Finite Group Databasebeta

← #30638739

Subgroups — #30638739

Order 16256.

356 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16861171C211yesmaximal, characteristic
2C2×Dic127⋊C8C211yesmaximal, characteristic, complemented
3C2×#16861131C211yesmaximal, characteristic
4C2×#16861129C211yesmaximal, characteristic
5#22549741C211yesmaximal, characteristic, complemented
6#22549638C211yesmaximal, characteristic, complemented
7#340×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C4×D127—21no—
10C22×C127⋊C8C2211yescharacteristic
11#16861171—22no—
12#16861171C414yescomplemented
13C2×C4⋊Dic127C2211yescharacteristic
14C2×(C2×D127)⋊C4—21no—
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17#16861131—22no—
18#16861129—22no—
19C127⋊#95—21no—
20C127⋊#95—21no—
21C22×C8×C127C2211yescharacteristic
22C2×C4⋊C4×C127C2211yescharacteristic
23C2×C4⋊C4×C127C2211yescharacteristic
24C23×D127—21no—
25C2×D508—21no—
26C2×D508—21no—
27C2×D508—21no—
28C2×D508C2×C412yes—
29C2×D508D412yes—
30C2×D508Q812yes—
31C22×Dic127—21no—
32C22×Dic127D411yescharacteristic
33C22×Dic127D411yescharacteristic
34C2×C4×D127—24no—
35C2×C4×D127—42no—
36C2×C4×D127—42no—
37C2×C127⋊C8—21no—
38C2×C127⋊C8—21no—
39C2×C127⋊C8—21no—
40C4⋊Dic127—21no—
41C4⋊Dic127—21no—
42(C2×D127)⋊C4—24no—
43C4×Dic127C2×C412yes—
44C4×Dic127D412yes—
45C22×C4×C127—21no—
46C22×C4×C127—21no—
47C22×C4×C127C2311yescharacteristic
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C2×C8×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127—21no—
53C4⋊C4×C127—21no—
54C4⋊C4×C127C2×C412yes—
55C22×D127—22no—
56C22×D127—22no—
57C22×D127—22no—
58C22×D127—41no—
59C22×D127—41no—
60C22×D127—41no—
61C22×D127—41no—
62D508—22no—
63D508—22no—
64D508—22no—
65D508C4⋊C414yes—
66C2×Dic127—21no—
67C2×Dic127—21no—
68C2×Dic127—21no—
69C2×Dic127—21no—
70C2×Dic127—41no—
71C2×Dic127—41no—
72C2×Dic127—41no—
73C2×Dic127C22⋊C412yes—
74C2×Dic127C22⋊C412yes—
75C4×D127—48no—
76C127⋊C8—41no—
77C23×C127C2×D411yescharacteristic
78C2×C4×C127—22no—
79C2×C4×C127—41no—
80C2×C4×C127—41no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127C22×C411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×Q811yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—41no—
91C22×C127C2×C22⋊C411yescharacteristic
92C22×C127#12111yescharacteristic
93C22×C127C22⋊Q811yescharacteristic
94C22×C127#13511yescharacteristic
95C22×C127#13511yescharacteristic
96C22×C127AGL(1,ℤ/8)11yescharacteristic
97C22×C127AGL(1,ℤ/8)11yescharacteristic
98C2×D127—24no—
99C2×D127—42no—
100C2×D127—42no—
101C2×D127—42no—
102C2×D127—42no—
103C2×D127—42no—
104C2×D127—42no—
105C4×C127—42no—
106C4×C127—81no—
107C4×C127C2×C4⋊C411yescharacteristic
108C4×C127#11711yescharacteristic
109C4×C127#12111yescharacteristic
110C4×C127C22⋊Q811yescharacteristic
111Dic127—22no—
112Dic127—22no—
113Dic127—81no—
114C2×C127#38011yescharacteristic
115C2×C127#41611yescharacteristic
116C2×C127#41811yescharacteristic
117C2×C127#46311yescharacteristic
118C2×C127#46311yescharacteristic
119C2×C127#47811yescharacteristic
120C2×C127#47811yescharacteristic
121D127—44no—
122#1804—1271nomaximal, Sylow
123C127#180411yescharacteristic, complemented, Sylow
