Finite Group Databasebeta

← C127×#2200 #30635152

Subgroups — C127×#2200 #30635152

Order 16256.

260 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 56 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
1#560×C127C211yesmaximal, characteristic, complemented
2#458×C127C211yesmaximal, characteristic, complemented
3B(4,𝔽2)×C127C211yesmaximal, characteristic, complemented
4#453×C127C211yesmaximal, characteristic, complemented
5#353×C127C211yesmaximal, characteristic, complemented
6#352×C127C211yesmaximal, characteristic, complemented
7AΣL(1,𝔽16)×C127C211yesmaximal, characteristic, complemented
8H(5,𝔽2)×C127C2211yescharacteristic
9AGL(1,ℤ/8)×C127—21no—
10#127×C127C2211yescharacteristic
11#126×C127—21no—
12#123×C127—21no—
13#123×C127—21no—
14#123×C127C2211yescharacteristic
15#121×C127—21no—
16#121×C127—21no—
17#121×C127—21no—
18#120×C127—21no—
19#120×C127—21no—
20#120×C127C2211yescharacteristic
21#104×C127—21no—
22#100×C127C2211yescharacteristic
23#99×C127—21no—
24#99×C127—21no—
25#99×C127C2211yescharacteristic
26#99×C127C2211yescharacteristic
27C24×C127D411yescharacteristic
28#41×C127—21no—
29#41×C127—41no—
30C2×D4×C127—21no—
31C2×D4×C127—21no—
32C2×D4×C127—21no—
33C2×D4×C127—21no—
34C2×D4×C127—21no—
35C2×D4×C127—21no—
36C2×D4×C127—21no—
37C2×D4×C127—41no—
38C2×D4×C127—41no—
39C2×D4×C127D411yescharacteristic, complemented
40C2×D4×C127C2311yescharacteristic
41C22×C4×C127—21no—
42C22×C4×C127D411yescharacteristic, complemented
43SD16×C127—41no—
44D8×C127—41no—
45M16×C127—21no—
46C4⋊C4×C127—21no—
47C4⋊C4×C127—21no—
48C4⋊C4×C127—21no—
49C4⋊C4×C127—41no—
50C22⋊C4×C127—21no—
51C22⋊C4×C127—21no—
52C22⋊C4×C127—21no—
53C22⋊C4×C127—21no—
54C22⋊C4×C127—21no—
55C22⋊C4×C127—21no—
56C22⋊C4×C127—21no—
57C22⋊C4×C127—41no—
58C22⋊C4×C127—41no—
59C22⋊C4×C127D411yescharacteristic
60C22⋊C4×C127D411yescharacteristic, complemented
61C42×C127D411yescharacteristic, complemented
62C23×C127—21no—
63C23×C127—21no—
64C23×C127—21no—
65C23×C127—21no—
66C23×C127—41no—
67C23×C127—81no—
68C23×C127C2×D411yescharacteristic
69C23×C127C2×D411yescharacteristic
70Q8×C127—21no—
71D4×C127—21no—
72D4×C127—41no—
73D4×C127—41no—
74D4×C127—41no—
75D4×C127—41no—
76D4×C127—41no—
77D4×C127—81no—
78D4×C127—81no—
79D4×C127—81no—
80D4×C127—81no—
81D4×C127—81no—
82D4×C127—82no—
83C2×C4×C127—21no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—21no—
88C2×C4×C127—21no—
89C2×C4×C127—41no—
90C2×C4×C127—41no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—81no—
94C2×C4×C127C2×D411yescharacteristic
95C8×C127—41no—
96C22×C127—21no—
97C22×C127—21no—
98C22×C127—41no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—41no—
102C22×C127—41no—
103C22×C127—41no—
104C22×C127—42no—
105C22×C127—41no—
106C22×C127—41no—
107C22×C127—41no—
108C22×C127—81no—
109C22×C127—81no—
110C22×C127—81no—
111C22×C127#12011yescharacteristic
112C4×C127—21no—
113C4×C127—41no—
114C4×C127—41no—
115C4×C127—41no—
116C4×C127—41no—
117C4×C127—41no—
118C4×C127—81no—
119C4×C127—81no—
120C2×C127—21no—
121C2×C127—41no—
122C2×C127—41no—
123C2×C127—81no—
124C2×C127—81no—
125C2×C127—81no—
126C2×C127B(4,𝔽2)11yescharacteristic
127#2200C12711yesmaximal, characteristic, direct product factor, Sylow
128C127#220011yescharacteristic, direct product factor, Sylow
129#560C2×C12711yescharacteristic, complemented
