Finite Group Databasebeta

← C127×#7191 #44570864

Subgroups — C127×#7191 #44570864

Order 32512.

342 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 110 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×#349×C127C212yesmaximal
2C127×#1728C211yesmaximal, characteristic
3C22×M16×C127C411yescharacteristic
4C22×M16×C127C2211yescharacteristic
5C22×C42×C127C411yescharacteristic
6C2×M32×C127—22no—
7#349×C127—22no—
8#349×C127C414yescomplemented
9#336×C127—22no—
10C23×C4×C127C812yes—
11C23×C4×C127C2×C411yescharacteristic
12C2×M16×C127—21no—
13C2×M16×C127—41no—
14C2×M16×C127—41no—
15C2×M16×C127—41no—
16C2×M16×C127—41no—
17C2×M16×C127C812yes—
18C2×M16×C127C2×C412yes—
19C2×M16×C127D411yescharacteristic
20C2×M16×C127Q811yescharacteristic
21C22×C8×C127—21no—
22C22×C8×C127D412yes—
23C2×C42×C127—21no—
24C2×C42×C127—21no—
25C2×C42×C127—41no—
26C2×C42×C127—41no—
27M32×C127—24no—
28M32×C127—42no—
29C2×C16×C127—42no—
30C24×C127C2×C811yescharacteristic
31C22×C4×C127—21no—
32C22×C4×C127—21no—
33C22×C4×C127—21no—
34C22×C4×C127—21no—
35C22×C4×C127—22no—
36C22×C4×C127—44no—
37C22×C4×C127—42no—
38C22×C4×C127C4211yescharacteristic
39C22×C4×C127C22⋊C411yescharacteristic
40C22×C4×C127C4⋊C411yescharacteristic
41C22×C4×C127C2×C811yescharacteristic
42C22×C4×C127M1611yescharacteristic
43C22×C4×C127M1611yescharacteristic
44M16×C127—42no—
45M16×C127—41no—
46M16×C127—41no—
47M16×C127—42no—
48M16×C127—42no—
49C2×C8×C127—22no—
50C2×C8×C127—22no—
51C2×C8×C127—22no—
52C2×C8×C127—41no—
53C2×C8×C127—41no—
54C2×C8×C127C22⋊C412yes—
55C2×C8×C127C4⋊C412yes—
56C42×C127—42no—
57C42×C127—41no—
58C42×C127—41no—
59C16×C127—44no—
60C23×C127—21no—
61C23×C127—21no—
62C23×C127—42no—
63C23×C127C4×C811yescharacteristic
64C23×C127C22⋊C811yescharacteristic
65C23×C127C4⋊C811yescharacteristic
66C2×C4×C127—21no—
67C2×C4×C127—21no—
68C2×C4×C127—21no—
69C2×C4×C127—21no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—21no—
74C2×C4×C127—21no—
75C2×C4×C127—21no—
76C2×C4×C127—24no—
77C2×C4×C127—42no—
78C2×C4×C127—44no—
79C2×C4×C127—42no—
80C2×C4×C127—44no—
81C2×C4×C127#9511yescharacteristic
82C2×C4×C127#9711yescharacteristic
83C2×C4×C127C22⋊C811yescharacteristic
84C2×C4×C127#9911yescharacteristic
85C2×C4×C127#9911yescharacteristic
86C2×C4×C127#10011yescharacteristic
87C2×C4×C127#10111yescharacteristic
88C2×C4×C127C4⋊C811yescharacteristic
89C8×C127—22no—
90C8×C127—22no—
91C8×C127—42no—
92C22×C127—22no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—42no—
98C22×C127—42no—
99C22×C127#32311yescharacteristic
100C22×C127#32411yescharacteristic
101C22×C127#33611yescharacteristic
102C4×C127—21no—
103C4×C127—21no—
104C4×C127—44no—
105C4×C127#32311yescharacteristic
106C4×C127#32411yescharacteristic
107C4×C127#34211yescharacteristic
108C4×C127#34311yescharacteristic
109#7191C12711yesmaximal, characteristic, direct product factor, Sylow
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—41no—
113C2×C127—41no—
114C2×C127#128011yescharacteristic
