Finite Group Databasebeta

← #30638698

Subgroups — #30638698

Order 16256.

588 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 97 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×#16861255C211yesmaximal, characteristic
2#22549894C212yesmaximal, complemented
3#22549880C212yesmaximal, complemented
4#22549860C211yesmaximal, characteristic, complemented
5#410×C127C211yesmaximal, characteristic, complemented
6#16861255—24no—
7#16861255—24no—
8C2×#14523249—21no—
9C2×#14523248C412yescomplemented
10C2×#14523248C2211yescharacteristic
11C22×D508C412yescomplemented
12C22×D508C2211yescharacteristic
13C2×D4×D127—22no—
14C2×D4×D127—21no—
15C2×#14523232C2212yescomplemented
16#16861137—24no—
17#16861129—24no—
18#16861128—21no—
19#16861126—21no—
20C2×#41×C127C2211yescharacteristic
21#117×C127C2212yescomplemented
22#99×C127—22no—
23C23×D127—22no—
24C23×D127—21no—
25#14523249—28no—
26#14523248—24no—
27#14523248—24no—
28#14523248—22no—
29#14523248D414yes—
30C2×Dic254C2×C411yescharacteristic
31C2×D508—24no—
32C2×D508—24no—
33C2×D508—22no—
34C2×D508—21no—
35C2×D508—22no—
36C2×D508C2×C411yescharacteristic
37C2×D508C2×C412yes—
38C2×D508D414yes—
39C2×C22⋊D127—22no—
40C2×C22⋊D127—21no—
41D4×D127—216no—
42D4×D127—28no—
43C2×C4×D127—22no—
44C2×C4×D127—21no—
45#14523232—22no—
46#14523232—22no—
47C2×C127⋊C8—22no—
48#41×C127—22no—
49#41×C127—22no—
50C2×Q8×C127D411yescharacteristic
51C2×D4×C127D412yes—
52C2×D4×C127D411yescharacteristic
53C22×C4×C127C2×C412yes—
54C22×C4×C127C2311yescharacteristic
55C4⋊C4×C127—22no—
56C22⋊C4×C127—22no—
57C42×C127—22no—
58C22×D127—22no—
59C22×D127—21no—
60C22×D127—28no—
61C22×D127—24no—
62C22×D127—82no—
63C22×D127—84no—
64Dic254—21no—
65Dic254C22⋊C412yes—
66D508—28no—
67D508—22no—
68D508—21no—
69D508—24no—
70D508C22⋊C412yes—
71D508C22⋊C414yes—
72C22⋊D127—28no—
73C22⋊D127—24no—
74C2×Dic127—21no—
75C4×D127—28no—
76C4×D127—24no—
77C127⋊C8—42no—
78C23×C127C22⋊C412yes—
79C23×C127C2×D411yescharacteristic
80Q8×C127—22no—
81D4×C127—24no—
82D4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—42no—
86C2×C4×C127C22⋊C412yes—
87C2×C4×C127C22×C411yescharacteristic
88C2×C4×C127C2×D412yes—
89C2×C4×C127C2×D412yes—
90C2×C4×C127C2×D411yescharacteristic
91C22×C127—22no—
92C22×C127—41no—
93C22×C127—42no—
94C22×C127C2×C22⋊C411yescharacteristic
95C22×C127#12012yes—
96C2×D127—24no—
97C2×D127—22no—
98C2×D127—44no—
99C2×D127—42no—
100C2×D127—84no—
101C2×D127—88no—
102C4×C127—22no—
103C4×C127—82no—
104C4×C127C2×C22⋊C411yescharacteristic
105C4×C127C2×C22⋊C411yescharacteristic
106C4×C127#12012yes—
107Dic127—22no—
108C2×C127—22no—
109C2×C127—21no—
110C2×C127—42no—
111C2×C127#37911yescharacteristic
112D127—44no—
113D127—42no—
114#1795—1271nomaximal, Sylow
115C127#179511yescharacteristic, complemented, Sylow
116C2×H(5,𝔽2)—1271no—
