prime radicalC24×C83C23
hypercenter#380D83
hypocenterC83#2667
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. #19127634C2semidirect
3. #19127634C2semidirect
5. #19127598C2semidirect
6. #19127584C2semidirect
108. C83#2667semidirect

Maximal subgroups

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

subgroupindex
#191276372
#191276342
#191276242
#191275982
#191275842
#191275642
C4×C22⋊Dic832
#380×C832
#266783