Finite Group Databasebeta

← C127×#12865 #44576538

Subgroups — C127×#12865 #44576538

Order 32512.

334 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2923C211yesmaximal, characteristic
2C127×#1827C212yesmaximal
3C127×#1395C212yesmaximal
4C127×#1392C211yesmaximal, characteristic
5C127×#1391C211yesmaximal, characteristic
6C4×#41×C127C2211yescharacteristic
7#433×C127—22no—
8#432×C127—22no—
9#424×C127C411yescharacteristic, complemented
10#424×C127C411yescharacteristic, complemented
11#413×C127C2212yes—
12#410×C127C2212yes—
13#408×C127C2211yescharacteristic
14#408×C127C2211yescharacteristic
15C4×M16×C127C411yescharacteristic, complemented
16C2×#97×C127C411yescharacteristic, complemented
17#339×C127—22no—
18H(3,ℤ/4)×C127—22no—
19C2×#41×C127C2311yescharacteristic
20C2×M16×C127—22no—
21C2×M16×C127C2×C411yescharacteristic
22C2×M16×C127C2×C411yescharacteristic
23C22×C8×C127C2×C411yescharacteristic
24C22×C8×C127C2×C411yescharacteristic
25C4×Q8×C127—21no—
26C4×D4×C127—21no—
27C4×D4×C127—22no—
28#117×C127—22no—
29#117×C127C2×C411yescharacteristic
30#117×C127D412yes—
31C2×C42×C127C2×C411yescharacteristic
32C2×C42×C127D412yes—
33C4⋊C8×C127—22no—
34C4⋊C8×C127—22no—
35#101×C127—22no—
36#100×C127—22no—
37#99×C127—22no—
38#99×C127—22no—
39C22⋊C8×C127—22no—
40C22⋊C8×C127—22no—
41#97×C127—21no—
42#97×C127—21no—
43#97×C127—21no—
44C4×C8×C127—21no—
45#41×C127—41no—
46#41×C127—41no—
47C2×Q8×C127#4111yescharacteristic
48C2×D4×C127#4111yescharacteristic
49C2×D4×C127#4112yes—
50C22×C4×C127—21no—
51C22×C4×C127—22no—
52C22×C4×C127C22×C411yescharacteristic
53C22×C4×C127C2×D412yes—
54M16×C127—41no—
55M16×C127—41no—
56M16×C127—42no—
57M16×C127—42no—
58C2×C8×C127—21no—
59C2×C8×C127—21no—
60C2×C8×C127—21no—
61C2×C8×C127—21no—
62C2×C8×C127—41no—
63C2×C8×C127—41no—
64C2×C8×C127—42no—
65C4⋊C4×C127—21no—
66C4⋊C4×C127—22no—
67C4⋊C4×C127—42no—
68C22⋊C4×C127—21no—
69C22⋊C4×C127—22no—
70C22⋊C4×C127—42no—
71C42×C127—21no—
72C42×C127—22no—
73C42×C127—42no—
74C42×C127C22⋊C411yescharacteristic
75C42×C127C22⋊C412yes—
76C42×C127C22⋊C411yescharacteristic
77C23×C127#11711yescharacteristic
78C23×C127#12312yes—
79Q8×C127—41no—
80D4×C127—42no—
81D4×C127—41no—
82C2×C4×C127—22no—
83C2×C4×C127—21no—
84C2×C4×C127—22no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—22no—
88C2×C4×C127—21no—
89C2×C4×C127—81no—
90C2×C4×C127—82no—
91C2×C4×C127—82no—
92C2×C4×C127C2×C22⋊C411yescharacteristic
93C2×C4×C127#11711yescharacteristic
94C2×C4×C127#12312yes—
95C8×C127—41no—
96C8×C127—41no—
97C8×C127—41no—
98C8×C127—41no—
99C8×C127—82no—
100C22×C127—22no—
101C22×C127—21no—
102C22×C127—41no—
103C22×C127—42no—
104C22×C127#38111yescharacteristic
105C4×C127—22no—
106C4×C127—21no—
107C4×C127—42no—
108C4×C127—41no—
109C4×C127—41no—
110C4×C127—162no—
111C4×C127#41011yescharacteristic
112C4×C127#41011yescharacteristic
113#12865C12711yesmaximal, characteristic, direct product factor, Sylow
114C2×C127—21no—
115C2×C127—42no—
116C2×C127—41no—
117C2×C127#182011yescharacteristic
118#2923C2×C12711yescharacteristic
119#1827C2×C12712yes—
120#1395C2×C12712yes—
121#1392C2×C12711yescharacteristic
