Finite Group Databasebeta

← C127×#12329 #44576002

Subgroups — C127×#12329 #44576002

Order 32512.

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

normal subgroups only

This group has 340 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×#3396C211yesmaximal, characteristic, complemented
2C127×#2246C211yesmaximal, characteristic
3C127×#2207C211yesmaximal, characteristic, complemented
4C127×#1668C211yesmaximal, characteristic, complemented
5C127×#1361C211yesmaximal, characteristic, complemented
6C127×Q8⋊C16C211yesmaximal, characteristic, complemented
7C127×#1332C211yesmaximal, characteristic, complemented
8#559×C127—21no—
9C2×SD32×C127—21no—
10C2×D16×C127—22no—
11#495×C127—21no—
12#493×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic
14#463×C127—22no—
15#460×C127—21no—
16#460×C127—21no—
17C8×Q8×C127C2211yescharacteristic
18C4×Q16×C127C2211yescharacteristic
19C4×D8×C127—21no—
20C4⋊C16×C127C2211yescharacteristic
21#357×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#331×C127—21no—
25C2×Q16×C127D411yescharacteristic
26C2×SD16×C127—22no—
27C2×SD16×C127—41no—
28C2×D8×C127—21no—
29C2×D8×C127—21no—
30C2×D8×C127—21no—
31C2×D8×C127D412yes—
32#127×C127—21no—
33#127×C127—21no—
34#124×C127—22no—
35#124×C127—41no—
36#123×C127—42no—
37C4×Q8×C127—21no—
38C4×Q8×C127D411yescharacteristic
39C4×D4×C127—22no—
40C4×D4×C127—41no—
41SD32×C127—42no—
42D16×C127—44no—
43C2×C16×C127—21no—
44C2×C16×C127—21no—
45#107×C127D411yescharacteristic
46C4⋊C8×C127—21no—
47C4⋊C8×C127—21no—
48C4⋊C8×C127D411yescharacteristic
49#103×C127—21no—
50#102×C127—22no—
51#102×C127—41no—
52C4×C8×C127—21no—
53C4×C8×C127C2311yescharacteristic
54C2×Q8×C127—21no—
55C2×Q8×C127D811yescharacteristic
56C2×D4×C127—22no—
57C2×D4×C127—41no—
58C2×D4×C127—41no—
59C2×D4×C127—41no—
60C2×D4×C127—41no—
61C2×D4×C127—41no—
62C22×C4×C127—42no—
63Q16×C127—21no—
64Q16×C127—21no—
65SD16×C127—82no—
66SD16×C127—82no—
67D8×C127—42no—
68D8×C127—42no—
69D8×C127—41no—
70D8×C127—41no—
71D8×C127—41no—
72D8×C127—41no—
73C2×C8×C127—21no—
74C2×C8×C127—41no—
75C2×C8×C127C2×D411yescharacteristic
76C2×C8×C127C2×D411yescharacteristic
77C4⋊C4×C127—21no—
78C4⋊C4×C127—21no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127D811yescharacteristic
81C22⋊C4×C127—42no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—42no—
84C42×C127—21no—
85C42×C127—41no—
86C42×C127C2×D411yescharacteristic
87C16×C127—22no—
88C16×C127—41no—
89C23×C127—42no—
90C23×C127—81no—
91Q8×C127—21no—
92Q8×C127—21no—
93Q8×C127—41no—
94Q8×C127—41no—
95D4×C127—42no—
96D4×C127—42no—
97D4×C127—81no—
98D4×C127—81no—
99D4×C127—81no—
100D4×C127—81no—
101D4×C127—81no—
102D4×C127—81no—
103D4×C127—161no—
104D4×C127—161no—
105C2×C4×C127—21no—
106C2×C4×C127—21no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—162no—
110C2×C4×C127#12011yescharacteristic
111C2×C4×C127C2×D811yescharacteristic
112C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
113C8×C127—21no—
114C8×C127—41no—
115C8×C127—81no—
116C8×C127#13511yescharacteristic
117C8×C127AGL(1,ℤ/8)11yescharacteristic
118C22×C127—82no—
119C22×C127—82no—
120C22×C127—82no—
121C22×C127—161no—
122C22×C127—161no—
123C22×C127—161no—
124C22×C127#44711yescharacteristic
125C4×C127—21no—
126C4×C127—41no—
127C4×C127—41no—
128C4×C127—81no—
129C4×C127—81no—
130C4×C127#44911yescharacteristic
131C4×C127#45311yescharacteristic
132C4×C127#50811yescharacteristic
133C4×C127#50911yescharacteristic
134#12329C12711yesmaximal, characteristic, direct product factor, Sylow
135C2×C127—162no—
136C2×C127—321no—
137C2×C127#162111yescharacteristic
138C2×C127#218611yescharacteristic
