Finite Group Databasebeta

← C127×#13683 #44577356

Subgroups — C127×#13683 #44577356

Order 32512.

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

normal subgroups only

This group has 392 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×#2205C212yesmaximal, complemented
3C127×#1623C211yesmaximal, characteristic, complemented
4C127×#1410C211yesmaximal, characteristic, complemented
5C127×#1405C212yesmaximal, complemented
6#577×C127—22no—
7Q82×C127C2211yescharacteristic, complemented
8#492×C127C2211yescharacteristic
9#471×C127—22no—
10#460×C127—21no—
11#456×C127—22no—
12#456×C127C2212yes—
13#453×C127—21no—
14#443×C127—21no—
15#356×C127—22no—
16#356×C127C2212yescomplemented
17#333×C127C2211yescharacteristic, complemented
18Q8⋊C8×C127—21no—
192−1+4×C127D412yes—
20H(5,𝔽2)×C127—21no—
21#137×C127—22no—
22#137×C127—42no—
23#137×C127—42no—
24AGL(1,ℤ/8)×C127—22no—
25AGL(1,ℤ/8)×C127—42no—
26#135×C127—21no—
27#135×C127—42no—
28C2×SD16×C127—22no—
29C2×SD16×C127—42no—
30C2×SD16×C127—41no—
31#131×C127—22no—
32C4⋊Q8×C127—22no—
33C4⋊Q8×C127—41no—
34C4⋊Q8×C127C2311yescharacteristic
35#127×C127D411yescharacteristic
36C4×Q8×C127—21no—
37C4×Q8×C127—41no—
38#108×C127—21no—
39C4⋊C8×C127—21no—
40#104×C127—22no—
41#104×C127—42no—
42#102×C127—41no—
43#101×C127—22no—
44#101×C127D412yes—
45#100×C127—21no—
46C4×C8×C127D411yescharacteristic
47#41×C127—22no—
48#41×C127—42no—
49#41×C127—42no—
50#41×C127—42no—
51C2×Q8×C127—21no—
52C2×Q8×C127—22no—
53C2×Q8×C127—42no—
54C2×Q8×C127C2×D412yes—
55C2×D4×C127—21no—
56C2×D4×C127—41no—
57C2×D4×C127—42no—
58C2×D4×C127—81no—
59Q16×C127—21no—
60Q16×C127—42no—
61Q16×C127—82no—
62SD16×C127—22no—
63SD16×C127—42no—
64SD16×C127—42no—
65SD16×C127—42no—
66SD16×C127—42no—
67SD16×C127—82no—
68SD16×C127—82no—
69D8×C127—21no—
70D8×C127—42no—
71M16×C127—22no—
72M16×C127—42no—
73M16×C127—42no—
74C2×C8×C127—21no—
75C2×C8×C127—42no—
76C2×C8×C127—41no—
77C4⋊C4×C127—21no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—42no—
80C4⋊C4×C127—41no—
81C42×C127—22no—
82C42×C127—41no—
83C42×C127C2×D411yescharacteristic
84C23×C127—41no—
85C23×C127—81no—
86Q8×C127—22no—
87Q8×C127—22no—
88Q8×C127—42no—
89Q8×C127—41no—
90Q8×C127—42no—
91Q8×C127—82no—
92D4×C127—22no—
93D4×C127—42no—
94D4×C127—42no—
95D4×C127—42no—
96D4×C127—81no—
97D4×C127—81no—
98D4×C127—82no—
99C2×C4×C127—21no—
100C2×C4×C127—22no—
101C2×C4×C127—42no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—82no—
105C2×C4×C127—81no—
106C2×C4×C127#12011yescharacteristic
107C8×C127—21no—
108C8×C127—21no—
109C8×C127—42no—
110C8×C127—81no—
111C8×C127—82no—
112C22×C127—42no—
113C22×C127—81no—
114C22×C127—161no—
115C22×C127—321no—
116C22×C127B(4,𝔽2)11yescharacteristic
117C4×C127—21no—
118C4×C127—21no—
119C4×C127—21no—
120C4×C127—42no—
121C4×C127—42no—
122C4×C127—42no—
123C4×C127—81no—
124#13683C12711yesmaximal, characteristic, direct product factor, Sylow
125C2×C127—21no—
126C2×C127—82no—
127C2×C127—161no—
128C2×C127#219611yescharacteristic
129#3291C2×C12711yescharacteristic
