Finite Group Databasebeta

← #46838012

Subgroups — #46838012

Order 34688.

266 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 55 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#31774462C211yesmaximal, characteristic
2#31773878C211yesmaximal, characteristic, complemented
3#31773869C211yesmaximal, characteristic, complemented
4#31773685C211yesmaximal, characteristic, complemented
5#31773684C211yesmaximal, characteristic, complemented
6#31773683C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C271C211yesmaximal, characteristic, complemented
8C2×#16864653C2211yescharacteristic
9C22×C22⋊D271C411yescharacteristic, complemented
10M16×D271—21no—
11#22576156—21no—
12#22576146—21no—
13#22576133—21no—
14#22576132—21no—
15#22576130C411yescharacteristic, complemented
16#22576129C411yescharacteristic, complemented
17#22576128C2211yescharacteristic
18#22576116—21no—
19#22576116C411yescharacteristic, complemented
20#22576078—21no—
21#22576077C2211yescharacteristic
22#22576075C2211yescharacteristic
23#22576074—21no—
24#120×C271C2211yescharacteristic
25#100×C271C2211yescharacteristic
26#99×C271C2211yescharacteristic
27C23×D271C2×C411yescharacteristic
28#16864653—41no—
29#16864653—41no—
30C2×Dic542D411yescharacteristic
31C2×D1084—21no—
32C2×C22⋊D271—21no—
33C2×C22⋊D271—21no—
34C2×C22⋊D271—21no—
35C2×C22⋊D271—21no—
36C2×C22⋊D271—41no—
37C2×C22⋊D271—41no—
38C2×C22⋊D271C2×C411yescharacteristic
39C2×C22⋊D271D411yescharacteristic
40C22×Dic271C2×C411yescharacteristic
41C22×Dic271D411yescharacteristic
42C2×C4×D271D411yescharacteristic
43#16864638—21no—
44#16864637—41no—
45C8×D271—41no—
46Dic271⋊C4—21no—
47Dic271⋊C4—41no—
48C22⋊Dic271—21no—
49C22⋊Dic271—21no—
50C22⋊Dic271C2×C411yescharacteristic
51(C2×D271)⋊C4—21no—
52(C2×D271)⋊C4—21no—
53C4×Dic271—21no—
54C4×Dic271—41no—
55C24×C271C2×C411yescharacteristic
56C2×D4×C271—21no—
57C2×D4×C271C2311yescharacteristic
58M16×C271—21no—
59C22⋊C4×C271—21no—
60C22⋊C4×C271—21no—
61C22⋊C4×C271C2×C411yescharacteristic
62C22×D271—21no—
63C22×D271—41no—
64C22×D271—81no—
65C22×D271C22⋊C411yescharacteristic
66Dic542—41no—
67D1084—81no—
68C22⋊D271—41no—
69C22⋊D271—41no—
70C22⋊D271—42no—
71C22⋊D271—41no—
72C22⋊D271—41no—
73C22⋊D271—81no—
74C22⋊D271—81no—
75C2×Dic271—21no—
76C2×Dic271—21no—
77C2×Dic271—21no—
78C2×Dic271—21no—
79C2×Dic271—21no—
80C2×Dic271—41no—
81C2×Dic271C22⋊C411yescharacteristic
82C2×Dic271C22⋊C411yescharacteristic
83C2×Dic271C22⋊C411yescharacteristic
84C4×D271—21no—
85C4×D271—21no—
86C271⋊C8—41no—
87C23×C271—21no—
88C23×C271—41no—
89C23×C271—81no—
90C23×C271C22×C411yescharacteristic
91C23×C271C2×D411yescharacteristic
92D4×C271—41no—
93D4×C271—81no—
94C2×C4×C271—21no—
95C2×C4×C271—41no—
96C2×C4×C271C2×D411yescharacteristic
97C8×C271—41no—
98C22×C271—21no—
99C22×C271—21no—
100C22×C271—41no—
101C22×C271—41no—
102C22×C271—41no—
103C22×C271—41no—
104C22×C271—41no—
105C22×C271—82no—
106C22×C271C2×C22⋊C411yescharacteristic
107C2×D271—21no—
108C2×D271—41no—
109C2×D271—41no—
110C2×D271—41no—
111C2×D271—41no—
112C2×D271—82no—
113C4×C271—21no—
114C4×C271—41no—
115C4×C271—81no—
116Dic271—21no—
117Dic271—21no—
118Dic271—41no—
119Dic271—81no—
120Dic271#9911yescharacteristic
121Dic271#9911yescharacteristic
122C2×C271—21no—
123C2×C271—41no—
124C2×C271—41no—
125C2×C271—81no—
126C2×C271C2×#9911yescharacteristic
