Finite Group Databasebeta

← #45663318

Subgroups — #45663318

Order 32896.

216 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 49 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#30657332C211yesmaximal, characteristic
2#30657068C211yesmaximal, characteristic, complemented
3#30657048C211yesmaximal, characteristic
4#30656669C211yesmaximal, characteristic, complemented
5#30656473C211yesmaximal, characteristic, complemented
6#30656462C211yesmaximal, characteristic, complemented
7#327×C257C211yesmaximal, characteristic
8#22551585—21no—
9#22551582—21no—
10#22551573—21no—
11#22551573—21no—
12#22551571C2211yescharacteristic
13#22551569—21no—
14D4×Dic257—21no—
15C2×C4⋊Dic257—21no—
16C2×C4⋊Dic257C2211yescharacteristic
17C2×#16861597C2211yescharacteristic
18#22551525—21no—
19#22551524—21no—
20#22551517—21no—
21#22551513—21no—
22C22⋊#14523339C2211yescharacteristic
23#22551499—21no—
24#121×C257C2211yescharacteristic
25C22⋊C8×C257C2211yescharacteristic
26#95×C257C2211yescharacteristic
27C2×Dic514D411yescharacteristic
28C22×Dic257—21no—
29C22×Dic257—21no—
30C22×Dic257D411yescharacteristic
31C22×Dic257D411yescharacteristic
32C2×#14523339—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257—21no—
35C4⋊Dic257—21no—
36C4⋊Dic257—42no—
37C4⋊Dic257D411yescharacteristic
38#16861597—21no—
39#16861597—21no—
40#16861597—21no—
41#16861597—21no—
42#16861597—21no—
43C22⋊Dic257—21no—
44C22⋊Dic257—21no—
45C22⋊Dic257—21no—
46C22⋊Dic257—21no—
47C4×Dic257—21no—
48C2×D4×C257—21no—
49C2×D4×C257D411yescharacteristic
50C22×C4×C257—21no—
51C22×C4×C257C2311yescharacteristic
52C2×C8×C257—21no—
53C4⋊C4×C257D411yescharacteristic
54C22⋊C4×C257—21no—
55Dic514—41no—
56C2×Dic257—21no—
57C2×Dic257—21no—
58C2×Dic257—21no—
59C2×Dic257—21no—
60C2×Dic257—21no—
61C2×Dic257—21no—
62C2×Dic257—41no—
63C2×Dic257—41no—
64C2×Dic257—41no—
65C2×Dic257—41no—
66C2×Dic257—81no—
67#14523339—41no—
68C23×C257—21no—
69C23×C257C2×D411yescharacteristic
70D4×C257—41no—
71D4×C257—41no—
72D4×C257—41no—
73C2×C4×C257—22no—
74C2×C4×C257—21no—
75C2×C4×C257—41no—
76C2×C4×C257—41no—
77C2×C4×C257—41no—
78C2×C4×C257C2×D411yescharacteristic
79C2×C4×C257C2×D411yescharacteristic
80C8×C257—41no—
81C22×C257—21no—
82C22×C257—21no—
83C22×C257—41no—
84C22×C257—41no—
85C22×C257—41no—
86C22×C257#12011yescharacteristic
87C22×C257#13511yescharacteristic
88C22×C257AGL(1,ℤ/8)11yescharacteristic
89C4×C257—21no—
90C4×C257—21no—
91C4×C257—42no—
92C4×C257—41no—
93Dic257—41no—
94Dic257—41no—
95Dic257—41no—
96Dic257—41no—
97Dic257—81no—
98C2×C257—21no—
99C2×C257—21no—
100C2×C257—81no—
101C2×C257#44911yescharacteristic
102C2×C257#45611yescharacteristic
103C2×C257#45811yescharacteristic
104C257#160911yescharacteristic, complemented, Sylow
105#1609—2571nomaximal, Sylow
106#539—2571no—
107#483—2571no—
108#480—2571no—
109#398—2571no—
110#328—2571no—
111#327D25711yescharacteristic
112#324—2571no—
113#126—5141no—
114#125—5141no—
115#123—5141no—
116#123—5141no—
117C22⋊Q8—2571no—
118C22⋊Q8—5141no—
119#121C2×D25711yescharacteristic
120C4×D4—5141no—
121C2×C4⋊C4—2571no—
122C2×C4⋊C4—2571no—
123C2×C4⋊C4—5141no—
124#107—5141no—
125#106—5141no—
126#103—5141no—
127#102—5141no—
128C22⋊C8—2571no—
129C22⋊C8C2×D25711yescharacteristic
130#95—5141no—
131#95C2×D25711yescharacteristic
132C2×Q8—2571no—
133C2×D4—21no—
134C2×D4D102811yescharacteristic
135C22×C4—2571no—
136C22×C4—2571no—
137C22×C4—21no—
138C22×C4—5141no—
139C22×C4—5141no—
140C22×C4C22×D25711yescharacteristic
141C2×C8—21no—
142C2×C8—5141no—
143C4⋊C4—2571no—
144C4⋊C4—5141no—
145C4⋊C4—5141no—
146C4⋊C4—5141no—
147C4⋊C4—5141no—
148C4⋊C4—5141no—
149C4⋊C4—5141no—
150C4⋊C4—5141no—
151C4⋊C4—5141no—
152C4⋊C4—10282no—
153C4⋊C4D102811yescharacteristic
154C22⋊C4—21no—
155C22⋊C4—5141no—
156C22⋊C4—5141no—
157C22⋊C4—5141no—
158C22⋊C4—5141no—
159C42—5141no—
160C23—21no—
161C23D4×D25711yescharacteristic
162Q8—10281no—
163D4—41no—
164D4—41no—
165D4—41no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—5141no—
169C2×C4—5141no—
170C2×C4—5141no—
171C2×C4—5141no—
172C2×C4—5141no—
173C2×C4—5141no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—10281no—
178C2×C4—10281no—
179C2×C4—10281no—
180C2×C4—10281no—
181C2×C4—20561no—
182C2×C4D4×D25711yescharacteristic
183C2×C4C2×D102811yescharacteristic
184C8—41no—
185C8—10281no—
186C22—21no—
187C22—21no—
188C22—41no—
189C22—41no—
190C22—41no—
191C22#2255156211yescharacteristic
192C22#2255161311yescharacteristic
193C22#2255161611yescharacteristic
194C4—21no—
195C4—21no—
196C4—42no—
197C4—41no—
198C4—10281no—
199C4—10281no—
200C4—10281no—
201C4—10281no—
202C4—20561no—
203C2—21no—
204C2—21no—
205C2—81no—
206C2#3065688311yescharacteristic
207C2#3065692111yescharacteristic
208C2#3065692811yescharacteristic
subgroupG/H
centerC22#22551562
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#327×C257C2
socleC22×C257#120
prime radical#121×C257C22
hypercenter#327D257
hypocenterC257#1609
LCSG ▷ C22×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #30657068C2semidirect
4. #30656669C2semidirect
5. #30656473C2semidirect
6. #30656462C2semidirect
104. C257#1609semidirect

Maximal subgroups

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

subgroupindex
#306573322
#306570682
#306570482
#306566692
#306564732
#306564622
#327×C2572
#1609257