Finite Group Databasebeta

← #30639568

Subgroups — #30639568

Order 16256.

396 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 87 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×#16861205C211yesmaximal, characteristic
2#22549899C211yesmaximal, characteristic, complemented
3C2×#16861126C211yesmaximal, characteristic, complemented
4C2×#16861125C211yesmaximal, characteristic, complemented
5C2×#16861125C211yesmaximal, characteristic, complemented
6M16×Dic127C211yesmaximal, characteristic, complemented
7C2×#100×C127C211yesmaximal, characteristic, complemented
8C22×D508C2211yescharacteristic
9C22×C22⋊D127C414yes—
10C2×#14523232C2211yescharacteristic
11C2×#14523232C2211yescharacteristic
12#16861205—21no—
13#16861205—21no—
14#16861205—22no—
15C2×C4×Dic127C2211yescharacteristic
16Dic127⋊C8—22no—
17#16861126—22no—
18#16861125—22no—
19#16861125—22no—
20#16861119—21no—
21C8×Dic127—21no—
22C22×D4×C127C2211yescharacteristic
23C2×M16×C127C2211yescharacteristic
24C2×M16×C127C2211yescharacteristic
25#100×C127—22no—
26C23×D127C2×C412yes—
27C2×D508—21no—
28C2×D508—21no—
29C2×D508—21no—
30C2×D508—21no—
31C2×D508—21no—
32C2×D508—21no—
33C2×C22⋊D127—24no—
34C2×C22⋊D127—24no—
35C2×C22⋊D127—48no—
36C22×Dic127C2×C412yes—
37#14523232—21no—
38#14523232—21no—
39#14523232—21no—
40#14523232D412yescomplemented
41C2×C127⋊C8—21no—
42C2×C127⋊C8—21no—
43C4×Dic127D411yescharacteristic
44C4×Dic127D411yescharacteristic
45C4×Dic127D412yes—
46C24×C127C2×C412yes—
47C2×D4×C127—21no—
48C2×D4×C127—21no—
49C2×D4×C127—21no—
50C2×D4×C127—21no—
51C2×D4×C127—21no—
52C2×D4×C127—21no—
53C22×C4×C127C2311yescharacteristic
54M16×C127—21no—
55M16×C127—21no—
56M16×C127—21no—
57M16×C127D412yescomplemented
58C2×C8×C127—21no—
59C2×C8×C127—21no—
60C22×D127—22no—
61C22×D127—22no—
62C22×D127—22no—
63C22×D127—44no—
64D508—41no—
65D508—41no—
66D508—41no—
67D508—41no—
68C22⋊D127—48no—
69C22⋊D127—48no—
70C2×Dic127—24no—
71C2×Dic127C22⋊C412yes—
72C2×Dic127C22⋊C412yes—
73C127⋊C8—22no—
74C127⋊C8—41no—
75C23×C127—22no—
76C23×C127—22no—
77C23×C127—22no—
78C23×C127—44no—
79C23×C127C22×C411yescharacteristic
80D4×C127—41no—
81D4×C127—41no—
82D4×C127—41no—
83D4×C127—41no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127#4111yescharacteristic
89C2×C4×C127#4111yescharacteristic
90C8×C127—22no—
91C8×C127—41no—
92C22×C127—22no—
93C22×C127—42no—
94C22×C127—44no—
95C22×C127—82no—
96C22×C127—82no—
97C22×C127C2×C22⋊C411yescharacteristic
98C22×C127C4×D411yescharacteristic
99C22×C127C4×D411yescharacteristic
100C2×D127—42no—
101C2×D127—44no—
102C2×D127—82no—
103C2×D127—82no—
104C4×C127#12111yescharacteristic
105C4×C127#12111yescharacteristic
106C4×C127#12411yescharacteristic
107C4×C127#12711yescharacteristic
108Dic127—22no—
109Dic127#10014yescomplemented
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—82no—
113C2×C127#39011yescharacteristic
114C2×C127C2×#10012yes—
115D127—82no—
116#1965—1271nomaximal, Sylow
117C127#196511yescharacteristic, complemented, Sylow
