Finite Group Databasebeta

← C127×#12102 #44575775

Subgroups — C127×#12102 #44575775

Order 32512.

528 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 82 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×#3186C211yesmaximal, characteristic, complemented
2C127×#2184C212yesmaximal, complemented
3C127×#1999C211yesmaximal, characteristic, complemented
4C127×#1347C212yesmaximal, complemented
5C127×#1314C211yesmaximal, characteristic, complemented
6C22×D8×C127C2211yescharacteristic, complemented
7C2×#121×C127—21no—
8C2×#121×C127—21no—
9C2×D16×C127—22no—
10#480×C127—22no—
11#466×C127—21no—
12#466×C127C2212yes—
13#447×C127—22no—
14C2×#102×C127—21no—
15C2×C22⋊C8×C127C2211yescharacteristic
16#357×C127—22no—
17C22⋊C16×C127C2212yescomplemented
18#340×C127C2211yescharacteristic, complemented
19C22×D4×C127—21no—
20C22×D4×C127—21no—
21C22×D4×C127—41no—
22C22×D4×C127—41no—
23C23×C4×C127D411yescharacteristic
24C2×D8×C127—21no—
25C2×D8×C127—22no—
26C2×D8×C127—42no—
27C2×D8×C127D412yescomplemented
28C22×C8×C127D411yescharacteristic
29C22×C8×C127C2311yescharacteristic
30#121×C127—22no—
31#121×C127—22no—
32#121×C127—42no—
33#121×C127—41no—
34#121×C127—42no—
35#121×C127—41no—
36C2×C4⋊C4×C127—21no—
37C2×C4⋊C4×C127—21no—
38C2×C22⋊C4×C127—41no—
39C2×C22⋊C4×C127—41no—
40D16×C127—44no—
41C2×C16×C127—22no—
42#107×C127—21no—
43#107×C127D412yescomplemented
44#102×C127—22no—
45#102×C127—22no—
46#102×C127—42no—
47C22⋊C8×C127—22no—
48C24×C127—41no—
49C24×C127—41no—
50C24×C127D811yescharacteristic
51C2×D4×C127—22no—
52C2×D4×C127—22no—
53C2×D4×C127—41no—
54C2×D4×C127—41no—
55C2×D4×C127—41no—
56C2×D4×C127—41no—
57C2×D4×C127—42no—
58C2×D4×C127—44no—
59C2×D4×C127—42no—
60C2×D4×C127—41no—
61C2×D4×C127—41no—
62C2×D4×C127—81no—
63C2×D4×C127—81no—
64C2×D4×C127—82no—
65C2×D4×C127—82no—
66C2×D4×C127—82no—
67C2×D4×C127—82no—
68C22×C4×C127—21no—
69C22×C4×C127—21no—
70C22×C4×C127—41no—
71C22×C4×C127—41no—
72C22×C4×C127—41no—
73C22×C4×C127—41no—
74C22×C4×C127D811yescharacteristic
75C22×C4×C127C2×D411yescharacteristic
76D8×C127—24no—
77D8×C127—42no—
78D8×C127—42no—
79D8×C127—81no—
80C2×C8×C127—21no—
81C2×C8×C127—41no—
82C2×C8×C127—41no—
83C2×C8×C127C2×D412yes—
84C4⋊C4×C127—22no—
85C4⋊C4×C127—22no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127—41no—
88C22⋊C4×C127—42no—
89C22⋊C4×C127—44no—
90C22⋊C4×C127—42no—
91C16×C127—42no—
92C23×C127—21no—
93C23×C127—21no—
94C23×C127—41no—
95C23×C127—41no—
96C23×C127—42no—
97C23×C127—42no—
98C23×C127—81no—
99C23×C127—81no—
100C23×C127—82no—
101C23×C127—81no—
102C23×C127—82no—
103C23×C127—82no—
104C23×C127—81no—
105C23×C127—81no—
106C23×C127—81no—
107C23×C127D1612yescomplemented
108C23×C127C2×D811yescharacteristic
109D4×C127—42no—
110D4×C127—42no—
111D4×C127—42no—
112D4×C127—42no—
113D4×C127—81no—
114D4×C127—82no—
115D4×C127—84no—
116D4×C127—88no—
117D4×C127—82no—
118D4×C127—82no—
119D4×C127—81no—
120C2×C4×C127—21no—
121C2×C4×C127—21no—
122C2×C4×C127—21no—
123C2×C4×C127—41no—
124C2×C4×C127—41no—
125C2×C4×C127—41no—
126C2×C4×C127—42no—
127C2×C4×C127—42no—
128C2×C4×C127—81no—
129C2×C4×C127—81no—
130C2×C4×C127—82no—
131C2×C4×C127—81no—
132C2×C4×C127#12011yescharacteristic
133C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
134C8×C127—22no—
135C8×C127—41no—
136C22×C127—21no—
137C22×C127—21no—
138C22×C127—22no—
139C22×C127—42no—
140C22×C127—41no—
141C22×C127—41no—
142C22×C127—81no—
143C22×C127—82no—
144C22×C127—82no—
145C22×C127—82no—
146C22×C127—82no—
147C22×C127—82no—
148C22×C127—82no—
149C22×C127—161no—
150C22×C127—161no—
151C22×C127—164no—
152C22×C127—161no—
153C22×C127—161no—
154C22×C127#44711yescharacteristic
155C22×C127C2×D1611yescharacteristic
156C22×C127#50911yescharacteristic
157C4×C127—21no—
158C4×C127—41no—
159C4×C127—82no—
160C4×C127—82no—
161C4×C127#45311yescharacteristic
162C4×C127B(4,𝔽2)11yescharacteristic
163#12102C12711yesmaximal, characteristic, direct product factor, Sylow
164C2×C127—21no—
165C2×C127—21no—
166C2×C127—42no—
