Finite Group Databasebeta

← C127×#2742 #30635694

Subgroups — C127×#2742 #30635694

Order 16256.

416 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#400×C127C212yesmaximal, complemented
2#399×C127C211yesmaximal, characteristic
3#395×C127C212yesmaximal, complemented
4#393×C127C211yesmaximal, characteristic
5#389×C127C212yesmaximal, complemented
6#388×C127C212yesmaximal
7#387×C127C212yesmaximal, complemented
8#385×C127C211yesmaximal, characteristic
9#380×C127C212yesmaximal
10C22×Q8×C127C2211yescharacteristic
11C23×C4×C127C2211yescharacteristic
12C2×C4⋊C4×C127C2212yes—
13C2×C4⋊C4×C127C2212yes—
14C2×C4⋊C4×C127C2212yes—
15C2×C4⋊C4×C127C2212yes—
16C2×C4⋊C4×C127C2212yes—
17C2×C4⋊C4×C127C2211yescharacteristic
18C2×C22⋊C4×C127C2212yes—
19C2×C22⋊C4×C127C2212yes—
20C2×C22⋊C4×C127C2211yescharacteristic
21C2×C22⋊C4×C127C2211yescharacteristic
22C2×C42×C127C2212yes—
23#95×C127C2212yes—
24#95×C127C2212yes—
25#95×C127C2212yes—
26#95×C127C2211yescharacteristic
27#95×C127C2212yes—
28#95×C127C2212yes—
29#95×C127C2212yes—
30#95×C127C2211yescharacteristic
31C24×C127C2311yescharacteristic
32C2×Q8×C127—42no—
33C2×Q8×C127—41no—
34C22×C4×C127—24no—
35C22×C4×C127—41no—
36C22×C4×C127C2312yes—
37C22×C4×C127C2312yes—
38C22×C4×C127C2311yescharacteristic
39C22×C4×C127C2312yes—
40C22×C4×C127C2311yescharacteristic
41C22×C4×C127C2312yes—
42C22×C4×C127C2312yes—
43C22×C4×C127C2312yes—
44C4⋊C4×C127—24no—
45C4⋊C4×C127—24no—
46C4⋊C4×C127—42no—
47C4⋊C4×C127—42no—
48C4⋊C4×C127—42no—
49C4⋊C4×C127—41no—
50C22⋊C4×C127—42no—
51C22⋊C4×C127—42no—
52C22⋊C4×C127—41no—
53C22⋊C4×C127Q814yescomplemented
54C42×C127—42no—
55C23×C127—22no—
56C23×C127—22no—
57C23×C127—21no—
58C23×C127—21no—
59C23×C127C2×Q812yes—
60C23×C127C2411yescharacteristic
61Q8×C127—44no—
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—22no—
67C2×C4×C127—22no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—21no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—21no—
76C2×C4×C127—22no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—22no—
82C2×C4×C127—44no—
83C2×C4×C127C2×Q814yes—
84C2×C4×C127#4114yes—
85C22×C127—22no—
86C22×C127—24no—
87C22×C127—42no—
88C22×C127—41no—
89C22×C127—41no—
90C22×C127C22×Q811yescharacteristic
91C22×C127C2×#4112yes—
92C22×C127H(5,𝔽2)12yes—
93C22×C127H(5,𝔽2)11yescharacteristic
94C22×C1272−1+411yescharacteristic
95C4×C127—24no—
96C4×C127—24no—
97C4×C127—42no—
98C4×C127—42no—
99C4×C127—41no—
100C4×C127—42no—
101C4×C127—42no—
102C4×C127—41no—
103C2×C127—42no—
104C2×C127#54212yes—
105C2×C127#54411yescharacteristic
106C2×C127#55111yescharacteristic
107C2×C127#55412yes—
108C2×C127#56011yescharacteristic
109#2742C12711yesmaximal, characteristic, direct product factor, Sylow
110C127#274211yescharacteristic, direct product factor, Sylow
111#400C2×C12712yescomplemented
112#399C2×C12711yescharacteristic
113#395C2×C12712yescomplemented
