Finite Group Databasebeta

← C127×#13440 #44577113

Subgroups — C127×#13440 #44577113

Order 32512.

398 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×SD32×C127C211yesmaximal, characteristic
2C127×#2151C211yesmaximal, characteristic, complemented
3C2×#362×C127C211yesmaximal, characteristic, complemented
4C2×#361×C127C211yesmaximal, characteristic
5C127×#1979C211yesmaximal, characteristic, complemented
6C127×#1978C211yesmaximal, characteristic
7C127×#1381C211yesmaximal, characteristic, complemented
8C22×Q16×C127C2211yescharacteristic
9C22×D8×C127C2211yescharacteristic
10C2×SD32×C127—22no—
11C2×SD32×C127—22no—
12C2×SD32×C127—22no—
13C2×M32×C127C2211yescharacteristic
14C22×C16×C127C2211yescharacteristic
15C2×#108×C127C2211yescharacteristic
16C2×#108×C127C2211yescharacteristic
17#419×C127—21no—
18#418×C127—21no—
19C2×#101×C127—21no—
20C2×#100×C127—21no—
21#405×C127C2211yescharacteristic
22C4⋊C16×C127—22no—
23#362×C127—22no—
24#361×C127—22no—
25C22×Q8×C127—21no—
26C22×D4×C127—21no—
27C2×Q16×C127—21no—
28C2×Q16×C127—21no—
29C2×Q16×C127—21no—
30C2×Q16×C127—21no—
31C2×Q16×C127—21no—
32C2×Q16×C127D412yes—
33C2×D8×C127—21no—
34C2×D8×C127—21no—
35C2×D8×C127—21no—
36C2×D8×C127—21no—
37C2×D8×C127—21no—
38C2×D8×C127D412yes—
39C2×M16×C127—21no—
40C2×M16×C127—21no—
41C2×M16×C127D411yescharacteristic
42C22×C8×C127C2311yescharacteristic
43#117×C127D411yescharacteristic
44SD32×C127—28no—
45M32×C127—21no—
46M32×C127—21no—
47C2×C16×C127—21no—
48C2×C16×C127—21no—
49C2×C16×C127—21no—
50C2×C16×C127D412yes—
51#108×C127—21no—
52#108×C127—21no—
53#108×C127—21no—
54#108×C127—21no—
55#103×C127—42no—
56#102×C127—42no—
57#101×C127—42no—
58#100×C127—42no—
59#97×C127—21no—
60C4×C8×C127—21no—
61C24×C127—41no—
62C2×Q8×C127—22no—
63C2×Q8×C127—41no—
64C2×Q8×C127—41no—
65C2×Q8×C127—41no—
66C2×Q8×C127—41no—
67C2×Q8×C127—41no—
68C2×D4×C127—22no—
69C2×D4×C127—41no—
70C2×D4×C127—41no—
71C2×D4×C127—41no—
72C2×D4×C127—41no—
73C2×D4×C127—41no—
74C22×C4×C127—41no—
75C22×C4×C127C2×D411yescharacteristic
76Q16×C127—22no—
77Q16×C127—22no—
78Q16×C127—41no—
79Q16×C127—41no—
80D8×C127—22no—
81D8×C127—22no—
82D8×C127—41no—
83D8×C127—41no—
84M16×C127—21no—
85M16×C127—21no—
86M16×C127—41no—
87M16×C127—41no—
88M16×C127—41no—
89M16×C127—41no—
90C2×C8×C127—21no—
91C2×C8×C127—41no—
92C2×C8×C127—41no—
93C2×C8×C127C2×D411yescharacteristic
94C2×C8×C127C2×D411yescharacteristic
95C2×C8×C127C2×D411yescharacteristic
96C2×C8×C127#4111yescharacteristic
97C2×C8×C127#4111yescharacteristic
98C2×C8×C127#4111yescharacteristic
99C4⋊C4×C127D812yes—
100C22⋊C4×C127—21no—
101C42×C127—21no—
102C16×C127—22no—
103C16×C127—41no—
104C23×C127—42no—
105C23×C127—81no—
106C23×C127—81no—
107C23×C127—162no—
108C23×C127#13511yescharacteristic
109Q8×C127—42no—
110Q8×C127—42no—
111Q8×C127—81no—
112Q8×C127—81no—
113D4×C127—42no—
114D4×C127—42no—
115D4×C127—81no—
116D4×C127—81no—
117C2×C4×C127—22no—
118C2×C4×C127—42no—
119C2×C4×C127—81no—
120C2×C4×C127—81no—
121C2×C4×C127#12111yescharacteristic
122C2×C4×C127#12111yescharacteristic
123C2×C4×C127#12111yescharacteristic
124C2×C4×C127C2×D811yescharacteristic
125C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
127C8×C127—41no—
128C8×C127—81no—
129C8×C127—81no—
130C8×C127#12011yescharacteristic
131C8×C127#12311yescharacteristic
132C8×C127#12311yescharacteristic
133C8×C127#12411yescharacteristic
134C22×C127—22no—
135C22×C127—82no—
136C22×C127—162no—
137C22×C127—162no—
138C22×C127—162no—
139C22×C127#46311yescharacteristic
140C22×C127#46311yescharacteristic
141C22×C127#46611yescharacteristic
142C4×C127—42no—
143C4×C127—82no—
144C4×C127#39411yescharacteristic
145C4×C127#45911yescharacteristic
146C4×C127#45911yescharacteristic
147C4×C127#46911yescharacteristic
148#13440C12711yesmaximal, characteristic, direct product factor, Sylow
149C2×C127—21no—
150C2×C127—21no—
151C2×C127—162no—
152C2×C127#205411yescharacteristic
153C2×C127#222312yes—
