Finite Group Databasebeta

← #44550336

Subgroups — #44550336

Order 32128.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#22548761C211yesmaximal, characteristic
2#30631703C212yesmaximal, complemented
3#30631693C212yesmaximal, complemented
4#30631661C211yesmaximal, characteristic, complemented
5#413×C251C211yesmaximal, characteristic, complemented
6#22548761—24no—
7#22548761—24no—
8C2×#16860811—21no—
9C2×#16860810C412yescomplemented
10C2×#16860810C2211yescharacteristic
11C22×D1004C412yescomplemented
12C22×D1004C2211yescharacteristic
13C2×D4×D251—21no—
14C2×D4×D251—22no—
15#22548674C2212yescomplemented
16#22548641—24no—
17#22548637—24no—
18#22548630—22no—
19C2×#41×C251C2211yescharacteristic
20C2×M16×C251C2212yescomplemented
21#101×C251—21no—
22#100×C251—21no—
23C23×D251—22no—
24C23×D251—21no—
25#16860811—28no—
26#16860810—24no—
27#16860810—24no—
28#16860810—22no—
29#16860810D414yes—
30C2×Dic502C2×C411yescharacteristic
31C2×D1004—24no—
32C2×D1004—24no—
33C2×D1004—22no—
34C2×D1004—21no—
35C2×D1004—22no—
36C2×D1004C2×C411yescharacteristic
37C2×D1004C2×C412yes—
38C2×D1004D414yes—
39C2×C22⋊D251—22no—
40C2×C22⋊D251—21no—
41D4×D251—216no—
42D4×D251—28no—
43C2×C4×D251—22no—
44C2×C4×D251—21no—
45C4⋊Dic251—22no—
46C22⋊Dic251—22no—
47C4×Dic251—22no—
48#41×C251—22no—
49#41×C251—22no—
50C2×Q8×C251D411yescharacteristic
51C2×D4×C251D411yescharacteristic
52C2×D4×C251D412yes—
53C22×C4×C251C2×C412yes—
54C22×C4×C251C2311yescharacteristic
55M16×C251—22no—
56M16×C251—22no—
57C2×C8×C251—22no—
58C22×D251—22no—
59C22×D251—21no—
60C22×D251—28no—
61C22×D251—24no—
62C22×D251—82no—
63C22×D251—84no—
64Dic502—21no—
65Dic502C22⋊C412yes—
66D1004—28no—
67D1004—22no—
68D1004—21no—
69D1004—24no—
70D1004C22⋊C412yes—
71D1004C22⋊C414yes—
72C22⋊D251—28no—
73C22⋊D251—24no—
74C2×Dic251—21no—
75C2×Dic251—42no—
76C4×D251—28no—
77C4×D251—24no—
78C23×C251C22⋊C412yes—
79C23×C251C2×D411yescharacteristic
80Q8×C251—22no—
81D4×C251—24no—
82D4×C251—22no—
83C2×C4×C251—22no—
84C2×C4×C251—22no—
85C2×C4×C251C22⋊C412yes—
86C2×C4×C251C22×C411yescharacteristic
87C2×C4×C251C2×D412yes—
88C2×C4×C251C2×D412yes—
89C2×C4×C251C2×D411yescharacteristic
90C8×C251—42no—
91C22×C251—22no—
92C22×C251—41no—
93C22×C251—42no—
94C22×C251C2×C22⋊C411yescharacteristic
95C22×C251#12012yes—
96C2×D251—24no—
97C2×D251—22no—
98C2×D251—44no—
99C2×D251—42no—
100C2×D251—84no—
101C2×D251—88no—
102C4×C251—22no—
103C4×C251C2×C22⋊C411yescharacteristic
104C4×C251C2×C22⋊C411yescharacteristic
105C4×C251#12012yes—
106Dic251—22no—
107Dic251—82no—
108C2×C251—22no—
109C2×C251—21no—
110C2×C251—42no—
111C2×C251#37911yescharacteristic
112D251—44no—
113D251—42no—
114C251#179511yescharacteristic, complemented, Sylow
115#1795—2511nomaximal, Sylow
116C2×H(5,𝔽2)—2511no—
