Finite Group Databasebeta

← C127×#7321 #44570994

Subgroups — C127×#7321 #44570994

Order 32512.

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

normal subgroups only

This group has 328 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×#1800C211yesmaximal, characteristic
3C127×#1465C211yesmaximal, characteristic
4C22×M16×C127C411yescharacteristic
5C2×C4×D4×C127C2211yescharacteristic
6#423×C127—21no—
7C2×#100×C127C412yes—
8#407×C127—21no—
9C2×C22⋊C8×C127—21no—
10C2×C22⋊C8×C127C411yescharacteristic
11#343×C127—21no—
12#336×C127—21no—
13#324×C127—21no—
14#323×C127C412yes—
15C22×D4×C127C2×C411yescharacteristic
16C23×C4×C127C2×C411yescharacteristic
17C23×C4×C127D411yescharacteristic
18C2×M16×C127—22no—
19C2×M16×C127—42no—
20C2×M16×C127—41no—
21C2×M16×C127—41no—
22C22×C8×C127—21no—
23C22×C8×C127—21no—
24C4×D4×C127—41no—
25C4×D4×C127—41no—
26C2×C4⋊C4×C127D411yescharacteristic
27C2×C22⋊C4×C127C2×C411yescharacteristic
28C2×C22⋊C4×C127Q811yescharacteristic
29C2×C42×C127D411yescharacteristic
30C4⋊C8×C127—41no—
31C4⋊C8×C127—41no—
32#100×C127—42no—
33C22⋊C8×C127—22no—
34C22⋊C8×C127—42no—
35C22⋊C8×C127—41no—
36C22⋊C8×C127—41no—
37C24×C127C4211yescharacteristic
38C24×C127C4⋊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×C127C22⋊C411yescharacteristic
52C22×C4×C127C22⋊C411yescharacteristic
53C22×C4×C127C22⋊C411yescharacteristic
54C22×C4×C127C4⋊C411yescharacteristic
55C22×C4×C127C4⋊C411yescharacteristic
56M16×C127—42no—
57M16×C127—42no—
58M16×C127—42no—
59M16×C127—42no—
60M16×C127—81no—
61M16×C127—81no—
62C2×C8×C127—22no—
63C2×C8×C127—41no—
64C2×C8×C127—41no—
65C2×C8×C127—42no—
66C2×C8×C127—41no—
67C2×C8×C127—42no—
68C2×C8×C127—41no—
69C2×C8×C127—41no—
70C4⋊C4×C127—41no—
71C22⋊C4×C127—41no—
72C22⋊C4×C127—41no—
73C42×C127—21no—
74C42×C127—21no—
75C23×C127—21no—
76C23×C127—21no—
77C23×C127—21no—
78C23×C127—21no—
79C23×C127—21no—
80C23×C127—21no—
81C23×C127—41no—
82C23×C127—41no—
83C23×C127—41no—
84C23×C127—41no—
85C23×C127#9511yescharacteristic
86D4×C127—42no—
87D4×C127—42no—
88C2×C4×C127—21no—
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—42no—
96C2×C4×C127—41no—
97C2×C4×C127—41no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—41no—
103C2×C4×C127—81no—
104C2×C4×C127#9911yescharacteristic
105C2×C4×C127#9911yescharacteristic
106C2×C4×C127#10011yescharacteristic
107C2×C4×C127#10111yescharacteristic
108C8×C127—42no—
109C8×C127—81no—
110C8×C127—82no—
111C8×C127—82no—
112C22×C127—21no—
113C22×C127—21no—
114C22×C127—42no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—41no—
119C22×C127—41no—
120C22×C127—41no—
121C22×C127—81no—
122C22×C127—81no—
123C22×C127—81no—
124C22×C127#34211yescharacteristic
125C22×C127#34311yescharacteristic
126C22×C127#34411yescharacteristic
127C4×C127—21no—
128C4×C127—21no—
129C4×C127—21no—
130C4×C127—41no—
131C4×C127—41no—
132C4×C127—41no—
133C4×C127AΣL(1,𝔽16)11yescharacteristic
134C4×C127#35211yescharacteristic
135#7321C12711yesmaximal, characteristic, direct product factor, Sylow
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—81no—
