Finite Group Databasebeta

← #44552028

Subgroups — #44552028

Order 32128.

258 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 65 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#30632233C211yesmaximal, characteristic
2#30631661C211yesmaximal, characteristic, complemented
3#30631481C212yesmaximal, complemented
4#30631479C212yesmaximal, complemented
5#410×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C411yescharacteristic, complemented
7#22548713C411yescharacteristic, complemented
8#22548711C2211yescharacteristic
9#22548704C411yescharacteristic, complemented
10#22548702C2211yescharacteristic
11#22548687—21no—
12#22548687—21no—
13C4×C22⋊D251—21no—
14C4×C22⋊D251—21no—
15#22548674—21no—
16C2×C4×Dic251C411yescharacteristic, complemented
17#22548630—21no—
18#22548630C2212yes—
19C2×#41×C251C2211yescharacteristic
20#117×C251—21no—
21#99×C251—21no—
22#99×C251C2212yes—
23#16860810—41no—
24#16860810—41no—
25C2×Dic502C2×C411yescharacteristic
26C2×D1004C2×C411yescharacteristic
27C2×C22⋊D251—21no—
28C2×C22⋊D251—21no—
29C2×C22⋊D251—21no—
30C22×Dic251—21no—
31C2×C4×D251—21no—
32C4⋊Dic251—41no—
33Dic251⋊C4—21no—
34Dic251⋊C4—21no—
35C22⋊Dic251—21no—
36C22⋊Dic251—21no—
37C22⋊Dic251—22no—
38(C2×D251)⋊C4—21no—
39(C2×D251)⋊C4—21no—
40C4×Dic251—21no—
41C4×Dic251—41no—
42C4×Dic251C2×C411yescharacteristic
43C4×Dic251C2×C411yescharacteristic
44#41×C251—41no—
45#41×C251—41no—
46C2×Q8×C251C2×C411yescharacteristic
47C2×D4×C251D411yescharacteristic
48C2×D4×C251D411yescharacteristic
49C2×D4×C251C2311yescharacteristic
50C22×C4×C251C2×C411yescharacteristic
51C22×C4×C251D411yescharacteristic
52C22×C4×C251D411yescharacteristic
53C4⋊C4×C251—41no—
54C22⋊C4×C251—22no—
55C42×C251—41no—
56C22×D251—21no—
57Dic502—41no—
58D1004—41no—
59C22⋊D251—42no—
60C22⋊D251—42no—
61C22⋊D251—42no—
62C2×Dic251—21no—
63C2×Dic251—21no—
64C2×Dic251—21no—
65C2×Dic251—42no—
66C2×Dic251—42no—
67C4×D251—41no—
68C4×D251—41no—
69C23×C251C22⋊C411yescharacteristic
70C23×C251C2×D411yescharacteristic
71C23×C251C2×D411yescharacteristic
72Q8×C251—41no—
73D4×C251—41no—
74D4×C251—41no—
75D4×C251—41no—
76C2×C4×C251—21no—
77C2×C4×C251—21no—
78C2×C4×C251—21no—
79C2×C4×C251—21no—
80C2×C4×C251—21no—
81C2×C4×C251—21no—
82C2×C4×C251—42no—
83C2×C4×C251C22⋊C411yescharacteristic
84C2×C4×C251C22⋊C411yescharacteristic
85C2×C4×C251C22⋊C411yescharacteristic
86C2×C4×C251C22×C411yescharacteristic
87C22×C251—21no—
88C22×C251—21no—
89C22×C251—21no—
90C22×C251—41no—
91C22×C251—41no—
92C22×C251—41no—
93C22×C251C2×C22⋊C411yescharacteristic
94C2×D251—41no—
95C2×D251—41no—
96C2×D251—41no—
97C4×C251—21no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—82no—
101C4×C251#9911yescharacteristic
102C4×C251#9911yescharacteristic
103Dic251—22no—
104Dic251—41no—
105Dic251—41no—
106Dic251—82no—
107C2×C251—21no—
108C2×C251—41no—
109C2×C251—41no—
110C2×C251—41no—
111C2×C251C2×#9911yescharacteristic
