Finite Group Databasebeta

← C127×#12322 #44575995

Subgroups — C127×#12322 #44575995

Order 32512.

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

normal subgroups only

This group has 368 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×Q16×C127C211yesmaximal, characteristic
2C127×#2247C211yesmaximal, characteristic, complemented
3C127×#2209C211yesmaximal, characteristic, complemented
4C127×#1669C211yesmaximal, characteristic
5C127×#1362C211yesmaximal, characteristic, complemented
6C127×#1332C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic
8C22×Q16×C127—21no—
9D4×Q8×C127—21no—
10C2×Q32×C127—21no—
11C2×Q32×C127—22no—
12#494×C127C2211yescharacteristic
13#494×C127C2211yescharacteristic
14#474×C127—21no—
15#467×C127—21no—
16#462×C127—21no—
17#451×C127—22no—
18C4×Q16×C127C2211yescharacteristic
19C8×D4×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#358×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#332×C127—21no—
25C22×Q8×C127—22no—
26C2×Q16×C127—22no—
27C2×Q16×C127—21no—
28C2×Q16×C127—21no—
29C2×Q16×C127—21no—
30C2×Q16×C127—42no—
31C2×Q16×C127—42no—
32C2×Q16×C127D411yescharacteristic
33C2×Q16×C127D412yes—
34C22×C8×C127—21no—
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127—21no—
37C4⋊Q8×C127—41no—
38C22⋊Q8×C127—22no—
39C22⋊Q8×C127—42no—
40C4×Q8×C127—21no—
41C4×D4×C127—41no—
42C4×D4×C127D411yescharacteristic
43Q32×C127—44no—
44Q32×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#103×C127—22no—
52#103×C127—21no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C4×C8×C127C2311yescharacteristic
56C2×Q8×C127—21no—
57C2×Q8×C127—22no—
58C2×Q8×C127—42no—
59C2×Q8×C127—44no—
60C2×Q8×C127—41no—
61C2×Q8×C127—82no—
62C2×D4×C127D811yescharacteristic
63C22×C4×C127—21no—
64C22×C4×C127—42no—
65Q16×C127—21no—
66Q16×C127—42no—
67Q16×C127—42no—
68Q16×C127—44no—
69Q16×C127—41no—
70Q16×C127—41no—
71Q16×C127—41no—
72Q16×C127—41no—
73Q16×C127D812yes—
74C2×C8×C127—21no—
75C2×C8×C127—41no—
76C2×C8×C127—41no—
77C2×C8×C127—41no—
78C2×C8×C127C2×D411yescharacteristic
79C2×C8×C127C2×D411yescharacteristic
80C4⋊C4×C127—21no—
81C4⋊C4×C127—42no—
82C4⋊C4×C127—41no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—41no—
85C4⋊C4×C127—41no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127D811yescharacteristic
88C22⋊C4×C127—21no—
89C22⋊C4×C127—42no—
90C42×C127—41no—
91C42×C127C2×D411yescharacteristic
92C16×C127—22no—
93C16×C127—41no—
94C23×C127—21no—
95Q8×C127—22no—
96Q8×C127—41no—
97Q8×C127—42no—
98Q8×C127—42no—
99Q8×C127—84no—
100Q8×C127—81no—
101Q8×C127—81no—
102D4×C127—21no—
103D4×C127Q3212yescomplemented
104C2×C4×C127—21no—
105C2×C4×C127—41no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—42no—
110C2×C4×C127—84no—
111C2×C4×C127—81no—
112C2×C4×C127#12011yescharacteristic
113C2×C4×C127C2×D811yescharacteristic
114C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
115C8×C127—21no—
116C8×C127—41no—
117C8×C127—81no—
118C8×C127C2×D811yescharacteristic
119C8×C127AGL(1,ℤ/8)11yescharacteristic
120C22×C127—22no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127#44711yescharacteristic
124C4×C127—21no—
125C4×C127—41no—
126C4×C127—42no—
127C4×C127—82no—
128C4×C127—81no—
129C4×C127—161no—
130C4×C127#44711yescharacteristic
131C4×C127#45311yescharacteristic
132C4×C127C2×Q3211yescharacteristic
133C4×C127#51011yescharacteristic
134#12322C12711yesmaximal, characteristic, direct product factor, Sylow
135C2×C127—42no—
136C2×C127#161911yescharacteristic
137C2×C127#218811yescharacteristic
138C2×C127#219511yescharacteristic
