Finite Group Databasebeta

← #44548857

Subgroups — #44548857

Order 32128.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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
2C4×C22⋊Dic251C211yesmaximal, characteristic, complemented
3#30631425C212yesmaximal, complemented
4#30631422C212yesmaximal, complemented
5C2×C22⋊C8×C251C211yesmaximal, characteristic, complemented
6C2×#16860810C411yescharacteristic, complemented
7#22548698—21no—
8#22548694C411yescharacteristic, complemented
9#22548694C2211yescharacteristic
10#22548689C411yescharacteristic, complemented
11#22548689C2211yescharacteristic
12#22548684—21no—
13C4×C22⋊D251—21no—
14C4×C22⋊D251—21no—
15#22548674C411yescharacteristic, complemented
16C2×C22⋊Dic251—21no—
17C2×C4×Dic251—21no—
18C251⋊#95C2212yes—
19C23×C4×C251C2211yescharacteristic
20C22×C8×C251—21no—
21C22⋊C8×C251—21no—
22C22⋊C8×C251C2212yes—
23#16860810—42no—
24C2×Dic502C2×C411yescharacteristic
25C2×D1004C2×C411yescharacteristic
26C2×C22⋊D251—21no—
27C2×C22⋊D251—21no—
28C2×C22⋊D251—21no—
29C22×Dic251—22no—
30C2×C4×D251—21no—
31C4⋊Dic251C2×C411yescharacteristic
32C4⋊Dic251C2×C411yescharacteristic
33Dic251⋊C4—21no—
34Dic251⋊C4—21no—
35C22⋊Dic251—24no—
36C22⋊Dic251—21no—
37C22⋊Dic251—21no—
38C22⋊Dic251—21no—
39(C2×D251)⋊C4—21no—
40(C2×D251)⋊C4—21no—
41C4×Dic251—21no—
42C4×Dic251—42no—
43C24×C251D411yescharacteristic
44C22×C4×C251—42no—
45C22×C4×C251C2×C411yescharacteristic
46C22×C4×C251C2×C411yescharacteristic
47C22×C4×C251D411yescharacteristic
48C22×C4×C251D411yescharacteristic
49C22×C4×C251D411yescharacteristic
50C22×C4×C251C2311yescharacteristic
51C2×C8×C251—22no—
52C2×C8×C251—42no—
53C22×D251—21no—
54Dic502—41no—
55D1004—41no—
56C22⋊D251—42no—
57C22⋊D251—42no—
58C22⋊D251—42no—
59C2×Dic251—24no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—21no—
63C2×Dic251—42no—
64C2×Dic251—42no—
65C4×D251—42no—
66C23×C251—22no—
67C23×C251—22no—
68C23×C251—41no—
69C23×C251C22⋊C411yescharacteristic
70C23×C251C22⋊C411yescharacteristic
71C23×C251C2×D411yescharacteristic
72C2×C4×C251—22no—
73C2×C4×C251—22no—
74C2×C4×C251—22no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251C22⋊C411yescharacteristic
79C2×C4×C251C22⋊C411yescharacteristic
80C2×C4×C251C22×C411yescharacteristic
81C2×C4×C251C2×D411yescharacteristic
82C8×C251—42no—
83C22×C251—21no—
84C22×C251—21no—
85C22×C251—21no—
86C22×C251—21no—
87C22×C251—21no—
88C22×C251—21no—
89C22×C251—21no—
90C22×C251—44no—
91C22×C251#10412yes—
92C22×C251C2×C22⋊C411yescharacteristic
93C22×C251#13512yes—
94C2×D251—41no—
95C2×D251—41no—
96C2×D251—41no—
97C4×C251—21no—
98C4×C251—21no—
99C4×C251—21no—
100C4×C251#9912yes—
101Dic251—44no—
102Dic251—41no—
103Dic251—41no—
104Dic251—41no—
105C2×C251—22no—
106C2×C251—22no—
107C2×C251—41no—
108C2×C251C2×#9911yescharacteristic
109C2×C251#41611yescharacteristic
110C2×C251C2×#10411yescharacteristic
111D251—81no—
112C251#150111yescharacteristic, complemented, Sylow
