Finite Group Databasebeta

← #30648384

Subgroups — #30648384

Order 16256.

460 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C22×#14523238C211yesmaximal, characteristic, complemented
2C2×#16861217C211yesmaximal, characteristic
3#22550452C211yesmaximal, characteristic
4#22550450C211yesmaximal, characteristic
5#22550133C214yesmaximal, complemented
6#22550116C214yesmaximal, complemented
7#22549912C211yesmaximal, characteristic, complemented
8#22549886C211yesmaximal, characteristic, complemented
9#419×C127C211yesmaximal, characteristic, complemented
10C2×#14523248C2211yescharacteristic
11C22×Dic254C2211yescharacteristic
12C2×#14523238—24no—
13C2×#14523238C2212yes—
14C2×#14523238C2212yes—
15#16861217—24no—
16C2×#14523231C2212yes—
17C22×C127⋊C8C2211yescharacteristic
18C2×C8×D127C2212yes—
19#16861209—21no—
20#16861207—21no—
21#16861198—21no—
22#16861197—21no—
23#16861193—22no—
24#16861192—22no—
25#16861189C2214yes—
26#16861185C2214yes—
27#16861168C2211yescharacteristic
28#16861142C2212yes—
29#16861140C2212yes—
30#16861134C2214yes—
31C22×Q8×C127C2211yescharacteristic
32C2×M16×C127C2211yescharacteristic
33#117×C127C2211yescharacteristic
34#103×C127C2214yes—
35#14523248—22no—
36#14523248—22no—
37C2×Dic254—21no—
38C2×Dic254—22no—
39C2×Dic254—22no—
40C2×Dic254D411yescharacteristic
41C2×Dic254C2312yes—
42C2×D508D411yescharacteristic
43C2×C22⋊D127D412yes—
44C22×Dic127—21no—
45C2×C4×D127C2312yes—
46#14523238—24no—
47#14523238—24no—
48#14523231—24no—
49C2×C127⋊C8D414yes—
50C2×C127⋊C8C2311yescharacteristic
51C2×C127⋊C8C2311yescharacteristic
52C8×D127—24no—
53C4⋊Dic127—24no—
54C4⋊Dic127C2312yes—
55Dic127⋊C4—24no—
56C22⋊Dic127—21no—
57(C2×D127)⋊C4—24no—
58(C2×D127)⋊C4—24no—
59C4×Dic127—21no—
60C2×Q8×C127—21no—
61C2×Q8×C127—22no—
62C2×Q8×C127—22no—
63C2×Q8×C127C2312yes—
64C22×C4×C127—21no—
65C22×C4×C127C2311yescharacteristic
66M16×C127—22no—
67C2×C8×C127C2312yes—
68C4⋊C4×C127C2312yes—
69C22⋊C4×C127—21no—
70C42×C127—21no—
71C22×D127C2×D412yes—
72Dic254—22no—
73Dic254—22no—
74Dic254—22no—
75Dic254—42no—
76D508—22no—
77C22⋊D127—24no—
78C2×Dic127—22no—
79C2×Dic127—22no—
80C2×Dic127—42no—
81C2×Dic127C2×D412yes—
82C4×D127—22no—
83C4×D127—22no—
84C127⋊C8C2×D412yes—
85C127⋊C8C2×D412yes—
86C23×C127C2×D411yescharacteristic
87Q8×C127—22no—
88Q8×C127—22no—
89Q8×C127—42no—
90C2×C4×C127—22no—
91C2×C4×C127—22no—
92C2×C4×C127—42no—
93C2×C4×C127C2×D411yescharacteristic
94C2×C4×C127C2×D412yes—
95C2×C4×C127#4112yes—
96C2×C4×C127C2411yescharacteristic
97C8×C127—22no—
98C22×C127—22no—
99C22×C127#12112yes—
100C22×C127C22×D411yescharacteristic
101C2×D127—22no—
102C2×D127—24no—
103C4×C127—42no—
104C4×C127—42no—
105C4×C127#12112yes—
106C4×C127C22×D411yescharacteristic
107C4×C127C2×#4111yescharacteristic
108Dic127—22no—
109Dic127—42no—
110Dic127—42no—
111C2×C127—22no—
112C2×C127C2×#12111yescharacteristic
113C2×C127#57812yes—