127D271—41no—
128D271—81no—
129C271#212211yescharacteristic, complemented, Sylow
130#2122—2711nomaximal, Sylow
131#535—2711no—
132#413—2711no—
133#410—2711no—
134AΣL(1,𝔽16)—2711no—
135AΣL(1,𝔽16)—2711no—
136AΣL(1,𝔽16)—2711no—
137AΣL(1,𝔽16)D27111yescharacteristic, complemented
138C2×#41—2711no—
139C22×D4—2711no—
140C2×M16—5421no—
141#127—5421no—
142#124—5421no—
143#121—5421no—
144#121—5421no—
145#120—2711no—
146#120—2711no—
147#120—2711no—
148#120C2×D27111yescharacteristic
149#117—2711no—
150#117—5421no—
151#101—5421no—
152#100—2711no—
153#100C2×D27111yescharacteristic
154#99—2711no—
155#99—5421no—
156#99C2×D27111yescharacteristic
157C24—2711no—
158C24C4×D27111yescharacteristic
159#41—10841no—
160#41—10841no—
161C2×Q8—2711no—
162C2×D4—21no—
163C2×D4—2711no—
164C2×D4—2711no—
165C2×D4—5421no—
166C2×D4—5421no—
167C2×D4—5421no—
168C2×D4—5421no—
169C2×D4—5421no—
170C2×D4—10841no—
171C2×D4—10841no—
172C2×D4C22×D27111yescharacteristic
173C22×C4—2711no—
174C22×C4—2711no—
175C22×C4—2711no—
176M16—21no—
177M16—5421no—
178M16—10841no—
179C2×C8—10841no—
180C4⋊C4—5421no—
181C4⋊C4—10841no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—2711no—
185C22⋊C4—5421no—
186C22⋊C4—5421no—
187C22⋊C4—5421no—
188C22⋊C4—5421no—
189C22⋊C4C4×D27111yescharacteristic
190C42—5421no—
191C42—10841no—
192C23—21no—
193C23—41no—
194C23—81no—
195C23—2711no—
196C23—5421no—
197C23—10841no—
198C23—21681no—
199C23C2×C4×D27111yescharacteristic
200C23D4×D27111yescharacteristic
201Q8—10841no—
202D4—41no—
203D4—81no—
204D4—10841no—
205D4—10841no—
206D4—10841no—
207D4—10841no—
208D4—10842no—
209D4—21681no—
210D4—21681no—
211D4—21681no—
212C2×C4—21no—
213C2×C4—41no—
214C2×C4—2711no—
215C2×C4—2711no—
216C2×C4—2711no—
217C2×C4—5421no—
218C2×C4—5421no—
219C2×C4—5421no—
220C2×C4—5421no—
221C2×C4—5421no—
222C2×C4—5421no—
223C2×C4—5421no—
224C2×C4—10841no—
225C2×C4D4×D27111yescharacteristic
226C8—41no—
227C8—10841no—
228C22—21no—
229C22—21no—
230C22—41no—
231C22—41no—
232C22—41no—
233C22—41no—
234C22—41no—
235C22—82no—
236C22—5421no—
237C22—10841no—
238C22—10841no—
239C22—10841no—
240C22—10841no—
241C22—21682no—
242C22C22⋊C4×D27111yescharacteristic
243C4—21no—
244C4—41no—
245C4—81no—
246C4—2711no—
247C4—2711no—
248C4—5421no—
249C4—5421no—
250C4—10841no—
251C4—21681no—
252C2—21no—
253C2—41no—
254C2—41no—
255C2—81no—
256C2—10841no—
257C2—21681no—
258C2#99×D27111yescharacteristic
subgroupG/H
centerC2#99×D271
commutatorC23×C271C22×C4
FrattiniC2×D4C22×D271
FittingAΣL(1,𝔽16)×C271C2
socleC2×C271C2×#99
prime radical#31774462C2
cosocleC2×D4×C271C23
hypercenterAΣL(1,𝔽16)D271
hypocenterC271#2122
LCSG ▷ C23×C271 ▷ C22×C271 ▷ C2×C271 ▷ C271
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ AΣL(1,𝔽16)

Complements

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

NH ≅ G/Ntype
2. #31773878C2semidirect
3. #31773869C2semidirect
4. #31773685C2semidirect
5. #31773684C2semidirect
6. #31773683C2semidirect
7. AΣL(1,𝔽16)×C271C2semidirect
9. C22×C22⋊D271C4semidirect
15. #22576130C4semidirect
16. #22576129C4semidirect
19. #22576116C4semidirect
129. C271#2122semidirect
137. AΣL(1,𝔽16)D271semidirect

Maximal subgroups

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

subgroupindex
#317744622
#317738782
#317738692
#317736852
#317736842
#317736832
AΣL(1,𝔽16)×C2712
#2122271