Finite Group Databasebeta

← #45662948

Subgroups — #45662948

Order 32896.

200 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 57 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#30657266C211yesmaximal, characteristic
2#30656792C211yesmaximal, characteristic, complemented
3#30656792C211yesmaximal, characteristic, complemented
4#30656478C211yesmaximal, characteristic, complemented
5#30656478C211yesmaximal, characteristic, complemented
6#30656477C211yesmaximal, characteristic, complemented
7#329×C257C211yesmaximal, characteristic, complemented
8#22551585—21no—
9#22551584C2211yescharacteristic
10#22551574—21no—
11#22551568C411yescharacteristic
12#22551563C411yescharacteristic
13Q8×Dic257C411yescharacteristic
14D4×Dic257C411yescharacteristic
15#22551552—21no—
16#22551548—21no—
17C42×D257C2211yescharacteristic
18#22551521—21no—
19#22551521—21no—
20#22551504—21no—
21#22551504—21no—
22#22551503C2211yescharacteristic
23#22551503C2211yescharacteristic
24#124×C257C2211yescharacteristic
25#97×C257C2211yescharacteristic
26#97×C257C2211yescharacteristic
27C2×C22⋊D257—21no—
28C22×Dic257—21no—
29C2×C4×D257C2×C411yescharacteristic
30C2×C4×D257D411yescharacteristic
31C2×C4×D257D411yescharacteristic
32C2×#14523339—21no—
33C2×#14523339—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257C2×C411yescharacteristic
36C4⋊Dic257C2×C411yescharacteristic
37#16861597—21no—
38#16861597—21no—
39#16861597—21no—
40C22⋊Dic257—21no—
41C22⋊Dic257—21no—
42#16861595—21no—
43C4×Dic257—21no—
44C4×Dic257C2×C411yescharacteristic
45C4×Dic257D411yescharacteristic
46C4×Dic257D411yescharacteristic
47C2×Q8×C257C2×C411yescharacteristic
48C2×D4×C257C2×C411yescharacteristic
49C2×C8×C257—21no—
50C2×C8×C257—21no—
51C22⋊C4×C257—21no—
52C22⋊C4×C257—21no—
53C42×C257C2311yescharacteristic
54C22×D257C22⋊C411yescharacteristic
55C22⋊D257—42no—
56C2×Dic257—21no—
57C2×Dic257—21no—
58C2×Dic257—42no—
59C2×Dic257C22⋊C411yescharacteristic
60C2×Dic257C22⋊C411yescharacteristic
61C2×Dic257C22⋊C411yescharacteristic
62C4×D257—41no—
63C4×D257—41no—
64C4×D257—41no—
65#14523339—41no—
66#14523339—41no—
67C23×C257—21no—
68Q8×C257—41no—
69D4×C257—41no—
70C2×C4×C257—21no—
71C2×C4×C257C22×C411yescharacteristic
72C2×C4×C257C2×D411yescharacteristic
73C2×C4×C257C2×D411yescharacteristic
74C8×C257—41no—
75C8×C257—41no—
76C22×C257—41no—
77C22×C257—41no—
78C22×C257—41no—
79C22×C257C2×C22⋊C411yescharacteristic
80C2×D257—21no—
81C2×D257—21no—
82C2×D257—21no—
83C4×C257—21no—
84C4×C257—21no—
85C4×C257—21no—
86C4×C257—41no—
87Dic257—21no—
88Dic257—21no—
89Dic257—21no—
90Dic257—41no—
91Dic257—41no—
92C2×C257—81no—
93C2×C257#41311yescharacteristic
94C2×C257#42111yescharacteristic
95C2×C257#42111yescharacteristic
96D257—41no—
97C257#153511yescharacteristic, complemented, Sylow
98#1535—2571nomaximal, Sylow
99#529—2571no—
100#432—2571no—
101#432—2571no—
102#330—2571no—
103#330—2571no—
104#329—2571no—
105#329D25711yescharacteristic, complemented
106#126—5141no—
107#125—2571no—
108#124C2×D25711yescharacteristic
109#123—5141no—
110C22⋊Q8—2571no—
111#121—2571no—
112C4×Q8—2571no—
113C4×D4—2571no—
114C4×D4—5141no—
115#117—5141no—
116C2×C42—2571no—
117C4⋊C8—5141no—
118C4⋊C8—5141no—
119C22⋊C8—5141no—
120C22⋊C8—5141no—
121#97—2571no—
122#97—2571no—
123#97C2×D25711yescharacteristic
124#97C2×D25711yescharacteristic
125C2×Q8C4×D25711yescharacteristic
126C2×D4—5141no—
127C2×D4C4×D25711yescharacteristic
128C22×C4—2571no—
129C22×C4—2571no—
130C22×C4—2571no—
131C22×C4—5141no—
132C2×C8—21no—
133C2×C8—21no—
134C2×C8—5141no—
135C2×C8—5141no—
136C4⋊C4—2571no—
137C4⋊C4—2571no—
138C4⋊C4—5141no—
139C4⋊C4—5141no—
140C4⋊C4—5141no—
141C4⋊C4—5141no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—5141no—
145C22⋊C4—5141no—
146C22⋊C4—5141no—
147C42—2571no—
148C42—2571no—
149C42—2571no—
150C42—5141no—
151C42C22×D25711yescharacteristic
152C23—2571no—
153C23—21no—
154Q8—41no—
155D4—41no—
156D4—10282no—
157C2×C4—2571no—
158C2×C4—2571no—
159C2×C4—2571no—
160C2×C4—21no—
161C2×C4—5141no—
162C2×C4—5141no—
163C2×C4—10281no—
164C2×C4—10281no—
165C2×C4—10281no—
166C2×C4—10282no—
167C2×C4C2×C4×D25711yescharacteristic
168C2×C4D4×D25711yescharacteristic
169C2×C4D4×D25711yescharacteristic
170C8—41no—
171C8—41no—
172C8—10281no—
173C8—10281no—
174C22—5141no—
175C22—5141no—
176C22—5141no—
177C22—41no—
178C22—41no—
179C22—41no—
180C22C22⋊C4×D25711yescharacteristic
181C4—21no—
182C4—21no—
183C4—21no—
184C4—5141no—
185C4—5141no—
186C4—5141no—
187C4—41no—
188C4—10281no—
189C4—10281no—
190C2—10281no—
191C2—81no—
192C2#3065671811yescharacteristic
193C2#3065675211yescharacteristic
194C2#3065675211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D257
commutatorC2×C4×C257C22×C4
FrattiniC42C22×D257
Fitting#329×C257C2
socleC22×C257C2×C22⋊C4
prime radical#22551563C4
cosocleC42×C257C23
hypercenter#329D257
hypocenterC257#1535
LCSG ▷ C2×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #30656792C2semidirect
3. #30656792C2semidirect
4. #30656478C2semidirect
5. #30656478C2semidirect
6. #30656477C2semidirect
7. #329×C257C2semidirect
97. C257#1535semidirect
105. #329D257semidirect

Maximal subgroups

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

subgroupindex
#306572662
#306567922
#306564782
#306564772
#329×C2572
#1535257