Finite Group Databasebeta

← C2×C127×#1780 #44584251

Subgroups — C2×C127×#1780 #44584251

Order 32512.

4070 conjugacy classes of subgroups; only the 702 normal subgroups are shown.

normal subgroups only

This group has 4070 conjugacy classes of subgroups — too many to draw. Only its 702 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
1C23×#41×C127C211yesmaximal, characteristic
2C22×C22⋊C8×C127C216yesmaximal, complemented
3C127×#1780C218yesmaximal, direct product factor
4C22×#41×C127—224no—
5C22×#41×C127C414yes—
6C22×#41×C127C2214yes—
7C23×Q8×C127C411yescharacteristic
8C23×D4×C127C413yes—
9C24×C4×C127C2213yes—
10C23×C8×C127C2214yes—
11C2×C22⋊C8×C127—224no—
12C2×C22⋊C8×C127C22124yescomplemented
13C25×C127C2×C413yes—
14C2×#41×C127—2128no—
15C2×#41×C127—248no—
16C2×#41×C127—248no—
17C22×Q8×C127—212no—
18C22×Q8×C127C2×C414yes—
19C22×D4×C127—224no—
20C22×D4×C127—212no—
21C22×D4×C127C2×C4112yes—
22C23×C4×C127—224no—
23C23×C4×C127C2×C413yes—
24C23×C4×C127C2×C416yes—
25C23×C4×C127D4124yes—
26C23×C4×C127C2316yes—
27C23×C4×C127C2311yescharacteristic
28C22×C8×C127—224no—
29C22×C8×C127C2318yes—
30C22⋊C8×C127—248no—
31C24×C127—212no—
32C24×C127—224no—
33C24×C127C22⋊C4112yes—
34C24×C127C22×C416yes—
35C24×C127C22×C411yescharacteristic
36#41×C127—2128no—
37#41×C127—2128no—
38C2×Q8×C127—232no—
39C2×Q8×C127—224no—
40C2×D4×C127—296no—
41C2×D4×C127—248no—
42C2×D4×C127—224no—
43C22×C4×C127—212no—
44C22×C4×C127—224no—
45C22×C4×C127—296no—
46C22×C4×C127—224no—
47C22×C4×C127C22⋊C4112yes—
48C22×C4×C127C22⋊C4124yes—
49C22×C4×C127C22×C416yes—
50C22×C4×C127C22×C411yescharacteristic
51C22×C4×C127C2×D4124yes—
52C22×C4×C127C2×D4112yes—
53C22×C4×C127C2411yescharacteristic
54C2×C8×C127—224no—
55C2×C8×C127—416no—
56C23×C127—212no—
57C23×C127—224no—
58C23×C127—224no—
59C23×C127—248no—
60C23×C127—448no—
61C23×C127C2×C22⋊C4112yes—
62C23×C127C2×C22⋊C416yes—
63C23×C127#13118yes—
64C23×C127C23×C411yescharacteristic
65Q8×C127—232no—
66D4×C127—296no—
67C2×C4×C127—212no—
68C2×C4×C127—224no—
69C2×C4×C127—248no—
70C2×C4×C127—248no—
71C2×C4×C127C2×C22⋊C4112yes—
72C2×C4×C127C2×C22⋊C416yes—
73C2×C4×C127#120116yes—
74C2×C4×C127C22×D416yes—
75C8×C127—48no—
76C22×C127—224no—
77C22×C127—212no—
78C22×C127—424no—
79C22×C127—448no—
80C22×C127#37914yes—
81C22×C127#408124yes—
82C22×C127C22×C22⋊C413yes—
83C22×C127C2×#13114yes—
84C4×C127—212no—
85C4×C127#37914yes—
86C4×C127C2×#12014yes—
87C2×#1780C12711yesmaximal, characteristic, direct product factor, Sylow
88C2×C127—412no—
89C2×C127#178018yesdirect product factor
90C2×C127C2×#37911yescharacteristic
91C2×C127C2×#40816yes—
92C23×#41C2×C12711yescharacteristic
