Finite Group Databasebeta

← C127×#10491 #44574164

Subgroups — C127×#10491 #44574164

Order 32512.

366 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 366 conjugacy classes of subgroups — too many to draw. Only its 98 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×#3227C211yesmaximal, characteristic
2C127×#1997C211yesmaximal, characteristic, complemented
3C127×#1574C211yesmaximal, characteristic, complemented
4C127×#1560C211yesmaximal, characteristic
5C127×#1546C211yesmaximal, characteristic, complemented
6C127×#1484C211yesmaximal, characteristic
7C127×#1299C211yesmaximal, characteristic, complemented
8#532×C127C2211yescharacteristic
9C2×C4⋊Q8×C127C2211yescharacteristic
10#498×C127—21no—
11#486×C127—21no—
12#481×C127—22no—
13#476×C127—21no—
14#476×C127—21no—
15#431×C127C2211yescharacteristic
16#431×C127C2211yescharacteristic
17#423×C127C2211yescharacteristic
18#423×C127C2211yescharacteristic
19C4×M16×C127C2211yescharacteristic
20#343×C127—22no—
21#335×C127—21no—
22#334×C127—21no—
23#333×C127—21no—
24#332×C127—22no—
25#331×C127—21no—
26D4⋊C8×C127—22no—
27C2×#41×C127—21no—
28C22×Q8×C127—21no—
29C2×M16×C127—21no—
30C2×M16×C127—21no—
31C2×M16×C127—21no—
32C4⋊Q8×C127—21no—
33C4⋊Q8×C127—21no—
34C4⋊Q8×C127—21no—
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127D411yescharacteristic
37#127×C127D411yescharacteristic
38#124×C127—21no—
39#121×C127—21no—
40#121×C127—21no—
41C4×D4×C127—21no—
42C4×D4×C127—21no—
43C2×C4⋊C4×C127—21no—
44C2×C4⋊C4×C127D412yes—
45C2×C42×C127C2311yescharacteristic
46#106×C127—22no—
47#106×C127—22no—
48C4⋊C8×C127—21no—
49C4⋊C8×C127—21no—
50C4⋊C8×C127—21no—
51C4⋊C8×C127—21no—
52C4⋊C8×C127D412yes—
53C4⋊C8×C127D411yescharacteristic
54C4⋊C8×C127D411yescharacteristic
55#102×C127—22no—
56#102×C127—22no—
57C22⋊C8×C127—21no—
58C22⋊C8×C127—21no—
59#97×C127—21no—
60C4×C8×C127—21no—
61C4×C8×C127—21no—
62C4×C8×C127—21no—
63#41×C127—82no—
64C2×Q8×C127—21no—
65C2×Q8×C127—41no—
66C2×Q8×C127—41no—
67C2×Q8×C127—81no—
68C2×Q8×C127—81no—
69C2×D4×C127—21no—
70C2×D4×C127—21no—
71C2×D4×C127—21no—
72C2×D4×C127—41no—
73C22×C4×C127—22no—
74C22×C4×C127—41no—
75C22×C4×C127C2×D411yescharacteristic
76C22×C4×C127#4111yescharacteristic
77C22×C4×C127#4111yescharacteristic
78M16×C127—22no—
79M16×C127—41no—
80M16×C127—42no—
81M16×C127—81no—
82C2×C8×C127—21no—
83C2×C8×C127—21no—
84C2×C8×C127—22no—
85C2×C8×C127—41no—
86C2×C8×C127—41no—
87C2×C8×C127—41no—
88C4⋊C4×C127—22no—
89C4⋊C4×C127—21no—
90C4⋊C4×C127—22no—
91C4⋊C4×C127—21no—
92C4⋊C4×C127—41no—
93C4⋊C4×C127—41no—
94C22⋊C4×C127—41no—
95C22⋊C4×C127—41no—
96C42×C127C2×D411yescharacteristic
97C42×C127C2×D411yescharacteristic
98C42×C127C2×D411yescharacteristic
99C42×C127#4111yescharacteristic
100C23×C127—41no—
101C23×C127#12311yescharacteristic
102Q8×C127—81no—
103Q8×C127—82no—
104Q8×C127—81no—
105Q8×C127—81no—
106D4×C127—41no—
107D4×C127—41no—
108D4×C127—41no—
109D4×C127—41no—
110D4×C127—81no—
111D4×C127—82no—
112C2×C4×C127—21no—
113C2×C4×C127—21no—
114C2×C4×C127—21no—
115C2×C4×C127—22no—
116C2×C4×C127—21no—
117C2×C4×C127—41no—
118C2×C4×C127—42no—
119C2×C4×C127—82no—
120C2×C4×C127—82no—
121C2×C4×C127#12011yescharacteristic
122C2×C4×C127#12111yescharacteristic
123C2×C4×C127#12111yescharacteristic
124C2×C4×C127#12111yescharacteristic
125C2×C4×C127#12311yescharacteristic
126C2×C4×C127#12411yescharacteristic
127C8×C127—42no—
128C8×C127—41no—
129C8×C127—41no—
130C8×C127—42no—
131C8×C127—81no—
132C8×C127—81no—
133C22×C127—21no—
134C22×C127—21no—
135C22×C127—21no—
136C22×C127—81no—
137C22×C127—81no—
138C22×C127—81no—
139C22×C127#39411yescharacteristic
140C4×C127—21no—
141C4×C127—21no—
142C4×C127—21no—
143C4×C127—82no—
144C4×C127—81no—
145C4×C127#45411yescharacteristic
146C4×C127#45611yescharacteristic
147C4×C127#47111yescharacteristic
148C4×C127#47112yes—
149C4×C127#47211yescharacteristic
