Finite Group Databasebeta

← C127×#12174 #44575847

Subgroups — C127×#12174 #44575847

Order 32512.

340 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 70 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×#3255C211yesmaximal, characteristic, complemented
2C127×#2191C212yesmaximal
3C127×#2050C211yesmaximal, characteristic
4C127×#1342C212yesmaximal, complemented
5C127×#1327C211yesmaximal, characteristic, complemented
6#575×C127C2211yescharacteristic
7#533×C127—21no—
8#510×C127—22no—
9#475×C127—22no—
10#470×C127—21no—
11#470×C127C2212yes—
12#462×C127—22no—
13#423×C127C2211yescharacteristic
14C2×#104×C127—21no—
15#413×C127—21no—
16#360×C127—22no—
17#349×C127C2212yes—
18#338×C127C2211yescharacteristic
19C2×#41×C127—21no—
20#137×C127—41no—
21AGL(1,ℤ/8)×C127—41no—
22#135×C127—22no—
23C2×Q16×C127—21no—
24C2×Q16×C127D412yes—
25C2×M16×C127—21no—
26C2×M16×C127D411yescharacteristic
27C2×M16×C127C2311yescharacteristic
28C4⋊Q8×C127—21no—
29C4⋊Q8×C127—21no—
30#125×C127—41no—
31C22⋊Q8×C127—22no—
32C22⋊Q8×C127—41no—
33C4×Q8×C127—22no—
34C4×Q8×C127—41no—
35#117×C127—41no—
36C2×C42×C127D411yescharacteristic
37Q32×C127—42no—
38SD32×C127—42no—
39M32×C127—22no—
40#106×C127—21no—
41#106×C127D412yes—
42C4⋊C8×C127—22no—
43#104×C127—41no—
44#104×C127—41no—
45#103×C127—22no—
46#103×C127—22no—
47#101×C127—41no—
48#100×C127—41no—
49#41×C127—41no—
50#41×C127—41no—
51#41×C127—41no—
52#41×C127—41no—
53C2×Q8×C127—21no—
54C2×Q8×C127—22no—
55C2×Q8×C127—41no—
56C2×D4×C127—21no—
57C2×D4×C127—41no—
58C22×C4×C127—21no—
59C22×C4×C127—41no—
60C22×C4×C127C2×D411yescharacteristic
61Q16×C127—22no—
62Q16×C127—42no—
63Q16×C127—81no—
64SD16×C127—42no—
65D8×C127—22no—
66M16×C127—41no—
67M16×C127—41no—
68M16×C127—41no—
69M16×C127—41no—
70C2×C8×C127—21no—
71C2×C8×C127—41no—
72C2×C8×C127C2×D412yes—
73C4⋊C4×C127—22no—
74C4⋊C4×C127—41no—
75C4⋊C4×C127—41no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—42no—
79C22⋊C4×C127—42no—
80C42×C127—21no—
81C42×C127—42no—
82C42×C127D811yescharacteristic
83C42×C127D811yescharacteristic
84C16×C127—42no—
85C23×C127—41no—
86C23×C127AGL(1,ℤ/8)11yescharacteristic
87Q8×C127—41no—
88Q8×C127—42no—
89Q8×C127—41no—
90Q8×C127—42no—
91Q8×C127—82no—
92D4×C127—41no—
93D4×C127—41no—
94D4×C127—81no—
95D4×C127—81no—
96C2×C4×C127—21no—
97C2×C4×C127—21no—
98C2×C4×C127—21no—
99C2×C4×C127—21no—
100C2×C4×C127—42no—
101C2×C4×C127—41no—
102C2×C4×C127—42no—
103C2×C4×C127—42no—
104C2×C4×C127—81no—
105C2×C4×C127—81no—
106C2×C4×C127#12011yescharacteristic
107C2×C4×C127C2×D811yescharacteristic
108C8×C127—22no—
109C8×C127—41no—
110C8×C127—81no—
111C22×C127—21no—
112C22×C127—41no—
113C22×C127—81no—
114C22×C127—161no—
115C22×C127#44711yescharacteristic
116C4×C127—22no—
117C4×C127—21no—
118C4×C127—41no—
119C4×C127—82no—
120C4×C127—81no—
121C4×C127—82no—
122C4×C127#45411yescharacteristic
123C4×C127B(4,𝔽2)11yescharacteristic
124#12174C12711yesmaximal, characteristic, direct product factor, Sylow
125C2×C127—21no—
126C2×C127—41no—
