Finite Group Databasebeta

← #44550823

Subgroups — #44550823

Order 32128.

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

normal subgroups only

This group has 378 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
1C2×#22548698C211yesmaximal, characteristic
2C2×#22548629C212yesmaximal
3#30631550C211yesmaximal, characteristic, complemented
4#30631453C212yesmaximal, complemented
5C2×#117×C251C211yesmaximal, characteristic, complemented
6C22×C22⋊D251C2211yescharacteristic
7#22548698—21no—
8#22548698—21no—
9#22548698—22no—
10C2×Dic251⋊C4C412yescomplemented
11C2×C22⋊Dic251C2212yes—
12C2×C22⋊Dic251C2211yescharacteristic
13C2×(C2×D251)⋊C4C412yescomplemented
14#22548629—24no—
15C251⋊#95—22no—
16C23×C4×C251C2211yescharacteristic
17#117×C251—24no—
18C2×C4⋊C4×C251—21no—
19C2×C22⋊C4×C251C2212yes—
20C2×C42×C251—21no—
21C23×D251C2×C411yescharacteristic
22C2×C22⋊D251—22no—
23C2×C22⋊D251—21no—
24C2×C22⋊D251—21no—
25C2×C22⋊D251—22no—
26C22×Dic251—22no—
27C22×Dic251C2×C411yescharacteristic
28C22×Dic251C2×C412yes—
29Dic251⋊C4—22no—
30Dic251⋊C4—22no—
31C22⋊Dic251—21no—
32C22⋊Dic251—21no—
33C22⋊Dic251—22no—
34C22⋊Dic251—22no—
35(C2×D251)⋊C4—22no—
36(C2×D251)⋊C4—22no—
37C24×C251C2311yescharacteristic
38C22×C4×C251—22no—
39C22×C4×C251—22no—
40C22×C4×C251—22no—
41C22×C4×C251C2×C412yes—
42C22×C4×C251D411yescharacteristic
43C22×C4×C251D411yescharacteristic
44C22×C4×C251D412yes—
45C4⋊C4×C251—24no—
46C22⋊C4×C251—22no—
47C22⋊C4×C251D414yes—
48C42×C251—24no—
49C22×D251—21no—
50C22×D251—22no—
51C22×D251—42no—
52C22⋊D251—41no—
53C22⋊D251—42no—
54C22⋊D251—41no—
55C2×Dic251—22no—
56C2×Dic251—21no—
57C2×Dic251—22no—
58C2×Dic251—42no—
59C2×Dic251—44no—
60C2×Dic251—42no—
61C23×C251—42no—
62C23×C251C22×C411yescharacteristic
63C23×C251C2×D411yescharacteristic
64C23×C251C2×D412yes—
65C23×C251C2×D411yescharacteristic
66C23×C251#4112yes—
67C2×C4×C251—24no—
68C2×C4×C251—21no—
69C2×C4×C251—21no—
70C2×C4×C251—24no—
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251—24no—
74C2×C4×C251—44no—
75C2×C4×C251C22⋊C412yes—
76C2×C4×C251C22⋊C412yes—
77C22×C251—22no—
78C22×C251—22no—
79C22×C251—22no—
80C22×C251—42no—
81C22×C251—42no—
82C22×C251C2×C22⋊C411yescharacteristic
83C22×C251C4×D412yes—
84C22×C251#12011yescharacteristic
85C22×C251#12112yes—
86C22×C251#12311yescharacteristic
87C2×D251—41no—
88C2×D251—42no—
89C2×D251—82no—
90C4×C251—22no—
91C4×C251—22no—
92C4×C251—44no—
93Dic251—41no—
94Dic251—42no—
95Dic251—82no—
96C2×C251—22no—
97C2×C251—21no—
98C2×C251—22no—
99C2×C251—21no—
100C2×C251#38611yescharacteristic
101C2×C251#41012yes—
102D251—81no—
103C251#189911yescharacteristic, complemented, Sylow
104#1899—2511nomaximal, Sylow
105C2×#123—2511no—
106C2×#117D25111yescharacteristic, complemented
107C2×#99—2512no—
108#381—2511no—
109#342—2512no—
110C22×D4—2511no—
111C23×C4C2×D25111yescharacteristic
112#123—5021no—
113#123—5021no—
114#123—5022no—
115#117—24no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—2512no—
118C2×C22⋊C4—2512no—
119C2×C22⋊C4—2511no—
120C2×C22⋊C4—2512no—
121C2×C22⋊C4C2×D25112yes—
122C2×C42—21no—
123#99—5024no—
124#95—5022no—
125C24—2511no—
126C24C22×D25111yescharacteristic
127C2×D4—5022no—
128C2×D4—5021no—
129C2×D4—5021no—
130C2×D4—5022no—
131C22×C4—22no—
132C22×C4—22no—
133C22×C4—22no—
134C22×C4—5022no—
135C22×C4—2512no—
136C22×C4—2511no—
137C22×C4C4×D25112yes—
138C22×C4C22⋊D25111yescharacteristic
139C22×C4C22⋊D25111yescharacteristic
140C22×C4D100412yes—
141C4⋊C4—24no—
142C4⋊C4—5022no—
143C4⋊C4—5022no—
144C22⋊C4—22no—
145C22⋊C4—5022no—
146C22⋊C4—5022no—
147C22⋊C4—5021no—
148C22⋊C4—5021no—
149C22⋊C4—5022no—
150C22⋊C4—5022no—
151C22⋊C4C22⋊D25114yes—
152C42—24no—
153C23—42no—
154C23—10042no—
155C23—5022no—
156C23—5021no—
157C23C2×C4×D25111yescharacteristic
158C23C2×C22⋊D25111yescharacteristic
159C23C2×C22⋊D25112yes—
160C23C2×D100411yescharacteristic
161C23#1686081012yes—
162D4—10041no—
163D4—10042no—
164D4—10041no—
165C2×C4—24no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—24no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—24no—
172C2×C4—44no—
173C2×C4—10044no—
174C2×C4—10042no—
175C2×C4—10042no—
176C2×C4—5022no—
177C2×C4—5021no—
178C2×C4—5022no—
179C2×C4(C2×D251)⋊C412yes—
180C2×C4(C2×D251)⋊C412yes—
181C22—22no—
182C22—22no—
183C22—22no—
184C22—42no—
185C22—42no—
186C22—20082no—
187C22—10041no—
188C22—10042no—
189C22C2×(C2×D251)⋊C411yescharacteristic
190C22C4×C22⋊D25112yes—
191C22#2254868411yescharacteristic
192C22#2254868912yes—
193C22#2254869811yescharacteristic
194C4—22no—
195C4—22no—
196C4—44no—
197C4—20082no—
198C4—10041no—
199C4—10042no—
200C2—21no—
201C2—22no—
202C2—22no—
203C2—21no—
204C2—20081no—
205C2#3063156611yescharacteristic
206C2#3063166012yes—
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC23×C251C22×C4
FrattiniC24C22×D251
FittingC2×#117×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC22×C22⋊D251C22
cosocleC24×C251C23
hypercenterC2×#117D251
hypocenterC251#1899
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
3. #30631550C2semidirect
4. #30631453C2semidirect
5. C2×#117×C251C2semidirect
10. C2×Dic251⋊C4C4semidirect
13. C2×(C2×D251)⋊C4C4semidirect
103. C251#1899semidirect
106. C2×#117D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486982
C2×#225486292
#306315502
#306314532
C2×#117×C2512
#1899251