Finite Group Databasebeta

← C127×#13492 #44577165

Subgroups — C127×#13492 #44577165

Order 32512.

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

normal subgroups only

This group has 342 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
1C2×#366×C127C211yesmaximal, characteristic, complemented
2C2×C22⋊C16×C127C211yesmaximal, characteristic
3C127×#1808C212yesmaximal
4C127×#1380C212yesmaximal, complemented
5C127×#1380C211yesmaximal, characteristic, complemented
6C23×C8×C127C2211yescharacteristic
7C22×C16×C127C2211yescharacteristic
8C22×C16×C127C2211yescharacteristic
9C2×#107×C127C2212yes—
10C2×#107×C127C2212yes—
11#380×C127—22no—
12#366×C127—22no—
13C22⋊C16×C127—22no—
14#340×C127—22no—
15C23×C4×C127D411yescharacteristic
16C22×C8×C127—21no—
17C22×C8×C127—21no—
18C22×C8×C127—22no—
19C22×C8×C127—22no—
20C22×C8×C127D411yescharacteristic
21C22×C8×C127C2311yescharacteristic
22C2×C4⋊C4×C127—22no—
23C2×C4⋊C4×C127—22no—
24C2×C22⋊C4×C127—42no—
25C2×C16×C127—22no—
26C2×C16×C127—21no—
27C2×C16×C127—22no—
28C2×C16×C127D412yes—
29#107×C127—22no—
30#107×C127—22no—
31#107×C127—22no—
32#107×C127—22no—
33#95×C127—42no—
34C24×C127D811yescharacteristic
35C22×C4×C127—21no—
36C22×C4×C127—22no—
37C22×C4×C127—21no—
38C22×C4×C127—22no—
39C22×C4×C127—42no—
40C22×C4×C127—42no—
41C22×C4×C127D811yescharacteristic
42C22×C4×C127C2×D411yescharacteristic
43C2×C8×C127—21no—
44C2×C8×C127—22no—
45C2×C8×C127—41no—
46C2×C8×C127—41no—
47C2×C8×C127—42no—
48C2×C8×C127C2×D411yescharacteristic
49C2×C8×C127#4112yes—
50C2×C8×C127#4112yes—
51C2×C8×C127#4111yescharacteristic
52C4⋊C4×C127—42no—
53C4⋊C4×C127—42no—
54C4⋊C4×C127—42no—
55C4⋊C4×C127—42no—
56C22⋊C4×C127—84no—
57C16×C127—22no—
58C16×C127—41no—
59C23×C127—21no—
60C23×C127—22no—
61C23×C127—21no—
62C23×C127—22no—
63C23×C127D1612yes—
64C23×C127C2×D811yescharacteristic
65C2×C4×C127—21no—
66C2×C4×C127—22no—
67C2×C4×C127—41no—
68C2×C4×C127—41no—
69C2×C4×C127—42no—
70C2×C4×C127—82no—
71C2×C4×C127—82no—
72C2×C4×C127—82no—
73C2×C4×C127—82no—
74C2×C4×C127—82no—
75C2×C4×C127—82no—
76C2×C4×C127#12111yescharacteristic
77C2×C4×C127#12312yes—
78C2×C4×C127#13511yescharacteristic
79C2×C4×C127#13712yes—
80C8×C127—41no—
81C8×C127#12311yescharacteristic
82C8×C127#12411yescharacteristic
83C8×C127#12612yes—
84C22×C127—22no—
85C22×C127—24no—
86C22×C127—41no—
87C22×C127—41no—
88C22×C127—42no—
89C22×C127#46611yescharacteristic
90C22×C127#48012yes—
91C22×C127C2×D1611yescharacteristic
92C22×C127#50811yescharacteristic
93C22×C127#51012yes—
94C4×C127—41no—
95C4×C127—162no—
96C4×C127—162no—
97C4×C127#39711yescharacteristic
98C4×C127#47011yescharacteristic
99C4×C127#48512yes—
100#13492C12711yesmaximal, characteristic, direct product factor, Sylow
101C2×C127—42no—
102C2×C127#207511yescharacteristic
103C2×C127#221511yescharacteristic
104C2×C127#221911yescharacteristic
105C2×C127#223212yes—
106C2×C127#223712yes—
