Finite Group Databasebeta

← C127×#12342 #44576015

Subgroups — C127×#12342 #44576015

Order 32512.

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

normal subgroups only

This group has 314 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×#3338C211yesmaximal, characteristic, complemented
2C127×#2240C211yesmaximal, characteristic, complemented
3C127×#2207C211yesmaximal, characteristic, complemented
4C127×#1667C211yesmaximal, characteristic
5C127×#1365C211yesmaximal, characteristic, complemented
6C127×#1332C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8#554×C127—21no—
9C2×SD32×C127—21no—
10#500×C127C2211yescharacteristic
11#493×C127C2211yescharacteristic
12#484×C127—22no—
13#477×C127—21no—
14#467×C127—21no—
15#459×C127—21no—
16C4×Q16×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×Q16×C127D411yescharacteristic
26C2×D8×C127—21no—
27C2×D8×C127—21no—
28C2×D8×C127—41no—
29C22×C8×C127—21no—
30C4⋊Q8×C127—21no—
31#127×C127—21no—
32#125×C127—41no—
33C22⋊Q8×C127—22no—
34C22⋊Q8×C127—42no—
35C4×Q8×C127—21no—
36C4×D4×C127—41no—
37C4×D4×C127D411yescharacteristic
38C2×C4⋊C4×C127—22no—
39SD32×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#103×C127—22no—
49#103×C127—21no—
50#102×C127—41no—
51C22⋊C8×C127—21no—
52C4×C8×C127C2311yescharacteristic
53C2×Q8×C127—21no—
54C2×Q8×C127—41no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—41no—
58C2×D4×C127D811yescharacteristic
59C22×C4×C127—21no—
60C22×C4×C127—42no—
61Q16×C127—21no—
62Q16×C127—21no—
63D8×C127—42no—
64D8×C127—41no—
65D8×C127—41no—
66D8×C127—82no—
67C2×C8×C127—21no—
68C2×C8×C127—41no—
69C2×C8×C127—41no—
70C2×C8×C127—41no—
71C2×C8×C127C2×D411yescharacteristic
72C2×C8×C127C2×D411yescharacteristic
73C4⋊C4×C127—21no—
74C4⋊C4×C127—22no—
75C4⋊C4×C127—42no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127D811yescharacteristic
80C22⋊C4×C127—21no—
81C22⋊C4×C127—42no—
82C42×C127—41no—
83C42×C127C2×D411yescharacteristic
84C16×C127—41no—
85C16×C127—41no—
86C23×C127—21no—
87C23×C127—81no—
88Q8×C127—41no—
89Q8×C127—41no—
90Q8×C127—161no—
91D4×C127—21no—
92D4×C127—81no—
93D4×C127—81no—
94D4×C127—81no—
95D4×C127—81no—
96D4×C127—161no—
97D4×C127SD3212yescomplemented
98C2×C4×C127—21no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—42no—
104C2×C4×C127—162no—
105C2×C4×C127#12011yescharacteristic
106C2×C4×C127C2×D811yescharacteristic
107C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
108C8×C127—21no—
109C8×C127—41no—
110C8×C127—81no—
111C8×C127#13511yescharacteristic
112C8×C127AGL(1,ℤ/8)11yescharacteristic
113C22×C127—22no—
114C22×C127—41no—
115C22×C127—41no—
116C22×C127—161no—
117C22×C127—161no—
118C22×C127—161no—
119C22×C127#44711yescharacteristic
120C4×C127—21no—
121C4×C127—41no—
122C4×C127—81no—
123C4×C127—81no—
124C4×C127—82no—
125C4×C127#44911yescharacteristic
126C4×C127#45411yescharacteristic
127C4×C127C2×SD3211yescharacteristic
128C4×C127#51011yescharacteristic
129#12342C12711yesmaximal, characteristic, direct product factor, Sylow
130C2×C127—42no—
131C2×C127—321no—
132C2×C127#162511yescharacteristic
133C2×C127#218511yescharacteristic
134C2×C127#219211yescharacteristic
135#3338C2×C12711yescharacteristic, complemented
136#2240C2×C12711yescharacteristic, complemented
137#2207C2×C12711yescharacteristic, complemented
138#1667C2×C12711yescharacteristic
