Finite Group Databasebeta

← C127×#13524 #44577197

Subgroups — C127×#13524 #44577197

Order 32512.

288 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 102 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×#2161C211yesmaximal, characteristic
2C127×#2155C211yesmaximal, characteristic
3C2×#362×C127C211yesmaximal, characteristic
4C127×#2141C211yesmaximal, characteristic
5C127×#1979C211yesmaximal, characteristic
6C127×#1833C211yesmaximal, characteristic
7C127×#1378C211yesmaximal, characteristic
8C22×Q16×C127C2211yescharacteristic
9C2×M32×C127C2211yescharacteristic
10C2×M32×C127C2211yescharacteristic
11C2×#108×C127C2211yescharacteristic
12C2×#108×C127C2211yescharacteristic
13#427×C127C2211yescharacteristic
14#419×C127—21no—
15C2×#101×C127—21no—
16#405×C127C2211yescharacteristic
17#368×C127—22no—
18#365×C127—22no—
19#362×C127—22no—
20#358×C127—22no—
21#344×C127—21no—
22H(3,ℤ/4)×C127—21no—
23C22×Q8×C127—21no—
24C2×Q16×C127—21no—
25C2×Q16×C127—21no—
26C2×Q16×C127—21no—
27C2×Q16×C127—21no—
28C2×Q16×C127—21no—
29C2×Q16×C127Q812yes—
30C2×M16×C127—21no—
31C2×M16×C127—21no—
32C2×M16×C127D411yescharacteristic
33C22×C8×C127C2311yescharacteristic
34#117×C127—21no—
35#117×C127D411yescharacteristic
36M32×C127—21no—
37M32×C127—21no—
38M32×C127—21no—
39M32×C127D412yes—
40C2×C16×C127—21no—
41C2×C16×C127D412yes—
42#108×C127—21no—
43#108×C127—21no—
44#108×C127—21no—
45#108×C127—21no—
46#107×C127—21no—
47#106×C127—21no—
48#103×C127—42no—
49#101×C127—42no—
50#97×C127—21no—
51C4×C8×C127—21no—
52C2×Q8×C127—22no—
53C2×Q8×C127—41no—
54C2×Q8×C127—41no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×Q8×C127—41no—
58C22×C4×C127—41no—
59C22×C4×C127C2×D411yescharacteristic
60Q16×C127—22no—
61Q16×C127—22no—
62Q16×C127—41no—
63Q16×C127—41no—
64M16×C127—21no—
65M16×C127—21no—
66M16×C127—41no—
67M16×C127—41no—
68M16×C127—41no—
69M16×C127—41no—
70C2×C8×C127—21no—
71C2×C8×C127—41no—
72C2×C8×C127—41no—
73C2×C8×C127C2×D411yescharacteristic
74C2×C8×C127C2×D411yescharacteristic
75C2×C8×C127C2×Q811yescharacteristic
76C2×C8×C127#4111yescharacteristic
77C2×C8×C127#4111yescharacteristic
78C2×C8×C127#4111yescharacteristic
79C4⋊C4×C127—21no—
80C4⋊C4×C127—41no—
81C22⋊C4×C127—21no—
82C22⋊C4×C127—41no—
83C42×C127—21no—
84C42×C127—41no—
85C16×C127—22no—
86C16×C127—22no—
87C23×C127#13511yescharacteristic
88Q8×C127—42no—
89Q8×C127—42no—
90Q8×C127—81no—
91Q8×C127—81no—
92C2×C4×C127—42no—
93C2×C4×C127—41no—
94C2×C4×C127—81no—
95C2×C4×C127—81no—
96C2×C4×C127—81no—
97C2×C4×C127C22⋊Q811yescharacteristic
98C2×C4×C127#12311yescharacteristic
99C2×C4×C127#12711yescharacteristic
100C2×C4×C127#13511yescharacteristic
101C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
102C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
103C8×C127—41no—
104C8×C127—81no—
105C8×C127—81no—
106C8×C127C22⋊Q811yescharacteristic
107C8×C127C22⋊Q811yescharacteristic
108C8×C127#12311yescharacteristic
109C8×C127#12311yescharacteristic
110C22×C127—41no—
111C22×C127#47811yescharacteristic
112C22×C127#48211yescharacteristic
113C22×C127#49511yescharacteristic
114C4×C127—81no—
115C4×C127—82no—
116C4×C127—161no—
117C4×C127#39911yescharacteristic
118C4×C127#47811yescharacteristic
119C4×C127#48211yescharacteristic
120C4×C127#49611yescharacteristic
121#13524C12711yesmaximal, characteristic, direct product factor, Sylow
122C2×C127—21no—
123C2×C127—21no—
124C2×C127—21no—
