Finite Group Databasebeta

← #44548678

Subgroups — #44548678

Order 32128.

264 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 79 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×#22548673C211yesmaximal, characteristic
2#30631737C212yesmaximal, complemented
3#30631413C211yesmaximal, characteristic, complemented
4#30631410C212yesmaximal
5#324×C251C211yesmaximal, characteristic, complemented
6C22×C4×D251C411yescharacteristic
7#22548673—22no—
8#22548673—22no—
9C2×C4⋊Dic251C411yescharacteristic
10C2×(C2×D251)⋊C4C411yescharacteristic
11C2×(C2×D251)⋊C4C2211yescharacteristic
12C2×C4×Dic251C411yescharacteristic
13C2×C4×Dic251C2211yescharacteristic
14Dic251⋊C8—22no—
15C22⋊(C251⋊C8)C2212yes—
16C8×Dic251—22no—
17C2×C4⋊C4×C251C2211yescharacteristic
18C22⋊C8×C251C2212yes—
19C23×D251C2×C411yescharacteristic
20C22×Dic251C2×C411yescharacteristic
21C22×Dic251C2×C411yescharacteristic
22C22×Dic251C2×C411yescharacteristic
23C2×C4×D251—21no—
24C2×C4×D251—21no—
25C2×C4×D251—22no—
26C2×C4×D251C814yescomplemented
27C2×C251⋊C8—22no—
28C4⋊Dic251—22no—
29(C2×D251)⋊C4—22no—
30(C2×D251)⋊C4—22no—
31C4×Dic251—21no—
32C4×Dic251—21no—
33C4×Dic251D412yes—
34C4×Dic251D412yes—
35C22×C4×C251C2×C411yescharacteristic
36C22×C4×C251C2×C411yescharacteristic
37C22×C4×C251C2311yescharacteristic
38C2×C8×C251—22no—
39C4⋊C4×C251—22no—
40C22×D251—21no—
41C22×D251—21no—
42C22×D251—41no—
43C22×D251—41no—
44C22×D251C2×C812yes—
45C2×Dic251—21no—
46C2×Dic251—41no—
47C2×Dic251—41no—
48C2×Dic251C22⋊C412yes—
49C2×Dic251C22⋊C411yescharacteristic
50C2×Dic251C22⋊C411yescharacteristic
51C2×Dic251C2×C812yes—
52C2×Dic251M1612yes—
53C4×D251—24no—
54C4×D251—24no—
55C251⋊C8—42no—
56C23×C251C22×C411yescharacteristic
57C2×C4×C251—21no—
58C2×C4×C251—21no—
59C2×C4×C251—21no—
60C2×C4×C251—41no—
61C2×C4×C251—41no—
62C2×C4×C251C2×C812yes—
63C2×C4×C251C2×D412yes—
64C8×C251—42no—
65C22×C251—21no—
66C22×C251—21no—
67C22×C251C2×C22⋊C411yescharacteristic
68C22×C251C22×C811yescharacteristic
69C22×C251C2×M1611yescharacteristic
70C2×D251—22no—
71C2×D251—42no—
72C2×D251—41no—
73C2×D251—41no—
74C2×D251—42no—
75C4×C251—22no—
76C4×C251—22no—
77C4×C251—41no—
78Dic251—22no—
79Dic251—41no—
80Dic251C22⋊C812yes—
81Dic251C22⋊C812yes—
82C2×C251—21no—
83C2×C251—21no—
84C2×C251C2×C22⋊C811yescharacteristic
85C2×C251#41011yescharacteristic
86C2×C251#41311yescharacteristic
87D251—42no—
88C251#145811yescharacteristic, complemented, Sylow
89#1458—2511nomaximal, Sylow
90C2×#117—2511no—
91#431—2512no—
92#324—2511no—
93#324D25111yescharacteristic, complemented
94#323—2512no—
95C23×C4—2511no—
96#117—5022no—
97#117—5022no—
98C2×C4⋊C4—2511no—
99C2×C4⋊C4C2×D25111yescharacteristic
100C2×C22⋊C4—2511no—
101C2×C22⋊C4—2511no—
102C2×C42—2511no—
103C2×C42—2511no—
104C4⋊C8—5022no—
105C22⋊C8—2512no—
106C22⋊C8C2×D25112yes—
107C4×C8—5022no—
108C24—2511no—
109C22×C4—5021no—
110C22×C4—5021no—
111C22×C4—5022no—
112C22×C4—2511no—
113C22×C4—2511no—
114C22×C4—2514no—
115C22×C4—2511no—
116C22×C4C4×D25111yescharacteristic
117C22×C4C4×D25111yescharacteristic
118C22×C4C22×D25111yescharacteristic
119C2×C8—22no—
120C2×C8—5022no—
121C4⋊C4—22no—
122C4⋊C4—5022no—
123C22⋊C4—5022no—
124C22⋊C4—5022no—
125C42—5021no—
126C42—5021no—
127C42—2512no—
128C42—2512no—
129C23—10041no—
130C23—10041no—
131C23—5021no—
132C23—5021no—
133C23—2512no—
134C23C2×C4×D25111yescharacteristic
135C2×C4—21no—
136C2×C4—21no—
137C2×C4—21no—
138C2×C4—41no—
139C2×C4—41no—
140C2×C4—10041no—
141C2×C4—10041no—
142C2×C4—5024no—
143C2×C4—5024no—
144C2×C4—5021no—
145C2×C4—2512no—
146C2×C4—2511no—
147C2×C4—2511no—
148C2×C4—2512no—
149C2×C4—2512no—
150C2×C4C8×D25112yes—
151C2×C4D4×D25112yes—
152C8—42no—
153C8—10042no—
154C22—21no—
155C22—21no—
156C22—10042no—
157C22—10042no—
158C22—10041no—
159C22—10041no—
160C22—5022no—
161C22C22⋊C4×D25111yescharacteristic
162C22C2×C8×D25111yescharacteristic
163C22M16×D25111yescharacteristic
164C4—22no—
165C4—22no—
166C4—41no—
167C4—10041no—
168C4—5022no—
169C4—2512no—
170C4—2512no—
171C2—21no—
172C2—21no—
173C2—10042no—
174C2C22⋊C8×D25111yescharacteristic
175C2#3063165911yescharacteristic
176C2#3063167011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC22×C251C22×C8
FrattiniC22×C4C22×D251
Fitting#324×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC23×D251C2×C4
cosocleC22×C4×C251C23
hypercenter#324D251
hypocenterC251#1458
LCSG ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #30631737C2semidirect
3. #30631413C2semidirect
5. #324×C251C2semidirect
26. C2×C4×D251C8semidirect
88. C251#1458semidirect
93. #324D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486732
#306317372
#306314132
#306314102
#324×C2512
#1458251