Finite Group Databasebeta

← #45665423

Subgroups — #45665423

Order 32896.

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

normal subgroups only

This group has 350 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#30657535C212yesmaximal, complemented
2#30656705C211yesmaximal, characteristic, complemented
3#30656704C212yesmaximal, complemented
4#30656495C211yesmaximal, characteristic, complemented
5#575×C257C211yesmaximal, characteristic, complemented
6C2×#16861618C2211yescharacteristic
7#22551623—22no—
8#22551622—22no—
9#22551618—22no—
10#22551617—22no—
11#22551612—24no—
12C2×#16861602C2212yes—
13C22×#14523339C2211yescharacteristic
14C22⋊#14523339—24no—
15#22551504—22no—
16C2×#41×C257C2212yes—
17#137×C257—22no—
18AGL(1,ℤ/8)×C257—22no—
19#135×C257—22no—
20C2×M16×C257C2211yescharacteristic
21#16861618—22no—
22#16861618—22no—
23C2×Dic514D411yescharacteristic
24C2×D1028D411yescharacteristic
25C2×C22⋊D257—21no—
26C2×C4×D257—21no—
27#16861608—24no—
28#16861606—24no—
29#16861605—24no—
30#16861603—24no—
31#16861602—22no—
32#16861602—22no—
33C2×#14523339—21no—
34C2×#14523339—22no—
35C2×#14523339D414yes—
36#41×C257—22no—
37#41×C257—22no—
38#41×C257—22no—
39#41×C257—22no—
40C2×Q8×C257D412yes—
41C2×D4×C257—22no—
42C2×D4×C257D412yes—
43C22×C4×C257—22no—
44C22×C4×C257C2311yescharacteristic
45Q16×C257—22no—
46SD16×C257—24no—
47D8×C257—22no—
48M16×C257—22no—
49C2×C8×C257D412yes—
50C22×D257—21no—
51Dic514—22no—
52D1028—22no—
53C22⋊D257—42no—
54C2×Dic257—21no—
55C4×D257—41no—
56C4×D257—41no—
57#14523339—24no—
58#14523339—41no—
59C23×C257—22no—
60C23×C257#4111yescharacteristic
61Q8×C257—22no—
62Q8×C257—22no—
63D4×C257—22no—
64D4×C257—22no—
65D4×C257—42no—
66D4×C257—42no—
67C2×C4×C257—22no—
68C2×C4×C257—42no—
69C2×C4×C257—42no—
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D412yes—
72C2×C4×C257C2×D412yes—
73C2×C4×C257C2×D411yescharacteristic
74C8×C257—22no—
75C22×C257—41no—
76C22×C257—44no—
77C22×C257—42no—
78C22×C257#12111yescharacteristic
79C22×C257#12112yes—
80C2×D257—42no—
81C2×D257—41no—
82C4×C257—42no—
83C4×C257#12012yes—
84C4×C257#12011yescharacteristic
85C4×C257#12711yescharacteristic
86Dic257—41no—
87C2×C257—21no—
88C2×C257—22no—
89C2×C257—82no—
90C2×C257#39211yescharacteristic
91D257—81no—
92C257#201711yescharacteristic, complemented, Sylow
93#2017—2571nomaximal, Sylow
94#575—2572no—
95#575D25711yescharacteristic, complemented
96#408—2572no—
97#408—2571no—
98#338—2571no—
99C2×#41—2571no—
100C2×#41C2×D25712yes—
101#137—22no—
102#137—5142no—
103#137—5142no—
104AGL(1,ℤ/8)—22no—
105AGL(1,ℤ/8)—5142no—
106AGL(1,ℤ/8)—5142no—
107#135—22no—
108#135—5144no—
109C2×M16—2572no—
110C2×M16C2×D25711yescharacteristic
111C22×C8—2571no—
112C22⋊C8—5144no—
113C22⋊C8—5142no—
114#41—22no—
115#41—22no—
116#41—22no—
117#41—22no—
118#41—5142no—
119#41—5142no—
120C2×Q8—2571no—
121C2×Q8C22⋊D25712yes—
122C2×D4—2571no—
123C2×D4—22no—
124C2×D4—5141no—
125C2×D4C22⋊D25712yes—
126C22×C4—22no—
127C22×C4—5141no—
128C22×C4C22×D25711yescharacteristic
129Q16—22no—
130Q16—5144no—
131SD16—24no—
132SD16—5144no—
133SD16—5144no—
134D8—22no—
135D8—5144no—
136M16—22no—
137M16—5142no—
138M16—5142no—
139C2×C8—2574no—
140C2×C8—5142no—
141C2×C8—5141no—
142C2×C8C22⋊D25712yes—
143C23—22no—
144C23—5141no—
145C23#1686161711yescharacteristic
146Q8—22no—
147Q8—22no—
148Q8—5142no—
149D4—22no—
150D4—22no—
151D4—5142no—
152D4—42no—
153D4—42no—
154D4—10282no—
155C2×C4—22no—
156C2×C4—5141no—
157C2×C4—42no—
158C2×C4—42no—
159C2×C4—10281no—
160C2×C4—10281no—
161C2×C4D4×D25711yescharacteristic
162C2×C4D4×D25712yes—
163C2×C4C2×C22⋊D25712yes—
164C2×C4C2×C22⋊D25711yescharacteristic
165C8—22no—
166C8—5144no—
167C8—10281no—
168C22—41no—
169C22—44no—
170C22—42no—
171C22—10282no—
172C22—10281no—
173C22#2255156311yescharacteristic
174C22#2255156412yes—
175C4—42no—
176C4—10281no—
177C4#2255156012yes—
178C4#2255156111yescharacteristic
179C4#2255158811yescharacteristic
180C2—22no—
181C2—21no—
182C2—82no—
183C2—20561no—
184C2#3065664511yescharacteristic
subgroupG/H
centerC4#22551588
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#575×C257C2
socleC2×C257#392
hypercenter#575D257
hypocenterC257#2017
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. #30657535C2semidirect
2. #30656705C2semidirect
3. #30656704C2semidirect
4. #30656495C2semidirect
5. #575×C257C2semidirect
92. C257#2017semidirect
95. #575D257semidirect

Maximal subgroups

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

subgroupindex
#306575352
#306567052
#306567042
#306564952
#575×C2572
#2017257