Finite Group Databasebeta

← #30640720

Subgroups — #30640720

Order 16256.

258 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 45 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#22550065C211yesmaximal, characteristic, complemented
2#22550062C211yesmaximal, characteristic
3#22550053C211yesmaximal, characteristic, complemented
4#22549671C211yesmaximal, characteristic, complemented
5#22549666C211yesmaximal, characteristic, complemented
6#22549663C211yesmaximal, characteristic, complemented
7#551×C127C211yesmaximal, characteristic, complemented
8#16861254C2211yescharacteristic
9#16861240—21no—
10#16861192—21no—
11#16861178—21no—
12#16861137—21no—
13#16861126C2211yescharacteristic
14#16861124C2211yescharacteristic
15#16861124C2211yescharacteristic
16#16861123—21no—
17#16861123—21no—
18#126×C127—21no—
19#124×C127C2211yescharacteristic
20#123×C127—21no—
21#123×C127C2211yescharacteristic
22C22⋊Q8×C127—21no—
23#120×C127C2211yescharacteristic
24C4×D4×C127—21no—
25#117×C127—21no—
26C2×C22⋊C4×C127—21no—
27#14523248—21no—
28#14523247—41no—
29C2×C22⋊D127—21no—
30C2×C22⋊D127—21no—
31C2×C22⋊D127—41no—
32D4×D127—41no—
33#14523238—41no—
34#14523235—41no—
35#14523232—21no—
36Dic127⋊C4—41no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127—41no—
40(C2×D127)⋊C4—41no—
41C24×C127D411yescharacteristic
42C2×Q8×C127D411yescharacteristic, complemented
43C2×D4×C127—21no—
44C2×D4×C127C2311yescharacteristic
45C22×C4×C127—21no—
46C22×C4×C127—21no—
47C22×C4×C127D411yescharacteristic, complemented
48C4⋊C4×C127—21no—
49C4⋊C4×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C22⋊C4×C127—21no—
54C22⋊C4×C127—21no—
55C22⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—21no—
58C22⋊C4×C127D411yescharacteristic
59C22⋊C4×C127D411yescharacteristic, complemented
60C42×C127—21no—
61C42×C127D411yescharacteristic, complemented
62C22×D127—21no—
63C22×D127—21no—
64Dic254—21no—
65D508—21no—
66C22⋊D127—41no—
67C22⋊D127—41no—
68C22⋊D127—41no—
69C22⋊D127—82no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—41no—
73C2×Dic127—41no—
74C4×D127—41no—
75C127⋊C8—41no—
76C23×C127—21no—
77C23×C127—42no—
78C23×C127—41no—
79C23×C127C2×D411yescharacteristic
80C23×C127C2×D411yescharacteristic
81Q8×C127—41no—
82D4×C127—41no—
83D4×C127—42no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—42no—
90C2×C4×C127—41no—
91C2×C4×C127—81no—
92C2×C4×C127C2×D411yescharacteristic
93C22×C127—22no—
94C22×C127—21no—
95C22×C127—21no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—41no—
99C22×C127—42no—
100C22×C127—41no—
101C22×C127—81no—
102C22×C127#12011yescharacteristic
103C2×D127—41no—
104C2×D127—81no—
105C2×D127—81no—
106C4×C127—21no—
107C4×C127—41no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127—41no—
111C4×C127—41no—
112Dic127—41no—
113Dic127—81no—
114Dic127—81no—
115C2×C127—21no—
116C2×C127—41no—
117C2×C127—41no—
118C2×C127—42no—
119C2×C127B(4,𝔽2)11yescharacteristic
120D127—81no—
121#2197—1271nomaximal, Sylow
122C127#219711yescharacteristic, complemented, Sylow
