Finite Group Databasebeta

← #44550837

Subgroups — #44550837

Order 32128.

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

normal subgroups only

This group has 304 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×#22548748C211yesmaximal, characteristic
2#30632453C211yesmaximal, characteristic
3#30631705C211yesmaximal, characteristic
4#30631695C211yesmaximal, characteristic, complemented
5C2×#22548633C211yesmaximal, characteristic, complemented
6C251⋊#338C211yesmaximal, characteristic, complemented
7#344×C251C211yesmaximal, characteristic, complemented
8C2×#16860810C2211yescharacteristic
9C22×Dic502C2211yescharacteristic
10#22548748—22no—
11#22548748—22no—
12C2×Dic1004C412yescomplemented
13C2×#16860799C412yescomplemented
14#22548723—22no—
15#22548721—22no—
16C2×#16860794C2211yescharacteristic
17C2×#16860794C2211yescharacteristic
18M16×D251—22no—
19#22548643—22no—
20#22548638—22no—
21#22548633—22no—
22C2×M16×C251C2211yescharacteristic
23C2×M16×C251C2211yescharacteristic
24#117×C251C2211yescharacteristic
25#16860810—21no—
26#16860810—21no—
27#16860810D412yes—
28#16860810D412yes—
29C2×Dic502—21no—
30C2×Dic502—21no—
31C2×Dic502—21no—
32C2×Dic502—21no—
33C2×Dic502—21no—
34C2×Dic502C2×C412yes—
35C2×Dic502C2×C411yescharacteristic
36C2×D1004C2×C411yescharacteristic
37C2×C22⋊D251—21no—
38C22×Dic251—21no—
39C2×C4×D251—21no—
40Dic1004—24no—
41#16860799—24no—
42#16860794—21no—
43#16860794—21no—
44#16860794—21no—
45#16860794—21no—
46#16860793—24no—
47C2×C251⋊C8—21no—
48C2×C251⋊C8D412yes—
49C8×D251—24no—
50C22×C4×C251C2311yescharacteristic
51M16×C251—21no—
52M16×C251—21no—
53M16×C251—21no—
54M16×C251D412yes—
55C2×C8×C251—21no—
56C2×C8×C251C2×C412yes—
57C4⋊C4×C251—21no—
58C22⋊C4×C251—21no—
59C42×C251—21no—
60C22×D251—21no—
61Dic502—21no—
62Dic502—22no—
63Dic502—22no—
64Dic502—41no—
65Dic502—41no—
66Dic502C22⋊C412yes—
67D1004—21no—
68D1004C22⋊C412yes—
69C22⋊D251—22no—
70C22⋊D251—22no—
71C2×Dic251—21no—
72C2×Dic251—22no—
73C2×Dic251—41no—
74C2×Dic251—41no—
75C4×D251—22no—
76C4×D251—22no—
77C251⋊C8—22no—
78C251⋊C8—41no—
79C23×C251#4111yescharacteristic
80C2×C4×C251—41no—
81C2×C4×C251C22×C411yescharacteristic
82C2×C4×C251C2×D411yescharacteristic
83C2×C4×C251C2×D411yescharacteristic
84C2×C4×C251C2×D411yescharacteristic
85C2×C4×C251C2×D411yescharacteristic
86C2×C4×C251#4111yescharacteristic
87C8×C251—22no—
88C8×C251—22no—
89C22×C251—41no—
90C22×C251C4×D411yescharacteristic
91C22×C251#12111yescharacteristic
92C22×C251#12311yescharacteristic
93C2×D251—22no—
94C2×D251—81no—
95C4×C251—81no—
96C4×C251C2×C22⋊C411yescharacteristic
97C4×C251C4×D411yescharacteristic
98C4×C251#12011yescharacteristic
99C4×C251#12111yescharacteristic
100Dic251—22no—
101Dic251—42no—
102C2×C251—21no—
103C2×C251—21no—
104C2×C251—21no—
105C2×C251#38611yescharacteristic
106D251—42no—
