Finite Group Databasebeta

← C127×#12113 #44575786

Subgroups — C127×#12113 #44575786

Order 32512.

402 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 74 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×#3203C211yesmaximal, characteristic
2C127×#2188C211yesmaximal, characteristic, complemented
3C127×#2185C211yesmaximal, characteristic, complemented
4C127×#2020C211yesmaximal, characteristic, complemented
5C127×#1350C211yesmaximal, characteristic, complemented
6C127×#1349C211yesmaximal, characteristic, complemented
7C127×#1315C211yesmaximal, characteristic, complemented
8C2×#137×C127—21no—
9C22×Q16×C127C2211yescharacteristic
10#537×C127—21no—
11C2×Q32×C127—21no—
12C2×SD32×C127—21no—
13#484×C127—21no—
14#480×C127—21no—
15#468×C127—21no—
16#467×C127C2211yescharacteristic
17#466×C127C2211yescharacteristic
18#452×C127—21no—
19#449×C127—21no—
20C2×#101×C127—21no—
21#408×C127C2211yescharacteristic
22#358×C127—21no—
23#358×C127—21no—
24C22⋊C16×C127C2211yescharacteristic
25C22⋊C16×C127C2211yescharacteristic
26#341×C127C2211yescharacteristic
27C2×#41×C127—21no—
28C2×#41×C127D411yescharacteristic
29C22×Q8×C127—21no—
30#137×C127—42no—
31#137×C127—42no—
32C2×Q16×C127—21no—
33C2×Q16×C127—21no—
34C2×Q16×C127—42no—
35C2×Q16×C127—41no—
36C2×Q16×C127D411yescharacteristic
37C2×SD16×C127—21no—
38C2×SD16×C127—41no—
39C2×D8×C127D411yescharacteristic
40C2×M16×C127—21no—
41C2×M16×C127D411yescharacteristic
42C22×C8×C127C2311yescharacteristic
43C22⋊Q8×C127—21no—
44C22⋊Q8×C127—21no—
45C22⋊Q8×C127—41no—
46#121×C127—21no—
47#121×C127—21no—
48#121×C127—41no—
49#121×C127—41no—
50#121×C127—41no—
51C2×C4⋊C4×C127—21no—
52Q32×C127—42no—
53SD32×C127—42no—
54C2×C16×C127—21no—
55C2×C16×C127—21no—
56#107×C127—21no—
57#107×C127D411yescharacteristic
58#107×C127D411yescharacteristic
59#103×C127—21no—
60#103×C127—21no—
61#102×C127—21no—
62#102×C127—21no—
63#101×C127—42no—
64C22⋊C8×C127—21no—
65C22⋊C8×C127—21no—
66#41×C127—41no—
67#41×C127—41no—
68#41×C127—81no—
69#41×C127—81no—
70C2×Q8×C127—22no—
71C2×Q8×C127—21no—
72C2×Q8×C127—41no—
73C2×Q8×C127—41no—
74C2×Q8×C127—41no—
75C2×Q8×C127—41no—
76C2×Q8×C127—41no—
77C2×Q8×C127D811yescharacteristic
78C2×D4×C127—21no—
79C2×D4×C127—21no—
80C2×D4×C127—41no—
81C2×D4×C127—41no—
82C2×D4×C127—41no—
83C2×D4×C127D811yescharacteristic
84C22×C4×C127—21no—
85C22×C4×C127—41no—
86C22×C4×C127—41no—
87C22×C4×C127—41no—
88C22×C4×C127C2×D411yescharacteristic
89Q16×C127—22no—
90Q16×C127—22no—
91Q16×C127—41no—
92Q16×C127—41no—
93Q16×C127—41no—
94Q16×C127—44no—
95SD16×C127—44no—
96SD16×C127—81no—
97D8×C127—41no—
98M16×C127—22no—
99M16×C127—41no—
100M16×C127—41no—
101C2×C8×C127—21no—
102C2×C8×C127—41no—
103C2×C8×C127—41no—
104C2×C8×C127C2×D411yescharacteristic
105C2×C8×C127C2×D411yescharacteristic
106C4⋊C4×C127—21no—
107C4⋊C4×C127—21no—
108C4⋊C4×C127—41no—
109C4⋊C4×C127—41no—
110C4⋊C4×C127—41no—
111C22⋊C4×C127—41no—
112C22⋊C4×C127—41no—
113C22⋊C4×C127—41no—
114C22⋊C4×C127—41no—
115C16×C127—41no—
116C16×C127—41no—
117C23×C127—21no—
118C23×C127—41no—
119C23×C127AGL(1,ℤ/8)11yescharacteristic
120Q8×C127—41no—
121Q8×C127—42no—
122Q8×C127—41no—
123Q8×C127—42no—
124Q8×C127—41no—
125Q8×C127—81no—
126Q8×C127—81no—
127D4×C127—41no—
128D4×C127—41no—
129D4×C127—41no—
130D4×C127—81no—
131D4×C127—82no—
132D4×C127—161no—
133D4×C127—161no—
134C2×C4×C127—21no—
135C2×C4×C127—21no—
136C2×C4×C127—21no—
137C2×C4×C127—41no—
138C2×C4×C127—41no—
139C2×C4×C127—42no—
140C2×C4×C127—41no—
141C2×C4×C127—41no—
142C2×C4×C127—41no—
143C2×C4×C127—82no—
144C2×C4×C127—81no—
145C2×C4×C127—81no—
146C2×C4×C127—161no—
147C2×C4×C127#12011yescharacteristic
148C2×C4×C127C2×D811yescharacteristic
149C8×C127—21no—
150C8×C127—21no—
151C8×C127—42no—
152C8×C127—41no—
153C22×C127—21no—
154C22×C127—21no—
155C22×C127—21no—
156C22×C127—41no—
157C22×C127—41no—
158C22×C127—41no—
159C22×C127—81no—
160C22×C127—81no—
161C22×C127—81no—
162C22×C127#44711yescharacteristic
163C4×C127—21no—
164C4×C127—41no—
165C4×C127—81no—
166C4×C127—82no—
167C4×C127—81no—
168C4×C127—81no—
169C4×C127#45311yescharacteristic
170C4×C127B(4,𝔽2)11yescharacteristic