115C2×C127#132012yes—
116C2×#349C2×C12712yes—
117#1728C2×C12711yescharacteristic
118C127#719111yescharacteristic, direct product factor, Sylow
119C22×M16C4×C12711yescharacteristic
120C22×M16C22×C12711yescharacteristic
121C22×C42C4×C12711yescharacteristic
122C2×M32—22no—
123#349—22no—
124#349C4×C12714yescomplemented
125#336—22no—
126C23×C4C8×C12712yes—
127C23×C4C2×C4×C12711yescharacteristic
128C2×M16—21no—
129C2×M16—41no—
130C2×M16—41no—
131C2×M16—41no—
132C2×M16—41no—
133C2×M16C8×C12712yes—
134C2×M16C2×C4×C12712yes—
135C2×M16D4×C12711yescharacteristic
136C2×M16Q8×C12711yescharacteristic
137C22×C8—21no—
138C22×C8D4×C12712yes—
139C2×C42—21no—
140C2×C42—21no—
141C2×C42—41no—
142C2×C42—41no—
143M32—24no—
144M32—42no—
145C2×C16—42no—
146C24C2×C8×C12711yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—22no—
152C22×C4—44no—
153C22×C4—42no—
154C22×C4C42×C12711yescharacteristic
155C22×C4C22⋊C4×C12711yescharacteristic
156C22×C4C4⋊C4×C12711yescharacteristic
157C22×C4C2×C8×C12711yescharacteristic
158C22×C4M16×C12711yescharacteristic
159C22×C4M16×C12711yescharacteristic
160M16—42no—
161M16—41no—
162M16—41no—
163M16—42no—
164M16—42no—
165C2×C8—22no—
166C2×C8—22no—
167C2×C8—22no—
168C2×C8—41no—
169C2×C8—41no—
170C2×C8C22⋊C4×C12712yes—
171C2×C8C4⋊C4×C12712yes—
172C42—42no—
173C42—41no—
174C42—41no—
175C16—44no—
176C23—21no—
177C23—21no—
178C23—42no—
179C23C4×C8×C12711yescharacteristic
180C23C22⋊C8×C12711yescharacteristic
181C23C4⋊C8×C12711yescharacteristic
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—24no—
193C2×C4—42no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—44no—
197C2×C4#95×C12711yescharacteristic
198C2×C4#97×C12711yescharacteristic
199C2×C4C22⋊C8×C12711yescharacteristic
200C2×C4#99×C12711yescharacteristic
201C2×C4#99×C12711yescharacteristic
202C2×C4#100×C12711yescharacteristic
203C2×C4#101×C12711yescharacteristic
204C2×C4C4⋊C8×C12711yescharacteristic
205C8—22no—
206C8—22no—
207C8—42no—
208C22—22no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—42no—
214C22—42no—
215C22#323×C12711yescharacteristic
216C22#324×C12711yescharacteristic
217C22#336×C12711yescharacteristic
218C4—21no—
219C4—21no—
220C4—44no—
221C4#323×C12711yescharacteristic
222C4#324×C12711yescharacteristic
223C4#342×C12711yescharacteristic
224C4#343×C12711yescharacteristic
225C2—21no—
226C2—21no—
227C2—41no—
228C2—41no—
229C2C127×#128011yescharacteristic
230C2C127×#132012yes—
subgroupG/H
centerC2×C4×C127#99
commutatorC23C4×C8×C127
Frattini, cosocleC22×M16C22×C127
socleC22×C127#323
prime radicalC24×C127C2×C8
LCSG ▷ C23 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C127 ◁ C22×C4×C127 ◁ C22×M16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
8. #349×C127C4semidirect
109. #7191C127direct (N × H)
118. C127#7191direct (N × H)
124. #349C4×C127semidirect

Maximal subgroups

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

subgroupindex
C2×#349×C1272
C127×#17282
#7191127