139C2×C127—81no—
140C2×C127#130511yescharacteristic
141C2×C127#139011yescharacteristic
142C2×C127#139211yescharacteristic
143#1870C2×C12711yescharacteristic
144#1800C2×C12711yescharacteristic
145#1465C2×C12711yescharacteristic
146C127#732111yescharacteristic, direct product factor, Sylow
147C22×M16C4×C12711yescharacteristic
148C2×C4×D4C22×C12711yescharacteristic
149#423—21no—
150C2×#100C4×C12712yes—
151#407—21no—
152C2×C22⋊C8—21no—
153C2×C22⋊C8C4×C12711yescharacteristic
154#343—21no—
155#336—21no—
156#324—21no—
157#323C4×C12712yes—
158C22×D4C2×C4×C12711yescharacteristic
159C23×C4C2×C4×C12711yescharacteristic
160C23×C4D4×C12711yescharacteristic
161C2×M16—22no—
162C2×M16—42no—
163C2×M16—41no—
164C2×M16—41no—
165C22×C8—21no—
166C22×C8—21no—
167C4×D4—41no—
168C4×D4—41no—
169C2×C4⋊C4D4×C12711yescharacteristic
170C2×C22⋊C4C2×C4×C12711yescharacteristic
171C2×C22⋊C4Q8×C12711yescharacteristic
172C2×C42D4×C12711yescharacteristic
173C4⋊C8—41no—
174C4⋊C8—41no—
175#100—42no—
176C22⋊C8—22no—
177C22⋊C8—42no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C24C42×C12711yescharacteristic
181C24C4⋊C4×C12711yescharacteristic
182C2×D4—21no—
183C2×D4—21no—
184C2×D4—41no—
185C2×D4—41no—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—41no—
191C22×C4—41no—
192C22×C4—41no—
193C22×C4—41no—
194C22×C4C22⋊C4×C12711yescharacteristic
195C22×C4C22⋊C4×C12711yescharacteristic
196C22×C4C22⋊C4×C12711yescharacteristic
197C22×C4C4⋊C4×C12711yescharacteristic
198C22×C4C4⋊C4×C12711yescharacteristic
199M16—42no—
200M16—42no—
201M16—42no—
202M16—42no—
203M16—81no—
204M16—81no—
205C2×C8—22no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—41no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C4⋊C4—41no—
214C22⋊C4—41no—
215C22⋊C4—41no—
216C42—21no—
217C42—21no—
218C23—21no—
219C23—21no—
220C23—21no—
221C23—21no—
222C23—21no—
223C23—21no—
224C23—41no—
225C23—41no—
226C23—41no—
227C23—41no—
228C23#95×C12711yescharacteristic
229D4—42no—
230D4—42no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—42no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—81no—
247C2×C4#99×C12711yescharacteristic
248C2×C4#99×C12711yescharacteristic
249C2×C4#100×C12711yescharacteristic
250C2×C4#101×C12711yescharacteristic
251C8—42no—
252C8—81no—
253C8—82no—
254C8—82no—
255C22—21no—
256C22—21no—
257C22—42no—
258C22—41no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22—41no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22#342×C12711yescharacteristic
268C22#343×C12711yescharacteristic
269C22#344×C12711yescharacteristic
270C4—21no—
271C4—21no—
272C4—21no—
273C4—41no—
274C4—41no—
275C4—41no—
276C4AΣL(1,𝔽16)×C12711yescharacteristic
277C4#352×C12711yescharacteristic
278C2—21no—
279C2—21no—
280C2—81no—
281C2—81no—
282C2C127×#130511yescharacteristic
283C2C127×#139011yescharacteristic
284C2C127×#139211yescharacteristic
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
135. #7321C127direct (N × H)
146. C127#7321direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#18702
C127×#18002
C127×#14652
#7321127