Finite Group Databasebeta

← #44558403

Subgroups — #44558403

Order 32128.

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

normal subgroups only

This group has 444 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×#22548692C212yesmaximal
2#30631629C212yesmaximal, complemented
3#30631628C212yesmaximal, complemented
4#30631615C212yesmaximal
5#30631612C211yesmaximal, characteristic
6#30631608C212yesmaximal, complemented
7#30631607C212yesmaximal, complemented
8#30631548C211yesmaximal, characteristic
9#397×C251C211yesmaximal, characteristic
10C22×Dic502C2212yes—
11C23×Dic251C2211yescharacteristic
12#22548692—28no—
13C2×C4⋊Dic251C2212yes—
14C2×C4⋊Dic251C2211yescharacteristic
15C2×Dic251⋊C4C2212yes—
16C2×Dic251⋊C4C2212yes—
17C2×Dic251⋊C4C2212yes—
18C2×C22⋊Dic251C2212yes—
19C2×C22⋊Dic251C2211yescharacteristic
20C251⋊#95C2212yes—
21C251⋊#95C2212yes—
22C251⋊#95C2212yes—
23C251⋊#95C2212yes—
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C251⋊#95C2212yes—
27C2×C4⋊C4×C251C2211yescharacteristic
28C2×C22⋊C4×C251C2212yes—
29C2×C22⋊C4×C251C2211yescharacteristic
30#95×C251C2212yes—
31#95×C251C2211yescharacteristic
32C2×Dic502—24no—
33C2×Dic502—42no—
34C2×Dic502—42no—
35C22×Dic251—41no—
36C22×Dic251—41no—
37C22×Dic251D414yes—
38C22×Dic251C2312yes—
39C22×Dic251C2312yes—
40C22×Dic251C2312yes—
41C22×Dic251C2312yes—
42C4⋊Dic251—24no—
43C4⋊Dic251—41no—
44Dic251⋊C4—24no—
45Dic251⋊C4—24no—
46Dic251⋊C4—42no—
47C22⋊Dic251—24no—
48C22⋊Dic251—41no—
49C24×C251C2311yescharacteristic
50C22×C4×C251C2312yes—
51C22×C4×C251C2311yescharacteristic
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C4⋊C4×C251—41no—
55C22⋊C4×C251—24no—
56C22⋊C4×C251—41no—
57Dic502—48no—
58C2×Dic251—22no—
59C2×Dic251—22no—
60C2×Dic251—22no—
61C2×Dic251—22no—
62C2×Dic251—22no—
63C2×Dic251—22no—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C2×Dic251—22no—
69C2×Dic251—44no—
70C2×Dic251C2×D414yes—
71C23×C251—21no—
72C23×C251—21no—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251C2×D412yes—
76C23×C251C2411yescharacteristic
77C2×C4×C251—22no—
78C2×C4×C251—22no—
79C2×C4×C251—21no—
80C2×C4×C251—21no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—22no—
86C2×C4×C251—22no—
87C22×C251—22no—
88C22×C251—24no—
89C22×C251—41no—
90C22×C251—42no—
91C22×C251—41no—
92C22×C251C22×D411yescharacteristic
93C22×C251H(5,𝔽2)11yescharacteristic
94C22×C251H(5,𝔽2)11yescharacteristic
95C22×C2512−1+412yes—
96C22×C2512−1+412yes—
97C4×C251—42no—
98C4×C251—42no—
99C4×C251—41no—
100C4×C251—41no—
101Dic251—24no—
102Dic251—42no—
103Dic251—42no—
104Dic251—42no—
105C2×C251—42no—
106C2×C251#53411yescharacteristic
107C2×C251#53612yes—
108C2×C251#56311yescharacteristic
109C2×C251#56312yes—
110C2×C251#56311yescharacteristic
111C251#281711yescharacteristic, complemented, Sylow
112#2817—2511nomaximal, Sylow
113C2×C22⋊Q8—2512no—
114#400—2512no—
115#400—2512no—
116#397—2511no—
117#397—2512no—
118#397D25111yescharacteristic
119#395—2512no—
120#395—2512no—
121#381—2511no—
122C22×Q8—2512no—
123C23×C4—2511no—
124C22⋊Q8—5028no—
125C2×C4⋊C4—2512no—
126C2×C4⋊C4—2511no—
127C2×C4⋊C4—2512no—
128C2×C4⋊C4—2512no—
129C2×C4⋊C4—2512no—
130C2×C4⋊C4C2×D25111yescharacteristic
131C2×C22⋊C4—2511no—
132C2×C22⋊C4—2512no—
133C2×C22⋊C4C2×D25112yes—
134C2×C22⋊C4C2×D25111yescharacteristic
135#95—2512no—
136#95—2512no—
137#95—2512no—
138#95—2512no—
139#95—2512no—
140#95—2512no—
141#95—2511no—
142#95C2×D25112yes—
143#95C2×D25111yescharacteristic
144C24C22×D25111yescharacteristic
145C2×Q8—10042no—
146C2×Q8—10042no—
147C2×Q8—5024no—
148C22×C4—10041no—
149C22×C4—10041no—
150C22×C4—2514no—
151C22×C4—2512no—
152C22×C4—2512no—
153C22×C4—2512no—
154C22×C4—2512no—
155C22×C4C22×D25111yescharacteristic
156C22×C4C22×D25111yescharacteristic
157C22×C4C22×D25112yes—
158C22×C4C22×D25112yes—
159C4⋊C4—41no—
160C4⋊C4—10042no—
161C4⋊C4—10041no—
162C4⋊C4—5024no—
163C4⋊C4—5024no—
164C4⋊C4—5024no—
165C22⋊C4—24no—
166C22⋊C4—41no—
167C22⋊C4—10041no—
168C22⋊C4—5024no—
169C23—21no—
170C23—21no—
171C23—22no—
172C23—22no—
173C23D4×D25112yes—
174C23C23×D25111yescharacteristic
175Q8—10048no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—22no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—22no—
186C2×C4—10044no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—5022no—
191C2×C4—5022no—
192C2×C4—5022no—
193C2×C4—5022no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—2514no—
199C22—22no—
200C22—24no—
201C22—41no—
202C22—42no—
203C22—41no—
204C22C2×D4×D25111yescharacteristic
205C22#2254876011yescharacteristic
206C22#2254876111yescharacteristic
207C22#2254876212yes—
208C22#2254876212yes—
209C4—42no—
210C4—42no—
211C4—41no—
212C4—41no—
213C4—10042no—
214C4—10042no—
215C4—10042no—
216C4—5024no—
217C2—42no—
218C2#3063224511yescharacteristic
219C2#3063225912yes—
220C2#3063243711yescharacteristic
221C2#3063243912yes—
222C2#3063244011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#397×C251C2
prime radicalC24×C251C23
hypercenter#397D251
hypocenterC251#2817
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #397

Complements

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

NH ≅ G/Ntype
2. #30631629C2semidirect
3. #30631628C2semidirect
6. #30631608C2semidirect
7. #30631607C2semidirect
111. C251#2817semidirect

Maximal subgroups

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

subgroupindex
C2×#225486922
#306316292
#306316282
#306316152
#306316122
#306316082
#306316072
#306315482
#397×C2512
#2817251