Finite Group Databasebeta

← C127×#7078 #44570751

Subgroups — C127×#7078 #44570751

Order 32512.

286 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 54 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×#2001C211yesmaximal, characteristic
2C127×#1314C211yesmaximal, characteristic
3C127×#1289C211yesmaximal, characteristic
4C22×Q16×C127C411yescharacteristic, complemented
5C2×C22⋊Q8×C127—21no—
6C2×#103×C127—21no—
7C2×C22⋊C8×C127C2211yescharacteristic
8#380×C127—21no—
9C22⋊C16×C127—21no—
10#340×C127C411yescharacteristic, complemented
11C22×Q8×C127—21no—
12C23×C4×C127D411yescharacteristic
13C2×Q16×C127—41no—
14C2×Q16×C127—42no—
15C22×C8×C127C2×C411yescharacteristic
16C22×C8×C127D411yescharacteristic
17C22⋊Q8×C127—42no—
18C22⋊Q8×C127—42no—
19C2×C4⋊C4×C127—21no—
20C2×C4⋊C4×C127—41no—
21C2×C4⋊C4×C127—41no—
22C2×C22⋊C4×C127—41no—
23C2×C22⋊C4×C127—41no—
24C2×C16×C127—41no—
25#103×C127—42no—
26C22⋊C8×C127—41no—
27#95×C127—22no—
28C24×C127D811yescharacteristic
29C2×Q8×C127—22no—
30C2×Q8×C127—41no—
31C2×Q8×C127—41no—
32C2×Q8×C127—41no—
33C2×Q8×C127—41no—
34C2×Q8×C127—41no—
35C22×C4×C127—21no—
36C22×C4×C127—21no—
37C22×C4×C127—41no—
38C22×C4×C127—41no—
39C22×C4×C127—41no—
40C22×C4×C127—41no—
41C22×C4×C127—42no—
42C22×C4×C127C22⋊C411yescharacteristic
43C22×C4×C127SD1611yescharacteristic
44Q16×C127—42no—
45Q16×C127—42no—
46C2×C8×C127—21no—
47C2×C8×C127—21no—
48C2×C8×C127—41no—
49C2×C8×C127—41no—
50C4⋊C4×C127—22no—
51C4⋊C4×C127—41no—
52C4⋊C4×C127—44no—
53C4⋊C4×C127—81no—
54C4⋊C4×C127—81no—
55C22⋊C4×C127—44no—
56C22⋊C4×C127—81no—
57C22⋊C4×C127—81no—
58C16×C127—81no—
59C23×C127—21no—
60C23×C127—21no—
61C23×C127—41no—
62C23×C127—41no—
63C23×C127#10211yescharacteristic
64C23×C127SD3211yescharacteristic
65C23×C127Q3211yescharacteristic
66Q8×C127—42no—
67Q8×C127—42no—
68Q8×C127—81no—
69Q8×C127—81no—
70C2×C4×C127—21no—
71C2×C4×C127—21no—
72C2×C4×C127—21no—
73C2×C4×C127—41no—
74C2×C4×C127—41no—
75C2×C4×C127—41no—
76C2×C4×C127—42no—
77C2×C4×C127—42no—
78C2×C4×C127—81no—
79C2×C4×C127—81no—
80C2×C4×C127—81no—
81C2×C4×C127—81no—
82C2×C4×C127—81no—
83C2×C4×C127—82no—
84C2×C4×C127—82no—
85C2×C4×C127—82no—
86C2×C4×C127#9911yescharacteristic
87C2×C4×C127#10411yescharacteristic
88C8×C127—41no—
89C8×C127—41no—
90C22×C127—21no—
91C22×C127—21no—
92C22×C127—21no—
93C22×C127—21no—
94C22×C127—41no—
95C22×C127—41no—
96C22×C127—41no—
97C22×C127—41no—
98C22×C127—81no—
99C22×C127#32711yescharacteristic
100C22×C127#35811yescharacteristic
101C22×C127#36011yescharacteristic
102C4×C127—21no—
103C4×C127—41no—
104C4×C127—82no—
105C4×C127—82no—
106C4×C127—162no—
107C4×C127AΣL(1,𝔽16)11yescharacteristic
108C4×C127#35311yescharacteristic
109#7078C12711yesmaximal, characteristic, direct product factor, Sylow
110C2×C127—21no—
111C2×C127—21no—
112C2×C127—41no—
113C2×C127—41no—
114C2×C127#134011yescharacteristic
