Finite Group Databasebeta

← #30639174

Subgroups — #30639174

Order 16256.

442 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 105 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
1C22×C4×Dic127C211yesmaximal, characteristic, complemented
2C2×#16861136C212yesmaximal, complemented
3#22549840C211yesmaximal, characteristic, complemented
4#22549635C212yesmaximal, complemented
5#525×C127C211yesmaximal, characteristic
6C23×Dic127—21no—
7C2×#14523232C2212yes—
8C2×C4×Dic127—24no—
9C2×C4×Dic127—21no—
10C2×C4×Dic127C2212yes—
11#16861136—24no—
12C22⋊(C127⋊C8)—22no—
13C2×#41×C127C2211yescharacteristic
14C23×C4×C127C2211yescharacteristic
15#123×C127—21no—
16C22⋊Q8×C127C412yescomplemented
17#121×C127C412yescomplemented
18#120×C127—21no—
19C4×D4×C127—22no—
20#117×C127C2211yescharacteristic
21C22×Dic127—28no—
22C22×Dic127—24no—
23C22×Dic127—22no—
24#14523232—22no—
25#14523232—22no—
26C2×C127⋊C8—22no—
27C4×Dic127—24no—
28C4×Dic127—22no—
29C4×Dic127D414yescomplemented
30C24×C127D411yescharacteristic
31#41×C127—22no—
32#41×C127—22no—
33C2×Q8×C127C2×C411yescharacteristic, complemented
34C2×D4×C127—22no—
35C2×D4×C127—21no—
36C2×D4×C127C2×C411yescharacteristic, complemented
37C22×C4×C127—22no—
38C22×C4×C127—22no—
39C22×C4×C127—21no—
40C22×C4×C127C2×C412yes—
41C22×C4×C127D411yescharacteristic
42C22×C4×C127D412yes—
43C22×C4×C127C2311yescharacteristic
44C4⋊C4×C127—22no—
45C4⋊C4×C127C2×C412yescomplemented
46C22⋊C4×C127—22no—
47C22⋊C4×C127—21no—
48C22⋊C4×C127—22no—
49C42×C127—21no—
50C2×Dic127—216no—
51C2×Dic127—28no—
52C2×Dic127—24no—
53C127⋊C8—42no—
54C23×C127—24no—
55C23×C127—21no—
56C23×C127—21no—
57C23×C127—21no—
58C23×C127C22⋊C412yes—
59C23×C127C2×D411yescharacteristic
60Q8×C127—22no—
61D4×C127—22no—
62D4×C127—44no—
63D4×C127—41no—
64D4×C127—41no—
65C2×C4×C127—21no—
66C2×C4×C127—21no—
67C2×C4×C127—24no—
68C2×C4×C127—24no—
69C2×C4×C127—22no—
70C2×C4×C127—21no—
71C2×C4×C127—42no—
72C2×C4×C127C22⋊C412yes—
73C2×C4×C127C22×C411yescharacteristic
74C2×C4×C127C2×D412yes—
75C2×C4×C127C2×D411yescharacteristic
76C2×C4×C127#4112yes—
77C22×C127—24no—
78C22×C127—24no—
79C22×C127—22no—
80C22×C127—22no—
81C22×C127—22no—
82C22×C127—41no—
83C22×C127—42no—
84C22×C127#10414yescomplemented
85C22×C127C2×C22⋊C411yescharacteristic
86C22×C127#12011yescharacteristic
87C22×C127#12311yescharacteristic
88C4×C127—22no—
89C4×C127—41no—
90C4×C127—42no—
91C4×C127C4×D412yes—
92C4×C127#12112yes—
93Dic127—28no—
94C2×C127—21no—
95C2×C127—24no—
96C2×C127—21no—
97C2×C127—81no—
98C2×C127#38611yescharacteristic
99C2×C127C2×#10412yes—
100#1896—1271nomaximal, Sylow
101C127#189611yescharacteristic, complemented, Sylow
102#525D12711yescharacteristic
103C22×C42—1271no—
104C2×#104—1272no—
105#407—1271no—
106#339—1272no—
107C2×#41C2×D12711yescharacteristic
