Finite Group Databasebeta

← #30640136

Subgroups — #30640136

Order 16256.

274 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 83 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×#16861130C211yesmaximal, characteristic
2#22549779C211yesmaximal, characteristic
3#22549745C211yesmaximal, characteristic, complemented
4#22549639C211yesmaximal, characteristic, complemented
5#22549638C211yesmaximal, characteristic, complemented
6#22549630C211yesmaximal, characteristic, complemented
7C2×#102×C127C211yesmaximal, characteristic
8C22×C127⋊C8C2211yescharacteristic
9C2×C4⋊Dic127C2211yescharacteristic
10C2×C4⋊Dic127C2211yescharacteristic
11C2×Dic127⋊C4—21no—
12C2×Dic127⋊C4—21no—
13C2×C22⋊Dic127—21no—
14C2×C22⋊Dic127—21no—
15C2×C4×Dic127C2211yescharacteristic
16#16861130—22no—
17C22×D4×C127C2211yescharacteristic
18C22×C8×C127C2211yescharacteristic
19C2×C4⋊C4×C127C2211yescharacteristic
20#102×C127—22no—
21C22×Dic127—21no—
22C22×Dic127—21no—
23C22×Dic127D411yescharacteristic
24C22×Dic127D411yescharacteristic
25C2×C127⋊C8—21no—
26C2×C127⋊C8—21no—
27C2×C127⋊C8—21no—
28C4⋊Dic127—21no—
29C4⋊Dic127—21no—
30C4⋊Dic127—21no—
31C4⋊Dic127—21no—
32Dic127⋊C4—42no—
33Dic127⋊C4—42no—
34C22⋊Dic127—42no—
35C22⋊Dic127—42no—
36C4×Dic127—21no—
37C4×Dic127—21no—
38C24×C127—21no—
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41C2×D4×C127—21no—
42C2×D4×C127—21no—
43C2×D4×C127—21no—
44C2×D4×C127D412yes—
45C22×C4×C127—21no—
46C22×C4×C127C2311yescharacteristic
47C2×C8×C127—21no—
48C2×C8×C127—21no—
49C2×C8×C127—21no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C2×Dic127—21no—
53C2×Dic127—21no—
54C2×Dic127—21no—
55C2×Dic127—21no—
56C2×Dic127—41no—
57C2×Dic127—41no—
58C2×Dic127—41no—
59C2×Dic127—41no—
60C2×Dic127—41no—
61C2×Dic127—41no—
62C2×Dic127D812yes—
63C2×Dic127SD1612yes—
64C127⋊C8—41no—
65C23×C127—22no—
66C23×C127—41no—
67C23×C127—41no—
68C23×C127—41no—
69C23×C127—41no—
70C23×C127—41no—
71C23×C127—41no—
72C23×C127C2×D411yescharacteristic
73D4×C127—22no—
74D4×C127—22no—
75D4×C127—41no—
76D4×C127—41no—
77C2×C4×C127—41no—
78C2×C4×C127—41no—
79C2×C4×C127—41no—
80C2×C4×C127C2×D411yescharacteristic
81C2×C4×C127#4111yescharacteristic
82C2×C4×C127#4111yescharacteristic
83C2×C4×C127#4111yescharacteristic
84C2×C4×C127#4111yescharacteristic
85C2×C4×C127#4111yescharacteristic
86C8×C127—41no—
87C22×C127—42no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—81no—
91C22×C127—81no—
92C22×C127—81no—
93C22×C127—81no—
94C22×C127#12111yescharacteristic
95C22×C127#12311yescharacteristic
96C22×C127#12411yescharacteristic
97C22×C127C2×D811yescharacteristic
98C22×C127C2×SD1611yescharacteristic
99C22×C127AGL(1,ℤ/8)11yescharacteristic
100C22×C127#13711yescharacteristic
101C4×C127—81no—
102C4×C127#12311yescharacteristic
103C4×C127#12311yescharacteristic
104C4×C127#12611yescharacteristic
105C4×C127#12611yescharacteristic
106Dic127—22no—
107Dic127—22no—
108Dic127—81no—
109Dic127—81no—
110C2×C127—82no—
111C2×C127#39711yescharacteristic
112C2×C127#45911yescharacteristic
113C2×C127#46111yescharacteristic
114C2×C127#48011yescharacteristic
115C2×C127#48111yescharacteristic
