Finite Group Databasebeta

← #30638700

Subgroups — #30638700

Order 16256.

588 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 588 conjugacy classes of subgroups — too many to draw. Only its 121 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#22549893C212yesmaximal, complemented
2#22549881C212yesmaximal, complemented
3#22549860C211yesmaximal, characteristic, complemented
4#22549850C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C127C211yesmaximal, characteristic
6C2×#14523232C2212yescomplemented
7#16861168C2212yescomplemented
8#16861136—24no—
9#16861130—24no—
10#16861128—21no—
11#16861126—21no—
12#16861124—22no—
13H(5,𝔽2)×C127—24no—
14H(5,𝔽2)×C127—24no—
15C2×#41×C127—21no—
16C2×#41×C127C412yescomplemented
17C2×#41×C127C2211yescharacteristic
18C2×#41×C127C2211yescharacteristic
19C22×D4×C127—22no—
20C22×D4×C127—21no—
21C22×D4×C127C412yescomplemented
22C22×D4×C127C2211yescharacteristic
23#14523232—22no—
24#14523232—22no—
25C2×C127⋊C8—22no—
26C4⋊Dic127—22no—
27C22⋊Dic127—22no—
28C4×Dic127—22no—
29C24×C127—22no—
30C24×C127—21no—
31#41×C127—28no—
32#41×C127—24no—
33#41×C127—24no—
34#41×C127—22no—
35#41×C127—22no—
36#41×C127—22no—
37#41×C127D414yes—
38C2×Q8×C127C2×C411yescharacteristic
39C2×Q8×C127D411yescharacteristic
40C2×D4×C127—216no—
41C2×D4×C127—28no—
42C2×D4×C127—22no—
43C2×D4×C127—21no—
44C2×D4×C127—24no—
45C2×D4×C127—24no—
46C2×D4×C127—22no—
47C2×D4×C127—21no—
48C2×D4×C127—22no—
49C2×D4×C127C2×C411yescharacteristic
50C2×D4×C127C2×C412yes—
51C2×D4×C127D412yes—
52C2×D4×C127D411yescharacteristic
53C2×D4×C127D414yes—
54C22×C4×C127—22no—
55C22×C4×C127—21no—
56C22×C4×C127C2×C412yes—
57C22×C4×C127C2311yescharacteristic
58C2×Dic127—42no—
59C127⋊C8—42no—
60C23×C127—28no—
61C23×C127—22no—
62C23×C127—21no—
63C23×C127—24no—
64C23×C127—82no—
65C23×C127—84no—
66C23×C127C22⋊C412yes—
67C23×C127C2×D411yescharacteristic
68Q8×C127—21no—
69Q8×C127—22no—
70Q8×C127C22⋊C412yes—
71D4×C127—28no—
72D4×C127—28no—
73D4×C127—24no—
74D4×C127—24no—
75D4×C127—22no—
76D4×C127—21no—
77D4×C127—22no—
78D4×C127—24no—
79D4×C127C22⋊C412yes—
80D4×C127C22⋊C414yes—
81C2×C4×C127—28no—
82C2×C4×C127—21no—
83C2×C4×C127—22no—
84C2×C4×C127—22no—
85C2×C4×C127—24no—
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—24no—
93C22×C127—22no—
94C22×C127—41no—
95C22×C127—42no—
96C22×C127—44no—
97C22×C127—42no—
98C22×C127—84no—
99C22×C127—88no—
100C22×C127C2×C22⋊C411yescharacteristic
101C22×C127#12012yes—
102C4×C127—22no—
103C4×C127—22no—
104C4×C127C2×C22⋊C411yescharacteristic
105C4×C127C2×C22⋊C411yescharacteristic
106C4×C127#12012yes—
107Dic127—82no—
108C2×C127—22no—
109C2×C127—21no—
110C2×C127—44no—
111C2×C127—42no—
112C2×C127—42no—
113C2×C127#37911yescharacteristic
114#1795—1271nomaximal, Sylow
115C127#179511yescharacteristic, complemented, Sylow
116C2×H(5,𝔽2)D12711yescharacteristic
