Finite Group Databasebeta

← C127×#11654 #44575327

Subgroups — C127×#11654 #44575327

Order 32512.

354 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 122 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×#3169C211yesmaximal, characteristic
2C127×#2163C211yesmaximal, characteristic, complemented
3C2×#334×C127C211yesmaximal, characteristic, complemented
4C127×#1333C214yesmaximal, complemented
5#532×C127—21no—
6#500×C127C411yescharacteristic
7#493×C127C411yescharacteristic
8#486×C127—21no—
9#466×C127—21no—
10#466×C127C412yes—
11C4×D8×C127—21no—
12C4×D8×C127C2212yes—
13C2×C4⋊C8×C127—21no—
14C2×C4×C8×C127C2211yescharacteristic
15C4⋊C16×C127—21no—
16C22⋊C16×C127—21no—
17#346×C127C2212yes—
18#334×C127—21no—
19#334×C127C2212yes—
20C2×#41×C127—41no—
21C2×D8×C127—21no—
22C2×D8×C127C2×C412yes—
23C22×C8×C127—41no—
24C22×C8×C127C2×C411yescharacteristic
25C22×C8×C127D411yescharacteristic
26C4⋊Q8×C127—21no—
27#127×C127—21no—
28#124×C127—41no—
29#121×C127—22no—
30#121×C127—41no—
31C4×D4×C127—22no—
32C4×D4×C127—41no—
33C2×C42×C127D411yescharacteristic
34C2×C16×C127—22no—
35#107×C127—21no—
36#107×C127C2×C412yes—
37C4⋊C8×C127—22no—
38C4⋊C8×C127—41no—
39#102×C127—22no—
40#102×C127—22no—
41C4×C8×C127C2×C411yescharacteristic
42C4×C8×C127D411yescharacteristic
43C4×C8×C127D411yescharacteristic
44C4×C8×C127C2311yescharacteristic
45#41×C127—84no—
46C2×Q8×C127—41no—
47C2×D4×C127—22no—
48C2×D4×C127—42no—
49C2×D4×C127—41no—
50C2×D4×C127—41no—
51C22×C4×C127—42no—
52C22×C4×C127C22⋊C411yescharacteristic
53C22×C4×C127D811yescharacteristic
54C22×C4×C127SD1611yescharacteristic
55D8×C127—24no—
56D8×C127—42no—
57C2×C8×C127—22no—
58C2×C8×C127—41no—
59C2×C8×C127—42no—
60C2×C8×C127—82no—
61C2×C8×C127C22⋊C411yescharacteristic
62C2×C8×C127C22⋊C411yescharacteristic
63C2×C8×C127C22×C411yescharacteristic
64C2×C8×C127C2×D411yescharacteristic
65C4⋊C4×C127—22no—
66C4⋊C4×C127—41no—
67C22⋊C4×C127—42no—
68C22⋊C4×C127—42no—
69C42×C127C22⋊C411yescharacteristic
70C42×C127D811yescharacteristic
71C42×C127SD1611yescharacteristic
72C42×C127C2×D411yescharacteristic
73C16×C127—42no—
74C23×C127—42no—
75C23×C127#10211yescharacteristic
76Q8×C127—82no—
77D4×C127—42no—
78D4×C127—42no—
79D4×C127—82no—
80D4×C127—84no—
81D4×C127—82no—
82C2×C4×C127—22no—
83C2×C4×C127—22no—
84C2×C4×C127—21no—
85C2×C4×C127—21no—
86C2×C4×C127—42no—
87C2×C4×C127—84no—
88C2×C4×C127#10211yescharacteristic
89C2×C4×C127#10211yescharacteristic
90C2×C4×C127#10211yescharacteristic
91C2×C4×C127C2×C22⋊C411yescharacteristic
92C2×C4×C127C2×D811yescharacteristic
93C2×C4×C127C2×SD1611yescharacteristic
94C8×C127—21no—
95C8×C127—21no—
96C8×C127—21no—
97C8×C127—82no—
98C8×C127#13112yes—
99C22×C127—21no—
100C22×C127—21no—
101C22×C127—82no—
102C22×C127—82no—
103C22×C127—82no—
104C22×C127#36112yes—
105C22×C127C2×#10211yescharacteristic
106C4×C127—21no—
107C4×C127—21no—
108C4×C127—21no—
109C4×C127—21no—
110C4×C127—82no—
111C4×C127#36012yes—
112C4×C127#40811yescharacteristic
113C4×C127#42111yescharacteristic
114#11654C12711yesmaximal, characteristic, direct product factor, Sylow
115C2×C127—22no—
116C2×C127—162no—
117C2×C127#148011yescharacteristic
118C2×C127C2×#36011yescharacteristic
119C2×C127C2×#36111yescharacteristic
