Finite Group Databasebeta

← C127×#17114 #44580787

Subgroups — C127×#17114 #44580787

Order 32512.

1222 conjugacy classes of subgroups.

The subgroup lattice has 238 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#398×C127C216yesmaximal
2C127×#1444C211yesmaximal, characteristic
3C127×#1444C213yesmaximal
4C127×#1443C212yesmaximal
5C127×#1442C213yesmaximal
6C22×C4⋊C4×C127C2213yes—
7C22×C4⋊C4×C127C2216yes—
8C22×C4⋊C4×C127C2213yes—
9C22×C4⋊C4×C127C2211yescharacteristic
10#398×C127—224no—
11C2×#95×C127C2216yes—
12C2×#95×C127C2216yes—
13C2×#95×C127C2213yes—
14C2×#95×C127C2213yes—
15C2×#95×C127C2213yes—
16C2×#95×C127C2211yescharacteristic
17C23×C4×C127C2312yes—
18C23×C4×C127C2313yes—
19C23×C4×C127C2316yes—
20C23×C4×C127C2313yes—
21C23×C4×C127C2311yescharacteristic
22C2×C4⋊C4×C127—212no—
23C2×C4⋊C4×C127—212no—
24C2×C4⋊C4×C127—212no—
25C2×C4⋊C4×C127—212no—
26C2×C4⋊C4×C127—43no—
27C2×C4⋊C4×C127—43no—
28C2×C4⋊C4×C127—46no—
29C2×C4⋊C4×C127—43no—
30#95×C127—212no—
31#95×C127—212no—
32#95×C127—26no—
33#95×C127—43no—
34#95×C127D4112yes—
35#95×C127Q814yes—
36C24×C127C2411yescharacteristic
37C22×C4×C127—26no—
38C22×C4×C127—26no—
39C22×C4×C127—26no—
40C22×C4×C127—26no—
41C22×C4×C127—23no—
42C22×C4×C127—23no—
43C22×C4×C127—23no—
44C22×C4×C127—26no—
45C22×C4×C127—26no—
46C22×C4×C127—26no—
47C22×C4×C127—26no—
48C22×C4×C127—23no—
49C22×C4×C127—23no—
50C22×C4×C127—23no—
51C22×C4×C127—26no—
52C22×C4×C127—26no—
53C22×C4×C127—26no—
54C22×C4×C127—26no—
55C22×C4×C127—22no—
56C22×C4×C127C2×D4112yes—
57C22×C4×C127C2×D416yes—
58C22×C4×C127C2×Q816yes—
59C22×C4×C127#4112yes—
60C4⋊C4×C127—412no—
61C4⋊C4×C127—412no—
62C4⋊C4×C127—424no—
63C4⋊C4×C127—44no—
64C23×C127C22×D413yes—
65C23×C127C22×Q811yescharacteristic
66C23×C127C2×#4111yescharacteristic
67C23×C127H(5,𝔽2)13yes—
68C23×C1272−1+413yes—
69C23×C1272−1+413yes—
70C23×C1272−1+411yescharacteristic
71C2×C4×C127—212no—
72C2×C4×C127—26no—
73C2×C4×C127—26no—
74C2×C4×C127—26no—
75C2×C4×C127—212no—
76C2×C4×C127—212no—
77C2×C4×C127—212no—
78C2×C4×C127—212no—
79C2×C4×C127—46no—
80C2×C4×C127—46no—
81C2×C4×C127—46no—
82C2×C4×C127—46no—
83C2×C4×C127—43no—
84C2×C4×C127—43no—
85C2×C4×C127—46no—
86C2×C4×C127—41no—
87C2×C4×C127—43no—
88C2×C4×C127—43no—
89C2×C4×C127—43no—
90C2×C4×C127—46no—
91C2×C4×C127—46no—
92C2×C4×C127—46no—
93C2×C4×C127—42no—
94C22×C127#53616yes—
95C22×C127#53713yes—
96C22×C127#53913yes—
97C22×C127#54413yes—
98C22×C127D4213yes—
99C22×C127#54713yes—
100C22×C127D4×Q813yes—
101C22×C127#55711yescharacteristic
102C22×C127#56011yescharacteristic
103C22×C127#56313yes—
104C22×C127#56313yes—
105C22×C127#56413yes—
106C4×C127—42no—
107C4×C127—46no—
108C4×C127—46no—
109C4×C127—412no—
110C4×C127—82no—
111#17114C12711yesmaximal, characteristic, direct product factor, Sylow
112C2×C127#267511yescharacteristic
113C2×C127#267713yes—
114C2×C127#268013yes—
115C2×C127#272613yes—
116C2×C127#282213yes—
117C2×C127#283411yescharacteristic