167C2×C127—162no—
168C2×C127—162no—
169C2×C127#159511yescharacteristic
170C2×C127#218411yescharacteristic
171C2×C127#219011yescharacteristic
172#3186C2×C12711yescharacteristic, complemented
173#2184C2×C12712yescomplemented
174#1999C2×C12711yescharacteristic, complemented
175#1347C2×C12712yescomplemented
176#1314C2×C12711yescharacteristic, complemented
177C127#1210211yescharacteristic, direct product factor, Sylow
178C22×D8C22×C12711yescharacteristic, complemented
179C2×#121—21no—
180C2×#121—21no—
181C2×D16—22no—
182#480—22no—
183#466—21no—
184#466C22×C12712yes—
185#447—22no—
186C2×#102—21no—
187C2×C22⋊C8C22×C12711yescharacteristic
188#357—22no—
189C22⋊C16C22×C12712yescomplemented
190#340C22×C12711yescharacteristic, complemented
191C22×D4—21no—
192C22×D4—21no—
193C22×D4—41no—
194C22×D4—41no—
195C23×C4D4×C12711yescharacteristic
196C2×D8—21no—
197C2×D8—22no—
198C2×D8—42no—
199C2×D8D4×C12712yescomplemented
200C22×C8D4×C12711yescharacteristic
201C22×C8C23×C12711yescharacteristic
202#121—22no—
203#121—22no—
204#121—42no—
205#121—41no—
206#121—42no—
207#121—41no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212D16—44no—
213C2×C16—22no—
214#107—21no—
215#107D4×C12712yescomplemented
216#102—22no—
217#102—22no—
218#102—42no—
219C22⋊C8—22no—
220C24—41no—
221C24—41no—
222C24D8×C12711yescharacteristic
223C2×D4—22no—
224C2×D4—22no—
225C2×D4—41no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—41no—
229C2×D4—42no—
230C2×D4—44no—
231C2×D4—42no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—81no—
235C2×D4—81no—
236C2×D4—82no—
237C2×D4—82no—
238C2×D4—82no—
239C2×D4—82no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4D8×C12711yescharacteristic
247C22×C4C2×D4×C12711yescharacteristic
248D8—24no—
249D8—42no—
250D8—42no—
251D8—81no—
252C2×C8—21no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8C2×D4×C12712yes—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—42no—
263C16—42no—
264C23—21no—
265C23—21no—
266C23—41no—
267C23—41no—
268C23—42no—
269C23—42no—
270C23—81no—
271C23—81no—
272C23—82no—
273C23—81no—
274C23—82no—
275C23—82no—
276C23—81no—
277C23—81no—
278C23—81no—
279C23D16×C12712yescomplemented
280C23C2×D8×C12711yescharacteristic
281D4—42no—
282D4—42no—
283D4—42no—
284D4—42no—
285D4—81no—
286D4—82no—
287D4—84no—
288D4—88no—
289D4—82no—
290D4—82no—
291D4—81no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—82no—
303C2×C4—81no—
304C2×C4#120×C12711yescharacteristic
305C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
306C8—22no—
307C8—41no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—81no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—161no—
322C22—161no—
323C22—164no—
324C22—161no—
325C22—161no—
326C22#447×C12711yescharacteristic
327C22C2×D16×C12711yescharacteristic
328C22#509×C12711yescharacteristic
329C4—21no—
330C4—41no—
331C4—82no—
332C4—82no—
333C4#453×C12711yescharacteristic
334C4B(4,𝔽2)×C12711yescharacteristic
335C2—21no—
336C2—21no—
337C2—42no—
338C2—162no—
339C2—162no—
340C2C127×#159511yescharacteristic
341C2C127×#218411yescharacteristic
342C2C127×#219011yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C22⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3186C2semidirect
2. C127×#2184C2semidirect
3. C127×#1999C2semidirect
4. C127×#1347C2semidirect
5. C127×#1314C2semidirect
6. C22×D8×C127C22semidirect
17. C22⋊C16×C127C22semidirect
18. #340×C127C22semidirect
27. C2×D8×C127D4semidirect
43. #107×C127D4semidirect
107. C23×C127D16semidirect
163. #12102C127direct (N × H)
172. #3186C2×C127semidirect
173. #2184C2×C127semidirect
174. #1999C2×C127semidirect
175. #1347C2×C127semidirect
176. #1314C2×C127semidirect
177. C127#12102direct (N × H)
178. C22×D8C22×C127semidirect
189. C22⋊C16C22×C127semidirect
190. #340C22×C127semidirect
199. C2×D8D4×C127semidirect
215. #107D4×C127semidirect
279. C23D16×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#31862
C127×#21842
C127×#19992
C127×#13472
C127×#13142
#12102127