139#1365C2×C12711yescharacteristic, complemented
140#1332C2×C12711yescharacteristic, complemented
141D4⋊C16C2×C12711yescharacteristic, complemented
142C127#1234211yescharacteristic, direct product factor, Sylow
143#554—21no—
144C2×SD32—21no—
145#500C22×C12711yescharacteristic
146#493C22×C12711yescharacteristic
147#484—22no—
148#477—21no—
149#467—21no—
150#459—21no—
151C4×Q16C22×C12711yescharacteristic
152C8×D4C22×C12711yescharacteristic
153C2×#107—21no—
154C4⋊C16C22×C12711yescharacteristic
155#357—21no—
156#357—22no—
157C4×C16C22×C12711yescharacteristic
158#335C22×C12711yescharacteristic
159#331—21no—
160C2×Q16D4×C12711yescharacteristic
161C2×D8—21no—
162C2×D8—21no—
163C2×D8—41no—
164C22×C8—21no—
165C4⋊Q8—21no—
166#127—21no—
167#125—41no—
168C22⋊Q8—22no—
169C22⋊Q8—42no—
170C4×Q8—21no—
171C4×D4—41no—
172C4×D4D4×C12711yescharacteristic
173C2×C4⋊C4—22no—
174SD32—42no—
175C2×C16—21no—
176C2×C16—21no—
177#107—22no—
178#107—21no—
179#107D4×C12711yescharacteristic
180#107D4×C12712yes—
181C4⋊C8—21no—
182C4⋊C8D4×C12711yescharacteristic
183#103—22no—
184#103—21no—
185#102—41no—
186C22⋊C8—21no—
187C4×C8C23×C12711yescharacteristic
188C2×Q8—21no—
189C2×Q8—41no—
190C2×D4—41no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4D8×C12711yescharacteristic
194C22×C4—21no—
195C22×C4—42no—
196Q16—21no—
197Q16—21no—
198D8—42no—
199D8—41no—
200D8—41no—
201D8—82no—
202C2×C8—21no—
203C2×C8—41no—
204C2×C8—41no—
205C2×C8—41no—
206C2×C8C2×D4×C12711yescharacteristic
207C2×C8C2×D4×C12711yescharacteristic
208C4⋊C4—21no—
209C4⋊C4—22no—
210C4⋊C4—42no—
211C4⋊C4—42no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4D8×C12711yescharacteristic
215C22⋊C4—21no—
216C22⋊C4—42no—
217C42—41no—
218C42C2×D4×C12711yescharacteristic
219C16—41no—
220C16—41no—
221C23—21no—
222C23—81no—
223Q8—41no—
224Q8—41no—
225Q8—161no—
226D4—21no—
227D4—81no—
228D4—81no—
229D4—81no—
230D4—81no—
231D4—161no—
232D4SD32×C12712yescomplemented
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—42no—
239C2×C4—162no—
240C2×C4#120×C12711yescharacteristic
241C2×C4C2×D8×C12711yescharacteristic
242C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
243C8—21no—
244C8—41no—
245C8—81no—
246C8#135×C12711yescharacteristic
247C8AGL(1,ℤ/8)×C12711yescharacteristic
248C22—22no—
249C22—41no—
250C22—41no—
251C22—161no—
252C22—161no—
253C22—161no—
254C22#447×C12711yescharacteristic
255C4—21no—
256C4—41no—
257C4—81no—
258C4—81no—
259C4—82no—
260C4#449×C12711yescharacteristic
261C4#454×C12711yescharacteristic
262C4C2×SD32×C12711yescharacteristic
263C4#510×C12711yescharacteristic
264C2—42no—
265C2—321no—
266C2C127×#162511yescharacteristic
267C2C127×#218511yescharacteristic
268C2C127×#219211yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#1667C2
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×#3338C2semidirect
2. C127×#2240C2semidirect
3. C127×#2207C2semidirect
5. C127×#1365C2semidirect
6. C127×#1332C2semidirect
7. C127×D4⋊C16C2semidirect
97. D4×C127SD32semidirect
129. #12342C127direct (N × H)
135. #3338C2×C127semidirect
136. #2240C2×C127semidirect
137. #2207C2×C127semidirect
139. #1365C2×C127semidirect
140. #1332C2×C127semidirect
141. D4⋊C16C2×C127semidirect
142. C127#12342direct (N × H)
232. D4SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#33382
C127×#22402
C127×#22072
C127×#16672
C127×#13652
C127×#13322
C127×D4⋊C162
#12342127