Finite Group Databasebeta

← #30648523

Subgroups — #30648523

Order 16256.

440 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 145 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×D508C211yesmaximal, characteristic
2#22550173C214yesmaximal, complemented
3#22550163C212yesmaximal, complemented
4#22550153C212yesmaximal, complemented
5#22549919C212yesmaximal, complemented
6#22549900C211yesmaximal, characteristic, complemented
7#22549880C212yesmaximal
8#533×C127C211yesmaximal, characteristic, complemented
9C22×D508C2211yescharacteristic
10C22×C4×D127—21no—
11C2×#14523232C2212yes—
12C4×D508—21no—
13C4×D508—21no—
14C4×D508C2212yes—
15C4×D508C2212yes—
16C2×C4⋊Dic127C2211yescharacteristic
17C2×(C2×D127)⋊C4—21no—
18#16861142C2214yescomplemented
19#16861140C2214yescomplemented
20C4⋊(C127⋊C8)C2212yescomplemented
21C4⋊(C127⋊C8)C2212yescomplemented
22#16861129C2214yes—
23#16861129C2214yes—
24C4⋊Q8×C127C2211yescharacteristic, complemented
25C4⋊Q8×C127C2211yescharacteristic, complemented
26#125×C127C2212yescomplemented
27C22⋊Q8×C127—21no—
28C22⋊Q8×C127—21no—
29C4×Q8×C127—21no—
30C4×Q8×C127—21no—
31#117×C127C2212yes—
32C2×C42×C127C2211yescharacteristic
33C23×D127—21no—
34C2×D508—21no—
35C2×D508—22no—
36C2×D508Q814yescomplemented
37C2×D508C2312yes—
38C22×Dic127—21no—
39C2×C4×D127—22no—
40C2×C4×D127—22no—
41C2×C4×D127—44no—
42#14523232—24no—
43C2×C127⋊C8C2314yes—
44C4⋊Dic127—21no—
45C4⋊Dic127C2312yes—
46(C2×D127)⋊C4—22no—
47(C2×D127)⋊C4—22no—
48C2×Q8×C127—21no—
49C2×Q8×C127—21no—
50C22×C4×C127D411yescharacteristic
51C22×C4×C127D411yescharacteristic
52C22×C4×C127C2311yescharacteristic
53C4⋊C4×C127—22no—
54C4⋊C4×C127—22no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127—21no—
57C4⋊C4×C127C2312yes—
58C4⋊C4×C127C2312yes—
59C22⋊C4×C127—22no—
60C42×C127—22no—
61C42×C127D411yescharacteristic
62C42×C127D411yescharacteristic
63C42×C127C2311yescharacteristic
64C42×C127C2311yescharacteristic
65C22×D127—24no—
66C22×D127—22no—
67C22×D127—42no—
68C22×D127—42no—
69D508—24no—
70D508—22no—
71D508C2×Q814yes—
72C2×Dic127—22no—
73C2×Dic127—42no—
74C4×D127—44no—
75C4×D127—44no—
76C127⋊C8—24no—
77C23×C127C2×D411yescharacteristic
78Q8×C127—42no—
79Q8×C127—42no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—22no—
83C2×C4×C127—41no—
84C2×C4×C127C2×D411yescharacteristic
85C2×C4×C127C2×D411yescharacteristic
86C2×C4×C127C2×D411yescharacteristic
87C2×C4×C127C2×D411yescharacteristic
88C2×C4×C127C2×D411yescharacteristic
89C2×C4×C127C2×Q812yes—
90C2×C4×C127#4112yes—
91C2×C4×C127C2411yescharacteristic
92C22×C127—21no—
93C22×C127—21no—
94C22×C127C22⋊Q812yes—
95C22×C127C22×D411yescharacteristic
96C2×D127—24no—
97C2×D127—48no—
98C2×D127—42no—
99C2×D127—42no—
100C4×C127—21no—
101C4×C127—21no—
102C4×C127—21no—
103C4×C127—42no—
104C4×C127—42no—
105C4×C127C22⋊Q812yes—
106C4×C127AGL(1,ℤ/8)12yes—
107C4×C127C22×Q811yescharacteristic
108C4×C127C2×#4111yescharacteristic
109Dic127—42no—
110C2×C127—22no—
111C2×C127C2×C22⋊Q811yescharacteristic
112C2×C127C2×AGL(1,ℤ/8)11yescharacteristic
113C2×C127#57511yescharacteristic
114D127—44no—
115#3077—1271nomaximal, Sylow
116C127#307711yescharacteristic, complemented, Sylow
117#533D12711yescharacteristic, complemented
118C2×C4×D4—1271no—
119#478—1274no—
120#476—1272no—
121#474—1272no—
