Finite Group Databasebeta

← C2×#22549613 #30637211

Subgroups — C2×#22549613 #30637211

Order 16256.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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×C22⋊(C127⋊C8)C211yesmaximal, characteristic
2C2×C127⋊#95C211yesmaximal, characteristic
3#22549613C214yesmaximal, direct product factor
4C2×C22⋊Q8×C127C211yesmaximal, characteristic
5C23×Dic127—21no—
6C22×C127⋊C8—21no—
7C22⋊(C127⋊C8)—21no—
8C22⋊(C127⋊C8)C2212yes—
9C127⋊#95—21no—
10C127⋊#95C2212yes—
11C22×Q8×C127C411yescharacteristic, complemented
12C23×C4×C127C2211yescharacteristic
13C22⋊Q8×C127—21no—
14C22⋊Q8×C127—21no—
15C22⋊Q8×C127C412yescomplemented
16C22⋊Q8×C127C2212yes—
17C2×C4⋊C4×C127—21no—
18C2×C4⋊C4×C127C411yescharacteristic, complemented
19C2×C22⋊C4×C127—21no—
20C22×Dic127—24no—
21C22×Dic127—22no—
22C22×Dic127—42no—
23C22×Dic127—42no—
24C2×C127⋊C8—22no—
25C2×C127⋊C8—42no—
26C24×C127D411yescharacteristic
27C2×Q8×C127—21no—
28C2×Q8×C127—42no—
29C2×Q8×C127C2×C412yescomplemented
30C22×C4×C127—21no—
31C22×C4×C127—21no—
32C22×C4×C127—42no—
33C22×C4×C127C2×C411yescharacteristic
34C22×C4×C127C2×C411yescharacteristic
35C22×C4×C127D411yescharacteristic
36C22×C4×C127D411yescharacteristic
37C22×C4×C127D411yescharacteristic
38C22×C4×C127C2311yescharacteristic
39C4⋊C4×C127—22no—
40C4⋊C4×C127—21no—
41C4⋊C4×C127—22no—
42C4⋊C4×C127C2×C412yescomplemented
43C22⋊C4×C127—22no—
44C22⋊C4×C127—22no—
45C2×Dic127—24no—
46C2×Dic127—48no—
47C2×Dic127—42no—
48C2×Dic127—42no—
49C127⋊C8—42no—
50C23×C127—22no—
51C23×C127—22no—
52C23×C127C22⋊C411yescharacteristic
53C23×C127C22⋊C411yescharacteristic
54C23×C127SD1612yes—
55C23×C127Q1612yes—
56C23×C127C2×D411yescharacteristic
57Q8×C127—42no—
58Q8×C127—42no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—22no—
62C2×C4×C127—22no—
63C2×C4×C127—41no—
64C2×C4×C127—41no—
65C2×C4×C127—41no—
66C2×C4×C127—41no—
67C2×C4×C127—42no—
68C2×C4×C127—42no—
69C2×C4×C127C22⋊C411yescharacteristic
70C2×C4×C127C22⋊C411yescharacteristic
71C2×C4×C127C22×C411yescharacteristic
72C2×C4×C127C2×D411yescharacteristic
73C22×C127—24no—
74C22×C127—24no—
75C22×C127—21no—
76C22×C127—21no—
77C22×C127—21no—
78C22×C127—21no—
79C22×C127#9912yes—
80C22×C127#10312yes—
81C22×C127#10312yes—
82C22×C127#10412yes—
83C22×C127C2×C22⋊C411yescharacteristic
84C22×C127C2×SD1611yescharacteristic
85C22×C127C2×Q1611yescharacteristic
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—21no—
89C4×C127—21no—
90C4×C127—42no—
91C4×C127—42no—
92Dic127—44no—
93C2×C127—21no—
94C2×C127—22no—
95C2×C127—21no—
96C2×C127#32814yescomplemented
97C2×C127C2×#9911yescharacteristic
98C2×C127C2×#10311yescharacteristic
99C2×C127C2×#10411yescharacteristic
100C2×#328—1271nomaximal, Sylow
101C127C2×#32811yescharacteristic, complemented, Sylow
102C2×C22⋊Q8D12711yescharacteristic
103C2×C22⋊C8—1271no—
104C2×#95—1271no—
105#328—1274no—
106C22×Q8Dic12711yescharacteristic, complemented
