Finite Group Databasebeta

← C127×#12340 #44576013

Subgroups — C127×#12340 #44576013

Order 32512.

346 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3334C211yesmaximal, characteristic, complemented
2C127×#2240C211yesmaximal, characteristic, complemented
3C127×#2206C211yesmaximal, characteristic, complemented
4C127×#1667C211yesmaximal, characteristic, complemented
5C127×#1364C211yesmaximal, characteristic, complemented
6C127×#1331C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8#547×C127—21no—
9C2×D16×C127—21no—
10#500×C127C2211yescharacteristic
11#493×C127C2211yescharacteristic
12#480×C127—22no—
13#474×C127—21no—
14#466×C127—21no—
15#459×C127—21no—
16C4×D8×C127C2211yescharacteristic
17C8×D4×C127C2211yescharacteristic
18C2×#107×C127—21no—
19C4⋊C16×C127C2211yescharacteristic
20#357×C127—21no—
21#357×C127—22no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#331×C127—21no—
25C2×#41×C127—41no—
26C2×D8×C127—21no—
27C2×D8×C127—21no—
28C2×D8×C127—41no—
29C2×D8×C127D411yescharacteristic
30C22×C8×C127—21no—
31C4⋊Q8×C127—21no—
32#127×C127—21no—
33#123×C127—42no—
34C22⋊Q8×C127—41no—
35#121×C127—22no—
36C4×D4×C127—21no—
37C4×D4×C127D411yescharacteristic
38C2×C4⋊C4×C127—22no—
39D16×C127—42no—
40C2×C16×C127—21no—
41C2×C16×C127—21no—
42#107×C127—22no—
43#107×C127—21no—
44#107×C127D412yes—
45#107×C127D411yescharacteristic
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
53#41×C127—82no—
54#41×C127—82no—
55C2×Q8×C127—41no—
56C2×D4×C127—21no—
57C2×D4×C127—42no—
58C2×D4×C127—41no—
59C2×D4×C127—41no—
60C2×D4×C127—41no—
61C2×D4×C127D811yescharacteristic
62C22×C4×C127—21no—
63C22×C4×C127—42no—
64C22×C4×C127—41no—
65D8×C127—21no—
66D8×C127—21no—
67D8×C127—42no—
68D8×C127—41no—
69D8×C127—41no—
70D8×C127—82no—
71C2×C8×C127—21no—
72C2×C8×C127—41no—
73C2×C8×C127—41no—
74C2×C8×C127—41no—
75C2×C8×C127C2×D411yescharacteristic
76C2×C8×C127C2×D411yescharacteristic
77C4⋊C4×C127—21no—
78C4⋊C4×C127—22no—
79C4⋊C4×C127—42no—
80C4⋊C4×C127—41no—
81C4⋊C4×C127D811yescharacteristic
82C22⋊C4×C127—21no—
83C22⋊C4×C127—42no—
84C22⋊C4×C127—41no—
85C42×C127C2×D411yescharacteristic
86C16×C127—41no—
87C16×C127—41no—
88C23×C127—21no—
89C23×C127—41no—
90C23×C127—81no—
91Q8×C127—81no—
92Q8×C127—81no—
93D4×C127—21no—
94D4×C127—41no—
95D4×C127—41no—
96D4×C127—84no—
97D4×C127—81no—
98D4×C127—81no—
99D4×C127—81no—
100D4×C127—81no—
101D4×C127—161no—
102D4×C127D1612yescomplemented
103C2×C4×C127—21no—
104C2×C4×C127—41no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127—42no—
108C2×C4×C127—84no—
109C2×C4×C127—81no—
110C2×C4×C127—81no—
111C2×C4×C127#12011yescharacteristic
112C2×C4×C127C2×D811yescharacteristic
113C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
114C8×C127—21no—
115C8×C127—41no—
116C8×C127—81no—
117C8×C127#13511yescharacteristic
118C8×C127AGL(1,ℤ/8)11yescharacteristic
119C22×C127—22no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127—81no—
123C22×C127—81no—
124C22×C127—81no—
125C22×C127—161no—
126C22×C127—161no—
127C22×C127—161no—
128C22×C127#44711yescharacteristic
129C4×C127—21no—
130C4×C127—41no—
131C4×C127—81no—
132C4×C127—82no—
133C4×C127#44911yescharacteristic
134C4×C127#45411yescharacteristic
135C4×C127C2×D1611yescharacteristic
136C4×C127#50911yescharacteristic
137#12340C12711yesmaximal, characteristic, direct product factor, Sylow
138C2×C127—42no—
139C2×C127—161no—
140C2×C127—321no—
141C2×C127#162511yescharacteristic
142C2×C127#218411yescharacteristic
143C2×C127#219311yescharacteristic
144#3334C2×C12711yescharacteristic, complemented
145#2240C2×C12711yescharacteristic, complemented
