Finite Group Databasebeta

← C127×#7194 #44570867

Subgroups — C127×#7194 #44570867

Order 32512.

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

normal subgroups only

This group has 310 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×#1729C211yesmaximal, characteristic
3C22×M16×C127C2211yescharacteristic
4C23×C8×C127C411yescharacteristic
5C2×#117×C127C411yescharacteristic
6C2×M32×C127—22no—
7#349×C127—22no—
8#349×C127C414yescomplemented
9#336×C127—22no—
10C23×C4×C127C2×C411yescharacteristic
11C2×M16×C127—41no—
12C2×M16×C127—41no—
13C2×M16×C127C2×C412yes—
14C2×M16×C127D411yescharacteristic
15C2×M16×C127Q811yescharacteristic
16C22×C8×C127—21no—
17C22×C8×C127—21no—
18C22×C8×C127—41no—
19C22×C8×C127—41no—
20C22×C8×C127C812yes—
21C22×C8×C127D412yes—
22#117×C127—41no—
23#117×C127—41no—
24C2×C4⋊C4×C127C812yes—
25C2×C22⋊C4×C127—21no—
26C2×C42×C127—21no—
27M32×C127—24no—
28M32×C127—42no—
29C2×C16×C127—42no—
30C24×C127M1611yescharacteristic
31C22×C4×C127—21no—
32C22×C4×C127—21no—
33C22×C4×C127—21no—
34C22×C4×C127—21no—
35C22×C4×C127—22no—
36C22×C4×C127C4211yescharacteristic
37C22×C4×C127C22⋊C411yescharacteristic
38C22×C4×C127C4⋊C411yescharacteristic
39C22×C4×C127C2×C811yescharacteristic
40C22×C4×C127C2×C811yescharacteristic
41C22×C4×C127M1611yescharacteristic
42M16×C127—42no—
43M16×C127—41no—
44M16×C127—41no—
45C2×C8×C127—22no—
46C2×C8×C127—22no—
47C2×C8×C127—22no—
48C2×C8×C127—42no—
49C2×C8×C127—41no—
50C2×C8×C127—41no—
51C2×C8×C127—42no—
52C2×C8×C127C22⋊C412yes—
53C2×C8×C127C4⋊C412yes—
54C4⋊C4×C127—42no—
55C22⋊C4×C127—42no—
56C42×C127—41no—
57C42×C127—41no—
58C16×C127—44no—
59C23×C127—21no—
60C23×C127—21no—
61C23×C127—42no—
62C23×C127#9711yescharacteristic
63C23×C127C22⋊C811yescharacteristic
64C23×C127C4⋊C811yescharacteristic
65C2×C4×C127—21no—
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—44no—
76C2×C4×C127—42no—
77C2×C4×C127#9511yescharacteristic
78C2×C4×C127C4×C811yescharacteristic
79C2×C4×C127C22⋊C811yescharacteristic
80C2×C4×C127#9911yescharacteristic
81C2×C4×C127#9911yescharacteristic
82C2×C4×C127#10011yescharacteristic
83C2×C4×C127#10111yescharacteristic
84C2×C4×C127C4⋊C811yescharacteristic
85C8×C127—22no—
86C8×C127—22no—
87C8×C127—42no—
88C22×C127—22no—
89C22×C127—21no—
90C22×C127—21no—
91C22×C127—41no—
92C22×C127—41no—
93C22×C127—42no—
94C22×C127—42no—
95C22×C127#32311yescharacteristic
96C22×C127#32411yescharacteristic
97C22×C127#33611yescharacteristic
98C4×C127—21no—
99C4×C127—21no—
100C4×C127—82no—
101C4×C127#32311yescharacteristic
102C4×C127#32411yescharacteristic
103C4×C127#34211yescharacteristic
104C4×C127#34311yescharacteristic
105#7194C12711yesmaximal, characteristic, direct product factor, Sylow
106C2×C127—21no—
107C2×C127—21no—
108C2×C127—41no—
109C2×C127—41no—
