Finite Group Databasebeta

← C127×#12320 #44575993

Subgroups — C127×#12320 #44575993

Order 32512.

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

normal subgroups only

This group has 442 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
1D4×D8×C127C211yesmaximal, characteristic
2C127×#2248C211yesmaximal, characteristic, complemented
3C127×#2208C211yesmaximal, characteristic, complemented
4C127×#1669C211yesmaximal, characteristic, complemented
5C127×#1361C211yesmaximal, characteristic, complemented
6C127×#1331C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8C22×D8×C127—21no—
9D42×C127—21no—
10C2×SD32×C127—21no—
11C2×SD32×C127—22no—
12#494×C127C2211yescharacteristic, complemented
13#493×C127C2211yescharacteristic
14#474×C127—21no—
15#466×C127—21no—
16#459×C127—21no—
17#447×C127—22no—
18C4×D8×C127C2211yescharacteristic
19C8×D4×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic, complemented
21#358×C127—21no—
22C4×C16×C127C2211yescharacteristic, complemented
23#335×C127C2211yescharacteristic, complemented
24#332×C127—21no—
25C22×D4×C127—22no—
26C22×D4×C127—41no—
27C2×Q16×C127—21no—
28C2×Q16×C127—21no—
29C2×D8×C127—22no—
30C2×D8×C127—21no—
31C2×D8×C127—42no—
32C2×D8×C127—42no—
33C2×D8×C127D411yescharacteristic
34C2×D8×C127D412yes—
35C22×C8×C127—21no—
36C4⋊Q8×C127—21no—
37#127×C127—21no—
38#121×C127—22no—
39#121×C127—41no—
40#120×C127—42no—
41C4×D4×C127—21no—
42C4×D4×C127D411yescharacteristic
43SD32×C127—44no—
44SD32×C127—42no—
45C2×C16×C127—21no—
46C2×C16×C127—21no—
47#107×C127—41no—
48#107×C127D411yescharacteristic
49C4⋊C8×C127—21no—
50C4⋊C8×C127D411yescharacteristic
51#102×C127—22no—
52#102×C127—21no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C4×C8×C127C2311yescharacteristic
56C24×C127—42no—
57C2×Q8×C127—41no—
58C2×D4×C127—21no—
59C2×D4×C127—22no—
60C2×D4×C127—42no—
61C2×D4×C127—44no—
62C2×D4×C127—42no—
63C2×D4×C127—41no—
64C2×D4×C127—41no—
65C2×D4×C127—82no—
66C2×D4×C127—82no—
67C2×D4×C127—82no—
68C2×D4×C127D811yescharacteristic
69C22×C4×C127—21no—
70C22×C4×C127—41no—
71Q16×C127—42no—
72Q16×C127—41no—
73Q16×C127—41no—
74D8×C127—21no—
75D8×C127—42no—
76D8×C127—44no—
77D8×C127—41no—
78D8×C127—41no—
79D8×C127D812yes—
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×C127—41no—
89C4⋊C4×C127D811yescharacteristic
90C22⋊C4×C127—21no—
91C22⋊C4×C127—42no—
92C22⋊C4×C127—41no—
93C42×C127C2×D411yescharacteristic
94C16×C127—22no—
95C16×C127—41no—
96C23×C127—21no—
97C23×C127—41no—
98C23×C127—42no—
99C23×C127—84no—
100C23×C127—84no—
101C23×C127—162no—
102Q8×C127—81no—
103Q8×C127—81no—
104D4×C127—22no—
105D4×C127—21no—
106D4×C127—42no—
107D4×C127—41no—
108D4×C127—42no—
109D4×C127—84no—
110D4×C127—84no—
111D4×C127—81no—
112D4×C127—81no—
113D4×C127—161no—
114D4×C127SD3212yescomplemented
115C2×C4×C127—21no—
116C2×C4×C127—41no—
117C2×C4×C127—41no—
118C2×C4×C127—41no—
119C2×C4×C127—81no—
120C2×C4×C127—81no—
121C2×C4×C127—81no—
122C2×C4×C127#12011yescharacteristic
123C2×C4×C127C2×D811yescharacteristic
124C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
125C8×C127—21no—
126C8×C127—41no—
127C8×C127—81no—
128C8×C127C2×D811yescharacteristic
129C8×C127AGL(1,ℤ/8)11yescharacteristic
130C22×C127—22no—
131C22×C127—41no—
132C22×C127—41no—
133C22×C127—42no—
134C22×C127—81no—
135C22×C127—81no—
136C22×C127—82no—
137C22×C127—82no—
138C22×C127—82no—
139C22×C127—168no—
140C22×C127#44711yescharacteristic
141C4×C127—21no—
142C4×C127—41no—
143C4×C127—81no—
144C4×C127—161no—
145C4×C127#44711yescharacteristic
146C4×C127#45311yescharacteristic
147C4×C127C2×SD3211yescharacteristic
148C4×C127#50911yescharacteristic
149#12320C12711yesmaximal, characteristic, direct product factor, Sylow
150C2×C127—42no—
151C2×C127—82no—
152C2×C127—162no—
153C2×C127#161911yescharacteristic
154C2×C127#218711yescharacteristic
155C2×C127#219411yescharacteristic
156D4×D8C2×C12711yescharacteristic
157#2248C2×C12711yescharacteristic, complemented
158#2208C2×C12711yescharacteristic, complemented
159#1669C2×C12711yescharacteristic, complemented