130#2205C2×C12712yescomplemented
131#1623C2×C12711yescharacteristic, complemented
132#1410C2×C12711yescharacteristic, complemented
133#1405C2×C12712yescomplemented
134C127#1368311yescharacteristic, direct product factor, Sylow
135#577—22no—
136Q82C22×C12711yescharacteristic, complemented
137#492C22×C12711yescharacteristic
138#471—22no—
139#460—21no—
140#456—22no—
141#456C22×C12712yes—
142#453—21no—
143#443—21no—
144#356—22no—
145#356C22×C12712yescomplemented
146#333C22×C12711yescharacteristic, complemented
147Q8⋊C8—21no—
1482−1+4D4×C12712yes—
149H(5,𝔽2)—21no—
150#137—22no—
151#137—42no—
152#137—42no—
153AGL(1,ℤ/8)—22no—
154AGL(1,ℤ/8)—42no—
155#135—21no—
156#135—42no—
157C2×SD16—22no—
158C2×SD16—42no—
159C2×SD16—41no—
160#131—22no—
161C4⋊Q8—22no—
162C4⋊Q8—41no—
163C4⋊Q8C23×C12711yescharacteristic
164#127D4×C12711yescharacteristic
165C4×Q8—21no—
166C4×Q8—41no—
167#108—21no—
168C4⋊C8—21no—
169#104—22no—
170#104—42no—
171#102—41no—
172#101—22no—
173#101D4×C12712yes—
174#100—21no—
175C4×C8D4×C12711yescharacteristic
176#41—22no—
177#41—42no—
178#41—42no—
179#41—42no—
180C2×Q8—21no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8C2×D4×C12712yes—
184C2×D4—21no—
185C2×D4—41no—
186C2×D4—42no—
187C2×D4—81no—
188Q16—21no—
189Q16—42no—
190Q16—82no—
191SD16—22no—
192SD16—42no—
193SD16—42no—
194SD16—42no—
195SD16—42no—
196SD16—82no—
197SD16—82no—
198D8—21no—
199D8—42no—
200M16—22no—
201M16—42no—
202M16—42no—
203C2×C8—21no—
204C2×C8—42no—
205C2×C8—41no—
206C4⋊C4—21no—
207C4⋊C4—41no—
208C4⋊C4—42no—
209C4⋊C4—41no—
210C42—22no—
211C42—41no—
212C42C2×D4×C12711yescharacteristic
213C23—41no—
214C23—81no—
215Q8—22no—
216Q8—22no—
217Q8—42no—
218Q8—41no—
219Q8—42no—
220Q8—82no—
221D4—22no—
222D4—42no—
223D4—42no—
224D4—42no—
225D4—81no—
226D4—81no—
227D4—82no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—42no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—82no—
234C2×C4—81no—
235C2×C4#120×C12711yescharacteristic
236C8—21no—
237C8—21no—
238C8—42no—
239C8—81no—
240C8—82no—
241C22—42no—
242C22—81no—
243C22—161no—
244C22—321no—
245C22B(4,𝔽2)×C12711yescharacteristic
246C4—21no—
247C4—21no—
248C4—21no—
249C4—42no—
250C4—42no—
251C4—42no—
252C4—81no—
253C2—21no—
254C2—82no—
255C2—161no—
256C2C127×#219611yescharacteristic
subgroupG/H
center, socleC2×C127#2196
commutator, Frattini, cosocleC4⋊Q8C23×C127
prime radicalC127×#3291C2
LCSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ C4⋊Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2205C2semidirect
3. C127×#1623C2semidirect
4. C127×#1410C2semidirect
5. C127×#1405C2semidirect
7. Q82×C127C22semidirect
16. #356×C127C22semidirect
17. #333×C127C22semidirect
124. #13683C127direct (N × H)
130. #2205C2×C127semidirect
131. #1623C2×C127semidirect
132. #1410C2×C127semidirect
133. #1405C2×C127semidirect
134. C127#13683direct (N × H)
136. Q82C22×C127semidirect
145. #356C22×C127semidirect
146. #333C22×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32912
C127×#22052
C127×#16232
C127×#14102
C127×#14052
#13683127