Finite Group Databasebeta

← #44550813

Subgroups — #44550813

Order 32128.

358 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 93 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#30632198C211yesmaximal, characteristic
2C2×#22548643C212yesmaximal, complemented
3#30631651C211yesmaximal, characteristic, complemented
4#30631447C212yesmaximal, complemented
5C2×#117×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C2211yescharacteristic
7C2×#16860794C2212yes—
8#22548698—21no—
9#22548694C412yescomplemented
10#22548689C412yescomplemented
11#22548684—21no—
12C4×C22⋊D251—22no—
13#22548674C2211yescharacteristic
14#22548643—24no—
15C22⋊(C251⋊C8)—22no—
16C23×C4×C251C2211yescharacteristic
17#117×C251—24no—
18C2×C4⋊C4×C251—21no—
19C2×C22⋊C4×C251—21no—
20C2×C42×C251C2212yes—
21#16860810—22no—
22#16860810—22no—
23C2×Dic502C2×C411yescharacteristic
24C2×D1004C2×C411yescharacteristic
25C2×C22⋊D251—22no—
26C2×C22⋊D251—21no—
27C2×C4×D251—21no—
28#16860794—22no—
29#16860794—22no—
30C2×C251⋊C8—22no—
31C4⋊Dic251C2×C412yes—
32Dic251⋊C4—22no—
33C22⋊Dic251—22no—
34C22⋊Dic251—21no—
35(C2×D251)⋊C4—22no—
36C4×Dic251—21no—
37C24×C251D411yescharacteristic
38C22×C4×C251—22no—
39C22×C4×C251—22no—
40C22×C4×C251—22no—
41C22×C4×C251C2×C412yes—
42C22×C4×C251D411yescharacteristic
43C22×C4×C251D412yes—
44C22×C4×C251C2311yescharacteristic
45C4⋊C4×C251—24no—
46C22⋊C4×C251—24no—
47C42×C251—22no—
48C42×C251D414yescomplemented
49C22×D251—21no—
50Dic502—22no—
51D1004—22no—
52C22⋊D251—44no—
53C22⋊D251—41no—
54C22⋊D251—41no—
55C2×Dic251—22no—
56C2×Dic251—21no—
57C4×D251—42no—
58C251⋊C8—42no—
59C23×C251—21no—
60C23×C251—21no—
61C23×C251—42no—
62C23×C251C22⋊C412yes—
63C23×C251C2×D411yescharacteristic
64C2×C4×C251—21no—
65C2×C4×C251—21no—
66C2×C4×C251—24no—
67C2×C4×C251—24no—
68C2×C4×C251—24no—
69C2×C4×C251—44no—
70C2×C4×C251C22⋊C412yes—
71C2×C4×C251C22×C411yescharacteristic
72C2×C4×C251C2×D412yes—
73C2×C4×C251C2×D411yescharacteristic
74C2×C4×C251#4112yes—
75C22×C251—22no—
76C22×C251—22no—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—42no—
80C22×C251—42no—
81C22×C251C2×C22⋊C411yescharacteristic
82C22×C251#12011yescharacteristic
83C22×C251#12311yescharacteristic
84C2×D251—41no—
85C2×D251—42no—
86C4×C251—22no—
87C4×C251—44no—
88C4×C251C4×D412yes—
89C4×C251#12112yes—
90Dic251—41no—
91Dic251—42no—
92C2×C251—21no—
93C2×C251—21no—
94C2×C251—42no—
95C2×C251#38611yescharacteristic
96C2×C251#42112yes—
97D251—81no—
98C251#189711yescharacteristic, complemented, Sylow
99#1897—2511nomaximal, Sylow
100#525—2511no—
101C2×#117D25111yescharacteristic, complemented
102C2×#104—2512no—
103#407—2511no—
104#339—2512no—
105C2×#41—2511no—
106C23×C4C2×D25111yescharacteristic
107C2×M16—2512no—
108#123—5021no—
109C22⋊Q8—2512no—
110#121—2512no—
111#120—5021no—
112C4×D4—5022no—
113#117—24no—
114#117—2511no—
115C2×C4⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C42C2×D25112yes—
118#104—5024no—
119C22⋊C8—5022no—
120C24C22⋊D25111yescharacteristic
121#41—5022no—
122#41—5022no—
123C2×Q8—2511no—
124C2×D4—5022no—
125C2×D4—5021no—
126C2×D4—2511no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—22no—
130C22×C4—5021no—
131C22×C4C4×D25112yes—
132C22×C4C22⋊D25111yescharacteristic
133C22×C4D100412yes—
134C22×C4C22×D25111yescharacteristic
135M16—5022no—
136M16—5022no—
137C2×C8—5022no—
138C4⋊C4—24no—
139C4⋊C4—5022no—
140C4⋊C4—2512no—
141C22⋊C4—24no—
142C22⋊C4—5022no—
143C22⋊C4—5021no—
144C22⋊C4—5022no—
145C42—22no—
146C42—5021no—
147C42C22⋊D25114yescomplemented
148C23—21no—
149C23—21no—
150C23—42no—
151C23—5021no—
152C23(C2×D251)⋊C412yes—
153C23C2×C22⋊D25111yescharacteristic
154Q8—5022no—
155D4—10044no—
156D4—10041no—
157D4—10041no—
158D4—5022no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—24no—
164C2×C4—44no—
165C2×C4—10042no—
166C2×C4—5022no—
167C2×C4—5021no—
168C2×C4(C2×D251)⋊C412yes—
169C2×C4C2×C4×D25111yescharacteristic
170C2×C4C2×C22⋊D25112yes—
171C2×C4C2×D100411yescharacteristic
172C2×C4#1686081012yes—
173C8—10042no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—10042no—
181C22—10041no—
182C22C2×(C2×D251)⋊C411yescharacteristic
183C22#2254868411yescharacteristic
184C22#2254869811yescharacteristic
185C4—22no—
186C4—44no—
187C4—10042no—
188C4—10041no—
189C4C4×C22⋊D25112yes—
190C4#2254868912yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—20081no—
195C2#3063156611yescharacteristic
196C2#3063170512yes—
subgroupG/H
centerC2×C4C2×D1004
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC2×#117×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenterC2×#117D251
hypocenterC251#1897
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. C2×#22548643C2semidirect
3. #30631651C2semidirect
4. #30631447C2semidirect
5. C2×#117×C251C2semidirect
9. #22548694C4semidirect
10. #22548689C4semidirect
48. C42×C251D4semidirect
98. C251#1897semidirect
101. C2×#117D251semidirect
147. C42C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C2×#225486432
#306316512
#306314472
C2×#117×C2512
#1897251