113#1501—2511nomaximal, Sylow
114#525—2511no—
115C2×C22⋊C8D25111yescharacteristic, complemented
116C4×C22⋊C4—2511no—
117#328—2512no—
118#327—2512no—
119C2×#41—2511no—
120C23×C4C2×D25111yescharacteristic
121C22×C8—21no—
122#123—5021no—
123C22⋊Q8—2511no—
124C22⋊Q8—2511no—
125#121—2511no—
126#121—2511no—
127#120—5021no—
128C4×D4—5021no—
129C4×D4—5021no—
130#117—2511no—
131C2×C22⋊C4—5021no—
132C2×C42—5021no—
133C22⋊C8—21no—
134C22⋊C8C2×D25112yes—
135#95—2512no—
136C24D100411yescharacteristic
137#41—10042no—
138C2×Q8—2511no—
139C2×D4—5021no—
140C2×D4—5021no—
141C2×D4—5021no—
142C2×D4—2511no—
143C22×C4—42no—
144C22×C4—5022no—
145C22×C4—5021no—
146C22×C4C4×D25111yescharacteristic
147C22×C4C4×D25111yescharacteristic
148C22×C4C22⋊D25111yescharacteristic
149C22×C4C22⋊D25111yescharacteristic
150C22×C4D100411yescharacteristic
151C22×C4C22×D25111yescharacteristic
152C2×C8—22no—
153C2×C8—42no—
154C4⋊C4—5021no—
155C4⋊C4—5021no—
156C4⋊C4—2511no—
157C4⋊C4—2511no—
158C22⋊C4—5024no—
159C22⋊C4—5021no—
160C22⋊C4—5021no—
161C22⋊C4—5021no—
162C22⋊C4—5021no—
163C22⋊C4—5021no—
164C42—10042no—
165C42—5021no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—5021no—
170C23(C2×D251)⋊C411yescharacteristic
171C23(C2×D251)⋊C411yescharacteristic
172C23C2×D100411yescharacteristic
173Q8—10041no—
174D4—10041no—
175D4—10042no—
176D4—10042no—
177D4—10042no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—10042no—
185C2×C4—10042no—
186C2×C4—10042no—
187C2×C4—5024no—
188C2×C4—5021no—
189C2×C4—5021no—
190C2×C4—5021no—
191C2×C4(C2×D251)⋊C411yescharacteristic
192C2×C4(C2×D251)⋊C411yescharacteristic
193C2×C4C2×C4×D25111yescharacteristic
194C2×C4C2×C22⋊D25111yescharacteristic
195C8—42no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—10041no—
205C22—10041no—
206C22—10041no—
207C22#2254864112yes—
208C22C2×(C2×D251)⋊C411yescharacteristic
209C22#2254873712yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—10044no—
214C4—10041no—
215C4—10041no—
216C4—10041no—
217C4#2254862912yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—20081no—
222C2C2×#2254862911yescharacteristic
223C2#3063168411yescharacteristic
224C2C2×#2254864111yescharacteristic
subgroupG/H
centerC2×C4(C2×D251)⋊C4
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
FittingC2×C22⋊C8×C251C2
socleC22×C251C2×C22⋊C4
prime radical#30632198C2
cosocleC22×C4×C251C23
hypercenterC2×C22⋊C8D251
hypocenterC251#1501
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C4×C22⋊Dic251C2semidirect
3. #30631425C2semidirect
4. #30631422C2semidirect
5. C2×C22⋊C8×C251C2semidirect
6. C2×#16860810C4semidirect
8. #22548694C4semidirect
10. #22548689C4semidirect
15. #22548674C4semidirect
112. C251#1501semidirect
115. C2×C22⋊C8D251semidirect

Maximal subgroups

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

subgroupindex
#306321982
C4×C22⋊Dic2512
#306314252
#306314222
C2×C22⋊C8×C2512
#1501251