115C2×C127#134311yescharacteristic
116C2×C127#134811yescharacteristic
117#2001C2×C12711yescharacteristic
118#1314C2×C12711yescharacteristic
119#1289C2×C12711yescharacteristic
120C127#707811yescharacteristic, direct product factor, Sylow
121C22×Q16C4×C12711yescharacteristic, complemented
122C2×C22⋊Q8—21no—
123C2×#103—21no—
124C2×C22⋊C8C22×C12711yescharacteristic
125#380—21no—
126C22⋊C16—21no—
127#340C4×C12711yescharacteristic, complemented
128C22×Q8—21no—
129C23×C4D4×C12711yescharacteristic
130C2×Q16—41no—
131C2×Q16—42no—
132C22×C8C2×C4×C12711yescharacteristic
133C22×C8D4×C12711yescharacteristic
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C22⋊C4—41no—
140C2×C22⋊C4—41no—
141C2×C16—41no—
142#103—42no—
143C22⋊C8—41no—
144#95—22no—
145C24D8×C12711yescharacteristic
146C2×Q8—22no—
147C2×Q8—41no—
148C2×Q8—41no—
149C2×Q8—41no—
150C2×Q8—41no—
151C2×Q8—41no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—41no—
155C22×C4—41no—
156C22×C4—41no—
157C22×C4—41no—
158C22×C4—42no—
159C22×C4C22⋊C4×C12711yescharacteristic
160C22×C4SD16×C12711yescharacteristic
161Q16—42no—
162Q16—42no—
163C2×C8—21no—
164C2×C8—21no—
165C2×C8—41no—
166C2×C8—41no—
167C4⋊C4—22no—
168C4⋊C4—41no—
169C4⋊C4—44no—
170C4⋊C4—81no—
171C4⋊C4—81no—
172C22⋊C4—44no—
173C22⋊C4—81no—
174C22⋊C4—81no—
175C16—81no—
176C23—21no—
177C23—21no—
178C23—41no—
179C23—41no—
180C23#102×C12711yescharacteristic
181C23SD32×C12711yescharacteristic
182C23Q32×C12711yescharacteristic
183Q8—42no—
184Q8—42no—
185Q8—81no—
186Q8—81no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—81no—
196C2×C4—81no—
197C2×C4—81no—
198C2×C4—81no—
199C2×C4—81no—
200C2×C4—82no—
201C2×C4—82no—
202C2×C4—82no—
203C2×C4#99×C12711yescharacteristic
204C2×C4#104×C12711yescharacteristic
205C8—41no—
206C8—41no—
207C22—21no—
208C22—21no—
209C22—21no—
210C22—21no—
211C22—41no—
212C22—41no—
213C22—41no—
214C22—41no—
215C22—81no—
216C22#327×C12711yescharacteristic
217C22#358×C12711yescharacteristic
218C22#360×C12711yescharacteristic
219C4—21no—
220C4—41no—
221C4—82no—
222C4—82no—
223C4—162no—
224C4AΣL(1,𝔽16)×C12711yescharacteristic
225C4#353×C12711yescharacteristic
226C2—21no—
227C2—21no—
228C2—41no—
229C2—41no—
230C2C127×#134011yescharacteristic
231C2C127×#134311yescharacteristic
232C2C127×#134811yescharacteristic
subgroupG/H
center, socleC22×C127#327
commutatorC22×C8C2×C4×C127
Frattini, cosocleC2×C22⋊C8C22×C127
prime radicalC24×C127D8
LCSG ▷ C22×C8 ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C127 ◁ C22×C4×C127 ◁ C2×C22⋊C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
4. C22×Q16×C127C4semidirect
10. #340×C127C4semidirect
109. #7078C127direct (N × H)
120. C127#7078direct (N × H)
121. C22×Q16C4×C127semidirect
127. #340C4×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#20012
C127×#13142
C127×#12892
#7078127