Finite Group Databasebeta

← C127×#12429 #44576102

Subgroups — C127×#12429 #44576102

Order 32512.

240 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 82 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×#2209C211yesmaximal, characteristic, complemented
2C127×#2207C211yesmaximal, characteristic
3C127×#1709C211yesmaximal, characteristic
4C8×Q16×C127C211yesmaximal, characteristic, complemented
5C127×#1370C211yesmaximal, characteristic, complemented
6C127×#1362C211yesmaximal, characteristic, complemented
7C127×#1361C211yesmaximal, characteristic
8C2×Q32×C127—21no—
9C2×SD32×C127—21no—
10#494×C127C2211yescharacteristic
11#493×C127C2211yescharacteristic
12#486×C127—21no—
13#486×C127—21no—
14C8×Q8×C127—21no—
15C4×Q16×C127C2211yescharacteristic
16C4⋊C16×C127C2211yescharacteristic
17C4⋊C16×C127C2211yescharacteristic
18#358×C127—21no—
19#357×C127—21no—
20#335×C127C2211yescharacteristic
21Q8⋊C8×C127—21no—
22C82×C127C2211yescharacteristic
23C2×Q16×C127—21no—
24C2×Q16×C127—21no—
25C2×Q16×C127D411yescharacteristic
26C2×D8×C127—21no—
27C2×D8×C127—21no—
28C4⋊Q8×C127—21no—
29#127×C127—21no—
30C4×Q8×C127—21no—
31Q32×C127—42no—
32SD32×C127—42no—
33C2×C16×C127—21no—
34C2×C16×C127—21no—
35#107×C127D411yescharacteristic
36C4⋊C8×C127—21no—
37C4⋊C8×C127—41no—
38#103×C127—21no—
39#102×C127—42no—
40#102×C127—41no—
41#102×C127—41no—
42C4×C8×C127—41no—
43C4×C8×C127D411yescharacteristic
44C4×C8×C127D411yescharacteristic
45C4×C8×C127C2311yescharacteristic
46C2×Q8×C127—21no—
47C2×Q8×C127—41no—
48C2×D4×C127—41no—
49C2×D4×C127—41no—
50C2×D4×C127—41no—
51Q16×C127—21no—
52Q16×C127—41no—
53Q16×C127—41no—
54Q16×C127—81no—
55Q16×C127D812yes—
56D8×C127—41no—
57D8×C127—41no—
58D8×C127—81no—
59C2×C8×C127—21no—
60C2×C8×C127—41no—
61C2×C8×C127D811yescharacteristic
62C2×C8×C127D811yescharacteristic
63C2×C8×C127C2×D411yescharacteristic
64C2×C8×C127#4111yescharacteristic
65C4⋊C4×C127—21no—
66C4⋊C4×C127—41no—
67C4⋊C4×C127—41no—
68C4⋊C4×C127—41no—
69C42×C127—41no—
70C42×C127C2×D411yescharacteristic
71C16×C127—41no—
72C16×C127—41no—
73C23×C127—81no—
74Q8×C127—22no—
75Q8×C127—41no—
76Q8×C127—81no—
77Q8×C127—81no—
78D4×C127—82no—
79D4×C127—81no—
80D4×C127—81no—
81D4×C127—81no—
82D4×C127—81no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—81no—
86C2×C4×C127#12111yescharacteristic
87C2×C4×C127C2×D811yescharacteristic
88C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
89C8×C127—21no—
90C8×C127—21no—
91C8×C127—21no—
92C8×C127—41no—
93C8×C127—81no—
94C8×C127C2×D811yescharacteristic
95C8×C127#13511yescharacteristic
96C22×C127—161no—
97C22×C127—161no—
98C22×C127—161no—
99C22×C127#45911yescharacteristic
100C4×C127—21no—
101C4×C127—42no—
102C4×C127—81no—
103C4×C127—161no—
104C4×C127#46611yescharacteristic
105C4×C127#47211yescharacteristic
106C4×C127#50811yescharacteristic
107C4×C127#51011yescharacteristic
108#12429C12711yesmaximal, characteristic, direct product factor, Sylow
109C2×C127—321no—
110C2×C127#166711yescharacteristic
111C2×C127#221111yescharacteristic
112C2×C127#221211yescharacteristic
113#2209C2×C12711yescharacteristic, complemented