160#1361C2×C12711yescharacteristic, complemented
161#1331C2×C12711yescharacteristic, complemented
162D4⋊C16C2×C12711yescharacteristic, complemented
163C127#1232011yescharacteristic, direct product factor, Sylow
164C22×D8—21no—
165D42—21no—
166C2×SD32—21no—
167C2×SD32—22no—
168#494C22×C12711yescharacteristic, complemented
169#493C22×C12711yescharacteristic
170#474—21no—
171#466—21no—
172#459—21no—
173#447—22no—
174C4×D8C22×C12711yescharacteristic
175C8×D4C22×C12711yescharacteristic
176C4⋊C16C22×C12711yescharacteristic, complemented
177#358—21no—
178C4×C16C22×C12711yescharacteristic, complemented
179#335C22×C12711yescharacteristic, complemented
180#332—21no—
181C22×D4—22no—
182C22×D4—41no—
183C2×Q16—21no—
184C2×Q16—21no—
185C2×D8—22no—
186C2×D8—21no—
187C2×D8—42no—
188C2×D8—42no—
189C2×D8D4×C12711yescharacteristic
190C2×D8D4×C12712yes—
191C22×C8—21no—
192C4⋊Q8—21no—
193#127—21no—
194#121—22no—
195#121—41no—
196#120—42no—
197C4×D4—21no—
198C4×D4D4×C12711yescharacteristic
199SD32—44no—
200SD32—42no—
201C2×C16—21no—
202C2×C16—21no—
203#107—41no—
204#107D4×C12711yescharacteristic
205C4⋊C8—21no—
206C4⋊C8D4×C12711yescharacteristic
207#102—22no—
208#102—21no—
209#102—41no—
210C22⋊C8—21no—
211C4×C8C23×C12711yescharacteristic
212C24—42no—
213C2×Q8—41no—
214C2×D4—21no—
215C2×D4—22no—
216C2×D4—42no—
217C2×D4—44no—
218C2×D4—42no—
219C2×D4—41no—
220C2×D4—41no—
221C2×D4—82no—
222C2×D4—82no—
223C2×D4—82no—
224C2×D4D8×C12711yescharacteristic
225C22×C4—21no—
226C22×C4—41no—
227Q16—42no—
228Q16—41no—
229Q16—41no—
230D8—21no—
231D8—42no—
232D8—44no—
233D8—41no—
234D8—41no—
235D8D8×C12712yes—
236C2×C8—21no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8C2×D4×C12711yescharacteristic
241C2×C8C2×D4×C12711yescharacteristic
242C4⋊C4—21no—
243C4⋊C4—41no—
244C4⋊C4—41no—
245C4⋊C4D8×C12711yescharacteristic
246C22⋊C4—21no—
247C22⋊C4—42no—
248C22⋊C4—41no—
249C42C2×D4×C12711yescharacteristic
250C16—22no—
251C16—41no—
252C23—21no—
253C23—41no—
254C23—42no—
255C23—84no—
256C23—84no—
257C23—162no—
258Q8—81no—
259Q8—81no—
260D4—21no—
261D4—22no—
262D4—42no—
263D4—41no—
264D4—42no—
265D4—84no—
266D4—84no—
267D4—81no—
268D4—81no—
269D4—161no—
270D4SD32×C12712yescomplemented
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—81no—
276C2×C4—81no—
277C2×C4—81no—
278C2×C4#120×C12711yescharacteristic
279C2×C4C2×D8×C12711yescharacteristic
280C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
281C8—21no—
282C8—41no—
283C8—81no—
284C8C2×D8×C12711yescharacteristic
285C8AGL(1,ℤ/8)×C12711yescharacteristic
286C22—22no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—81no—
291C22—81no—
292C22—82no—
293C22—82no—
294C22—82no—
295C22—168no—
296C22#447×C12711yescharacteristic
297C4—21no—
298C4—41no—
299C4—81no—
300C4—161no—
301C4#447×C12711yescharacteristic
302C4#453×C12711yescharacteristic
303C4C2×SD32×C12711yescharacteristic
304C4#509×C12711yescharacteristic
305C2—42no—
306C2—82no—
307C2—162no—
308C2C127×#161911yescharacteristic
309C2C127×#218711yescharacteristic
310C2C127×#219411yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalD4×D8×C127C2
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
2. C127×#2248C2semidirect
3. C127×#2208C2semidirect
4. C127×#1669C2semidirect
5. C127×#1361C2semidirect
6. C127×#1331C2semidirect
7. C127×D4⋊C16C2semidirect
12. #494×C127C22semidirect
20. C4⋊C16×C127C22semidirect
22. C4×C16×C127C22semidirect
23. #335×C127C22semidirect
114. D4×C127SD32semidirect
149. #12320C127direct (N × H)
157. #2248C2×C127semidirect
158. #2208C2×C127semidirect
159. #1669C2×C127semidirect
160. #1361C2×C127semidirect
161. #1331C2×C127semidirect
162. D4⋊C16C2×C127semidirect
163. C127#12320direct (N × H)
168. #494C22×C127semidirect
176. C4⋊C16C22×C127semidirect
178. C4×C16C22×C127semidirect
179. #335C22×C127semidirect
270. D4SD32×C127semidirect

Maximal subgroups

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

subgroupindex
D4×D8×C1272
C127×#22482
C127×#22082
C127×#16692
C127×#13612
C127×#13312
C127×D4⋊C162
#12320127