Finite Group Databasebeta

← #44558416

Subgroups — #44558416

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×#22548707C211yesmaximal, characteristic, complemented
2#30631623C212yesmaximal, complemented
3#30631612C212yesmaximal, complemented
4#30631610C214yesmaximal, complemented
5#30631606C212yesmaximal, complemented
6#30631601C212yesmaximal
7#30631553C211yesmaximal, characteristic, complemented
8C2×#124×C251C211yesmaximal, characteristic
9#22548707—24no—
10C2×C4⋊Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2212yes—
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2214yes—
16C2×C22⋊Dic251C2214yes—
17C2×C22⋊Dic251C2214yes—
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2214yes—
21C22×Q8×C251C2211yescharacteristic
22C22×D4×C251C2211yescharacteristic
23#124×C251—24no—
24C2×C22⋊C4×C251C2212yes—
25C2×C22⋊C4×C251C2212yes—
26C2×C42×C251C2211yescharacteristic
27C22×Dic251C2314yes—
28C22×Dic251C2312yes—
29C22×Dic251C2312yes—
30C4⋊Dic251—22no—
31C4⋊Dic251—22no—
32C4⋊Dic251—42no—
33Dic251⋊C4—24no—
34Dic251⋊C4—24no—
35Dic251⋊C4—44no—
36C22⋊Dic251—44no—
37C22⋊Dic251—44no—
38C24×C251C2312yes—
39C2×Q8×C251—22no—
40C2×Q8×C251—42no—
41C2×D4×C251—22no—
42C2×D4×C251—42no—
43C22×C4×C251C2312yes—
44C22×C4×C251C2311yescharacteristic
45C22×C4×C251C2311yescharacteristic
46C22×C4×C251C2311yescharacteristic
47C22⋊C4×C251—24no—
48C22⋊C4×C251—24no—
49C42×C251D414yes—
50C2×Dic251—22no—
51C2×Dic251—22no—
52C2×Dic251—22no—
53C2×Dic251—24no—
54C2×Dic251—24no—
55C2×Dic251—24no—
56C2×Dic251—22no—
57C2×Dic251—24no—
58C23×C251—22no—
59C23×C251—24no—
60C23×C251—22no—
61C23×C251—24no—
62C23×C251—22no—
63C23×C251C2411yescharacteristic
64Q8×C251—44no—
65D4×C251—44no—
66C2×C4×C251—22no—
67C2×C4×C251—21no—
68C2×C4×C251—22no—
69C2×C4×C251—21no—
70C2×C4×C251—22no—
71C2×C4×C251—24no—
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D412yes—
74C2×C4×C251C2×D412yes—
75C22×C251—42no—
76C22×C251—42no—
77C22×C251—42no—
78C22×C251—44no—
79C22×C251—44no—
80C22×C251C22×D411yescharacteristic
81C22×C251H(5,𝔽2)12yes—
82C22×C251H(5,𝔽2)11yescharacteristic
83C22×C2512−1+412yes—
84C22×C2512−1+411yescharacteristic
85C4×C251—22no—
86C4×C251—22no—
87C4×C251—22no—
88C4×C251—42no—
89Dic251—42no—
90Dic251—42no—
91Dic251—44no—
92C2×C251—82no—
93C2×C251#53511yescharacteristic
94C2×C251#53611yescharacteristic
95C2×C251#53711yescharacteristic
96C2×C251#56212yes—
97C2×C251#56312yes—
98C251#282011yescharacteristic, complemented, Sylow
99#2820—2511nomaximal, Sylow
100C2×#125—2511no—
101C2×#124D25111yescharacteristic
102#399—2512no—
103#397—2512no—
104#396—2514no—
105#395—2512no—
106#394—2512no—
107#382—2511no—
108C22×Q8C2×D25111yescharacteristic
109C22×D4C2×D25111yescharacteristic
110#125—5024no—
111#124—24no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2514no—
114C2×C4⋊C4—2512no—
115C2×C4⋊C4—2512no—
116C2×C4⋊C4—2511no—
117C2×C4⋊C4—2511no—
118C2×C22⋊C4—2514no—
119C2×C22⋊C4—2514no—
120C2×C22⋊C4C2×D25112yes—
121C2×C22⋊C4C2×D25112yes—
122C2×C42C2×D25111yescharacteristic
123#95—2512no—
124#95—2512no—
125#95—2514no—
126C24C22×D25112yes—
127C2×Q8—22no—
128C2×Q8—42no—
129C2×D4—22no—
130C2×D4—42no—
131C22×C4—2514no—
132C22×C4—2512no—
133C22×C4—2512no—
134C22×C4C22×D25112yes—
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25111yescharacteristic
137C22×C4C22×D25111yescharacteristic
138C4⋊C4—10044no—
139C4⋊C4—10042no—
140C4⋊C4—5024no—
141C4⋊C4—5024no—
142C4⋊C4—5022no—
143C4⋊C4—5022no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—10044no—
147C22⋊C4—10044no—
148C42C22⋊D25114yes—
149C23—22no—
150C23—24no—
151C23—22no—
152C23—24no—
153C23—22no—
154C23C23×D25111yescharacteristic
155Q8—44no—
156D4—44no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—24no—
163C2×C4—5022no—
164C2×C4—5022no—
165C2×C4—5024no—
166C2×C4—5024no—
167C2×C4—5022no—
168C2×C4—5024no—
169C2×C4—5022no—
170C2×C4—5024no—
171C2×C4C2×C22⋊D25112yes—
172C2×C4C2×C22⋊D25112yes—
173C2×C4C2×C22⋊D25112yes—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—44no—
178C22—44no—
179C22C22×C22⋊D25111yescharacteristic
180C22#2254876012yes—
181C22#2254876111yescharacteristic
182C22#2254876212yes—
183C22#2254876211yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—10042no—
189C4—10042no—
190C4—10044no—
191C2—82no—
192C2#3063225411yescharacteristic
193C2#3063225811yescharacteristic
194C2#3063226511yescharacteristic
195C2#3063243112yes—
196C2#3063243712yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#124×C251C2
prime radicalC22×D4×C251C22
hypercenterC2×#124D251
hypocenterC251#2820
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#124

Complements

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

NH ≅ G/Ntype
1. C2×#22548707C2semidirect
2. #30631623C2semidirect
3. #30631612C2semidirect
4. #30631610C2semidirect
5. #30631606C2semidirect
7. #30631553C2semidirect
98. C251#2820semidirect

Maximal subgroups

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

subgroupindex
C2×#225487072
#306316232
#306316122
#306316102
#306316062
#306316012
#306315532
C2×#124×C2512
#2820251