Finite Group Databasebeta

← C127×#10306 #44573979

Subgroups — C127×#10306 #44573979

Order 32512.

670 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 170 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
1C2×#494×C127C211yesmaximal, characteristic, complemented
2C2×#486×C127C212yesmaximal
3C127×#1990C212yesmaximal, complemented
4C127×#1957C211yesmaximal, characteristic
5C127×#1273C211yesmaximal, characteristic, complemented
6C22×Q16×C127—22no—
7C2×C4⋊Q8×C127C2212yes—
8C2×#127×C127C2211yescharacteristic
9#494×C127—24no—
10#486×C127—24no—
11#486×C127—24no—
12C2×#102×C127—24no—
13C2×#102×C127—22no—
14C2×C4×C8×C127C2212yes—
15C2×C4×C8×C127C2211yescharacteristic
16#384×C127C2211yescharacteristic
17#340×C127—24no—
18C22×Q8×C127—22no—
19C22×D4×C127—21no—
20C22×D4×C127—22no—
21C2×Q16×C127—28no—
22C2×Q16×C127—44no—
23C2×Q16×C127—44no—
24C22×C8×C127D414yes—
25C22×C8×C127D412yes—
26C4⋊Q8×C127—22no—
27C4⋊Q8×C127—22no—
28C4⋊Q8×C127—22no—
29C4⋊Q8×C127D414yes—
30#127×C127—22no—
31#127×C127—21no—
32#127×C127D412yes—
33C2×C4⋊C4×C127—22no—
34C2×C4⋊C4×C127—22no—
35C2×C4⋊C4×C127—22no—
36C2×C4⋊C4×C127—21no—
37C2×C42×C127C2311yescharacteristic
38#102×C127—216no—
39#102×C127—44no—
40C4×C8×C127—22no—
41C4×C8×C127—22no—
42C4×C8×C127—22no—
43C24×C127—41no—
44C2×Q8×C127—24no—
45C2×Q8×C127—42no—
46C2×Q8×C127—44no—
47C2×Q8×C127—44no—
48C2×D4×C127—24no—
49C2×D4×C127—22no—
50C2×D4×C127—41no—
51C2×D4×C127—42no—
52C2×D4×C127—42no—
53C2×D4×C127—44no—
54C2×D4×C127—42no—
55C2×D4×C127—44no—
56C22×C4×C127—42no—
57C22×C4×C127—41no—
58C22×C4×C127C2×D411yescharacteristic
59C22×C4×C127C2×D412yes—
60Q16×C127—416no—
61C2×C8×C127—22no—
62C2×C8×C127—22no—
63C2×C8×C127—24no—
64C2×C8×C127—24no—
65C2×C8×C127—24no—
66C2×C8×C127D814yes—
67C4⋊C4×C127—24no—
68C4⋊C4×C127—24no—
69C4⋊C4×C127—42no—
70C4⋊C4×C127—42no—
71C4⋊C4×C127—42no—
72C4⋊C4×C127—44no—
73C42×C127C2×D412yes—
74C42×C127C2×D411yescharacteristic
75C42×C127#4111yescharacteristic
76C23×C127—42no—
77C23×C127—81no—
78C23×C127—82no—
79C23×C127—81no—
80C23×C127—82no—
81C23×C127#12711yescharacteristic
82Q8×C127—48no—
83Q8×C127—84no—
84D4×C127—48no—
85D4×C127—44no—
86D4×C127—82no—
87D4×C127—84no—
88C2×C4×C127—44no—
89C2×C4×C127—82no—
90C2×C4×C127—82no—
91C2×C4×C127—81no—
92C2×C4×C127—82no—
93C2×C4×C127#12011yescharacteristic
94C2×C4×C127#12012yes—
95C2×C4×C127#12111yescharacteristic
96C2×C4×C127#12112yes—
97C2×C4×C127C2×D812yes—
98C2×C4×C127#13514yes—
99C2×C4×C127AGL(1,ℤ/8)12yes—
100C2×C4×C127#13714yes—
101C8×C127—24no—
102C8×C127—44no—
103C22×C127—82no—
104C22×C127—84no—
105C22×C127—161no—
106C22×C127—162no—
107C22×C127—161no—
108C22×C127#39211yescharacteristic
109C22×C127#49311yescharacteristic
110C22×C127#49512yes—
111C22×C127#49611yescharacteristic
112C22×C127#49712yes—
113C4×C127—84no—
114C4×C127—161no—
115C4×C127#44712yes—
116C4×C127#45214yes—
117C4×C127#45912yes—
118C4×C127#46414yes—
119#10306C12711yesmaximal, characteristic, direct product factor, Sylow
120C2×C127—162no—
121C2×C127#170911yescharacteristic
122C2×C127#172012yes—
123C2×C127#172211yescharacteristic