125C2×C127#209211yescharacteristic
126#2161C2×C12711yescharacteristic
127#2155C2×C12711yescharacteristic
128C2×#362C2×C12711yescharacteristic
129#2141C2×C12711yescharacteristic
130#1979C2×C12711yescharacteristic
131#1833C2×C12711yescharacteristic
132#1378C2×C12711yescharacteristic
133C127#1352411yescharacteristic, direct product factor, Sylow
134C22×Q16C22×C12711yescharacteristic
135C2×M32C22×C12711yescharacteristic
136C2×M32C22×C12711yescharacteristic
137C2×#108C22×C12711yescharacteristic
138C2×#108C22×C12711yescharacteristic
139#427C22×C12711yescharacteristic
140#419—21no—
141C2×#101—21no—
142#405C22×C12711yescharacteristic
143#368—22no—
144#365—22no—
145#362—22no—
146#358—22no—
147#344—21no—
148H(3,ℤ/4)—21no—
149C22×Q8—21no—
150C2×Q16—21no—
151C2×Q16—21no—
152C2×Q16—21no—
153C2×Q16—21no—
154C2×Q16—21no—
155C2×Q16Q8×C12712yes—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16D4×C12711yescharacteristic
159C22×C8C23×C12711yescharacteristic
160#117—21no—
161#117D4×C12711yescharacteristic
162M32—21no—
163M32—21no—
164M32—21no—
165M32D4×C12712yes—
166C2×C16—21no—
167C2×C16D4×C12712yes—
168#108—21no—
169#108—21no—
170#108—21no—
171#108—21no—
172#107—21no—
173#106—21no—
174#103—42no—
175#101—42no—
176#97—21no—
177C4×C8—21no—
178C2×Q8—22no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—41no—
182C2×Q8—41no—
183C2×Q8—41no—
184C22×C4—41no—
185C22×C4C2×D4×C12711yescharacteristic
186Q16—22no—
187Q16—22no—
188Q16—41no—
189Q16—41no—
190M16—21no—
191M16—21no—
192M16—41no—
193M16—41no—
194M16—41no—
195M16—41no—
196C2×C8—21no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8C2×D4×C12711yescharacteristic
200C2×C8C2×D4×C12711yescharacteristic
201C2×C8C2×Q8×C12711yescharacteristic
202C2×C8#41×C12711yescharacteristic
203C2×C8#41×C12711yescharacteristic
204C2×C8#41×C12711yescharacteristic
205C4⋊C4—21no—
206C4⋊C4—41no—
207C22⋊C4—21no—
208C22⋊C4—41no—
209C42—21no—
210C42—41no—
211C16—22no—
212C16—22no—
213C23#135×C12711yescharacteristic
214Q8—42no—
215Q8—42no—
216Q8—81no—
217Q8—81no—
218C2×C4—42no—
219C2×C4—41no—
220C2×C4—81no—
221C2×C4—81no—
222C2×C4—81no—
223C2×C4C22⋊Q8×C12711yescharacteristic
224C2×C4#123×C12711yescharacteristic
225C2×C4#127×C12711yescharacteristic
226C2×C4#135×C12711yescharacteristic
227C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
228C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
229C8—41no—
230C8—81no—
231C8—81no—
232C8C22⋊Q8×C12711yescharacteristic
233C8C22⋊Q8×C12711yescharacteristic
234C8#123×C12711yescharacteristic
235C8#123×C12711yescharacteristic
236C22—41no—
237C22#478×C12711yescharacteristic
238C22#482×C12711yescharacteristic
239C22#495×C12711yescharacteristic
240C4—81no—
241C4—82no—
242C4—161no—
243C4#399×C12711yescharacteristic
244C4#478×C12711yescharacteristic
245C4#482×C12711yescharacteristic
246C4#496×C12711yescharacteristic
247C2—21no—
248C2—21no—
249C2—21no—
250C2C127×#209211yescharacteristic
subgroupG/H
center, socleC2×C127#2092
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC23×C127#135
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C4×C127 ◁ #405×C127 ◁ G

Complements

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

NH ≅ G/Ntype
121. #13524C127direct (N × H)
133. C127#13524direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#21612
C127×#21552
C2×#362×C1272
C127×#21412
C127×#19792
C127×#18332
C127×#13782
#13524127