Finite Group Databasebeta

← #45664739

Subgroups — #45664739

Order 32896.

342 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 93 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#30657504C211yesmaximal, characteristic, complemented
2#30656721C211yesmaximal, characteristic, complemented
3#30656701C212yesmaximal, complemented
4#30656507C212yesmaximal, complemented
5#525×C257C211yesmaximal, characteristic
6#22551599—24no—
7C2×#16861602—22no—
8C2×#16861602C2212yes—
9C2×#16861602C2212yes—
10#22551511—21no—
11#22551509—21no—
12C22⋊#14523339—24no—
13C2×#41×C257C2211yescharacteristic
14C23×C4×C257C2211yescharacteristic
15#123×C257C412yes—
16C22⋊Q8×C257—21no—
17#121×C257—21no—
18#120×C257C412yes—
19C4×D4×C257—22no—
20#117×C257C2211yescharacteristic
21#16861602—24no—
22#16861602—22no—
23#16861602—22no—
24#16861602—22no—
25#16861602—22no—
26C2×#14523339—24no—
27C2×#14523339—22no—
28C2×#14523339D414yescomplemented
29C24×C257C2×C411yescharacteristic
30#41×C257—22no—
31#41×C257—22no—
32C2×Q8×C257D411yescharacteristic
33C2×D4×C257—22no—
34C2×D4×C257C2×C412yes—
35C2×D4×C257D411yescharacteristic
36C22×C4×C257—21no—
37C22×C4×C257—21no—
38C22×C4×C257—22no—
39C22×C4×C257—22no—
40C22×C4×C257C2×C411yescharacteristic
41C22×C4×C257D412yes—
42C22×C4×C257C2311yescharacteristic
43C4⋊C4×C257—22no—
44C4⋊C4×C257—21no—
45C22⋊C4×C257—22no—
46C22⋊C4×C257—22no—
47C22⋊C4×C257C2×C412yes—
48C42×C257—21no—
49#14523339—24no—
50#14523339—42no—
51C23×C257—21no—
52C23×C257—22no—
53C23×C257—42no—
54C23×C257C22⋊C412yes—
55C23×C257C22×C411yescharacteristic
56Q8×C257—22no—
57D4×C257—24no—
58D4×C257—22no—
59D4×C257—44no—
60C2×C4×C257—22no—
61C2×C4×C257—22no—
62C2×C4×C257—22no—
63C2×C4×C257—21no—
64C2×C4×C257—22no—
65C2×C4×C257—41no—
66C2×C4×C257—42no—
67C2×C4×C257—41no—
68C2×C4×C257C22⋊C412yes—
69C2×C4×C257C2×D412yes—
70C2×C4×C257C2×D411yescharacteristic
71C2×C4×C257C2×D411yescharacteristic
72C2×C4×C257#4112yes—
73C22×C257—22no—
74C22×C257—24no—
75C22×C257—22no—
76C22×C257—41no—
77C22×C257—42no—
78C22×C257—41no—
79C22×C257—42no—
80C22×C257—41no—
81C22×C257C2×C22⋊C411yescharacteristic
82C22×C257C4×D412yes—
83C4×C257—22no—
84C4×C257—42no—
85C4×C257—41no—
86C4×C257#12011yescharacteristic
87C4×C257#12112yes—
88C4×C257#12311yescharacteristic
89C2×C257—21no—
90C2×C257—22no—
91C2×C257—42no—
92C2×C257—42no—
93C2×C257#38611yescharacteristic
94C257#189111yescharacteristic, complemented, Sylow
95#1891—2571nomaximal, Sylow
96#568—2571no—
97#525D25711yescharacteristic
98#413—2571no—
99#407—2572no—
100#344—2572no—
101C2×#41C2×D25711yescharacteristic
102C23×C4C2×D25711yescharacteristic
103#131—5144no—
104C2×M16—2572no—
105C2×M16—2572no—
106C2×M16—5142no—
107#123Dic25712yes—
108C22⋊Q8—21no—
109#121—21no—
110#120Dic25712yes—
111C4×D4—22no—
112#117C2×D25711yescharacteristic
113#101—5141no—
114#100—5141no—
115C22⋊C8—5144no—
116C24C2×Dic25711yescharacteristic
117#41—22no—
118#41—22no—
119C2×Q8C22⋊D25711yescharacteristic
120C2×D4—22no—
121C2×D4C2×Dic25712yes—
122C2×D4C22⋊D25711yescharacteristic
123C22×C4—22no—
124C22×C4—21no—
125C22×C4—21no—
126C22×C4—22no—
127C22×C4C2×Dic25711yescharacteristic
128C22×C4C22⋊D25712yes—
129C22×C4C22×D25711yescharacteristic
130M16—5144no—
131M16—5142no—
132M16—5142no—
133M16—5142no—
134M16—5142no—
135C2×C8—2574no—
136C2×C8—5144no—
137C2×C8—5142no—
138C4⋊C4—22no—
139C4⋊C4—21no—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4C2×Dic25712yes—
143C42—21no—
144C23—22no—
145C23—21no—
146C23—42no—
147C23C22⋊Dic25712yes—
148C23C22×Dic25711yescharacteristic
149Q8—22no—
150D4—24no—
151D4—22no—
152D4—44no—
153C2×C4—22no—
154C2×C4—22no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—21no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—41no—
161C2×C4C22⋊Dic25712yes—
162C2×C4D4×D25712yes—
163C2×C4C2×C22⋊D25711yescharacteristic
164C2×C4C2×C22⋊D25711yescharacteristic
165C2×C4#1686161712yes—
166C8—5144no—
167C8—10282no—
168C22—22no—
169C22—24no—
170C22—22no—
171C22—41no—
172C22—42no—
173C22—41no—
174C22—42no—
175C22—41no—
176C22C2×C22⋊Dic25711yescharacteristic
177C22D4×Dic25712yes—
178C4—22no—
179C4—42no—
180C4—41no—
181C4#2255156011yescharacteristic
182C4#2255156412yes—
183C4#2255157311yescharacteristic
184C2—21no—
185C2—22no—
186C2—42no—
187C2—42no—
188C2#3065661711yescharacteristic
subgroupG/H
centerC4#22551560
commutatorC23×C257C22×C4
FrattiniC22×C4C22×D257
Fitting, prime radical#525×C257C2
socleC2×C257#386
cosocleC22×C4×C257C23
hypercenter#525D257
hypocenterC257#1891
LCSG ▷ C23×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C4 ◁ C22×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. #30657504C2semidirect
2. #30656721C2semidirect
3. #30656701C2semidirect
4. #30656507C2semidirect
28. C2×#14523339D4semidirect
94. C257#1891semidirect

Maximal subgroups

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

subgroupindex
#306575042
#306567212
#306567012
#306565072
#525×C2572
#1891257