Finite Group Databasebeta

← #45665366

Subgroups — #45665366

Order 32896.

370 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 73 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
1C2×#22551623C212yesmaximal, complemented
2#30657318C211yesmaximal, characteristic
3C2×#22551520C212yesmaximal, complemented
4#30656506C211yesmaximal, characteristic, complemented
5#391×C257C211yesmaximal, characteristic, complemented
6C2×#16861618C2212yes—
7#22551623—24no—
8#22551623—24no—
9C2×#16861608—22no—
10C2×#16861605—22no—
11C2×#16861602C2212yes—
12#22551572—22no—
13#22551567—22no—
14#22551566—22no—
15#22551520—24no—
16C22×Q8×C257C2211yescharacteristic
17C2×C4⋊C4×C257C2211yescharacteristic
18C2×C42×C257C2211yescharacteristic
19#95×C257—22no—
20#16861618—22no—
21#16861618—22no—
22#16861618—22no—
23#16861618—22no—
24C2×Dic514D412yes—
25C2×D1028—22no—
26C2×D1028D412yes—
27C2×C22⋊D257—22no—
28C2×C4×D257—22no—
29#16861608—28no—
30#16861605—28no—
31#16861602—22no—
32#16861602D414yescomplemented
33C2×#14523339—22no—
34C4⋊Dic257—22no—
35#16861595—24no—
36C2×Q8×C257—21no—
37C2×Q8×C257—22no—
38C2×Q8×C257—22no—
39C2×Q8×C257D412yescomplemented
40C22×C4×C257—21no—
41C22×C4×C257—21no—
42C22×C4×C257D412yes—
43C22×C4×C257C2311yescharacteristic
44C4⋊C4×C257—22no—
45C42×C257—22no—
46C22×D257—22no—
47Dic514—22no—
48Dic514—22no—
49D1028—22no—
50D1028—22no—
51D1028—82no—
52C22⋊D257—42no—
53C22⋊D257—42no—
54C2×Dic257—22no—
55C4×D257—42no—
56C4×D257—42no—
57#14523339—24no—
58C23×C257C2×D411yescharacteristic
59Q8×C257—24no—
60Q8×C257—42no—
61C2×C4×C257—22no—
62C2×C4×C257—22no—
63C2×C4×C257—22no—
64C2×C4×C257—41no—
65C2×C4×C257—41no—
66C2×C4×C257—41no—
67C2×C4×C257—41no—
68C2×C4×C257—42no—
69C2×C4×C257C2×D412yes—
70C2×C4×C257C2×D412yes—
71C2×C4×C257C2×D411yescharacteristic
72C2×C4×C257#4111yescharacteristic
73C22×C257—21no—
74C22×C257—21no—
75C22×C257#12011yescharacteristic
76C22×C257#12111yescharacteristic
77C22×C257#12711yescharacteristic
78C2×D257—42no—
79C2×D257—42no—
80C2×D257—42no—
81C4×C257—42no—
82C4×C257—42no—
83C4×C257—42no—
84C4×C257#12012yes—
85C4×C257#12112yes—
86Dic257—42no—
87C2×C257—21no—
88C2×C257—21no—
89C2×C257#39211yescharacteristic
90C2×C257#45511yescharacteristic
91C2×C257#45611yescharacteristic
92D257—82no—
93C257#200711yescharacteristic, complemented, Sylow
94#2007—2571nomaximal, Sylow
95C2×#137—2572no—
96#537—2571no—
97C2×#104—2572no—
98#391D25711yescharacteristic, complemented
99#343—2571no—
100C2×#41—2572no—
101C22×Q8C2×D25711yescharacteristic
102#137—5144no—
103#137—5144no—
104C2×Q16—5142no—
105C2×SD16—5142no—
106C2×M16—2572no—
107C22⋊Q8—5142no—
108#121—5142no—
109#121—5142no—
110C2×C4⋊C4C2×D25711yescharacteristic
111C2×C42C2×D25711yescharacteristic
112#104—5144no—
113#95—22no—
114#41—5142no—
115#41—5142no—
116#41—5142no—
117#41—5142no—
118C2×Q8—2572no—
119C2×Q8—21no—
120C2×Q8—22no—
121C2×Q8—22no—
122C2×Q8D102812yescomplemented
123C2×D4—2572no—
124C2×D4—5142no—
125C2×D4—5142no—
126C22×C4—21no—
127C22×C4—21no—
128C22×C4—5142no—
129C22×C4C22⋊D25712yes—
130C22×C4C22×D25711yescharacteristic
131Q16—5148no—
132SD16—5148no—
133M16—2574no—
134M16—5142no—
135C2×C8—5142no—
136C4⋊C4—22no—
137C4⋊C4—5142no—
138C22⋊C4—5144no—
139C42—22no—
140C23—5142no—
141C23C2×C22⋊D25711yescharacteristic
142Q8—24no—
143Q8—5142no—
144Q8—5142no—
145Q8—42no—
146D4—5142no—
147D4—5142no—
148D4—10282no—
149D4—10282no—
150D4—20562no—
151C2×C4—22no—
152C2×C4—22no—
153C2×C4—22no—
154C2×C4—5142no—
155C2×C4—41no—
156C2×C4—41no—
157C2×C4—41no—
158C2×C4—41no—
159C2×C4—42no—
160C2×C4—10282no—
161C2×C4—10282no—
162C2×C4D4×D25712yes—
163C2×C4D4×D25712yes—
164C2×C4C2×D102811yescharacteristic
165C2×C4#1686161811yescharacteristic
166C8—5144no—
167C22—21no—
168C22—21no—
169C22—10282no—
170C22—10282no—
171C22—10282no—
172C22#2255156011yescharacteristic
173C22#2255156611yescharacteristic
174C22#2255158811yescharacteristic
175C4—42no—
176C4—42no—
177C4—42no—
178C4—10282no—
179C4#2255156212yes—
180C4#2255156512yes—
181C2—21no—
182C2—21no—
183C2—20562no—
184C2#3065664611yescharacteristic
185C2#3065691411yescharacteristic
186C2#3065691911yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#391×C257C2
socleC22×C257#120
prime radical#30657318C2
hypercenter#391D257
hypocenterC257#2007
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#22551623C2semidirect
3. C2×#22551520C2semidirect
4. #30656506C2semidirect
5. #391×C257C2semidirect
32. #16861602D4semidirect
39. C2×Q8×C257D4semidirect
93. C257#2007semidirect
98. #391D257semidirect
122. C2×Q8D1028semidirect

Maximal subgroups

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

subgroupindex
C2×#225516232
#306573182
C2×#225515202
#306565062
#391×C2572
#2007257