122#1391C2×C12711yescharacteristic
123C127#1286511yescharacteristic, direct product factor, Sylow
124C4×#41C22×C12711yescharacteristic
125#433—22no—
126#432—22no—
127#424C4×C12711yescharacteristic, complemented
128#424C4×C12711yescharacteristic, complemented
129#413C22×C12712yes—
130#410C22×C12712yes—
131#408C22×C12711yescharacteristic
132#408C22×C12711yescharacteristic
133C4×M16C4×C12711yescharacteristic, complemented
134C2×#97C4×C12711yescharacteristic, complemented
135#339—22no—
136H(3,ℤ/4)—22no—
137C2×#41C23×C12711yescharacteristic
138C2×M16—22no—
139C2×M16C2×C4×C12711yescharacteristic
140C2×M16C2×C4×C12711yescharacteristic
141C22×C8C2×C4×C12711yescharacteristic
142C22×C8C2×C4×C12711yescharacteristic
143C4×Q8—21no—
144C4×D4—21no—
145C4×D4—22no—
146#117—22no—
147#117C2×C4×C12711yescharacteristic
148#117D4×C12712yes—
149C2×C42C2×C4×C12711yescharacteristic
150C2×C42D4×C12712yes—
151C4⋊C8—22no—
152C4⋊C8—22no—
153#101—22no—
154#100—22no—
155#99—22no—
156#99—22no—
157C22⋊C8—22no—
158C22⋊C8—22no—
159#97—21no—
160#97—21no—
161#97—21no—
162C4×C8—21no—
163#41—41no—
164#41—41no—
165C2×Q8#41×C12711yescharacteristic
166C2×D4#41×C12711yescharacteristic
167C2×D4#41×C12712yes—
168C22×C4—21no—
169C22×C4—22no—
170C22×C4C22×C4×C12711yescharacteristic
171C22×C4C2×D4×C12712yes—
172M16—41no—
173M16—41no—
174M16—42no—
175M16—42no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—21no—
179C2×C8—21no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—42no—
183C4⋊C4—21no—
184C4⋊C4—22no—
185C4⋊C4—42no—
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—42no—
189C42—21no—
190C42—22no—
191C42—42no—
192C42C22⋊C4×C12711yescharacteristic
193C42C22⋊C4×C12712yes—
194C42C22⋊C4×C12711yescharacteristic
195C23#117×C12711yescharacteristic
196C23#123×C12712yes—
197Q8—41no—
198D4—42no—
199D4—41no—
200C2×C4—22no—
201C2×C4—22no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—22no—
207C2×C4—81no—
208C2×C4—82no—
209C2×C4—82no—
210C2×C4C2×C22⋊C4×C12711yescharacteristic
211C2×C4#117×C12711yescharacteristic
212C2×C4#123×C12712yes—
213C8—41no—
214C8—41no—
215C8—41no—
216C8—41no—
217C8—82no—
218C22—22no—
219C22—21no—
220C22—41no—
221C22—42no—
222C22#381×C12711yescharacteristic
223C4—22no—
224C4—21no—
225C4—41no—
226C4—41no—
227C4—42no—
228C4—162no—
229C4#410×C12711yescharacteristic
230C4#410×C12711yescharacteristic
231C2—21no—
232C2—42no—
233C2—41no—
234C2C127×#182011yescharacteristic
subgroupG/H
centerC4×C127#410
commutatorC22×C4C22×C4×C127
Frattini, cosocleC2×#41C23×C127
socleC2×C127#1820
prime radicalC2×#41×C127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C127 ◁ C42×C127 ◁ C4×#41×C127 ◁ G

Complements

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

NH ≅ G/Ntype
9. #424×C127C4semidirect
10. #424×C127C4semidirect
15. C4×M16×C127C4semidirect
16. C2×#97×C127C4semidirect
113. #12865C127direct (N × H)
123. C127#12865direct (N × H)
127. #424C4×C127semidirect
128. #424C4×C127semidirect
133. C4×M16C4×C127semidirect
134. C2×#97C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#29232
C127×#18272
C127×#13952
C127×#13922
C127×#13912
#12865127