Finite Group Databasebeta

← #44550862

Subgroups — #44550862

Order 32128.

310 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 79 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×#22548745C211yesmaximal, characteristic
2#30632164C211yesmaximal, characteristic, complemented
3#30631704C211yesmaximal, characteristic
4#30631695C211yesmaximal, characteristic, complemented
5#30631457C211yesmaximal, characteristic, complemented
6#30631443C211yesmaximal, characteristic, complemented
7#410×C251C211yesmaximal, characteristic
8C22×Dic502C2211yescharacteristic
9#22548745—22no—
10#22548745—22no—
11C2×#16860800C412yescomplemented
12C2×#16860798C412yescomplemented
13C2×#16860794C2211yescharacteristic
14C2×#16860794C2211yescharacteristic
15C4×Dic502—22no—
16#22548680—22no—
17#22548674C2211yescharacteristic
18#22548671—22no—
19#22548642—22no—
20#22548638—22no—
21C2×#41×C251C2211yescharacteristic
22#117×C251C2211yescharacteristic
23#117×C251C2211yescharacteristic
24#99×C251—22no—
25C2×Dic502—21no—
26C2×Dic502—21no—
27C2×Dic502—21no—
28C2×Dic502C2×C412yes—
29C2×Dic502D412yes—
30C2×Dic502D412yes—
31C22×Dic251—21no—
32#16860800—24no—
33#16860798—24no—
34#16860794—21no—
35#16860794—21no—
36#16860794—21no—
37#16860794—21no—
38C2×C251⋊C8—21no—
39C2×C251⋊C8C2×C412yes—
40C4⋊Dic251—21no—
41Dic251⋊C4—24no—
42C22⋊Dic251—21no—
43C4×Dic251—24no—
44C4×Dic251D412yes—
45#41×C251—21no—
46#41×C251—21no—
47#41×C251—21no—
48#41×C251—21no—
49C2×Q8×C251C2×C411yescharacteristic
50C2×D4×C251—21no—
51C2×D4×C251C2×C411yescharacteristic
52C22×C4×C251—21no—
53C22×C4×C251C2311yescharacteristic
54C4⋊C4×C251—21no—
55C4⋊C4×C251—21no—
56C22⋊C4×C251—21no—
57C22⋊C4×C251D412yes—
58C42×C251—21no—
59C42×C251—21no—
60Dic502—22no—
61Dic502—22no—
62Dic502—22no—
63Dic502C22⋊C414yes—
64C2×Dic251—22no—
65C2×Dic251—22no—
66C2×Dic251—22no—
67C2×Dic251—22no—
68C251⋊C8—22no—
69C251⋊C8—41no—
70C23×C251—21no—
71C23×C251C2×D411yescharacteristic
72Q8×C251—21no—
73Q8×C251—21no—
74D4×C251—21no—
75D4×C251—21no—
76D4×C251—41no—
77D4×C251—41no—
78C2×C4×C251—21no—
79C2×C4×C251—22no—
80C2×C4×C251—41no—
81C2×C4×C251—41no—
82C2×C4×C251—41no—
83C2×C4×C251C22×C411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251C2×D411yescharacteristic
87C2×C4×C251#4111yescharacteristic
88C2×C4×C251#4111yescharacteristic
89C22×C251—41no—
90C22×C251—41no—
91C22×C251—81no—
92C22×C251C4×D411yescharacteristic
93C22×C251#12011yescharacteristic
94C22×C251#12111yescharacteristic
95C4×C251—42no—
96C4×C251—41no—
97C4×C251—81no—
98C4×C251C2×C22⋊C411yescharacteristic
99C4×C251C4×D411yescharacteristic
100C4×C251#12111yescharacteristic
101C4×C251#12311yescharacteristic
102Dic251—24no—
103Dic251—42no—
104C2×C251—21no—
105C2×C251—21no—
106C2×C251—21no—
107C2×C251—81no—
108C2×C251#38611yescharacteristic