122#428—1272no—
123#423—1271no—
124#418—1272no—
125C22×D4—1271no—
126C23×C4—2541no—
127C2×M16—1272no—
128C4⋊Q8C2×D12711yescharacteristic, complemented
129C4⋊Q8C2×D12711yescharacteristic, complemented
130#125C2×D12712yescomplemented
131C22⋊Q8—21no—
132C22⋊Q8—21no—
133C4×Q8—21no—
134C4×Q8—21no—
135C4×D4—1272no—
136C4×D4—1272no—
137C4×D4—2541no—
138C4×D4—2541no—
139#117C2×D12712yes—
140C2×C4⋊C4—1271no—
141C2×C22⋊C4—2541no—
142C2×C42C2×D12711yescharacteristic
143#107—1274no—
144#106—1274no—
145C4⋊C8—1272no—
146C4⋊C8—1272no—
147#102—1274no—
148#102—1274no—
149C24—2541no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×D4—1274no—
153C2×D4—1272no—
154C2×D4—2541no—
155C2×D4—2542no—
156C22×C4—5084no—
157C22×C4—2541no—
158C22×C4—2542no—
159C22×C4—2542no—
160C22×C4C22⋊D12711yescharacteristic
161C22×C4C22⋊D12711yescharacteristic
162C22×C4C22×D12711yescharacteristic
163M16—2544no—
164C2×C8—1274no—
165C4⋊C4—22no—
166C4⋊C4—22no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—1272no—
170C4⋊C4—2541no—
171C4⋊C4C22×D12712yes—
172C4⋊C4C22×D12712yes—
173C22⋊C4—22no—
174C22⋊C4—2542no—
175C22⋊C4—2542no—
176C42—22no—
177C42C22⋊D12711yescharacteristic
178C42C22⋊D12711yescharacteristic
179C42C22×D12711yescharacteristic
180C42C22×D12711yescharacteristic
181C23—5082no—
182C23—5082no—
183C23—2544no—
184C23—2542no—
185C23C2×C22⋊D12711yescharacteristic
186Q8—42no—
187Q8—42no—
188D4—1274no—
189D4—2544no—
190D4—2542no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—41no—
195C2×C4—5084no—
196C2×C4—5082no—
197C2×C4—5084no—
198C2×C4—2542no—
199C2×C4C2×C22⋊D12711yescharacteristic
200C2×C4C2×C22⋊D12711yescharacteristic
201C2×C4C2×C22⋊D12711yescharacteristic
202C2×C4C2×C22⋊D12711yescharacteristic
203C2×C4C2×C22⋊D12711yescharacteristic
204C2×C4Q8×D12712yes—
205C2×C4#1452324912yes—
206C2×C4C23×D12711yescharacteristic
207C8—2544no—
208C22—21no—
209C22—21no—
210C22—5088no—
211C22—5082no—
212C22—5082no—
213C22—2544no—
214C22#1686118512yes—
215C22C22×C22⋊D12711yescharacteristic
216C4—21no—
217C4—21no—
218C4—21no—
219C4—42no—
220C4—42no—
221C4—5082no—
222C4#1686118512yes—
223C4#1686123412yes—
224C4C2×Q8×D12711yescharacteristic
225C4C2×#1452324911yescharacteristic
226C2—22no—
227C2—5084no—
228C2C2×#1686118511yescharacteristic
229C2C2×#1686123411yescharacteristic
230C2#2255067411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
Fitting#533×C127C2
socleC22×C127C22×D4
prime radicalC22×D508C22
hypercenter#533D127
hypocenterC127#3077
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C22 ◁ C2×C42 ◁ #533

Complements

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

NH ≅ G/Ntype
2. #22550173C2semidirect
3. #22550163C2semidirect
4. #22550153C2semidirect
5. #22549919C2semidirect
6. #22549900C2semidirect
8. #533×C127C2semidirect
18. #16861142C22semidirect
19. #16861140C22semidirect
20. C4⋊(C127⋊C8)C22semidirect
21. C4⋊(C127⋊C8)C22semidirect
24. C4⋊Q8×C127C22semidirect
25. C4⋊Q8×C127C22semidirect
26. #125×C127C22semidirect
36. C2×D508Q8semidirect
116. C127#3077semidirect
117. #533D127semidirect
128. C4⋊Q8C2×D127semidirect
129. C4⋊Q8C2×D127semidirect
130. #125C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D5082
#225501732
#225501632
#225501532
#225499192
#225499002
#225498802
#533×C1272
#3077127