117#421—2512no—
118#418—2512no—
119#413D25111yescharacteristic, complemented
120#410—2511no—
121H(5,𝔽2)—5024no—
122H(5,𝔽2)—5024no—
123C2×#41—5021no—
124C2×#41—2512no—
125C2×#41—2511no—
126C2×#41C2×D25111yescharacteristic
127C22×D4—5021no—
128C22×D4—5022no—
129C22×D4—2512no—
130C22×D4—2511no—
131C2×M16C2×D25112yescomplemented
132#117—2512no—
133#104—5024no—
134#102—5024no—
135#101—21no—
136#100—21no—
137#99—5022no—
138C24—5022no—
139C24—5021no—
140#41—22no—
141#41—22no—
142#41—5024no—
143#41—5024no—
144#41—5028no—
145#41—5022no—
146#41—2514no—
147C2×Q8—2511no—
148C2×Q8C22⋊D25111yescharacteristic
149C2×D4—5024no—
150C2×D4—5024no—
151C2×D4—50216no—
152C2×D4—5028no—
153C2×D4—5022no—
154C2×D4—5021no—
155C2×D4—5022no—
156C2×D4—5022no—
157C2×D4—5021no—
158C2×D4—2511no—
159C2×D4—2512no—
160C2×D4—2514no—
161C2×D4C22⋊D25111yescharacteristic
162C2×D4D100412yes—
163C22×C4—5022no—
164C22×C4—5021no—
165C22×C4C4×D25112yes—
166C22×C4C22×D25111yescharacteristic
167M16—22no—
168M16—22no—
169C2×C8—22no—
170C4⋊C4—5022no—
171C22⋊C4—5022no—
172C42—5022no—
173C23—20082no—
174C23—20084no—
175C23—5028no—
176C23—5024no—
177C23—5022no—
178C23—5021no—
179C23(C2×D251)⋊C412yes—
180C23C2×D100411yescharacteristic
181Q8—22no—
182Q8—5021no—
183Q8—2512no—
184D4—24no—
185D4—22no—
186D4—5028no—
187D4—5028no—
188D4—5024no—
189D4—5022no—
190D4—5021no—
191D4—5024no—
192D4—2512no—
193D4—2514no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—10042no—
197C2×C4—5028no—
198C2×C4—5021no—
199C2×C4—5024no—
200C2×C4(C2×D251)⋊C412yes—
201C2×C4C2×C4×D25111yescharacteristic
202C2×C4D4×D25112yes—
203C2×C4D4×D25112yes—
204C2×C4C2×C22⋊D25111yescharacteristic
205C8—42no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—20084no—
210C22—20088no—
211C22—10044no—
212C22—10042no—
213C22—5024no—
214C22—5022no—
215C22C2×(C2×D251)⋊C411yescharacteristic
216C22#2254868512yes—
217C4—22no—
218C4—20082no—
219C4—5022no—
220C4C22⋊C4×D25111yescharacteristic
221C4C22⋊C4×D25111yescharacteristic
222C4#2254868312yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—10044no—
227C2—10042no—
228C2#3063154211yescharacteristic
subgroupG/H
centerC2#30631542
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#413×C251C2
socleC2×C251#379
prime radicalC2×#22548761C2
cosocleC22×C4×C251C23
hypercenter#413D251
hypocenterC251#1795
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #30631703C2semidirect
3. #30631693C2semidirect
4. #30631661C2semidirect
5. #413×C251C2semidirect
9. C2×#16860810C4semidirect
11. C22×D1004C4semidirect
15. #22548674C22semidirect
20. C2×M16×C251C22semidirect
114. C251#1795semidirect
119. #413D251semidirect
131. C2×M16C2×D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487612
#306317032
#306316932
#306316612
#413×C2512
#1795251