Finite Group Databasebeta

← C127×#12298 #44575971

Subgroups — C127×#12298 #44575971

Order 32512.

376 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3192C211yesmaximal, characteristic
2C127×#2238C212yesmaximal, complemented
3C127×#2091C211yesmaximal, characteristic, complemented
4C127×#1341C212yesmaximal, complemented
5C127×#1320C211yesmaximal, characteristic, complemented
6C2×#121×C127—21no—
7#481×C127—22no—
8#469×C127—21no—
9#469×C127C2212yes—
10#447×C127—22no—
11#430×C127—21no—
12#430×C127C2211yescharacteristic, complemented
13#407×C127—21no—
14#407×C127C2211yescharacteristic
15#365×C127—22no—
16#361×C127—22no—
17#349×C127C2212yescomplemented
18#338×C127C2211yescharacteristic, complemented
19C22×D4×C127—21no—
20C22×D4×C127—41no—
21C23×C4×C127D411yescharacteristic
22C2×D8×C127—21no—
23C2×D8×C127D412yes—
24C2×M16×C127—21no—
25C2×M16×C127—21no—
26C2×M16×C127D411yescharacteristic
27C2×M16×C127C2311yescharacteristic
28#121×C127—22no—
29#121×C127—22no—
30#121×C127—41no—
31#121×C127—41no—
32C2×C4⋊C4×C127—21no—
33C2×C22⋊C4×C127—41no—
34M32×C127—22no—
35#108×C127—22no—
36#108×C127—42no—
37#106×C127—21no—
38#106×C127D412yes—
39#102×C127—22no—
40#102×C127—22no—
41C22⋊C8×C127—22no—
42C22⋊C8×C127—42no—
43C24×C127—41no—
44C24×C127SD1611yescharacteristic
45C2×D4×C127—22no—
46C2×D4×C127—41no—
47C2×D4×C127—42no—
48C2×D4×C127—42no—
49C2×D4×C127—81no—
50C2×D4×C127—81no—
51C2×D4×C127—84no—
52C22×C4×C127—21no—
53C22×C4×C127—21no—
54C22×C4×C127—41no—
55C22×C4×C127—41no—
56C22×C4×C127—41no—
57C22×C4×C127SD1611yescharacteristic
58C22×C4×C127C2×D411yescharacteristic
59D8×C127—42no—
60D8×C127—81no—
61M16×C127—41no—
62M16×C127—41no—
63M16×C127—41no—
64M16×C127—41no—
65M16×C127—41no—
66M16×C127—41no—
67C2×C8×C127—22no—
68C2×C8×C127—21no—
69C2×C8×C127—41no—
70C2×C8×C127C2×D412yes—
71C4⋊C4×C127—22no—
72C4⋊C4×C127—41no—
73C22⋊C4×C127—42no—
74C22⋊C4×C127—42no—
75C16×C127—42no—
76C23×C127—21no—
77C23×C127—41no—
78C23×C127—42no—
79C23×C127—42no—
80C23×C127—82no—
81C23×C127—82no—
82C23×C127—82no—
83C23×C127C2×SD1611yescharacteristic
84D4×C127—42no—
85D4×C127—42no—
86D4×C127—82no—
87D4×C127—84no—
88D4×C127—84no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—41no—
93C2×C4×C127—41no—
94C2×C4×C127—41no—
95C2×C4×C127—42no—
96C2×C4×C127—81no—
97C2×C4×C127—82no—
98C2×C4×C127#12011yescharacteristic
99C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
100C8×C127—22no—
101C8×C127—42no—
102C8×C127—41no—
103C8×C127—81no—
104C22×C127—22no—
105C22×C127—21no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127—42no—
109C22×C127—41no—
110C22×C127—82no—
111C22×C127—82no—
112C22×C127—81no—
113C22×C127—82no—
114C22×C127—164no—
115C22×C127#45011yescharacteristic
116C4×C127—21no—
117C4×C127—41no—
118C4×C127—82no—
119C4×C127#45311yescharacteristic
120C4×C127#45811yescharacteristic
121#12298C12711yesmaximal, characteristic, direct product factor, Sylow
122C2×C127—21no—
123C2×C127—41no—
124C2×C127—42no—
125C2×C127—162no—
126C2×C127#161311yescharacteristic
