Finite Group Databasebeta

← #30637653

Subgroups — #30637653

Order 16256.

316 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 57 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#22550026C212yesmaximal, complemented
2#22549785C211yesmaximal, characteristic
3#22549613C212yesmaximal, complemented
4#22549600C211yesmaximal, characteristic, complemented
5#543×C127C211yesmaximal, characteristic, complemented
6C22×D508C2211yescharacteristic
7C22×C22⋊D127—21no—
8C2×#14523235—22no—
9C2×(C2×D127)⋊C4—21no—
10#16861129—22no—
11C22⋊(C127⋊C8)C2212yes—
12C127⋊#95C2211yescharacteristic
13C22⋊Q8×C127—22no—
14C22⋊Q8×C127—21no—
15C22⋊Q8×C127—22no—
16C22⋊Q8×C127C2212yes—
17C2×C22⋊C4×C127—21no—
18C2×C22⋊C4×C127C2211yescharacteristic
19C23×D127—21no—
20C2×D508—22no—
21C2×D508—42no—
22C2×C22⋊D127—42no—
23C2×C22⋊D127—42no—
24C2×C22⋊D127—42no—
25C22×Dic127—21no—
26#14523235—44no—
27C2×C127⋊C8—22no—
28(C2×D127)⋊C4—42no—
29C24×C127D411yescharacteristic
30C2×Q8×C127—21no—
31C2×Q8×C127D412yescomplemented
32C22×C4×C127—21no—
33C22×C4×C127—21no—
34C22×C4×C127D411yescharacteristic
35C22×C4×C127C2311yescharacteristic
36C4⋊C4×C127—21no—
37C4⋊C4×C127—22no—
38C4⋊C4×C127—22no—
39C4⋊C4×C127D412yescomplemented
40C22⋊C4×C127—22no—
41C22⋊C4×C127—22no—
42C22⋊C4×C127—22no—
43C22×D127—22no—
44C22×D127—41no—
45C22×D127—41no—
46C22×D127—41no—
47C22×D127—41no—
48C22×D127—41no—
49C22×D127—41no—
50D508—24no—
51D508—42no—
52C22⋊D127—48no—
53C2×Dic127—22no—
54C2×Dic127—41no—
55C2×Dic127—41no—
56C127⋊C8—42no—
57C23×C127—21no—
58C23×C127—21no—
59C23×C127—41no—
60C23×C127—41no—
61C23×C127SD1612yescomplemented
62C23×C127C2×D411yescharacteristic
63Q8×C127—42no—
64Q8×C127—81no—
65C2×C4×C127—22no—
66C2×C4×C127—21no—
67C2×C4×C127—22no—
68C2×C4×C127—41no—
69C2×C4×C127—41no—
70C2×C4×C127—81no—
71C2×C4×C127—81no—
72C2×C4×C127C2×D412yes—
73C22×C127—21no—
74C22×C127—21no—
75C22×C127—22no—
76C22×C127—42no—
77C22×C127—41no—
78C22×C127—41no—
79C22×C127—81no—
80C22×C127#12011yescharacteristic
81C22×C127C2×SD1611yescharacteristic
82C22×C127#13711yescharacteristic
83C2×D127—42no—
84C2×D127—42no—
85C2×D127—42no—
86C2×D127—81no—
87C2×D127—81no—
88C2×D127—81no—
89C2×D127—81no—
90C4×C127—22no—
91C4×C127—41no—
92C4×C127—42no—
93C4×C127—42no—
94Dic127—42no—
95C2×C127—21no—
96C2×C127—21no—
97C2×C127—42no—
98C2×C127#44811yescharacteristic
99C2×C127#45311yescharacteristic
100C2×C127B(4,𝔽2)11yescharacteristic
101D127—82no—
102#1596—1271nomaximal, Sylow
103C127#159611yescharacteristic, complemented, Sylow
104#543D12711yescharacteristic, complemented
105#450—1272no—
106#392—1271no—
107#328—1272no—
108#323—1271no—
109C22×D4—1271no—
110C22×D4—2541no—
111C2×SD16—2542no—
112C22⋊Q8—22no—
113C22⋊Q8—21no—
114C22⋊Q8—22no—
115C22⋊Q8C2×D12712yes—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—2541no—
118C2×C22⋊C4C2×D12711yescharacteristic
119#102—2542no—
120C22⋊C8—1272no—
121#95—1271no—
122C24—2541no—
123C24C22⋊D12711yescharacteristic
124C2×Q8—21no—
125C2×Q8C22⋊D12712yescomplemented
126C2×D4—5082no—
127C2×D4—5082no—
128C2×D4—5082no—
129C2×D4—5082no—
130C2×D4—2542no—
131C22×C4—21no—
132C22×C4—21no—
133C22×C4—2541no—
134C22×C4C22⋊D12711yescharacteristic
135C22×C4C22×D12711yescharacteristic
136SD16—5084no—
137C2×C8—2542no—
138C4⋊C4—21no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4C22⋊D12712yescomplemented
142C22⋊C4—22no—
143C22⋊C4—22no—
144C22⋊C4—22no—
145C22⋊C4—5082no—
146C23—21no—
147C23—21no—
148C23—41no—
149C23—41no—
150C23—5081no—
151C23—5081no—
152C23—5081no—
153C23—5081no—
154C23—5081no—
155C23—5081no—
156C23—2542no—
157C23#1452323512yescomplemented
158C23C2×C22⋊D12711yescharacteristic
159Q8—42no—
160Q8—81no—
161D4—5082no—
162D4—5088no—
163D4—2544no—
164C2×C4—22no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—81no—
170C2×C4—81no—
171C2×C4—5081no—
172C2×C4—5081no—
173C2×C4—2542no—
174C2×C4C2×C22⋊D12712yes—
175C8—5082no—
176C22—21no—
177C22—21no—
178C22—22no—
179C22—42no—
180C22—41no—
181C22—41no—
182C22—81no—
183C22—10161no—
184C22—10161no—
185C22—10161no—
186C22—10161no—
187C22—5082no—
188C22—5082no—
189C22—5082no—
190C22#1686117811yescharacteristic
191C22C2×#1452323511yescharacteristic
192C22#1686124011yescharacteristic
193C4—22no—
194C4—41no—
195C4—42no—
196C4—42no—
197C4—5082no—
198C2—21no—
199C2—21no—
200C2—42no—
201C2—10162no—
202C2#2255001511yescharacteristic
203C2#2255004411yescharacteristic
204C2#2255006311yescharacteristic
subgroupG/H
centerC22#16861178
commutator, cosocleC22×C4×C127C23
FrattiniC22×C4C22×D127
Fitting#543×C127C2
socleC22×C127#120
prime radical#22549785C2
hypercenter#543D127
hypocenterC127#1596
LCSG ▷ C22×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C22 ◁ C22×C4 ◁ #543

Complements

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

NH ≅ G/Ntype
1. #22550026C2semidirect
3. #22549613C2semidirect
4. #22549600C2semidirect
5. #543×C127C2semidirect
31. C2×Q8×C127D4semidirect
39. C4⋊C4×C127D4semidirect
61. C23×C127SD16semidirect
103. C127#1596semidirect
104. #543D127semidirect
125. C2×Q8C22⋊D127semidirect
141. C4⋊C4C22⋊D127semidirect
157. C23#14523235semidirect

Maximal subgroups

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

subgroupindex
#225500262
#225497852
#225496132
#225496002
#543×C1272
#1596127