Finite Group Databasebeta

← C127×#12337 #44576010

Subgroups — C127×#12337 #44576010

Order 32512.

324 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3402C211yesmaximal, characteristic, complemented
2C127×#2250C211yesmaximal, characteristic
3C127×#2207C211yesmaximal, characteristic, complemented
4C127×#1688C211yesmaximal, characteristic, complemented
5C127×#1360C211yesmaximal, characteristic, complemented
6C127×#1338C211yesmaximal, characteristic, complemented
7C127×#1334C211yesmaximal, characteristic, complemented
8#556×C127—21no—
9C2×SD32×C127—21no—
10C2×SD32×C127—21no—
11C2×D16×C127—21no—
12#497×C127—21no—
13#495×C127C2211yescharacteristic
14#493×C127C2211yescharacteristic
15#464×C127—21no—
16#463×C127—21no—
17#462×C127—21no—
18#460×C127—21no—
19#446×C127C2211yescharacteristic
20#440×C127—21no—
21C4×Q16×C127C2211yescharacteristic
22C4⋊C16×C127C2211yescharacteristic
23#357×C127—21no—
24#346×C127C2211yescharacteristic
25#334×C127C2211yescharacteristic
26#331×C127—21no—
27C2×Q16×C127—21no—
28C2×Q16×C127D411yescharacteristic
29C2×Q16×C127D411yescharacteristic
30C2×SD16×C127—21no—
31C2×SD16×C127—21no—
32C2×SD16×C127—41no—
33C2×D8×C127—21no—
34C2×D8×C127—21no—
35C2×D8×C127D411yescharacteristic
36C4⋊Q8×C127—21no—
37#127×C127—21no—
38#126×C127—41no—
39#126×C127—41no—
40#124×C127—21no—
41#124×C127—21no—
42C22⋊Q8×C127—41no—
43C4×Q8×C127—21no—
44C4×Q8×C127—21no—
45C4×Q8×C127D411yescharacteristic
46C4×D4×C127—21no—
47#117×C127—41no—
48SD32×C127—42no—
49SD32×C127—42no—
50D16×C127—42no—
51C2×C16×C127—21no—
52C2×C16×C127—21no—
53#107×C127D411yescharacteristic
54C4⋊C8×C127—21no—
55C4⋊C8×C127—21no—
56C4⋊C8×C127D411yescharacteristic
57#103×C127—21no—
58#103×C127—21no—
59#102×C127—21no—
60#102×C127—41no—
61#97×C127—21no—
62C4×C8×C127C2311yescharacteristic
63C2×Q8×C127—21no—
64C2×Q8×C127—21no—
65C2×Q8×C127D811yescharacteristic
66C2×D4×C127—21no—
67C2×D4×C127—41no—
68C2×D4×C127—41no—
69C2×D4×C127—41no—
70C22×C4×C127—41no—
71Q16×C127—21no—
72Q16×C127—21no—
73Q16×C127—41no—
74Q16×C127—81no—
75SD16×C127—41no—
76SD16×C127—41no—
77SD16×C127—81no—
78SD16×C127—82no—
79D8×C127—41no—
80D8×C127—41no—
81D8×C127—41no—
82D8×C127—41no—
83D8×C127—41no—
84C2×C8×C127—21no—
85C2×C8×C127—41no—
86C2×C8×C127C2×D411yescharacteristic
87C2×C8×C127C2×D411yescharacteristic
88C4⋊C4×C127—21no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—41no—
91C4⋊C4×C127—41no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C4⋊C4×C127D811yescharacteristic
95C22⋊C4×C127—41no—
96C22⋊C4×C127—41no—
97C22⋊C4×C127—41no—
98C22⋊C4×C127—41no—
99C22⋊C4×C127—41no—
100C42×C127—21no—
101C42×C127—41no—
102C42×C127—41no—
103C42×C127C2×D411yescharacteristic
104C16×C127—22no—
105C16×C127—41no—
106C23×C127—41no—
107C23×C127—81no—
108Q8×C127—41no—
109Q8×C127—41no—
110Q8×C127—41no—
111Q8×C127—41no—
112Q8×C127—41no—
113D4×C127—41no—
114D4×C127—41no—
115D4×C127—81no—
116D4×C127—81no—
117D4×C127—81no—
118D4×C127—81no—
119D4×C127—161no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—82no—
126C2×C4×C127#12011yescharacteristic
127C2×C4×C127C2×D811yescharacteristic
128C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
129C8×C127—21no—
130C8×C127—41no—
131C8×C127—81no—
132C8×C127#13511yescharacteristic
133C8×C127AGL(1,ℤ/8)11yescharacteristic
134C22×C127—81no—
135C22×C127—81no—
136C22×C127—81no—
137C22×C127—161no—
138C22×C127—161no—
139C22×C127—161no—
140C22×C127#44711yescharacteristic
141C4×C127—21no—
142C4×C127—21no—
143C4×C127—41no—
144C4×C127—41no—
145C4×C127—81no—
146C4×C127—81no—
147C4×C127—81no—
148C4×C127#44911yescharacteristic
149C4×C127#45311yescharacteristic
150#12337C12711yesmaximal, characteristic, direct product factor, Sylow
151C2×C127—161no—
152C2×C127—321no—
153C2×C127#162111yescharacteristic
