Finite Group Databasebeta

← #2197×C257 #45660739

Subgroups — #2197×C257 #45660739

Order 32896.

258 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 56 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#551×C257C211yesmaximal, characteristic, complemented
2#458×C257C211yesmaximal, characteristic, complemented
3B(4,𝔽2)×C257C211yesmaximal, characteristic
4#455×C257C211yesmaximal, characteristic, complemented
5#354×C257C211yesmaximal, characteristic, complemented
6#352×C257C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C257C211yesmaximal, characteristic, complemented
8H(5,𝔽2)×C257C2211yescharacteristic
9#137×C257—21no—
10#126×C257—21no—
11#124×C257C2211yescharacteristic
12#123×C257—21no—
13#123×C257—21no—
14#123×C257C2211yescharacteristic
15C22⋊Q8×C257—21no—
16#120×C257—21no—
17#120×C257C2211yescharacteristic
18C4×D4×C257—21no—
19#117×C257—21no—
20C2×C22⋊C4×C257—21no—
21#104×C257—21no—
22#100×C257C2211yescharacteristic
23#99×C257—21no—
24#99×C257—21no—
25#99×C257C2211yescharacteristic
26#99×C257C2211yescharacteristic
27C24×C257D411yescharacteristic
28#41×C257—21no—
29#41×C257—41no—
30C2×Q8×C257D411yescharacteristic, complemented
31C2×D4×C257—21no—
32C2×D4×C257—21no—
33C2×D4×C257—21no—
34C2×D4×C257—41no—
35C2×D4×C257—41no—
36C2×D4×C257C2311yescharacteristic
37C22×C4×C257—21no—
38C22×C4×C257—21no—
39C22×C4×C257D411yescharacteristic, complemented
40Q16×C257—41no—
41SD16×C257—41no—
42M16×C257—21no—
43C4⋊C4×C257—21no—
44C4⋊C4×C257—21no—
45C4⋊C4×C257—21no—
46C4⋊C4×C257—21no—
47C4⋊C4×C257—41no—
48C22⋊C4×C257—21no—
49C22⋊C4×C257—21no—
50C22⋊C4×C257—21no—
51C22⋊C4×C257—21no—
52C22⋊C4×C257—21no—
53C22⋊C4×C257—21no—
54C22⋊C4×C257—21no—
55C22⋊C4×C257—21no—
56C22⋊C4×C257—41no—
57C22⋊C4×C257—41no—
58C22⋊C4×C257D411yescharacteristic
59C22⋊C4×C257D411yescharacteristic, complemented
60C42×C257—21no—
61C42×C257D411yescharacteristic, complemented
62C23×C257—21no—
63C23×C257—21no—
64C23×C257—21no—
65C23×C257—41no—
66C23×C257—42no—
67C23×C257C2×D411yescharacteristic
68C23×C257C2×D411yescharacteristic
69Q8×C257—21no—
70Q8×C257—41no—
71D4×C257—21no—
72D4×C257—41no—
73D4×C257—42no—
74D4×C257—41no—
75D4×C257—41no—
76D4×C257—41no—
77D4×C257—82no—
78C2×C4×C257—21no—
79C2×C4×C257—21no—
80C2×C4×C257—21no—
81C2×C4×C257—21no—
82C2×C4×C257—21no—
83C2×C4×C257—21no—
84C2×C4×C257—21no—
85C2×C4×C257—41no—
86C2×C4×C257—42no—
87C2×C4×C257—41no—
88C2×C4×C257—41no—
89C2×C4×C257—41no—
90C2×C4×C257—81no—
91C2×C4×C257C2×D411yescharacteristic
92C8×C257—41no—
93C22×C257—22no—
94C22×C257—21no—
95C22×C257—21no—
96C22×C257—41no—
97C22×C257—41no—
98C22×C257—41no—
99C22×C257—42no—
100C22×C257—41no—
101C22×C257—41no—
102C22×C257—81no—
103C22×C257—81no—
104C22×C257—81no—
105C22×C257#12011yescharacteristic
106C4×C257—21no—
107C4×C257—41no—
108C4×C257—41no—
109C4×C257—41no—
110C4×C257—41no—
111C4×C257—41no—
112C4×C257—41no—
113C4×C257—81no—
114C4×C257—81no—
115C2×C257—21no—
116C2×C257—41no—
117C2×C257—41no—
118C2×C257—42no—
119C2×C257—81no—
120C2×C257B(4,𝔽2)11yescharacteristic
121C257#219711yescharacteristic, direct product factor, Sylow
122#2197C25711yesmaximal, characteristic, direct product factor, Sylow
