Finite Group Databasebeta

← #30638715

Subgroups — #30638715

Order 16256.

380 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C2×C4×C22⋊D127C211yesmaximal, characteristic
2C2×#16861123C212yesmaximal
3#22549736C211yesmaximal, characteristic, complemented
4#22549642C212yesmaximal, complemented
5#380×C127C211yesmaximal, characteristic, complemented
6C22×C22⋊D127C2211yescharacteristic
7C22×C4×D127C411yescharacteristic, complemented
8C4×C22⋊D127—21no—
9C4×C22⋊D127—21no—
10C4×C22⋊D127—21no—
11C4×C22⋊D127—21no—
12C2×C4⋊Dic127—21no—
13C2×Dic127⋊C4C411yescharacteristic, complemented
14C2×C22⋊Dic127C2212yes—
15C2×C22⋊Dic127C2211yescharacteristic
16C2×(C2×D127)⋊C4C411yescharacteristic, complemented
17C2×C4×Dic127C411yescharacteristic, complemented
18#16861123—24no—
19C127⋊#95—21no—
20C23×C4×C127C2211yescharacteristic
21C2×C4⋊C4×C127—21no—
22C2×C22⋊C4×C127C2212yes—
23#95×C127—21no—
24C23×D127C2×C411yescharacteristic
25C2×C22⋊D127—21no—
26C2×C22⋊D127—21no—
27C2×C22⋊D127—21no—
28C2×C22⋊D127—21no—
29C2×C22⋊D127D412yes—
30C2×C22⋊D127Q812yes—
31C22×Dic127—22no—
32C22×Dic127C2×C411yescharacteristic
33C22×Dic127C2×C411yescharacteristic
34C22×Dic127C2×C411yescharacteristic
35C2×C4×D127—21no—
36C2×C4×D127—21no—
37C2×C4×D127—42no—
38C4⋊Dic127—42no—
39Dic127⋊C4—21no—
40Dic127⋊C4—21no—
41C22⋊Dic127—21no—
42C22⋊Dic127—21no—
43C22⋊Dic127—22no—
44C22⋊Dic127—22no—
45(C2×D127)⋊C4—21no—
46(C2×D127)⋊C4—21no—
47C4×Dic127—21no—
48C4×Dic127—21no—
49C24×C127C2311yescharacteristic
50C22×C4×C127—22no—
51C22×C4×C127—42no—
52C22×C4×C127C2×C411yescharacteristic
53C22×C4×C127C2×C411yescharacteristic
54C22×C4×C127D411yescharacteristic
55C22×C4×C127D411yescharacteristic
56C22×C4×C127D411yescharacteristic
57C22×C4×C127D411yescharacteristic
58C4⋊C4×C127—42no—
59C22⋊C4×C127—22no—
60C22⋊C4×C127—22no—
61C22×D127—21no—
62C22×D127—21no—
63C22×D127—41no—
64C22×D127—41no—
65C22×D127C4⋊C412yescomplemented
66C22⋊D127—22no—
67C22⋊D127—22no—
68C22⋊D127—22no—
69C22⋊D127—22no—
70C2×Dic127—21no—
71C2×Dic127—21no—
72C2×Dic127—21no—
73C2×Dic127—21no—
74C2×Dic127—41no—
75C2×Dic127—41no—
76C2×Dic127—42no—
77C2×Dic127—42no—
78C2×Dic127—42no—
79C2×Dic127C4⋊C412yescomplemented
80C4×D127—42no—
81C4×D127—42no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127C22×C411yescharacteristic
85C23×C127C2×D411yescharacteristic
86C23×C127C2×D411yescharacteristic
87C23×C127C2×D411yescharacteristic
88C23×C127C2×Q811yescharacteristic
89C23×C127#4111yescharacteristic
90C23×C127#4111yescharacteristic
91C2×C4×C127—22no—
92C2×C4×C127—22no—
93C2×C4×C127—22no—
94C2×C4×C127—41no—
95C2×C4×C127—41no—
96C2×C4×C127—41no—
97C2×C4×C127—42no—
98C2×C4×C127—42no—
99C2×C4×C127—42no—
100C2×C4×C127C22⋊C411yescharacteristic
101C2×C4×C127C22⋊C411yescharacteristic
102C2×C4×C127C22⋊C411yescharacteristic
103C2×C4×C127C22⋊C411yescharacteristic
104C22×C127—21no—
105C22×C127—21no—
106C22×C127—21no—
107C22×C127—21no—
108C22×C127—21no—
109C22×C127—21no—
110C22×C127—41no—
111C22×C127—41no—
112C22×C127—41no—
113C22×C127—41no—
114C22×C127C2×C22⋊C411yescharacteristic
115C22×C127C2×C4⋊C411yescharacteristic
116C22×C127#11711yescharacteristic
117C22×C127#12111yescharacteristic
118C22×C127#12111yescharacteristic
119C22×C127C22⋊Q811yescharacteristic
120C22×C127C22⋊Q811yescharacteristic
121C2×D127—22no—
122C2×D127—42no—
123C2×D127—41no—
124C2×D127—41no—
125C2×D127—42no—
126C4×C127—21no—
127C4×C127—21no—
128C4×C127—21no—
129C4×C127—82no—
130C4×C127#9912yescomplemented
131Dic127—22no—
132Dic127—41no—
133Dic127—41no—
134Dic127—82no—
135C2×C127—21no—
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—21no—
139C2×C127—21no—
140C2×C127—21no—
141C2×C127#38011yescharacteristic
142C2×C127C2×#9911yescharacteristic
143C2×C127#41011yescharacteristic
