Finite Group Databasebeta

← #44548899

Subgroups — #44548899

Order 32128.

274 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 73 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×#22548694C211yesmaximal, characteristic
2#30631550C211yesmaximal, characteristic
3#30631425C214yesmaximal, complemented
4#407×C251C211yesmaximal, characteristic
5C22×Dic502C411yescharacteristic, complemented
6#22548694—21no—
7#22548694—21no—
8#22548694C412yescomplemented
9#22548694C2212yes—
10C2×C4⋊Dic251C411yescharacteristic, complemented
11C2×Dic251⋊C4—21no—
12C2×C22⋊Dic251—21no—
13C2×C22⋊Dic251—21no—
14C251⋊#95—21no—
15C251⋊#95C2212yes—
16C23×C4×C251C2211yescharacteristic
17C2×M16×C251—21no—
18C22⋊C8×C251—21no—
19C22⋊C8×C251C2212yes—
20C2×Dic502—21no—
21C2×Dic502—42no—
22C2×Dic502C2×C412yes—
23C22×Dic251—21no—
24C22×Dic251—21no—
25C22×Dic251—22no—
26C4⋊Dic251—21no—
27C4⋊Dic251C2×C412yes—
28Dic251⋊C4—22no—
29Dic251⋊C4—22no—
30C22⋊Dic251—22no—
31C22⋊Dic251—22no—
32C22⋊Dic251—42no—
33C24×C251D411yescharacteristic
34C22×C4×C251—42no—
35C22×C4×C251C2×C411yescharacteristic
36C22×C4×C251C2×C411yescharacteristic
37C22×C4×C251D411yescharacteristic
38C22×C4×C251D411yescharacteristic
39C22×C4×C251D411yescharacteristic
40C22×C4×C251C2311yescharacteristic
41M16×C251—42no—
42C2×C8×C251—22no—
43Dic502—42no—
44Dic502—42no—
45C2×Dic251—22no—
46C2×Dic251—22no—
47C2×Dic251—42no—
48C2×Dic251—42no—
49C2×Dic251—42no—
50C2×Dic251—42no—
51C2×Dic251—42no—
52C23×C251—22no—
53C23×C251—22no—
54C23×C251—41no—
55C23×C251C22⋊C411yescharacteristic
56C23×C251C22⋊C411yescharacteristic
57C23×C251C2×D411yescharacteristic
58C2×C4×C251—22no—
59C2×C4×C251—22no—
60C2×C4×C251—41no—
61C2×C4×C251—41no—
62C2×C4×C251—41no—
63C2×C4×C251—41no—
64C2×C4×C251C22⋊C411yescharacteristic
65C2×C4×C251C22⋊C411yescharacteristic
66C2×C4×C251C22×C411yescharacteristic
67C2×C4×C251C2×D411yescharacteristic
68C8×C251—42no—
69C22×C251—21no—
70C22×C251—21no—
71C22×C251—21no—
72C22×C251—21no—
73C22×C251—21no—
74C22×C251—21no—
75C22×C251—21no—
76C22×C251—44no—
77C22×C251#9912yes—
78C22×C251C2×C22⋊C411yescharacteristic
79C22×C251#13712yes—
80C4×C251—21no—
81C4×C251—21no—
82C4×C251—21no—
83C4×C251—21no—
84Dic251—42no—
85Dic251—42no—
86Dic251—82no—
87C2×C251—22no—
88C2×C251—22no—
89C2×C251—41no—
90C2×C251C2×#9911yescharacteristic
91C2×C251#41911yescharacteristic
92C2×C251#42111yescharacteristic
93C251#151111yescharacteristic, complemented, Sylow
94#1511—2511nomaximal, Sylow
95C2×C22⋊Q8—2511no—
96#407D25111yescharacteristic
97#381—2511no—
98#328—2514no—
99C22×Q8—2511no—
100C23×C4C2×D25111yescharacteristic
101C2×M16—21no—
102C22⋊Q8—5021no—
103C22⋊Q8—5021no—
104C22⋊Q8—2512no—
105C22⋊Q8—2512no—
106C2×C4⋊C4—5021no—
107C2×C4⋊C4—2511no—
108C2×C22⋊C4—5021no—
109C2×C22⋊C4—5021no—
110C22⋊C8—21no—
111C22⋊C8C2×D25112yes—
112#95—5021no—
113#95—2512no—
114C24D100411yescharacteristic
115C2×Q8—10042no—
116C2×Q8—5021no—
117C2×Q8—2512no—
118C22×C4—42no—
119C22×C4—5022no—
120C22×C4—5021no—
121C22×C4—5021no—
122C22×C4C4×D25111yescharacteristic
123C22×C4C4×D25111yescharacteristic
124C22×C4C22⋊D25111yescharacteristic
125C22×C4C22⋊D25111yescharacteristic
126C22×C4D100411yescharacteristic
127C22×C4C22×D25111yescharacteristic
128M16—42no—
129C2×C8—22no—
130C4⋊C4—5022no—
131C4⋊C4—5021no—
132C4⋊C4—5022no—
133C4⋊C4—2512no—
134C22⋊C4—10042no—
135C22⋊C4—5022no—
136C22⋊C4—5022no—
137C23—22no—
138C23—22no—
139C23—41no—
140C23(C2×D251)⋊C411yescharacteristic
141C23(C2×D251)⋊C411yescharacteristic
142C23C2×D100411yescharacteristic
143Q8—10042no—
144Q8—10042no—
145C2×C4—22no—
146C2×C4—22no—
147C2×C4—41no—
148C2×C4—41no—
149C2×C4—41no—
150C2×C4—41no—
151C2×C4—10042no—
152C2×C4—10042no—
153C2×C4—10042no—
154C2×C4—10042no—
155C2×C4—10042no—
156C2×C4—5022no—
157C2×C4—5022no—
158C2×C4(C2×D251)⋊C411yescharacteristic
159C2×C4(C2×D251)⋊C411yescharacteristic
160C2×C4C2×C4×D25111yescharacteristic
161C2×C4C2×C22⋊D25111yescharacteristic
162C8—42no—
163C22—21no—
164C22—21no—
165C22—21no—
166C22—21no—
167C22—21no—
168C22—21no—
169C22—21no—
170C22—44no—
171C22#2254862912yes—
172C22C2×(C2×D251)⋊C411yescharacteristic
173C22#2254874812yes—
174C4—21no—
175C4—21no—
176C4—21no—
177C4—21no—
178C4—20082no—
179C4—10042no—
180C4—10042no—
181C2—22no—
182C2—22no—
183C2—41no—
184C2C2×#2254862911yescharacteristic
185C2#3063169711yescharacteristic
186C2#3063170311yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#407×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC24×C251D4
cosocleC22×C4×C251C23
hypercenter#407D251
hypocenterC251#1511
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
3. #30631425C2semidirect
5. C22×Dic502C4semidirect
8. #22548694C4semidirect
10. C2×C4⋊Dic251C4semidirect
93. C251#1511semidirect

Maximal subgroups

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

subgroupindex
C2×#225486942
#306315502
#306314252
#407×C2512
#1511251