Finite Group Databasebeta

← C127×#15198 #44578871

Subgroups — C127×#15198 #44578871

Order 32512.

1066 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C127×#1443C216yesmaximal
2C127×#1443C212yesmaximal
3C127×C4⋊#95C213yesmaximal
4C127×C4⋊#95C213yesmaximal
5C4×#95×C127C211yesmaximal, characteristic
6C22×C4⋊C4×C127C2213yes—
7C22×C4⋊C4×C127C2211yescharacteristic
8C22×C42×C127C2213yes—
9C2×#95×C127C2213yes—
10C2×#95×C127C2216yes—
11C2×#95×C127C2216yes—
12C2×#95×C127C2216yes—
13C2×#95×C127C2216yes—
14C2×#95×C127C2211yescharacteristic
15C23×C4×C127C2316yes—
16C23×C4×C127C2312yes—
17C23×C4×C127C2313yes—
18C23×C4×C127C2313yes—
19C23×C4×C127C2311yescharacteristic
20C2×C4⋊C4×C127—26no—
21C2×C4⋊C4×C127—26no—
22C2×C4⋊C4×C127—26no—
23C2×C4⋊C4×C127—26no—
24C2×C42×C127—26no—
25C2×C42×C127—26no—
26C2×C42×C127—26no—
27#95×C127—26no—
28#95×C127—212no—
29#95×C127—46no—
30#95×C127—46no—
31#95×C127—46no—
32#95×C127Q814yes—
33C24×C127C2411yescharacteristic
34C22×C4×C127—23no—
35C22×C4×C127—23no—
36C22×C4×C127—23no—
37C22×C4×C127—23no—
38C22×C4×C127—23no—
39C22×C4×C127—23no—
40C22×C4×C127—23no—
41C22×C4×C127—23no—
42C22×C4×C127—23no—
43C22×C4×C127—23no—
44C22×C4×C127—23no—
45C22×C4×C127—23no—
46C22×C4×C127—26no—
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—22no—
56C22×C4×C127C2×Q816yes—
57C22×C4×C127#4116yes—
58C22×C4×C127#4116yes—
59C22×C4×C127#4116yes—
60C22×C4×C127#4112yes—
61C4⋊C4×C127—44no—
62C4⋊C4×C127—412no—
63C42×C127—412no—
64C23×C127C22×Q811yescharacteristic
65C23×C127C2×#4113yes—
66C23×C127C2×#4113yes—
67C23×C127C2×#4111yescharacteristic
68C23×C127C2×#4113yes—
69C23×C127H(5,𝔽2)13yes—
70C23×C1272−1+411yescharacteristic
71C2×C4×C127—26no—
72C2×C4×C127—26no—
73C2×C4×C127—26no—
74C2×C4×C127—26no—
75C2×C4×C127—26no—
76C2×C4×C127—26no—
77C2×C4×C127—43no—
78C2×C4×C127—43no—
79C2×C4×C127—43no—
80C2×C4×C127—43no—
81C2×C4×C127—43no—
82C2×C4×C127—43no—
83C2×C4×C127—43no—
84C2×C4×C127—43no—
85C2×C4×C127—42no—
86C2×C4×C127—46no—
87C2×C4×C127—46no—
88C2×C4×C127—46no—
89C2×C4×C127—46no—
90C2×C4×C127—46no—
91C2×C4×C127—46no—
92C2×C4×C127—46no—
93C2×C4×C127—46no—
94C2×C4×C127—46no—
95C2×C4×C127#126112yes—
96C2×C4×C127#12614yes—
97C2×C4×C127#126112yes—
98C22×C127C2×#12611yescharacteristic
99C22×C127C2×#12613yes—
100C22×C127C2×#12613yes—
101C22×C127#52913yes—
102C22×C127#53313yes—
103C22×C127#54013yes—
104C22×C127#54213yes—
105C22×C127#55213yes—
106C22×C127#55313yes—
107C22×C127#55413yes—
108C22×C127#55513yes—
109C22×C127#55711yescharacteristic
110C22×C127#55913yes—
111C4×C127—46no—
112C4×C127—46no—
113C4×C127—82no—
114C4×C127—86no—
115C4×C127#40118yes—
116#15198C12711yesmaximal, characteristic, direct product factor, Sylow
117C2×C127C2×#40111yescharacteristic
118C2×C127#249513yes—
119C2×C127#249713yes—
120C2×C127#251113yes—
121C2×C127#251313yes—
122C2×C127#264411yescharacteristic
