Finite Group Databasebeta

← C127×#12317 #44575990

Subgroups — C127×#12317 #44575990

Order 32512.

400 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 86 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×#3287C211yesmaximal, characteristic, complemented
2C127×#2246C211yesmaximal, characteristic, complemented
3C127×#2207C211yesmaximal, characteristic, complemented
4C127×#1667C211yesmaximal, characteristic, complemented
5C127×#1361C211yesmaximal, characteristic, complemented
6C127×#1332C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8C2×#135×C127—21no—
9#550×C127—21no—
10C2×SD32×C127—21no—
11C2×D16×C127—22no—
12#493×C127C2211yescharacteristic
13#493×C127C2211yescharacteristic
14#466×C127—21no—
15#460×C127—21no—
16#459×C127—21no—
17#449×C127—22no—
18C4×Q16×C127C2211yescharacteristic
19C8×D4×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#357×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#331×C127—21no—
25C2×#41×C127—22no—
26#135×C127—42no—
27#135×C127—42no—
28C2×Q16×C127D411yescharacteristic
29C2×SD16×C127—22no—
30C2×D8×C127—21no—
31C2×D8×C127—21no—
32C2×D8×C127—21no—
33C2×D8×C127—41no—
34C2×D8×C127D412yes—
35C22×C8×C127—21no—
36#127×C127—21no—
37#127×C127—21no—
38#127×C127—41no—
39#121×C127—22no—
40#121×C127—42no—
41C4×Q8×C127—21no—
42C4×D4×C127—41no—
43C4×D4×C127D411yescharacteristic
44SD32×C127—42no—
45D16×C127—44no—
46C2×C16×C127—21no—
47C2×C16×C127—21no—
48#107×C127D411yescharacteristic
49C4⋊C8×C127—21no—
50C4⋊C8×C127D411yescharacteristic
51#103×C127—21no—
52#102×C127—22no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C4×C8×C127C2311yescharacteristic
56#41×C127—44no—
57#41×C127—82no—
58C2×Q8×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127—42no—
61C2×D4×C127—42no—
62C2×D4×C127—41no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—41no—
66C2×D4×C127—41no—
67C2×D4×C127D811yescharacteristic
68C22×C4×C127—21no—
69C22×C4×C127—42no—
70Q16×C127—21no—
71Q16×C127D812yescomplemented
72SD16×C127—44no—
73D8×C127—42no—
74D8×C127—42no—
75D8×C127—41no—
76D8×C127—41no—
77D8×C127—41no—
78D8×C127—41no—
79D8×C127—82no—
80C2×C8×C127—21no—
81C2×C8×C127—41no—
82C2×C8×C127—41no—
83C2×C8×C127—41no—
84C2×C8×C127C2×D411yescharacteristic
85C2×C8×C127C2×D411yescharacteristic
86C4⋊C4×C127—21no—
87C4⋊C4×C127—41no—
88C4⋊C4×C127D811yescharacteristic
89C22⋊C4×C127—21no—
90C22⋊C4×C127—42no—
91C42×C127—41no—
92C42×C127C2×D411yescharacteristic
93C16×C127—22no—
94C16×C127—41no—
95C23×C127—21no—
96C23×C127—42no—
97C23×C127—81no—
98Q8×C127—22no—
99Q8×C127—41no—
100D4×C127—21no—
101D4×C127—21no—
102D4×C127—42no—
103D4×C127—42no—
104D4×C127—84no—
105D4×C127—81no—
106D4×C127—81no—
107D4×C127—81no—
108D4×C127—81no—
109D4×C127—81no—
110D4×C127—81no—
111D4×C127—161no—
112D4×C127—161no—
113D4×C127—162no—
114C2×C4×C127—21no—
115C2×C4×C127—41no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—84no—
120C2×C4×C127#12011yescharacteristic
121C2×C4×C127C2×D811yescharacteristic
122C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
123C8×C127—21no—
124C8×C127—41no—
125C8×C127—81no—
126C8×C127C2×D811yescharacteristic
127C8×C127AGL(1,ℤ/8)11yescharacteristic
128C22×C127—41no—
129C22×C127—41no—
130C22×C127—41no—
131C22×C127—82no—
132C22×C127—82no—
133C22×C127—82no—
134C22×C127—161no—
135C22×C127—161no—
136C22×C127—161no—
137C22×C127#44711yescharacteristic
138C4×C127—21no—
139C4×C127—41no—
140C4×C127—42no—
141C4×C127—81no—
142C4×C127#44711yescharacteristic
143C4×C127#45311yescharacteristic
144C4×C127#50811yescharacteristic
145C4×C127#50911yescharacteristic
146#12317C12711yesmaximal, characteristic, direct product factor, Sylow
147C2×C127—81no—
148C2×C127—162no—
149C2×C127—321no—
150C2×C127#161911yescharacteristic
151C2×C127#218611yescharacteristic
152C2×C127#219211yescharacteristic