93C22×C22⋊C8C2×C12716yescomplemented
94#1780C2×C12718yesdirect product factor
95C127C2×#178011yescharacteristic, direct product factor, Sylow
96C22×#41—224no—
97C22×#41C4×C12714yes—
98C22×#41C22×C12714yes—
99C23×Q8C4×C12711yescharacteristic
100C23×D4C4×C12713yes—
101C24×C4C22×C12713yes—
102C23×C8C22×C12714yes—
103C2×C22⋊C8—224no—
104C2×C22⋊C8C22×C127124yescomplemented
105C25C2×C4×C12713yes—
106C2×#41—2128no—
107C2×#41—248no—
108C2×#41—248no—
109C22×Q8—212no—
110C22×Q8C2×C4×C12714yes—
111C22×D4—224no—
112C22×D4—212no—
113C22×D4C2×C4×C127112yes—
114C23×C4—224no—
115C23×C4C2×C4×C12713yes—
116C23×C4C2×C4×C12716yes—
117C23×C4D4×C127124yes—
118C23×C4C23×C12716yes—
119C23×C4C23×C12711yescharacteristic
120C22×C8—224no—
121C22×C8C23×C12718yes—
122C22⋊C8—248no—
123C24—212no—
124C24—224no—
125C24C22⋊C4×C127112yes—
126C24C22×C4×C12716yes—
127C24C22×C4×C12711yescharacteristic
128#41—2128no—
129#41—2128no—
130C2×Q8—232no—
131C2×Q8—224no—
132C2×D4—296no—
133C2×D4—248no—
134C2×D4—224no—
135C22×C4—212no—
136C22×C4—224no—
137C22×C4—296no—
138C22×C4—224no—
139C22×C4C22⋊C4×C127112yes—
140C22×C4C22⋊C4×C127124yes—
141C22×C4C22×C4×C12716yes—
142C22×C4C22×C4×C12711yescharacteristic
143C22×C4C2×D4×C127124yes—
144C22×C4C2×D4×C127112yes—
145C22×C4C24×C12711yescharacteristic
146C2×C8—224no—
147C2×C8—416no—
148C23—212no—
149C23—224no—
150C23—224no—
151C23—248no—
152C23—448no—
153C23C2×C22⋊C4×C127112yes—
154C23C2×C22⋊C4×C12716yes—
155C23#131×C12718yes—
156C23C23×C4×C12711yescharacteristic
157Q8—232no—
158D4—296no—
159C2×C4—212no—
160C2×C4—224no—
161C2×C4—248no—
162C2×C4—248no—
163C2×C4C2×C22⋊C4×C127112yes—
164C2×C4C2×C22⋊C4×C12716yes—
165C2×C4#120×C127116yes—
166C2×C4C22×D4×C12716yes—
167C8—48no—
168C22—224no—
169C22—212no—
170C22—424no—
171C22—448no—
172C22#379×C12714yes—
173C22#408×C127124yes—
174C22C22×C22⋊C4×C12713yes—
175C22C2×#131×C12714yes—
176C4—212no—
177C4#379×C12714yes—
178C4C2×#120×C12714yes—
179C2—412no—
180C2C127×#178018yesdirect product factor
181C2C2×#379×C12711yescharacteristic
182C2C2×#408×C12716yes—
subgroupG/H
centerC23×C4×C127C23
commutatorC23C23×C4×C127
Frattini, cosocleC22×C4C24×C127
socleC24×C127C22×C4
prime radicalC23×#41×C127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊C8×C127C2semidirect
3. C127×#1780C2direct (N × H)
12. C2×C22⋊C8×C127C22semidirect
87. C2×#1780C127direct (N × H)
89. C2×C127#1780direct (N × H)
93. C22×C22⋊C8C2×C127semidirect
94. #1780C2×C127direct (N × H)
95. C127C2×#1780direct (N × H)
104. C2×C22⋊C8C22×C127semidirect
180. C2C127×#1780direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#41×C1272
C22×C22⋊C8×C1272
C127×#17802
C2×#1780127