Finite Group Databasebeta

← #44558550

Subgroups — #44558550

Order 32128.

372 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 117 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
1C251⋊#401C212yesmaximal, complemented
2#30631623C211yesmaximal, characteristic
3#30631621C212yesmaximal, complemented
4#30631619C212yesmaximal, complemented
5#30631617C212yesmaximal, complemented
6#30631615C212yesmaximal
7#30631562C212yesmaximal
8#30631559C211yesmaximal, characteristic
9#397×C251C211yesmaximal, characteristic
10C23×Dic251C2211yescharacteristic
11C2×C4⋊Dic251C2212yes—
12C2×C4⋊Dic251C2212yes—
13C2×C4⋊Dic251C2211yescharacteristic
14C2×Dic251⋊C4C2212yes—
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×C22⋊Dic251C2211yescharacteristic
18C2×C22⋊Dic251C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C2×C4⋊C4×C251C2211yescharacteristic
27C2×C22⋊C4×C251C2212yes—
28C2×C22⋊C4×C251C2211yescharacteristic
29#95×C251C2212yes—
30#95×C251C2211yescharacteristic
31C22×Dic251—41no—
32C22×Dic251—42no—
33C22×Dic251C2312yes—
34C22×Dic251C2312yes—
35C22×Dic251C2312yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—41no—
38C4⋊Dic251—42no—
39C4⋊Dic251—42no—
40Dic251⋊C4—42no—
41Dic251⋊C4—42no—
42Dic251⋊C4—42no—
43C22⋊Dic251—22no—
44C22⋊Dic251—42no—
45C24×C251C2311yescharacteristic
46C22×C4×C251C2311yescharacteristic
47C22×C4×C251C2312yes—
48C22×C4×C251C2312yes—
49C22×C4×C251C2311yescharacteristic
50C4⋊C4×C251—41no—
51C22⋊C4×C251—24no—
52C22⋊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—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—44no—
66C23×C251—21no—
67C23×C251—22no—
68C23×C251—22no—
69C23×C251—21no—
70C23×C251#4112yes—
71C23×C251C2411yescharacteristic
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—21no—
75C2×C4×C251—21no—
76C2×C4×C251—22no—
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—22no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C22×C251—24no—
83C22×C251—22no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251—42no—
87C22×C251C2×#4111yescharacteristic
88C22×C251H(5,𝔽2)11yescharacteristic
89C22×C251H(5,𝔽2)12yes—
90C22×C2512−1+412yes—
91C22×C2512−1+411yescharacteristic
92C4×C251—42no—
93C4×C251—42no—
94C4×C251—41no—
95C4×C251—41no—
96Dic251—42no—
97Dic251—42no—
98Dic251—42no—
99Dic251—42no—
100C2×C251—42no—
101C2×C251#53912yes—
102C2×C251#53911yescharacteristic
103C2×C251#56011yescharacteristic
104C2×C251#56311yescharacteristic
105C2×C251#56312yes—
106C251#283411yescharacteristic, complemented, Sylow
107#2834—2511nomaximal, Sylow
108#401—2512no—
109#399—2511no—
110#398—2512no—
111#398—2512no—
112#398—2512no—
113#397—2512no—
114#397D25111yescharacteristic
115#385—2511no—
116#385—2512no—
117C23×C4—2511no—
118C2×C4⋊C4—2512no—
119C2×C4⋊C4—2512no—
120C2×C4⋊C4—2512no—
121C2×C4⋊C4—2512no—
122C2×C4⋊C4—2512no—
123C2×C4⋊C4—2511no—
124C2×C4⋊C4C2×D25111yescharacteristic
125C2×C22⋊C4—2512no—
126C2×C22⋊C4—2511no—
127C2×C22⋊C4C2×D25112yes—
128C2×C22⋊C4C2×D25111yescharacteristic
129#95—2512no—
130#95—2512no—
131#95—2512no—
132#95—2512no—
133#95—2512no—
134#95—2512no—
135#95—2511no—
136#95C2×D25111yescharacteristic
137#95C2×D25112yes—
138C24C22×D25111yescharacteristic
139C22×C4—10041no—
140C22×C4—10042no—
141C22×C4—2512no—
142C22×C4—2512no—
143C22×C4—2512no—
144C22×C4—2512no—
145C22×C4C22×D25111yescharacteristic
146C22×C4C22×D25112yes—
147C22×C4C22×D25112yes—
148C22×C4C22×D25111yescharacteristic
149C4⋊C4—41no—
150C4⋊C4—10042no—
151C4⋊C4—10042no—
152C4⋊C4—10042no—
153C4⋊C4—10041no—
154C4⋊C4—10042no—
155C4⋊C4—10042no—
156C22⋊C4—24no—
157C22⋊C4—41no—
158C22⋊C4—10042no—
159C22⋊C4—5022no—
160C23—21no—
161C23—22no—
162C23—22no—
163C23—21no—
164C23#1686080912yes—
165C23C23×D25111yescharacteristic
166C2×C4—22no—
167C2×C4—22no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—10044no—
177C2×C4—5022no—
178C2×C4—5022no—
179C2×C4—5022no—
180C2×C4—5022no—
181C2×C4—5022no—
182C2×C4—5022no—
183C2×C4—5022no—
184C2×C4—5022no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4—5022no—
188C2×C4—5022no—
189C22—24no—
190C22—22no—
191C22—41no—
192C22—41no—
193C22—42no—
194C22C2×#1686080911yescharacteristic
195C22#2254876011yescharacteristic
196C22#2254876112yes—
197C22#2254876212yes—
198C22#2254876211yescharacteristic
199C4—42no—
200C4—42no—
201C4—41no—
202C4—41no—
203C4—10042no—
204C4—10042no—
205C4—10042no—
206C4—10042no—
207C2—42no—
208C2#3063228112yes—
209C2#3063228211yescharacteristic
210C2#3063242711yescharacteristic
211C2#3063243711yescharacteristic
212C2#3063243912yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#397×C251C2
prime radicalC24×C251C23
hypercenter#397D251
hypocenterC251#2834
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
1. C251⋊#401C2semidirect
3. #30631621C2semidirect
4. #30631619C2semidirect
5. #30631617C2semidirect
106. C251#2834semidirect

Maximal subgroups

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

subgroupindex
C251⋊#4012
#306316232
#306316212
#306316192
#306316172
#306316152
#306315622
#306315592
#397×C2512
#2834251