117#421—1272no—
118#418—1272no—
119#413—1271no—
120#410—1271no—
121H(5,𝔽2)—24no—
122H(5,𝔽2)—24no—
123C2×#41—21no—
124C2×#41Dic12712yescomplemented
125C2×#41C2×D12711yescharacteristic
126C2×#41C2×D12711yescharacteristic
127C22×D4—22no—
128C22×D4—21no—
129C22×D4Dic12712yescomplemented
130C22×D4C2×D12711yescharacteristic
131C2×M16—1272no—
132#117—1272no—
133#104—2544no—
134#102—2544no—
135#101—2541no—
136#100—2541no—
137#99—2542no—
138C24—22no—
139C24—21no—
140#41—28no—
141#41—24no—
142#41—24no—
143#41—22no—
144#41—22no—
145#41—22no—
146#41C22⋊D12714yes—
147C2×Q8C2×Dic12711yescharacteristic
148C2×Q8C22⋊D12711yescharacteristic
149C2×D4—216no—
150C2×D4—28no—
151C2×D4—22no—
152C2×D4—21no—
153C2×D4—24no—
154C2×D4—24no—
155C2×D4—22no—
156C2×D4—21no—
157C2×D4—22no—
158C2×D4C2×Dic12711yescharacteristic
159C2×D4C2×Dic12712yes—
160C2×D4C22⋊D12712yes—
161C2×D4C22⋊D12711yescharacteristic
162C2×D4C22⋊D12714yes—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4C2×Dic12712yes—
166C22×C4C22×D12711yescharacteristic
167M16—2542no—
168M16—2542no—
169C2×C8—2542no—
170C4⋊C4—2542no—
171C22⋊C4—2542no—
172C42—2542no—
173C23—28no—
174C23—22no—
175C23—21no—
176C23—24no—
177C23—82no—
178C23—84no—
179C23C22⋊Dic12712yes—
180C23C2×C22⋊D12711yescharacteristic
181Q8—21no—
182Q8—22no—
183Q8C22⋊Dic12712yes—
184D4—28no—
185D4—28no—
186D4—24no—
187D4—24no—
188D4—22no—
189D4—21no—
190D4—22no—
191D4—24no—
192D4C22⋊Dic12712yes—
193D4C22⋊Dic12714yes—
194C2×C4—28no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—24no—
199C2×C4—5082no—
200C2×C4C22⋊Dic12712yes—
201C2×C4C22×Dic12711yescharacteristic
202C2×C4C2×C22⋊D12712yes—
203C2×C4C2×C22⋊D12712yes—
204C2×C4C2×C22⋊D12711yescharacteristic
205C8—5082no—
206C22—22no—
207C22—24no—
208C22—22no—
209C22—41no—
210C22—42no—
211C22—44no—
212C22—42no—
213C22—84no—
214C22—88no—
215C22C2×C22⋊Dic12711yescharacteristic
216C22#1686117812yes—
217C4—22no—
218C4—22no—
219C4—10162no—
220C4C2×C22⋊Dic12711yescharacteristic
221C4C2×C22⋊Dic12711yescharacteristic
222C4#1686117812yes—
223C2—22no—
224C2—21no—
225C2—44no—
226C2—42no—
227C2—42no—
228C2#2254973211yescharacteristic
subgroupG/H
centerC2#22549732
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radicalC2×H(5,𝔽2)×C127C2
socleC2×C127#379
cosocleC22×C4×C127C23
hypercenterC2×H(5,𝔽2)D127
hypocenterC127#1795
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. #22549893C2semidirect
2. #22549881C2semidirect
3. #22549860C2semidirect
4. #22549850C2semidirect
6. C2×#14523232C22semidirect
7. #16861168C22semidirect
16. C2×#41×C127C4semidirect
21. C22×D4×C127C4semidirect
115. C127#1795semidirect
124. C2×#41Dic127semidirect
129. C22×D4Dic127semidirect

Maximal subgroups

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

subgroupindex
#225498932
#225498812
#225498602
#225498502
C2×H(5,𝔽2)×C1272
#1795127