107C2×#366C2×C12711yescharacteristic, complemented
108C2×C22⋊C16C2×C12711yescharacteristic
109#1808C2×C12712yes—
110#1380C2×C12711yescharacteristic, complemented
111#1380C2×C12712yescomplemented
112C127#1349211yescharacteristic, direct product factor, Sylow
113C23×C8C22×C12711yescharacteristic
114C22×C16C22×C12711yescharacteristic
115C22×C16C22×C12711yescharacteristic
116C2×#107C22×C12712yes—
117C2×#107C22×C12712yes—
118#380—22no—
119#366—22no—
120C22⋊C16—22no—
121#340—22no—
122C23×C4D4×C12711yescharacteristic
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8D4×C12711yescharacteristic
128C22×C8C23×C12711yescharacteristic
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C22⋊C4—42no—
132C2×C16—22no—
133C2×C16—21no—
134C2×C16—22no—
135C2×C16D4×C12712yes—
136#107—22no—
137#107—22no—
138#107—22no—
139#107—22no—
140#95—42no—
141C24D8×C12711yescharacteristic
142C22×C4—21no—
143C22×C4—22no—
144C22×C4—21no—
145C22×C4—22no—
146C22×C4—42no—
147C22×C4—42no—
148C22×C4D8×C12711yescharacteristic
149C22×C4C2×D4×C12711yescharacteristic
150C2×C8—21no—
151C2×C8—22no—
152C2×C8—41no—
153C2×C8—41no—
154C2×C8—42no—
155C2×C8C2×D4×C12711yescharacteristic
156C2×C8#41×C12712yes—
157C2×C8#41×C12712yes—
158C2×C8#41×C12711yescharacteristic
159C4⋊C4—42no—
160C4⋊C4—42no—
161C4⋊C4—42no—
162C4⋊C4—42no—
163C22⋊C4—84no—
164C16—22no—
165C16—41no—
166C23—21no—
167C23—22no—
168C23—21no—
169C23—22no—
170C23D16×C12712yes—
171C23C2×D8×C12711yescharacteristic
172C2×C4—21no—
173C2×C4—22no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—42no—
177C2×C4—82no—
178C2×C4—82no—
179C2×C4—82no—
180C2×C4—82no—
181C2×C4—82no—
182C2×C4—82no—
183C2×C4#121×C12711yescharacteristic
184C2×C4#123×C12712yes—
185C2×C4#135×C12711yescharacteristic
186C2×C4#137×C12712yes—
187C8—41no—
188C8#123×C12711yescharacteristic
189C8#124×C12711yescharacteristic
190C8#126×C12712yes—
191C22—22no—
192C22—24no—
193C22—41no—
194C22—41no—
195C22—42no—
196C22#466×C12711yescharacteristic
197C22#480×C12712yes—
198C22C2×D16×C12711yescharacteristic
199C22#508×C12711yescharacteristic
200C22#510×C12712yes—
201C4—41no—
202C4—162no—
203C4—162no—
204C4#397×C12711yescharacteristic
205C4#470×C12711yescharacteristic
206C4#485×C12712yes—
207C2—42no—
208C2C127×#207511yescharacteristic
209C2C127×#221511yescharacteristic
210C2C127×#221911yescharacteristic
211C2C127×#223212yes—
212C2C127×#223712yes—
subgroupG/H
center, socleC23×C127C2×D8
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC24×C127D8
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C23×C4×C127 ◁ C23×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#366×C127C2semidirect
4. C127×#1380C2semidirect
5. C127×#1380C2semidirect
100. #13492C127direct (N × H)
107. C2×#366C2×C127semidirect
110. #1380C2×C127semidirect
111. #1380C2×C127semidirect
112. C127#13492direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#366×C1272
C2×C22⋊C16×C1272
C127×#18082
C127×#13802
#13492127