Finite Group Databasebeta

← C127×#7344 #44571017

Subgroups — C127×#7344 #44571017

Order 32512.

376 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 50 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×#1792C211yesmaximal, characteristic
2C127×#1318C211yesmaximal, characteristic, complemented
3C127×#1270C211yesmaximal, characteristic, complemented
4C2×H(5,𝔽2)×C127C411yescharacteristic
5C2×#104×C127—21no—
6#417×C127—21no—
7C2×#100×C127C411yescharacteristic
8#390×C127C2211yescharacteristic
9#336×C127—21no—
10#324×C127—21no—
11H(5,𝔽2)×C127—21no—
12H(5,𝔽2)×C127—41no—
13H(5,𝔽2)×C127—42no—
14H(5,𝔽2)×C127C812yescomplemented
15C2×#41×C127—21no—
16C2×#41×C127—41no—
17C22×D4×C127—41no—
18C22×D4×C127—41no—
19C22×D4×C127C2×C411yescharacteristic
20C2×M16×C127—21no—
21C22×C8×C127—41no—
22C2×C4⋊C4×C127D411yescharacteristic
23C2×C22⋊C4×C127—41no—
24C2×C42×C127D411yescharacteristic
25#104×C127—42no—
26#103×C127—41no—
27#102×C127—41no—
28#100×C127—41no—
29C22⋊C8×C127—41no—
30C24×C127—21no—
31C24×C127—41no—
32#41×C127—22no—
33#41×C127—22no—
34#41×C127—41no—
35#41×C127—42no—
36#41×C127—44no—
37#41×C127—41no—
38#41×C127—42no—
39C2×Q8×C127—21no—
40C2×D4×C127—21no—
41C2×D4×C127—21no—
42C2×D4×C127—41no—
43C2×D4×C127—41no—
44C2×D4×C127—42no—
45C2×D4×C127—42no—
46C2×D4×C127—42no—
47C2×D4×C127—42no—
48C2×D4×C127—44no—
49C2×D4×C127—42no—
50C2×D4×C127—41no—
51C2×D4×C127—42no—
52C2×D4×C127—42no—
53C2×D4×C127C2×C811yescharacteristic
54C2×D4×C127M1611yescharacteristic
55C22×C4×C127—21no—
56C22×C4×C127—21no—
57C22×C4×C127—41no—
58C22×C4×C127—41no—
59C22×C4×C127C22⋊C411yescharacteristic
60M16×C127—41no—
61M16×C127—41no—
62C2×C8×C127—41no—
63C2×C8×C127—42no—
64C2×C8×C127—81no—
65C2×C8×C127—81no—
66C2×C8×C127—81no—
67C4⋊C4×C127—41no—
68C22⋊C4×C127—81no—
69C22⋊C4×C127—81no—
70C42×C127—41no—
71C23×C127—21no—
72C23×C127—21no—
73C23×C127—42no—
74C23×C127—41no—
75C23×C127—41no—
76C23×C127—42no—
77C23×C127—41no—
78C23×C127—81no—
79C23×C127—81no—
80C23×C127—161no—
81C23×C127—161no—
82C23×C127#9911yescharacteristic
83Q8×C127—22no—
84Q8×C127—41no—
85D4×C127—22no—
86D4×C127—41no—
87D4×C127—41no—
88D4×C127—41no—
89D4×C127—42no—
90D4×C127—44no—
91D4×C127—44no—
92D4×C127—42no—
93D4×C127—42no—
94C2×C4×C127—21no—
95C2×C4×C127—21no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—42no—
99C2×C4×C127—42no—
100C2×C4×C127—41no—
101C2×C4×C127—42no—
102C2×C4×C127—41no—
103C2×C4×C127—41no—
104C2×C4×C127—42no—
105C2×C4×C127—41no—
106C2×C4×C127—81no—
107C2×C4×C127C22⋊C811yescharacteristic
108C2×C4×C127#10011yescharacteristic
109C8×C127—81no—
110C8×C127—82no—
111C8×C127—161no—
112C22×C127—21no—
113C22×C127—21no—
114C22×C127—42no—
115C22×C127—41no—
116C22×C127—41no—
117C22×C127—81no—
118C22×C127—81no—
119C22×C127—81no—
120C22×C127—81no—
121C22×C127—81no—
122C22×C127—81no—
123C22×C127—161no—
124C22×C127—162no—
125C22×C127—162no—
126C22×C127#32311yescharacteristic
127C22×C127AΣL(1,𝔽16)11yescharacteristic
128C22×C127AΣL(1,𝔽16)11yescharacteristic
129C4×C127—21no—
130C4×C127—21no—
131C4×C127—41no—
132C4×C127—41no—
133C4×C127—42no—
134C4×C127—81no—
135#7344C12711yesmaximal, characteristic, direct product factor, Sylow
136C2×C127—21no—
137C2×C127—21no—
138C2×C127—81no—
