Finite Group Databasebeta

← C127×#13689 #44577362

Subgroups — C127×#13689 #44577362

Order 32512.

216 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 74 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×SD64×C127C211yesmaximal, characteristic, complemented
2C2×D32×C127C211yesmaximal, characteristic
3C127×#2249C211yesmaximal, characteristic, complemented
4C4×D16×C127C211yesmaximal, characteristic, complemented
5C127×C4⋊C32C211yesmaximal, characteristic, complemented
6C127×#1416C211yesmaximal, characteristic, complemented
7C127×#1415C211yesmaximal, characteristic, complemented
8C2×Q32×C127C2211yescharacteristic
9C2×SD32×C127—21no—
10C2×D16×C127C2211yescharacteristic
11C2×D16×C127C2211yescharacteristic
12#495×C127—21no—
13C4×D8×C127—21no—
14SD64×C127—22no—
15D32×C127—22no—
16C2×C32×C127C2211yescharacteristic
17C2×C32×C127C2211yescharacteristic
18#366×C127C2211yescharacteristic
19#357×C127—21no—
20C4×C16×C127C2211yescharacteristic
21C2×Q16×C127—21no—
22C2×SD16×C127—41no—
23C2×D8×C127—21no—
24C2×D8×C127—21no—
25#124×C127—41no—
26C4×D4×C127—41no—
27Q32×C127—21no—
28Q32×C127—21no—
29SD32×C127—42no—
30D16×C127—21no—
31D16×C127—21no—
32D16×C127—21no—
33D16×C127D412yes—
34C2×C16×C127D411yescharacteristic
35C2×C16×C127C2311yescharacteristic
36#107×C127—21no—
37#102×C127—41no—
38C4×C8×C127D411yescharacteristic
39C32×C127—21no—
40C32×C127—21no—
41C2×Q8×C127—41no—
42C2×D4×C127—41no—
43C2×D4×C127—41no—
44C22×C4×C127—81no—
45Q16×C127—41no—
46Q16×C127—41no—
47SD16×C127—82no—
48D8×C127—22no—
49D8×C127—41no—
50D8×C127—41no—
51D8×C127—41no—
52C2×C8×C127D811yescharacteristic
53C2×C8×C127C2×D411yescharacteristic
54C4⋊C4×C127—41no—
55C22⋊C4×C127—81no—
56C22⋊C4×C127—81no—
57C22⋊C4×C127—81no—
58C42×C127D811yescharacteristic
59C16×C127—21no—
60C16×C127C2×D411yescharacteristic
61C16×C127#4111yescharacteristic
62C23×C127—81no—
63C23×C127—81no—
64Q8×C127—81no—
65Q8×C127—81no—
66D4×C127—42no—
67D4×C127—81no—
68D4×C127—81no—
69D4×C127—81no—
70C2×C4×C127—81no—
71C2×C4×C127—81no—
72C2×C4×C127—162no—
73C2×C4×C127D1611yescharacteristic
74C2×C4×C127D1611yescharacteristic
75C2×C4×C127C2×D811yescharacteristic
76C8×C127—21no—
77C8×C127#12111yescharacteristic
78C8×C127AGL(1,ℤ/8)11yescharacteristic
79C22×C127—82no—
80C22×C127—161no—
81C22×C127—161no—
82C22×C127—161no—
83C22×C127—161no—
84C22×C127—161no—
85C22×C127C2×D1611yescharacteristic
86C4×C127—21no—
87C4×C127—21no—
88C4×C127—161no—
89C4×C127—161no—
90C4×C127#45911yescharacteristic
91C4×C127#50911yescharacteristic
92#13689C12711yesmaximal, characteristic, direct product factor, Sylow
93C2×C127—162no—
94C2×C127—321no—
95C2×C127#220611yescharacteristic
96C2×C127#226211yescharacteristic
97C2×C127#226311yescharacteristic
98C2×SD64C2×C12711yescharacteristic, complemented
99C2×D32C2×C12711yescharacteristic
100#2249C2×C12711yescharacteristic, complemented
101C4×D16C2×C12711yescharacteristic, complemented
