Finite Group Databasebeta

← #24912659

Subgroups — #24912659

Order 10624.

548 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 169 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×#14581784C211yesmaximal, characteristic, complemented
2#19127624C214yesmaximal, complemented
3#19127604C211yesmaximal, characteristic
4#19127604C211yesmaximal, characteristic
5#19127574C214yesmaximal, complemented
6#19127570C211yesmaximal, characteristic, complemented
7#19127566C211yesmaximal, characteristic, complemented
8C4×C4⋊Dic83C211yesmaximal, characteristic, complemented
9C2×C4×D4×C83C211yesmaximal, characteristic
10#14581784—24no—
11C2×C4⋊Dic83C2212yes—
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2211yescharacteristic
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2214yes—
17C2×C22⋊Dic83C2214yes—
18C2×C22⋊Dic83C2214yes—
19C2×C4×Dic83C2211yescharacteristic
20C2×C4×Dic83C2211yescharacteristic
21C83⋊#95C2212yes—
22C83⋊#95C2214yes—
23C22×D4×C83C2211yescharacteristic
24C23×C4×C83C2212yes—
25C4×D4×C83—24no—
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C22⋊C4×C83C2212yes—
28C2×C42×C83C2211yescharacteristic
29C22×Dic83C2312yes—
30C22×Dic83C2312yes—
31C22×Dic83C2312yes—
32C22×Dic83C2312yes—
33C4⋊Dic83—22no—
34C4⋊Dic83—22no—
35C4⋊Dic83—24no—
36Dic83⋊C4—24no—
37Dic83⋊C4—24no—
38Dic83⋊C4—44no—
39C22⋊Dic83—28no—
40C22⋊Dic83—44no—
41C4×Dic83—22no—
42C4×Dic83—22no—
43C24×C83C2312yes—
44C2×D4×C83—22no—
45C2×D4×C83—22no—
46C2×D4×C83Q814yes—
47C22×C4×C83—24no—
48C22×C4×C83—44no—
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C22×C4×C83C2311yescharacteristic
52C22×C4×C83C2311yescharacteristic
53C4⋊C4×C83—22no—
54C22⋊C4×C83—24no—
55C42×C83D414yes—
56C23×C83—22no—
57C23×C83—22no—
58C23×C83—24no—
59C23×C83—24no—
60C23×C83C2×Q814yes—
61C23×C83C2411yescharacteristic
62D4×C83—28no—
63C2×C4×C83—22no—
64C2×C4×C83—22no—
65C2×C4×C83—22no—
66C2×C4×C83—24no—
67C2×C4×C83—48no—
68C2×C4×C83C2×D412yes—
69C2×C4×C83C2×D412yes—
70C2×C4×C83C2×D412yes—
71C2×C4×C83C2×Q812yes—
72C2×C4×C83#4112yes—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—22no—
78C2×Dic83—24no—
79C2×Dic83—22no—
80C2×Dic83—22no—
81C2×Dic83—24no—
82C2×Dic83#4114yes—
83C22×C83—24no—
84C22×C83—28no—
85C22×C83—42no—
86C22×C83—42no—
87C22×C83—44no—
88C22×C83C22×D411yescharacteristic
89C22×C83C22×Q811yescharacteristic
90C22×C83C2×#4112yes—
91C22×C83C2×#4111yescharacteristic
92C22×C83H(5,𝔽2)12yes—
93C4×C83—22no—
94C4×C83—22no—
95C4×C83—42no—
96C4×C83C22⋊Q814yes—
97Dic83—24no—
98Dic83—42no—
99Dic83—42no—
100Dic83—42no—
101C2×C83—44no—
102C2×C83C2×C22⋊Q811yescharacteristic
103C2×C83#53211yescharacteristic
104C2×C83#53511yescharacteristic
105C2×C83#55312yes—
106C2×C83#55412yes—
