Finite Group Databasebeta

← #1867×C251 #44546454

Subgroups — #1867×C251 #44546454

Order 32128.

374 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 112 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
1#518×C251C211yesmaximal, characteristic
2C2×#99×C251C211yesmaximal, characteristic
3#386×C251C211yesmaximal, characteristic
4#385×C251C212yesmaximal, complemented
5#342×C251C212yesmaximal, complemented
6C22×D4×C251C2211yescharacteristic
7C23×C4×C251C2211yescharacteristic
8C4×D4×C251—22no—
9C4×D4×C251—22no—
10#117×C251—21no—
11C2×C4⋊C4×C251—22no—
12C2×C22⋊C4×C251—21no—
13C2×C22⋊C4×C251—21no—
14C2×C22⋊C4×C251C2211yescharacteristic
15C2×C22⋊C4×C251C2212yes—
16C2×C22⋊C4×C251C2212yes—
17#99×C251C414yescomplemented
18#95×C251—22no—
19#95×C251—21no—
20C24×C251D411yescharacteristic
21C24×C251C2311yescharacteristic
22C2×D4×C251—22no—
23C2×D4×C251—22no—
24C2×D4×C251C2×C412yes—
25C2×D4×C251Q812yes—
26C22×C4×C251—21no—
27C22×C4×C251—22no—
28C22×C4×C251—22no—
29C22×C4×C251—21no—
30C22×C4×C251—21no—
31C22×C4×C251—21no—
32C22×C4×C251—42no—
33C22×C4×C251D411yescharacteristic
34C22×C4×C251D412yes—
35C22×C4×C251D412yes—
36C4⋊C4×C251—21no—
37C4⋊C4×C251—21no—
38C4⋊C4×C251—44no—
39C22⋊C4×C251—22no—
40C22⋊C4×C251—22no—
41C22⋊C4×C251—22no—
42C22⋊C4×C251—22no—
43C22⋊C4×C251—22no—
44C22⋊C4×C251—42no—
45C22⋊C4×C251C2×C414yescomplemented
46C42×C251—22no—
47C23×C251—21no—
48C23×C251—21no—
49C23×C251—41no—
50C23×C251—41no—
51C23×C251—41no—
52C23×C251—41no—
53C23×C251C4⋊C412yescomplemented
54C23×C251C22×C411yescharacteristic
55C23×C251C2×D411yescharacteristic
56C23×C251C2×D412yes—
57C23×C251C2×Q811yescharacteristic
58C23×C251#4112yes—
59D4×C251—24no—
60D4×C251—24no—
61C2×C4×C251—21no—
62C2×C4×C251—21no—
63C2×C4×C251—22no—
64C2×C4×C251—24no—
65C2×C4×C251—24no—
66C2×C4×C251—22no—
67C2×C4×C251—22no—
68C2×C4×C251—42no—
69C2×C4×C251—42no—
70C2×C4×C251—42no—
71C2×C4×C251—41no—
72C2×C4×C251—41no—
73C2×C4×C251—42no—
74C2×C4×C251—42no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251—81no—
79C2×C4×C251—81no—
80C2×C4×C251C4⋊C412yescomplemented
81C22×C251—22no—
82C22×C251—21no—
83C22×C251—22no—
84C22×C251—21no—
85C22×C251—22no—
86C22×C251—41no—
87C22×C251—41no—
88C22×C251—42no—
89C22×C251—42no—
90C22×C251—41no—
91C22×C251—41no—
92C22×C251—81no—
93C22×C251C2×C4⋊C411yescharacteristic
94C22×C251C4×D412yes—
95C22×C251#12011yescharacteristic
96C22×C251C22⋊Q812yes—
97C22×C251#12311yescharacteristic
98C4×C251—22no—
99C4×C251—44no—
100C4×C251—42no—
101C4×C251—42no—
102C4×C251—42no—
103C4×C251—81no—
104C2×C251—22no—
105C2×C251—21no—
106C2×C251—21no—
107C2×C251—22no—
108C2×C251—42no—
109C2×C251#38511yescharacteristic
110C2×C251B(4,𝔽2)11yescharacteristic
111C2×C251#45811yescharacteristic
112C251#186711yescharacteristic, direct product factor, Sylow
113#1867C25111yesmaximal, characteristic, direct product factor, Sylow
114#518C2×C25111yescharacteristic
