Finite Group Databasebeta

← #24912669

Subgroups — #24912669

Order 10624.

500 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 177 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×#14581791C211yesmaximal, characteristic
2C2×#14581757C211yesmaximal, characteristic
3#19127621C214yesmaximal
4#19127607C211yesmaximal, characteristic
5#19127606C211yesmaximal, characteristic
6#19127596C214yesmaximal
7#19127570C211yesmaximal, characteristic
8C4×Dic83⋊C4C211yesmaximal, characteristic
9#384×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C22×Dic166C2212yes—
12#14581791—24no—
13#14581757—24no—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×Dic83⋊C4C2212yes—
16C2×Dic83⋊C4C2212yes—
17C2×Dic83⋊C4C2214yes—
18C2×Dic83⋊C4C2212yes—
19C2×C4×Dic83C2211yescharacteristic
20C2×C4×Dic83C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2214yes—
24C83⋊#95C2214yes—
25C2×C4⋊C4×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2214yes—
28C2×C42×C83C2211yescharacteristic
29C2×Dic166—22no—
30C2×Dic166—22no—
31C2×Dic166—24no—
32C2×Dic166—44no—
33C2×Dic166Q814yescomplemented
34C22×Dic83C2312yes—
35C22×Dic83C2311yescharacteristic
36C22×Dic83C2312yes—
37C22×Dic83C2312yes—
38C22×Dic83C2311yescharacteristic
39C4⋊Dic83—22no—
40Dic83⋊C4—24no—
41Dic83⋊C4—24no—
42Dic83⋊C4—44no—
43Dic83⋊C4Q818yescomplemented
44C4×Dic83—22no—
45C4×Dic83—24no—
46C4×Dic83D414yes—
47C22×C4×C83C2312yes—
48C22×C4×C83C2312yes—
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—22no—
52C4⋊C4×C83—22no—
53C4⋊C4×C83—44no—
54C42×C83—22no—
55C23×C83C2411yescharacteristic
56C2×C4×C83—22no—
57C2×C4×C83—22no—
58C2×C4×C83—22no—
59C2×C4×C83—22no—
60C2×C4×C83—24no—
61C2×C4×C83—24no—
62C2×C4×C83C2×D412yes—
63C2×C4×C83C2×Q814yes—
64C2×C4×C83C2×Q812yes—
65C2×C4×C83#4112yes—
66C2×Dic83—22no—
67C2×Dic83—22no—
68C2×Dic83—22no—
69C2×Dic83—22no—
70C2×Dic83—22no—
71C2×Dic83—22no—
72C2×Dic83—24no—
73C2×Dic83C2×D412yes—
74C2×Dic83C2×D412yes—
75C2×Dic83C2×Q814yes—
76C2×Dic83C2×Q814yes—
77C2×Dic83C2×Q814yes—
78Dic166—28no—
79Dic166—48no—
80C22×C83C22×D411yescharacteristic
81C22×C83C22×Q811yescharacteristic
82C22×C83C22×Q812yes—
83C22×C83C2×#4111yescharacteristic
84C22×C83H(5,𝔽2)12yes—
85C4×C83—24no—
86C4×C83—42no—
87C4×C83—42no—
88C4×C83C22⋊Q814yes—
89Dic83—24no—
90Dic83—22no—
91Dic83—24no—
92Dic83—42no—
93Dic83C4⋊Q814yes—
94C2×C83C2×C22⋊Q811yescharacteristic
95C2×C83C2×C4⋊Q811yescharacteristic
96C2×C83#53511yescharacteristic
97C2×C83#55412yes—
98C2×C83Q8212yes—
99#2567—831nomaximal, Sylow
100C83#256711yescharacteristic, complemented, Sylow
101C2×C4⋊Q8—831no—
102C2×C4×Q8—831no—
103#395—834no—
104#391—831no—
105#391—831no—
106#389—834no—
107#384—831no—
108#384D8311yescharacteristic
109C4×C4⋊C4—831no—
110C22×Q8—832no—
111C22×Q8—831no—
112C4⋊Q8—1664no—
113C4×Q8—1664no—
114C2×C4⋊C4—834no—
115C2×C4⋊C4—832no—
116C2×C4⋊C4—832no—
117C2×C4⋊C4—831no—
118C2×C4⋊C4—832no—
119C2×C4⋊C4C2×D8311yescharacteristic
120C2×C4⋊C4C2×D8311yescharacteristic
121C2×C4⋊C4C2×D8314yes—
122C2×C42—831no—
123C2×C42—832no—
124C2×C42—831no—
125C2×C42C2×D8311yescharacteristic
126#95—832no—
127#95—834no—
128#95—834no—
129C2×Q8—3324no—
130C2×Q8—834no—
131C2×Q8—1664no—
132C2×Q8—1662no—
133C2×Q8—1662no—
134C22×C4—831no—
135C22×C4—832no—
136C22×C4—832no—
137C22×C4—831no—
138C22×C4—832no—
139C22×C4C22×D8312yes—
140C22×C4C22×D8312yes—
141C22×C4C22×D8312yes—
142C22×C4C22×D8311yescharacteristic
143C4⋊C4—22no—
144C4⋊C4—22no—
145C4⋊C4—44no—
146C4⋊C4—3324no—
147C4⋊C4—838no—
148C4⋊C4—1664no—
149C4⋊C4—1664no—
150C4⋊C4—1662no—
151C42—22no—
152C42—834no—
153C42—1664no—
154C42—1662no—
155C23C23×D8311yescharacteristic
156Q8—3328no—
157Q8—1668no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—832no—
165C2×C4—834no—
166C2×C4—832no—
167C2×C4—834no—
168C2×C4—834no—
169C2×C4—1662no—
170C2×C4—1664no—
171C2×C4—1662no—
172C2×C4—1662no—
173C2×C4—1662no—
174C2×C4—1662no—
175C2×C4—1662no—
176C2×C4D4×D8312yes—
177C2×C4Q8×D8314yes—
178C2×C4Q8×D8312yes—
179C2×C4#1305059012yes—
180C22C2×D4×D8311yescharacteristic
181C22C2×Q8×D8312yes—
182C22C2×Q8×D8311yescharacteristic
183C22C2×#1305059011yescharacteristic
184C22#1458183712yes—
185C4—24no—
186C4—42no—
187C4—42no—
188C4—3322no—
189C4—834no—
190C4—1664no—
191C4—1664no—
192C4—1662no—
193C4#1458177014yes—
194C2C2×#1458177011yescharacteristic
195C2C4⋊Q8×D8311yescharacteristic
196C2#1912826411yescharacteristic
197C2#1912840612yes—
198C2#1912843112yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, prime radical, cosocleC23×C83C24
Fitting#384×C83C2
hypercenter#384D83
hypocenterC83#2567
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #384

Complements

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

NH ≅ G/Ntype
33. C2×Dic166Q8semidirect
43. Dic83⋊C4Q8semidirect
100. C83#2567semidirect

Maximal subgroups

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

subgroupindex
C2×#145817912
C2×#145817572
#191276212
#191276072
#191276062
#191275962
#191275702
C4×Dic83⋊C42
#384×C832
#256783