Finite Group Databasebeta

← C127×#7320 #44570993

Subgroups — C127×#7320 #44570993

Order 32512.

342 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 78 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×#1870C211yesmaximal, characteristic
2C127×#1799C211yesmaximal, characteristic
3C127×#1466C211yesmaximal, characteristic
4C22×M16×C127C411yescharacteristic, complemented
5C2×C4×D4×C127C2211yescharacteristic
6#432×C127—21no—
7C2×#99×C127C412yes—
8C2×C22⋊C8×C127—21no—
9C2×C22⋊C8×C127C411yescharacteristic, complemented
10#380×C127—21no—
11#342×C127—21no—
12#336×C127—21no—
13#323×C127—21no—
14#323×C127C412yescomplemented
15C22×D4×C127C2×C411yescharacteristic
16C23×C4×C127C2×C411yescharacteristic
17C23×C4×C127D411yescharacteristic
18C2×M16×C127—42no—
19C2×M16×C127—41no—
20C22×C8×C127—21no—
21C22×C8×C127—21no—
22C4×D4×C127—41no—
23C4×D4×C127—41no—
24C2×C4⋊C4×C127—41no—
25C2×C4⋊C4×C127Q811yescharacteristic
26C2×C22⋊C4×C127—22no—
27C2×C22⋊C4×C127C2×C411yescharacteristic
28C2×C22⋊C4×C127D411yescharacteristic
29C2×C42×C127D411yescharacteristic
30C4⋊C8×C127—41no—
31#99×C127—42no—
32C22⋊C8×C127—22no—
33C22⋊C8×C127—42no—
34C22⋊C8×C127—41no—
35#97×C127—41no—
36#95×C127—41no—
37C24×C127C4211yescharacteristic
38C24×C127C22⋊C411yescharacteristic
39C2×D4×C127—21no—
40C2×D4×C127—21no—
41C2×D4×C127—41no—
42C2×D4×C127—41no—
43C22×C4×C127—21no—
44C22×C4×C127—21no—
45C22×C4×C127—21no—
46C22×C4×C127—21no—
47C22×C4×C127—41no—
48C22×C4×C127—41no—
49C22×C4×C127—41no—
50C22×C4×C127—41no—
51C22×C4×C127—42no—
52C22×C4×C127C22⋊C411yescharacteristic
53C22×C4×C127C22⋊C411yescharacteristic
54C22×C4×C127C4⋊C411yescharacteristic
55C22×C4×C127C4⋊C411yescharacteristic
56C22×C4×C127C4⋊C411yescharacteristic
57M16×C127—42no—
58M16×C127—42no—
59C2×C8×C127—22no—
60C2×C8×C127—41no—
61C2×C8×C127—41no—
62C2×C8×C127—42no—
63C2×C8×C127—41no—
64C2×C8×C127—41no—
65C2×C8×C127—41no—
66C4⋊C4×C127—41no—
67C4⋊C4×C127—81no—
68C4⋊C4×C127—81no—
69C22⋊C4×C127—41no—
70C22⋊C4×C127—41no—
71C22⋊C4×C127—42no—
72C22⋊C4×C127—42no—
73C42×C127—21no—
74C42×C127—21no—
75C23×C127—21no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—21no—
79C23×C127—21no—
80C23×C127—41no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127#9511yescharacteristic
85C23×C127#10011yescharacteristic
86C23×C127#10111yescharacteristic
87D4×C127—42no—
88D4×C127—42no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—21no—
92C2×C4×C127—21no—
93C2×C4×C127—21no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—42no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—41no—
105C2×C4×C127—81no—
106C2×C4×C127—82no—
107C2×C4×C127—82no—
108C2×C4×C127—82no—
109C2×C4×C127#9911yescharacteristic
110C2×C4×C127#9911yescharacteristic
111C8×C127—42no—
112C8×C127—81no—
113C8×C127—82no—
114C22×C127—21no—
115C22×C127—21no—
116C22×C127—21no—
117C22×C127—21no—
118C22×C127—41no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127—41no—
122C22×C127—41no—
123C22×C127—41no—
124C22×C127—41no—
125C22×C127—81no—
126C22×C127—81no—
127C22×C127—81no—
128C22×C127#34211yescharacteristic
129C22×C127#34311yescharacteristic
130C22×C127#34411yescharacteristic
131C4×C127—21no—
132C4×C127—41no—
133C4×C127—41no—
134C4×C127—41no—
135C4×C127—41no—
136C4×C127—162no—
137C4×C127AΣL(1,𝔽16)11yescharacteristic
138C4×C127#35211yescharacteristic
139#7320C12711yesmaximal, characteristic, direct product factor, Sylow
140C2×C127—21no—
141C2×C127—21no—