112D251—81no—
113C251#212611yescharacteristic, complemented, Sylow
114#2126—2511nomaximal, Sylow
115#532—2511no—
116#410—2511no—
117#410D25111yescharacteristic, complemented
118#354—2512no—
119#353—2512no—
120C2×#41—2511no—
121C2×#41C2×D25111yescharacteristic
122C4⋊Q8—2511no—
123#127—2511no—
124#124—2511no—
125#124—2511no—
126#121—5021no—
127#121—5021no—
128C4×D4—5021no—
129C4×D4—5021no—
130#117—21no—
131#117—5021no—
132C2×C42—2511no—
133#99—21no—
134#99—5021no—
135#99—2512no—
136#99C2×D25112yes—
137#41—41no—
138#41—41no—
139#41—10041no—
140#41—10041no—
141C2×Q8—2511no—
142C2×Q8C4×D25111yescharacteristic
143C2×D4—5021no—
144C2×D4—5021no—
145C2×D4—5021no—
146C2×D4—2511no—
147C2×D4C22⋊D25111yescharacteristic
148C2×D4D100411yescharacteristic
149C2×D4C22×D25111yescharacteristic
150C22×C4—5021no—
151C22×C4—5021no—
152C22×C4C4×D25111yescharacteristic
153C22×C4C22⋊D25111yescharacteristic
154C22×C4D100411yescharacteristic
155C4⋊C4—41no—
156C4⋊C4—10041no—
157C4⋊C4—5021no—
158C4⋊C4—5021no—
159C22⋊C4—22no—
160C22⋊C4—5022no—
161C22⋊C4—5021no—
162C22⋊C4—5021no—
163C22⋊C4—5021no—
164C22⋊C4—5021no—
165C42—41no—
166C42—10041no—
167C42—5021no—
168C42—2511no—
169C42—2511no—
170C23—5021no—
171C23(C2×D251)⋊C411yescharacteristic
172C23C2×C22⋊D25111yescharacteristic
173C23C2×D100411yescharacteristic
174Q8—41no—
175Q8—10041no—
176D4—41no—
177D4—41no—
178D4—41no—
179D4—10041no—
180D4—10042no—
181D4—10042no—
182D4—10042no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—10042no—
191C2×C4—10042no—
192C2×C4—10041no—
193C2×C4—10041no—
194C2×C4—5021no—
195C2×C4—5021no—
196C2×C4—5021no—
197C2×C4(C2×D251)⋊C411yescharacteristic
198C2×C4(C2×D251)⋊C411yescharacteristic
199C2×C4(C2×D251)⋊C411yescharacteristic
200C2×C4C2×C4×D25111yescharacteristic
201C22—21no—
202C22—21no—
203C22—21no—
204C22—41no—
205C22—41no—
206C22—41no—
207C22—10041no—
208C22—10041no—
209C22—10041no—
210C22C2×(C2×D251)⋊C411yescharacteristic
211C4—21no—
212C4—21no—
213C4—21no—
214C4—82no—
215C4—20082no—
216C4—10041no—
217C4—10041no—
218C4—5022no—
219C4#2254862911yescharacteristic
220C4#2254862911yescharacteristic
221C2—21no—
222C2—41no—
223C2—41no—
224C2—41no—
225C2—20081no—
226C2C2×#2254862911yescharacteristic
subgroupG/H
centerC4#22548629
commutatorC2×C4×C251C22×C4
FrattiniC2×D4C22×D251
Fitting#410×C251C2
socleC2×C251C2×#99
prime radical#30632233C2
cosocleC2×D4×C251C23
hypercenter#410D251
hypocenterC251#2126
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #30631661C2semidirect
3. #30631481C2semidirect
4. #30631479C2semidirect
5. #410×C251C2semidirect
6. C2×#16860810C4semidirect
7. #22548713C4semidirect
9. #22548704C4semidirect
16. C2×C4×Dic251C4semidirect
113. C251#2126semidirect
117. #410D251semidirect

Maximal subgroups

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

subgroupindex
#306322332
#306316612
#306314812
#306314792
#410×C2512
#2126251