127C2×C127—161no—
128C2×C127#159811yescharacteristic
129#3255C2×C12711yescharacteristic, complemented
130#2191C2×C12712yes—
131#2050C2×C12711yescharacteristic
132#1342C2×C12712yescomplemented
133#1327C2×C12711yescharacteristic, complemented
134C127#1217411yescharacteristic, direct product factor, Sylow
135#575C22×C12711yescharacteristic
136#533—21no—
137#510—22no—
138#475—22no—
139#470—21no—
140#470C22×C12712yes—
141#462—22no—
142#423C22×C12711yescharacteristic
143C2×#104—21no—
144#413—21no—
145#360—22no—
146#349C22×C12712yes—
147#338C22×C12711yescharacteristic
148C2×#41—21no—
149#137—41no—
150AGL(1,ℤ/8)—41no—
151#135—22no—
152C2×Q16—21no—
153C2×Q16D4×C12712yes—
154C2×M16—21no—
155C2×M16D4×C12711yescharacteristic
156C2×M16C23×C12711yescharacteristic
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159#125—41no—
160C22⋊Q8—22no—
161C22⋊Q8—41no—
162C4×Q8—22no—
163C4×Q8—41no—
164#117—41no—
165C2×C42D4×C12711yescharacteristic
166Q32—42no—
167SD32—42no—
168M32—22no—
169#106—21no—
170#106D4×C12712yes—
171C4⋊C8—22no—
172#104—41no—
173#104—41no—
174#103—22no—
175#103—22no—
176#101—41no—
177#100—41no—
178#41—41no—
179#41—41no—
180#41—41no—
181#41—41no—
182C2×Q8—21no—
183C2×Q8—22no—
184C2×Q8—41no—
185C2×D4—21no—
186C2×D4—41no—
187C22×C4—21no—
188C22×C4—41no—
189C22×C4C2×D4×C12711yescharacteristic
190Q16—22no—
191Q16—42no—
192Q16—81no—
193SD16—42no—
194D8—22no—
195M16—41no—
196M16—41no—
197M16—41no—
198M16—41no—
199C2×C8—21no—
200C2×C8—41no—
201C2×C8C2×D4×C12712yes—
202C4⋊C4—22no—
203C4⋊C4—41no—
204C4⋊C4—41no—
205C4⋊C4—42no—
206C4⋊C4—41no—
207C4⋊C4—42no—
208C22⋊C4—42no—
209C42—21no—
210C42—42no—
211C42D8×C12711yescharacteristic
212C42D8×C12711yescharacteristic
213C16—42no—
214C23—41no—
215C23AGL(1,ℤ/8)×C12711yescharacteristic
216Q8—41no—
217Q8—42no—
218Q8—41no—
219Q8—42no—
220Q8—82no—
221D4—41no—
222D4—41no—
223D4—81no—
224D4—81no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—42no—
230C2×C4—41no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—81no—
234C2×C4—81no—
235C2×C4#120×C12711yescharacteristic
236C2×C4C2×D8×C12711yescharacteristic
237C8—22no—
238C8—41no—
239C8—81no—
240C22—21no—
241C22—41no—
242C22—81no—
243C22—161no—
244C22#447×C12711yescharacteristic
245C4—22no—
246C4—21no—
247C4—41no—
248C4—82no—
249C4—81no—
250C4—82no—
251C4#454×C12711yescharacteristic
252C4B(4,𝔽2)×C12711yescharacteristic
253C2—21no—
254C2—41no—
255C2—161no—
256C2C127×#159811yescharacteristic
subgroupG/H
center, socleC2×C127#1598
commutator, Frattini, cosocleC2×M16C23×C127
prime radical#575×C127C22
LCSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C2×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3255C2semidirect
4. C127×#1342C2semidirect
5. C127×#1327C2semidirect
124. #12174C127direct (N × H)
129. #3255C2×C127semidirect
132. #1342C2×C127semidirect
133. #1327C2×C127semidirect
134. C127#12174direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32552
C127×#21912
C127×#20502
C127×#13422
C127×#13272
#12174127