Finite Group Databasebeta

← #44548733

Subgroups — #44548733

Order 32128.

214 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 61 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×#22548710C211yesmaximal, characteristic
2#30631741C211yesmaximal, characteristic, complemented
3#30631737C211yesmaximal, characteristic, complemented
4#30631413C211yesmaximal, characteristic, complemented
5#30631410C211yesmaximal, characteristic
6#30631410C211yesmaximal, characteristic
7#324×C251C211yesmaximal, characteristic, complemented
8#22548710—21no—
9#22548710—21no—
10#22548710—21no—
11#22548710—21no—
12C2×C4⋊Dic251C411yescharacteristic
13C2×Dic251⋊C4C411yescharacteristic
14C2×(C2×D251)⋊C4C411yescharacteristic
15C2×(C2×D251)⋊C4C411yescharacteristic
16C2×(C2×D251)⋊C4C2211yescharacteristic
17C2×C4×Dic251C2211yescharacteristic
18Dic251⋊C8—21no—
19Dic251⋊C8—21no—
20C22⋊(C251⋊C8)C2211yescharacteristic
21C22⋊(C251⋊C8)C2211yescharacteristic
22#22548625—21no—
23C8×Dic251—21no—
24C2×C4⋊C4×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26C22⋊C8×C251C2211yescharacteristic
27C23×D251C2×C411yescharacteristic
28C22×Dic251C2×C411yescharacteristic
29C22×Dic251C2×C411yescharacteristic
30C22×Dic251C2×C411yescharacteristic
31C2×C251⋊C8—21no—
32C2×C251⋊C8—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251—21no—
35Dic251⋊C4—21no—
36Dic251⋊C4—21no—
37(C2×D251)⋊C4—21no—
38(C2×D251)⋊C4—21no—
39(C2×D251)⋊C4—21no—
40(C2×D251)⋊C4—21no—
41(C2×D251)⋊C4—21no—
42(C2×D251)⋊C4—21no—
43C4×Dic251D411yescharacteristic
44C4×Dic251D411yescharacteristic
45C4×Dic251D411yescharacteristic
46C4×Dic251D411yescharacteristic
47C22×C4×C251C2×C411yescharacteristic
48C22×C4×C251C2×C411yescharacteristic
49C22×C4×C251C2311yescharacteristic
50C2×C8×C251—21no—
51C2×C8×C251—21no—
52C4⋊C4×C251—21no—
53C4⋊C4×C251—21no—
54C22×D251—21no—
55C22×D251—21no—
56C22×D251—21no—
57C22×D251—41no—
58C22×D251—41no—
59C2×Dic251—21no—
60C2×Dic251—21no—
61C2×Dic251—21no—
62C2×Dic251—41no—
63C2×Dic251—41no—
64C2×Dic251C22⋊C411yescharacteristic
65C2×Dic251C22⋊C411yescharacteristic
66C2×Dic251C22⋊C411yescharacteristic
67C2×Dic251C22⋊C411yescharacteristic
68C251⋊C8—41no—
69C251⋊C8—41no—
70C23×C251C22×C411yescharacteristic
71C2×C4×C251—21no—
72C2×C4×C251—21no—
73C2×C4×C251—21no—
74C2×C4×C251—21no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251C2×D411yescharacteristic
78C2×C4×C251C2×D411yescharacteristic
79C8×C251—41no—
80C8×C251—41no—
81C22×C251—21no—
82C22×C251—21no—
83C22×C251C2×C22⋊C411yescharacteristic
84C22×C251#13111yescharacteristic
85C22×C251#13111yescharacteristic
86C2×D251—41no—
87C2×D251—41no—
88C2×D251—41no—
89C2×D251—81no—
90C2×D251—81no—
91C4×C251—21no—
92C4×C251—21no—
93C4×C251—41no—
94C4×C251—41no—
95Dic251—21no—
96Dic251—21no—
97Dic251—21no—
98Dic251—21no—
99Dic251—41no—
100C2×C251—21no—
101C2×C251—21no—
102C2×C251#40811yescharacteristic
103C2×C251#41011yescharacteristic
104C2×C251#41311yescharacteristic
105D251—81no—
106C251#147311yescharacteristic, complemented, Sylow
107#1473—2511nomaximal, Sylow
108C2×#126—2511no—
109#432—2511no—
110#431—2511no—
111#324—2511no—
112#324D25111yescharacteristic, complemented
113#323—2511no—
114#323—2511no—
115#126—5021no—
116#126—5021no—
117#126—5021no—
118#126—5021no—
119C2×C4⋊C4—2511no—
120C2×C4⋊C4—2511no—
121C2×C4⋊C4C2×D25111yescharacteristic
122C2×C22⋊C4—2511no—
123C2×C22⋊C4—2511no—
124C2×C22⋊C4—2511no—
125C2×C42—2511no—
126C4⋊C8—5021no—
127C4⋊C8—5021no—
128C22⋊C8—2511no—
129C22⋊C8—2511no—
130C22⋊C8C2×D25111yescharacteristic
131C22⋊C8C2×D25111yescharacteristic
132#97—5021no—
133C4×C8—5021no—
134C24—2511no—
135C22×C4—2511no—
136C22×C4—2511no—
137C22×C4—2511no—
138C22×C4C4×D25111yescharacteristic
139C22×C4C4×D25111yescharacteristic
140C22×C4C22×D25111yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—5021no—
144C2×C8—5021no—
145C4⋊C4—21no—
146C4⋊C4—21no—
147C4⋊C4—5021no—
148C4⋊C4—5021no—
149C4⋊C4—5021no—
150C4⋊C4—5021no—
151C22⋊C4—5021no—
152C22⋊C4—5021no—
153C22⋊C4—5021no—
154C22⋊C4—5021no—
155C22⋊C4—5021no—
156C22⋊C4—5021no—
157C42—2511no—
158C42—2511no—
159C42—2511no—
160C42—2511no—
161C23—10041no—
162C23—10041no—
163C23—5021no—
164C23—5021no—
165C23—5021no—
166C23C2×C4×D25111yescharacteristic
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—10041no—
174C2×C4—10041no—
175C2×C4—5021no—
176C2×C4—5021no—
177C2×C4—5021no—
178C2×C4—2511no—
179C2×C4—2511no—
180C2×C4—2511no—
181C2×C4—2511no—
182C2×C4D4×D25111yescharacteristic
183C2×C4D4×D25111yescharacteristic
184C8—41no—
185C8—41no—
186C8—10041no—
187C8—10041no—
188C22—21no—
189C22—21no—
190C22—20081no—
191C22—20081no—
192C22—10041no—
193C22—10041no—
194C22—10041no—
195C22C22⋊C4×D25111yescharacteristic
196C22#2254872111yescharacteristic
197C22#2254872311yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—41no—
202C4—10041no—
203C4—5021no—
204C4—5021no—
205C4—5021no—
206C4—5021no—
207C2—21no—
208C2—21no—
209C2—20081no—
210C2#3063165211yescharacteristic
211C2#3063165911yescharacteristic
212C2#3063167011yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC23×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#324×C251C2
socleC22×C251C2×C22⋊C4
prime radicalC23×D251C2×C4
cosocleC22×C4×C251C23
hypercenter#324D251
hypocenterC251#1473
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
2. #30631741C2semidirect
3. #30631737C2semidirect
4. #30631413C2semidirect
7. #324×C251C2semidirect
106. C251#1473semidirect
112. #324D251semidirect

Maximal subgroups

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

subgroupindex
C2×#225487102
#306317412
#306317372
#306314132
#306314102
#324×C2512
#1473251