114D127—42no—
115#3059—1271nomaximal, Sylow
116C127#305911yescharacteristic, complemented, Sylow
117C22×Q16—1271no—
118C2×#131—1271no—
119#536—1271no—
120#536—1271no—
121#470—1274no—
122#467—1274no—
123#427—1271no—
124#419—1271no—
125#419D12711yescharacteristic, complemented
126C2×#41—1271no—
127C22×Q8—1271no—
128C22×Q8C2×D12711yescharacteristic
129C2×Q16—1272no—
130C2×Q16—1272no—
131C2×Q16—2544no—
132#131—2544no—
133C2×M16—1272no—
134C2×M16C2×D12711yescharacteristic
135C22×C8—1271no—
136C22×C8—1272no—
137C4⋊Q8—2541no—
138C4⋊Q8—2541no—
139#124—2541no—
140#124—2541no—
141#123—2542no—
142#123—2542no—
143C22⋊Q8—1274no—
144C22⋊Q8—1274no—
145#117—1271no—
146#117C2×D12711yescharacteristic
147#107—1272no—
148#106—1272no—
149#103—1274no—
150#103C2×D12714yes—
151#41—2542no—
152#41—2542no—
153C2×Q8—21no—
154C2×Q8—22no—
155C2×Q8—22no—
156C2×Q8—1271no—
157C2×Q8—1272no—
158C2×Q8—2542no—
159C2×Q8—2542no—
160C2×Q8—2541no—
161C2×Q8C22×D12712yes—
162C2×D4—1272no—
163C2×D4—1271no—
164C22×C4—21no—
165C22×C4—1272no—
166C22×C4—2541no—
167C22×C4C22×D12711yescharacteristic
168Q16—2544no—
169Q16—2544no—
170M16—22no—
171M16—2544no—
172C2×C8—1274no—
173C2×C8—1271no—
174C2×C8—1271no—
175C2×C8—2544no—
176C2×C8C22×D12712yes—
177C4⋊C4—1272no—
178C4⋊C4—2544no—
179C4⋊C4—2544no—
180C4⋊C4C22×D12712yes—
181C22⋊C4—21no—
182C22⋊C4—2544no—
183C22⋊C4—2544no—
184C22⋊C4—2541no—
185C42—21no—
186C42—2541no—
187C23—1272no—
188C23D4×D12711yescharacteristic
189Q8—22no—
190Q8—22no—
191Q8—42no—
192Q8—5082no—
193Q8—2542no—
194Q8—2542no—
195Q8—2542no—
196D4—2544no—
197D4—2542no—
198C2×C4—22no—
199C2×C4—22no—
200C2×C4—42no—
201C2×C4—1272no—
202C2×C4—5082no—
203C2×C4—2542no—
204C2×C4—2542no—
205C2×C4—2542no—
206C2×C4—2542no—
207C2×C4D4×D12712yes—
208C2×C4D4×D12711yescharacteristic
209C2×C4#1452324812yes—
210C2×C4C23×D12711yescharacteristic
211C8—22no—
212C8—1272no—
213C8—1272no—
214C22—22no—
215C22—2542no—
216C22—2544no—
217C22#1686118212yes—
218C22C2×D4×D12711yescharacteristic
219C4—42no—
220C4—42no—
221C4—5082no—
222C4—5082no—
223C4—2542no—
224C4#1686118212yes—
225C4C2×D4×D12711yescharacteristic
226C4C2×#1452324811yescharacteristic
227C2—22no—
228C2—5082no—
229C2C2×#1686118211yescharacteristic
230C2#2255069212yes—
subgroupG/H
centerC22C2×D4×D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#419×C127C2
socleC22×C127C22×D4
prime radicalC2×#14523248C22
hypercenter#419D127
hypocenterC127#3059
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. C22×#14523238C2semidirect
5. #22550133C2semidirect
6. #22550116C2semidirect
7. #22549912C2semidirect
8. #22549886C2semidirect
9. #419×C127C2semidirect
116. C127#3059semidirect
125. #419D127semidirect

Maximal subgroups

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

subgroupindex
C22×#145232382
C2×#168612172
#225504522
#225504502
#225501332
#225501162
#225499122
#225498862
#419×C1272
#3059127