Finite Group Databasebeta

← C127×#12165 #44575838

Subgroups — C127×#12165 #44575838

Order 32512.

350 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 74 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×#2187C212yesmaximal
3C127×#2045C211yesmaximal, characteristic
4C127×#1350C212yesmaximal, complemented
5C127×#1323C211yesmaximal, characteristic, complemented
6C22×D8×C127C2211yescharacteristic
7#541×C127—21no—
8C2×SD32×C127—22no—
9#479×C127—22no—
10#470×C127—21no—
11#467×C127C2212yes—
12#462×C127—22no—
13#424×C127C2211yescharacteristic
14#418×C127—21no—
15C2×#100×C127—21no—
16#357×C127—22no—
17C22⋊C16×C127C2212yes—
18#341×C127C2211yescharacteristic
19C22×D4×C127—21no—
20C2×Q16×C127—21no—
21C2×Q16×C127D412yes—
22C2×D8×C127—22no—
23C2×D8×C127—42no—
24C2×M16×C127—21no—
25C2×M16×C127D411yescharacteristic
26C22×C8×C127C2311yescharacteristic
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×C127D411yescharacteristic
34SD32×C127—44no—
35C2×C16×C127—22no—
36#107×C127D412yes—
37#106×C127—21no—
38C4⋊C8×C127—22no—
39#103×C127—22no—
40#103×C127—22no—
41#102×C127—42no—
42#100×C127—42no—
43C24×C127—41no—
44C2×Q8×C127—22no—
45C2×D4×C127—22no—
46C2×D4×C127—41no—
47C2×D4×C127—41no—
48C2×D4×C127—41no—
49C2×D4×C127—41no—
50C2×D4×C127—41no—
51C22×C4×C127C2×D411yescharacteristic
52Q16×C127—42no—
53Q16×C127—81no—
54D8×C127—24no—
55D8×C127—42no—
56M16×C127—41no—
57M16×C127—41no—
58M16×C127—41no—
59C2×C8×C127—21no—
60C2×C8×C127—41no—
61C2×C8×C127—41no—
62C2×C8×C127C2×D412yes—
63C4⋊C4×C127—22no—
64C4⋊C4×C127—42no—
65C4⋊C4×C127—42no—
66C4⋊C4×C127—42no—
67C4⋊C4×C127—42no—
68C4⋊C4×C127D812yes—
69C22⋊C4×C127—21no—
70C22⋊C4×C127—42no—
71C42×C127—21no—
72C42×C127—42no—
73C16×C127—42no—
74C23×C127—42no—
75C23×C127—81no—
76C23×C127—81no—
77C23×C127—162no—
78C23×C127AGL(1,ℤ/8)11yescharacteristic
79Q8×C127—42no—
80Q8×C127—42no—
81D4×C127—42no—
82D4×C127—42no—
83D4×C127—81no—
84D4×C127—81no—
85C2×C4×C127—21no—
86C2×C4×C127—21no—
87C2×C4×C127—22no—
88C2×C4×C127—42no—
89C2×C4×C127—42no—
90C2×C4×C127#12011yescharacteristic
91C2×C4×C127C2×D811yescharacteristic
92C8×C127—22no—
93C8×C127—41no—
94C8×C127—81no—
95C22×C127—21no—
96C22×C127—22no—
97C22×C127—82no—
98C22×C127—162no—
99C22×C127—162no—
100C22×C127—162no—
101C22×C127#44711yescharacteristic
102C4×C127—21no—
103C4×C127—42no—
104C4×C127—82no—
105C4×C127—82no—
106C4×C127#45411yescharacteristic
107C4×C127B(4,𝔽2)11yescharacteristic
108#12165C12711yesmaximal, characteristic, direct product factor, Sylow
109C2×C127—41no—
110C2×C127—162no—
111C2×C127#159811yescharacteristic
112C2×C127#219312yes—
113#3261C2×C12711yescharacteristic, complemented
114#2187C2×C12712yes—
115#2045C2×C12711yescharacteristic
116#1350C2×C12712yescomplemented
117#1323C2×C12711yescharacteristic, complemented
118C127#1216511yescharacteristic, direct product factor, Sylow
119C22×D8C22×C12711yescharacteristic
120#541—21no—
121C2×SD32—22no—
122#479—22no—
123#470—21no—
124#467C22×C12712yes—
125#462—22no—
126#424C22×C12711yescharacteristic
127#418—21no—
128C2×#100—21no—
129#357—22no—
130C22⋊C16C22×C12712yes—
131#341C22×C12711yescharacteristic
132C22×D4—21no—
133C2×Q16—21no—
134C2×Q16D4×C12712yes—
135C2×D8—22no—
136C2×D8—42no—
137C2×M16—21no—
138C2×M16D4×C12711yescharacteristic
139C22×C8C23×C12711yescharacteristic
140C4⋊Q8—21no—
141#126—42no—
142#125—21no—
143#125—42no—
144C22⋊Q8—22no—
145C4×Q8—22no—
146#117D4×C12711yescharacteristic
147SD32—44no—
148C2×C16—22no—
149#107D4×C12712yes—
150#106—21no—
151C4⋊C8—22no—
152#103—22no—
153#103—22no—
154#102—42no—
155#100—42no—
156C24—41no—
157C2×Q8—22no—
158C2×D4—22no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164C22×C4C2×D4×C12711yescharacteristic
165Q16—42no—
166Q16—81no—
167D8—24no—
168D8—42no—
169M16—41no—
170M16—41no—
171M16—41no—
172C2×C8—21no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8C2×D4×C12712yes—
176C4⋊C4—22no—
177C4⋊C4—42no—
178C4⋊C4—42no—
179C4⋊C4—42no—
180C4⋊C4—42no—
181C4⋊C4D8×C12712yes—
182C22⋊C4—21no—
183C22⋊C4—42no—
184C42—21no—
185C42—42no—
186C16—42no—
187C23—42no—
188C23—81no—
189C23—81no—
190C23—162no—
191C23AGL(1,ℤ/8)×C12711yescharacteristic
192Q8—42no—
193Q8—42no—
194D4—42no—
195D4—42no—
196D4—81no—
197D4—81no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—42no—
202C2×C4—42no—
203C2×C4#120×C12711yescharacteristic
204C2×C4C2×D8×C12711yescharacteristic
205C8—22no—
206C8—41no—
207C8—81no—
208C22—21no—
209C22—22no—
210C22—82no—
211C22—162no—
212C22—162no—
213C22—162no—
214C22#447×C12711yescharacteristic
215C4—21no—
216C4—42no—
217C4—82no—
218C4—82no—
219C4#454×C12711yescharacteristic
220C4B(4,𝔽2)×C12711yescharacteristic
221C2—41no—
222C2—162no—
223C2C127×#159811yescharacteristic
224C2C127×#219312yes—
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC22×D8×C127C22
LCSG ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ #424×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#3261C2semidirect
4. C127×#1350C2semidirect
5. C127×#1323C2semidirect
108. #12165C127direct (N × H)
113. #3261C2×C127semidirect
116. #1350C2×C127semidirect
117. #1323C2×C127semidirect
118. C127#12165direct (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×#21872
C127×#20452
C127×#13502
C127×#13232
#12165127