Finite Group Databasebeta

← #45663271

Subgroups — #45663271

Order 32896.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#30657299C211yesmaximal, characteristic
2#30656876C212yesmaximal, complemented
3#30656470C212yesmaximal, complemented
4#30656461C211yesmaximal, characteristic, complemented
5#393×C257C211yesmaximal, characteristic, complemented
6C22×C22⋊D257—21no—
7C2×#16861605—22no—
8#22551576—22no—
9#22551566—21no—
10#22551566C2212yes—
11#22551562—22no—
12#22551516—22no—
13C22⋊#14523339C2212yescomplemented
14C22×Q8×C257C2211yescharacteristic, complemented
15C2×C22⋊C4×C257—21no—
16C2×C22⋊C4×C257C2211yescharacteristic
17#95×C257—21no—
18#95×C257C2211yescharacteristic, complemented
19C23×D257—21no—
20C2×D1028—21no—
21C2×D1028D412yes—
22C2×C22⋊D257—22no—
23C2×C22⋊D257—22no—
24C2×C22⋊D257—42no—
25C2×C22⋊D257—42no—
26C22×Dic257—21no—
27#16861605—44no—
28C2×#14523339—22no—
29C4⋊Dic257—21no—
30C4⋊Dic257D412yes—
31#16861595—22no—
32#16861595—22no—
33C24×C257D411yescharacteristic
34C2×Q8×C257—22no—
35C2×Q8×C257—42no—
36C22×C4×C257—21no—
37C22×C4×C257—21no—
38C22×C4×C257D411yescharacteristic
39C22×C4×C257C2311yescharacteristic
40C22⋊C4×C257—22no—
41C22⋊C4×C257—42no—
42C22×D257—22no—
43C22×D257—42no—
44C22×D257—81no—
45C22×D257—81no—
46D1028—42no—
47D1028—81no—
48C22⋊D257—44no—
49C22⋊D257—42no—
50C22⋊D257—42no—
51C2×Dic257—22no—
52C2×Dic257—42no—
53#14523339—42no—
54C23×C257—21no—
55C23×C257—21no—
56C23×C257—41no—
57C23×C257—41no—
58C23×C257SD1612yescomplemented
59C23×C257C2×D411yescharacteristic
60Q8×C257—24no—
61Q8×C257—42no—
62C2×C4×C257—22no—
63C2×C4×C257—21no—
64C2×C4×C257—41no—
65C2×C4×C257—41no—
66C2×C4×C257—41no—
67C2×C4×C257—41no—
68C2×C4×C257—41no—
69C2×C4×C257—41no—
70C2×C4×C257—41no—
71C2×C4×C257C2×D412yes—
72C22×C257—21no—
73C22×C257—21no—
74C22×C257—22no—
75C22×C257—42no—
76C22×C257—41no—
77C22×C257—41no—
78C22×C257—81no—
79C22×C257#12011yescharacteristic
80C22×C257C2×SD1611yescharacteristic
81C22×C257AGL(1,ℤ/8)11yescharacteristic
82C2×D257—42no—
83C2×D257—42no—
84C2×D257—42no—
85C2×D257—84no—
86C4×C257—22no—
87C4×C257—41no—
88C4×C257—42no—
89C4×C257—81no—
90Dic257—42no—
91C2×C257—21no—
92C2×C257—21no—
93C2×C257—42no—
94C2×C257#45011yescharacteristic
95C2×C257#45511yescharacteristic
96C2×C257B(4,𝔽2)11yescharacteristic
97D257—82no—
98C257#160111yescharacteristic, complemented, Sylow
99#1601—2571nomaximal, Sylow
100#534—2571no—
101#448—2572no—
102#393D25711yescharacteristic, complemented
103#327—2572no—
104#323—2571no—
105C22×Q8C2×D25711yescharacteristic, complemented
106C22×D4—5141no—
107C2×SD16—5142no—
108#123—5142no—
109#121—2572no—
110#121—5141no—
111#120—5142no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C2×D25711yescharacteristic
114#103—5142no—
115C22⋊C8—2572no—
116#95—21no—
117#95C2×D25711yescharacteristic, complemented
118C24—5141no—
119C24C22⋊D25711yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—2572no—
123C2×D4—5141no—
124C2×D4—5142no—
125C2×D4—5142no—
126C2×D4—10282no—
127C2×D4—10282no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—5141no—
131C22×C4C22⋊D25711yescharacteristic
132C22×C4C22×D25711yescharacteristic
133SD16—10284no—
134C2×C8—5142no—
135C4⋊C4—2572no—
136C4⋊C4—5141no—
137C22⋊C4—22no—
138C22⋊C4—5142no—
139C22⋊C4—5142no—
140C22⋊C4—42no—
141C23—21no—
142C23—21no—
143C23—5142no—
144C23—41no—
145C23—41no—
146C23—10282no—
147C23—20561no—
148C23—20561no—
149C23#1686160512yescomplemented
150C23C2×C22⋊D25711yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—10282no—
154D4—10284no—
155D4—10282no—
156D4—10282no—
157D4—20561no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—5142no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—41no—
168C2×C4—10282no—
169C2×C4D4×D25712yes—
170C8—10282no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—10282no—
178C22—10282no—
179C22—10282no—
180C22—81no—
181C22—20564no—
182C22#2255156011yescharacteristic
183C22C2×#1686160511yescharacteristic
184C22#2255161711yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—10282no—
189C4—81no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—20562no—
194C2#3065688711yescharacteristic
195C2#3065691411yescharacteristic
196C2#3065692311yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#393×C257C2
socleC22×C257#120
prime radical#30657299C2
hypercenter#393D257
hypocenterC257#1601
LCSG ▷ C22×C4×C257 ▷ C22×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #30656876C2semidirect
3. #30656470C2semidirect
4. #30656461C2semidirect
5. #393×C257C2semidirect
13. C22⋊#14523339C22semidirect
14. C22×Q8×C257C22semidirect
18. #95×C257C22semidirect
58. C23×C257SD16semidirect
98. C257#1601semidirect
102. #393D257semidirect
105. C22×Q8C2×D257semidirect
117. #95C2×D257semidirect
149. C23#16861605semidirect

Maximal subgroups

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

subgroupindex
#306572992
#306568762
#306564702
#306564612
#393×C2572
#1601257