Finite Group Databasebeta

← C2×#22550180 #30648504

Subgroups — C2×#22550180 #30648504

Order 16256.

384 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 157 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×C4×Dic254C211yesmaximal, characteristic
2#22550180C218yesmaximal, direct product factor
3C2×#16861142C211yesmaximal, characteristic
4C2×#16861140C211yesmaximal, characteristic
5C2×C4⋊(C127⋊C8)C211yesmaximal, characteristic
6C2×#16861132C211yesmaximal, characteristic
7C2×#16861132C211yesmaximal, characteristic
8C2×#125×C127C211yesmaximal, characteristic
9C22×Dic254C2211yescharacteristic
10C22×C127⋊C8C2211yescharacteristic
11C22×C127⋊C8C2211yescharacteristic
12C4×Dic254—22no—
13C4×Dic254C2214yes—
14C2×C4⋊Dic127C2211yescharacteristic
15C2×Dic127⋊C4—21no—
16C2×C4×Dic127—21no—
17#16861142C2214yes—
18#16861140C2214yes—
19C4⋊(C127⋊C8)C2214yes—
20#16861132C2214yes—
21#16861132C2214yes—
22#125×C127—22no—
23#125×C127C2214yes—
24C2×C4⋊C4×C127—21no—
25C2×C4⋊C4×C127—21no—
26C2×C4⋊C4×C127C2211yescharacteristic
27C2×C4⋊C4×C127C2211yescharacteristic
28C2×C42×C127C2211yescharacteristic
29C2×Dic254—21no—
30C2×Dic254—22no—
31C2×Dic254Q814yes—
32C2×Dic254C2312yes—
33C22×Dic127—21no—
34C22×Dic127—21no—
35C2×C127⋊C8—22no—
36C2×C127⋊C8—22no—
37C2×C127⋊C8C2312yes—
38C2×C127⋊C8C2312yes—
39C4⋊Dic127—21no—
40C4⋊Dic127C2312yes—
41Dic127⋊C4—24no—
42C4×Dic127—24no—
43C22×C4×C127—21no—
44C22×C4×C127—21no—
45C22×C4×C127D411yescharacteristic
46C22×C4×C127D411yescharacteristic
47C22×C4×C127C2311yescharacteristic
48C4⋊C4×C127—24no—
49C4⋊C4×C127—24no—
50C4⋊C4×C127—21no—
51C4⋊C4×C127—21no—
52C4⋊C4×C127C2312yes—
53C4⋊C4×C127C2312yes—
54C42×C127D412yes—
55C42×C127C2312yes—
56Dic254—24no—
57Dic254—22no—
58Dic254C2×Q814yes—
59C2×Dic127—24no—
60C2×Dic127—22no—
61C2×Dic127—22no—
62C2×Dic127—42no—
63C127⋊C8—22no—
64C127⋊C8—22no—
65C23×C127C2×D411yescharacteristic
66C2×C4×C127—22no—
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127—42no—
71C2×C4×C127—42no—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D412yes—
74C2×C4×C127C2×D411yescharacteristic
75C2×C4×C127C2×Q812yes—
76C2×C4×C127#4112yes—
77C2×C4×C127C2411yescharacteristic
78C22×C127C22⋊Q812yes—
79C22×C127#13512yes—
80C22×C127#13712yes—
81C22×C127C22×D411yescharacteristic
82C4×C127—22no—
83C4×C127—22no—
84C4×C127—42no—
85C4×C127—42no—
86C4×C127C22⋊Q812yes—
87C4×C127C22×Q811yescharacteristic
88C4×C127C2×#4111yescharacteristic
89Dic127—24no—
90Dic127—42no—
91C2×C127#47914yescomplemented
92C2×C127C2×C22⋊Q811yescharacteristic
93C2×C127C2×#13511yescharacteristic
94C2×C127C2×#13711yescharacteristic