123#551D12711yescharacteristic, complemented
124#458—1271no—
125B(4,𝔽2)—1271no—
126#455—1271no—
127#354—1271no—
128#352—1271no—
129AΣL(1,𝔽16)—1271no—
130H(5,𝔽2)—1271no—
131#137—2541no—
132#126—21no—
133#124C2×D12711yescharacteristic
134#123—21no—
135#123—2541no—
136#123C2×D12711yescharacteristic
137C22⋊Q8—21no—
138#120—2541no—
139#120C2×D12711yescharacteristic
140C4×D4—21no—
141#117—21no—
142C2×C22⋊C4—21no—
143#104—2541no—
144#100—1271no—
145#99—1271no—
146#99—1271no—
147#99—2541no—
148#99—2541no—
149C24C22⋊D12711yescharacteristic
150#41—5081no—
151#41—2541no—
152C2×Q8C22⋊D12711yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—5081no—
155C2×D4—5081no—
156C2×D4—2541no—
157C2×D4—2541no—
158C2×D4C22×D12711yescharacteristic
159C22×C4—21no—
160C22×C4—21no—
161C22×C4C22⋊D12711yescharacteristic, complemented
162Q16—5081no—
163SD16—5081no—
164M16—2541no—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—5081no—
170C22⋊C4—21no—
171C22⋊C4—21no—
172C22⋊C4—21no—
173C22⋊C4—21no—
174C22⋊C4—21no—
175C22⋊C4—21no—
176C22⋊C4—5081no—
177C22⋊C4—5081no—
178C22⋊C4—2541no—
179C22⋊C4—2541no—
180C22⋊C4C22⋊D12711yescharacteristic
181C22⋊C4C22⋊D12711yescharacteristic, complemented
182C42—21no—
183C42C22⋊D12711yescharacteristic, complemented
184C23—21no—
185C23—41no—
186C23—42no—
187C23—2541no—
188C23—2541no—
189C23C2×C22⋊D12711yescharacteristic
190C23C2×C22⋊D12711yescharacteristic
191Q8—41no—
192Q8—2541no—
193D4—41no—
194D4—42no—
195D4—10162no—
196D4—5081no—
197D4—5081no—
198D4—5081no—
199D4—2541no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—5081no—
209C2×C4—5081no—
210C2×C4—5081no—
211C2×C4—2541no—
212C2×C4—2541no—
213C2×C4C2×C22⋊D12711yescharacteristic
214C8—5081no—
215C22—22no—
216C22—21no—
217C22—21no—
218C22—41no—
219C22—41no—
220C22—41no—
221C22—42no—
222C22—41no—
223C22—81no—
224C22—10161no—
225C22—10161no—
226C22—5081no—
227C22#1686117811yescharacteristic
228C4—21no—
229C4—41no—
230C4—41no—
231C4—41no—
232C4—41no—
233C4—41no—
234C4—10161no—
235C4—10161no—
236C4—5081no—
237C2—21no—
238C2—41no—
239C2—41no—
240C2—42no—
241C2—10161no—
242C2#2255006311yescharacteristic
subgroupG/H
centerC2#22550063
commutator, cosocleC2×D4×C127C23
FrattiniC2×D4C22×D127
Fitting#551×C127C2
socleC2×C127B(4,𝔽2)
prime radical#22550062C2
hypercenter#551D127
hypocenterC127#2197
LCSG ▷ C2×D4×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #551

Complements

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

NH ≅ G/Ntype
1. #22550065C2semidirect
3. #22550053C2semidirect
4. #22549671C2semidirect
5. #22549666C2semidirect
6. #22549663C2semidirect
7. #551×C127C2semidirect
42. C2×Q8×C127D4semidirect
47. C22×C4×C127D4semidirect
59. C22⋊C4×C127D4semidirect
61. C42×C127D4semidirect
122. C127#2197semidirect
123. #551D127semidirect
152. C2×Q8C22⋊D127semidirect
161. C22×C4C22⋊D127semidirect
181. C22⋊C4C22⋊D127semidirect
183. C42C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
#225500652
#225500622
#225500532
#225496712
#225496662
#225496632
#551×C1272
#2197127