Finite Group Databasebeta

← C127×#2820 #30635772

Subgroups — C127×#2820 #30635772

Order 16256.

464 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 184 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×#125×C127C211yesmaximal, characteristic, complemented
2C2×#124×C127C211yesmaximal, characteristic
3#399×C127C212yesmaximal, complemented
4#397×C127C212yesmaximal, complemented
5#396×C127C214yesmaximal, complemented
6#395×C127C212yesmaximal, complemented
7#394×C127C212yesmaximal
8#382×C127C211yesmaximal, characteristic, complemented
9C22×Q8×C127C2211yescharacteristic
10C22×D4×C127C2211yescharacteristic
11#125×C127—24no—
12#124×C127—24no—
13C2×C4⋊C4×C127C2212yes—
14C2×C4⋊C4×C127C2212yes—
15C2×C4⋊C4×C127C2211yescharacteristic
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C4⋊C4×C127C2212yes—
18C2×C4⋊C4×C127C2214yes—
19C2×C22⋊C4×C127C2212yes—
20C2×C22⋊C4×C127C2212yes—
21C2×C22⋊C4×C127C2214yes—
22C2×C22⋊C4×C127C2214yes—
23C2×C42×C127C2211yescharacteristic
24#95×C127C2212yes—
25#95×C127C2212yes—
26#95×C127C2214yes—
27C24×C127C2312yes—
28C2×Q8×C127—22no—
29C2×Q8×C127—42no—
30C2×D4×C127—22no—
31C2×D4×C127—42no—
32C22×C4×C127C2314yes—
33C22×C4×C127C2312yes—
34C22×C4×C127C2312yes—
35C22×C4×C127C2312yes—
36C22×C4×C127C2311yescharacteristic
37C22×C4×C127C2311yescharacteristic
38C22×C4×C127C2311yescharacteristic
39C4⋊C4×C127—24no—
40C4⋊C4×C127—22no—
41C4⋊C4×C127—22no—
42C4⋊C4×C127—24no—
43C4⋊C4×C127—42no—
44C4⋊C4×C127—44no—
45C22⋊C4×C127—24no—
46C22⋊C4×C127—24no—
47C22⋊C4×C127—44no—
48C22⋊C4×C127—44no—
49C42×C127D414yes—
50C23×C127—22no—
51C23×C127—22no—
52C23×C127—24no—
53C23×C127—24no—
54C23×C127—22no—
55C23×C127C2411yescharacteristic
56Q8×C127—44no—
57D4×C127—44no—
58C2×C4×C127—21no—
59C2×C4×C127—22no—
60C2×C4×C127—22no—
61C2×C4×C127—21no—
62C2×C4×C127—22no—
63C2×C4×C127—22no—
64C2×C4×C127—22no—
65C2×C4×C127—22no—
66C2×C4×C127—24no—
67C2×C4×C127—24no—
68C2×C4×C127—24no—
69C2×C4×C127—24no—
70C2×C4×C127—22no—
71C2×C4×C127—24no—
72C2×C4×C127C2×D412yes—
73C2×C4×C127C2×D412yes—
74C2×C4×C127C2×D412yes—
75C22×C127—42no—
76C22×C127—42no—
77C22×C127—42no—
78C22×C127—44no—
79C22×C127—44no—
80C22×C127C22×D411yescharacteristic
81C22×C127H(5,𝔽2)12yes—
82C22×C127H(5,𝔽2)11yescharacteristic
83C22×C1272−1+412yes—
84C22×C1272−1+411yescharacteristic
85C4×C127—22no—
86C4×C127—22no—
87C4×C127—22no—
88C4×C127—42no—
89C4×C127—42no—
90C4×C127—42no—
91C4×C127—44no—
92C2×C127—82no—
93C2×C127#53511yescharacteristic
94C2×C127#53611yescharacteristic
95C2×C127#53711yescharacteristic
96C2×C127#56212yes—
97C2×C127#56312yes—
98#2820C12711yesmaximal, characteristic, direct product factor, Sylow
99C127#282011yescharacteristic, direct product factor, Sylow
100C2×#125C2×C12711yescharacteristic, complemented