142C2×C127—41no—
143C2×C127—41no—
144C2×C127—81no—
145C2×C127#130511yescharacteristic
146C2×C127#139011yescharacteristic
147C2×C127#139111yescharacteristic
148#1870C2×C12711yescharacteristic
149#1799C2×C12711yescharacteristic
150#1466C2×C12711yescharacteristic
151C127#732011yescharacteristic, direct product factor, Sylow
152C22×M16C4×C12711yescharacteristic, complemented
153C2×C4×D4C22×C12711yescharacteristic
154#432—21no—
155C2×#99C4×C12712yes—
156C2×C22⋊C8—21no—
157C2×C22⋊C8C4×C12711yescharacteristic, complemented
158#380—21no—
159#342—21no—
160#336—21no—
161#323—21no—
162#323C4×C12712yescomplemented
163C22×D4C2×C4×C12711yescharacteristic
164C23×C4C2×C4×C12711yescharacteristic
165C23×C4D4×C12711yescharacteristic
166C2×M16—42no—
167C2×M16—41no—
168C22×C8—21no—
169C22×C8—21no—
170C4×D4—41no—
171C4×D4—41no—
172C2×C4⋊C4—41no—
173C2×C4⋊C4Q8×C12711yescharacteristic
174C2×C22⋊C4—22no—
175C2×C22⋊C4C2×C4×C12711yescharacteristic
176C2×C22⋊C4D4×C12711yescharacteristic
177C2×C42D4×C12711yescharacteristic
178C4⋊C8—41no—
179#99—42no—
180C22⋊C8—22no—
181C22⋊C8—42no—
182C22⋊C8—41no—
183#97—41no—
184#95—41no—
185C24C42×C12711yescharacteristic
186C24C22⋊C4×C12711yescharacteristic
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—41no—
190C2×D4—41no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—41no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—42no—
200C22×C4C22⋊C4×C12711yescharacteristic
201C22×C4C22⋊C4×C12711yescharacteristic
202C22×C4C4⋊C4×C12711yescharacteristic
203C22×C4C4⋊C4×C12711yescharacteristic
204C22×C4C4⋊C4×C12711yescharacteristic
205M16—42no—
206M16—42no—
207C2×C8—22no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C4⋊C4—41no—
215C4⋊C4—81no—
216C4⋊C4—81no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—21no—
222C42—21no—
223C23—21no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—41no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23#95×C12711yescharacteristic
233C23#100×C12711yescharacteristic
234C23#101×C12711yescharacteristic
235D4—42no—
236D4—42no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—81no—
254C2×C4—82no—
255C2×C4—82no—
256C2×C4—82no—
257C2×C4#99×C12711yescharacteristic
258C2×C4#99×C12711yescharacteristic
259C8—42no—
260C8—81no—
261C8—82no—
262C22—21no—
263C22—21no—
264C22—21no—
265C22—21no—
266C22—41no—
267C22—41no—
268C22—41no—
269C22—41no—
270C22—41no—
271C22—41no—
272C22—41no—
273C22—81no—
274C22—81no—
275C22—81no—
276C22#342×C12711yescharacteristic
277C22#343×C12711yescharacteristic
278C22#344×C12711yescharacteristic
279C4—21no—
280C4—41no—
281C4—41no—
282C4—41no—
283C4—41no—
284C4—162no—
285C4AΣL(1,𝔽16)×C12711yescharacteristic
286C4#352×C12711yescharacteristic
287C2—21no—
288C2—21no—
289C2—41no—
290C2—41no—
291C2—81no—
292C2C127×#130511yescharacteristic
293C2C127×#139011yescharacteristic
294C2C127×#139111yescharacteristic
subgroupG/H
center, socleC22×C127#342
commutatorC24C42×C127
Frattini, cosocleC2×C4×D4C22×C127
prime radicalC22×D4×C127C2×C4
LCSG ▷ C24 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C4×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C22×M16×C127C4semidirect
9. C2×C22⋊C8×C127C4semidirect
14. #323×C127C4semidirect
139. #7320C127direct (N × H)
151. C127#7320direct (N × H)
152. C22×M16C4×C127semidirect
157. C2×C22⋊C8C4×C127semidirect
162. #323C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#18702
C127×#17992
C127×#14662
#7320127