Finite Group Databasebeta

← #44548919

Subgroups — #44548919

Order 32128.

284 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 73 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×C22⋊(C251⋊C8)C211yesmaximal, characteristic, complemented
3#30631423C212yesmaximal, complemented
4#30631420C212yesmaximal, complemented
5#381×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C411yescharacteristic, complemented
7C22×C251⋊C8—21no—
8#22548698—21no—
9#22548694C411yescharacteristic, complemented
10#22548694C2211yescharacteristic
11#22548689C411yescharacteristic, complemented
12#22548689C2211yescharacteristic
13#22548684—21no—
14C4×C22⋊D251—21no—
15C4×C22⋊D251—21no—
16#22548674C411yescharacteristic, complemented
17C22⋊(C251⋊C8)—21no—
18C22⋊(C251⋊C8)C2212yes—
19C23×C4×C251C2211yescharacteristic
20C2×C22⋊C4×C251—21no—
21#95×C251—21no—
22#95×C251C2212yes—
23#16860810—42no—
24C2×Dic502C2×C411yescharacteristic
25C2×D1004C2×C411yescharacteristic
26C2×C22⋊D251—21no—
27C2×C22⋊D251—21no—
28C2×C22⋊D251—21no—
29C2×C4×D251—21no—
30C2×C251⋊C8—22no—
31C2×C251⋊C8—42no—
32C4⋊Dic251C2×C411yescharacteristic
33C4⋊Dic251C2×C411yescharacteristic
34Dic251⋊C4—21no—
35Dic251⋊C4—21no—
36C22⋊Dic251—21no—
37C22⋊Dic251—21no—
38C22⋊Dic251—21no—
39(C2×D251)⋊C4—21no—
40(C2×D251)⋊C4—21no—
41C4×Dic251—21no—
42C24×C251D411yescharacteristic
43C22×C4×C251—22no—
44C22×C4×C251—42no—
45C22×C4×C251C2×C411yescharacteristic
46C22×C4×C251C2×C411yescharacteristic
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D411yescharacteristic
49C22×C4×C251D411yescharacteristic
50C22×C4×C251C2311yescharacteristic
51C22⋊C4×C251—42no—
52C22×D251—21no—
53Dic502—41no—
54D1004—41no—
55C22⋊D251—42no—
56C22⋊D251—42no—
57C22⋊D251—42no—
58C2×Dic251—21no—
59C2×Dic251—21no—
60C2×Dic251—21no—
61C4×D251—42no—
62C251⋊C8—42no—
63C23×C251—22no—
64C23×C251—22no—
65C23×C251—41no—
66C23×C251C22⋊C411yescharacteristic
67C23×C251C22⋊C411yescharacteristic
68C23×C251C2×D411yescharacteristic
69C2×C4×C251—22no—
70C2×C4×C251—22no—
71C2×C4×C251—41no—
72C2×C4×C251—41no—
73C2×C4×C251—41no—
74C2×C4×C251—41no—
75C2×C4×C251—42no—
76C2×C4×C251—42no—
77C2×C4×C251—42no—
78C2×C4×C251C22⋊C411yescharacteristic
79C2×C4×C251C22⋊C411yescharacteristic
80C2×C4×C251C22×C411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C22×C251—21no—
83C22×C251—21no—
84C22×C251—21no—
85C22×C251—21no—
86C22×C251—21no—
87C22×C251—21no—
88C22×C251—21no—
89C22×C251—44no—
90C22×C251#9912yes—
91C22×C251C2×C22⋊C411yescharacteristic
92C22×C251#13512yes—
93C2×D251—41no—
94C2×D251—41no—
95C2×D251—41no—
96C4×C251—21no—
97C4×C251—21no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251—82no—
101Dic251—41no—
102Dic251—41no—
103Dic251—41no—
104C2×C251—22no—
105C2×C251—22no—
106C2×C251—41no—
107C2×C251C2×#9911yescharacteristic
108C2×C251#41611yescharacteristic
109C2×C251#42111yescharacteristic
110D251—81no—