118C2×C127#284311yescharacteristic
119C2×#398C2×C12716yes—
120#1444C2×C12711yescharacteristic
121#1444C2×C12713yes—
122#1443C2×C12712yes—
123#1442C2×C12713yes—
124C127#1711411yescharacteristic, direct product factor, Sylow
125C22×C4⋊C4C22×C12713yes—
126C22×C4⋊C4C22×C12716yes—
127C22×C4⋊C4C22×C12713yes—
128C22×C4⋊C4C22×C12711yescharacteristic
129#398—224no—
130C2×#95C22×C12716yes—
131C2×#95C22×C12716yes—
132C2×#95C22×C12713yes—
133C2×#95C22×C12713yes—
134C2×#95C22×C12713yes—
135C2×#95C22×C12711yescharacteristic
136C23×C4C23×C12712yes—
137C23×C4C23×C12713yes—
138C23×C4C23×C12716yes—
139C23×C4C23×C12713yes—
140C23×C4C23×C12711yescharacteristic
141C2×C4⋊C4—212no—
142C2×C4⋊C4—212no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4—212no—
145C2×C4⋊C4—43no—
146C2×C4⋊C4—43no—
147C2×C4⋊C4—46no—
148C2×C4⋊C4—43no—
149#95—212no—
150#95—212no—
151#95—26no—
152#95—43no—
153#95D4×C127112yes—
154#95Q8×C12714yes—
155C24C24×C12711yescharacteristic
156C22×C4—26no—
157C22×C4—26no—
158C22×C4—26no—
159C22×C4—26no—
160C22×C4—23no—
161C22×C4—23no—
162C22×C4—23no—
163C22×C4—26no—
164C22×C4—26no—
165C22×C4—26no—
166C22×C4—26no—
167C22×C4—23no—
168C22×C4—23no—
169C22×C4—23no—
170C22×C4—26no—
171C22×C4—26no—
172C22×C4—26no—
173C22×C4—26no—
174C22×C4—22no—
175C22×C4C2×D4×C127112yes—
176C22×C4C2×D4×C12716yes—
177C22×C4C2×Q8×C12716yes—
178C22×C4#41×C12712yes—
179C4⋊C4—412no—
180C4⋊C4—412no—
181C4⋊C4—424no—
182C4⋊C4—44no—
183C23C22×D4×C12713yes—
184C23C22×Q8×C12711yescharacteristic
185C23C2×#41×C12711yescharacteristic
186C23H(5,𝔽2)×C12713yes—
187C232−1+4×C12713yes—
188C232−1+4×C12713yes—
189C232−1+4×C12711yescharacteristic
190C2×C4—212no—
191C2×C4—26no—
192C2×C4—26no—
193C2×C4—26no—
194C2×C4—212no—
195C2×C4—212no—
196C2×C4—212no—
197C2×C4—212no—
198C2×C4—46no—
199C2×C4—46no—
200C2×C4—46no—
201C2×C4—46no—
202C2×C4—43no—
203C2×C4—43no—
204C2×C4—46no—
205C2×C4—41no—
206C2×C4—43no—
207C2×C4—43no—
208C2×C4—43no—
209C2×C4—46no—
210C2×C4—46no—
211C2×C4—46no—
212C2×C4—42no—
213C22#536×C12716yes—
214C22#537×C12713yes—
215C22#539×C12713yes—
216C22#544×C12713yes—
217C22D42×C12713yes—
218C22#547×C12713yes—
219C22D4×Q8×C12713yes—
220C22#557×C12711yescharacteristic
221C22#560×C12711yescharacteristic
222C22#563×C12713yes—
223C22#563×C12713yes—
224C22#564×C12713yes—
225C4—42no—
226C4—46no—
227C4—46no—
228C4—412no—
229C4—82no—
230C2C127×#267511yescharacteristic
231C2C127×#267713yes—
232C2C127×#268013yes—
233C2C127×#272613yes—
234C2C127×#282213yes—
235C2C127×#283411yescharacteristic
236C2C127×#284311yescharacteristic
subgroupG/H
center, socle, prime radicalC24×C127C24
commutator, Frattini, cosocleC24C24×C127
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24×C127 ◁ G

Complements

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

NH ≅ G/Ntype
111. #17114C127direct (N × H)
124. C127#17114direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#398×C1272
C127×#14442
C127×#14432
C127×#14422
#17114127