Finite Group Databasebeta

← C127×#17230 #44580903

Subgroups — C127×#17230 #44580903

Order 32512.

1010 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#401×C127C212yesmaximal
2C127×#1445C213yesmaximal
3C127×#1444C211yesmaximal, characteristic
4C127×#1442C213yesmaximal
5C127×#1441C216yesmaximal
6C22×C4⋊C4×C127C2213yes—
7C22×C4⋊C4×C127C2213yes—
8C22×C4⋊C4×C127C2211yescharacteristic
9#401×C127—28no—
10C2×#95×C127C2216yes—
11C2×#95×C127C2216yes—
12C2×#95×C127C2213yes—
13C2×#95×C127C2216yes—
14C2×#95×C127C2213yes—
15C2×#95×C127C2213yes—
16C2×#95×C127C2211yescharacteristic
17C23×C4×C127C2316yes—
18C23×C4×C127C2312yes—
19C23×C4×C127C2313yes—
20C23×C4×C127C2313yes—
21C23×C4×C127C2311yescharacteristic
22C2×C4⋊C4×C127—212no—
23C2×C4⋊C4×C127—26no—
24C2×C4⋊C4×C127—43no—
25C2×C4⋊C4×C127—43no—
26C2×C4⋊C4×C127—43no—
27C2×C4⋊C4×C127—43no—
28#95×C127—212no—
29#95×C127—212no—
30#95×C127—26no—
31#95×C127—43no—
32#95×C127—46no—
33#95×C127—43no—
34#95×C127D414yes—
35C24×C127C2411yescharacteristic
36C22×C4×C127—26no—
37C22×C4×C127—26no—
38C22×C4×C127—23no—
39C22×C4×C127—23no—
40C22×C4×C127—23no—
41C22×C4×C127—23no—
42C22×C4×C127—26no—
43C22×C4×C127—23no—
44C22×C4×C127—23no—
45C22×C4×C127—26no—
46C22×C4×C127—22no—
47C22×C4×C127—26no—
48C22×C4×C127—26no—
49C22×C4×C127—26no—
50C22×C4×C127—26no—
51C22×C4×C127—26no—
52C22×C4×C127—26no—
53C22×C4×C127—26no—
54C22×C4×C127—26no—
55C22×C4×C127—26no—
56C22×C4×C127C2×D416yes—
57C22×C4×C127#4112yes—
58C22×C4×C127#4116yes—
59C4⋊C4×C127—412no—
60C4⋊C4×C127—412no—
61C4⋊C4×C127—44no—
62C23×C127C22×D411yescharacteristic
63C23×C127C2×#4113yes—
64C23×C127C2×#4111yescharacteristic
65C23×C127H(5,𝔽2)13yes—
66C23×C127H(5,𝔽2)11yescharacteristic
67C23×C1272−1+413yes—
68C23×C1272−1+413yes—
69C2×C4×C127—26no—
70C2×C4×C127—26no—
71C2×C4×C127—26no—
72C2×C4×C127—212no—
73C2×C4×C127—212no—
74C2×C4×C127—46no—
75C2×C4×C127—46no—
76C2×C4×C127—43no—
77C2×C4×C127—43no—
78C2×C4×C127—41no—
79C2×C4×C127—43no—
80C2×C4×C127—43no—
81C2×C4×C127—43no—
82C2×C4×C127—46no—
83C2×C4×C127—46no—
84C2×C4×C127—46no—
85C2×C4×C127—46no—
86C2×C4×C127—46no—
87C2×C4×C127—46no—
88C2×C4×C127—46no—
89C2×C4×C127—42no—
90C2×C4×C127—46no—
91C2×C4×C127—46no—
92C22×C127#53713yes—
93C22×C127#53813yes—
94C22×C127#53913yes—
95C22×C127#54116yes—
96C22×C127#54713yes—
97C22×C127#55011yescharacteristic
98C22×C127#55113yes—
99C22×C127#55613yes—
100C22×C127#56011yescharacteristic
101C22×C127#56313yes—
102C22×C127#56313yes—
103C22×C127#56313yes—
104C4×C127—42no—
105C4×C127—46no—
106C4×C127—46no—
107C4×C127—82no—
108C4×C127—86no—
109#17230C12711yesmaximal, characteristic, direct product factor, Sylow
110C2×C127#272213yes—
111C2×C127#272713yes—
112C2×C127#276011yescharacteristic
113C2×C127#278913yes—