154C2×C127#219011yescharacteristic
155C2×C127#219311yescharacteristic
156#3402C2×C12711yescharacteristic, complemented
157#2250C2×C12711yescharacteristic
158#2207C2×C12711yescharacteristic, complemented
159#1688C2×C12711yescharacteristic, complemented
160#1360C2×C12711yescharacteristic, complemented
161#1338C2×C12711yescharacteristic, complemented
162#1334C2×C12711yescharacteristic, complemented
163C127#1233711yescharacteristic, direct product factor, Sylow
164#556—21no—
165C2×SD32—21no—
166C2×SD32—21no—
167C2×D16—21no—
168#497—21no—
169#495C22×C12711yescharacteristic
170#493C22×C12711yescharacteristic
171#464—21no—
172#463—21no—
173#462—21no—
174#460—21no—
175#446C22×C12711yescharacteristic
176#440—21no—
177C4×Q16C22×C12711yescharacteristic
178C4⋊C16C22×C12711yescharacteristic
179#357—21no—
180#346C22×C12711yescharacteristic
181#334C22×C12711yescharacteristic
182#331—21no—
183C2×Q16—21no—
184C2×Q16D4×C12711yescharacteristic
185C2×Q16D4×C12711yescharacteristic
186C2×SD16—21no—
187C2×SD16—21no—
188C2×SD16—41no—
189C2×D8—21no—
190C2×D8—21no—
191C2×D8D4×C12711yescharacteristic
192C4⋊Q8—21no—
193#127—21no—
194#126—41no—
195#126—41no—
196#124—21no—
197#124—21no—
198C22⋊Q8—41no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8D4×C12711yescharacteristic
202C4×D4—21no—
203#117—41no—
204SD32—42no—
205SD32—42no—
206D16—42no—
207C2×C16—21no—
208C2×C16—21no—
209#107D4×C12711yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8D4×C12711yescharacteristic
213#103—21no—
214#103—21no—
215#102—21no—
216#102—41no—
217#97—21no—
218C4×C8C23×C12711yescharacteristic
219C2×Q8—21no—
220C2×Q8—21no—
221C2×Q8D8×C12711yescharacteristic
222C2×D4—21no—
223C2×D4—41no—
224C2×D4—41no—
225C2×D4—41no—
226C22×C4—41no—
227Q16—21no—
228Q16—21no—
229Q16—41no—
230Q16—81no—
231SD16—41no—
232SD16—41no—
233SD16—81no—
234SD16—82no—
235D8—41no—
236D8—41no—
237D8—41no—
238D8—41no—
239D8—41no—
240C2×C8—21no—
241C2×C8—41no—
242C2×C8C2×D4×C12711yescharacteristic
243C2×C8C2×D4×C12711yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4D8×C12711yescharacteristic
251C22⋊C4—41no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—41no—
255C22⋊C4—41no—
256C42—21no—
257C42—41no—
258C42—41no—
259C42C2×D4×C12711yescharacteristic
260C16—22no—
261C16—41no—
262C23—41no—
263C23—81no—
264Q8—41no—
265Q8—41no—
266Q8—41no—
267Q8—41no—
268Q8—41no—
269D4—41no—
270D4—41no—
271D4—81no—
272D4—81no—
273D4—81no—
274D4—81no—
275D4—161no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—82no—
282C2×C4#120×C12711yescharacteristic
283C2×C4C2×D8×C12711yescharacteristic
284C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
285C8—21no—
286C8—41no—
287C8—81no—
288C8#135×C12711yescharacteristic
289C8AGL(1,ℤ/8)×C12711yescharacteristic
290C22—81no—
291C22—81no—
292C22—81no—
293C22—161no—
294C22—161no—
295C22—161no—
296C22#447×C12711yescharacteristic
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4#449×C12711yescharacteristic
305C4#453×C12711yescharacteristic
306C2—161no—
307C2—321no—
308C2C127×#162111yescharacteristic
309C2C127×#219011yescharacteristic
310C2C127×#219311yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#2250C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ #446×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3402C2semidirect
3. C127×#2207C2semidirect
4. C127×#1688C2semidirect
5. C127×#1360C2semidirect
6. C127×#1338C2semidirect
7. C127×#1334C2semidirect
150. #12337C127direct (N × H)
156. #3402C2×C127semidirect
158. #2207C2×C127semidirect
159. #1688C2×C127semidirect
160. #1360C2×C127semidirect
161. #1338C2×C127semidirect
162. #1334C2×C127semidirect
163. C127#12337direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#34022
C127×#22502
C127×#22072
C127×#16882
C127×#13602
C127×#13382
C127×#13342
#12337127