Finite Group Databasebeta

← #44548939

Subgroups — #44548939

Order 32128.

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

normal subgroups only

This group has 218 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
1#30632218C211yesmaximal, characteristic
2#30631737C211yesmaximal, characteristic, complemented
3#30631554C211yesmaximal, characteristic, complemented
4#30631425C211yesmaximal, characteristic, complemented
5#30631423C211yesmaximal, characteristic, complemented
6#30631421C211yesmaximal, characteristic
7#327×C251C211yesmaximal, characteristic
8#22548708—21no—
9#22548706C411yescharacteristic, complemented
10#22548702C411yescharacteristic, complemented
11#22548696—21no—
12#22548694C2211yescharacteristic
13#22548680C411yescharacteristic, complemented
14D4×Dic251—21no—
15D4×Dic251C411yescharacteristic, complemented
16#22548671—21no—
17C2×C4×Dic251C2211yescharacteristic
18Dic251⋊C8—21no—
19C22⋊(C251⋊C8)C2211yescharacteristic
20C8×Dic251—21no—
21C251⋊#95—21no—
22C251⋊#95—21no—
23C251⋊#95C2211yescharacteristic
24#121×C251C2211yescharacteristic
25C22⋊C8×C251C2211yescharacteristic
26#95×C251C2211yescharacteristic
27C2×Dic502C2×C411yescharacteristic
28C22×Dic251—21no—
29C22×Dic251—21no—
30C22×Dic251D411yescharacteristic
31C22×Dic251D411yescharacteristic
32C2×C251⋊C8—21no—
33C4⋊Dic251—21no—
34C4⋊Dic251C2×C411yescharacteristic
35Dic251⋊C4—21no—
36Dic251⋊C4—21no—
37Dic251⋊C4—21no—
38C22⋊Dic251—21no—
39C22⋊Dic251—21no—
40C22⋊Dic251—21no—
41C22⋊Dic251—21no—
42C4×Dic251—21no—
43C4×Dic251C2×C411yescharacteristic
44C4×Dic251C2×C411yescharacteristic
45C4×Dic251D411yescharacteristic
46C4×Dic251D411yescharacteristic
47C2×D4×C251—21no—
48C2×D4×C251C2×C411yescharacteristic
49C22×C4×C251—21no—
50C22×C4×C251C2311yescharacteristic
51C2×C8×C251—21no—
52C4⋊C4×C251C2×C411yescharacteristic
53C22⋊C4×C251—21no—
54Dic502—41no—
55C2×Dic251—21no—
56C2×Dic251—21no—
57C2×Dic251—21no—
58C2×Dic251—21no—
59C2×Dic251—41no—
60C2×Dic251—42no—
61C2×Dic251—41no—
62C2×Dic251—41no—
63C2×Dic251—41no—
64C2×Dic251C22⋊C411yescharacteristic
65C2×Dic251C22⋊C411yescharacteristic
66C2×Dic251C22⋊C411yescharacteristic
67C2×Dic251C22⋊C411yescharacteristic
68C251⋊C8—41no—
69C23×C251—21no—
70C23×C251C2×D411yescharacteristic
71D4×C251—41no—
72D4×C251—41no—
73D4×C251—41no—
74C2×C4×C251—21no—
75C2×C4×C251—41no—
76C2×C4×C251—41no—
77C2×C4×C251—41no—
78C2×C4×C251—41no—
79C2×C4×C251C22×C411yescharacteristic
80C2×C4×C251C2×D411yescharacteristic
81C8×C251—41no—
82C22×C251—21no—
83C22×C251—21no—
84C22×C251—41no—
85C22×C251—41no—
86C22×C251—41no—
87C22×C251C2×C22⋊C411yescharacteristic
88C22×C251#13511yescharacteristic
89C22×C251#13511yescharacteristic
90C4×C251—21no—
91C4×C251—21no—
92C4×C251—41no—
93C4×C251—81no—
94Dic251—21no—
95Dic251—21no—
96Dic251—21no—
97Dic251—21no—
98Dic251—41no—
99Dic251—41no—
100Dic251—81no—
101C2×C251—21no—