120#3169C2×C12711yescharacteristic
121#2163C2×C12711yescharacteristic, complemented
122C2×#334C2×C12711yescharacteristic, complemented
123#1333C2×C12714yescomplemented
124C127#1165411yescharacteristic, direct product factor, Sylow
125#532—21no—
126#500C4×C12711yescharacteristic
127#493C4×C12711yescharacteristic
128#486—21no—
129#466—21no—
130#466C4×C12712yes—
131C4×D8—21no—
132C4×D8C22×C12712yes—
133C2×C4⋊C8—21no—
134C2×C4×C8C22×C12711yescharacteristic
135C4⋊C16—21no—
136C22⋊C16—21no—
137#346C22×C12712yes—
138#334—21no—
139#334C22×C12712yes—
140C2×#41—41no—
141C2×D8—21no—
142C2×D8C2×C4×C12712yes—
143C22×C8—41no—
144C22×C8C2×C4×C12711yescharacteristic
145C22×C8D4×C12711yescharacteristic
146C4⋊Q8—21no—
147#127—21no—
148#124—41no—
149#121—22no—
150#121—41no—
151C4×D4—22no—
152C4×D4—41no—
153C2×C42D4×C12711yescharacteristic
154C2×C16—22no—
155#107—21no—
156#107C2×C4×C12712yes—
157C4⋊C8—22no—
158C4⋊C8—41no—
159#102—22no—
160#102—22no—
161C4×C8C2×C4×C12711yescharacteristic
162C4×C8D4×C12711yescharacteristic
163C4×C8D4×C12711yescharacteristic
164C4×C8C23×C12711yescharacteristic
165#41—84no—
166C2×Q8—41no—
167C2×D4—22no—
168C2×D4—42no—
169C2×D4—41no—
170C2×D4—41no—
171C22×C4—42no—
172C22×C4C22⋊C4×C12711yescharacteristic
173C22×C4D8×C12711yescharacteristic
174C22×C4SD16×C12711yescharacteristic
175D8—24no—
176D8—42no—
177C2×C8—22no—
178C2×C8—41no—
179C2×C8—42no—
180C2×C8—82no—
181C2×C8C22⋊C4×C12711yescharacteristic
182C2×C8C22⋊C4×C12711yescharacteristic
183C2×C8C22×C4×C12711yescharacteristic
184C2×C8C2×D4×C12711yescharacteristic
185C4⋊C4—22no—
186C4⋊C4—41no—
187C22⋊C4—42no—
188C22⋊C4—42no—
189C42C22⋊C4×C12711yescharacteristic
190C42D8×C12711yescharacteristic
191C42SD16×C12711yescharacteristic
192C42C2×D4×C12711yescharacteristic
193C16—42no—
194C23—42no—
195C23#102×C12711yescharacteristic
196Q8—82no—
197D4—42no—
198D4—42no—
199D4—82no—
200D4—84no—
201D4—82no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—21no—
206C2×C4—42no—
207C2×C4—84no—
208C2×C4#102×C12711yescharacteristic
209C2×C4#102×C12711yescharacteristic
210C2×C4#102×C12711yescharacteristic
211C2×C4C2×C22⋊C4×C12711yescharacteristic
212C2×C4C2×D8×C12711yescharacteristic
213C2×C4C2×SD16×C12711yescharacteristic
214C8—21no—
215C8—21no—
216C8—21no—
217C8—82no—
218C8#131×C12712yes—
219C22—21no—
220C22—21no—
221C22—82no—
222C22—82no—
223C22—82no—
224C22#361×C12712yes—
225C22C2×#102×C12711yescharacteristic
226C4—21no—
227C4—21no—
228C4—21no—
229C4—21no—
230C4—82no—
231C4#360×C12712yes—
232C4#408×C12711yescharacteristic
233C4#421×C12711yescharacteristic
234C2—22no—
235C2—162no—
236C2C127×#148011yescharacteristic
237C2C2×#360×C12711yescharacteristic
238C2C2×#361×C12711yescharacteristic
subgroupG/H
center, socleC22×C127C2×#102
commutatorC2×C8C22×C4×C127
Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3169C2
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2163C2semidirect
3. C2×#334×C127C2semidirect
4. C127×#1333C2semidirect
114. #11654C127direct (N × H)
121. #2163C2×C127semidirect
122. C2×#334C2×C127semidirect
123. #1333C2×C127semidirect
124. C127#11654direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#31692
C127×#21632
C2×#334×C1272
C127×#13332
#11654127