109C251#190511yescharacteristic, complemented, Sylow
110#1905—2511nomaximal, Sylow
111C2×#137—2511no—
112#519—2511no—
113#421—2511no—
114#419—2511no—
115#410D25111yescharacteristic
116#344—2511no—
117H(3,ℤ/4)—2511no—
118C2×#41C2×D25111yescharacteristic
119C22×Q8—2511no—
120#137—5022no—
121#137—5022no—
122C2×Q16—2512no—
123C2×SD16—2512no—
124C2×M16—2511no—
125C2×M16—2511no—
126C4×Q8—5022no—
127C4×Q8—5022no—
128#117—5022no—
129#117—2511no—
130#117C2×D25111yescharacteristic
131#117C2×D25111yescharacteristic
132#104—5022no—
133#103—5022no—
134#99—22no—
135#41—21no—
136#41—21no—
137#41—21no—
138#41—21no—
139C2×Q8—5021no—
140C2×Q8—5021no—
141C2×Q8—5021no—
142C2×Q8—2512no—
143C2×Q8—2512no—
144C2×Q8—2512no—
145C2×Q8C4×D25111yescharacteristic
146C2×D4—21no—
147C2×D4C4×D25111yescharacteristic
148C22×C4—21no—
149C22×C4—5021no—
150C22×C4C22×D25111yescharacteristic
151Q16—5024no—
152SD16—5024no—
153M16—5021no—
154M16—5021no—
155M16—5021no—
156M16—5021no—
157C2×C8—5021no—
158C2×C8—2512no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—5024no—
162C4⋊C4—5021no—
163C22⋊C4—21no—
164C22⋊C4—5021no—
165C22⋊C4C22⋊D25112yes—
166C42—21no—
167C42—21no—
168C42—5024no—
169C42—2512no—
170C23—21no—
171C23C2×C22⋊D25111yescharacteristic
172Q8—21no—
173Q8—21no—
174Q8—5022no—
175Q8—5022no—
176Q8—5022no—
177Q8—2514no—
178D4—21no—
179D4—21no—
180D4—41no—
181D4—41no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—41no—
187C2×C4—5022no—
188C2×C4—5022no—
189C2×C4—5022no—
190C2×C4—5022no—
191C2×C4C2×C4×D25111yescharacteristic
192C2×C4D4×D25111yescharacteristic
193C2×C4D4×D25111yescharacteristic
194C2×C4D4×D25111yescharacteristic
195C2×C4#1686080911yescharacteristic
196C2×C4#1686081011yescharacteristic
197C8—10041no—
198C8—5022no—
199C22—41no—
200C22—41no—
201C22—81no—
202C22C4×C22⋊D25111yescharacteristic
203C22#2254868311yescharacteristic
204C22#2254868611yescharacteristic
205C4—42no—
206C4—41no—
207C4—81no—
208C4—10042no—
209C4—5024no—
210C4C22⋊C4×D25111yescharacteristic
211C4(C22⋊D251)⋊C411yescharacteristic
212C4#2254868811yescharacteristic
213C4#2254869711yescharacteristic
214C2—21no—
215C2—21no—
216C2—21no—
217C2—81no—
218C2#3063156311yescharacteristic
subgroupG/H
centerC2#30631563
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#410×C251C2
socleC2×C251#386
prime radicalC2×#41×C251C22
cosocleC22×C4×C251C23
hypercenter#410D251
hypocenterC251#1905
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #30632164C2semidirect
4. #30631695C2semidirect
5. #30631457C2semidirect
6. #30631443C2semidirect
11. C2×#16860800C4semidirect
12. C2×#16860798C4semidirect
109. C251#1905semidirect

Maximal subgroups

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

subgroupindex
C2×#225487452
#306321642
#306317042
#306316952
#306314572
#306314432
#410×C2512
#1905251