Finite Group Databasebeta

← C127×#13342 #44577015

Subgroups — C127×#13342 #44577015

Order 32512.

558 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 122 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×#510×C127C212yesmaximal, complemented
2C127×#3150C211yesmaximal, characteristic
3C2×#360×C127C212yesmaximal, complemented
4C127×#1971C211yesmaximal, characteristic, complemented
5C127×#1377C211yesmaximal, characteristic, complemented
6C2×#135×C127—21no—
7C2×#135×C127C2212yes—
8C22×Q16×C127C2211yescharacteristic
9#532×C127—21no—
10#510×C127—24no—
11#510×C127—24no—
12C2×Q32×C127—22no—
13C2×SD32×C127—22no—
14C2×M32×C127C2212yes—
15#497×C127—22no—
16#496×C127—22no—
17C2×#106×C127C2211yescharacteristic
18C2×#103×C127—21no—
19C2×#97×C127C2211yescharacteristic
20#391×C127—21no—
21#360×C127—24no—
22C2×#41×C127—22no—
23C22×Q8×C127—21no—
24#135×C127—22no—
25#135×C127—22no—
26#135×C127—22no—
27#135×C127—22no—
28#135×C127—42no—
29#135×C127—42no—
30C2×Q16×C127—21no—
31C2×Q16×C127—22no—
32C2×Q16×C127—22no—
33C2×Q16×C127—21no—
34C2×Q16×C127D412yes—
35C2×Q16×C127D412yescomplemented
36C2×SD16×C127—22no—
37C2×SD16×C127—22no—
38C2×D8×C127—21no—
39C2×D8×C127D412yes—
40C22×C8×C127D411yescharacteristic
41C22×C8×C127C2311yescharacteristic
42C4⋊Q8×C127—21no—
43#127×C127—21no—
44#124×C127—22no—
45#121×C127—42no—
46C4×D4×C127—42no—
47C2×C4⋊C4×C127—21no—
48C2×C42×C127D411yescharacteristic
49Q32×C127—28no—
50SD32×C127—28no—
51M32×C127—22no—
52M32×C127D414yescomplemented
53C2×C16×C127—22no—
54#106×C127—22no—
55#103×C127—42no—
56#97×C127—22no—
57#95×C127—42no—
58#41×C127—42no—
59#41×C127—42no—
60#41×C127—42no—
61#41×C127—42no—
62C2×Q8×C127—22no—
63C2×Q8×C127—22no—
64C2×Q8×C127—41no—
65C2×Q8×C127—41no—
66C2×Q8×C127—41no—
67C2×Q8×C127—41no—
68C2×Q8×C127—41no—
69C2×D4×C127—22no—
70C2×D4×C127—42no—
71C2×D4×C127—42no—
72C22×C4×C127—41no—
73C22×C4×C127—42no—
74C22×C4×C127—41no—
75C22×C4×C127D812yes—
76C22×C4×C127C2×D411yescharacteristic
77Q16×C127—24no—
78Q16×C127—22no—
79Q16×C127—22no—
80Q16×C127—42no—
81Q16×C127—42no—
82SD16×C127—44no—
83SD16×C127—42no—
84SD16×C127—42no—
85D8×C127—22no—
86D8×C127—22no—
87D8×C127—42no—
88C2×C8×C127—21no—
89C2×C8×C127—21no—
90C2×C8×C127D812yes—
91C2×C8×C127C2×D411yescharacteristic
92C2×C8×C127C2×D412yes—
93C2×C8×C127C2×D412yes—
94C2×C8×C127#4111yescharacteristic
95C4⋊C4×C127—42no—
96C4⋊C4×C127—41no—
97C4⋊C4×C127—41no—
98C22⋊C4×C127—44no—
99C42×C127—21no—
100C42×C127—21no—
101C16×C127—24no—
102C23×C127—42no—
103C23×C127C2×D811yescharacteristic
104Q8×C127—42no—
105Q8×C127—42no—
106Q8×C127—42no—
107Q8×C127—42no—
108Q8×C127—81no—
109Q8×C127—81no—
110D4×C127—42no—
111D4×C127—42no—
112D4×C127—84no—
113D4×C127—82no—
114D4×C127—82no—
115C2×C4×C127—22no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—42no—
119C2×C4×C127—81no—
120C2×C4×C127—81no—
121C2×C4×C127—82no—
122C2×C4×C127—82no—
123C2×C4×C127—82no—
124C2×C4×C127—81no—
125C2×C4×C127#12011yescharacteristic
126C2×C4×C127#12111yescharacteristic
127C2×C4×C127#12711yescharacteristic
128C2×C4×C127C2×D811yescharacteristic
129C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
130C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
131C8×C127—22no—
132C8×C127#12012yes—
133C8×C127#12112yes—
134C22×C127—22no—
135C22×C127—84no—
136C22×C127—82no—
137C22×C127#44711yescharacteristic
138C22×C127#45911yescharacteristic
139C22×C127#49311yescharacteristic
140C4×C127—42no—
141C4×C127—82no—
142C4×C127—82no—
143C4×C127—161no—
144C4×C127#39211yescharacteristic
145C4×C127#44711yescharacteristic
146C4×C127#45911yescharacteristic
147C4×C127#49611yescharacteristic
148#13342C12711yesmaximal, characteristic, direct product factor, Sylow
149C2×C127—21no—
150C2×C127—21no—
