Finite Group Databasebeta

← C127×#2631 #30635583

Subgroups — C127×#2631 #30635583

Order 16256.

628 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 628 conjugacy classes of subgroups — too many to draw. Only its 192 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×#124×C127C211yesmaximal, characteristic, complemented
2C2×C4×D4×C127C211yesmaximal, characteristic, complemented
3#397×C127C212yesmaximal, complemented
4#393×C127C212yesmaximal, complemented
5#392×C127C212yesmaximal, complemented
6#392×C127C212yesmaximal, complemented
7#386×C127C214yesmaximal, complemented
8#383×C127C211yesmaximal, characteristic, complemented
9C22×Q8×C127C2211yescharacteristic
10C22×D4×C127C2211yescharacteristic
11C22×D4×C127C2211yescharacteristic
12C22×D4×C127C2214yes—
13C23×C4×C127C2212yes—
14#124×C127—24no—
15C4×D4×C127—24no—
16C2×C4⋊C4×C127C2211yescharacteristic
17C2×C22⋊C4×C127C2212yes—
18C2×C22⋊C4×C127C2212yes—
19C2×C22⋊C4×C127C2212yes—
20C2×C22⋊C4×C127C2214yes—
21C2×C22⋊C4×C127C2214yes—
22C2×C42×C127C2211yescharacteristic
23#95×C127C2212yes—
24#95×C127C2212yes—
25#95×C127C2212yes—
26#95×C127C2214yes—
27C24×C127C2312yes—
28C24×C127C2312yes—
29C2×Q8×C127—22no—
30C2×Q8×C127—42no—
31C2×D4×C127—24no—
32C2×D4×C127—28no—
33C2×D4×C127—22no—
34C2×D4×C127—22no—
35C2×D4×C127—42no—
36C2×D4×C127—44no—
37C2×D4×C127—44no—
38C22×C4×C127—24no—
39C22×C4×C127—44no—
40C22×C4×C127C2314yes—
41C22×C4×C127C2312yes—
42C22×C4×C127C2312yes—
43C22×C4×C127C2311yescharacteristic
44C22×C4×C127C2311yescharacteristic
45C22×C4×C127C2311yescharacteristic
46C4⋊C4×C127—22no—
47C22⋊C4×C127—24no—
48C22⋊C4×C127—24no—
49C22⋊C4×C127—24no—
50C22⋊C4×C127—44no—
51C22⋊C4×C127—44no—
52C42×C127D414yescomplemented
53C23×C127—22no—
54C23×C127—22no—
55C23×C127—24no—
56C23×C127—24no—
57C23×C127—22no—
58C23×C127—24no—
59C23×C127—22no—
60C23×C127—24no—
61C23×C127—22no—
62C23×C127#4114yes—
63C23×C127C2411yescharacteristic
64Q8×C127—44no—
65D4×C127—28no—
66D4×C127—44no—
67D4×C127—416no—
68C2×C4×C127—22no—
69C2×C4×C127—22no—
70C2×C4×C127—22no—
71C2×C4×C127—22no—
72C2×C4×C127—22no—
73C2×C4×C127—24no—
74C2×C4×C127—24no—
75C2×C4×C127—24no—
76C2×C4×C127—24no—
77C2×C4×C127—24no—
78C2×C4×C127—48no—
79C2×C4×C127C2×D412yes—
80C2×C4×C127C2×D412yes—
81C2×C4×C127C2×D412yes—
82C22×C127—24no—
83C22×C127—28no—
84C22×C127—42no—
85C22×C127—42no—
86C22×C127—44no—
87C22×C127—42no—
88C22×C127—42no—
89C22×C127—42no—
90C22×C127—44no—
91C22×C127—44no—
92C22×C127C22×D411yescharacteristic
93C22×C127C2×#4112yes—
94C22×C127H(5,𝔽2)12yes—
95C22×C127H(5,𝔽2)12yes—
96C4×C127—22no—
97C4×C127—22no—
98C4×C127—22no—
99C4×C127—42no—
100C4×C127—42no—
101C4×C127—44no—
102C2×C127—44no—
103C2×C127—82no—
104C2×C127#52511yescharacteristic
105C2×C127#53511yescharacteristic
106C2×C127#53511yescharacteristic
107C2×C127#53814yes—
108#2631C12711yesmaximal, characteristic, direct product factor, Sylow
109C127#263111yescharacteristic, direct product factor, Sylow
110C2×#124C2×C12711yescharacteristic, complemented