124C2×C4×D4—1271no—
125C2×C4⋊C8—1271no—
126C2×#102—1271no—
127C2×#102—1271no—
128#382—1271no—
129#340—1271no—
130#340D12711yescharacteristic, complemented
131C22×D4—1271no—
132C23×C4—2541no—
133C22×C8—1271no—
134C22×C8C2×D12711yescharacteristic
135C4×D4—1274no—
136C4×D4—2542no—
137C2×C4⋊C4—1271no—
138C2×C4⋊C4C2×D12711yescharacteristic
139C2×C4⋊C4C2×D12711yescharacteristic
140C2×C22⋊C4—2541no—
141C2×C42—1271no—
142C4⋊C8—2542no—
143#102—2542no—
144#102—2542no—
145#95—2541no—
146#95—2541no—
147C24—2541no—
148C2×D4—1272no—
149C2×D4—1272no—
150C2×D4—1272no—
151C2×D4—2541no—
152C2×D4—2541no—
153C2×D4—2541no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—1271no—
157C22×C4—1271no—
158C22×C4—5082no—
159C22×C4—5082no—
160C22×C4—2544no—
161C22×C4—2541no—
162C22×C4C22×D12711yescharacteristic
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—2541no—
167C2×C8—2541no—
168C2×C8—2541no—
169C4⋊C4—21no—
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—2541no—
173C4⋊C4—2541no—
174C4⋊C4C4×D12712yes—
175C22⋊C4—2544no—
176C42—1272no—
177C42—1272no—
178C23—5081no—
179C23—5081no—
180C23—5081no—
181C23—5081no—
182C23—2542no—
183C23—2542no—
184C23—2542no—
185C23D4×D12711yescharacteristic
186D4—1274no—
187D4—2542no—
188D4—2542no—
189D4—2542no—
190C2×C4—22no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—41no—
195C2×C4—41no—
196C2×C4—1272no—
197C2×C4—1272no—
198C2×C4—5088no—
199C2×C4—5081no—
200C2×C4—5081no—
201C2×C4—5081no—
202C2×C4—2541no—
203C2×C4—2541no—
204C2×C4—2541no—
205C2×C4—2541no—
206C2×C4C2×C4×D12711yescharacteristic
207C2×C4D4×D12711yescharacteristic
208C2×C4D4×D12711yescharacteristic
209C2×C4Q8×D12711yescharacteristic
210C2×C4#1452324811yescharacteristic
211C2×C4#1452324811yescharacteristic
212C8—41no—
213C8—5081no—
214C22—5082no—
215C22—5082no—
216C22—5082no—
217C22—5082no—
218C22—5082no—
219C22—5082no—
220C22—2544no—
221C22C22⋊C4×D12711yescharacteristic
222C22#1686118211yescharacteristic
223C22#1686118711yescharacteristic
224C22#1686123011yescharacteristic
225C22#1686123211yescharacteristic
226C22#1686123311yescharacteristic
227C22#1686123611yescharacteristic
228C4—42no—
229C4—81no—
230C4—10161no—
231C4—2542no—
232C4—2542no—
233C4C4⋊C4×D12711yescharacteristic
234C4#1686116611yescharacteristic
235C4#1686118211yescharacteristic
236C4#1686118711yescharacteristic
237C2—5084no—
238C2#2254973511yescharacteristic
239C2#2254987111yescharacteristic
240C2#2254987911yescharacteristic
241C2#2255009011yescharacteristic
242C2#2255009411yescharacteristic
243C2#2255017511yescharacteristic
244C2#2255017911yescharacteristic
subgroupG/H
centerC23D4×D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#340×C127C2
socleC23×C127C2×D4
prime radicalC22×D508C22
cosocleC22×C4×C127C23
hypercenter#340D127
hypocenterC127#1804
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×Dic127⋊C8C2semidirect
5. #22549741C2semidirect
6. #22549638C2semidirect
7. #340×C127C2semidirect
12. #16861171C4semidirect
123. C127#1804semidirect
130. #340D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168611712
C2×Dic127⋊C82
C2×#168611312
C2×#168611292
#225497412
#225496382
#340×C1272
#1804127