Finite Group Databasebeta

← C127×#14337 #44578010

Subgroups — C127×#14337 #44578010

Order 32512.

4390 conjugacy classes of subgroups; only the 702 normal subgroups are shown.

normal subgroups only

This group has 4390 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
2C2×#380×C127C216yesmaximal, complemented
3C127×#1446C212yesmaximal, complemented
4C127×#1438C216yesmaximal, complemented
5C22×#41×C127—224no—
6C22×#41×C127C418yescomplemented
7C23×Q8×C127C2211yescharacteristic
8C23×D4×C127C2213yes—
9C24×C4×C127C2213yes—
10C22×C4⋊C4×C127C2214yes—
11C22×C22⋊C4×C127C22112yes—
12#380×C127—224no—
13C2×#95×C127C22112yes—
14C25×C127C2313yes—
15C2×#41×C127—2128no—
16C2×#41×C127—296no—
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×C4112yes—
24C23×C4×C127D4124yes—
25C23×C4×C127C2318yes—
26C23×C4×C127C2313yes—
27C23×C4×C127C2311yescharacteristic
28C2×C4⋊C4×C127—224no—
29C2×C22⋊C4×C127—224no—
30C2×C22⋊C4×C127—424no—
31#95×C127—224no—
32C24×C127—212no—
33C24×C127—212no—
34C24×C127—212no—
35C24×C127C22×C416yes—
36C24×C127C2×D4112yes—
37C24×C127C2411yescharacteristic
38#41×C127—2256no—
39C2×Q8×C127—232no—
40C2×Q8×C127—224no—
41C2×D4×C127—296no—
42C2×D4×C127—248no—
43C2×D4×C127—224no—
44C22×C4×C127—212no—
45C22×C4×C127—248no—
46C22×C4×C127—296no—
47C22×C4×C127—224no—
48C22×C4×C127—224no—
49C22×C4×C127—28no—
50C22×C4×C127C22⋊C4148yescomplemented
51C22×C4×C127C22×C416yes—
52C22×C4×C127C22×C412yes—
53C22×C4×C127C2×D4112yes—
54C22×C4×C127C2×D4112yes—
55C4⋊C4×C127—416no—
56C22⋊C4×C127—448no—
57C23×C127—212no—
58C23×C127—224no—
59C23×C127—224no—
60C23×C127—212no—
61C23×C127—212no—
62C23×C127—224no—
63C23×C127—412no—
64C23×C127—412no—
65C23×C127—424no—
66C23×C127C2×C22⋊C4112yes—
67C23×C127C23×C411yescharacteristic
68C23×C127C22×D416yes—
69C23×C127H(5,𝔽2)14yes—
70C23×C1272−1+414yes—
71Q8×C127—232no—
72D4×C127—296no—
73C2×C4×C127—212no—
74C2×C4×C127—224no—
75C2×C4×C127—296no—
76C2×C4×C127—48no—
77C2×C4×C127—424no—
78C2×C4×C127—424no—
79C2×C4×C127C2×C22⋊C4112yes—
80C2×C4×C127C2×C22⋊C4112yes—
81C2×C4×C127#120116yes—
82C22×C127—224no—
83C22×C127—212no—
84C22×C127—424no—
85C22×C127—424no—
86C22×C127—424no—
87C22×C127C22×C22⋊C413yes—
88C22×C127#52014yes—
89C22×C127C2×#12014yes—
90C22×C127#53516yes—
91C22×C127#53616yes—
92C22×C127#537112yes—
93C4×C127—212no—
94C4×C127—88no—
95C4×C127#37918yes—
96#14337C12711yesmaximal, characteristic, direct product factor, Sylow
97C2×C127—412no—
98C2×C127C2×#37911yescharacteristic
99C2×C127#231116yes—
100C2×C127#261514yes—
101C2×C127#261614yes—
102C23×#41C2×C12711yescharacteristic
103C2×#380C2×C12716yescomplemented
104#1446C2×C12712yescomplemented
105#1438C2×C12716yescomplemented
