Finite Group Databasebeta

← #45665384

Subgroups — #45665384

Order 32896.

394 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 71 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#30657244C212yesmaximal, complemented
2#30656754C212yesmaximal, complemented
3#30656700C211yesmaximal, characteristic, complemented
4#30656494C211yesmaximal, characteristic, complemented
5#575×C257C211yesmaximal, characteristic, complemented
6C22×C4×D257C2211yescharacteristic
7C2×#16861602C2211yescharacteristic
8#22551573—22no—
9#22551568—22no—
10#22551563—22no—
11#22551560—22no—
12C4×C22⋊D257—24no—
13#22551551C2212yes—
14#22551519—24no—
15#22551504—22no—
16C2×#41×C257C2212yes—
17#137×C257—22no—
18AGL(1,ℤ/8)×C257—22no—
19#135×C257—22no—
20C2×M16×C257C2211yescharacteristic
21C23×D257D411yescharacteristic
22C2×C22⋊D257—24no—
23C22×Dic257D411yescharacteristic
24C2×C4×D257—22no—
25C2×C4×D257—22no—
26C2×C4×D257—21no—
27C2×C4×D257—21no—
28#16861602—22no—
29C2×#14523339—21no—
30C4⋊Dic257—22no—
31#16861597—24no—
32C22⋊Dic257—24no—
33C22⋊Dic257—22no—
34#16861595—24no—
35C4×Dic257D414yescomplemented
36#41×C257—22no—
37#41×C257—22no—
38#41×C257—22no—
39#41×C257—22no—
40C2×Q8×C257D412yescomplemented
41C2×D4×C257—22no—
42C2×D4×C257D412yescomplemented
43C22×C4×C257—22no—
44C22×C4×C257C2311yescharacteristic
45Q16×C257—22no—
46SD16×C257—24no—
47D8×C257—22no—
48M16×C257—22no—
49C2×C8×C257D412yescomplemented
50C22×D257—22no—
51C22×D257—21no—
52C22×D257—42no—
53C22⋊D257—48no—
54C2×Dic257—24no—
55C2×Dic257—22no—
56C2×Dic257—21no—
57C4×D257—41no—
58C4×D257—42no—
59C4×D257—41no—
60#14523339—41no—
61C23×C257—22no—
62C23×C257C2×D411yescharacteristic
63Q8×C257—22no—
64Q8×C257—22no—
65D4×C257—22no—
66D4×C257—22no—
67D4×C257—42no—
68D4×C257—42no—
69C2×C4×C257—22no—
70C2×C4×C257—42no—
71C2×C4×C257—42no—
72C2×C4×C257C2×D412yes—
73C2×C4×C257C2×D412yes—
74C2×C4×C257C2×D411yescharacteristic
75C2×C4×C257#4111yescharacteristic
76C8×C257—22no—
77C22×C257—41no—
78C22×C257—44no—
79C22×C257—42no—
80C22×C257#12012yes—
81C22×C257#12111yescharacteristic
82C2×D257—24no—
83C2×D257—22no—
84C2×D257—42no—
85C2×D257—41no—
86C2×D257—41no—
87C4×C257—42no—
88C4×C257#12011yescharacteristic
89C4×C257#12112yes—
90C4×C257#12711yescharacteristic
91Dic257—44no—
92Dic257—41no—
93C2×C257—22no—
94C2×C257—21no—
95C2×C257—82no—
96C2×C257#39211yescharacteristic
97D257—42no—
98C257#201011yescharacteristic, complemented, Sylow
99#2010—2571nomaximal, Sylow
100#575D25711yescharacteristic, complemented
101#525—2572no—
102#421—2572no—
103#407—2571no—
104H(3,ℤ/4)—2571no—
105C2×#41C2×D25712yes—
106C23×C4—2571no—
107#137—22no—
108AGL(1,ℤ/8)—22no—
109#135—22no—
110C2×M16—2571no—
111C2×M16C2×D25711yescharacteristic
112#123—5142no—
113C22⋊Q8—5142no—
114#121—5142no—
115#120—5142no—
116C4×D4—5144no—
117#117—2572no—
118#104—5144no—
119C22⋊C8—5142no—
120C24—2571no—
121#41—22no—
122#41—22no—
123#41—22no—
124#41—22no—
125C2×Q8C22⋊D25712yescomplemented
126C2×D4—22no—
127C2×D4—5144no—
128C2×D4C22⋊D25712yescomplemented
129C22×C4—2571no—
130C22×C4—22no—
131C22×C4—5142no—
132C22×C4—5141no—
133C22×C4—5142no—
134C22×C4—5141no—
135C22×C4C22×D25711yescharacteristic
136Q16—22no—
137SD16—24no—
138D8—22no—
139M16—22no—
140M16—5142no—
141C2×C8—5141no—
142C2×C8C22⋊D25712yescomplemented
143C4⋊C4—5144no—
144C4⋊C4—5142no—
145C22⋊C4—5144no—
146C22⋊C4—5144no—
147C22⋊C4—5142no—
148C42—2574no—
149C23—22no—
150C23—5142no—
151C23—5141no—
152C23—10282no—
153C23D4×D25711yescharacteristic
154Q8—22no—
155Q8—22no—
156D4—22no—
157D4—22no—
158D4—42no—
159D4—42no—
160D4—10288no—
161C2×C4—22no—
162C2×C4—5144no—
163C2×C4—5142no—
164C2×C4—5141no—
165C2×C4—42no—
166C2×C4—42no—
167C2×C4—10281no—
168C2×C4—10282no—
169C2×C4—10281no—
170C2×C4D4×D25712yes—
171C2×C4C2×C22⋊D25712yes—
172C2×C4C2×C22⋊D25711yescharacteristic
173C2×C4#1686161711yescharacteristic
174C8—22no—
175C8—10281no—
176C22—5144no—
177C22—5142no—
178C22—41no—
179C22—44no—
180C22—42no—
181C22—10282no—
182C22—10281no—
183C22—10281no—
184C22#2255156012yes—
185C22#2255156311yescharacteristic
186C4—42no—
187C4—10284no—
188C4—10281no—
189C4#2255156111yescharacteristic
190C4#2255156412yes—
191C4#2255158811yescharacteristic
192C2—22no—
193C2—21no—
194C2—10282no—
195C2—82no—
196C2#3065664511yescharacteristic
subgroupG/H
centerC4#22551588
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#575×C257C2
socleC2×C257#392
hypercenter#575D257
hypocenterC257#2010
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C4 ◁ C22×C4 ◁ #575

Complements

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

NH ≅ G/Ntype
1. #30657244C2semidirect
2. #30656754C2semidirect
3. #30656700C2semidirect
4. #30656494C2semidirect
5. #575×C257C2semidirect
35. C4×Dic257D4semidirect
40. C2×Q8×C257D4semidirect
42. C2×D4×C257D4semidirect
49. C2×C8×C257D4semidirect
98. C257#2010semidirect
100. #575D257semidirect
125. C2×Q8C22⋊D257semidirect
128. C2×D4C22⋊D257semidirect
142. C2×C8C22⋊D257semidirect

Maximal subgroups

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

subgroupindex
#306572442
#306567542
#306567002
#306564942
#575×C2572
#2010257