130#458C2×C12711yescharacteristic, complemented
131B(4,𝔽2)C2×C12711yescharacteristic, complemented
132#453C2×C12711yescharacteristic, complemented
133#353C2×C12711yescharacteristic, complemented
134#352C2×C12711yescharacteristic, complemented
135AΣL(1,𝔽16)C2×C12711yescharacteristic, complemented
136H(5,𝔽2)C22×C12711yescharacteristic
137AGL(1,ℤ/8)—21no—
138#127C22×C12711yescharacteristic
139#126—21no—
140#123—21no—
141#123—21no—
142#123C22×C12711yescharacteristic
143#121—21no—
144#121—21no—
145#121—21no—
146#120—21no—
147#120—21no—
148#120C22×C12711yescharacteristic
149#104—21no—
150#100C22×C12711yescharacteristic
151#99—21no—
152#99—21no—
153#99C22×C12711yescharacteristic
154#99C22×C12711yescharacteristic
155C24D4×C12711yescharacteristic
156#41—21no—
157#41—41no—
158C2×D4—21no—
159C2×D4—21no—
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—21no—
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4D4×C12711yescharacteristic, complemented
168C2×D4C23×C12711yescharacteristic
169C22×C4—21no—
170C22×C4D4×C12711yescharacteristic, complemented
171SD16—41no—
172D8—41no—
173M16—21no—
174C4⋊C4—21no—
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—41no—
178C22⋊C4—21no—
179C22⋊C4—21no—
180C22⋊C4—21no—
181C22⋊C4—21no—
182C22⋊C4—21no—
183C22⋊C4—21no—
184C22⋊C4—21no—
185C22⋊C4—41no—
186C22⋊C4—41no—
187C22⋊C4D4×C12711yescharacteristic
188C22⋊C4D4×C12711yescharacteristic, complemented
189C42D4×C12711yescharacteristic, complemented
190C23—21no—
191C23—21no—
192C23—21no—
193C23—21no—
194C23—41no—
195C23—81no—
196C23C2×D4×C12711yescharacteristic
197C23C2×D4×C12711yescharacteristic
198Q8—21no—
199D4—21no—
200D4—41no—
201D4—41no—
202D4—41no—
203D4—41no—
204D4—41no—
205D4—81no—
206D4—81no—
207D4—81no—
208D4—81no—
209D4—81no—
210D4—82no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—81no—
222C2×C4C2×D4×C12711yescharacteristic
223C8—41no—
224C22—21no—
225C22—21no—
226C22—41no—
227C22—41no—
228C22—41no—
229C22—41no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—41no—
234C22—41no—
235C22—41no—
236C22—81no—
237C22—81no—
238C22—81no—
239C22#120×C12711yescharacteristic
240C4—21no—
241C4—41no—
242C4—41no—
243C4—41no—
244C4—41no—
245C4—41no—
246C4—81no—
247C4—81no—
248C2—21no—
249C2—41no—
250C2—41no—
251C2—81no—
252C2—81no—
253C2—81no—
254C2B(4,𝔽2)×C12711yescharacteristic
subgroupG/H
center, socleC2×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C23×C127
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. #560×C127C2semidirect
2. #458×C127C2semidirect
3. B(4,𝔽2)×C127C2semidirect
4. #453×C127C2semidirect
5. #353×C127C2semidirect
6. #352×C127C2semidirect
7. AΣL(1,𝔽16)×C127C2semidirect
39. C2×D4×C127D4semidirect
42. C22×C4×C127D4semidirect
60. C22⋊C4×C127D4semidirect
61. C42×C127D4semidirect
127. #2200C127direct (N × H)
128. C127#2200direct (N × H)
129. #560C2×C127semidirect
130. #458C2×C127semidirect
131. B(4,𝔽2)C2×C127semidirect
132. #453C2×C127semidirect
133. #353C2×C127semidirect
134. #352C2×C127semidirect
135. AΣL(1,𝔽16)C2×C127semidirect
167. C2×D4D4×C127semidirect
170. C22×C4D4×C127semidirect
188. C22⋊C4D4×C127semidirect
189. C42D4×C127semidirect

Maximal subgroups

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

subgroupindex
#560×C1272
#458×C1272
B(4,𝔽2)×C1272
#453×C1272
#353×C1272
#352×C1272
AΣL(1,𝔽16)×C1272
#2200127