Finite Group Databasebeta

← C127×#12118 #44575791

Subgroups — C127×#12118 #44575791

Order 32512.

392 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3202C211yesmaximal, characteristic, complemented
2C127×#2191C212yesmaximal, complemented
3C127×#2017C211yesmaximal, characteristic
4C127×#1342C212yesmaximal, complemented
5C127×#1321C211yesmaximal, characteristic, complemented
6#575×C127—21no—
7#575×C127C2211yescharacteristic
8#536×C127—21no—
9#510×C127—22no—
10#485×C127—22no—
11#470×C127C2212yes—
12#467×C127—21no—
13#451×C127—22no—
14#408×C127—21no—
15#408×C127C2211yescharacteristic
16#360×C127—22no—
17#349×C127C2212yes—
18#338×C127C2211yescharacteristic
19C2×#41×C127—21no—
20C2×#41×C127D411yescharacteristic
21C22×Q8×C127—21no—
22#137×C127—41no—
23#137×C127—41no—
24#137×C127—41no—
25AGL(1,ℤ/8)×C127—41no—
26AGL(1,ℤ/8)×C127—41no—
27AGL(1,ℤ/8)×C127—41no—
28#135×C127—22no—
29#135×C127—42no—
30C2×Q16×C127—21no—
31C2×Q16×C127D412yes—
32C2×M16×C127—21no—
33C2×M16×C127C2311yescharacteristic
34C22×C8×C127D411yescharacteristic
35C4⋊Q8×C127—41no—
36#124×C127—41no—
37#123×C127—41no—
38#123×C127—41no—
39C22⋊Q8×C127—22no—
40C22⋊Q8×C127—22no—
41#117×C127—21no—
42Q32×C127—42no—
43SD32×C127—42no—
44M32×C127—22no—
45#107×C127—21no—
46#106×C127D412yes—
47#103×C127—22no—
48#103×C127—22no—
49C22⋊C8×C127—22no—
50C22⋊C8×C127—42no—
51#41×C127—41no—
52#41×C127—41no—
53#41×C127—41no—
54#41×C127—41no—
55#41×C127—41no—
56#41×C127—41no—
57C2×Q8×C127—21no—
58C2×Q8×C127—22no—
59C2×Q8×C127—41no—
60C2×Q8×C127—81no—
61C2×Q8×C127—81no—
62C2×Q8×C127D811yescharacteristic
63C2×D4×C127—21no—
64C2×D4×C127—21no—
65C2×D4×C127—41no—
66C2×D4×C127D811yescharacteristic
67C22×C4×C127—21no—
68C22×C4×C127—41no—
69C22×C4×C127—41no—
70C22×C4×C127C2×D411yescharacteristic
71Q16×C127—22no—
72Q16×C127—42no—
73Q16×C127—42no—
74Q16×C127—81no—
75SD16×C127—42no—
76SD16×C127—42no—
77SD16×C127—42no—
78D8×C127—22no—
79D8×C127—42no—
80M16×C127—41no—
81M16×C127—41no—
82M16×C127—41no—
83C2×C8×C127—22no—
84C2×C8×C127—21no—
85C2×C8×C127—41no—
86C2×C8×C127C2×D412yes—
87C4⋊C4×C127—22no—
88C4⋊C4×C127—42no—
89C4⋊C4×C127—42no—
90C22⋊C4×C127—41no—
91C22⋊C4×C127—42no—
92C22⋊C4×C127—42no—
93C42×C127—41no—
94C16×C127—42no—
95C23×C127—21no—
96C23×C127—41no—
97C23×C127AGL(1,ℤ/8)11yescharacteristic
98Q8×C127—41no—
99Q8×C127—41no—
100Q8×C127—42no—
101Q8×C127—41no—
102Q8×C127—42no—
103Q8×C127—82no—
104D4×C127—41no—
105D4×C127—41no—
106D4×C127—41no—
107D4×C127—81no—
108D4×C127—81no—
109D4×C127—81no—
110C2×C4×C127—21no—
111C2×C4×C127—21no—
112C2×C4×C127—21no—
113C2×C4×C127—41no—
114C2×C4×C127—41no—
115C2×C4×C127—41no—
116C2×C4×C127—42no—
117C2×C4×C127—42no—
118C2×C4×C127—81no—
119C2×C4×C127—82no—
120C2×C4×C127—81no—
121C2×C4×C127#12011yescharacteristic
122C2×C4×C127C2×D811yescharacteristic
123C8×C127—22no—
124C8×C127—42no—
125C8×C127—41no—
126C22×C127—21no—
127C22×C127—41no—
128C22×C127—42no—
129C22×C127—41no—
130C22×C127—82no—
131C22×C127—81no—
132C22×C127#44711yescharacteristic
133C4×C127—21no—
134C4×C127—41no—
135C4×C127—82no—
136C4×C127—81no—
137C4×C127—82no—
138C4×C127#45311yescharacteristic
139C4×C127B(4,𝔽2)11yescharacteristic
140#12118C12711yesmaximal, characteristic, direct product factor, Sylow