116C2×C127#48611yescharacteristic
117C2×C127#48911yescharacteristic
118#2071—1271nomaximal, Sylow
119C127#207111yescharacteristic, complemented, Sylow
120C2×#102—1271no—
121C2×#102D12711yescharacteristic
122#390—1271no—
123#384—1271no—
124#340—1271no—
125#340—1271no—
126#336—1271no—
127C22×D4C2×D12711yescharacteristic
128C22×C8—1271no—
129C22×C8C2×D12711yescharacteristic
130C2×C4⋊C4—1271no—
131C2×C4⋊C4—1271no—
132C2×C4⋊C4—2541no—
133C2×C4⋊C4—2541no—
134C2×C4⋊C4C2×D12711yescharacteristic
135C2×C22⋊C4—2541no—
136C2×C22⋊C4—2541no—
137C2×C42—1271no—
138#102—22no—
139#102—2542no—
140C24—21no—
141C2×D4—21no—
142C2×D4—21no—
143C2×D4—21no—
144C2×D4—21no—
145C2×D4—21no—
146C2×D4C22⋊D12712yes—
147C22×C4—21no—
148C22×C4—1271no—
149C22×C4—1271no—
150C22×C4—2541no—
151C22×C4—2541no—
152C22×C4C22×D12711yescharacteristic
153C2×C8—21no—
154C2×C8—21no—
155C2×C8—21no—
156C2×C8—2541no—
157C2×C8—2541no—
158C2×C8—2541no—
159C4⋊C4—21no—
160C4⋊C4—21no—
161C4⋊C4—5082no—
162C4⋊C4—5082no—
163C4⋊C4—2541no—
164C4⋊C4—2541no—
165C4⋊C4—2541no—
166C4⋊C4—2541no—
167C22⋊C4—5082no—
168C22⋊C4—5082no—
169C42—2541no—
170C42—2541no—
171C23—22no—
172C23—41no—
173C23—41no—
174C23—41no—
175C23—41no—
176C23—41no—
177C23—41no—
178C23D4×D12711yescharacteristic
179D4—22no—
180D4—22no—
181D4—41no—
182D4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—41no—
186C2×C4—1272no—
187C2×C4—1272no—
188C2×C4—5081no—
189C2×C4—5081no—
190C2×C4—5081no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4—2541no—
198C2×C4C2×C22⋊D12711yescharacteristic
199C2×C4#1452324711yescharacteristic
200C2×C4#1452324711yescharacteristic
201C2×C4#1452324711yescharacteristic
202C2×C4#1452324811yescharacteristic
203C2×C4#1452324811yescharacteristic
204C8—41no—
205C8—5081no—
206C22—42no—
207C22—42no—
208C22—42no—
209C22—81no—
210C22—81no—
211C22—81no—
212C22—81no—
213C22#1686118111yescharacteristic
214C22#1686119111yescharacteristic
215C22#1686119511yescharacteristic
216C22D8×D12711yescharacteristic
217C22SD16×D12711yescharacteristic
218C22#1686123311yescharacteristic
219C22#1686123811yescharacteristic
220C4—81no—
221C4—10161no—
222C4—10161no—
223C4—2542no—
224C4—2542no—
225C4#1686119011yescharacteristic
226C4#1686119211yescharacteristic
227C4#1686120211yescharacteristic
228C4#1686120211yescharacteristic
229C2—82no—
230C2#2254980411yescharacteristic
231C2#2255006911yescharacteristic
232C2#2255007911yescharacteristic
233C2#2255018811yescharacteristic
234C2#2255019311yescharacteristic
235C2#2255022111yescharacteristic
236C2#2255023411yescharacteristic
subgroupG/H
centerC23D4×D127
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
FittingC2×#102×C127C2
socleC23×C127C2×D4
prime radicalC22×D4×C127C22
hypercenterC2×#102D127
hypocenterC127#2071
LCSG ▷ C22×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
3. #22549745C2semidirect
4. #22549639C2semidirect
5. #22549638C2semidirect
6. #22549630C2semidirect
119. C127#2071semidirect

Maximal subgroups

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

subgroupindex
C2×#168611302
#225497792
#225497452
#225496392
#225496382
#225496302
C2×#102×C1272
#2071127