95C2×#479—1271nomaximal, Sylow
96C127C2×#47911yescharacteristic, complemented, Sylow
97C2×#125D12711yescharacteristic
98C2×C4×Q8—1271no—
99#479—1278no—
100C2×#107—1271no—
101C2×#106—1271no—
102C2×C4⋊C8—1271no—
103C2×#103—1271no—
104C2×#103—1271no—
105C22×Q8—1271no—
106C22×C8—1271no—
107C22×C8—1271no—
108#125—22no—
109#125C2×D12714yes—
110C4×Q8—1274no—
111C4×Q8—2542no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—1271no—
115C2×C4⋊C4—2541no—
116C2×C4⋊C4C2×D12711yescharacteristic
117C2×C4⋊C4C2×D12711yescharacteristic
118C2×C42—2541no—
119C2×C42C2×D12711yescharacteristic
120#107—1274no—
121#106—1274no—
122C4⋊C8—1274no—
123#103—1274no—
124#103—1274no—
125C2×Q8—1274no—
126C2×Q8—1272no—
127C2×Q8—2541no—
128C2×Q8—2542no—
129C22×C4—21no—
130C22×C4—21no—
131C22×C4—2541no—
132C22×C4—2541no—
133C22×C4C22⋊D12711yescharacteristic
134C22×C4C22⋊D12711yescharacteristic
135C22×C4C22×D12711yescharacteristic
136C2×C8—1272no—
137C2×C8—1272no—
138C2×C8—2542no—
139C2×C8—2542no—
140C4⋊C4—24no—
141C4⋊C4—24no—
142C4⋊C4—21no—
143C4⋊C4—21no—
144C4⋊C4—1272no—
145C4⋊C4—2544no—
146C4⋊C4—2541no—
147C4⋊C4C22×D12712yes—
148C4⋊C4C22×D12712yes—
149C42—2544no—
150C42C22⋊D12712yes—
151C42C22×D12712yes—
152C23C2×C22⋊D12711yescharacteristic
153Q8—1274no—
154Q8—2544no—
155Q8—2542no—
156C2×C4—22no—
157C2×C4—22no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—42no—
161C2×C4—42no—
162C2×C4—5082no—
163C2×C4—2542no—
164C2×C4—2544no—
165C2×C4—2542no—
166C2×C4C2×C22⋊D12712yes—
167C2×C4C2×C22⋊D12712yes—
168C2×C4C2×C22⋊D12711yescharacteristic
169C2×C4Q8×D12712yes—
170C2×C4#1452324912yes—
171C2×C4C23×D12711yescharacteristic
172C8—2542no—
173C8—2542no—
174C22#1686118512yes—
175C22#1686122912yes—
176C22#1686123912yes—
177C22C22×C22⋊D12711yescharacteristic
178C4—22no—
179C4—22no—
180C4—42no—
181C4—42no—
182C4—5082no—
183C4—2544no—
184C4#1686118512yes—
185C4C2×Q8×D12711yescharacteristic
186C4C2×#1452324911yescharacteristic
187C2#2255018014yesdirect product factor
188C2C2×#1686118511yescharacteristic
189C2C2×#1686122911yescharacteristic
190C2C2×#1686123911yescharacteristic
subgroupG/H
centerC23C2×C22⋊D127
commutator, cosocleC2×C4×C127C24
FrattiniC2×C4C23×D127
FittingC2×#125×C127C2
socle, prime radicalC23×C127C2×D4
hypercenterC2×#125D127
hypocenterC127C2×#479
LCSG ▷ C2×C4×C127 ▷ C2×C127 ▷ C127
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #22550180C2direct (N × H)
91. C2×C127#479semidirect
96. C127C2×#479semidirect
187. C2#22550180direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×Dic2542
#225501802
C2×#168611422
C2×#168611402
C2×C4⋊(C127⋊C8)2
C2×#168611322
C2×#125×C1272
C2×#479127