Finite Group Databasebeta

← #30640369

Subgroups — #30640369

Order 16256.

264 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#22550383C211yesmaximal, characteristic
2#22549860C211yesmaximal, characteristic, complemented
3#22549665C212yesmaximal, complemented
4#22549662C212yesmaximal, complemented
5#410×C127C211yesmaximal, characteristic, complemented
6C22×C4×D127C411yescharacteristic, complemented
7C2×#14523232—21no—
8#16861190C411yescharacteristic, complemented
9#16861190C2211yescharacteristic
10#16861185—21no—
11#16861180—21no—
12#16861177C411yescharacteristic, complemented
13#16861177C2211yescharacteristic
14C4×C22⋊D127—21no—
15C4×C22⋊D127—21no—
16#16861168C411yescharacteristic, complemented
17#16861128—21no—
18#16861126C2212yes—
19C2×#41×C127C2211yescharacteristic
20#117×C127—21no—
21#99×C127—21no—
22#99×C127C2212yes—
23C23×D127C2×C411yescharacteristic
24C2×C22⋊D127—21no—
25C2×C22⋊D127—21no—
26C22×Dic127C2×C411yescharacteristic
27C2×C4×D127—21no—
28C2×C4×D127—21no—
29C2×C4×D127—41no—
30C2×C4×D127—41no—
31#14523232—22no—
32#14523232—41no—
33C2×C127⋊C8—41no—
34C4⋊Dic127—21no—
35Dic127⋊C4—21no—
36Dic127⋊C4—21no—
37C22⋊Dic127—21no—
38C22⋊Dic127—21no—
39C22⋊Dic127C2×C411yescharacteristic
40C22⋊Dic127C2×C411yescharacteristic
41(C2×D127)⋊C4—21no—
42(C2×D127)⋊C4—21no—
43C4×Dic127—21no—
44#41×C127—41no—
45#41×C127—41no—
46C2×Q8×C127D411yescharacteristic
47C2×D4×C127C2×C411yescharacteristic
48C2×D4×C127D411yescharacteristic
49C2×D4×C127C2311yescharacteristic
50C22×C4×C127C2×C411yescharacteristic
51C22×C4×C127D411yescharacteristic
52C22×C4×C127D411yescharacteristic
53C4⋊C4×C127—41no—
54C22⋊C4×C127—22no—
55C42×C127—41no—
56C22×D127—21no—
57C22×D127—42no—
58C22×D127—41no—
59C22⋊D127—42no—
60C22⋊D127—42no—
61C2×Dic127—21no—
62C2×Dic127—21no—
63C2×Dic127—21no—
64C2×Dic127—41no—
65C4×D127—42no—
66C4×D127—41no—
67C4×D127—41no—
68C127⋊C8—42no—
69C23×C127C22⋊C411yescharacteristic
70C23×C127C22×C411yescharacteristic
71C23×C127C2×D411yescharacteristic
72Q8×C127—41no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—21no—
79C2×C4×C127—21no—
80C2×C4×C127—21no—
81C2×C4×C127—21no—
82C2×C4×C127—42no—
83C2×C4×C127C22⋊C411yescharacteristic
84C2×C4×C127C22⋊C411yescharacteristic
85C2×C4×C127C22⋊C411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C22×C127—21no—
88C22×C127—21no—
89C22×C127—21no—
90C22×C127—41no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127C2×C22⋊C411yescharacteristic
94C2×D127—22no—
95C2×D127—41no—
96C2×D127—42no—
97C2×D127—41no—
98C2×D127—42no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—82no—
103C4×C127#9911yescharacteristic
104C4×C127#9911yescharacteristic
105Dic127—41no—
106Dic127—41no—
107Dic127—41no—
108C2×C127—21no—
109C2×C127—41no—
110C2×C127—41no—
111C2×C127—41no—
112C2×C127C2×#9911yescharacteristic
113D127—42no—
114#2120—1271nomaximal, Sylow
115C127#212011yescharacteristic, complemented, Sylow
