Finite Group Databasebeta

← C127×#10497 #44574170

Subgroups — C127×#10497 #44574170

Order 32512.

348 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 90 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×#3234C211yesmaximal, characteristic
2C127×#1998C211yesmaximal, characteristic, complemented
3C127×#1569C211yesmaximal, characteristic, complemented
4C127×#1557C211yesmaximal, characteristic
5C127×#1528C211yesmaximal, characteristic, complemented
6C127×#1525C211yesmaximal, characteristic
7C127×#1291C211yesmaximal, characteristic, complemented
8#532×C127C2211yescharacteristic
9C2×#125×C127C2211yescharacteristic
10#501×C127—21no—
11#490×C127—21no—
12#481×C127—21no—
13#480×C127—21no—
14#478×C127—21no—
15#478×C127—21no—
16#432×C127C2211yescharacteristic
17#432×C127C2211yescharacteristic
18#423×C127C2211yescharacteristic
19C4×M16×C127C2211yescharacteristic
20C4×M16×C127C2211yescharacteristic
21#343×C127—21no—
22#343×C127—21no—
23#335×C127—21no—
24#334×C127—21no—
25#332×C127—21no—
26#331×C127—21no—
27#330×C127—21no—
28#330×C127—21no—
29#329×C127—21no—
30#329×C127—21no—
31C2×#41×C127—21no—
32C2×M16×C127—21no—
33C2×M16×C127—21no—
34C2×M16×C127—21no—
35C4⋊Q8×C127D411yescharacteristic
36#127×C127D411yescharacteristic
37#125×C127—21no—
38#125×C127—21no—
39#125×C127—21no—
40#125×C127—21no—
41#124×C127—21no—
42#121×C127—21no—
43#121×C127—21no—
44C4×D4×C127—21no—
45C4×D4×C127—21no—
46C2×C4⋊C4×C127—21no—
47C2×C4⋊C4×C127—21no—
48C2×C4⋊C4×C127D411yescharacteristic
49C2×C4⋊C4×C127D411yescharacteristic
50C2×C42×C127C2311yescharacteristic
51#107×C127—21no—
52#107×C127—21no—
53#106×C127—21no—
54#106×C127—21no—
55C4⋊C8×C127—21no—
56C4⋊C8×C127—21no—
57C4⋊C8×C127—21no—
58C4⋊C8×C127D411yescharacteristic
59C4⋊C8×C127D411yescharacteristic
60#102×C127—21no—
61#102×C127—21no—
62#102×C127—21no—
63#102×C127—21no—
64C22⋊C8×C127—21no—
65C22⋊C8×C127—21no—
66#97×C127—21no—
67#97×C127—21no—
68#97×C127—21no—
69#97×C127D411yescharacteristic
70#97×C127D411yescharacteristic
71C4×C8×C127—21no—
72C4×C8×C127—21no—
73#41×C127—82no—
74C2×Q8×C127—21no—
75C2×D4×C127—21no—
76C2×D4×C127—21no—
77C2×D4×C127—21no—
78C2×D4×C127—41no—
79C22×C4×C127—21no—
80C22×C4×C127—21no—
81C22×C4×C127—41no—
82C22×C4×C127C2×D411yescharacteristic
83C22×C4×C127#4111yescharacteristic
84C22×C4×C127#4111yescharacteristic
85M16×C127—22no—
86M16×C127—42no—
87M16×C127—41no—
88M16×C127—42no—
89C2×C8×C127—21no—
90C2×C8×C127—21no—
91C2×C8×C127—21no—
92C2×C8×C127—21no—
93C2×C8×C127—41no—
94C2×C8×C127—41no—
95C2×C8×C127—41no—
96C4⋊C4×C127—21no—
97C4⋊C4×C127—21no—
98C4⋊C4×C127—21no—
99C4⋊C4×C127—21no—
100C4⋊C4×C127—21no—
101C4⋊C4×C127—21no—
102C4⋊C4×C127—41no—
103C4⋊C4×C127—41no—
104C4⋊C4×C127—41no—
105C4⋊C4×C127—41no—
106C22⋊C4×C127—41no—
107C22⋊C4×C127—41no—
108C42×C127C2×D411yescharacteristic
109C42×C127C2×D411yescharacteristic
110C42×C127C2×D411yescharacteristic
111C42×C127#4111yescharacteristic
112C23×C127—41no—
113C23×C127#12311yescharacteristic
114Q8×C127—81no—
115D4×C127—41no—
116D4×C127—41no—
117D4×C127—81no—
118D4×C127—81no—
119D4×C127—82no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—41no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—41no—
127C2×C4×C127—82no—
128C2×C4×C127—81no—
129C2×C4×C127—81no—
130C2×C4×C127#12011yescharacteristic
131C2×C4×C127#12111yescharacteristic
132C2×C4×C127#12111yescharacteristic
133C2×C4×C127#12111yescharacteristic
134C2×C4×C127#12311yescharacteristic
135C2×C4×C127#12411yescharacteristic
136C8×C127—22no—
137C8×C127—41no—
138C8×C127—42no—
139C8×C127—41no—
140C8×C127—41no—
141C8×C127—81no—
142C8×C127—81no—
143C22×C127—21no—
144C22×C127—21no—
145C22×C127—21no—
146C22×C127—81no—
147C22×C127—81no—
148C22×C127—81no—
149C22×C127#39411yescharacteristic
150C4×C127—21no—
151C4×C127—21no—
152C4×C127—21no—
153C4×C127—21no—
154C4×C127—21no—
155C4×C127—81no—
156C4×C127—81no—
157C4×C127—81no—
158C4×C127#45411yescharacteristic