150#10491C12711yesmaximal, characteristic, direct product factor, Sylow
151C2×C127—41no—
152C2×C127—161no—
153C2×C127#203811yescharacteristic
154C2×C127#204611yescharacteristic
155C2×C127#205111yescharacteristic
156#3227C2×C12711yescharacteristic
157#1997C2×C12711yescharacteristic, complemented
158#1574C2×C12711yescharacteristic, complemented
159#1560C2×C12711yescharacteristic
160#1546C2×C12711yescharacteristic, complemented
161#1484C2×C12711yescharacteristic
162#1299C2×C12711yescharacteristic, complemented
163C127#1049111yescharacteristic, direct product factor, Sylow
164#532C22×C12711yescharacteristic
165C2×C4⋊Q8C22×C12711yescharacteristic
166#498—21no—
167#486—21no—
168#481—22no—
169#476—21no—
170#476—21no—
171#431C22×C12711yescharacteristic
172#431C22×C12711yescharacteristic
173#423C22×C12711yescharacteristic
174#423C22×C12711yescharacteristic
175C4×M16C22×C12711yescharacteristic
176#343—22no—
177#335—21no—
178#334—21no—
179#333—21no—
180#332—22no—
181#331—21no—
182D4⋊C8—22no—
183C2×#41—21no—
184C22×Q8—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8D4×C12711yescharacteristic
193#127D4×C12711yescharacteristic
194#124—21no—
195#121—21no—
196#121—21no—
197C4×D4—21no—
198C4×D4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4D4×C12712yes—
201C2×C42C23×C12711yescharacteristic
202#106—22no—
203#106—22no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8D4×C12712yes—
209C4⋊C8D4×C12711yescharacteristic
210C4⋊C8D4×C12711yescharacteristic
211#102—22no—
212#102—22no—
213C22⋊C8—21no—
214C22⋊C8—21no—
215#97—21no—
216C4×C8—21no—
217C4×C8—21no—
218C4×C8—21no—
219#41—82no—
220C2×Q8—21no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—81no—
224C2×Q8—81no—
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—41no—
231C22×C4C2×D4×C12711yescharacteristic
232C22×C4#41×C12711yescharacteristic
233C22×C4#41×C12711yescharacteristic
234M16—22no—
235M16—41no—
236M16—42no—
237M16—81no—
238C2×C8—21no—
239C2×C8—21no—
240C2×C8—22no—
241C2×C8—41no—
242C2×C8—41no—
243C2×C8—41no—
244C4⋊C4—22no—
245C4⋊C4—21no—
246C4⋊C4—22no—
247C4⋊C4—21no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C22⋊C4—41no—
251C22⋊C4—41no—
252C42C2×D4×C12711yescharacteristic
253C42C2×D4×C12711yescharacteristic
254C42C2×D4×C12711yescharacteristic
255C42#41×C12711yescharacteristic
256C23—41no—
257C23#123×C12711yescharacteristic
258Q8—81no—
259Q8—82no—
260Q8—81no—
261Q8—81no—
262D4—41no—
263D4—41no—
264D4—41no—
265D4—41no—
266D4—81no—
267D4—82no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—82no—
276C2×C4—82no—
277C2×C4#120×C12711yescharacteristic
278C2×C4#121×C12711yescharacteristic
279C2×C4#121×C12711yescharacteristic
280C2×C4#121×C12711yescharacteristic
281C2×C4#123×C12711yescharacteristic
282C2×C4#124×C12711yescharacteristic
283C8—42no—
284C8—41no—
285C8—41no—
286C8—42no—
287C8—81no—
288C8—81no—
289C22—21no—
290C22—21no—
291C22—21no—
292C22—81no—
293C22—81no—
294C22—81no—
295C22#394×C12711yescharacteristic
296C4—21no—
297C4—21no—
298C4—21no—
299C4—81no—
300C4—82no—
301C4#454×C12711yescharacteristic
302C4#456×C12711yescharacteristic
303C4#471×C12712yes—
304C4#471×C12711yescharacteristic
305C4#472×C12711yescharacteristic
306C2—41no—
307C2—161no—
308C2C127×#203811yescharacteristic
309C2C127×#204611yescharacteristic
310C2C127×#205111yescharacteristic
subgroupG/H
center, socleC22×C127#394
commutator, Frattini, cosocleC2×C42C23×C127
prime radical#532×C127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1997C2semidirect
3. C127×#1574C2semidirect
5. C127×#1546C2semidirect
7. C127×#1299C2semidirect
150. #10491C127direct (N × H)
157. #1997C2×C127semidirect
158. #1574C2×C127semidirect
160. #1546C2×C127semidirect
162. #1299C2×C127semidirect
163. C127#10491direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32272
C127×#19972
C127×#15742
C127×#15602
C127×#15462
C127×#14842
C127×#12992
#10491127