Finite Group Databasebeta

← C127×#12343 #44576016

Subgroups — C127×#12343 #44576016

Order 32512.

318 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3341C211yesmaximal, characteristic, complemented
2C127×#2240C211yesmaximal, characteristic, complemented
3C127×#2206C211yesmaximal, characteristic, complemented
4C127×#1667C211yesmaximal, characteristic, complemented
5C127×#1365C211yesmaximal, characteristic, complemented
6C127×#1331C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8#555×C127—21no—
9C2×D16×C127—21no—
10#500×C127C2211yescharacteristic
11#493×C127C2211yescharacteristic
12#482×C127—22no—
13#476×C127—21no—
14#466×C127—21no—
15#459×C127—21no—
16C4×D8×C127C2211yescharacteristic
17C8×D4×C127C2211yescharacteristic
18#427×C127—21no—
19C4⋊C16×C127C2211yescharacteristic
20#357×C127—21no—
21#357×C127—22no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#331×C127—21no—
25C2×D8×C127—21no—
26C2×D8×C127—21no—
27C2×D8×C127—41no—
28C2×D8×C127D411yescharacteristic
29C22×C8×C127—21no—
30C4⋊Q8×C127—21no—
31#127×C127—21no—
32#124×C127—42no—
33#121×C127—22no—
34#121×C127—41no—
35C4×D4×C127—21no—
36C4×D4×C127D411yescharacteristic
37#117×C127—22no—
38#117×C127—41no—
39D16×C127—42no—
40C2×C16×C127—21no—
41C2×C16×C127—21no—
42#107×C127—21no—
43#107×C127D412yes—
44#107×C127D411yescharacteristic
45#106×C127—22no—
46C4⋊C8×C127—21no—
47C4⋊C8×C127D411yescharacteristic
48#102×C127—22no—
49#102×C127—21no—
50#102×C127—41no—
51C22⋊C8×C127—21no—
52C4×C8×C127C2311yescharacteristic
53C2×Q8×C127—41no—
54C2×D4×C127—21no—
55C2×D4×C127—42no—
56C2×D4×C127—41no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127D811yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127—41no—
62D8×C127—21no—
63D8×C127—21no—
64D8×C127—42no—
65D8×C127—41no—
66D8×C127—41no—
67D8×C127—82no—
68C2×C8×C127—21no—
69C2×C8×C127—41no—
70C2×C8×C127—41no—
71C2×C8×C127—41no—
72C2×C8×C127C2×D411yescharacteristic
73C2×C8×C127C2×D411yescharacteristic
74C4⋊C4×C127—21no—
75C4⋊C4×C127—22no—
76C4⋊C4×C127—41no—
77C4⋊C4×C127D811yescharacteristic
78C22⋊C4×C127—21no—
79C22⋊C4×C127—42no—
80C22⋊C4×C127—42no—
81C22⋊C4×C127—41no—
82C42×C127—42no—
83C42×C127C2×D411yescharacteristic
84C16×C127—41no—
85C16×C127—41no—
86C23×C127—21no—
87C23×C127—41no—
88C23×C127—81no—
89Q8×C127—161no—
90D4×C127—21no—
91D4×C127—21no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—81no—
95D4×C127—81no—
96D4×C127—81no—
97D4×C127—81no—
98D4×C127—161no—
99D4×C127—162no—
100C2×C4×C127—21no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—42no—
105C2×C4×C127—81no—
106C2×C4×C127—81no—
107C2×C4×C127#12011yescharacteristic
108C2×C4×C127C2×D811yescharacteristic
109C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
110C8×C127—21no—
111C8×C127—41no—
112C8×C127—81no—
113C8×C127#13511yescharacteristic
114C8×C127AGL(1,ℤ/8)11yescharacteristic
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—81no—
119C22×C127—81no—
120C22×C127—81no—
121C22×C127—161no—
122C22×C127—161no—
123C22×C127—161no—
124C22×C127#44711yescharacteristic
125C4×C127—21no—
126C4×C127—41no—
127C4×C127—81no—
128C4×C127—82no—
129C4×C127#44911yescharacteristic
130C4×C127#45411yescharacteristic
131C4×C127#50811yescharacteristic
132C4×C127#50911yescharacteristic
133#12343C12711yesmaximal, characteristic, direct product factor, Sylow
134C2×C127—81no—
135C2×C127—161no—
136C2×C127—321no—
137C2×C127#162511yescharacteristic
138C2×C127#218611yescharacteristic
139C2×C127#219211yescharacteristic