123C2×C127#286311yescharacteristic
124#1443C2×C12716yes—
125#1443C2×C12712yes—
126C4⋊#95C2×C12713yes—
127C4⋊#95C2×C12713yes—
128C4×#95C2×C12711yescharacteristic
129C127#1519811yescharacteristic, direct product factor, Sylow
130C22×C4⋊C4C22×C12713yes—
131C22×C4⋊C4C22×C12711yescharacteristic
132C22×C42C22×C12713yes—
133C2×#95C22×C12713yes—
134C2×#95C22×C12716yes—
135C2×#95C22×C12716yes—
136C2×#95C22×C12716yes—
137C2×#95C22×C12716yes—
138C2×#95C22×C12711yescharacteristic
139C23×C4C23×C12716yes—
140C23×C4C23×C12712yes—
141C23×C4C23×C12713yes—
142C23×C4C23×C12713yes—
143C23×C4C23×C12711yescharacteristic
144C2×C4⋊C4—26no—
145C2×C4⋊C4—26no—
146C2×C4⋊C4—26no—
147C2×C4⋊C4—26no—
148C2×C42—26no—
149C2×C42—26no—
150C2×C42—26no—
151#95—26no—
152#95—212no—
153#95—46no—
154#95—46no—
155#95—46no—
156#95Q8×C12714yes—
157C24C24×C12711yescharacteristic
158C22×C4—23no—
159C22×C4—23no—
160C22×C4—23no—
161C22×C4—23no—
162C22×C4—23no—
163C22×C4—23no—
164C22×C4—23no—
165C22×C4—23no—
166C22×C4—23no—
167C22×C4—23no—
168C22×C4—23no—
169C22×C4—23no—
170C22×C4—26no—
171C22×C4—26no—
172C22×C4—26no—
173C22×C4—26no—
174C22×C4—26no—
175C22×C4—26no—
176C22×C4—26no—
177C22×C4—26no—
178C22×C4—26no—
179C22×C4—22no—
180C22×C4C2×Q8×C12716yes—
181C22×C4#41×C12716yes—
182C22×C4#41×C12716yes—
183C22×C4#41×C12716yes—
184C22×C4#41×C12712yes—
185C4⋊C4—44no—
186C4⋊C4—412no—
187C42—412no—
188C23C22×Q8×C12711yescharacteristic
189C23C2×#41×C12713yes—
190C23C2×#41×C12713yes—
191C23C2×#41×C12711yescharacteristic
192C23C2×#41×C12713yes—
193C23H(5,𝔽2)×C12713yes—
194C232−1+4×C12711yescharacteristic
195C2×C4—26no—
196C2×C4—26no—
197C2×C4—26no—
198C2×C4—26no—
199C2×C4—26no—
200C2×C4—26no—
201C2×C4—43no—
202C2×C4—43no—
203C2×C4—43no—
204C2×C4—43no—
205C2×C4—43no—
206C2×C4—43no—
207C2×C4—43no—
208C2×C4—43no—
209C2×C4—42no—
210C2×C4—46no—
211C2×C4—46no—
212C2×C4—46no—
213C2×C4—46no—
214C2×C4—46no—
215C2×C4—46no—
216C2×C4—46no—
217C2×C4—46no—
218C2×C4—46no—
219C2×C4#126×C127112yes—
220C2×C4#126×C12714yes—
221C2×C4#126×C127112yes—
222C22C2×#126×C12711yescharacteristic
223C22C2×#126×C12713yes—
224C22C2×#126×C12713yes—
225C22#529×C12713yes—
226C22#533×C12713yes—
227C22#540×C12713yes—
228C22#542×C12713yes—
229C22#552×C12713yes—
230C22#553×C12713yes—
231C22#554×C12713yes—
232C22#555×C12713yes—
233C22#557×C12711yescharacteristic
234C22#559×C12713yes—
235C4—46no—
236C4—46no—
237C4—82no—
238C4—86no—
239C4#401×C12718yes—
240C2C2×#401×C12711yescharacteristic
241C2C127×#249513yes—
242C2C127×#249713yes—
243C2C127×#251113yes—
244C2C127×#251313yes—
245C2C127×#264411yescharacteristic
246C2C127×#286311yescharacteristic
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
116. #15198C127direct (N × H)
129. C127#15198direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#14432
C127×C4⋊#952
C4×#95×C1272
#15198127