114#2207C2×C12711yescharacteristic
115#1709C2×C12711yescharacteristic
116C8×Q16C2×C12711yescharacteristic, complemented
117#1370C2×C12711yescharacteristic, complemented
118#1362C2×C12711yescharacteristic, complemented
119#1361C2×C12711yescharacteristic
120C127#1242911yescharacteristic, direct product factor, Sylow
121C2×Q32—21no—
122C2×SD32—21no—
123#494C22×C12711yescharacteristic
124#493C22×C12711yescharacteristic
125#486—21no—
126#486—21no—
127C8×Q8—21no—
128C4×Q16C22×C12711yescharacteristic
129C4⋊C16C22×C12711yescharacteristic
130C4⋊C16C22×C12711yescharacteristic
131#358—21no—
132#357—21no—
133#335C22×C12711yescharacteristic
134Q8⋊C8—21no—
135C82C22×C12711yescharacteristic
136C2×Q16—21no—
137C2×Q16—21no—
138C2×Q16D4×C12711yescharacteristic
139C2×D8—21no—
140C2×D8—21no—
141C4⋊Q8—21no—
142#127—21no—
143C4×Q8—21no—
144Q32—42no—
145SD32—42no—
146C2×C16—21no—
147C2×C16—21no—
148#107D4×C12711yescharacteristic
149C4⋊C8—21no—
150C4⋊C8—41no—
151#103—21no—
152#102—42no—
153#102—41no—
154#102—41no—
155C4×C8—41no—
156C4×C8D4×C12711yescharacteristic
157C4×C8D4×C12711yescharacteristic
158C4×C8C23×C12711yescharacteristic
159C2×Q8—21no—
160C2×Q8—41no—
161C2×D4—41no—
162C2×D4—41no—
163C2×D4—41no—
164Q16—21no—
165Q16—41no—
166Q16—41no—
167Q16—81no—
168Q16D8×C12712yes—
169D8—41no—
170D8—41no—
171D8—81no—
172C2×C8—21no—
173C2×C8—41no—
174C2×C8D8×C12711yescharacteristic
175C2×C8D8×C12711yescharacteristic
176C2×C8C2×D4×C12711yescharacteristic
177C2×C8#41×C12711yescharacteristic
178C4⋊C4—21no—
179C4⋊C4—41no—
180C4⋊C4—41no—
181C4⋊C4—41no—
182C42—41no—
183C42C2×D4×C12711yescharacteristic
184C16—41no—
185C16—41no—
186C23—81no—
187Q8—22no—
188Q8—41no—
189Q8—81no—
190Q8—81no—
191D4—82no—
192D4—81no—
193D4—81no—
194D4—81no—
195D4—81no—
196C2×C4—41no—
197C2×C4—41no—
198C2×C4—81no—
199C2×C4#121×C12711yescharacteristic
200C2×C4C2×D8×C12711yescharacteristic
201C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
202C8—21no—
203C8—21no—
204C8—21no—
205C8—41no—
206C8—81no—
207C8C2×D8×C12711yescharacteristic
208C8#135×C12711yescharacteristic
209C22—161no—
210C22—161no—
211C22—161no—
212C22#459×C12711yescharacteristic
213C4—21no—
214C4—42no—
215C4—81no—
216C4—161no—
217C4#466×C12711yescharacteristic
218C4#472×C12711yescharacteristic
219C4#508×C12711yescharacteristic
220C4#510×C12711yescharacteristic
221C2—321no—
222C2C127×#166711yescharacteristic
223C2C127×#221111yescharacteristic
224C2C127×#221211yescharacteristic
subgroupG/H
center, socleC22×C127#459
commutator, Frattini, cosocleC4×C8C23×C127
prime radical#493×C127C22
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C82×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2209C2semidirect
4. C8×Q16×C127C2semidirect
5. C127×#1370C2semidirect
6. C127×#1362C2semidirect
108. #12429C127direct (N × H)
113. #2209C2×C127semidirect
116. C8×Q16C2×C127semidirect
117. #1370C2×C127semidirect
118. #1362C2×C127semidirect
120. C127#12429direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#22092
C127×#22072
C127×#17092
C8×Q16×C1272
C127×#13702
C127×#13622
C127×#13612
#12429127