139D4×Q16C2×C12711yescharacteristic
140#2247C2×C12711yescharacteristic, complemented
141#2209C2×C12711yescharacteristic, complemented
142#1669C2×C12711yescharacteristic
143#1362C2×C12711yescharacteristic, complemented
144#1332C2×C12711yescharacteristic, complemented
145D4⋊C16C2×C12711yescharacteristic
146C127#1232211yescharacteristic, direct product factor, Sylow
147C22×Q16—21no—
148D4×Q8—21no—
149C2×Q32—21no—
150C2×Q32—22no—
151#494C22×C12711yescharacteristic
152#494C22×C12711yescharacteristic
153#474—21no—
154#467—21no—
155#462—21no—
156#451—22no—
157C4×Q16C22×C12711yescharacteristic
158C8×D4C22×C12711yescharacteristic
159C4⋊C16C22×C12711yescharacteristic
160#358—21no—
161C4×C16C22×C12711yescharacteristic
162#335C22×C12711yescharacteristic
163#332—21no—
164C22×Q8—22no—
165C2×Q16—22no—
166C2×Q16—21no—
167C2×Q16—21no—
168C2×Q16—21no—
169C2×Q16—42no—
170C2×Q16—42no—
171C2×Q16D4×C12711yescharacteristic
172C2×Q16D4×C12712yes—
173C22×C8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—41no—
177C22⋊Q8—22no—
178C22⋊Q8—42no—
179C4×Q8—21no—
180C4×D4—41no—
181C4×D4D4×C12711yescharacteristic
182Q32—44no—
183Q32—42no—
184C2×C16—21no—
185C2×C16—21no—
186#107—41no—
187#107D4×C12711yescharacteristic
188C4⋊C8—21no—
189C4⋊C8D4×C12711yescharacteristic
190#103—22no—
191#103—21no—
192#102—41no—
193C22⋊C8—21no—
194C4×C8C23×C12711yescharacteristic
195C2×Q8—21no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—44no—
199C2×Q8—41no—
200C2×Q8—82no—
201C2×D4D8×C12711yescharacteristic
202C22×C4—21no—
203C22×C4—42no—
204Q16—21no—
205Q16—42no—
206Q16—42no—
207Q16—44no—
208Q16—41no—
209Q16—41no—
210Q16—41no—
211Q16—41no—
212Q16D8×C12712yes—
213C2×C8—21no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8C2×D4×C12711yescharacteristic
218C2×C8C2×D4×C12711yescharacteristic
219C4⋊C4—21no—
220C4⋊C4—42no—
221C4⋊C4—41no—
222C4⋊C4—41no—
223C4⋊C4—41no—
224C4⋊C4—41no—
225C4⋊C4—41no—
226C4⋊C4D8×C12711yescharacteristic
227C22⋊C4—21no—
228C22⋊C4—42no—
229C42—41no—
230C42C2×D4×C12711yescharacteristic
231C16—22no—
232C16—41no—
233C23—21no—
234Q8—22no—
235Q8—41no—
236Q8—42no—
237Q8—42no—
238Q8—84no—
239Q8—81no—
240Q8—81no—
241D4—21no—
242D4Q32×C12712yescomplemented
243C2×C4—21no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—84no—
250C2×C4—81no—
251C2×C4#120×C12711yescharacteristic
252C2×C4C2×D8×C12711yescharacteristic
253C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
254C8—21no—
255C8—41no—
256C8—81no—
257C8C2×D8×C12711yescharacteristic
258C8AGL(1,ℤ/8)×C12711yescharacteristic
259C22—22no—
260C22—41no—
261C22—41no—
262C22#447×C12711yescharacteristic
263C4—21no—
264C4—41no—
265C4—42no—
266C4—82no—
267C4—81no—
268C4—161no—
269C4#447×C12711yescharacteristic
270C4#453×C12711yescharacteristic
271C4C2×Q32×C12711yescharacteristic
272C4#510×C12711yescharacteristic
273C2—42no—
274C2C127×#161911yescharacteristic
275C2C127×#218811yescharacteristic
276C2C127×#219511yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC2×D4×C127D8
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×#2247C2semidirect
3. C127×#2209C2semidirect
5. C127×#1362C2semidirect
6. C127×#1332C2semidirect
103. D4×C127Q32semidirect
134. #12322C127direct (N × H)
140. #2247C2×C127semidirect
141. #2209C2×C127semidirect
143. #1362C2×C127semidirect
144. #1332C2×C127semidirect
146. C127#12322direct (N × H)
242. D4Q32×C127semidirect

Maximal subgroups

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

subgroupindex
D4×Q16×C1272
C127×#22472
C127×#22092
C127×#16692
C127×#13622
C127×#13322
C127×D4⋊C162
#12322127