Finite Group Databasebeta

← C127×#12313 #44575986

Subgroups — C127×#12313 #44575986

Order 32512.

356 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 78 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×#3293C211yesmaximal, characteristic
2C127×#2214C212yesmaximal, complemented
3C127×#1713C211yesmaximal, characteristic, complemented
4C127×#1358C211yesmaximal, characteristic, complemented
5C127×#1335C212yesmaximal, complemented
6#578×C127—22no—
7#510×C127—22no—
8#500×C127—22no—
9#494×C127C2211yescharacteristic
10#494×C127C2211yescharacteristic
11#473×C127—22no—
12#456×C127—22no—
13#443×C127C2212yes—
14#364×C127C2212yes—
15#362×C127—22no—
16C82×C127C2211yescharacteristic
172−1+4×C127—22no—
18#137×C127—24no—
19#137×C127—44no—
20#135×C127—42no—
21#135×C127D412yes—
22C2×Q16×C127—22no—
23C2×Q16×C127—22no—
24C2×Q16×C127—42no—
25#131×C127—22no—
26C4⋊Q8×C127—21no—
27C4⋊Q8×C127D412yes—
28Q32×C127—42no—
29SD32×C127—42no—
30M32×C127—22no—
31#108×C127D412yes—
32#107×C127—44no—
33#104×C127—22no—
34#101×C127—22no—
35C4×C8×C127—21no—
36C4×C8×C127C2311yescharacteristic
37#41×C127—22no—
38#41×C127—44no—
39#41×C127—82no—
40C2×Q8×C127—22no—
41C2×Q8×C127—44no—
42C2×Q8×C127—41no—
43Q16×C127—44no—
44Q16×C127—42no—
45Q16×C127—42no—
46Q16×C127—42no—
47Q16×C127—42no—
48Q16×C127D812yes—
49SD16×C127—22no—
50SD16×C127—44no—
51D8×C127D812yes—
52M16×C127—22no—
53M16×C127—42no—
54C2×C8×C127—22no—
55C2×C8×C127—42no—
56C2×C8×C127C2×D412yes—
57C4⋊C4×C127—42no—
58C4⋊C4×C127—42no—
59C42×C127C2×D411yescharacteristic
60C16×C127—42no—
61Q8×C127—22no—
62Q8×C127—44no—
63Q8×C127—82no—
64Q8×C127—82no—
65D4×C127—22no—
66D4×C127—42no—
67D4×C127—82no—
68C2×C4×C127—21no—
69C2×C4×C127—42no—
70C2×C4×C127—42no—
71C2×C4×C127—84no—
72C2×C4×C127—81no—
73C2×C4×C127#12011yescharacteristic
74C8×C127—24no—
75C8×C127—42no—
76C8×C127C2×D812yes—
77C8×C127AGL(1,ℤ/8)12yes—
78C22×C127—42no—
79C22×C127#45311yescharacteristic
80C4×C127—22no—
81C4×C127—42no—
82C4×C127—82no—
83C4×C127—161no—
84C4×C127#44712yes—
85#12313C12711yesmaximal, characteristic, direct product factor, Sylow
86C2×C127—21no—
87C2×C127—82no—
88C2×C127#161911yescharacteristic
89#3293C2×C12711yescharacteristic
90#2214C2×C12712yescomplemented
91#1713C2×C12711yescharacteristic, complemented
92#1358C2×C12711yescharacteristic, complemented
93#1335C2×C12712yescomplemented
94C127#1231311yescharacteristic, direct product factor, Sylow
95#578—22no—
96#510—22no—
97#500—22no—
98#494C22×C12711yescharacteristic
99#494C22×C12711yescharacteristic
100#473—22no—
101#456—22no—
102#443C22×C12712yes—
103#364C22×C12712yes—
104#362—22no—
105C82C22×C12711yescharacteristic
1062−1+4—22no—
107#137—24no—
108#137—44no—
109#135—42no—
110#135D4×C12712yes—
111C2×Q16—22no—
112C2×Q16—22no—
113C2×Q16—42no—
114#131—22no—
115C4⋊Q8—21no—
116C4⋊Q8D4×C12712yes—
117Q32—42no—
118SD32—42no—
119M32—22no—
120#108D4×C12712yes—
121#107—44no—
122#104—22no—
123#101—22no—
124C4×C8—21no—
125C4×C8C23×C12711yescharacteristic
126#41—22no—
127#41—44no—
128#41—82no—
129C2×Q8—22no—
130C2×Q8—44no—
131C2×Q8—41no—
132Q16—44no—
133Q16—42no—
134Q16—42no—
135Q16—42no—
136Q16—42no—
137Q16D8×C12712yes—
138SD16—22no—
139SD16—44no—
140D8D8×C12712yes—
141M16—22no—
142M16—42no—
143C2×C8—22no—
144C2×C8—42no—
145C2×C8C2×D4×C12712yes—
146C4⋊C4—42no—
147C4⋊C4—42no—
148C42C2×D4×C12711yescharacteristic
149C16—42no—
150Q8—22no—
151Q8—44no—
152Q8—82no—
153Q8—82no—
154D4—22no—
155D4—42no—
156D4—82no—
157C2×C4—21no—
158C2×C4—42no—
159C2×C4—42no—
160C2×C4—84no—
161C2×C4—81no—
162C2×C4#120×C12711yescharacteristic
163C8—24no—
164C8—42no—
165C8C2×D8×C12712yes—
166C8AGL(1,ℤ/8)×C12712yes—
167C22—42no—
168C22#453×C12711yescharacteristic
169C4—22no—
170C4—42no—
171C4—82no—
172C4—161no—
173C4#447×C12712yes—
174C2—21no—
175C2—82no—
176C2C127×#161911yescharacteristic
subgroupG/H
center, socleC2×C127#1619
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3293C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C2×C4×C127 ◁ C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2214C2semidirect
3. C127×#1713C2semidirect
4. C127×#1358C2semidirect
5. C127×#1335C2semidirect
85. #12313C127direct (N × H)
90. #2214C2×C127semidirect
91. #1713C2×C127semidirect
92. #1358C2×C127semidirect
93. #1335C2×C127semidirect
94. C127#12313direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32932
C127×#22142
C127×#17132
C127×#13582
C127×#13352
#12313127