Finite Group Databasebeta

← C127×#15172 #44578845

Subgroups — C127×#15172 #44578845

Order 32512.

2826 conjugacy classes of subgroups; only the 478 normal subgroups are shown.

normal subgroups only

This group has 2826 conjugacy classes of subgroups — too many to draw. Only its 478 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×#399×C127C214yesmaximal, complemented
2C2×#392×C127C214yesmaximal, complemented
3C2×#390×C127C216yesmaximal, complemented
4C4×#95×C127C211yesmaximal, characteristic, complemented
5C23×D4×C127C2216yes—
6C22×C4⋊C4×C127C2216yes—
7C22×C22⋊C4×C127C22112yes—
8C22×C22⋊C4×C127C2214yes—
9C22×C42×C127C2213yes—
10#399×C127—216no—
11#392×C127—216no—
12#390×C127—224no—
13C2×#95×C127C2214yes—
14C25×C127C2314yes—
15C22×D4×C127—224no—
16C22×D4×C127—212no—
17C22×D4×C127—224no—
18C22×D4×C127—412no—
19C23×C4×C127C2314yes—
20C23×C4×C127C2311yescharacteristic
21C23×C4×C127C2316yes—
22C2×C4⋊C4×C127—212no—
23C2×C4⋊C4×C127—224no—
24C2×C22⋊C4×C127—224no—
25C2×C22⋊C4×C127—224no—
26C2×C22⋊C4×C127—224no—
27C2×C22⋊C4×C127—224no—
28C2×C42×C127—46no—
29C2×C42×C127D4112yescomplemented
30#95×C127D4116yescomplemented
31C24×C127—24no—
32C24×C127—212no—
33C24×C127—212no—
34C24×C127—212no—
35C24×C127—212no—
36C24×C127—24no—
37C24×C127#4118yes—
38C24×C127C2411yescharacteristic
39C2×D4×C127—248no—
40C2×D4×C127—496no—
41C2×D4×C127—424no—
42C2×D4×C127—424no—
43C22×C4×C127—212no—
44C22×C4×C127—23no—
45C22×C4×C127—212no—
46C22×C4×C127—21no—
47C22×C4×C127—24no—
48C22×C4×C127—212no—
49C22×C4×C127—212no—
50C22×C4×C127C2×D4124yes—
51C22×C4×C127C2×D416yes—
52C22×C4×C127C2×D4112yes—
53C4⋊C4×C127—424no—
54C22⋊C4×C127—416no—
55C22⋊C4×C127—448no—
56C42×C127—412no—
57C23×C127—224no—
58C23×C127—224no—
59C23×C127—28no—
60C23×C127—412no—
61C23×C127—44no—
62C23×C127—412no—
63C23×C127—412no—
64C23×C127—412no—
65C23×C127—412no—
66C23×C127—424no—
67C23×C127—412no—
68C23×C127—412no—
69C23×C127C22×D413yes—
70C23×C127C22×D414yes—
71C23×C127C2×#4114yes—
72C23×C127H(5,𝔽2)13yes—
73C23×C1272−1+411yescharacteristic
74D4×C127—496no—
75C2×C4×C127—212no—
76C2×C4×C127—26no—
77C2×C4×C127—212no—
78C2×C4×C127—26no—
79C2×C4×C127—248no—
80C2×C4×C127—44no—
81C2×C4×C127—412no—
82C2×C4×C127—412no—
83C2×C4×C127#12714yes—
84C2×C4×C127#127124yes—
85C22×C127—48no—
86C22×C127—424no—
87C22×C127—424no—
88C22×C127—84no—
89C22×C127—84no—
90C22×C127—812no—
91C22×C127—812no—
92C22×C127#52516yes—
93C22×C127C2×#12711yescharacteristic
94C22×C127C2×#12716yes—
95C22×C127#53413yes—
96C22×C127#53713yes—
97C22×C127#546112yes—
98C22×C127#54814yes—
99C4×C127—24no—
100C4×C127—224no—
101C4×C127—84no—
102#15172C12711yesmaximal, characteristic, direct product factor, Sylow
103C2×C127—88no—
104C2×C127#239711yescharacteristic
105C2×C127#253916yes—
106C2×C127#254016yes—
107C2×C127#265611yescharacteristic
