Finite Group Databasebeta

← #44558436

Subgroups — #44558436

Order 32128.

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

normal subgroups only

This group has 456 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×#22548708C212yesmaximal, complemented
2#30631621C212yesmaximal, complemented
3#30631617C211yesmaximal, characteristic, complemented
4#30631610C212yesmaximal, complemented
5#30631610C212yesmaximal, complemented
6#30631610C212yesmaximal, complemented
7#30631601C212yesmaximal
8#30631556C211yesmaximal, characteristic, complemented
9#394×C251C211yesmaximal, characteristic
10#22548708—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×C22⋊Dic251C2212yes—
20C2×C22⋊Dic251C2212yes—
21C2×C22⋊Dic251C2212yes—
22C2×C22⋊Dic251C2212yes—
23C2×C4×Dic251C2211yescharacteristic
24C251⋊#95C2212yes—
25C251⋊#95C2211yescharacteristic
26C251⋊#95C2212yes—
27C22×D4×C251C2211yescharacteristic
28C2×C4⋊C4×C251C2211yescharacteristic
29C2×C22⋊C4×C251C2212yes—
30C2×C22⋊C4×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—42no—
42Dic251⋊C4—42no—
43Dic251⋊C4—42no—
44C22⋊Dic251—24no—
45C22⋊Dic251—24no—
46C22⋊Dic251—42no—
47C22⋊Dic251—42no—
48C4×Dic251D414yescomplemented
49C24×C251C2312yes—
50C2×D4×C251—41no—
51C2×D4×C251—42no—
52C22×C4×C251C2312yes—
53C22×C4×C251C2311yescharacteristic
54C22×C4×C251C2311yescharacteristic
55C22×C4×C251C2311yescharacteristic
56C4⋊C4×C251—41no—
57C22⋊C4×C251—24no—
58C22⋊C4×C251—42no—
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—22no—
70C2×Dic251C2×D414yes—
71C23×C251—22no—
72C23×C251—22no—
73C23×C251—22no—
74C23×C251—22no—
75C23×C251—22no—
76C23×C251—22no—
77C23×C251—22no—
78C23×C251C2411yescharacteristic
79D4×C251—44no—
80C2×C4×C251—21no—
81C2×C4×C251—21no—
82C2×C4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251—21no—
86C2×C4×C251—21no—
87C2×C4×C251—22no—
88C2×C4×C251—22no—
89C2×C4×C251C2×D412yes—
90C22×C251—42no—
91C22×C251—42no—
92C22×C251—42no—
93C22×C251—42no—
94C22×C251—42no—
95C22×C251—42no—
96C22×C251—42no—
97C22×C251C22×D411yescharacteristic
98C22×C251H(5,𝔽2)11yescharacteristic
99C22×C251H(5,𝔽2)12yes—
100C22×C2512−1+411yescharacteristic
101C22×C2512−1+412yes—
102C4×C251—22no—
103C4×C251—42no—
104C4×C251—41no—
105C4×C251—41no—
106Dic251—24no—
107Dic251—42no—
108Dic251—42no—
109Dic251—42no—
110C2×C251—82no—
111C2×C251#53711yescharacteristic
112C2×C251#53712yes—
113C2×C251#56011yescharacteristic
114C2×C251#56311yescharacteristic
115C2×C251#56312yes—
116C251#282311yescharacteristic, complemented, Sylow
117#2823—2511nomaximal, Sylow
118C2×#126—2512no—
119#398—2511no—
120#398—2512no—
121#396—2512no—
122#396—2512no—
123#396—2512no—
124#394—2512no—
125#394D25111yescharacteristic
126#384—2511no—
127C22×D4C2×D25111yescharacteristic
128#126—5028no—
129C2×C4⋊C4—2512no—
130C2×C4⋊C4—2512no—
131C2×C4⋊C4—2511no—
132C2×C4⋊C4—2512no—
133C2×C4⋊C4—2511no—
134C2×C4⋊C4—2512no—
135C2×C4⋊C4—2512no—
136C2×C4⋊C4—2512no—
137C2×C4⋊C4C2×D25111yescharacteristic
138C2×C22⋊C4—2512no—
139C2×C22⋊C4—2512no—
140C2×C22⋊C4—2512no—
141C2×C22⋊C4—2512no—
142C2×C22⋊C4C2×D25112yes—
143C2×C22⋊C4C2×D25112yes—
144C2×C42—2511no—
145#95—2512no—
146#95—2511no—
147#95—2512no—
148#95C2×D25111yescharacteristic
149C24C22×D25112yes—
150C2×D4—41no—
151C2×D4—42no—
152C22×C4—2512no—
153C22×C4—2512no—
154C22×C4—2512no—
155C22×C4—2512no—
156C22×C4C22×D25111yescharacteristic
157C22×C4C22×D25112yes—
158C22×C4C22×D25111yescharacteristic
159C22×C4C22×D25111yescharacteristic
160C4⋊C4—41no—
161C4⋊C4—10042no—
162C4⋊C4—10042no—
163C4⋊C4—10042no—
164C4⋊C4—10041no—
165C4⋊C4—10041no—
166C4⋊C4—5024no—
167C4⋊C4—5024no—
168C4⋊C4—5024no—
169C22⋊C4—24no—
170C22⋊C4—42no—
171C22⋊C4—10042no—
172C22⋊C4—10042no—
173C22⋊C4—5024no—
174C22⋊C4—5024no—
175C42—2514no—
176C23—22no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—22no—
181C23—22no—
182C23—22no—
183C23C23×D25111yescharacteristic
184D4—44no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—5022no—
195C2×C4—5022no—
196C2×C4—5022no—
197C2×C4—5022no—
198C2×C4—5022no—
199C2×C4—5022no—
200C2×C4—5022no—
201C2×C4—5022no—
202C2×C4—5022no—
203C2×C4—5022no—
204C2×C4—5022no—
205C2×C4—2514no—
206C2×C4D4×D25112yes—
207C22—42no—
208C22—42no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—42no—
213C22—42no—
214C22C2×D4×D25111yescharacteristic
215C22#2254876011yescharacteristic
216C22#2254876012yes—
217C22#2254876211yescharacteristic
218C22#2254876212yes—
219C4—22no—
220C4—42no—
221C4—41no—
222C4—41no—
223C4—10042no—
224C4—10042no—
225C4—10042no—
226C4—5024no—
227C2—82no—
228C2#3063226411yescharacteristic
229C2#3063226612yes—
230C2#3063242611yescharacteristic
231C2#3063243711yescharacteristic
232C2#3063244012yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
Fitting#394×C251C2
prime radicalC22×D4×C251C22
hypercenter#394D251
hypocenterC251#2823
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
1. C2×#22548708C2semidirect
2. #30631621C2semidirect
3. #30631617C2semidirect
4. #30631610C2semidirect
5. #30631610C2semidirect
6. #30631610C2semidirect
8. #30631556C2semidirect
48. C4×Dic251D4semidirect
116. C251#2823semidirect

Maximal subgroups

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

subgroupindex
C2×#225487082
#306316212
#306316172
#306316102
#306316012
#306315562
#394×C2512
#2823251