140#3341C2×C12711yescharacteristic, complemented
141#2240C2×C12711yescharacteristic, complemented
142#2206C2×C12711yescharacteristic, complemented
143#1667C2×C12711yescharacteristic, complemented
144#1365C2×C12711yescharacteristic, complemented
145#1331C2×C12711yescharacteristic, complemented
146D4⋊C16C2×C12711yescharacteristic, complemented
147C127#1234311yescharacteristic, direct product factor, Sylow
148#555—21no—
149C2×D16—21no—
150#500C22×C12711yescharacteristic
151#493C22×C12711yescharacteristic
152#482—22no—
153#476—21no—
154#466—21no—
155#459—21no—
156C4×D8C22×C12711yescharacteristic
157C8×D4C22×C12711yescharacteristic
158#427—21no—
159C4⋊C16C22×C12711yescharacteristic
160#357—21no—
161#357—22no—
162C4×C16C22×C12711yescharacteristic
163#335C22×C12711yescharacteristic
164#331—21no—
165C2×D8—21no—
166C2×D8—21no—
167C2×D8—41no—
168C2×D8D4×C12711yescharacteristic
169C22×C8—21no—
170C4⋊Q8—21no—
171#127—21no—
172#124—42no—
173#121—22no—
174#121—41no—
175C4×D4—21no—
176C4×D4D4×C12711yescharacteristic
177#117—22no—
178#117—41no—
179D16—42no—
180C2×C16—21no—
181C2×C16—21no—
182#107—21no—
183#107D4×C12711yescharacteristic
184#107D4×C12712yes—
185#106—22no—
186C4⋊C8—21no—
187C4⋊C8D4×C12711yescharacteristic
188#102—22no—
189#102—21no—
190#102—41no—
191C22⋊C8—21no—
192C4×C8C23×C12711yescharacteristic
193C2×Q8—41no—
194C2×D4—21no—
195C2×D4—42no—
196C2×D4—41no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4D8×C12711yescharacteristic
200C22×C4—21no—
201C22×C4—41no—
202D8—21no—
203D8—21no—
204D8—42no—
205D8—41no—
206D8—41no—
207D8—82no—
208C2×C8—21no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—41no—
212C2×C8C2×D4×C12711yescharacteristic
213C2×C8C2×D4×C12711yescharacteristic
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—41no—
217C4⋊C4D8×C12711yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C42—42no—
223C42C2×D4×C12711yescharacteristic
224C16—41no—
225C16—41no—
226C23—21no—
227C23—41no—
228C23—81no—
229Q8—161no—
230D4—21no—
231D4—21no—
232D4—41no—
233D4—41no—
234D4—81no—
235D4—81no—
236D4—81no—
237D4—81no—
238D4—161no—
239D4—162no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—81no—
246C2×C4—81no—
247C2×C4#120×C12711yescharacteristic
248C2×C4C2×D8×C12711yescharacteristic
249C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
250C8—21no—
251C8—41no—
252C8—81no—
253C8#135×C12711yescharacteristic
254C8AGL(1,ℤ/8)×C12711yescharacteristic
255C22—41no—
256C22—41no—
257C22—41no—
258C22—81no—
259C22—81no—
260C22—81no—
261C22—161no—
262C22—161no—
263C22—161no—
264C22#447×C12711yescharacteristic
265C4—21no—
266C4—41no—
267C4—81no—
268C4—82no—
269C4#449×C12711yescharacteristic
270C4#454×C12711yescharacteristic
271C4#508×C12711yescharacteristic
272C4#509×C12711yescharacteristic
273C2—81no—
274C2—161no—
275C2—321no—
276C2C127×#162511yescharacteristic
277C2C127×#218611yescharacteristic
278C2C127×#219211yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3341C2semidirect
2. C127×#2240C2semidirect
3. C127×#2206C2semidirect
4. C127×#1667C2semidirect
5. C127×#1365C2semidirect
6. C127×#1331C2semidirect
7. C127×D4⋊C16C2semidirect
133. #12343C127direct (N × H)
140. #3341C2×C127semidirect
141. #2240C2×C127semidirect
142. #2206C2×C127semidirect
143. #1667C2×C127semidirect
144. #1365C2×C127semidirect
145. #1331C2×C127semidirect
146. D4⋊C16C2×C127semidirect
147. C127#12343direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#33412
C127×#22402
C127×#22062
C127×#16672
C127×#13652
C127×#13312
C127×D4⋊C162
#12343127