110C2×C127#128011yescharacteristic
111C2×C127#132112yes—
112C2×#349C2×C12712yes—
113#1729C2×C12711yescharacteristic
114C127#719411yescharacteristic, direct product factor, Sylow
115C22×M16C22×C12711yescharacteristic
116C23×C8C4×C12711yescharacteristic
117C2×#117C4×C12711yescharacteristic
118C2×M32—22no—
119#349—22no—
120#349C4×C12714yescomplemented
121#336—22no—
122C23×C4C2×C4×C12711yescharacteristic
123C2×M16—41no—
124C2×M16—41no—
125C2×M16C2×C4×C12712yes—
126C2×M16D4×C12711yescharacteristic
127C2×M16Q8×C12711yescharacteristic
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—41no—
131C22×C8—41no—
132C22×C8C8×C12712yes—
133C22×C8D4×C12712yes—
134#117—41no—
135#117—41no—
136C2×C4⋊C4C8×C12712yes—
137C2×C22⋊C4—21no—
138C2×C42—21no—
139M32—24no—
140M32—42no—
141C2×C16—42no—
142C24M16×C12711yescharacteristic
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—22no—
148C22×C4C42×C12711yescharacteristic
149C22×C4C22⋊C4×C12711yescharacteristic
150C22×C4C4⋊C4×C12711yescharacteristic
151C22×C4C2×C8×C12711yescharacteristic
152C22×C4C2×C8×C12711yescharacteristic
153C22×C4M16×C12711yescharacteristic
154M16—42no—
155M16—41no—
156M16—41no—
157C2×C8—22no—
158C2×C8—22no—
159C2×C8—22no—
160C2×C8—42no—
161C2×C8—41no—
162C2×C8—41no—
163C2×C8—42no—
164C2×C8C22⋊C4×C12712yes—
165C2×C8C4⋊C4×C12712yes—
166C4⋊C4—42no—
167C22⋊C4—42no—
168C42—41no—
169C42—41no—
170C16—44no—
171C23—21no—
172C23—21no—
173C23—42no—
174C23#97×C12711yescharacteristic
175C23C22⋊C8×C12711yescharacteristic
176C23C4⋊C8×C12711yescharacteristic
177C2×C4—21no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—44no—
188C2×C4—42no—
189C2×C4#95×C12711yescharacteristic
190C2×C4C4×C8×C12711yescharacteristic
191C2×C4C22⋊C8×C12711yescharacteristic
192C2×C4#99×C12711yescharacteristic
193C2×C4#99×C12711yescharacteristic
194C2×C4#100×C12711yescharacteristic
195C2×C4#101×C12711yescharacteristic
196C2×C4C4⋊C8×C12711yescharacteristic
197C8—22no—
198C8—22no—
199C8—42no—
200C22—22no—
201C22—21no—
202C22—21no—
203C22—41no—
204C22—41no—
205C22—42no—
206C22—42no—
207C22#323×C12711yescharacteristic
208C22#324×C12711yescharacteristic
209C22#336×C12711yescharacteristic
210C4—21no—
211C4—21no—
212C4—82no—
213C4#323×C12711yescharacteristic
214C4#324×C12711yescharacteristic
215C4#342×C12711yescharacteristic
216C4#343×C12711yescharacteristic
217C2—21no—
218C2—21no—
219C2—41no—
220C2—41no—
221C2C127×#128011yescharacteristic
222C2C127×#132112yes—
subgroupG/H
centerC2×C4×C127#99
commutatorC2×C4C4×C8×C127
Frattini, cosocleC22×M16C22×C127
socleC22×C127#323
prime radicalC24×C127M16
LCSG ▷ C2×C4 ▷ 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
105. #7194C127direct (N × H)
114. C127#7194direct (N × H)
120. #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×#17292
#7194127