153#3287C2×C12711yescharacteristic, complemented
154#2246C2×C12711yescharacteristic, complemented
155#2207C2×C12711yescharacteristic, complemented
156#1667C2×C12711yescharacteristic, complemented
157#1361C2×C12711yescharacteristic, complemented
158#1332C2×C12711yescharacteristic, complemented
159D4⋊C16C2×C12711yescharacteristic, complemented
160C127#1231711yescharacteristic, direct product factor, Sylow
161C2×#135—21no—
162#550—21no—
163C2×SD32—21no—
164C2×D16—22no—
165#493C22×C12711yescharacteristic
166#493C22×C12711yescharacteristic
167#466—21no—
168#460—21no—
169#459—21no—
170#449—22no—
171C4×Q16C22×C12711yescharacteristic
172C8×D4C22×C12711yescharacteristic
173C4⋊C16C22×C12711yescharacteristic
174#357—21no—
175C4×C16C22×C12711yescharacteristic
176#335C22×C12711yescharacteristic
177#331—21no—
178C2×#41—22no—
179#135—42no—
180#135—42no—
181C2×Q16D4×C12711yescharacteristic
182C2×SD16—22no—
183C2×D8—21no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—41no—
187C2×D8D4×C12712yes—
188C22×C8—21no—
189#127—21no—
190#127—21no—
191#127—41no—
192#121—22no—
193#121—42no—
194C4×Q8—21no—
195C4×D4—41no—
196C4×D4D4×C12711yescharacteristic
197SD32—42no—
198D16—44no—
199C2×C16—21no—
200C2×C16—21no—
201#107D4×C12711yescharacteristic
202C4⋊C8—21no—
203C4⋊C8D4×C12711yescharacteristic
204#103—21no—
205#102—22no—
206#102—41no—
207C22⋊C8—21no—
208C4×C8C23×C12711yescharacteristic
209#41—44no—
210#41—82no—
211C2×Q8—21no—
212C2×D4—22no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—41no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—41no—
220C2×D4D8×C12711yescharacteristic
221C22×C4—21no—
222C22×C4—42no—
223Q16—21no—
224Q16D8×C12712yescomplemented
225SD16—44no—
226D8—42no—
227D8—42no—
228D8—41no—
229D8—41no—
230D8—41no—
231D8—41no—
232D8—82no—
233C2×C8—21no—
234C2×C8—41no—
235C2×C8—41no—
236C2×C8—41no—
237C2×C8C2×D4×C12711yescharacteristic
238C2×C8C2×D4×C12711yescharacteristic
239C4⋊C4—21no—
240C4⋊C4—41no—
241C4⋊C4D8×C12711yescharacteristic
242C22⋊C4—21no—
243C22⋊C4—42no—
244C42—41no—
245C42C2×D4×C12711yescharacteristic
246C16—22no—
247C16—41no—
248C23—21no—
249C23—42no—
250C23—81no—
251Q8—22no—
252Q8—41no—
253D4—21no—
254D4—21no—
255D4—42no—
256D4—42no—
257D4—84no—
258D4—81no—
259D4—81no—
260D4—81no—
261D4—81no—
262D4—81no—
263D4—81no—
264D4—161no—
265D4—161no—
266D4—162no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—84no—
273C2×C4#120×C12711yescharacteristic
274C2×C4C2×D8×C12711yescharacteristic
275C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
276C8—21no—
277C8—41no—
278C8—81no—
279C8C2×D8×C12711yescharacteristic
280C8AGL(1,ℤ/8)×C12711yescharacteristic
281C22—41no—
282C22—41no—
283C22—41no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22—161no—
288C22—161no—
289C22—161no—
290C22#447×C12711yescharacteristic
291C4—21no—
292C4—41no—
293C4—42no—
294C4—81no—
295C4#447×C12711yescharacteristic
296C4#453×C12711yescharacteristic
297C4#508×C12711yescharacteristic
298C4#509×C12711yescharacteristic
299C2—81no—
300C2—162no—
301C2—321no—
302C2C127×#161911yescharacteristic
303C2C127×#218611yescharacteristic
304C2C127×#219211yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3287C2semidirect
2. C127×#2246C2semidirect
3. C127×#2207C2semidirect
4. C127×#1667C2semidirect
5. C127×#1361C2semidirect
6. C127×#1332C2semidirect
7. C127×D4⋊C16C2semidirect
71. Q16×C127D8semidirect
146. #12317C127direct (N × H)
153. #3287C2×C127semidirect
154. #2246C2×C127semidirect
155. #2207C2×C127semidirect
156. #1667C2×C127semidirect
157. #1361C2×C127semidirect
158. #1332C2×C127semidirect
159. D4⋊C16C2×C127semidirect
160. C127#12317direct (N × H)
224. Q16D8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#32872
C127×#22462
C127×#22072
C127×#16672
C127×#13612
C127×#13322
C127×D4⋊C162
#12317127