101C2×#124C2×C12711yescharacteristic
102#399C2×C12712yescomplemented
103#397C2×C12712yescomplemented
104#396C2×C12714yescomplemented
105#395C2×C12712yescomplemented
106#394C2×C12712yes—
107#382C2×C12711yescharacteristic, complemented
108C22×Q8C22×C12711yescharacteristic
109C22×D4C22×C12711yescharacteristic
110#125—24no—
111#124—24no—
112C2×C4⋊C4C22×C12712yes—
113C2×C4⋊C4C22×C12712yes—
114C2×C4⋊C4C22×C12711yescharacteristic
115C2×C4⋊C4C22×C12711yescharacteristic
116C2×C4⋊C4C22×C12712yes—
117C2×C4⋊C4C22×C12714yes—
118C2×C22⋊C4C22×C12712yes—
119C2×C22⋊C4C22×C12712yes—
120C2×C22⋊C4C22×C12714yes—
121C2×C22⋊C4C22×C12714yes—
122C2×C42C22×C12711yescharacteristic
123#95C22×C12712yes—
124#95C22×C12712yes—
125#95C22×C12714yes—
126C24C23×C12712yes—
127C2×Q8—22no—
128C2×Q8—42no—
129C2×D4—22no—
130C2×D4—42no—
131C22×C4C23×C12714yes—
132C22×C4C23×C12712yes—
133C22×C4C23×C12712yes—
134C22×C4C23×C12712yes—
135C22×C4C23×C12711yescharacteristic
136C22×C4C23×C12711yescharacteristic
137C22×C4C23×C12711yescharacteristic
138C4⋊C4—24no—
139C4⋊C4—22no—
140C4⋊C4—22no—
141C4⋊C4—24no—
142C4⋊C4—42no—
143C4⋊C4—44no—
144C22⋊C4—24no—
145C22⋊C4—24no—
146C22⋊C4—44no—
147C22⋊C4—44no—
148C42D4×C12714yes—
149C23—22no—
150C23—22no—
151C23—24no—
152C23—24no—
153C23—22no—
154C23C24×C12711yescharacteristic
155Q8—44no—
156D4—44no—
157C2×C4—21no—
158C2×C4—22no—
159C2×C4—22no—
160C2×C4—21no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—22no—
165C2×C4—24no—
166C2×C4—24no—
167C2×C4—24no—
168C2×C4—24no—
169C2×C4—22no—
170C2×C4—24no—
171C2×C4C2×D4×C12712yes—
172C2×C4C2×D4×C12712yes—
173C2×C4C2×D4×C12712yes—
174C22—42no—
175C22—42no—
176C22—42no—
177C22—44no—
178C22—44no—
179C22C22×D4×C12711yescharacteristic
180C22H(5,𝔽2)×C12712yes—
181C22H(5,𝔽2)×C12711yescharacteristic
182C222−1+4×C12712yes—
183C222−1+4×C12711yescharacteristic
184C4—22no—
185C4—22no—
186C4—22no—
187C4—42no—
188C4—42no—
189C4—42no—
190C4—44no—
191C2—82no—
192C2#535×C12711yescharacteristic
193C2#536×C12711yescharacteristic
194C2#537×C12711yescharacteristic
195C2#562×C12712yes—
196C2#563×C12712yes—
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
prime radicalC22×D4×C127C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#125×C127C2semidirect
3. #399×C127C2semidirect
4. #397×C127C2semidirect
5. #396×C127C2semidirect
6. #395×C127C2semidirect
8. #382×C127C2semidirect
98. #2820C127direct (N × H)
99. C127#2820direct (N × H)
100. C2×#125C2×C127semidirect
102. #399C2×C127semidirect
103. #397C2×C127semidirect
104. #396C2×C127semidirect
105. #395C2×C127semidirect
107. #382C2×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#125×C1272
C2×#124×C1272
#399×C1272
#397×C1272
#396×C1272
#395×C1272
#394×C1272
#382×C1272
#2820127