108C2×C127#286711yescharacteristic
109C2×#399C2×C12714yescomplemented
110C2×#392C2×C12714yescomplemented
111C2×#390C2×C12716yescomplemented
112C4×#95C2×C12711yescharacteristic, complemented
113C127#1517211yescharacteristic, direct product factor, Sylow
114C23×D4C22×C12716yes—
115C22×C4⋊C4C22×C12716yes—
116C22×C22⋊C4C22×C127112yes—
117C22×C22⋊C4C22×C12714yes—
118C22×C42C22×C12713yes—
119#399—216no—
120#392—216no—
121#390—224no—
122C2×#95C22×C12714yes—
123C25C23×C12714yes—
124C22×D4—224no—
125C22×D4—212no—
126C22×D4—224no—
127C22×D4—412no—
128C23×C4C23×C12714yes—
129C23×C4C23×C12711yescharacteristic
130C23×C4C23×C12716yes—
131C2×C4⋊C4—212no—
132C2×C4⋊C4—224no—
133C2×C22⋊C4—224no—
134C2×C22⋊C4—224no—
135C2×C22⋊C4—224no—
136C2×C22⋊C4—224no—
137C2×C42—46no—
138C2×C42D4×C127112yescomplemented
139#95D4×C127116yescomplemented
140C24—24no—
141C24—212no—
142C24—212no—
143C24—212no—
144C24—212no—
145C24—24no—
146C24#41×C12718yes—
147C24C24×C12711yescharacteristic
148C2×D4—248no—
149C2×D4—496no—
150C2×D4—424no—
151C2×D4—424no—
152C22×C4—212no—
153C22×C4—23no—
154C22×C4—212no—
155C22×C4—21no—
156C22×C4—24no—
157C22×C4—212no—
158C22×C4—212no—
159C22×C4C2×D4×C127124yes—
160C22×C4C2×D4×C12716yes—
161C22×C4C2×D4×C127112yes—
162C4⋊C4—424no—
163C22⋊C4—416no—
164C22⋊C4—448no—
165C42—412no—
166C23—224no—
167C23—224no—
168C23—28no—
169C23—412no—
170C23—44no—
171C23—412no—
172C23—412no—
173C23—412no—
174C23—412no—
175C23—424no—
176C23—412no—
177C23—412no—
178C23C22×D4×C12713yes—
179C23C22×D4×C12714yes—
180C23C2×#41×C12714yes—
181C23H(5,𝔽2)×C12713yes—
182C232−1+4×C12711yescharacteristic
183D4—496no—
184C2×C4—212no—
185C2×C4—212no—
186C2×C4—26no—
187C2×C4—26no—
188C2×C4—248no—
189C2×C4—44no—
190C2×C4—412no—
191C2×C4—412no—
192C2×C4#127×C12714yes—
193C2×C4#127×C127124yes—
194C22—48no—
195C22—424no—
196C22—424no—
197C22—84no—
198C22—84no—
199C22—812no—
200C22—812no—
201C22#525×C12716yes—
202C22C2×#127×C12711yescharacteristic
203C22C2×#127×C12716yes—
204C22#534×C12713yes—
205C22#537×C12713yes—
206C22#546×C127112yes—
207C22#548×C12714yes—
208C4—24no—
209C4—224no—
210C4—84no—
211C2—88no—
212C2C127×#239711yescharacteristic
213C2C127×#253916yes—
214C2C127×#254016yes—
215C2C127×#265611yescharacteristic
216C2C127×#286711yescharacteristic
subgroupG/H
center, socleC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#399×C127C2semidirect
2. C2×#392×C127C2semidirect
3. C2×#390×C127C2semidirect
4. C4×#95×C127C2semidirect
29. C2×C42×C127D4semidirect
30. #95×C127D4semidirect
102. #15172C127direct (N × H)
109. C2×#399C2×C127semidirect
110. C2×#392C2×C127semidirect
111. C2×#390C2×C127semidirect
112. C4×#95C2×C127semidirect
113. C127#15172direct (N × H)
138. C2×C42D4×C127semidirect
139. #95D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#399×C1272
C2×#392×C1272
C2×#390×C1272
C4×#95×C1272
#15172127