159C4×C127#45611yescharacteristic
160#10497C12711yesmaximal, characteristic, direct product factor, Sylow
161C2×C127—41no—
162C2×C127—161no—
163C2×C127#203811yescharacteristic
164C2×C127#204811yescharacteristic
165C2×C127#205011yescharacteristic
166#3234C2×C12711yescharacteristic
167#1998C2×C12711yescharacteristic, complemented
168#1569C2×C12711yescharacteristic, complemented
169#1557C2×C12711yescharacteristic
170#1528C2×C12711yescharacteristic, complemented
171#1525C2×C12711yescharacteristic
172#1291C2×C12711yescharacteristic, complemented
173C127#1049711yescharacteristic, direct product factor, Sylow
174#532C22×C12711yescharacteristic
175C2×#125C22×C12711yescharacteristic
176#501—21no—
177#490—21no—
178#481—21no—
179#480—21no—
180#478—21no—
181#478—21no—
182#432C22×C12711yescharacteristic
183#432C22×C12711yescharacteristic
184#423C22×C12711yescharacteristic
185C4×M16C22×C12711yescharacteristic
186C4×M16C22×C12711yescharacteristic
187#343—21no—
188#343—21no—
189#335—21no—
190#334—21no—
191#332—21no—
192#331—21no—
193#330—21no—
194#330—21no—
195#329—21no—
196#329—21no—
197C2×#41—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C4⋊Q8D4×C12711yescharacteristic
202#127D4×C12711yescharacteristic
203#125—21no—
204#125—21no—
205#125—21no—
206#125—21no—
207#124—21no—
208#121—21no—
209#121—21no—
210C4×D4—21no—
211C4×D4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4D4×C12711yescharacteristic
215C2×C4⋊C4D4×C12711yescharacteristic
216C2×C42C23×C12711yescharacteristic
217#107—21no—
218#107—21no—
219#106—21no—
220#106—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8D4×C12711yescharacteristic
225C4⋊C8D4×C12711yescharacteristic
226#102—21no—
227#102—21no—
228#102—21no—
229#102—21no—
230C22⋊C8—21no—
231C22⋊C8—21no—
232#97—21no—
233#97—21no—
234#97—21no—
235#97D4×C12711yescharacteristic
236#97D4×C12711yescharacteristic
237C4×C8—21no—
238C4×C8—21no—
239#41—82no—
240C2×Q8—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—41no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4C2×D4×C12711yescharacteristic
249C22×C4#41×C12711yescharacteristic
250C22×C4#41×C12711yescharacteristic
251M16—22no—
252M16—42no—
253M16—41no—
254M16—42no—
255C2×C8—21no—
256C2×C8—21no—
257C2×C8—21no—
258C2×C8—21no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8—41no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42C2×D4×C12711yescharacteristic
275C42C2×D4×C12711yescharacteristic
276C42C2×D4×C12711yescharacteristic
277C42#41×C12711yescharacteristic
278C23—41no—
279C23#123×C12711yescharacteristic
280Q8—81no—
281D4—41no—
282D4—41no—
283D4—81no—
284D4—81no—
285D4—82no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—82no—
294C2×C4—81no—
295C2×C4—81no—
296C2×C4#120×C12711yescharacteristic
297C2×C4#121×C12711yescharacteristic
298C2×C4#121×C12711yescharacteristic
299C2×C4#121×C12711yescharacteristic
300C2×C4#123×C12711yescharacteristic
301C2×C4#124×C12711yescharacteristic
302C8—22no—
303C8—41no—
304C8—42no—
305C8—41no—
306C8—41no—
307C8—81no—
308C8—81no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—81no—
313C22—81no—
314C22—81no—
315C22#394×C12711yescharacteristic
316C4—21no—
317C4—21no—
318C4—21no—
319C4—21no—
320C4—21no—
321C4—81no—
322C4—81no—
323C4—81no—
324C4#454×C12711yescharacteristic
325C4#456×C12711yescharacteristic
326C2—41no—
327C2—161no—
328C2C127×#203811yescharacteristic
329C2C127×#204811yescharacteristic
330C2C127×#205011yescharacteristic
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×#1998C2semidirect
3. C127×#1569C2semidirect
5. C127×#1528C2semidirect
7. C127×#1291C2semidirect
160. #10497C127direct (N × H)
167. #1998C2×C127semidirect
168. #1569C2×C127semidirect
170. #1528C2×C127semidirect
172. #1291C2×C127semidirect
173. C127#10497direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32342
C127×#19982
C127×#15692
C127×#15572
C127×#15282
C127×#15252
C127×#12912
#10497127