139C2×C127#219211yescharacteristic
140#3396C2×C12711yescharacteristic, complemented
141#2246C2×C12711yescharacteristic
142#2207C2×C12711yescharacteristic, complemented
143#1668C2×C12711yescharacteristic, complemented
144#1361C2×C12711yescharacteristic, complemented
145Q8⋊C16C2×C12711yescharacteristic, complemented
146#1332C2×C12711yescharacteristic, complemented
147C127#1232911yescharacteristic, direct product factor, Sylow
148#559—21no—
149C2×SD32—21no—
150C2×D16—22no—
151#495—21no—
152#493C22×C12711yescharacteristic
153#493C22×C12711yescharacteristic
154#463—22no—
155#460—21no—
156#460—21no—
157C8×Q8C22×C12711yescharacteristic
158C4×Q16C22×C12711yescharacteristic
159C4×D8—21no—
160C4⋊C16C22×C12711yescharacteristic
161#357—21no—
162C4×C16C22×C12711yescharacteristic
163#335C22×C12711yescharacteristic
164#331—21no—
165C2×Q16D4×C12711yescharacteristic
166C2×SD16—22no—
167C2×SD16—41no—
168C2×D8—21no—
169C2×D8—21no—
170C2×D8—21no—
171C2×D8D4×C12712yes—
172#127—21no—
173#127—21no—
174#124—22no—
175#124—41no—
176#123—42no—
177C4×Q8—21no—
178C4×Q8D4×C12711yescharacteristic
179C4×D4—22no—
180C4×D4—41no—
181SD32—42no—
182D16—44no—
183C2×C16—21no—
184C2×C16—21no—
185#107D4×C12711yescharacteristic
186C4⋊C8—21no—
187C4⋊C8—21no—
188C4⋊C8D4×C12711yescharacteristic
189#103—21no—
190#102—22no—
191#102—41no—
192C4×C8—21no—
193C4×C8C23×C12711yescharacteristic
194C2×Q8—21no—
195C2×Q8D8×C12711yescharacteristic
196C2×D4—22no—
197C2×D4—41no—
198C2×D4—41no—
199C2×D4—41no—
200C2×D4—41no—
201C2×D4—41no—
202C22×C4—42no—
203Q16—21no—
204Q16—21no—
205SD16—82no—
206SD16—82no—
207D8—42no—
208D8—42no—
209D8—41no—
210D8—41no—
211D8—41no—
212D8—41no—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8C2×D4×C12711yescharacteristic
216C2×C8C2×D4×C12711yescharacteristic
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—41no—
220C4⋊C4D8×C12711yescharacteristic
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C42—21no—
225C42—41no—
226C42C2×D4×C12711yescharacteristic
227C16—22no—
228C16—41no—
229C23—42no—
230C23—81no—
231Q8—21no—
232Q8—21no—
233Q8—41no—
234Q8—41no—
235D4—42no—
236D4—42no—
237D4—81no—
238D4—81no—
239D4—81no—
240D4—81no—
241D4—81no—
242D4—81no—
243D4—161no—
244D4—161no—
245C2×C4—21no—
246C2×C4—21no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—162no—
250C2×C4#120×C12711yescharacteristic
251C2×C4C2×D8×C12711yescharacteristic
252C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
253C8—21no—
254C8—41no—
255C8—81no—
256C8#135×C12711yescharacteristic
257C8AGL(1,ℤ/8)×C12711yescharacteristic
258C22—82no—
259C22—82no—
260C22—82no—
261C22—161no—
262C22—161no—
263C22—161no—
264C22#447×C12711yescharacteristic
265C4—21no—
266C4—41no—
267C4—41no—
268C4—81no—
269C4—81no—
270C4#449×C12711yescharacteristic
271C4#453×C12711yescharacteristic
272C4#508×C12711yescharacteristic
273C4#509×C12711yescharacteristic
274C2—162no—
275C2—321no—
276C2C127×#162111yescharacteristic
277C2C127×#218611yescharacteristic
278C2C127×#219211yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#2246C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3396C2semidirect
3. C127×#2207C2semidirect
4. C127×#1668C2semidirect
5. C127×#1361C2semidirect
6. C127×Q8⋊C16C2semidirect
7. C127×#1332C2semidirect
134. #12329C127direct (N × H)
140. #3396C2×C127semidirect
142. #2207C2×C127semidirect
143. #1668C2×C127semidirect
144. #1361C2×C127semidirect
145. Q8⋊C16C2×C127semidirect
146. #1332C2×C127semidirect
147. C127#12329direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#33962
C127×#22462
C127×#22072
C127×#16682
C127×#13612
C127×Q8⋊C162
C127×#13322
#12329127