107#2566—831nomaximal, Sylow
108C83#256611yescharacteristic, complemented, Sylow
109C2×#125—831no—
110C2×C4×D4D8311yescharacteristic
111#396—834no—
112#390—831no—
113#390—831no—
114#385—834no—
115#384—831no—
116#382—831no—
117C4×C4⋊C4—831no—
118C22×D4C2×D8311yescharacteristic
119C23×C4C2×D8312yes—
120#125—1664no—
121C4×D4—24no—
122C2×C4⋊C4—834no—
123C2×C4⋊C4—831no—
124C2×C4⋊C4—831no—
125C2×C4⋊C4—832no—
126C2×C4⋊C4—832no—
127C2×C4⋊C4—832no—
128C2×C4⋊C4C2×D8311yescharacteristic
129C2×C22⋊C4—834no—
130C2×C22⋊C4—834no—
131C2×C22⋊C4C2×D8312yes—
132C2×C42—831no—
133C2×C42—831no—
134C2×C42C2×D8311yescharacteristic
135#95—832no—
136#95—834no—
137C24C22×D8312yes—
138C2×D4—22no—
139C2×D4—22no—
140C2×D4Dic16614yes—
141C22×C4—24no—
142C22×C4—44no—
143C22×C4—832no—
144C22×C4—832no—
145C22×C4—832no—
146C22×C4—832no—
147C22×C4C22×D8312yes—
148C22×C4C22×D8311yescharacteristic
149C22×C4C22×D8311yescharacteristic
150C22×C4C22×D8311yescharacteristic
151C4⋊C4—22no—
152C4⋊C4—3324no—
153C4⋊C4—1664no—
154C4⋊C4—1664no—
155C4⋊C4—1664no—
156C4⋊C4—1662no—
157C4⋊C4—1662no—
158C22⋊C4—24no—
159C22⋊C4—3324no—
160C22⋊C4—1668no—
161C42—1662no—
162C42—1662no—
163C42C22⋊D8314yes—
164C23—22no—
165C23—24no—
166C23—22no—
167C23—24no—
168C23C2×Dic16614yes—
169C23C23×D8311yescharacteristic
170D4—28no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—24no—
175C2×C4—48no—
176C2×C4—834no—
177C2×C4—1662no—
178C2×C4—1662no—
179C2×C4—1664no—
180C2×C4—1662no—
181C2×C4—1662no—
182C2×C4—1662no—
183C2×C4—1664no—
184C2×C4—1662no—
185C2×C4—1662no—
186C2×C4C2×C22⋊D8312yes—
187C2×C4C2×C22⋊D8312yes—
188C2×C4C2×C22⋊D8312yes—
189C2×C4C2×Dic16612yes—
190C2×C4#1305059012yes—
191C22—28no—
192C22—24no—
193C22—44no—
194C22—42no—
195C22—42no—
196C22C22×C22⋊D8311yescharacteristic
197C22C22×Dic16611yescharacteristic
198C22#41×D8312yes—
199C22C2×#1305059011yescharacteristic
200C22#1458183712yes—
201C4—22no—
202C4—22no—
203C4—42no—
204C4—3322no—
205C4—3322no—
206C4—3322no—
207C4—1664no—
208C4#1458177114yes—
209C2—44no—
210C2C2×#1458177111yescharacteristic
211C2#1912824711yescharacteristic
212C2#1912826811yescharacteristic
213C2#1912840112yes—
214C2#1912841012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC2×C4×D4×C83C2
prime radicalC22×D4×C83C22
hypercenterC2×C4×D4D83
hypocenterC83#2566
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×#14581784C2semidirect
2. #19127624C2semidirect
5. #19127574C2semidirect
6. #19127570C2semidirect
7. #19127566C2semidirect
8. C4×C4⋊Dic83C2semidirect
108. C83#2566semidirect

Maximal subgroups

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

subgroupindex
C2×#145817842
#191276242
#191276042
#191275742
#191275702
#191275662
C4×C4⋊Dic832
C2×C4×D4×C832
#256683