114#393C2×C12711yescharacteristic
115#389C2×C12712yescomplemented
116#388C2×C12712yes—
117#387C2×C12712yescomplemented
118#385C2×C12711yescharacteristic
119#380C2×C12712yes—
120C22×Q8C22×C12711yescharacteristic
121C23×C4C22×C12711yescharacteristic
122C2×C4⋊C4C22×C12712yes—
123C2×C4⋊C4C22×C12712yes—
124C2×C4⋊C4C22×C12712yes—
125C2×C4⋊C4C22×C12712yes—
126C2×C4⋊C4C22×C12712yes—
127C2×C4⋊C4C22×C12711yescharacteristic
128C2×C22⋊C4C22×C12712yes—
129C2×C22⋊C4C22×C12712yes—
130C2×C22⋊C4C22×C12711yescharacteristic
131C2×C22⋊C4C22×C12711yescharacteristic
132C2×C42C22×C12712yes—
133#95C22×C12712yes—
134#95C22×C12712yes—
135#95C22×C12712yes—
136#95C22×C12711yescharacteristic
137#95C22×C12712yes—
138#95C22×C12712yes—
139#95C22×C12712yes—
140#95C22×C12711yescharacteristic
141C24C23×C12711yescharacteristic
142C2×Q8—42no—
143C2×Q8—41no—
144C22×C4—24no—
145C22×C4—41no—
146C22×C4C23×C12712yes—
147C22×C4C23×C12712yes—
148C22×C4C23×C12711yescharacteristic
149C22×C4C23×C12712yes—
150C22×C4C23×C12711yescharacteristic
151C22×C4C23×C12712yes—
152C22×C4C23×C12712yes—
153C22×C4C23×C12712yes—
154C4⋊C4—24no—
155C4⋊C4—24no—
156C4⋊C4—42no—
157C4⋊C4—42no—
158C4⋊C4—42no—
159C4⋊C4—41no—
160C22⋊C4—42no—
161C22⋊C4—42no—
162C22⋊C4—41no—
163C22⋊C4Q8×C12714yescomplemented
164C42—42no—
165C23—22no—
166C23—22no—
167C23—21no—
168C23—21no—
169C23C2×Q8×C12712yes—
170C23C24×C12711yescharacteristic
171Q8—44no—
172C2×C4—22no—
173C2×C4—22no—
174C2×C4—22no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—21no—
183C2×C4—22no—
184C2×C4—22no—
185C2×C4—21no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—22no—
189C2×C4—22no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—44no—
193C2×C4C2×Q8×C12714yes—
194C2×C4#41×C12714yes—
195C22—22no—
196C22—24no—
197C22—42no—
198C22—41no—
199C22—41no—
200C22C22×Q8×C12711yescharacteristic
201C22C2×#41×C12712yes—
202C22H(5,𝔽2)×C12712yes—
203C22H(5,𝔽2)×C12711yescharacteristic
204C222−1+4×C12711yescharacteristic
205C4—24no—
206C4—24no—
207C4—42no—
208C4—42no—
209C4—41no—
210C4—42no—
211C4—41no—
212C4—42no—
213C2—42no—
214C2#542×C12712yes—
215C2#544×C12711yescharacteristic
216C2#551×C12711yescharacteristic
217C2#554×C12712yes—
218C2#560×C12711yescharacteristic
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC24×C127C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. #400×C127C2semidirect
3. #395×C127C2semidirect
5. #389×C127C2semidirect
7. #387×C127C2semidirect
53. C22⋊C4×C127Q8semidirect
109. #2742C127direct (N × H)
110. C127#2742direct (N × H)
111. #400C2×C127semidirect
113. #395C2×C127semidirect
115. #389C2×C127semidirect
117. #387C2×C127semidirect
163. C22⋊C4Q8×C127semidirect

Maximal subgroups

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

subgroupindex
#400×C1272
#399×C1272
#395×C1272
#393×C1272
#389×C1272
#388×C1272
#387×C1272
#385×C1272
#380×C1272
#2742127