Finite Group Databasebeta

← #24913172

Subgroups — #24913172

Order 10624.

388 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 133 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#19127635C212yesmaximal
2#19127621C211yesmaximal, characteristic
3#19127605C212yesmaximal
4#19127596C212yesmaximal
5#19127583C212yesmaximal
6#19127583C212yesmaximal
7#19127566C211yesmaximal, characteristic
8#19127546C212yesmaximal
9#400×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic
12C2×Dic83⋊C4C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×C4×Dic83C2212yes—
17C2×C4×Dic83C2212yes—
18C2×C4×Dic83C2211yescharacteristic
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2212yes—
25C83⋊#95C2212yes—
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28#95×C83C2211yescharacteristic
29#95×C83C2212yes—
30#95×C83C2212yes—
31C2×Dic166—24no—
32C2×Dic166—41no—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C4⋊Dic83—41no—
38Dic83⋊C4—24no—
39Dic83⋊C4—24no—
40Dic83⋊C4—24no—
41Dic83⋊C4—42no—
42C4×Dic83—24no—
43C4×Dic83—24no—
44C4×Dic83Q814yescomplemented
45C22×C4×C83C2311yescharacteristic
46C22×C4×C83C2312yes—
47C22×C4×C83C2312yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2311yescharacteristic
50C4⋊C4×C83—41no—
51C4⋊C4×C83—41no—
52C23×C83C2411yescharacteristic
53C2×C4×C83—21no—
54C2×C4×C83—21no—
55C2×C4×C83—22no—
56C2×C4×C83—22no—
57C2×C4×C83—22no—
58C2×C4×C83—22no—
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83—21no—
62C2×C4×C83—22no—
63C2×C4×C83—21no—
64C2×C4×C83—22no—
65C2×C4×C83C2×Q812yes—
66C2×Dic83—22no—
67C2×Dic83—22no—
68C2×Dic83—22no—
69C2×Dic83—22no—
70C2×Dic83—22no—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83C2×Q814yes—
76C2×Dic83#4114yes—
77C2×Dic83#4114yes—
78Dic166—44no—
79C22×C83C22×Q811yescharacteristic
80C22×C83C2×#4112yes—
81C22×C83C2×#4112yes—
82C22×C83H(5,𝔽2)11yescharacteristic
83C22×C832−1+411yescharacteristic
84C4×C83—22no—
85C4×C83—41no—
86C4×C83—41no—
87C4×C83—42no—
88C4×C83—42no—
89Dic83—24no—
90Dic83—24no—
91Dic83—24no—
92Dic83—42no—
93C2×C83#53312yes—
94C2×C83#54411yescharacteristic
95C2×C83#55111yescharacteristic
96C2×C83#55212yes—
97C2×C83#55511yescharacteristic
98#2612—831nomaximal, Sylow
99C83#261211yescharacteristic, complemented, Sylow
100#400D8311yescharacteristic
101#398—832no—
102#395—831no—
103#391—832no—
104#389—832no—
105#387—832no—
106#387—832no—
107#382—831no—
108#376—832no—
109C22×Q8—831no—
110C2×C4⋊C4—831no—
111C2×C4⋊C4—832no—
112C2×C4⋊C4—832no—
113C2×C4⋊C4—832no—
114C2×C4⋊C4—832no—
115C2×C4⋊C4C2×D8311yescharacteristic
116C2×C4⋊C4C2×D8311yescharacteristic
117C2×C42—831no—
118C2×C42—832no—
119C2×C42—832no—
120#95—832no—
121#95—832no—
122#95—832no—
123#95—832no—
124#95—832no—
125#95—831no—
126#95—832no—
127#95C2×D8312yes—
128#95C2×D8311yescharacteristic
129#95C2×D8312yes—
130C2×Q8—3321no—
131C2×Q8—1664no—
132C22×C4—832no—
133C22×C4—832no—
134C22×C4—832no—
135C22×C4—832no—
136C22×C4C22×D8311yescharacteristic
137C22×C4C22×D8312yes—
138C22×C4C22×D8311yescharacteristic
139C22×C4C22×D8312yes—
140C22×C4C22×D8311yescharacteristic
141C4⋊C4—41no—
142C4⋊C4—41no—
143C4⋊C4—3322no—
144C4⋊C4—3321no—
145C4⋊C4—1664no—
146C4⋊C4—1664no—
147C4⋊C4—1664no—
148C42—834no—
149C42—1664no—
150C42—1664no—
151C23C23×D8311yescharacteristic
152Q8—3324no—
153C2×C4—21no—
154C2×C4—21no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—21no—
162C2×C4—22no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—834no—
166C2×C4—834no—
167C2×C4—834no—
168C2×C4—1662no—
169C2×C4—1662no—
170C2×C4—1662no—
171C2×C4—1662no—
172C2×C4—1662no—
173C2×C4—1662no—
174C2×C4—1662no—
175C2×C4—1662no—
176C2×C4—1662no—
177C2×C4Q8×D8312yes—
178C22C2×Q8×D8311yescharacteristic
179C22#41×D8312yes—
180C22#41×D8312yes—
181C22#1458183711yescharacteristic
182C22#1458183911yescharacteristic
183C4—22no—
184C4—41no—
185C4—41no—
186C4—42no—
187C4—42no—
188C4—3322no—
189C4—1664no—
190C4—1664no—
191C4—1664no—
192C2#1912825712yes—
193C2#1912833011yescharacteristic
194C2#1912837911yescharacteristic
195C2#1912838712yes—
196C2#1912841311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, prime radical, cosocleC23×C83C24
Fitting#400×C83C2
hypercenter#400D83
hypocenterC83#2612
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
44. C4×Dic83Q8semidirect
99. C83#2612semidirect

Maximal subgroups

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

subgroupindex
#191276352
#191276212
#191276052
#191275962
#191275832
#191275662
#191275462
#400×C832
#261283