117#421—1272no—
118#418—1272no—
119#413—1271no—
120#410D12711yescharacteristic, complemented
121H(5,𝔽2)—2544no—
122H(5,𝔽2)—2544no—
123C2×#41—1272no—
124C2×#41—1271no—
125C2×#41—2541no—
126C2×#41C2×D12711yescharacteristic
127C22×D4—1272no—
128C22×D4—1271no—
129C22×D4—2542no—
130C22×D4—2541no—
131C2×M16—1272no—
132#117C2×D12712yescomplemented
133#104—2544no—
134#102—2544no—
135#101—2541no—
136#100—2541no—
137#99—22no—
138C24—2542no—
139C24—2541no—
140#41—22no—
141#41—22no—
142#41—1274no—
143#41—2544no—
144#41—2544no—
145#41—2548no—
146#41—2542no—
147C2×Q8—1271no—
148C2×Q8D50811yescharacteristic
149C2×D4—1272no—
150C2×D4—1271no—
151C2×D4—1274no—
152C2×D4—2544no—
153C2×D4—2544no—
154C2×D4—25416no—
155C2×D4—2548no—
156C2×D4—2542no—
157C2×D4—2541no—
158C2×D4—2542no—
159C2×D4—2542no—
160C2×D4—2541no—
161C2×D4C22⋊D12712yes—
162C2×D4D50811yescharacteristic
163C22×C4—2542no—
164C22×C4—2541no—
165C22×C4C4×D12712yes—
166C22×C4C22×D12711yescharacteristic
167M16—2542no—
168M16—2542no—
169C2×C8—2542no—
170C4⋊C4—22no—
171C22⋊C4—22no—
172C42—22no—
173C23—10162no—
174C23—10164no—
175C23—2548no—
176C23—2544no—
177C23—2542no—
178C23—2541no—
179C23(C2×D127)⋊C412yes—
180C23C2×C22⋊D12711yescharacteristic
181Q8—22no—
182Q8—1272no—
183Q8—2541no—
184D4—24no—
185D4—22no—
186D4—1274no—
187D4—1272no—
188D4—2548no—
189D4—2548no—
190D4—2544no—
191D4—2541no—
192D4—2542no—
193D4—2544no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—42no—
197C2×C4—2548no—
198C2×C4—2541no—
199C2×C4—2544no—
200C2×C4(C2×D127)⋊C412yes—
201C2×C4C2×C4×D12711yescharacteristic
202C2×C4D4×D12712yes—
203C2×C4D4×D12712yes—
204C2×C4C2×D50811yescharacteristic
205C8—5082no—
206C22—22no—
207C22—41no—
208C22—42no—
209C22—10164no—
210C22—10168no—
211C22—5084no—
212C22—5082no—
213C22—2544no—
214C22—2542no—
215C22C2×(C2×D127)⋊C411yescharacteristic
216C22#1686117712yes—
217C4—22no—
218C4—82no—
219C4—2542no—
220C4C22⋊C4×D12711yescharacteristic
221C4C22⋊C4×D12711yescharacteristic
222C4#1686117912yes—
223C2—22no—
224C2—21no—
225C2—42no—
226C2—5084no—
227C2—5082no—
228C2#2254973111yescharacteristic
subgroupG/H
centerC2#22549731
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting#410×C127C2
socleC2×C127#379
prime radicalC2×#16861255C2
cosocleC22×C4×C127C23
hypercenter#410D127
hypocenterC127#1795
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #22549894C2semidirect
3. #22549880C2semidirect
4. #22549860C2semidirect
5. #410×C127C2semidirect
9. C2×#14523248C4semidirect
11. C22×D508C4semidirect
15. C2×#14523232C22semidirect
21. #117×C127C22semidirect
115. C127#1795semidirect
120. #410D127semidirect
132. #117C2×D127semidirect

Maximal subgroups

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

subgroupindex
C2×#168612552
#225498942
#225498802
#225498602
#410×C1272
#1795127