139C2×C127—81no—
140C2×C127—82no—
141C2×C127#131611yescharacteristic
142C2×C127#140211yescharacteristic
143C2×C127#140311yescharacteristic
144#1792C2×C12711yescharacteristic
145#1318C2×C12711yescharacteristic, complemented
146#1270C2×C12711yescharacteristic, complemented
147C127#734411yescharacteristic, direct product factor, Sylow
148C2×H(5,𝔽2)C4×C12711yescharacteristic
149C2×#104—21no—
150#417—21no—
151C2×#100C4×C12711yescharacteristic
152#390C22×C12711yescharacteristic
153#336—21no—
154#324—21no—
155H(5,𝔽2)—21no—
156H(5,𝔽2)—41no—
157H(5,𝔽2)—42no—
158H(5,𝔽2)C8×C12712yescomplemented
159C2×#41—21no—
160C2×#41—41no—
161C22×D4—41no—
162C22×D4—41no—
163C22×D4C2×C4×C12711yescharacteristic
164C2×M16—21no—
165C22×C8—41no—
166C2×C4⋊C4D4×C12711yescharacteristic
167C2×C22⋊C4—41no—
168C2×C42D4×C12711yescharacteristic
169#104—42no—
170#103—41no—
171#102—41no—
172#100—41no—
173C22⋊C8—41no—
174C24—21no—
175C24—41no—
176#41—22no—
177#41—22no—
178#41—41no—
179#41—42no—
180#41—44no—
181#41—41no—
182#41—42no—
183C2×Q8—21no—
184C2×D4—21no—
185C2×D4—21no—
186C2×D4—41no—
187C2×D4—41no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—42no—
192C2×D4—44no—
193C2×D4—42no—
194C2×D4—41no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4C2×C8×C12711yescharacteristic
198C2×D4M16×C12711yescharacteristic
199C22×C4—21no—
200C22×C4—21no—
201C22×C4—41no—
202C22×C4—41no—
203C22×C4C22⋊C4×C12711yescharacteristic
204M16—41no—
205M16—41no—
206C2×C8—41no—
207C2×C8—42no—
208C2×C8—81no—
209C2×C8—81no—
210C2×C8—81no—
211C4⋊C4—41no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—41no—
215C23—21no—
216C23—21no—
217C23—42no—
218C23—41no—
219C23—41no—
220C23—42no—
221C23—41no—
222C23—81no—
223C23—81no—
224C23—161no—
225C23—161no—
226C23#99×C12711yescharacteristic
227Q8—22no—
228Q8—41no—
229D4—22no—
230D4—41no—
231D4—41no—
232D4—41no—
233D4—42no—
234D4—44no—
235D4—44no—
236D4—42no—
237D4—42no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—81no—
251C2×C4C22⋊C8×C12711yescharacteristic
252C2×C4#100×C12711yescharacteristic
253C8—81no—
254C8—82no—
255C8—161no—
256C22—21no—
257C22—21no—
258C22—42no—
259C22—41no—
260C22—41no—
261C22—81no—
262C22—81no—
263C22—81no—
264C22—81no—
265C22—81no—
266C22—81no—
267C22—161no—
268C22—162no—
269C22—162no—
270C22#323×C12711yescharacteristic
271C22AΣL(1,𝔽16)×C12711yescharacteristic
272C22AΣL(1,𝔽16)×C12711yescharacteristic
273C4—21no—
274C4—21no—
275C4—41no—
276C4—41no—
277C4—42no—
278C4—81no—
279C2—21no—
280C2—21no—
281C2—81no—
282C2—81no—
283C2—82no—
284C2C127×#131611yescharacteristic
285C2C127×#140211yescharacteristic
286C2C127×#140311yescharacteristic
subgroupG/H
center, socleC22×C127AΣL(1,𝔽16)
commutatorC2×D4C2×C8×C127
Frattini, cosocle#390C22×C127
prime radicalC2×H(5,𝔽2)×C127C4
LCSG ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C22×C4×C127 ◁ #390×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#1318C2semidirect
3. C127×#1270C2semidirect
14. H(5,𝔽2)×C127C8semidirect
135. #7344C127direct (N × H)
145. #1318C2×C127semidirect
146. #1270C2×C127semidirect
147. C127#7344direct (N × H)
158. H(5,𝔽2)C8×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#17922
C127×#13182
C127×#12702
#7344127