Finite Group Databasebeta

← C127×#13676 #44577349

Subgroups — C127×#13676 #44577349

Order 32512.

394 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3291C211yesmaximal, characteristic
2C127×#2198C212yesmaximal, complemented
3C127×#1630C211yesmaximal, characteristic, complemented
4C127×#1409C211yesmaximal, characteristic, complemented
5C127×#1402C212yesmaximal, complemented
6#577×C127—22no—
7#559×C127C2211yescharacteristic
8#492×C127C2211yescharacteristic
9#477×C127—21no—
10#471×C127—22no—
11#458×C127—22no—
12#456×C127—21no—
13#453×C127C2212yes—
14#443×C127—21no—
15#353×C127C2212yes—
16#352×C127—22no—
17#331×C127C2211yescharacteristic
18Q8⋊C8×C127—21no—
192−1+4×C127—21no—
20H(5,𝔽2)×C127D412yes—
21#137×C127—22no—
22#137×C127—42no—
23AGL(1,ℤ/8)×C127—22no—
24AGL(1,ℤ/8)×C127—42no—
25#135×C127—21no—
26#135×C127—42no—
27C2×SD16×C127—22no—
28C2×SD16×C127—42no—
29#131×C127—22no—
30C4⋊Q8×C127D411yescharacteristic
31#127×C127C2311yescharacteristic
32#124×C127—41no—
33#123×C127—22no—
34#123×C127—42no—
35C4×Q8×C127—21no—
36C4×D4×C127—41no—
37#108×C127—21no—
38#107×C127—41no—
39C4⋊C8×C127—21no—
40#104×C127—22no—
41#103×C127—41no—
42#101×C127—21no—
43#100×C127D412yes—
44#99×C127—22no—
45#99×C127—42no—
46C4×C8×C127D411yescharacteristic
47#41×C127—22no—
48#41×C127—42no—
49#41×C127—42no—
50#41×C127—81no—
51C2×Q8×C127—21no—
52C2×Q8×C127—21no—
53C2×Q8×C127—42no—
54C2×D4×C127—41no—
55C2×D4×C127—42no—
56C2×D4×C127—42no—
57C2×D4×C127C2×D412yes—
58C22×C4×C127—22no—
59Q16×C127—21no—
60Q16×C127—42no—
61SD16×C127—22no—
62SD16×C127—42no—
63SD16×C127—42no—
64SD16×C127—42no—
65SD16×C127—42no—
66D8×C127—21no—
67D8×C127—42no—
68M16×C127—22no—
69M16×C127—42no—
70C2×C8×C127—21no—
71C2×C8×C127—42no—
72C2×C8×C127—41no—
73C4⋊C4×C127—21no—
74C4⋊C4×C127—41no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127—82no—
77C22⋊C4×C127—22no—
78C22⋊C4×C127—42no—
79C22⋊C4×C127—42no—
80C22⋊C4×C127—82no—
81C42×C127—41no—
82C42×C127C2×D411yescharacteristic
83C23×C127—22no—
84C23×C127—42no—
85Q8×C127—22no—
86Q8×C127—42no—
87Q8×C127—41no—
88Q8×C127—41no—
89Q8×C127—81no—
90D4×C127—22no—
91D4×C127—42no—
92D4×C127—42no—
93D4×C127—81no—
94D4×C127—81no—
95D4×C127—82no—
96D4×C127—81no—
97C2×C4×C127—21no—
98C2×C4×C127—42no—
99C2×C4×C127—42no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—82no—
104C2×C4×C127—82no—
105C2×C4×C127—82no—
106C2×C4×C127#12011yescharacteristic
107C8×C127—21no—
108C8×C127—21no—
109C8×C127—42no—
110C8×C127—81no—
111C22×C127—42no—
112C22×C127—42no—
113C22×C127—82no—
114C22×C127—162no—
115C22×C127B(4,𝔽2)11yescharacteristic
116C4×C127—21no—
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—42no—
120C4×C127—81no—
121C4×C127—81no—
122C4×C127—81no—
123C4×C127—162no—
124#13676C12711yesmaximal, characteristic, direct product factor, Sylow
125C2×C127—21no—
126C2×C127—82no—
127C2×C127—82no—
128C2×C127#219611yescharacteristic