115C2×#99C2×C25111yescharacteristic
116#386C2×C25111yescharacteristic
117#385C2×C25112yescomplemented
118#342C2×C25112yescomplemented
119C22×D4C22×C25111yescharacteristic
120C23×C4C22×C25111yescharacteristic
121C4×D4—22no—
122C4×D4—22no—
123#117—21no—
124C2×C4⋊C4—22no—
125C2×C22⋊C4—21no—
126C2×C22⋊C4—21no—
127C2×C22⋊C4C22×C25111yescharacteristic
128C2×C22⋊C4C22×C25112yes—
129C2×C22⋊C4C22×C25112yes—
130#99C4×C25114yescomplemented
131#95—22no—
132#95—21no—
133C24D4×C25111yescharacteristic
134C24C23×C25111yescharacteristic
135C2×D4—22no—
136C2×D4—22no—
137C2×D4C2×C4×C25112yes—
138C2×D4Q8×C25112yes—
139C22×C4—21no—
140C22×C4—22no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—42no—
146C22×C4D4×C25111yescharacteristic
147C22×C4D4×C25112yes—
148C22×C4D4×C25112yes—
149C4⋊C4—21no—
150C4⋊C4—21no—
151C4⋊C4—44no—
152C22⋊C4—22no—
153C22⋊C4—22no—
154C22⋊C4—22no—
155C22⋊C4—22no—
156C22⋊C4—22no—
157C22⋊C4—42no—
158C22⋊C4C2×C4×C25114yescomplemented
159C42—22no—
160C23—21no—
161C23—21no—
162C23—41no—
163C23—41no—
164C23—41no—
165C23—41no—
166C23C4⋊C4×C25112yescomplemented
167C23C22×C4×C25111yescharacteristic
168C23C2×D4×C25111yescharacteristic
169C23C2×D4×C25112yes—
170C23C2×Q8×C25111yescharacteristic
171C23#41×C25112yes—
172D4—24no—
173D4—24no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—41no—
184C2×C4—42no—
185C2×C4—41no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—41no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—81no—
192C2×C4—81no—
193C2×C4C4⋊C4×C25112yescomplemented
194C22—22no—
195C22—21no—
196C22—22no—
197C22—21no—
198C22—22no—
199C22—41no—
200C22—41no—
201C22—42no—
202C22—42no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22C2×C4⋊C4×C25111yescharacteristic
207C22C4×D4×C25112yes—
208C22#120×C25111yescharacteristic
209C22C22⋊Q8×C25112yes—
210C22#123×C25111yescharacteristic
211C4—22no—
212C4—44no—
213C4—42no—
214C4—42no—
215C4—42no—
216C4—81no—
217C2—22no—
218C2—21no—
219C2—21no—
220C2—22no—
221C2—42no—
222C2#385×C25111yescharacteristic
223C2B(4,𝔽2)×C25111yescharacteristic
224C2#458×C25111yescharacteristic
subgroupG/H
center, socleC22×C251#120
commutatorC23C22×C4×C251
Frattini, cosocleC24C23×C251
prime radicalC22×D4×C251C22
LCSG ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C251 ◁ C24×C251 ◁ G

Complements

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

NH ≅ G/Ntype
4. #385×C251C2semidirect
5. #342×C251C2semidirect
17. #99×C251C4semidirect
45. C22⋊C4×C251C2×C4semidirect
53. C23×C251C4⋊C4semidirect
80. C2×C4×C251C4⋊C4semidirect
112. C251#1867direct (N × H)
113. #1867C251direct (N × H)
117. #385C2×C251semidirect
118. #342C2×C251semidirect
130. #99C4×C251semidirect
158. C22⋊C4C2×C4×C251semidirect
166. C23C4⋊C4×C251semidirect
193. C2×C4C4⋊C4×C251semidirect

Maximal subgroups

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

subgroupindex
#518×C2512
C2×#99×C2512
#386×C2512
#385×C2512
#342×C2512
#1867251