Finite Group Databasebeta

← C127×#12171 #44575844

Subgroups — C127×#12171 #44575844

Order 32512.

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

normal subgroups only

This group has 332 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×#3261C211yesmaximal, characteristic, complemented
2C127×#2191C212yesmaximal
3C127×#2053C211yesmaximal, characteristic
4C127×#1340C212yesmaximal, complemented
5C127×#1328C211yesmaximal, characteristic, complemented
6#575×C127C2211yescharacteristic
7#541×C127—21no—
8#510×C127—22no—
9#479×C127—22no—
10#470×C127C2212yes—
11#467×C127—21no—
12#462×C127—22no—
13#424×C127C2211yescharacteristic
14#416×C127—21no—
15#410×C127—21no—
16#360×C127—22no—
17#349×C127C2212yes—
18H(3,ℤ/4)×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×C127C2311yescharacteristic
26C22×C8×C127D411yescharacteristic
27C4⋊Q8×C127—21no—
28#126×C127—42no—
29#125×C127—21no—
30#125×C127—42no—
31C22⋊Q8×C127—22no—
32C4×Q8×C127—22no—
33#117×C127—21no—
34#117×C127D411yescharacteristic
35Q32×C127—42no—
36SD32×C127—42no—
37M32×C127—22no—
38#107×C127—21no—
39#106×C127D412yes—
40C4⋊C8×C127—22no—
41#103×C127—22no—
42#103×C127—22no—
43#103×C127—41no—
44#102×C127—41no—
45#99×C127—42no—
46#41×C127—41no—
47#41×C127—41no—
48#41×C127—41no—
49#41×C127—41no—
50C2×Q8×C127—21no—
51C2×Q8×C127—22no—
52C2×D4×C127—21no—
53C2×D4×C127—41no—
54C22×C4×C127—41no—
55C22×C4×C127C2×D411yescharacteristic
56Q16×C127—22no—
57Q16×C127—42no—
58Q16×C127—81no—
59SD16×C127—42no—
60D8×C127—22no—
61M16×C127—41no—
62C2×C8×C127—21no—
63C2×C8×C127—41no—
64C2×C8×C127C2×D412yes—
65C4⋊C4×C127—21no—
66C4⋊C4×C127—22no—
67C4⋊C4×C127—42no—
68C4⋊C4×C127—42no—
69C4⋊C4×C127—42no—
70C4⋊C4×C127—41no—
71C4⋊C4×C127—42no—
72C22⋊C4×C127—42no—
73C22⋊C4×C127—41no—
74C22⋊C4×C127D812yes—
75C42×C127—21no—
76C42×C127—42no—
77C42×C127—41no—
78C16×C127—42no—
79C23×C127—41no—
80C23×C127C2×D811yescharacteristic
81Q8×C127—41no—
82Q8×C127—42no—
83Q8×C127—41no—
84Q8×C127—42no—
85D4×C127—41no—
86D4×C127—41no—
87D4×C127—81no—
88D4×C127—81no—
89C2×C4×C127—21no—
90C2×C4×C127—21no—
91C2×C4×C127—22no—
92C2×C4×C127—41no—
93C2×C4×C127—42no—
94C2×C4×C127—42no—
95C2×C4×C127—81no—
96C2×C4×C127—81no—
97C2×C4×C127—81no—
98C2×C4×C127#12011yescharacteristic
99C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
100C8×C127—22no—
101C8×C127—41no—
102C22×C127—21no—
103C22×C127—41no—
104C22×C127—81no—
105C22×C127—161no—
106C22×C127#44711yescharacteristic
107C4×C127—21no—
108C4×C127—42no—
109C4×C127—82no—
110C4×C127—81no—
111C4×C127—82no—
112C4×C127—161no—
113C4×C127#45411yescharacteristic
114C4×C127B(4,𝔽2)11yescharacteristic
115#12171C12711yesmaximal, characteristic, direct product factor, Sylow
116C2×C127—21no—
117C2×C127—41no—
118C2×C127—161no—
119C2×C127#159811yescharacteristic
120#3261C2×C12711yescharacteristic, complemented
121#2191C2×C12712yes—
122#2053C2×C12711yescharacteristic
123#1340C2×C12712yescomplemented
124#1328C2×C12711yescharacteristic, complemented
125C127#1217111yescharacteristic, direct product factor, Sylow
126#575C22×C12711yescharacteristic
127#541—21no—
128#510—22no—
129#479—22no—
130#470C22×C12712yes—
131#467—21no—
132#462—22no—
133#424C22×C12711yescharacteristic
134#416—21no—
135#410—21no—
136#360—22no—
137#349C22×C12712yes—
138H(3,ℤ/4)C22×C12711yescharacteristic
139C2×#41—21no—
140#137—41no—
141AGL(1,ℤ/8)—41no—
142#135—22no—
143C2×Q16—21no—
144C2×Q16D4×C12712yes—
145C2×M16C23×C12711yescharacteristic
146C22×C8D4×C12711yescharacteristic
147C4⋊Q8—21no—
148#126—42no—
149#125—21no—
150#125—42no—
151C22⋊Q8—22no—
152C4×Q8—22no—
153#117—21no—
154#117D4×C12711yescharacteristic
155Q32—42no—
156SD32—42no—
157M32—22no—
158#107—21no—
159#106D4×C12712yes—
160C4⋊C8—22no—
161#103—22no—
162#103—22no—
163#103—41no—
164#102—41no—
165#99—42no—
166#41—41no—
167#41—41no—
168#41—41no—
169#41—41no—
170C2×Q8—21no—
171C2×Q8—22no—
172C2×D4—21no—
173C2×D4—41no—
174C22×C4—41no—
175C22×C4C2×D4×C12711yescharacteristic
176Q16—22no—
177Q16—42no—
178Q16—81no—
179SD16—42no—
180D8—22no—
181M16—41no—
182C2×C8—21no—
183C2×C8—41no—
184C2×C8C2×D4×C12712yes—
185C4⋊C4—21no—
186C4⋊C4—22no—
187C4⋊C4—42no—
188C4⋊C4—42no—
189C4⋊C4—42no—
190C4⋊C4—41no—
191C4⋊C4—42no—
192C22⋊C4—42no—
193C22⋊C4—41no—
194C22⋊C4D8×C12712yes—
195C42—21no—
196C42—42no—
197C42—41no—
198C16—42no—
199C23—41no—
200C23C2×D8×C12711yescharacteristic
201Q8—41no—
202Q8—42no—
203Q8—41no—
204Q8—42no—
205D4—41no—
206D4—41no—
207D4—81no—
208D4—81no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—81no—
216C2×C4—81no—
217C2×C4—81no—
218C2×C4#120×C12711yescharacteristic
219C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
220C8—22no—
221C8—41no—
222C22—21no—
223C22—41no—
224C22—81no—
225C22—161no—
226C22#447×C12711yescharacteristic
227C4—21no—
228C4—42no—
229C4—82no—
230C4—81no—
231C4—82no—
232C4—161no—
233C4#454×C12711yescharacteristic
234C4B(4,𝔽2)×C12711yescharacteristic
235C2—21no—
236C2—41no—
237C2—161no—
238C2C127×#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×#3261C2semidirect
4. C127×#1340C2semidirect
5. C127×#1328C2semidirect
115. #12171C127direct (N × H)
120. #3261C2×C127semidirect
123. #1340C2×C127semidirect
124. #1328C2×C127semidirect
125. C127#12171direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32612
C127×#21912
C127×#20532
C127×#13402
C127×#13282
#12171127