102C4⋊C32C2×C12711yescharacteristic, complemented
103#1416C2×C12711yescharacteristic, complemented
104#1415C2×C12711yescharacteristic, complemented
105C127#1368911yescharacteristic, direct product factor, Sylow
106C2×Q32C22×C12711yescharacteristic
107C2×SD32—21no—
108C2×D16C22×C12711yescharacteristic
109C2×D16C22×C12711yescharacteristic
110#495—21no—
111C4×D8—21no—
112SD64—22no—
113D32—22no—
114C2×C32C22×C12711yescharacteristic
115C2×C32C22×C12711yescharacteristic
116#366C22×C12711yescharacteristic
117#357—21no—
118C4×C16C22×C12711yescharacteristic
119C2×Q16—21no—
120C2×SD16—41no—
121C2×D8—21no—
122C2×D8—21no—
123#124—41no—
124C4×D4—41no—
125Q32—21no—
126Q32—21no—
127SD32—42no—
128D16—21no—
129D16—21no—
130D16—21no—
131D16D4×C12712yes—
132C2×C16D4×C12711yescharacteristic
133C2×C16C23×C12711yescharacteristic
134#107—21no—
135#102—41no—
136C4×C8D4×C12711yescharacteristic
137C32—21no—
138C32—21no—
139C2×Q8—41no—
140C2×D4—41no—
141C2×D4—41no—
142C22×C4—81no—
143Q16—41no—
144Q16—41no—
145SD16—82no—
146D8—22no—
147D8—41no—
148D8—41no—
149D8—41no—
150C2×C8D8×C12711yescharacteristic
151C2×C8C2×D4×C12711yescharacteristic
152C4⋊C4—41no—
153C22⋊C4—81no—
154C22⋊C4—81no—
155C22⋊C4—81no—
156C42D8×C12711yescharacteristic
157C16—21no—
158C16C2×D4×C12711yescharacteristic
159C16#41×C12711yescharacteristic
160C23—81no—
161C23—81no—
162Q8—81no—
163Q8—81no—
164D4—42no—
165D4—81no—
166D4—81no—
167D4—81no—
168C2×C4—81no—
169C2×C4—81no—
170C2×C4—162no—
171C2×C4D16×C12711yescharacteristic
172C2×C4D16×C12711yescharacteristic
173C2×C4C2×D8×C12711yescharacteristic
174C8—21no—
175C8#121×C12711yescharacteristic
176C8AGL(1,ℤ/8)×C12711yescharacteristic
177C22—82no—
178C22—161no—
179C22—161no—
180C22—161no—
181C22—161no—
182C22—161no—
183C22C2×D16×C12711yescharacteristic
184C4—21no—
185C4—21no—
186C4—161no—
187C4—161no—
188C4#459×C12711yescharacteristic
189C4#509×C12711yescharacteristic
190C2—162no—
191C2—321no—
192C2C127×#220611yescharacteristic
193C2C127×#226211yescharacteristic
194C2C127×#226311yescharacteristic
subgroupG/H
center, socleC22×C127C2×D16
commutator, Frattini, cosocleC2×C16C23×C127
prime radicalC2×D32×C127C2
LCSG ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C4×C8×C127 ◁ C4×C16×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×SD64×C127C2semidirect
3. C127×#2249C2semidirect
4. C4×D16×C127C2semidirect
5. C127×C4⋊C32C2semidirect
6. C127×#1416C2semidirect
7. C127×#1415C2semidirect
92. #13689C127direct (N × H)
98. C2×SD64C2×C127semidirect
100. #2249C2×C127semidirect
101. C4×D16C2×C127semidirect
102. C4⋊C32C2×C127semidirect
103. #1416C2×C127semidirect
104. #1415C2×C127semidirect
105. C127#13689direct (N × H)

Maximal subgroups

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

subgroupindex
C2×SD64×C1272
C2×D32×C1272
C127×#22492
C4×D16×C1272
C127×C4⋊C322
C127×#14162
C127×#14152
#13689127