144D127—42no—
145#1799—1271nomaximal, Sylow
146C127#179911yescharacteristic, complemented, Sylow
147C2×C4×D4—1271no—
148C2×#99—1272no—
149#380—1271no—
150#380D12711yescharacteristic, complemented
151#342—1272no—
152C22×D4—1271no—
153C23×C4—1271no—
154C23×C4C2×D12711yescharacteristic
155C4×D4—2541no—
156C4×D4—2541no—
157C4×D4—2541no—
158C4×D4—2541no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—1271no—
161C2×C4⋊C4—2541no—
162C2×C22⋊C4—1271no—
163C2×C22⋊C4—1271no—
164C2×C22⋊C4—1272no—
165C2×C22⋊C4C2×D12712yes—
166C2×C42—1271no—
167#99—2544no—
168#95—21no—
169#95—2541no—
170C24—1271no—
171C24C22×D12711yescharacteristic
172C2×D4—1272no—
173C2×D4—1272no—
174C2×D4—2541no—
175C2×D4—2541no—
176C2×D4—2541no—
177C2×D4—2541no—
178C22×C4—22no—
179C22×C4—42no—
180C22×C4—1271no—
181C22×C4—1271no—
182C22×C4—1271no—
183C22×C4—5082no—
184C22×C4—2542no—
185C22×C4—2541no—
186C22×C4—2541no—
187C22×C4C4×D12711yescharacteristic
188C22×C4C4×D12711yescharacteristic
189C22×C4C22⋊D12711yescharacteristic
190C22×C4C22⋊D12711yescharacteristic
191C22×C4D50811yescharacteristic
192C22×C4D50811yescharacteristic
193C4⋊C4—42no—
194C4⋊C4—5082no—
195C4⋊C4—2541no—
196C4⋊C4—2541no—
197C22⋊C4—22no—
198C22⋊C4—22no—
199C22⋊C4—2542no—
200C22⋊C4—2542no—
201C22⋊C4—2541no—
202C22⋊C4—2541no—
203C22⋊C4—2541no—
204C22⋊C4—2541no—
205C42—2541no—
206C42—2541no—
207C23—41no—
208C23—41no—
209C23—1272no—
210C23—5081no—
211C23—5081no—
212C23—2541no—
213C23—2541no—
214C23C2×C4×D12711yescharacteristic
215C23D4×D12711yescharacteristic
216C23C2×C22⋊D12711yescharacteristic
217C23C2×D50811yescharacteristic
218C23Q8×D12711yescharacteristic
219C23#1452324711yescharacteristic
220C23#1452324911yescharacteristic
221D4—2542no—
222D4—2542no—
223D4—2542no—
224D4—2542no—
225C2×C4—22no—
226C2×C4—22no—
227C2×C4—22no—
228C2×C4—41no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—1272no—
235C2×C4—5082no—
236C2×C4—5082no—
237C2×C4—5082no—
238C2×C4—5082no—
239C2×C4—5082no—
240C2×C4—5081no—
241C2×C4—5081no—
242C2×C4—2541no—
243C2×C4—2541no—
244C2×C4—2541no—
245C2×C4—2541no—
246C2×C4(C2×D127)⋊C411yescharacteristic
247C2×C4(C2×D127)⋊C411yescharacteristic
248C2×C4(C2×D127)⋊C411yescharacteristic
249C2×C4(C2×D127)⋊C411yescharacteristic
250C22—21no—
251C22—21no—
252C22—21no—
253C22—21no—
254C22—21no—
255C22—21no—
256C22—41no—
257C22—41no—
258C22—41no—
259C22—41no—
260C22—5082no—
261C22—5082no—
262C22—5081no—
263C22—5081no—
264C22—2542no—
265C22C2×(C2×D127)⋊C411yescharacteristic
266C22C4⋊C4×D12711yescharacteristic
267C22#1686116711yescharacteristic
268C22#1686118011yescharacteristic
269C22#1686118411yescharacteristic
270C22#1686118511yescharacteristic
271C22#1686118911yescharacteristic
272C4—21no—
273C4—21no—
274C4—21no—
275C4—82no—
276C4—10162no—
277C4—5081no—
278C4—5081no—
279C4—2542no—
280C4#1686112312yescomplemented
281C2—21no—
282C2—21no—
283C2—21no—
284C2—21no—
285C2—21no—
286C2—21no—
287C2—5082no—
288C2#2254973311yescharacteristic
289C2C2×#1686112311yescharacteristic
290C2#2254984911yescharacteristic
subgroupG/H
centerC22C2×(C2×D127)⋊C4
commutatorC23×C127C22×C4
FrattiniC24C22×D127
Fitting#380×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC22×C22⋊D127C22
cosocleC24×C127C23
hypercenter#380D127
hypocenterC127#1799
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #22549736C2semidirect
4. #22549642C2semidirect
5. #380×C127C2semidirect
7. C22×C4×D127C4semidirect
13. C2×Dic127⋊C4C4semidirect
16. C2×(C2×D127)⋊C4C4semidirect
17. C2×C4×Dic127C4semidirect
65. C22×D127C4⋊C4semidirect
79. C2×Dic127C4⋊C4semidirect
130. C4×C127#99semidirect
146. C127#1799semidirect
150. #380D127semidirect
280. C4#16861123semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1272
C2×#168611232
#225497362
#225496422
#380×C1272
#1799127