Finite Group Databasebeta

← #24912656

Subgroups — #24912656

Order 10624.

452 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 165 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×#14581783C211yesmaximal, characteristic
2C2×Q8×Dic83C211yesmaximal, characteristic
3#19127633C214yesmaximal
4#19127608C211yesmaximal, characteristic
5#19127584C214yesmaximal
6#19127571C211yesmaximal, characteristic
7#19127566C211yesmaximal, characteristic
8C4×Dic83⋊C4C211yesmaximal, characteristic
9#391×C83C211yesmaximal, characteristic
10#14581783—24no—
11Q8×Dic83—24no—
12C2×C4⋊Dic83C2211yescharacteristic
13C2×C4⋊Dic83C2214yes—
14C2×C4⋊Dic83C2211yescharacteristic
15C2×C4⋊Dic83C2212yes—
16C2×C4⋊Dic83C2212yes—
17C2×Dic83⋊C4C2212yes—
18C2×Dic83⋊C4C2212yes—
19C2×C4×Dic83C2211yescharacteristic
20C2×C4×Dic83C2212yes—
21C2×C4×Dic83C2211yescharacteristic
22C83⋊#95C2212yes—
23C83⋊#95C2214yes—
24C83⋊#95C2214yes—
25C22×Q8×C83C2211yescharacteristic
26C2×C4⋊C4×C83C2211yescharacteristic
27C2×C42×C83C2211yescharacteristic
28#95×C83C2214yes—
29C22×Dic83C2312yes—
30C22×Dic83C2312yes—
31C22×Dic83C2312yes—
32C22×Dic83C2311yescharacteristic
33C22×Dic83C2311yescharacteristic
34C4⋊Dic83—22no—
35C4⋊Dic83—22no—
36C4⋊Dic83—28no—
37C4⋊Dic83—24no—
38C4⋊Dic83—24no—
39Dic83⋊C4—24no—
40Dic83⋊C4—24no—
41C4×Dic83—22no—
42C4×Dic83—24no—
43C4×Dic83D414yes—
44C2×Q8×C83—22no—
45C2×Q8×C83—22no—
46C2×Q8×C83Q814yes—
47C22×C4×C83C2312yes—
48C22×C4×C83C2312yes—
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—22no—
52C42×C83—22no—
53C23×C83C2411yescharacteristic
54Q8×C83—28no—
55C2×C4×C83—22no—
56C2×C4×C83—22no—
57C2×C4×C83—22no—
58C2×C4×C83—22no—
59C2×C4×C83—22no—
60C2×C4×C83—24no—
61C2×C4×C83C2×D412yes—
62C2×C4×C83C2×Q814yes—
63C2×C4×C83C2×Q812yes—
64C2×C4×C83#4112yes—
65C2×C4×C83#4114yes—
66C2×Dic83—22no—
67C2×Dic83—22no—
68C2×Dic83—22no—
69C2×Dic83—22no—
70C2×Dic83—22no—
71C2×Dic83—22no—
72C2×Dic83—24no—
73C2×Dic83—24no—
74C2×Dic83—22no—
75C2×Dic83C2×D412yes—
76C2×Dic83C2×D412yes—
77C22×C83C22×D411yescharacteristic
78C22×C83C22×Q811yescharacteristic
79C22×C83C2×#4111yescharacteristic
80C22×C83C2×#4112yes—
81C22×C832−1+412yes—
82C4×C83—24no—
83C4×C83—24no—
84C4×C83—42no—
85C4×C83C22⋊Q814yes—
86Dic83—22no—
87Dic83—22no—
88Dic83—42no—
89Dic83—42no—
90Dic83—42no—
91C2×C83C2×C22⋊Q811yescharacteristic
92C2×C83#53211yescharacteristic
93C2×C83#53611yescharacteristic
94C2×C83#55212yes—
95C2×C83#55712yes—
96#2565—831nomaximal, Sylow
97C83#256511yescharacteristic, complemented, Sylow
98C2×#125—831no—
99C2×C4×Q8—831no—
100#398—834no—
101#391—831no—
102#391D8311yescharacteristic
103#387—834no—
104#384—831no—
105#382—831no—
106C4×C4⋊C4—831no—
107C22×Q8C2×D8311yescharacteristic
108#125—1664no—
109C4×Q8—1664no—
110C2×C4⋊C4—831no—
111C2×C4⋊C4—832no—
112C2×C4⋊C4—831no—
113C2×C4⋊C4—832no—
114C2×C4⋊C4—832no—
115C2×C4⋊C4—832no—
116C2×C4⋊C4—834no—
117C2×C4⋊C4C2×D8311yescharacteristic
118C2×C42—831no—
119C2×C42—832no—
120C2×C42—831no—
121C2×C42C2×D8311yescharacteristic
122#95—832no—
123#95—834no—
124#95—834no—
125#95C2×D8314yes—
126C2×Q8—22no—
127C2×Q8—22no—
128C2×Q8Dic16614yes—
129C22×C4—831no—
130C22×C4—831no—
131C22×C4—832no—
132C22×C4—832no—
133C22×C4—832no—
134C22×C4C22×D8312yes—
135C22×C4C22×D8312yes—
136C22×C4C22×D8312yes—
137C22×C4C22×D8311yescharacteristic
138C4⋊C4—22no—
139C4⋊C4—1664no—
140C4⋊C4—1664no—
141C4⋊C4—1664no—
142C4⋊C4—1662no—
143C4⋊C4—1662no—
144C4⋊C4—1664no—
145C4⋊C4—1668no—
146C42—22no—
147C42—834no—
148C42—1664no—
149C42—1662no—
150C23C23×D8311yescharacteristic
151Q8—28no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—24no—
158C2×C4—832no—
159C2×C4—832no—
160C2×C4—1662no—
161C2×C4—1662no—
162C2×C4—1662no—
163C2×C4—1662no—
164C2×C4—1664no—
165C2×C4—1664no—
166C2×C4—1662no—
167C2×C4—1662no—
168C2×C4—1662no—
169C2×C4D4×D8312yes—
170C2×C4C2×Dic16614yes—
171C2×C4C2×Dic16612yes—
172C2×C4#1305058912yes—
173C2×C4#1305059114yes—
174C22C2×D4×D8311yescharacteristic
175C22C22×Dic16611yescharacteristic
176C22C2×#1305058911yescharacteristic
177C22C2×#1305059112yes—
178C22#1458183912yes—
179C4—24no—
180C4—24no—
181C4—42no—
182C4—3322no—
183C4—3322no—
184C4—3322no—
185C4—1662no—
186C4—1662no—
187C4#1458176914yes—
188C2C2×#1458176911yescharacteristic
189C2#1912825211yescharacteristic
190C2#1912827111yescharacteristic
191C2#1912839312yes—
192C2#1912843012yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, prime radical, cosocleC23×C83C24
Fitting#391×C83C2
hypercenter#391D83
hypocenterC83#2565
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
97. C83#2565semidirect

Maximal subgroups

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

subgroupindex
C2×#145817832
C2×Q8×Dic832
#191276332
#191276082
#191275842
#191275712
#191275662
C4×Dic83⋊C42
#391×C832
#256583