111C2×C4×D4C2×C12711yescharacteristic, complemented
112#397C2×C12712yescomplemented
113#393C2×C12712yescomplemented
114#392C2×C12712yescomplemented
115#392C2×C12712yescomplemented
116#386C2×C12714yescomplemented
117#383C2×C12711yescharacteristic, complemented
118C22×Q8C22×C12711yescharacteristic
119C22×D4C22×C12711yescharacteristic
120C22×D4C22×C12711yescharacteristic
121C22×D4C22×C12714yes—
122C23×C4C22×C12712yes—
123#124—24no—
124C4×D4—24no—
125C2×C4⋊C4C22×C12711yescharacteristic
126C2×C22⋊C4C22×C12712yes—
127C2×C22⋊C4C22×C12712yes—
128C2×C22⋊C4C22×C12712yes—
129C2×C22⋊C4C22×C12714yes—
130C2×C22⋊C4C22×C12714yes—
131C2×C42C22×C12711yescharacteristic
132#95C22×C12712yes—
133#95C22×C12712yes—
134#95C22×C12712yes—
135#95C22×C12714yes—
136C24C23×C12712yes—
137C24C23×C12712yes—
138C2×Q8—22no—
139C2×Q8—42no—
140C2×D4—24no—
141C2×D4—28no—
142C2×D4—22no—
143C2×D4—22no—
144C2×D4—42no—
145C2×D4—44no—
146C2×D4—44no—
147C22×C4—24no—
148C22×C4—44no—
149C22×C4C23×C12714yes—
150C22×C4C23×C12712yes—
151C22×C4C23×C12712yes—
152C22×C4C23×C12711yescharacteristic
153C22×C4C23×C12711yescharacteristic
154C22×C4C23×C12711yescharacteristic
155C4⋊C4—22no—
156C22⋊C4—24no—
157C22⋊C4—24no—
158C22⋊C4—24no—
159C22⋊C4—44no—
160C22⋊C4—44no—
161C42D4×C12714yescomplemented
162C23—22no—
163C23—22no—
164C23—24no—
165C23—24no—
166C23—22no—
167C23—24no—
168C23—22no—
169C23—24no—
170C23—22no—
171C23#41×C12714yes—
172C23C24×C12711yescharacteristic
173Q8—44no—
174D4—28no—
175D4—44no—
176D4—416no—
177C2×C4—22no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—24no—
183C2×C4—24no—
184C2×C4—24no—
185C2×C4—24no—
186C2×C4—24no—
187C2×C4—48no—
188C2×C4C2×D4×C12712yes—
189C2×C4C2×D4×C12712yes—
190C2×C4C2×D4×C12712yes—
191C22—24no—
192C22—28no—
193C22—42no—
194C22—42no—
195C22—44no—
196C22—42no—
197C22—42no—
198C22—42no—
199C22—44no—
200C22—44no—
201C22C22×D4×C12711yescharacteristic
202C22C2×#41×C12712yes—
203C22H(5,𝔽2)×C12712yes—
204C22H(5,𝔽2)×C12712yes—
205C4—22no—
206C4—22no—
207C4—22no—
208C4—42no—
209C4—42no—
210C4—44no—
211C2—44no—
212C2—82no—
213C2#525×C12711yescharacteristic
214C2#535×C12711yescharacteristic
215C2#535×C12711yescharacteristic
216C2#538×C12714yes—
subgroupG/H
center, socleC23×C127C24
commutator, Frattini, cosocleC23C24×C127
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#124×C127C2semidirect
2. C2×C4×D4×C127C2semidirect
3. #397×C127C2semidirect
4. #393×C127C2semidirect
5. #392×C127C2semidirect
6. #392×C127C2semidirect
7. #386×C127C2semidirect
8. #383×C127C2semidirect
52. C42×C127D4semidirect
108. #2631C127direct (N × H)
109. C127#2631direct (N × H)
110. C2×#124C2×C127semidirect
111. C2×C4×D4C2×C127semidirect
112. #397C2×C127semidirect
113. #393C2×C127semidirect
114. #392C2×C127semidirect
115. #392C2×C127semidirect
116. #386C2×C127semidirect
117. #383C2×C127semidirect
161. C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#124×C1272
C2×C4×D4×C1272
#397×C1272
#393×C1272
#392×C1272
#386×C1272
#383×C1272
#2631127