124C2×C127#201111yescharacteristic
125C2×C127#201512yes—
126C2×#494C2×C12711yescharacteristic, complemented
127C2×#486C2×C12712yes—
128#1990C2×C12712yescomplemented
129#1957C2×C12711yescharacteristic
130#1273C2×C12711yescharacteristic, complemented
131C127#1030611yescharacteristic, direct product factor, Sylow
132C22×Q16—22no—
133C2×C4⋊Q8C22×C12712yes—
134C2×#127C22×C12711yescharacteristic
135#494—24no—
136#486—24no—
137#486—24no—
138C2×#102—24no—
139C2×#102—22no—
140C2×C4×C8C22×C12712yes—
141C2×C4×C8C22×C12711yescharacteristic
142#384C22×C12711yescharacteristic
143#340—24no—
144C22×Q8—22no—
145C22×D4—21no—
146C22×D4—22no—
147C2×Q16—28no—
148C2×Q16—44no—
149C2×Q16—44no—
150C22×C8D4×C12714yes—
151C22×C8D4×C12712yes—
152C4⋊Q8—22no—
153C4⋊Q8—22no—
154C4⋊Q8—22no—
155C4⋊Q8D4×C12714yes—
156#127—22no—
157#127—21no—
158#127D4×C12712yes—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C42C23×C12711yescharacteristic
164#102—216no—
165#102—44no—
166C4×C8—22no—
167C4×C8—22no—
168C4×C8—22no—
169C24—41no—
170C2×Q8—24no—
171C2×Q8—42no—
172C2×Q8—44no—
173C2×Q8—44no—
174C2×D4—24no—
175C2×D4—22no—
176C2×D4—41no—
177C2×D4—42no—
178C2×D4—42no—
179C2×D4—44no—
180C2×D4—42no—
181C2×D4—44no—
182C22×C4—42no—
183C22×C4—41no—
184C22×C4C2×D4×C12711yescharacteristic
185C22×C4C2×D4×C12712yes—
186Q16—416no—
187C2×C8—22no—
188C2×C8—22no—
189C2×C8—24no—
190C2×C8—24no—
191C2×C8—24no—
192C2×C8D8×C12714yes—
193C4⋊C4—24no—
194C4⋊C4—24no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—42no—
198C4⋊C4—44no—
199C42C2×D4×C12712yes—
200C42C2×D4×C12711yescharacteristic
201C42#41×C12711yescharacteristic
202C23—42no—
203C23—81no—
204C23—82no—
205C23—81no—
206C23—82no—
207C23#127×C12711yescharacteristic
208Q8—48no—
209Q8—84no—
210D4—48no—
211D4—44no—
212D4—82no—
213D4—84no—
214C2×C4—44no—
215C2×C4—82no—
216C2×C4—82no—
217C2×C4—81no—
218C2×C4—82no—
219C2×C4#120×C12711yescharacteristic
220C2×C4#120×C12712yes—
221C2×C4#121×C12711yescharacteristic
222C2×C4#121×C12712yes—
223C2×C4C2×D8×C12712yes—
224C2×C4#135×C12714yes—
225C2×C4AGL(1,ℤ/8)×C12712yes—
226C2×C4#137×C12714yes—
227C8—24no—
228C8—44no—
229C22—82no—
230C22—84no—
231C22—161no—
232C22—162no—
233C22—161no—
234C22#392×C12711yescharacteristic
235C22#493×C12711yescharacteristic
236C22#495×C12712yes—
237C22#496×C12711yescharacteristic
238C22#497×C12712yes—
239C4—84no—
240C4—161no—
241C4#447×C12712yes—
242C4#452×C12714yes—
243C4#459×C12712yes—
244C4#464×C12714yes—
245C2—162no—
246C2C127×#170911yescharacteristic
247C2C127×#172012yes—
248C2C127×#172211yescharacteristic
249C2C127×#201111yescharacteristic
250C2C127×#201512yes—
subgroupG/H
center, socleC23×C127#127
commutator, Frattini, cosocleC2×C42C23×C127
prime radicalC2×#127×C127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#494×C127C2semidirect
3. C127×#1990C2semidirect
5. C127×#1273C2semidirect
119. #10306C127direct (N × H)
126. C2×#494C2×C127semidirect
128. #1990C2×C127semidirect
130. #1273C2×C127semidirect
131. C127#10306direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#494×C1272
C2×#486×C1272
C127×#19902
C127×#19572
C127×#12732
#10306127