171#12113C12711yesmaximal, characteristic, direct product factor, Sylow
172C2×C127—41no—
173C2×C127—81no—
174C2×C127—161no—
175C2×C127#159511yescharacteristic
176C2×C127#219411yescharacteristic
177C2×C127#219511yescharacteristic
178#3203C2×C12711yescharacteristic
179#2188C2×C12711yescharacteristic, complemented
180#2185C2×C12711yescharacteristic, complemented
181#2020C2×C12711yescharacteristic, complemented
182#1350C2×C12711yescharacteristic, complemented
183#1349C2×C12711yescharacteristic, complemented
184#1315C2×C12711yescharacteristic, complemented
185C127#1211311yescharacteristic, direct product factor, Sylow
186C2×#137—21no—
187C22×Q16C22×C12711yescharacteristic
188#537—21no—
189C2×Q32—21no—
190C2×SD32—21no—
191#484—21no—
192#480—21no—
193#468—21no—
194#467C22×C12711yescharacteristic
195#466C22×C12711yescharacteristic
196#452—21no—
197#449—21no—
198C2×#101—21no—
199#408C22×C12711yescharacteristic
200#358—21no—
201#358—21no—
202C22⋊C16C22×C12711yescharacteristic
203C22⋊C16C22×C12711yescharacteristic
204#341C22×C12711yescharacteristic
205C2×#41—21no—
206C2×#41D4×C12711yescharacteristic
207C22×Q8—21no—
208#137—42no—
209#137—42no—
210C2×Q16—21no—
211C2×Q16—21no—
212C2×Q16—42no—
213C2×Q16—41no—
214C2×Q16D4×C12711yescharacteristic
215C2×SD16—21no—
216C2×SD16—41no—
217C2×D8D4×C12711yescharacteristic
218C2×M16—21no—
219C2×M16D4×C12711yescharacteristic
220C22×C8C23×C12711yescharacteristic
221C22⋊Q8—21no—
222C22⋊Q8—21no—
223C22⋊Q8—41no—
224#121—21no—
225#121—21no—
226#121—41no—
227#121—41no—
228#121—41no—
229C2×C4⋊C4—21no—
230Q32—42no—
231SD32—42no—
232C2×C16—21no—
233C2×C16—21no—
234#107—21no—
235#107D4×C12711yescharacteristic
236#107D4×C12711yescharacteristic
237#103—21no—
238#103—21no—
239#102—21no—
240#102—21no—
241#101—42no—
242C22⋊C8—21no—
243C22⋊C8—21no—
244#41—41no—
245#41—41no—
246#41—81no—
247#41—81no—
248C2×Q8—22no—
249C2×Q8—21no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8D8×C12711yescharacteristic
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4D8×C12711yescharacteristic
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4C2×D4×C12711yescharacteristic
267Q16—22no—
268Q16—22no—
269Q16—41no—
270Q16—41no—
271Q16—41no—
272Q16—44no—
273SD16—44no—
274SD16—81no—
275D8—41no—
276M16—22no—
277M16—41no—
278M16—41no—
279C2×C8—21no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8C2×D4×C12711yescharacteristic
283C2×C8C2×D4×C12711yescharacteristic
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—41no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—41no—
292C22⋊C4—41no—
293C16—41no—
294C16—41no—
295C23—21no—
296C23—41no—
297C23AGL(1,ℤ/8)×C12711yescharacteristic
298Q8—41no—
299Q8—42no—
300Q8—41no—
301Q8—42no—
302Q8—41no—
303Q8—81no—
304Q8—81no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—81no—
309D4—82no—
310D4—161no—
311D4—161no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—82no—
322C2×C4—81no—
323C2×C4—81no—
324C2×C4—161no—
325C2×C4#120×C12711yescharacteristic
326C2×C4C2×D8×C12711yescharacteristic
327C8—21no—
328C8—21no—
329C8—42no—
330C8—41no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22#447×C12711yescharacteristic
341C4—21no—
342C4—41no—
343C4—81no—
344C4—82no—
345C4—81no—
346C4—81no—
347C4#453×C12711yescharacteristic
348C4B(4,𝔽2)×C12711yescharacteristic
349C2—41no—
350C2—81no—
351C2—161no—
352C2C127×#159511yescharacteristic
353C2C127×#219411yescharacteristic
354C2C127×#219511yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC127×#3203C2
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #408×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2188C2semidirect
3. C127×#2185C2semidirect
4. C127×#2020C2semidirect
5. C127×#1350C2semidirect
6. C127×#1349C2semidirect
7. C127×#1315C2semidirect
171. #12113C127direct (N × H)
179. #2188C2×C127semidirect
180. #2185C2×C127semidirect
181. #2020C2×C127semidirect
182. #1350C2×C127semidirect
183. #1349C2×C127semidirect
184. #1315C2×C127semidirect
185. C127#12113direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32032
C127×#21882
C127×#21852
C127×#20202
C127×#13502
C127×#13492
C127×#13152
#12113127