111C251#151611yescharacteristic, complemented, Sylow
112#1516—2511nomaximal, Sylow
113#525—2511no—
114C2×C22⋊C8—2511no—
115#381D25111yescharacteristic, complemented
116#328—2512no—
117#327—2512no—
118C2×#41—2511no—
119C23×C4C2×D25111yescharacteristic
120C22×C8—5021no—
121#123—5021no—
122C22⋊Q8—2511no—
123C22⋊Q8—2511no—
124#121—2511no—
125#121—2511no—
126#120—5021no—
127C4×D4—5021no—
128C4×D4—5021no—
129#117—2511no—
130C2×C22⋊C4—21no—
131C22⋊C8—5021no—
132C22⋊C8—2512no—
133#95—21no—
134#95C2×D25112yes—
135C24C22⋊D25111yescharacteristic
136#41—10042no—
137C2×Q8—2511no—
138C2×D4—5021no—
139C2×D4—5021no—
140C2×D4—5021no—
141C2×D4—2511no—
142C22×C4—22no—
143C22×C4—42no—
144C22×C4—5021no—
145C22×C4C4×D25111yescharacteristic
146C22×C4C4×D25111yescharacteristic
147C22×C4C22⋊D25111yescharacteristic
148C22×C4D100411yescharacteristic
149C22×C4D100411yescharacteristic
150C22×C4C22×D25111yescharacteristic
151C2×C8—10042no—
152C2×C8—5022no—
153C4⋊C4—5021no—
154C4⋊C4—5021no—
155C4⋊C4—2511no—
156C4⋊C4—2511no—
157C22⋊C4—42no—
158C22⋊C4—5021no—
159C22⋊C4—5021no—
160C22⋊C4—5021no—
161C22⋊C4—5021no—
162C22⋊C4—5021no—
163C42—5021no—
164C23—22no—
165C23—22no—
166C23—41no—
167C23—5021no—
168C23(C2×D251)⋊C411yescharacteristic
169C23(C2×D251)⋊C411yescharacteristic
170C23C2×C22⋊D25111yescharacteristic
171Q8—10041no—
172D4—10041no—
173D4—10042no—
174D4—10042no—
175D4—10042no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—41no—
179C2×C4—41no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—42no—
185C2×C4—10042no—
186C2×C4—5021no—
187C2×C4—5021no—
188C2×C4—5021no—
189C2×C4(C2×D251)⋊C411yescharacteristic
190C2×C4(C2×D251)⋊C411yescharacteristic
191C2×C4C2×C4×D25111yescharacteristic
192C2×C4C2×D100411yescharacteristic
193C8—10042no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—44no—
202C22—10041no—
203C22—10041no—
204C22—10041no—
205C22#2254862912yes—
206C22C2×(C2×D251)⋊C411yescharacteristic
207C22#2254873512yes—
208C4—21no—
209C4—21no—
210C4—21no—
211C4—21no—
212C4—82no—
213C4—10041no—
214C4—10041no—
215C4—10041no—
216C2—22no—
217C2—22no—
218C2—41no—
219C2—20081no—
220C2C2×#2254862911yescharacteristic
221C2#3063168111yescharacteristic
222C2#3063170511yescharacteristic
subgroupG/H
centerC22C2×(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#381×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenter#381D251
hypocenterC251#1516
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C23×C4 ◁ #381

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊(C251⋊C8)C2semidirect
3. #30631423C2semidirect
4. #30631420C2semidirect
5. #381×C251C2semidirect
6. C2×#16860810C4semidirect
9. #22548694C4semidirect
11. #22548689C4semidirect
16. #22548674C4semidirect
111. C251#1516semidirect
115. #381D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C2×C22⋊(C251⋊C8)2
#306314232
#306314202
#381×C2512
#1516251