141C2×C127—21no—
142C2×C127—41no—
143C2×C127—81no—
144C2×C127—161no—
145C2×C127#159511yescharacteristic
146#3202C2×C12711yescharacteristic, complemented
147#2191C2×C12712yescomplemented
148#2017C2×C12711yescharacteristic
149#1342C2×C12712yescomplemented
150#1321C2×C12711yescharacteristic, complemented
151C127#1211811yescharacteristic, direct product factor, Sylow
152#575—21no—
153#575C22×C12711yescharacteristic
154#536—21no—
155#510—22no—
156#485—22no—
157#470C22×C12712yes—
158#467—21no—
159#451—22no—
160#408—21no—
161#408C22×C12711yescharacteristic
162#360—22no—
163#349C22×C12712yes—
164#338C22×C12711yescharacteristic
165C2×#41—21no—
166C2×#41D4×C12711yescharacteristic
167C22×Q8—21no—
168#137—41no—
169#137—41no—
170#137—41no—
171AGL(1,ℤ/8)—41no—
172AGL(1,ℤ/8)—41no—
173AGL(1,ℤ/8)—41no—
174#135—22no—
175#135—42no—
176C2×Q16—21no—
177C2×Q16D4×C12712yes—
178C2×M16—21no—
179C2×M16C23×C12711yescharacteristic
180C22×C8D4×C12711yescharacteristic
181C4⋊Q8—41no—
182#124—41no—
183#123—41no—
184#123—41no—
185C22⋊Q8—22no—
186C22⋊Q8—22no—
187#117—21no—
188Q32—42no—
189SD32—42no—
190M32—22no—
191#107—21no—
192#106D4×C12712yes—
193#103—22no—
194#103—22no—
195C22⋊C8—22no—
196C22⋊C8—42no—
197#41—41no—
198#41—41no—
199#41—41no—
200#41—41no—
201#41—41no—
202#41—41no—
203C2×Q8—21no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—81no—
207C2×Q8—81no—
208C2×Q8D8×C12711yescharacteristic
209C2×D4—21no—
210C2×D4—21no—
211C2×D4—41no—
212C2×D4D8×C12711yescharacteristic
213C22×C4—21no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4C2×D4×C12711yescharacteristic
217Q16—22no—
218Q16—42no—
219Q16—42no—
220Q16—81no—
221SD16—42no—
222SD16—42no—
223SD16—42no—
224D8—22no—
225D8—42no—
226M16—41no—
227M16—41no—
228M16—41no—
229C2×C8—22no—
230C2×C8—21no—
231C2×C8—41no—
232C2×C8C2×D4×C12712yes—
233C4⋊C4—22no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C22⋊C4—41no—
237C22⋊C4—42no—
238C22⋊C4—42no—
239C42—41no—
240C16—42no—
241C23—21no—
242C23—41no—
243C23AGL(1,ℤ/8)×C12711yescharacteristic
244Q8—41no—
245Q8—41no—
246Q8—42no—
247Q8—41no—
248Q8—42no—
249Q8—82no—
250D4—41no—
251D4—41no—
252D4—41no—
253D4—81no—
254D4—81no—
255D4—81no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—82no—
266C2×C4—81no—
267C2×C4#120×C12711yescharacteristic
268C2×C4C2×D8×C12711yescharacteristic
269C8—22no—
270C8—42no—
271C8—41no—
272C22—21no—
273C22—41no—
274C22—42no—
275C22—41no—
276C22—82no—
277C22—81no—
278C22#447×C12711yescharacteristic
279C4—21no—
280C4—41no—
281C4—82no—
282C4—81no—
283C4—82no—
284C4#453×C12711yescharacteristic
285C4B(4,𝔽2)×C12711yescharacteristic
286C2—21no—
287C2—41no—
288C2—81no—
289C2—161no—
290C2C127×#159511yescharacteristic
subgroupG/H
center, socleC2×C127#1595
commutator, Frattini, cosocleC2×M16C23×C127
prime radicalC127×#2017C2
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3202C2semidirect
2. C127×#2191C2semidirect
4. C127×#1342C2semidirect
5. C127×#1321C2semidirect
140. #12118C127direct (N × H)
146. #3202C2×C127semidirect
147. #2191C2×C127semidirect
149. #1342C2×C127semidirect
150. #1321C2×C127semidirect
151. C127#12118direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32022
C127×#21912
C127×#20172
C127×#13422
C127×#13212
#12118127