118C2×#127—1271no—
119#423—1271no—
120C2×#100—1271no—
121C2×#100—1271no—
122C2×#100—1271no—
123C2×#100D12711yescharacteristic, complemented
124C4×M16—1271no—
125C22×D4—1274no—
126C22×D4—1271no—
127C22×D4C2×D12711yescharacteristic
128C2×M16—1271no—
129C2×M16—1271no—
130C2×M16C2×D12711yescharacteristic
131C2×M16C2×D12711yescharacteristic
132#127—2541no—
133#127—2541no—
134#127—2542no—
135C2×C42—1271no—
136C4⋊C8—2542no—
137#100—22no—
138#100—2542no—
139#100—2542no—
140#100—2542no—
141#97—2541no—
142C4×C8—2541no—
143C24—1272no—
144C24C4×D12712yes—
145C2×D4—21no—
146C2×D4—21no—
147C2×D4—21no—
148C2×D4—21no—
149C2×D4—21no—
150C2×D4—21no—
151C2×D4—5088no—
152C2×D4—2544no—
153C2×D4—2544no—
154C2×D4—2541no—
155C2×D4—2541no—
156C2×D4—2541no—
157C2×D4—2541no—
158C2×D4—2541no—
159C2×D4—2541no—
160C22×C4—1272no—
161C22×C4C22×D12711yescharacteristic
162M16—21no—
163M16—21no—
164M16—21no—
165M16—1272no—
166M16—2541no—
167M16—2541no—
168M16—2541no—
169M16C22⋊D12712yescomplemented
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—2541no—
173C2×C8—2541no—
174C42—1271no—
175C42—1271no—
176C42—1272no—
177C23—22no—
178C23—22no—
179C23—22no—
180C23—44no—
181C23—5084no—
182C23—2542no—
183C23—2542no—
184C23—2542no—
185C23C2×C4×D12711yescharacteristic
186D4—41no—
187D4—41no—
188D4—41no—
189D4—41no—
190D4—5088no—
191D4—5088no—
192D4—5081no—
193D4—5081no—
194D4—5081no—
195D4—5081no—
196C2×C4—1272no—
197C2×C4—1272no—
198C2×C4—2544no—
199C2×C4D4×D12711yescharacteristic
200C2×C4D4×D12711yescharacteristic
201C2×C4D4×D12711yescharacteristic
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4#1452324711yescharacteristic
204C2×C4#1452324811yescharacteristic
205C8—22no—
206C8—41no—
207C8—5081no—
208C8—2542no—
209C22—22no—
210C22—42no—
211C22—44no—
212C22—82no—
213C22—82no—
214C22—10162no—
215C22—10162no—
216C22—5082no—
217C22—5084no—
218C22C22⋊C4×D12711yescharacteristic
219C22(C22⋊D127)⋊C411yescharacteristic
220C22C4×C22⋊D12711yescharacteristic
221C4—1274no—
222C4—2542no—
223C4#1686118111yescharacteristic
224C4#1686118211yescharacteristic
225C4#1686119511yescharacteristic
226C4#1686120511yescharacteristic
227C2—21no—
228C2—21no—
229C2—82no—
230C2—10162no—
231C2#2254977611yescharacteristic
232C2#100×D12712yes—
subgroupG/H
centerC22C22⋊C4×D127
commutatorC23×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×#100×C127C2
socleC22×C127C2×C22⋊C4
prime radicalC2×#16861205C2
cosocleC22×C4×C127C23
hypercenterC2×#100D127
hypocenterC127#1965
LCSG ▷ C23×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. #22549899C2semidirect
3. C2×#16861126C2semidirect
4. C2×#16861125C2semidirect
5. C2×#16861125C2semidirect
6. M16×Dic127C2semidirect
7. C2×#100×C127C2semidirect
40. #14523232D4semidirect
57. M16×C127D4semidirect
109. Dic127#100semidirect
117. C127#1965semidirect
123. C2×#100D127semidirect
169. M16C22⋊D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612052
#225498992
C2×#168611262
C2×#168611252
M16×Dic1272
C2×#100×C1272
#1965127