102C2×C251—21no—
103C2×C251—81no—
104C2×C251#41011yescharacteristic
105C2×C251#41611yescharacteristic
106C2×C251#42111yescharacteristic
107C251#152111yescharacteristic, complemented, Sylow
108#1521—2511nomaximal, Sylow
109#529—2511no—
110#431—2511no—
111#383—2511no—
112#328—2511no—
113#328—2511no—
114#327—2511no—
115#327D25111yescharacteristic
116#126—5021no—
117#125—2511no—
118#124—2511no—
119#123—5021no—
120C22⋊Q8—2511no—
121#121C2×D25111yescharacteristic
122C4×Q8—2511no—
123C4×D4—5021no—
124C4×D4—2511no—
125#117—5021no—
126C2×C42—2511no—
127C4⋊C8—5021no—
128C22⋊C8—2511no—
129C22⋊C8C2×D25111yescharacteristic
130C4×C8—5021no—
131#95—5021no—
132#95—5021no—
133#95—2511no—
134#95C2×D25111yescharacteristic
135C2×Q8—2511no—
136C2×D4—21no—
137C2×D4C4×D25111yescharacteristic
138C22×C4—21no—
139C22×C4—5021no—
140C22×C4—5021no—
141C22×C4—2511no—
142C22×C4—2511no—
143C22×C4C22×D25111yescharacteristic
144C2×C8—21no—
145C2×C8—5021no—
146C4⋊C4—5021no—
147C4⋊C4—5021no—
148C4⋊C4—5021no—
149C4⋊C4—5021no—
150C4⋊C4—2511no—
151C4⋊C4C4×D25111yescharacteristic
152C22⋊C4—21no—
153C22⋊C4—5021no—
154C22⋊C4—5021no—
155C22⋊C4—5021no—
156C22⋊C4—5021no—
157C42—5021no—
158C42—2511no—
159C42—2511no—
160C42—2511no—
161C42—2511no—
162C23—21no—
163C23D4×D25111yescharacteristic
164Q8—10041no—
165D4—41no—
166D4—41no—
167D4—41no—
168C2×C4—21no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—10041no—
174C2×C4—10041no—
175C2×C4—10041no—
176C2×C4—10042no—
177C2×C4—10041no—
178C2×C4—5021no—
179C2×C4—5021no—
180C2×C4—5021no—
181C2×C4—5021no—
182C2×C4—2511no—
183C2×C4—2511no—
184C2×C4—2511no—
185C2×C4—2511no—
186C2×C4C2×C4×D25111yescharacteristic
187C2×C4D4×D25111yescharacteristic
188C8—41no—
189C8—10041no—
190C22—21no—
191C22—21no—
192C22—41no—
193C22—41no—
194C22—41no—
195C22C22⋊C4×D25111yescharacteristic
196C22#2254873411yescharacteristic
197C22#2254873611yescharacteristic
198C4—21no—
199C4—21no—
200C4—41no—
201C4—81no—
202C4—20081no—
203C4—10041no—
204C4—10041no—
205C4—5021no—
206C4—5021no—
207C4—5021no—
208C4—5021no—
209C2—21no—
210C2—21no—
211C2—81no—
212C2#3063165911yescharacteristic
213C2#3063168011yescharacteristic
214C2#3063170211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D251
commutatorC2×C4×C251C22×C4
FrattiniC22×C4C22×D251
Fitting#327×C251C2
socleC22×C251C2×C22⋊C4
prime radical#121×C251C22
cosocleC22×C4×C251C23
hypercenter#327D251
hypocenterC251#1521
LCSG ▷ C2×C4×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #30631737C2semidirect
3. #30631554C2semidirect
4. #30631425C2semidirect
5. #30631423C2semidirect
9. #22548706C4semidirect
10. #22548702C4semidirect
13. #22548680C4semidirect
15. D4×Dic251C4semidirect
107. C251#1521semidirect

Maximal subgroups

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

subgroupindex
#306322182
#306317372
#306315542
#306314252
#306314232
#306314212
#327×C2512
#1521251