129#3291C2×C12711yescharacteristic
130#2198C2×C12712yescomplemented
131#1630C2×C12711yescharacteristic, complemented
132#1409C2×C12711yescharacteristic, complemented
133#1402C2×C12712yescomplemented
134C127#1367611yescharacteristic, direct product factor, Sylow
135#577—22no—
136#559C22×C12711yescharacteristic
137#492C22×C12711yescharacteristic
138#477—21no—
139#471—22no—
140#458—22no—
141#456—21no—
142#453C22×C12712yes—
143#443—21no—
144#353C22×C12712yes—
145#352—22no—
146#331C22×C12711yescharacteristic
147Q8⋊C8—21no—
1482−1+4—21no—
149H(5,𝔽2)D4×C12712yes—
150#137—22no—
151#137—42no—
152AGL(1,ℤ/8)—22no—
153AGL(1,ℤ/8)—42no—
154#135—21no—
155#135—42no—
156C2×SD16—22no—
157C2×SD16—42no—
158#131—22no—
159C4⋊Q8D4×C12711yescharacteristic
160#127C23×C12711yescharacteristic
161#124—41no—
162#123—22no—
163#123—42no—
164C4×Q8—21no—
165C4×D4—41no—
166#108—21no—
167#107—41no—
168C4⋊C8—21no—
169#104—22no—
170#103—41no—
171#101—21no—
172#100D4×C12712yes—
173#99—22no—
174#99—42no—
175C4×C8D4×C12711yescharacteristic
176#41—22no—
177#41—42no—
178#41—42no—
179#41—81no—
180C2×Q8—21no—
181C2×Q8—21no—
182C2×Q8—42no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—42no—
186C2×D4C2×D4×C12712yes—
187C22×C4—22no—
188Q16—21no—
189Q16—42no—
190SD16—22no—
191SD16—42no—
192SD16—42no—
193SD16—42no—
194SD16—42no—
195D8—21no—
196D8—42no—
197M16—22no—
198M16—42no—
199C2×C8—21no—
200C2×C8—42no—
201C2×C8—41no—
202C4⋊C4—21no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—82no—
206C22⋊C4—22no—
207C22⋊C4—42no—
208C22⋊C4—42no—
209C22⋊C4—82no—
210C42—41no—
211C42C2×D4×C12711yescharacteristic
212C23—22no—
213C23—42no—
214Q8—22no—
215Q8—42no—
216Q8—41no—
217Q8—41no—
218Q8—81no—
219D4—22no—
220D4—42no—
221D4—42no—
222D4—81no—
223D4—81no—
224D4—82no—
225D4—81no—
226C2×C4—21no—
227C2×C4—42no—
228C2×C4—42no—
229C2×C4—41no—
230C2×C4—41no—
231C2×C4—41no—
232C2×C4—82no—
233C2×C4—82no—
234C2×C4—82no—
235C2×C4#120×C12711yescharacteristic
236C8—21no—
237C8—21no—
238C8—42no—
239C8—81no—
240C22—42no—
241C22—42no—
242C22—82no—
243C22—162no—
244C22B(4,𝔽2)×C12711yescharacteristic
245C4—21no—
246C4—21no—
247C4—21no—
248C4—42no—
249C4—81no—
250C4—81no—
251C4—81no—
252C4—162no—
253C2—21no—
254C2—82no—
255C2—82no—
256C2C127×#219611yescharacteristic
subgroupG/H
center, socleC2×C127#2196
commutator, Frattini, cosocle#127C23×C127
prime radicalC127×#3291C2
LCSG ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ #127×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2198C2semidirect
3. C127×#1630C2semidirect
4. C127×#1409C2semidirect
5. C127×#1402C2semidirect
124. #13676C127direct (N × H)
130. #2198C2×C127semidirect
131. #1630C2×C127semidirect
132. #1409C2×C127semidirect
133. #1402C2×C127semidirect
134. C127#13676direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32912
C127×#21982
C127×#16302
C127×#14092
C127×#14022
#13676127