107C23×C4—2541no—
108C23×C4C2×D12711yescharacteristic
109C22×C8—2541no—
110C22⋊Q8—21no—
111C22⋊Q8—21no—
112C22⋊Q8Dic12712yescomplemented
113C22⋊Q8C2×D12712yes—
114C2×C4⋊C4—21no—
115C2×C4⋊C4Dic12711yescharacteristic, complemented
116C2×C22⋊C4—21no—
117C22⋊C8—1272no—
118C22⋊C8—2541no—
119#95—1272no—
120#95—2541no—
121C24C22⋊D12711yescharacteristic
122C2×Q8—21no—
123C2×Q8—42no—
124C2×Q8C2×Dic12712yescomplemented
125C22×C4—21no—
126C22×C4—21no—
127C22×C4—42no—
128C22×C4—5082no—
129C22×C4—5082no—
130C22×C4—2544no—
131C22×C4—2542no—
132C22×C4C2×Dic12711yescharacteristic
133C22×C4C2×Dic12711yescharacteristic
134C22×C4C22⋊D12711yescharacteristic
135C22×C4C22⋊D12711yescharacteristic
136C22×C4C22⋊D12711yescharacteristic
137C22×C4C22×D12711yescharacteristic
138C2×C8—5082no—
139C2×C8—2542no—
140C4⋊C4—22no—
141C4⋊C4—21no—
142C4⋊C4—22no—
143C4⋊C4C2×Dic12712yescomplemented
144C22⋊C4—22no—
145C22⋊C4—22no—
146C23—22no—
147C23—22no—
148C23C22⋊Dic12711yescharacteristic
149C23C22⋊Dic12711yescharacteristic
150C23#1452323512yes—
151C23#1452323812yes—
152C23C2×C22⋊D12711yescharacteristic
153Q8—42no—
154Q8—42no—
155C2×C4—22no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—41no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—42no—
164C2×C4—42no—
165C2×C4—5082no—
166C2×C4—5088no—
167C2×C4—5082no—
168C2×C4—2544no—
169C2×C4C22⋊Dic12711yescharacteristic
170C2×C4C22⋊Dic12711yescharacteristic
171C2×C4C22×Dic12711yescharacteristic
172C2×C4C2×C22⋊D12711yescharacteristic
173C8—5082no—
174C22—21no—
175C22—21no—
176C22—24no—
177C22—24no—
178C22—21no—
179C22—21no—
180C22#1686112412yes—
181C22#1686113312yes—
182C22#1686113312yes—
183C22#1686113612yes—
184C22C2×C22⋊Dic12711yescharacteristic
185C22C2×#1452323511yescharacteristic
186C22C2×#1452323811yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—21no—
191C4—42no—
192C4—42no—
193C4—5084no—
194C2—21no—
195C2—21no—
196C2—22no—
197C2#2254961314yesdirect product factor
198C2C2×#1686112411yescharacteristic
199C2C2×#1686113311yescharacteristic
200C2C2×#1686113611yescharacteristic
subgroupG/H
centerC23C22⋊Dic127
commutatorC2×C4×C127C22×C4
FrattiniC22×C4C22×D127
FittingC2×C22⋊Q8×C127C2
socleC23×C127C22⋊C4
prime radicalC24×C127D4
cosocleC22×C4×C127C23
hypercenterC2×C22⋊Q8D127
hypocenterC127C2×#328
LCSG ▷ C2×C4×C127 ▷ C22×C127 ▷ C127
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #22549613C2direct (N × H)
11. C22×Q8×C127C4semidirect
15. C22⋊Q8×C127C4semidirect
18. C2×C4⋊C4×C127C4semidirect
29. C2×Q8×C127C2×C4semidirect
42. C4⋊C4×C127C2×C4semidirect
96. C2×C127#328semidirect
101. C127C2×#328semidirect
106. C22×Q8Dic127semidirect
112. C22⋊Q8Dic127semidirect
115. C2×C4⋊C4Dic127semidirect
124. C2×Q8C2×Dic127semidirect
143. C4⋊C4C2×Dic127semidirect
197. C2#22549613direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊(C127⋊C8)2
C2×C127⋊#952
#225496132
C2×C22⋊Q8×C1272
C2×#328127