127#3192C2×C12711yescharacteristic
128#2238C2×C12712yescomplemented
129#2091C2×C12711yescharacteristic, complemented
130#1341C2×C12712yescomplemented
131#1320C2×C12711yescharacteristic, complemented
132C127#1229811yescharacteristic, direct product factor, Sylow
133C2×#121—21no—
134#481—22no—
135#469—21no—
136#469C22×C12712yes—
137#447—22no—
138#430—21no—
139#430C22×C12711yescharacteristic, complemented
140#407—21no—
141#407C22×C12711yescharacteristic
142#365—22no—
143#361—22no—
144#349C22×C12712yescomplemented
145#338C22×C12711yescharacteristic, complemented
146C22×D4—21no—
147C22×D4—41no—
148C23×C4D4×C12711yescharacteristic
149C2×D8—21no—
150C2×D8D4×C12712yes—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16D4×C12711yescharacteristic
154C2×M16C23×C12711yescharacteristic
155#121—22no—
156#121—22no—
157#121—41no—
158#121—41no—
159C2×C4⋊C4—21no—
160C2×C22⋊C4—41no—
161M32—22no—
162#108—22no—
163#108—42no—
164#106—21no—
165#106D4×C12712yes—
166#102—22no—
167#102—22no—
168C22⋊C8—22no—
169C22⋊C8—42no—
170C24—41no—
171C24SD16×C12711yescharacteristic
172C2×D4—22no—
173C2×D4—41no—
174C2×D4—42no—
175C2×D4—42no—
176C2×D4—81no—
177C2×D4—81no—
178C2×D4—84no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4SD16×C12711yescharacteristic
185C22×C4C2×D4×C12711yescharacteristic
186D8—42no—
187D8—81no—
188M16—41no—
189M16—41no—
190M16—41no—
191M16—41no—
192M16—41no—
193M16—41no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8C2×D4×C12712yes—
198C4⋊C4—22no—
199C4⋊C4—41no—
200C22⋊C4—42no—
201C22⋊C4—42no—
202C16—42no—
203C23—21no—
204C23—41no—
205C23—42no—
206C23—42no—
207C23—82no—
208C23—82no—
209C23—82no—
210C23C2×SD16×C12711yescharacteristic
211D4—42no—
212D4—42no—
213D4—82no—
214D4—84no—
215D4—84no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—81no—
224C2×C4—82no—
225C2×C4#120×C12711yescharacteristic
226C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
227C8—22no—
228C8—42no—
229C8—41no—
230C8—81no—
231C22—22no—
232C22—21no—
233C22—41no—
234C22—41no—
235C22—42no—
236C22—41no—
237C22—82no—
238C22—82no—
239C22—81no—
240C22—82no—
241C22—164no—
242C22#450×C12711yescharacteristic
243C4—21no—
244C4—41no—
245C4—82no—
246C4#453×C12711yescharacteristic
247C4#458×C12711yescharacteristic
248C2—21no—
249C2—42no—
250C2—41no—
251C2—162no—
252C2C127×#161311yescharacteristic
subgroupG/H
center, socleC2×C127#1613
commutator, Frattini, cosocleC2×M16C23×C127
prime radicalC127×#3192C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2238C2semidirect
3. C127×#2091C2semidirect
4. C127×#1341C2semidirect
5. C127×#1320C2semidirect
12. #430×C127C22semidirect
17. #349×C127C22semidirect
18. #338×C127C22semidirect
121. #12298C127direct (N × H)
128. #2238C2×C127semidirect
129. #2091C2×C127semidirect
130. #1341C2×C127semidirect
131. #1320C2×C127semidirect
132. C127#12298direct (N × H)
139. #430C22×C127semidirect
144. #349C22×C127semidirect
145. #338C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31922
C127×#22382
C127×#20912
C127×#13412
C127×#13202
#12298127