146#2206C2×C12711yescharacteristic, complemented
147#1667C2×C12711yescharacteristic, complemented
148#1364C2×C12711yescharacteristic, complemented
149#1331C2×C12711yescharacteristic, complemented
150D4⋊C16C2×C12711yescharacteristic, complemented
151C127#1234011yescharacteristic, direct product factor, Sylow
152#547—21no—
153C2×D16—21no—
154#500C22×C12711yescharacteristic
155#493C22×C12711yescharacteristic
156#480—22no—
157#474—21no—
158#466—21no—
159#459—21no—
160C4×D8C22×C12711yescharacteristic
161C8×D4C22×C12711yescharacteristic
162C2×#107—21no—
163C4⋊C16C22×C12711yescharacteristic
164#357—21no—
165#357—22no—
166C4×C16C22×C12711yescharacteristic
167#335C22×C12711yescharacteristic
168#331—21no—
169C2×#41—41no—
170C2×D8—21no—
171C2×D8—21no—
172C2×D8—41no—
173C2×D8D4×C12711yescharacteristic
174C22×C8—21no—
175C4⋊Q8—21no—
176#127—21no—
177#123—42no—
178C22⋊Q8—41no—
179#121—22no—
180C4×D4—21no—
181C4×D4D4×C12711yescharacteristic
182C2×C4⋊C4—22no—
183D16—42no—
184C2×C16—21no—
185C2×C16—21no—
186#107—22no—
187#107—21no—
188#107D4×C12711yescharacteristic
189#107D4×C12712yes—
190C4⋊C8—21no—
191C4⋊C8D4×C12711yescharacteristic
192#102—22no—
193#102—21no—
194#102—41no—
195C22⋊C8—21no—
196C4×C8C23×C12711yescharacteristic
197#41—82no—
198#41—82no—
199C2×Q8—41no—
200C2×D4—21no—
201C2×D4—42no—
202C2×D4—41no—
203C2×D4—41no—
204C2×D4—41no—
205C2×D4D8×C12711yescharacteristic
206C22×C4—21no—
207C22×C4—42no—
208C22×C4—41no—
209D8—21no—
210D8—21no—
211D8—42no—
212D8—41no—
213D8—41no—
214D8—82no—
215C2×C8—21no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8C2×D4×C12711yescharacteristic
220C2×C8C2×D4×C12711yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—22no—
223C4⋊C4—42no—
224C4⋊C4—41no—
225C4⋊C4D8×C12711yescharacteristic
226C22⋊C4—21no—
227C22⋊C4—42no—
228C22⋊C4—41no—
229C42C2×D4×C12711yescharacteristic
230C16—41no—
231C16—41no—
232C23—21no—
233C23—41no—
234C23—81no—
235Q8—81no—
236Q8—81no—
237D4—21no—
238D4—41no—
239D4—41no—
240D4—84no—
241D4—81no—
242D4—81no—
243D4—81no—
244D4—81no—
245D4—161no—
246D4D16×C12712yescomplemented
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—84no—
253C2×C4—81no—
254C2×C4—81no—
255C2×C4#120×C12711yescharacteristic
256C2×C4C2×D8×C12711yescharacteristic
257C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
258C8—21no—
259C8—41no—
260C8—81no—
261C8#135×C12711yescharacteristic
262C8AGL(1,ℤ/8)×C12711yescharacteristic
263C22—22no—
264C22—41no—
265C22—41no—
266C22—81no—
267C22—81no—
268C22—81no—
269C22—161no—
270C22—161no—
271C22—161no—
272C22#447×C12711yescharacteristic
273C4—21no—
274C4—41no—
275C4—81no—
276C4—82no—
277C4#449×C12711yescharacteristic
278C4#454×C12711yescharacteristic
279C4C2×D16×C12711yescharacteristic
280C4#509×C12711yescharacteristic
281C2—42no—
282C2—161no—
283C2—321no—
284C2C127×#162511yescharacteristic
285C2C127×#218411yescharacteristic
286C2C127×#219311yescharacteristic
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×#3334C2semidirect
2. C127×#2240C2semidirect
3. C127×#2206C2semidirect
4. C127×#1667C2semidirect
5. C127×#1364C2semidirect
6. C127×#1331C2semidirect
7. C127×D4⋊C16C2semidirect
102. D4×C127D16semidirect
137. #12340C127direct (N × H)
144. #3334C2×C127semidirect
145. #2240C2×C127semidirect
146. #2206C2×C127semidirect
147. #1667C2×C127semidirect
148. #1364C2×C127semidirect
149. #1331C2×C127semidirect
150. D4⋊C16C2×C127semidirect
151. C127#12340direct (N × H)
246. D4D16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#33342
C127×#22402
C127×#22062
C127×#16672
C127×#13642
C127×#13312
C127×D4⋊C162
#12340127