Finite Group Databasebeta

← #44558387

Subgroups — #44558387

Order 32128.

644 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 133 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×#22548689C211yesmaximal, characteristic, complemented
2#30631613C212yesmaximal, complemented
3#30631612C212yesmaximal, complemented
4#30631596C212yesmaximal, complemented
5#30631595C214yesmaximal, complemented
6#30631595C212yesmaximal, complemented
7#30631550C211yesmaximal, characteristic, complemented
8C2×#121×C251C211yesmaximal, characteristic, complemented
9C22×D1004C2211yescharacteristic
10C22×C22⋊D251C2212yes—
11C22×C22⋊D251C2214yes—
12#22548689—24no—
13C2×C4⋊Dic251C2211yescharacteristic
14C2×C22⋊Dic251C2212yes—
15C2×C22⋊Dic251C2214yes—
16C2×(C2×D251)⋊C4C2212yes—
17C2×(C2×D251)⋊C4C2214yes—
18C251⋊#95C2212yes—
19C251⋊#95C2212yes—
20C251⋊#95C2214yes—
21C22×D4×C251C2212yes—
22C22×D4×C251C2211yescharacteristic
23C23×C4×C251C2211yescharacteristic
24#121×C251—24no—
25C2×C4⋊C4×C251C2211yescharacteristic
26C2×C22⋊C4×C251C2212yes—
27C23×D251C2312yes—
28C2×D1004—22no—
29C2×D1004—42no—
30C2×C22⋊D251—24no—
31C2×C22⋊D251—44no—
32C2×C22⋊D251—44no—
33C2×C22⋊D251—44no—
34C2×C22⋊D251—44no—
35C22×Dic251C2314yes—
36C22×Dic251C2312yes—
37C4⋊Dic251—22no—
38C22⋊Dic251—42no—
39C22⋊Dic251—44no—
40(C2×D251)⋊C4—24no—
41(C2×D251)⋊C4—44no—
42C24×C251C2312yes—
43C24×C251C2311yescharacteristic
44C2×D4×C251—24no—
45C2×D4×C251—22no—
46C2×D4×C251—42no—
47C2×D4×C251—44no—
48C22×C4×C251—42no—
49C22×C4×C251D414yescomplemented
50C22×C4×C251C2312yes—
51C22×C4×C251C2311yescharacteristic
52C22×C4×C251C2311yescharacteristic
53C4⋊C4×C251—22no—
54C22⋊C4×C251—24no—
55C22×D251—22no—
56C22×D251—22no—
57C22×D251—22no—
58C22×D251—24no—
59C22×D251—24no—
60D1004—44no—
61C22⋊D251—48no—
62C22⋊D251—416no—
63C2×Dic251—22no—
64C2×Dic251—24no—
65C2×Dic251—24no—
66C2×Dic251—24no—
67C2×Dic251—24no—
68C23×C251—22no—
69C23×C251—22no—
70C23×C251—22no—
71C23×C251—22no—
72C23×C251—24no—
73C23×C251—24no—
74C23×C251—22no—
75C23×C251—22no—
76C23×C251C2×D412yes—
77C23×C251C2411yescharacteristic
78D4×C251—44no—
79D4×C251—48no—
80C2×C4×C251—22no—
81C2×C4×C251—22no—
82C2×C4×C251—22no—
83C2×C4×C251—24no—
84C2×C4×C251—44no—
85C2×C4×C251C2×D412yes—
86C2×C4×C251C2×D412yes—
87C22×C251—22no—
88C22×C251—22no—
89C22×C251—22no—
90C22×C251—44no—
91C22×C251—42no—
92C22×C251—42no—
93C22×C251—42no—
94C22×C251—44no—
95C22×C251—44no—
96C22×C251C22×D411yescharacteristic
97C22×C251H(5,𝔽2)12yes—
98C22×C251H(5,𝔽2)12yes—
99C22×C251H(5,𝔽2)12yes—
100C2×D251—42no—
101C2×D251—42no—
102C2×D251—42no—
103C2×D251—44no—
104C2×D251—44no—
105C4×C251—22no—
106C4×C251—22no—
107C4×C251—42no—
108Dic251—42no—
109Dic251—44no—
110C2×C251—42no—
111C2×C251—82no—
112C2×C251#53411yescharacteristic
113C2×C251#53511yescharacteristic
114C2×C251#53511yescharacteristic
115C2×C251#56014yes—
116D251—82no—
117C251#281511yescharacteristic, complemented, Sylow