114C2×C127#282311yescharacteristic
115C2×C127#283313yes—
116C2×C127#283411yescharacteristic
117C2×#401C2×C12712yes—
118#1445C2×C12713yes—
119#1444C2×C12711yescharacteristic
120#1442C2×C12713yes—
121#1441C2×C12716yes—
122C127#1723011yescharacteristic, direct product factor, Sylow
123C22×C4⋊C4C22×C12713yes—
124C22×C4⋊C4C22×C12713yes—
125C22×C4⋊C4C22×C12711yescharacteristic
126#401—28no—
127C2×#95C22×C12716yes—
128C2×#95C22×C12716yes—
129C2×#95C22×C12713yes—
130C2×#95C22×C12716yes—
131C2×#95C22×C12713yes—
132C2×#95C22×C12713yes—
133C2×#95C22×C12711yescharacteristic
134C23×C4C23×C12716yes—
135C23×C4C23×C12712yes—
136C23×C4C23×C12713yes—
137C23×C4C23×C12713yes—
138C23×C4C23×C12711yescharacteristic
139C2×C4⋊C4—212no—
140C2×C4⋊C4—26no—
141C2×C4⋊C4—43no—
142C2×C4⋊C4—43no—
143C2×C4⋊C4—43no—
144C2×C4⋊C4—43no—
145#95—212no—
146#95—212no—
147#95—26no—
148#95—43no—
149#95—46no—
150#95—43no—
151#95D4×C12714yes—
152C24C24×C12711yescharacteristic
153C22×C4—26no—
154C22×C4—26no—
155C22×C4—23no—
156C22×C4—23no—
157C22×C4—23no—
158C22×C4—23no—
159C22×C4—26no—
160C22×C4—23no—
161C22×C4—23no—
162C22×C4—26no—
163C22×C4—22no—
164C22×C4—26no—
165C22×C4—26no—
166C22×C4—26no—
167C22×C4—26no—
168C22×C4—26no—
169C22×C4—26no—
170C22×C4—26no—
171C22×C4—26no—
172C22×C4—26no—
173C22×C4C2×D4×C12716yes—
174C22×C4#41×C12712yes—
175C22×C4#41×C12716yes—
176C4⋊C4—412no—
177C4⋊C4—412no—
178C4⋊C4—44no—
179C23C22×D4×C12711yescharacteristic
180C23C2×#41×C12713yes—
181C23C2×#41×C12711yescharacteristic
182C23H(5,𝔽2)×C12713yes—
183C23H(5,𝔽2)×C12711yescharacteristic
184C232−1+4×C12713yes—
185C232−1+4×C12713yes—
186C2×C4—26no—
187C2×C4—26no—
188C2×C4—26no—
189C2×C4—212no—
190C2×C4—212no—
191C2×C4—46no—
192C2×C4—46no—
193C2×C4—43no—
194C2×C4—43no—
195C2×C4—41no—
196C2×C4—43no—
197C2×C4—43no—
198C2×C4—43no—
199C2×C4—46no—
200C2×C4—46no—
201C2×C4—46no—
202C2×C4—46no—
203C2×C4—46no—
204C2×C4—46no—
205C2×C4—46no—
206C2×C4—42no—
207C2×C4—46no—
208C2×C4—46no—
209C22#537×C12713yes—
210C22#538×C12713yes—
211C22#539×C12713yes—
212C22#541×C12716yes—
213C22#547×C12713yes—
214C22#550×C12711yescharacteristic
215C22#551×C12713yes—
216C22#556×C12713yes—
217C22#560×C12711yescharacteristic
218C22#563×C12713yes—
219C22#563×C12713yes—
220C22#563×C12713yes—
221C4—42no—
222C4—46no—
223C4—46no—
224C4—86no—
225C4—82no—
226C2C127×#272213yes—
227C2C127×#272713yes—
228C2C127×#276011yescharacteristic
229C2C127×#278913yes—
230C2C127×#282311yescharacteristic
231C2C127×#283313yes—
232C2C127×#283411yescharacteristic
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
109. #17230C127direct (N × H)
122. C127#17230direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#401×C1272
C127×#14452
C127×#14442
C127×#14422
C127×#14412
#17230127