Finite Group Databasebeta

← #44558431

Subgroups — #44558431

Order 32128.

380 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 125 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×#22548705C212yesmaximal
2#30631629C212yesmaximal
3#30631628C212yesmaximal
4#30631621C212yesmaximal
5#30631620C212yesmaximal
6#30631620C212yesmaximal
7#30631617C211yesmaximal, characteristic
8#30631552C211yesmaximal, characteristic
9#398×C251C211yesmaximal, characteristic
10#22548705—28no—
11C2×C4⋊Dic251C2212yes—
12C2×C4⋊Dic251C2211yescharacteristic
13C2×C4⋊Dic251C2211yescharacteristic
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2212yes—
18C2×Dic251⋊C4C2212yes—
19C2×C4×Dic251C2211yescharacteristic
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2211yescharacteristic
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C251⋊#95C2212yes—
27C2×C4⋊C4×C251C2212yes—
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C4⋊C4×C251C2211yescharacteristic
30#95×C251C2212yes—
31#95×C251C2211yescharacteristic
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C4⋊Dic251—24no—
37C4⋊Dic251—41no—
38C4⋊Dic251—41no—
39Dic251⋊C4—24no—
40Dic251⋊C4—24no—
41Dic251⋊C4—24no—
42Dic251⋊C4—24no—
43Dic251⋊C4—42no—
44C4×Dic251D414yes—
45C22×C4×C251C2311yescharacteristic
46C22×C4×C251C2312yes—
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50C4⋊C4×C251—24no—
51C4⋊C4×C251—41no—
52C4⋊C4×C251—41no—
53C2×Dic251—22no—
54C2×Dic251—22no—
55C2×Dic251—22no—
56C2×Dic251—22no—
57C2×Dic251—22no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—24no—
63C2×Dic251C2×D414yes—
64C23×C251C2411yescharacteristic
65C2×C4×C251—21no—
66C2×C4×C251—21no—
67C2×C4×C251—22no—
68C2×C4×C251—22no—
69C2×C4×C251—21no—
70C2×C4×C251—22no—
71C2×C4×C251—21no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—22no—
76C2×C4×C251—22no—
77C2×C4×C251C2×D412yes—
78C22×C251C22×D411yescharacteristic
79C22×C251H(5,𝔽2)11yescharacteristic
80C22×C2512−1+412yes—
81C22×C2512−1+411yescharacteristic
82C22×C2512−1+412yes—
83C4×C251—22no—
84C4×C251—42no—
85C4×C251—42no—
86C4×C251—41no—
87C4×C251—41no—
88Dic251—24no—
89Dic251—42no—
90Dic251—42no—
91Dic251—42no—
92C2×C251#53612yes—
93C2×C251#53711yescharacteristic
94C2×C251#56311yescharacteristic
95C2×C251#56311yescharacteristic
96C2×C251#56412yes—
97C251#282211yescharacteristic, complemented, Sylow
98#2822—2511nomaximal, Sylow
99C2×#125—2512no—
100#400—2512no—
101#400—2512no—
102#398—2511no—
103#398—2512no—
104#398—2512no—
105#398—2512no—
106#398D25111yescharacteristic
107#382—2511no—
108#125—5028no—
109C2×C4⋊C4—2511no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4—2511no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2512no—
114C2×C4⋊C4—2512no—
115C2×C4⋊C4—2512no—
116C2×C4⋊C4—2512no—
117C2×C4⋊C4C2×D25112yes—
118C2×C4⋊C4C2×D25111yescharacteristic
119C2×C4⋊C4C2×D25111yescharacteristic
120C2×C42—2511no—
121#95—2512no—
122#95—2512no—
123#95—2512no—
124#95—2512no—
125#95—2512no—
126#95—2512no—
127#95—2511no—
128#95C2×D25112yes—
129#95C2×D25111yescharacteristic
130C22×C4—2512no—
131C22×C4—2512no—
132C22×C4—2512no—
133C22×C4—2512no—
134C22×C4C22×D25111yescharacteristic
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25112yes—
137C22×C4C22×D25112yes—
138C22×C4C22×D25111yescharacteristic
139C4⋊C4—24no—
140C4⋊C4—41no—
141C4⋊C4—41no—
142C4⋊C4—10042no—
143C4⋊C4—10041no—
144C4⋊C4—10041no—
145C4⋊C4—5024no—
146C4⋊C4—5024no—
147C4⋊C4—5024no—
148C4⋊C4—5024no—
149C4⋊C4—5024no—
150C42—2514no—
151C23C23×D25111yescharacteristic
152C2×C4—21no—
153C2×C4—21no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—5022no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—5022no—
170C2×C4—5022no—
171C2×C4—5024no—
172C2×C4—5022no—
173C2×C4—5022no—
174C2×C4—2514no—
175C2×C4D4×D25112yes—
176C22C2×D4×D25111yescharacteristic
177C22#2254876011yescharacteristic
178C22#2254876212yes—
179C22#2254876211yescharacteristic
180C22#2254876312yes—
181C4—22no—
182C4—42no—
183C4—42no—
184C4—41no—
185C4—41no—
186C4—10042no—
187C4—10042no—
188C4—10042no—
189C4—5024no—
190C2#3063226012yes—
191C2#3063226411yescharacteristic
192C2#3063243611yescharacteristic
193C2#3063244011yescharacteristic
194C2C251⋊#56412yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, prime radical, cosocleC23×C251C24
Fitting#398×C251C2
hypercenter#398D251
hypocenterC251#2822
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #398

Complements

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

NH ≅ G/Ntype
97. C251#2822semidirect

Maximal subgroups

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

subgroupindex
C2×#225487052
#306316292
#306316282
#306316212
#306316202
#306316172
#306315522
#398×C2512
#2822251