123#551C2×C25711yescharacteristic, complemented
124#458C2×C25711yescharacteristic, complemented
125B(4,𝔽2)C2×C25711yescharacteristic
126#455C2×C25711yescharacteristic, complemented
127#354C2×C25711yescharacteristic, complemented
128#352C2×C25711yescharacteristic, complemented
129AΣL(1,𝔽16)C2×C25711yescharacteristic, complemented
130H(5,𝔽2)C22×C25711yescharacteristic
131#137—21no—
132#126—21no—
133#124C22×C25711yescharacteristic
134#123—21no—
135#123—21no—
136#123C22×C25711yescharacteristic
137C22⋊Q8—21no—
138#120—21no—
139#120C22×C25711yescharacteristic
140C4×D4—21no—
141#117—21no—
142C2×C22⋊C4—21no—
143#104—21no—
144#100C22×C25711yescharacteristic
145#99—21no—
146#99—21no—
147#99C22×C25711yescharacteristic
148#99C22×C25711yescharacteristic
149C24D4×C25711yescharacteristic
150#41—21no—
151#41—41no—
152C2×Q8D4×C25711yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—21no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4C23×C25711yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4D4×C25711yescharacteristic, complemented
162Q16—41no—
163SD16—41no—
164M16—21no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—41no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—21no—
177C22⋊C4—21no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4D4×C25711yescharacteristic
181C22⋊C4D4×C25711yescharacteristic, complemented
182C42—21no—
183C42D4×C25711yescharacteristic, complemented
184C23—21no—
185C23—21no—
186C23—21no—
187C23—41no—
188C23—42no—
189C23C2×D4×C25711yescharacteristic
190C23C2×D4×C25711yescharacteristic
191Q8—21no—
192Q8—41no—
193D4—21no—
194D4—41no—
195D4—41no—
196D4—41no—
197D4—42no—
198D4—41no—
199D4—82no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—41no—
212C2×C4—81no—
213C2×C4C2×D4×C25711yescharacteristic
214C8—41no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—41no—
224C22—81no—
225C22—81no—
226C22—81no—
227C22#120×C25711yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—41no—
235C4—81no—
236C4—81no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—81no—
242C2B(4,𝔽2)×C25711yescharacteristic
subgroupG/H
center, socleC2×C257B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C257
prime radicalB(4,𝔽2)×C257C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C257 ◁ C22×C257 ◁ C2×D4×C257 ◁ G

Complements

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

NH ≅ G/Ntype
1. #551×C257C2semidirect
2. #458×C257C2semidirect
4. #455×C257C2semidirect
5. #354×C257C2semidirect
6. #352×C257C2semidirect
7. AΣL(1,𝔽16)×C257C2semidirect
30. C2×Q8×C257D4semidirect
39. C22×C4×C257D4semidirect
59. C22⋊C4×C257D4semidirect
61. C42×C257D4semidirect
121. C257#2197direct (N × H)
122. #2197C257direct (N × H)
123. #551C2×C257semidirect
124. #458C2×C257semidirect
126. #455C2×C257semidirect
127. #354C2×C257semidirect
128. #352C2×C257semidirect
129. AΣL(1,𝔽16)C2×C257semidirect
152. C2×Q8D4×C257semidirect
161. C22×C4D4×C257semidirect
181. C22⋊C4D4×C257semidirect
183. C42D4×C257semidirect

Maximal subgroups

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

subgroupindex
#551×C2572
#458×C2572
B(4,𝔽2)×C2572
#455×C2572
#354×C2572
#352×C2572
AΣL(1,𝔽16)×C2572
#2197257