151C2×C127—162no—
152C2×C127#201111yescharacteristic
153C2×C127#219112yes—
154C2×#510C2×C12712yescomplemented
155#3150C2×C12711yescharacteristic
156C2×#360C2×C12712yescomplemented
157#1971C2×C12711yescharacteristic, complemented
158#1377C2×C12711yescharacteristic, complemented
159C127#1334211yescharacteristic, direct product factor, Sylow
160C2×#135—21no—
161C2×#135C22×C12712yes—
162C22×Q16C22×C12711yescharacteristic
163#532—21no—
164#510—24no—
165#510—24no—
166C2×Q32—22no—
167C2×SD32—22no—
168C2×M32C22×C12712yes—
169#497—22no—
170#496—22no—
171C2×#106C22×C12711yescharacteristic
172C2×#103—21no—
173C2×#97C22×C12711yescharacteristic
174#391—21no—
175#360—24no—
176C2×#41—22no—
177C22×Q8—21no—
178#135—22no—
179#135—22no—
180#135—22no—
181#135—22no—
182#135—42no—
183#135—42no—
184C2×Q16—21no—
185C2×Q16—22no—
186C2×Q16—22no—
187C2×Q16—21no—
188C2×Q16D4×C12712yes—
189C2×Q16D4×C12712yescomplemented
190C2×SD16—22no—
191C2×SD16—22no—
192C2×D8—21no—
193C2×D8D4×C12712yes—
194C22×C8D4×C12711yescharacteristic
195C22×C8C23×C12711yescharacteristic
196C4⋊Q8—21no—
197#127—21no—
198#124—22no—
199#121—42no—
200C4×D4—42no—
201C2×C4⋊C4—21no—
202C2×C42D4×C12711yescharacteristic
203Q32—28no—
204SD32—28no—
205M32—22no—
206M32D4×C12714yescomplemented
207C2×C16—22no—
208#106—22no—
209#103—42no—
210#97—22no—
211#95—42no—
212#41—42no—
213#41—42no—
214#41—42no—
215#41—42no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×D4—22no—
224C2×D4—42no—
225C2×D4—42no—
226C22×C4—41no—
227C22×C4—42no—
228C22×C4—41no—
229C22×C4D8×C12712yes—
230C22×C4C2×D4×C12711yescharacteristic
231Q16—24no—
232Q16—22no—
233Q16—22no—
234Q16—42no—
235Q16—42no—
236SD16—44no—
237SD16—42no—
238SD16—42no—
239D8—22no—
240D8—22no—
241D8—42no—
242C2×C8—21no—
243C2×C8—21no—
244C2×C8D8×C12712yes—
245C2×C8C2×D4×C12711yescharacteristic
246C2×C8C2×D4×C12712yes—
247C2×C8C2×D4×C12712yes—
248C2×C8#41×C12711yescharacteristic
249C4⋊C4—42no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C22⋊C4—44no—
253C42—21no—
254C42—21no—
255C16—24no—
256C23—42no—
257C23C2×D8×C12711yescharacteristic
258Q8—42no—
259Q8—42no—
260Q8—42no—
261Q8—42no—
262Q8—81no—
263Q8—81no—
264D4—42no—
265D4—42no—
266D4—84no—
267D4—82no—
268D4—82no—
269C2×C4—22no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4—82no—
278C2×C4—81no—
279C2×C4#120×C12711yescharacteristic
280C2×C4#121×C12711yescharacteristic
281C2×C4#127×C12711yescharacteristic
282C2×C4C2×D8×C12711yescharacteristic
283C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
284C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
285C8—22no—
286C8#120×C12712yes—
287C8#121×C12712yes—
288C22—22no—
289C22—84no—
290C22—82no—
291C22#447×C12711yescharacteristic
292C22#459×C12711yescharacteristic
293C22#493×C12711yescharacteristic
294C4—42no—
295C4—82no—
296C4—82no—
297C4—161no—
298C4#392×C12711yescharacteristic
299C4#447×C12711yescharacteristic
300C4#459×C12711yescharacteristic
301C4#496×C12711yescharacteristic
302C2—21no—
303C2—21no—
304C2—162no—
305C2C127×#201111yescharacteristic
306C2C127×#219112yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3150C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×#97×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#510×C127C2semidirect
3. C2×#360×C127C2semidirect
4. C127×#1971C2semidirect
5. C127×#1377C2semidirect
35. C2×Q16×C127D4semidirect
52. M32×C127D4semidirect
148. #13342C127direct (N × H)
154. C2×#510C2×C127semidirect
156. C2×#360C2×C127semidirect
157. #1971C2×C127semidirect
158. #1377C2×C127semidirect
159. C127#13342direct (N × H)
189. C2×Q16D4×C127semidirect
206. M32D4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#510×C1272
C127×#31502
C2×#360×C1272
C127×#19712
C127×#13772
#13342127