118#2815—2511nomaximal, Sylow
119C2×#121—2511no—
120C2×#121D25111yescharacteristic, complemented
121#397—2512no—
122#397—2512no—
123#392—2512no—
124#392—2514no—
125#392—2512no—
126#381—2511no—
127C22×D4—2514no—
128C22×D4—2512no—
129C22×D4—2511no—
130C22×D4C2×D25112yes—
131C22×D4C2×D25111yescharacteristic
132C23×C4C2×D25111yescharacteristic
133#121—24no—
134#121—5024no—
135C2×C4⋊C4—2511no—
136C2×C4⋊C4C2×D25111yescharacteristic
137C2×C22⋊C4—2512no—
138C2×C22⋊C4—2514no—
139C2×C22⋊C4—2514no—
140C2×C22⋊C4—2512no—
141C2×C22⋊C4C2×D25112yes—
142#95—2512no—
143#95—2512no—
144#95—2514no—
145C24—2512no—
146C24C22×D25112yes—
147C24C22×D25111yescharacteristic
148C2×D4—24no—
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—44no—
152C2×D4—10042no—
153C2×D4—10044no—
154C2×D4—10044no—
155C2×D4—10044no—
156C2×D4—10044no—
157C2×D4—5024no—
158C2×D4—5022no—
159C22×C4—42no—
160C22×C4—2514no—
161C22×C4—2512no—
162C22×C4C22⋊D25114yescomplemented
163C22×C4C22×D25112yes—
164C22×C4C22×D25111yescharacteristic
165C22×C4C22×D25111yescharacteristic
166C4⋊C4—22no—
167C4⋊C4—5022no—
168C22⋊C4—24no—
169C22⋊C4—10044no—
170C22⋊C4—10042no—
171C22⋊C4—10044no—
172C22⋊C4—5024no—
173C23—22no—
174C23—22no—
175C23—22no—
176C23—22no—
177C23—24no—
178C23—24no—
179C23—22no—
180C23—22no—
181C23—5022no—
182C23—5022no—
183C23—5024no—
184C23—5022no—
185C23—5024no—
186C23C2×C22⋊D25112yes—
187C23C23×D25111yescharacteristic
188D4—44no—
189D4—48no—
190D4—10044no—
191D4—10048no—
192D4—100416no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—24no—
197C2×C4—44no—
198C2×C4—5022no—
199C2×C4—5024no—
200C2×C4—5024no—
201C2×C4—5024no—
202C2×C4—5024no—
203C2×C4C2×C22⋊D25112yes—
204C2×C4C2×C22⋊D25112yes—
205C22—22no—
206C22—22no—
207C22—22no—
208C22—44no—
209C22—42no—
210C22—42no—
211C22—42no—
212C22—44no—
213C22—44no—
214C22—10042no—
215C22—10042no—
216C22—10042no—
217C22—10044no—
218C22—10044no—
219C22C22×C22⋊D25111yescharacteristic
220C22#2254876012yes—
221C22#2254876012yes—
222C22#2254876112yes—
223C4—22no—
224C4—22no—
225C4—42no—
226C4—10042no—
227C4—10044no—
228C2—42no—
229C2—82no—
230C2—20082no—
231C2#3063224611yescharacteristic
232C2#3063225111yescharacteristic
233C2#3063225411yescharacteristic
234C2#3063242514yes—
subgroupG/H
center, FrattiniC23C23×D251
commutator, socle, cosocleC23×C251C24
FittingC2×#121×C251C2
hypercenterC2×#121D251
hypocenterC251#2815
LCSG ▷ C23×C251 ▷ C251
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×#22548689C2semidirect
2. #30631613C2semidirect
3. #30631612C2semidirect
4. #30631596C2semidirect
5. #30631595C2semidirect
6. #30631595C2semidirect
7. #30631550C2semidirect
8. C2×#121×C251C2semidirect
49. C22×C4×C251D4semidirect
117. C251#2815semidirect
120. C2×#121D251semidirect
162. C22×C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225486892
#306316132
#306316122
#306315962
#306315952
#306315502
C2×#121×C2512
#2815251