Finite Group Databasebeta

← #44558415

Subgroups — #44558415

Order 32128.

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

normal subgroups only

This group has 464 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
1C2×#22548703C211yesmaximal, characteristic
2#30631622C212yesmaximal, complemented
3#30631613C212yesmaximal, complemented
4#30631609C214yesmaximal, complemented
5#30631607C212yesmaximal, complemented
6#30631604C212yesmaximal
7#30631553C211yesmaximal, characteristic, complemented
8C2×#125×C251C211yesmaximal, characteristic, complemented
9C22×Dic502C2211yescharacteristic
10C22×D1004C2211yescharacteristic
11#22548703—24no—
12C2×C4⋊Dic251C2212yes—
13C2×Dic251⋊C4C2214yes—
14C2×(C2×D251)⋊C4C2212yes—
15C2×(C2×D251)⋊C4C2212yes—
16C2×(C2×D251)⋊C4C2214yes—
17C2×(C2×D251)⋊C4C2214yes—
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2214yes—
21#125×C251—24no—
22C2×C4⋊C4×C251C2212yes—
23C2×C4⋊C4×C251C2212yes—
24C2×C4⋊C4×C251C2211yescharacteristic
25C2×C4⋊C4×C251C2211yescharacteristic
26C2×C42×C251C2211yescharacteristic
27C23×D251C2312yes—
28C2×Dic502—22no—
29C2×Dic502—42no—
30C2×D1004—22no—
31C2×D1004—42no—
32C22×Dic251C2312yes—
33C22×Dic251C2314yes—
34C4⋊Dic251—42no—
35Dic251⋊C4—44no—
36(C2×D251)⋊C4—24no—
37(C2×D251)⋊C4—24no—
38(C2×D251)⋊C4—44no—
39(C2×D251)⋊C4—44no—
40C22×C4×C251C2312yes—
41C22×C4×C251C2312yes—
42C22×C4×C251C2311yescharacteristic
43C22×C4×C251C2311yescharacteristic
44C22×C4×C251C2311yescharacteristic
45C4⋊C4×C251—24no—
46C4⋊C4×C251—22no—
47C4⋊C4×C251—22no—
48C4⋊C4×C251—24no—
49C42×C251D414yes—
50C22×D251—22no—
51C22×D251—22no—
52C22×D251—24no—
53C22×D251—22no—
54C22×D251—24no—
55Dic502—44no—
56D1004—44no—
57C2×Dic251—22no—
58C2×Dic251—24no—
59C2×Dic251—24no—
60C2×Dic251—24no—
61C2×Dic251—24no—
62C23×C251C2411yescharacteristic
63C2×C4×C251—21no—
64C2×C4×C251—22no—
65C2×C4×C251—21no—
66C2×C4×C251—22no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—22no—
70C2×C4×C251—24no—
71C2×C4×C251—22no—
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D412yes—
74C2×C4×C251C2×D412yes—
75C22×C251C22×D411yescharacteristic
76C22×C251H(5,𝔽2)12yes—
77C22×C251H(5,𝔽2)11yescharacteristic
78C22×C2512−1+411yescharacteristic
79C22×C2512−1+412yes—
80C2×D251—42no—
81C2×D251—42no—
82C2×D251—42no—
83C2×D251—44no—
84C2×D251—44no—
85C4×C251—22no—
86C4×C251—22no—
87C4×C251—22no—
88C4×C251—42no—
89C4×C251—42no—
90Dic251—42no—
91Dic251—44no—
92C2×C251#53511yescharacteristic
93C2×C251#53611yescharacteristic
94C2×C251#53711yescharacteristic
95C2×C251#56212yes—
96C2×C251#56312yes—
97D251—82no—
98C251#282011yescharacteristic, complemented, Sylow
99#2820—2511nomaximal, Sylow
100C2×#125D25111yescharacteristic, complemented
101C2×#124—2511no—
102#399—2512no—
103#397—2512no—
104#396—2514no—
105#395—2512no—
106#394—2512no—
107#382—2511no—
108C22×Q8—2511no—
109C22×D4—2511no—
110#125—24no—
111#124—5024no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2514no—
114C2×C4⋊C4C2×D25112yes—
115C2×C4⋊C4C2×D25112yes—
116C2×C4⋊C4C2×D25111yescharacteristic
117C2×C4⋊C4C2×D25111yescharacteristic
118C2×C22⋊C4—2514no—
119C2×C22⋊C4—2514no—
120C2×C22⋊C4—2512no—
121C2×C22⋊C4—2512no—
122C2×C42C2×D25111yescharacteristic
123#95—2512no—
124#95—2512no—
125#95—2514no—
126C24—2512no—
127C2×Q8—10042no—
128C2×Q8—5022no—
129C2×D4—10042no—
130C2×D4—5022no—
131C22×C4—2514no—
132C22×C4—2512no—
133C22×C4C22×D25112yes—
134C22×C4C22×D25112yes—
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25111yescharacteristic
137C22×C4C22×D25111yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—10044no—
143C4⋊C4—10042no—
144C22⋊C4—10044no—
145C22⋊C4—10044no—
146C22⋊C4—5024no—
147C22⋊C4—5024no—
148C42C22⋊D25114yes—
149C23—5022no—
150C23—5022no—
151C23—5024no—
152C23—5022no—
153C23—5024no—
154C23C23×D25111yescharacteristic
155Q8—10044no—
156D4—10044no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—24no—
165C2×C4—22no—
166C2×C4—5022no—
167C2×C4—5024no—
168C2×C4—5024no—
169C2×C4—5024no—
170C2×C4—5024no—
171C2×C4C2×C22⋊D25112yes—
172C2×C4C2×C22⋊D25112yes—
173C2×C4C2×C22⋊D25112yes—
174C22—10042no—
175C22—10042no—
176C22—10042no—
177C22—10044no—
178C22—10044no—
179C22C22×C22⋊D25111yescharacteristic
180C22#2254876112yes—
181C22#2254876111yescharacteristic
182C22#2254876211yescharacteristic
183C22#2254876312yes—
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—10042no—
190C4—10044no—
191C2—20082no—
192C2#3063225111yescharacteristic
193C2#3063226111yescharacteristic
194C2#3063226511yescharacteristic
195C2#3063243512yes—
196C2#3063244112yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#125×C251C2
prime radicalC22×D1004C22
hypercenterC2×#125D251
hypocenterC251#2820
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #30631622C2semidirect
3. #30631613C2semidirect
4. #30631609C2semidirect
5. #30631607C2semidirect
7. #30631553C2semidirect
8. C2×#125×C251C2semidirect
98. C251#2820semidirect
100. C2×#125D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487032
#306316222
#306316132
#306316092
#306316072
#306316042
#306315532
C2×#125×C2512
#2820251