Finite Group Databasebeta

← C127×#12216 #44575889

Subgroups — C127×#12216 #44575889

Order 32512.

388 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 74 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
1C127×#2607C211yesmaximal, characteristic
2C127×#2030C211yesmaximal, characteristic
3C127×#1614C212yesmaximal
4C127×#1352C212yesmaximal
5C127×#1317C211yesmaximal, characteristic
6#543×C127C2211yescharacteristic
7#484×C127—22no—
8#399×C127C2212yes—
9#393×C127—21no—
10#391×C127—22no—
11#387×C127—22no—
12#381×C127—21no—
13#381×C127C2211yescharacteristic
14C4×C22⋊C4×C127—21no—
15#342×C127—21no—
16#342×C127C2211yescharacteristic
17#328×C127—22no—
18#323×C127C2212yes—
19C22×Q8×C127—21no—
20C23×C4×C127D411yescharacteristic
21C22⋊Q8×C127—22no—
22C22⋊Q8×C127—42no—
23C2×C4⋊C4×C127—22no—
24C2×C4⋊C4×C127D412yes—
25C2×C22⋊C4×C127—21no—
26C2×C22⋊C4×C127—22no—
27C2×C22⋊C4×C127—21no—
28C2×C22⋊C4×C127D411yescharacteristic
29C2×C22⋊C4×C127C2311yescharacteristic
30C2×C42×C127—21no—
31C2×C42×C127—21no—
32#107×C127—42no—
33#103×C127—42no—
34C22⋊C8×C127—22no—
35#95×C127—22no—
36#95×C127—21no—
37#95×C127—22no—
38#95×C127—22no—
39#95×C127—42no—
40#95×C127D412yes—
41C24×C127C2×D411yescharacteristic
42C2×Q8×C127—22no—
43C2×Q8×C127—44no—
44C2×Q8×C127—81no—
45C22×C4×C127—21no—
46C22×C4×C127—22no—
47C22×C4×C127—21no—
48C22×C4×C127—21no—
49C22×C4×C127—22no—
50C22×C4×C127—22no—
51C22×C4×C127—41no—
52C22×C4×C127—41no—
53C22×C4×C127—41no—
54C22×C4×C127Q1612yes—
55C22×C4×C127C2×D412yes—
56C2×C8×C127—42no—
57C4⋊C4×C127—22no—
58C4⋊C4×C127—42no—
59C4⋊C4×C127—44no—
60C4⋊C4×C127—42no—
61C22⋊C4×C127—41no—
62C22⋊C4×C127—44no—
63C22⋊C4×C127—41no—
64C22⋊C4×C127—41no—
65C22⋊C4×C127—41no—
66C22⋊C4×C127—41no—
67C22⋊C4×C127—41no—
68C42×C127—81no—
69C42×C127—81no—
70C23×C127—21no—
71C23×C127—21no—
72C23×C127—42no—
73C23×C127#12011yescharacteristic
74C23×C127C2×Q1611yescharacteristic
75C23×C127#13711yescharacteristic
76Q8×C127—84no—
77Q8×C127—82no—
78C2×C4×C127—24no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—42no—
82C2×C4×C127—41no—
83C2×C4×C127—42no—
84C2×C4×C127—41no—
85C2×C4×C127—42no—
86C2×C4×C127—42no—
87C2×C4×C127—42no—
88C2×C4×C127—42no—
89C2×C4×C127—42no—
90C2×C4×C127—41no—
91C2×C4×C127—42no—
92C2×C4×C127—42no—
93C2×C4×C127—81no—
94C2×C4×C127—82no—
95C2×C4×C127—81no—
96C2×C4×C127—82no—
97C8×C127—82no—
98C22×C127—22no—
99C22×C127—21no—
100C22×C127—21no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—42no—
104C22×C127—81no—
105C22×C127#45111yescharacteristic
106C22×C127#45511yescharacteristic
107C22×C127B(4,𝔽2)11yescharacteristic
108C4×C127—44no—
109C4×C127—42no—
110C4×C127—42no—
111C4×C127—81no—
112C4×C127—82no—
113C4×C127—82no—
114C4×C127—161no—
115#12216C12711yesmaximal, characteristic, direct product factor, Sylow
116C2×C127—21no—
117C2×C127—21no—
118C2×C127—41no—
119C2×C127—41no—
120C2×C127#160211yescharacteristic
121C2×C127#219812yes—
122#2607C2×C12711yescharacteristic
123#2030C2×C12711yescharacteristic
124#1614C2×C12712yes—
125#1352C2×C12712yes—
126#1317C2×C12711yescharacteristic
127C127#1221611yescharacteristic, direct product factor, Sylow
128#543C22×C12711yescharacteristic
129#484—22no—
130#399C22×C12712yes—
131#393—21no—
132#391—22no—
133#387—22no—
134#381—21no—
135#381C22×C12711yescharacteristic
136C4×C22⋊C4—21no—
137#342—21no—
138#342C22×C12711yescharacteristic
139#328—22no—
140#323C22×C12712yes—
141C22×Q8—21no—
142C23×C4D4×C12711yescharacteristic
143C22⋊Q8—22no—
144C22⋊Q8—42no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4D4×C12712yes—
147C2×C22⋊C4—21no—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—21no—
150C2×C22⋊C4D4×C12711yescharacteristic
151C2×C22⋊C4C23×C12711yescharacteristic
152C2×C42—21no—
153C2×C42—21no—
154#107—42no—
155#103—42no—
156C22⋊C8—22no—
157#95—22no—
158#95—21no—
159#95—22no—
160#95—22no—
161#95—42no—
162#95D4×C12712yes—
163C24C2×D4×C12711yescharacteristic
164C2×Q8—22no—
165C2×Q8—44no—
166C2×Q8—81no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4—41no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4Q16×C12712yes—
177C22×C4C2×D4×C12712yes—
178C2×C8—42no—
179C4⋊C4—22no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C22⋊C4—41no—
184C22⋊C4—44no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4—41no—
188C22⋊C4—41no—
189C22⋊C4—41no—
190C42—81no—
191C42—81no—
192C23—21no—
193C23—21no—
194C23—42no—
195C23#120×C12711yescharacteristic
196C23C2×Q16×C12711yescharacteristic
197C23#137×C12711yescharacteristic
198Q8—84no—
199Q8—82no—
200C2×C4—24no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—42no—
204C2×C4—41no—
205C2×C4—42no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—42no—
210C2×C4—42no—
211C2×C4—42no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—82no—
219C8—82no—
220C22—22no—
221C22—21no—
222C22—21no—
223C22—41no—
224C22—41no—
225C22—42no—
226C22—81no—
227C22#451×C12711yescharacteristic
228C22#455×C12711yescharacteristic
229C22B(4,𝔽2)×C12711yescharacteristic
230C4—44no—
231C4—42no—
232C4—42no—
233C4—81no—
234C4—82no—
235C4—82no—
236C4—161no—
237C2—21no—
238C2—21no—
239C2—41no—
240C2—41no—
241C2C127×#160211yescharacteristic
242C2C127×#219812yes—
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C22⋊C4C23×C127
prime radicalC24×C127C2×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C22⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
115. #12216C127direct (N × H)
127. C127#12216direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#26072
C127×#20302
C127×#16142
C127×#13522
C127×#13172
#12216127