Finite Group Databasebeta

← #44553470

Subgroups — #44553470

Order 32128.

448 conjugacy classes of subgroups; only the 185 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 185 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
2#30631572C212yesmaximal
3#30631571C212yesmaximal
4#30631570C212yesmaximal
5#30631569C212yesmaximal
6C4×Dic251⋊C4C212yesmaximal
7#30631538C212yesmaximal
8#30631533C211yesmaximal, characteristic
9#376×C251C211yesmaximal, characteristic
10#22548707C418yescomplemented
11C2×C4⋊Dic251C2211yescharacteristic
12C2×C4⋊Dic251C2211yescharacteristic
13C2×Dic251⋊C4C2212yes—
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×C4×Dic251C2212yes—
18C2×C4×Dic251C2212yes—
19C2×C4×Dic251C2212yes—
20C251⋊#95C2211yescharacteristic
21C251⋊#95C2212yes—
22C251⋊#95C2211yescharacteristic
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2212yes—
26C251⋊#95C2212yes—
27C2×C42×C251C2212yes—
28C2×C42×C251C2211yescharacteristic
29#95×C251C2211yescharacteristic
30#95×C251C2211yescharacteristic
31#95×C251C2212yes—
32C22×Dic251C2312yes—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C4⋊Dic251C2×C414yes—
37C4⋊Dic251C2×C414yes—
38Dic251⋊C4—24no—
39Dic251⋊C4—24no—
40Dic251⋊C4C2×C418yes—
41Dic251⋊C4C2×C418yes—
42C4×Dic251—24no—
43C4×Dic251—24no—
44C4×Dic251—24no—
45C22×C4×C251C2312yes—
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2311yescharacteristic
48C22×C4×C251C2311yescharacteristic
49C22×C4×C251C2312yes—
50C42×C251—24no—
51C42×C251C2×C414yes—
52C2×Dic251—22no—
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×Dic251C22×C414yes—
62C2×Dic251C22×C414yes—
63C2×Dic251#4114yes—
64C23×C251C2411yescharacteristic
65C2×C4×C251—22no—
66C2×C4×C251—22no—
67C2×C4×C251—22no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—22no—
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251C22×C412yes—
75C2×C4×C251C22×C412yes—
76C2×C4×C251C22×C412yes—
77C2×C4×C251#4114yes—
78C22×C251C23×C411yescharacteristic
79C22×C251C2×#4112yes—
80C22×C251C2×#4112yes—
81C22×C2512−1+411yescharacteristic
82C22×C2512−1+411yescharacteristic
83C4×C251—22no—
84C4×C251—22no—
85C4×C251—22no—
86C4×C251—24no—
87C4×C251—42no—
88Dic251—24no—
89Dic251—24no—
90Dic251—24no—
91Dic251—42no—
92C2×C251C4×#4111yescharacteristic
93C2×C251C4×#4111yescharacteristic
94C2×C251#51911yescharacteristic
95C2×C251#55212yes—
96C2×C251#55212yes—
97C251#237111yescharacteristic, complemented, Sylow
98#2371—2511nomaximal, Sylow
99C2×#125—2511no—
100#387—2512no—
101#387—2512no—
102#387—2512no—
103#387—2512no—
104C4×C4⋊C4—2512no—
105C4×C4⋊C4—2512no—
106#376—2511no—
107#376D25111yescharacteristic
108#125—2518no—
109C2×C4⋊C4—2512no—
110C2×C4⋊C4—2512no—
111C2×C4⋊C4—2512no—
112C2×C4⋊C4—2512no—
113C2×C4⋊C4—2511no—
114C2×C4⋊C4—2511no—
115C2×C42—2512no—
116C2×C42—2512no—
117C2×C42—2512no—
118C2×C42C2×D25112yes—
119C2×C42C2×D25111yescharacteristic
120#95—2511no—
121#95—2511no—
122#95—2512no—
123#95—2512no—
124#95—2512no—
125#95—2512no—
126#95—2512no—
127#95C2×D25111yescharacteristic
128#95C2×D25111yescharacteristic
129#95C2×D25112yes—
130C22×C4—2512no—
131C22×C4—2512no—
132C22×C4—2512no—
133C22×C4—2512no—
134C22×C4C22×D25112yes—
135C22×C4C22×D25111yescharacteristic
136C22×C4C22×D25111yescharacteristic
137C22×C4C22×D25111yescharacteristic
138C22×C4C22×D25112yes—
139C4⋊C4—5024no—
140C4⋊C4—5024no—
141C4⋊C4—2518no—
142C4⋊C4—2518no—
143C4⋊C4—2514no—
144C4⋊C4—2514no—
145C42—24no—
146C42—5024no—
147C42—5024no—
148C42—5024no—
149C42C4×D25114yes—
150C23C23×D25111yescharacteristic
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—21no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—5022no—
161C2×C4—5022no—
162C2×C4—5022no—
163C2×C4—5022no—
164C2×C4—5022no—
165C2×C4—5022no—
166C2×C4—5022no—
167C2×C4—5022no—
168C2×C4—5022no—
169C2×C4—2514no—
170C2×C4—2514no—
171C2×C4—2514no—
172C2×C4C2×C4×D25112yes—
173C2×C4C2×C4×D25112yes—
174C2×C4C2×C4×D25112yes—
175C2×C4#1686081014yes—
176C22C22×C4×D25111yescharacteristic
177C22#41×D25112yes—
178C22C2×#1686081012yes—
179C22#2254876211yescharacteristic
180C22#2254876211yescharacteristic
181C4—24no—
182C4—22no—
183C4—22no—
184C4—22no—
185C4—42no—
186C4—10042no—
187C4—5024no—
188C4—5024no—
189C4—5024no—
190C2#3063215511yescharacteristic
191C2C4×#1686081011yescharacteristic
192C2#3063216411yescharacteristic
193C2#3063237012yes—
194C2#3063237612yes—
subgroupG/H
center, FrattiniC23C23×D251
commutatorC22×C251C23×C4
Fitting#376×C251C2
socle, prime radical, cosocleC23×C251C24
hypercenter#376D251
hypocenterC251#2371
LCSG ▷ C22×C251 ▷ C251
UCSC1 ◁ C23 ◁ #376

Complements

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

NH ≅ G/Ntype
10. #22548707C4semidirect
97. C251#2371semidirect

Maximal subgroups

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

subgroupindex
C2×#225487072
#306315722
#306315712
#306315702
#306315692
C4×Dic251⋊C42
#306315382
#306315332
#376×C2512
#2371251