106C127#1433711yescharacteristic, direct product factor, Sylow
107C22×#41—224no—
108C22×#41C4×C12718yescomplemented
109C23×Q8C22×C12711yescharacteristic
110C23×D4C22×C12713yes—
111C24×C4C22×C12713yes—
112C22×C4⋊C4C22×C12714yes—
113C22×C22⋊C4C22×C127112yes—
114#380—224no—
115C2×#95C22×C127112yes—
116C25C23×C12713yes—
117C2×#41—2128no—
118C2×#41—296no—
119C22×Q8—212no—
120C22×Q8C2×C4×C12714yes—
121C22×D4—224no—
122C22×D4—212no—
123C22×D4C2×C4×C127112yes—
124C23×C4—224no—
125C23×C4C2×C4×C127112yes—
126C23×C4D4×C127124yes—
127C23×C4C23×C12718yes—
128C23×C4C23×C12713yes—
129C23×C4C23×C12711yescharacteristic
130C2×C4⋊C4—224no—
131C2×C22⋊C4—224no—
132C2×C22⋊C4—424no—
133#95—224no—
134C24—212no—
135C24—212no—
136C24—212no—
137C24C22×C4×C12716yes—
138C24C2×D4×C127112yes—
139C24C24×C12711yescharacteristic
140#41—2256no—
141C2×Q8—232no—
142C2×Q8—224no—
143C2×D4—296no—
144C2×D4—248no—
145C2×D4—224no—
146C22×C4—212no—
147C22×C4—248no—
148C22×C4—296no—
149C22×C4—224no—
150C22×C4—224no—
151C22×C4—28no—
152C22×C4C22⋊C4×C127148yescomplemented
153C22×C4C22×C4×C12716yes—
154C22×C4C22×C4×C12712yes—
155C22×C4C2×D4×C127112yes—
156C22×C4C2×D4×C127112yes—
157C4⋊C4—416no—
158C22⋊C4—448no—
159C23—212no—
160C23—224no—
161C23—224no—
162C23—212no—
163C23—212no—
164C23—224no—
165C23—412no—
166C23—412no—
167C23—424no—
168C23C2×C22⋊C4×C127112yes—
169C23C23×C4×C12711yescharacteristic
170C23C22×D4×C12716yes—
171C23H(5,𝔽2)×C12714yes—
172C232−1+4×C12714yes—
173Q8—232no—
174D4—296no—
175C2×C4—212no—
176C2×C4—224no—
177C2×C4—296no—
178C2×C4—48no—
179C2×C4—424no—
180C2×C4—424no—
181C2×C4C2×C22⋊C4×C127112yes—
182C2×C4C2×C22⋊C4×C127112yes—
183C2×C4#120×C127116yes—
184C22—212no—
185C22—224no—
186C22—424no—
187C22—424no—
188C22—424no—
189C22C22×C22⋊C4×C12713yes—
190C22#520×C12714yes—
191C22C2×#120×C12714yes—
192C22#535×C12716yes—
193C22#536×C12716yes—
194C22#537×C127112yes—
195C4—212no—
196C4—88no—
197C4#379×C12718yes—
198C2—412no—
199C2C2×#379×C12711yescharacteristic
200C2C127×#231116yes—
201C2C127×#261514yes—
202C2C127×#261614yes—
subgroupG/H
center, socleC24×C127C24
commutatorC23C23×C4×C127
Frattini, cosocleC24C24×C127
prime radicalC23×#41×C127C2
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#380×C127C2semidirect
3. C127×#1446C2semidirect
4. C127×#1438C2semidirect
6. C22×#41×C127C4semidirect
50. C22×C4×C127C22⋊C4semidirect
96. #14337C127direct (N × H)
103. C2×#380C2×C127semidirect
104. #1446C2×C127semidirect
105. #1438C2×C127semidirect
106. C127#14337direct (N × H)
108. C22×#41C4×C127semidirect
152. C22×C4C22⋊C4×C127semidirect

Maximal subgroups

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

subgroupindex
C23×#41×C1272
C2×#380×C1272
C127×#14462
C127×#14382
#14337127