116#525—1271no—
117#413—1271no—
118#410D12711yescharacteristic, complemented
119#352—1272no—
120AΣL(1,𝔽16)—1272no—
121C2×#41C2×D12711yescharacteristic
122C23×C4—1271no—
123C2×M16—2541no—
124#123—1271no—
125#123—1271no—
126C22⋊Q8—2541no—
127#121—2541no—
128#120—1271no—
129#120—1271no—
130C4×D4—2541no—
131C4×D4—2541no—
132#117—21no—
133#117—1271no—
134#101—2541no—
135#100—1272no—
136#99—21no—
137#99C2×D12712yes—
138C24—1271no—
139#41—41no—
140#41—41no—
141C2×Q8D50811yescharacteristic
142C2×D4—2541no—
143C2×D4—2541no—
144C2×D4C4×D12711yescharacteristic
145C2×D4C22⋊D12711yescharacteristic
146C2×D4C22×D12711yescharacteristic
147C22×C4—1271no—
148C22×C4—5081no—
149C22×C4—5081no—
150C22×C4—2541no—
151C22×C4—2541no—
152C22×C4C4×D12711yescharacteristic
153C22×C4C22⋊D12711yescharacteristic
154C22×C4D50811yescharacteristic
155M16—5081no—
156M16—2542no—
157C2×C8—5081no—
158C4⋊C4—41no—
159C4⋊C4—2541no—
160C4⋊C4—2541no—
161C4⋊C4—2541no—
162C22⋊C4—22no—
163C22⋊C4—1271no—
164C22⋊C4—1271no—
165C22⋊C4—2541no—
166C22⋊C4—2541no—
167C22⋊C4—2541no—
168C22⋊C4—2541no—
169C42—41no—
170C42—2541no—
171C23—5081no—
172C23—5082no—
173C23—2541no—
174C23(C2×D127)⋊C411yescharacteristic
175C23C2×C4×D12711yescharacteristic
176C23C2×C22⋊D12711yescharacteristic
177Q8—41no—
178D4—41no—
179D4—41no—
180D4—41no—
181D4—5082no—
182D4—5082no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—42no—
190C2×C4—5082no—
191C2×C4—5081no—
192C2×C4—5081no—
193C2×C4—5081no—
194C2×C4—2541no—
195C2×C4—2541no—
196C2×C4—2541no—
197C2×C4(C2×D127)⋊C411yescharacteristic
198C2×C4(C2×D127)⋊C411yescharacteristic
199C2×C4(C2×D127)⋊C411yescharacteristic
200C2×C4C2×D50811yescharacteristic
201C8—5082no—
202C22—21no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—5081no—
209C22—5082no—
210C22—5082no—
211C22—5081no—
212C22—2542no—
213C22C2×(C2×D127)⋊C411yescharacteristic
214C4—21no—
215C4—21no—
216C4—21no—
217C4—82no—
218C4—5081no—
219C4—5081no—
220C4—5081no—
221C4#1686112311yescharacteristic
222C4#1686112311yescharacteristic
223C2—21no—
224C2—41no—
225C2—41no—
226C2—41no—
227C2—5082no—
228C2C2×#1686112311yescharacteristic
subgroupG/H
centerC4#16861123
commutatorC23×C127C22×C4
FrattiniC2×D4C22×D127
Fitting#410×C127C2
socleC2×C127C2×#99
prime radical#22550383C2
cosocleC2×D4×C127C23
hypercenter#410D127
hypocenterC127#2120
LCSG ▷ C23×C127 ▷ C22×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C4 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549860C2semidirect
3. #22549665C2semidirect
4. #22549662C2semidirect
5. #410×C127C2semidirect
6. C22×C4×D127C4semidirect
8. #16861190C4semidirect
12. #16861177C4semidirect
16. #16861168C4semidirect
115. C127#2120semidirect
118. #410D127semidirect

Maximal subgroups

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

subgroupindex
#225503832
#225498602
#225496652
#225496622
#410×C1272
#2120127