Finite Group Databasebeta

← #45665479

Subgroups — #45665479

Order 32896.

246 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 69 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#30657365C211yesmaximal, characteristic
2#30656746C211yesmaximal, characteristic
3#30656644C211yesmaximal, characteristic
4#30656505C211yesmaximal, characteristic
5#30656498C211yesmaximal, characteristic
6#30656496C211yesmaximal, characteristic
7#419×C257C211yesmaximal, characteristic
8C2×#16861602C2211yescharacteristic
9#22551592—21no—
10#22551591—21no—
11#22551584—21no—
12#22551583—21no—
13#22551569—22no—
14#22551551C2211yescharacteristic
15C2×C4⋊Dic257C2211yescharacteristic
16C2×C4×Dic257C2211yescharacteristic
17#22551516—21no—
18#22551516—21no—
19#22551499—22no—
20C22×Q8×C257C2211yescharacteristic
21C2×M16×C257C2211yescharacteristic
22#117×C257C2211yescharacteristic
23#103×C257—21no—
24#103×C257—21no—
25C2×Dic514—22no—
26C22×Dic257—21no—
27C22×Dic257D412yes—
28#16861602—21no—
29#16861602—21no—
30C2×#14523339—21no—
31C4⋊Dic257—22no—
32C4⋊Dic257—21no—
33C4⋊Dic257—21no—
34C4⋊Dic257D411yescharacteristic
35C4⋊Dic257D411yescharacteristic
36#16861597—22no—
37#16861597—22no—
38C22⋊Dic257—21no—
39C4×Dic257—21no—
40C4×Dic257—21no—
41C4×Dic257—21no—
42C2×Q8×C257—21no—
43C2×Q8×C257—21no—
44C2×Q8×C257—21no—
45C2×Q8×C257—21no—
46C2×Q8×C257—21no—
47C2×Q8×C257Q812yescomplemented
48C22×C4×C257—21no—
49C22×C4×C257C2311yescharacteristic
50M16×C257—21no—
51M16×C257—21no—
52C2×C8×C257—21no—
53C4⋊C4×C257D411yescharacteristic
54C4⋊C4×C257D411yescharacteristic
55C22⋊C4×C257—21no—
56C42×C257—21no—
57Dic514—82no—
58C2×Dic257—22no—
59C2×Dic257—21no—
60C2×Dic257—21no—
61C2×Dic257—22no—
62C2×Dic257—41no—
63C2×Dic257—41no—
64C2×Dic257—42no—
65#14523339—41no—
66C23×C257C2×D411yescharacteristic
67Q8×C257—22no—
68Q8×C257—22no—
69Q8×C257—41no—
70Q8×C257—41no—
71C2×C4×C257—21no—
72C2×C4×C257—21no—
73C2×C4×C257—22no—
74C2×C4×C257—41no—
75C2×C4×C257—41no—
76C2×C4×C257C2×D411yescharacteristic
77C2×C4×C257C2×D411yescharacteristic
78C2×C4×C257C2×Q811yescharacteristic
79C2×C4×C257#4111yescharacteristic
80C2×C4×C257#4111yescharacteristic
81C2×C4×C257#4111yescharacteristic
82C8×C257—41no—
83C22×C257—21no—
84C22×C257—21no—
85C22×C257#12011yescharacteristic
86C22×C257C22⋊Q811yescharacteristic
87C22×C257#12411yescharacteristic
88C4×C257—42no—
89C4×C257—41no—
90C4×C257—41no—
91C4×C257C22⋊Q811yescharacteristic
92C4×C257C22⋊Q811yescharacteristic
93C4×C257#12411yescharacteristic
94C4×C257#12411yescharacteristic
95Dic257—42no—
96Dic257—42no—
97Dic257—41no—
98Dic257—41no—
99C2×C257—21no—
100C2×C257—21no—
101C2×C257#39311yescharacteristic
102C2×C257#45511yescharacteristic
103C2×C257#45611yescharacteristic
104C257#202811yescharacteristic, complemented, Sylow
105#2028—2571nomaximal, Sylow
106#544—2571no—
107#419—2571no—
108#419D25711yescharacteristic
109#391—2571no—
110#343—2571no—
111#339—2571no—
112#339—2571no—
113C22×Q8C2×D25711yescharacteristic
114C2×M16—2571no—
115C2×M16C2×D25711yescharacteristic
116C4⋊Q8—5141no—
117C4⋊Q8—5141no—
118#125—5141no—
119#125—5141no—
120C22⋊Q8—5142no—
121#117—2571no—
122#117C2×D25711yescharacteristic
123C2×C4⋊C4—2571no—
124C2×C42—2571no—
125#103—21no—
126#103—21no—
127#103—5141no—
128#103—5141no—
129#95—5142no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—5142no—
136C2×Q8Dic51412yescomplemented
137C22×C4—2572no—
138C22×C4—21no—
139C22×C4—5141no—
140C22×C4C22×D25711yescharacteristic
141M16—21no—
142M16—21no—
143M16—5141no—
144M16—5141no—
145C2×C8—21no—
146C2×C8—5141no—
147C4⋊C4—2571no—
148C4⋊C4—2571no—
149C4⋊C4—5142no—
150C4⋊C4—5142no—
151C4⋊C4—5142no—
152C4⋊C4—5141no—
153C4⋊C4—5141no—
154C4⋊C4C22⋊D25711yescharacteristic
155C4⋊C4C22⋊D25711yescharacteristic
156C22⋊C4—21no—
157C22⋊C4—5141no—
158C42—21no—
159C42—5141no—
160C42—5141no—
161C42—5141no—
162C23D4×D25711yescharacteristic
163Q8—22no—
164Q8—22no—
165Q8—41no—
166Q8—41no—
167Q8—20562no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—22no—
171C2×C4—5142no—
172C2×C4—5141no—
173C2×C4—5141no—
174C2×C4—5142no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—10281no—
178C2×C4—10281no—
179C2×C4—10282no—
180C2×C4D4×D25711yescharacteristic
181C2×C4C2×C22⋊D25711yescharacteristic
182C2×C4C2×Dic51411yescharacteristic
183C2×C4#1686161711yescharacteristic
184C2×C4#1686161711yescharacteristic
185C2×C4#1686161811yescharacteristic
186C8—41no—
187C8—10281no—
188C22—21no—
189C22—21no—
190C22#2255156011yescharacteristic
191C22#2255156911yescharacteristic
192C22#2255157811yescharacteristic
193C4—42no—
194C4—41no—
195C4—41no—
196C4—10282no—
197C4—10282no—
198C4—10281no—
199C4—10281no—
200C4#2255156911yescharacteristic
201C4#2255157111yescharacteristic
202C4#2255157811yescharacteristic
203C4#2255157911yescharacteristic
204C2—21no—
205C2—21no—
206C2#3065664911yescharacteristic
207C2#3065691311yescharacteristic
208C2#3065691811yescharacteristic
subgroupG/H
centerC22#22551560
commutator, cosocleC22×C4×C257C23
FrattiniC22×C4C22×D257
Fitting#419×C257C2
socleC22×C257#120
prime radicalC23×C257C2×D4
hypercenter#419D257
hypocenterC257#2028
LCSG ▷ C22×C4×C257 ▷ C2×C257 ▷ C257
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
47. C2×Q8×C257Q8semidirect
104. C257#2028semidirect
136. C2×Q8Dic514semidirect

Maximal subgroups

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

subgroupindex
#306573652
#306567462
#306566442
#306565052
#306564982
#306564962
#419×C2572
#2028257