Finite Group Databasebeta

← #44550324

Subgroups — #44550324

Order 32128.

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

normal subgroups only

This group has 552 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×#22548642C212yesmaximal, complemented
2#30631688C212yesmaximal, complemented
3C2×#22548634C211yesmaximal, characteristic, complemented
4#30631594C211yesmaximal, characteristic, complemented
5C2×2−1+4×C251C211yesmaximal, characteristic
6C2×#16860794C2212yes—
7C2×C4⋊Dic251C2211yescharacteristic
8C2×C4×Dic251C2211yescharacteristic
9#22548642—24no—
10#22548639—22no—
11#22548636—22no—
12#22548634—22no—
13C251⋊#95—22no—
142−1+4×C251—24no—
152−1+4×C251—22no—
162−1+4×C251C414yescomplemented
17C2×#41×C251—22no—
18C2×#41×C251—22no—
19C2×#41×C251C2212yes—
20C22×Q8×C251—22no—
21C22×Q8×C251C2211yescharacteristic
22C22×Dic251—21no—
23C22×Dic251—21no—
24#16860794—22no—
25#16860794—22no—
26C2×C251⋊C8—22no—
27C4⋊Dic251—22no—
28C4×Dic251—22no—
29#41×C251—28no—
30#41×C251—28no—
31#41×C251—28no—
32#41×C251—28no—
33#41×C251—22no—
34#41×C251—22no—
35#41×C251C2×C414yes—
36#41×C251D414yes—
37C2×Q8×C251—22no—
38C2×Q8×C251—28no—
39C2×Q8×C251—28no—
40C2×Q8×C251—22no—
41C2×Q8×C251—22no—
42C2×Q8×C251C2×C412yes—
43C2×Q8×C251D412yes—
44C2×Q8×C251D412yes—
45C2×D4×C251—22no—
46C2×D4×C251—22no—
47C2×D4×C251D412yes—
48C22×C4×C251—24no—
49C22×C4×C251—22no—
50C22×C4×C251D412yes—
51C22×C4×C251C2311yescharacteristic
52C2×Dic251—22no—
53C2×Dic251—41no—
54C2×Dic251—41no—
55C2×Dic251—42no—
56C2×Dic251—41no—
57C251⋊C8—42no—
58C23×C251—22no—
59C23×C251C2×D411yescharacteristic
60Q8×C251—28no—
61Q8×C251—24no—
62Q8×C251—22no—
63Q8×C251—24no—
64Q8×C251C22⋊C414yes—
65D4×C251—28no—
66D4×C251—24no—
67D4×C251—24no—
68D4×C251—22no—
69D4×C251C22⋊C414yes—
70C2×C4×C251—216no—
71C2×C4×C251—22no—
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—24no—
75C2×C4×C251—24no—
76C2×C4×C251C22⋊C414yes—
77C2×C4×C251C22×C411yescharacteristic
78C2×C4×C251C2×D412yes—
79C2×C4×C251C2×D412yes—
80C2×C4×C251C2×D411yescharacteristic
81C22×C251—21no—
82C22×C251—21no—
83C22×C251—24no—
84C22×C251—42no—
85C22×C251—42no—
86C22×C251C2×C22⋊C411yescharacteristic
87C22×C251#12011yescharacteristic
88C22×C251#12011yescharacteristic
89C4×C251—24no—
90C4×C251—24no—
91C4×C251C2×C22⋊C412yes—
92C4×C251#12012yes—
93Dic251—42no—
94Dic251—81no—
95C2×C251—21no—
96C2×C251—21no—
97C2×C251—44no—
98C2×C251#37911yescharacteristic
99C2×C251#45411yescharacteristic
100C2×C251#45611yescharacteristic
101C251#179311yescharacteristic, complemented, Sylow
102#1793—2511nomaximal, Sylow
103C2×2−1+4D25111yescharacteristic
104C2×#104—2512no—
105#417—2512no—
106C2×#101—2511no—
107#391—2511no—
1082−1+4—24no—
1092−1+4—22no—
1102−1+4Dic25114yescomplemented
111C2×#41—22no—
112C2×#41—22no—
113C2×#41C2×D25112yes—
114C22×Q8—22no—
115C22×Q8C2×D25111yescharacteristic
116C2×M16—2512no—
117C2×C4⋊C4—2511no—
118C2×C42—2511no—
119#104—5024no—
120#103—5022no—
121#102—5022no—
122#101—5022no—
123#95—5022no—
124#41—28no—
125#41—28no—
126#41—28no—
127#41—28no—
128#41—22no—
129#41—22no—
130#41C2×Dic25114yes—
131#41C22⋊D25114yes—
132C2×Q8—22no—
133C2×Q8—28no—
134C2×Q8—28no—
135C2×Q8—22no—
136C2×Q8—22no—
137C2×Q8C2×Dic25112yes—
138C2×Q8C22⋊D25112yes—
139C2×Q8C22⋊D25112yes—
140C2×D4—22no—
141C2×D4—22no—
142C2×D4C22⋊D25112yes—
143C22×C4—24no—
144C22×C4—22no—
145C22×C4—5021no—
146C22×C4—5021no—
147C22×C4C22⋊D25112yes—
148C22×C4C22×D25111yescharacteristic
149M16—5022no—
150M16—5022no—
151C2×C8—5022no—
152C4⋊C4—5022no—
153C42—5022no—
154C23—22no—
155C23C2×C22⋊D25111yescharacteristic
156Q8—28no—
157Q8—24no—
158Q8—22no—
159Q8—24no—
160Q8C22⋊Dic25114yes—
161D4—28no—
162D4—24no—
163D4—24no—
164D4—22no—
165D4C22⋊Dic25114yes—
166C2×C4—216no—
167C2×C4—22no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4—24no—
172C2×C4—10042no—
173C2×C4—10041no—
174C2×C4—10041no—
175C2×C4—10041no—
176C2×C4—5022no—
177C2×C4C22⋊Dic25114yes—
178C2×C4C22×Dic25111yescharacteristic
179C2×C4C2×C22⋊D25112yes—
180C2×C4C2×C22⋊D25112yes—
181C2×C4C2×C22⋊D25111yescharacteristic
182C8—10042no—
183C22—21no—
184C22—21no—
185C22—24no—
186C22—42no—
187C22—42no—
188C22C2×C22⋊Dic25111yescharacteristic
189C22#2254868411yescharacteristic
190C22#2254868411yescharacteristic
191C4—24no—
192C4—24no—
193C4—20081no—
194C4—10042no—
195C4C2×C22⋊Dic25112yes—
196C4#2254868412yes—
197C2—21no—
198C2—21no—
199C2—44no—
200C2#3063154311yescharacteristic
201C2#3063186011yescharacteristic
202C2#3063187011yescharacteristic
subgroupG/H
centerC22#22548684
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting, prime radicalC2×2−1+4×C251C2
socleC22×C251#120
cosocleC22×C4×C251C23
hypercenterC2×2−1+4D251
hypocenterC251#1793
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×2−1+4

Complements

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

NH ≅ G/Ntype
1. C2×#22548642C2semidirect
2. #30631688C2semidirect
3. C2×#22548634C2semidirect
4. #30631594C2semidirect
16. 2−1+4×C251C4semidirect
101. C251#1793semidirect
110. 2−1+4Dic251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486422
#306316882
C2×#225486342
#306315942
C2×2−1+4×C2512
#1793251