154C22×SD32C2×C12711yescharacteristic
155#2151C2×C12711yescharacteristic, complemented
156C2×#362C2×C12711yescharacteristic, complemented
157C2×#361C2×C12711yescharacteristic
158#1979C2×C12711yescharacteristic, complemented
159#1978C2×C12711yescharacteristic
160#1381C2×C12711yescharacteristic, complemented
161C127#1344011yescharacteristic, direct product factor, Sylow
162C22×Q16C22×C12711yescharacteristic
163C22×D8C22×C12711yescharacteristic
164C2×SD32—22no—
165C2×SD32—22no—
166C2×SD32—22no—
167C2×M32C22×C12711yescharacteristic
168C22×C16C22×C12711yescharacteristic
169C2×#108C22×C12711yescharacteristic
170C2×#108C22×C12711yescharacteristic
171#419—21no—
172#418—21no—
173C2×#101—21no—
174C2×#100—21no—
175#405C22×C12711yescharacteristic
176C4⋊C16—22no—
177#362—22no—
178#361—22no—
179C22×Q8—21no—
180C22×D4—21no—
181C2×Q16—21no—
182C2×Q16—21no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×Q16—21no—
186C2×Q16D4×C12712yes—
187C2×D8—21no—
188C2×D8—21no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8—21no—
192C2×D8D4×C12712yes—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16D4×C12711yescharacteristic
196C22×C8C23×C12711yescharacteristic
197#117D4×C12711yescharacteristic
198SD32—28no—
199M32—21no—
200M32—21no—
201C2×C16—21no—
202C2×C16—21no—
203C2×C16—21no—
204C2×C16D4×C12712yes—
205#108—21no—
206#108—21no—
207#108—21no—
208#108—21no—
209#103—42no—
210#102—42no—
211#101—42no—
212#100—42no—
213#97—21no—
214C4×C8—21no—
215C24—41no—
216C2×Q8—22no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—22no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C22×C4—41no—
229C22×C4C2×D4×C12711yescharacteristic
230Q16—22no—
231Q16—22no—
232Q16—41no—
233Q16—41no—
234D8—22no—
235D8—22no—
236D8—41no—
237D8—41no—
238M16—21no—
239M16—21no—
240M16—41no—
241M16—41no—
242M16—41no—
243M16—41no—
244C2×C8—21no—
245C2×C8—41no—
246C2×C8—41no—
247C2×C8C2×D4×C12711yescharacteristic
248C2×C8C2×D4×C12711yescharacteristic
249C2×C8C2×D4×C12711yescharacteristic
250C2×C8#41×C12711yescharacteristic
251C2×C8#41×C12711yescharacteristic
252C2×C8#41×C12711yescharacteristic
253C4⋊C4D8×C12712yes—
254C22⋊C4—21no—
255C42—21no—
256C16—22no—
257C16—41no—
258C23—42no—
259C23—81no—
260C23—81no—
261C23—162no—
262C23#135×C12711yescharacteristic
263Q8—42no—
264Q8—42no—
265Q8—81no—
266Q8—81no—
267D4—42no—
268D4—42no—
269D4—81no—
270D4—81no—
271C2×C4—22no—
272C2×C4—42no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4#121×C12711yescharacteristic
276C2×C4#121×C12711yescharacteristic
277C2×C4#121×C12711yescharacteristic
278C2×C4C2×D8×C12711yescharacteristic
279C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
280C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
281C8—41no—
282C8—81no—
283C8—81no—
284C8#120×C12711yescharacteristic
285C8#123×C12711yescharacteristic
286C8#123×C12711yescharacteristic
287C8#124×C12711yescharacteristic
288C22—22no—
289C22—82no—
290C22—162no—
291C22—162no—
292C22—162no—
293C22#463×C12711yescharacteristic
294C22#463×C12711yescharacteristic
295C22#466×C12711yescharacteristic
296C4—42no—
297C4—82no—
298C4#394×C12711yescharacteristic
299C4#459×C12711yescharacteristic
300C4#459×C12711yescharacteristic
301C4#469×C12711yescharacteristic
302C2—21no—
303C2—21no—
304C2—162no—
305C2C127×#205411yescharacteristic
306C2C127×#222312yes—
subgroupG/H
center, socleC22×C127#466
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC22×D8×C127C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2151C2semidirect
3. C2×#362×C127C2semidirect
5. C127×#1979C2semidirect
7. C127×#1381C2semidirect
148. #13440C127direct (N × H)
155. #2151C2×C127semidirect
156. C2×#362C2×C127semidirect
158. #1979C2×C127semidirect
160. #1381C2×C127semidirect
161. C127#13440direct (N × H)

Maximal subgroups

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

subgroupindex
C22×SD32×C1272
C127×#21512
C2×#362×C1272
C2×#361×C1272
C127×#19792
C127×#19782
C127×#13812
#13440127