107C251#190211yescharacteristic, complemented, Sylow
108#1902—2511nomaximal, Sylow
109C2×#137—2511no—
110#568—2511no—
111#421—2511no—
112#419—2511no—
113C2×#101—2511no—
114#344D25111yescharacteristic, complemented
115#338—2511no—
116C2×#41—2511no—
117C22×Q8—2511no—
118#137—5022no—
119#137—5022no—
120C2×Q16—2512no—
121C2×SD16—2512no—
122#131—5022no—
123#131—5022no—
124C2×M16—5022no—
125C2×M16—2511no—
126C2×M16—2511no—
127C2×M16C2×D25111yescharacteristic
128C2×M16C2×D25111yescharacteristic
129#117C2×D25111yescharacteristic
130#104—5022no—
131#103—5022no—
132#101—5022no—
133#41—5021no—
134#41—5021no—
135#41—2512no—
136#41—2512no—
137C2×Q8—5021no—
138C2×Q8—5021no—
139C2×Q8—5021no—
140C2×Q8—5021no—
141C2×Q8—5021no—
142C2×Q8—2512no—
143C2×Q8—2511no—
144C2×D4—5021no—
145C2×D4—2511no—
146C22×C4—5021no—
147C22×C4—5021no—
148C22×C4C22×D25111yescharacteristic
149Q16—5024no—
150SD16—5024no—
151M16—21no—
152M16—21no—
153M16—21no—
154M16—5024no—
155M16—5021no—
156M16—5021no—
157M16—5021no—
158M16—5021no—
159M16D100412yes—
160C2×C8—21no—
161C2×C8—5024no—
162C2×C8—5021no—
163C2×C8—2512no—
164C2×C8C4×D25112yes—
165C4⋊C4—21no—
166C22⋊C4—21no—
167C42—21no—
168C23—5021no—
169C23#1686081011yescharacteristic
170Q8—10041no—
171Q8—10041no—
172Q8—5022no—
173Q8—5022no—
174Q8—5021no—
175Q8—2512no—
176D4—5022no—
177D4—5022no—
178D4—5021no—
179D4—2512no—
180C2×C4—41no—
181C2×C4—10041no—
182C2×C4—10041no—
183C2×C4—5022no—
184C2×C4—5021no—
185C2×C4—5022no—
186C2×C4—5022no—
187C2×C4C2×C4×D25111yescharacteristic
188C2×C4D4×D25111yescharacteristic
189C2×C4D4×D25111yescharacteristic
190C2×C4D4×D25111yescharacteristic
191C2×C4C2×D100411yescharacteristic
192C2×C4#1686081111yescharacteristic
193C8—22no—
194C8—22no—
195C8—10041no—
196C8—5022no—
197C22—41no—
198C22—20081no—
199C22—5022no—
200C22C4×D100411yescharacteristic
201C22#2254868811yescharacteristic
202C22#2254870011yescharacteristic
203C4—81no—
204C4—10042no—
205C4—5022no—
206C4C22⋊C4×D25111yescharacteristic
207C4#2254867711yescharacteristic
208C4#2254868511yescharacteristic
209C4#2254869011yescharacteristic
210C2—21no—
211C2—21no—
212C2—21no—
213C2—10042no—
214C2#3063156511yescharacteristic
subgroupG/H
centerC2#30631565
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#344×C251C2
socleC2×C251#386
prime radicalC2×#16860810C22
cosocleC22×C4×C251C23
hypercenter#344D251
hypocenterC251#1902
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #344

Complements

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

NH ≅ G/Ntype
4. #30631695C2semidirect
5. C2×#22548633C2semidirect
6. C251⋊#338C2semidirect
7. #344×C251C2semidirect
12. C2×Dic1004C4semidirect
13. C2×#16860799C4semidirect
107. C251#1902semidirect
114. #344D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487482
#306324532
#306317052
#306316952
C2×#225486332
C251⋊#3382
#344×C2512
#1902251