108C23×C4—2541no—
109C23×C4C2×D12711yescharacteristic
110C2×M16—1272no—
111#123—21no—
112C22⋊Q8Dic12712yescomplemented
113#121Dic12712yescomplemented
114#120—21no—
115C4×D4—22no—
116#117C2×D12711yescharacteristic
117C2×C42—1272no—
118C2×C42—2544no—
119C2×C42—2541no—
120#104—2544no—
121C22⋊C8—2542no—
122C24C22⋊D12711yescharacteristic
123#41—22no—
124#41—22no—
125C2×Q8C2×Dic12711yescharacteristic, complemented
126C2×D4—22no—
127C2×D4—21no—
128C2×D4C2×Dic12711yescharacteristic, complemented
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—21no—
132C22×C4—2544no—
133C22×C4—2548no—
134C22×C4—2542no—
135C22×C4C2×Dic12712yes—
136C22×C4C22⋊D12711yescharacteristic
137C22×C4C22⋊D12712yes—
138C22×C4C22×D12711yescharacteristic
139M16—2542no—
140M16—2542no—
141C2×C8—2542no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic12712yescomplemented
144C22⋊C4—22no—
145C22⋊C4—21no—
146C22⋊C4—22no—
147C42—21no—
148C42—1274no—
149C42—2544no—
150C42—2542no—
151C23—21no—
152C23—24no—
153C23—21no—
154C23—21no—
155C23C22⋊Dic12712yes—
156C23C2×C22⋊D12711yescharacteristic
157Q8—22no—
158D4—22no—
159D4—41no—
160D4—44no—
161D4—41no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—22no—
167C2×C4—21no—
168C2×C4—42no—
169C2×C4—25416no—
170C2×C4—2548no—
171C2×C4—2544no—
172C2×C4C22⋊Dic12712yes—
173C2×C4C22×Dic12711yescharacteristic
174C2×C4D4×D12712yes—
175C2×C4C2×C22⋊D12711yescharacteristic
176C2×C4#1452324712yes—
177C8—5082no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—42no—
184C22—41no—
185C22#1686113614yescomplemented
186C22C2×C22⋊Dic12711yescharacteristic
187C22#1686117711yescharacteristic
188C22#1686119011yescharacteristic
189C4—22no—
190C4—42no—
191C4—41no—
192C4—2548no—
193C4D4×Dic12712yes—
194C4#1686118112yes—
195C2—21no—
196C2—21no—
197C2—24no—
198C2—81no—
199C2#2254975611yescharacteristic
200C2C2×#1686113612yes—
subgroupG/H
centerC2×C4C2×C22⋊D127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
Fitting, prime radical#525×C127C2
socleC22×C127C2×C22⋊C4
cosocleC22×C4×C127C23
hypercenter#525D127
hypocenterC127#1896
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C22×C4×Dic127C2semidirect
2. C2×#16861136C2semidirect
3. #22549840C2semidirect
4. #22549635C2semidirect
16. C22⋊Q8×C127C4semidirect
17. #121×C127C4semidirect
29. C4×Dic127D4semidirect
33. C2×Q8×C127C2×C4semidirect
36. C2×D4×C127C2×C4semidirect
45. C4⋊C4×C127C2×C4semidirect
84. C22×C127#104semidirect
101. C127#1896semidirect
112. C22⋊Q8Dic127semidirect
113. #121Dic127semidirect
125. C2×Q8C2×Dic127semidirect
128. C2×D4C2×Dic127semidirect
143. C4⋊C4C2×Dic127semidirect
185. C22#16861136semidirect

Maximal subgroups

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

subgroupindex
C22×C4×Dic1272
C2×#168611362
#225498402
#225496352
#525×C1272
#1896127