Finite Group Databasebeta

← C127×#12432 #44576105

Subgroups — C127×#12432 #44576105

Order 32512.

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 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×#2248C211yesmaximal, characteristic, complemented
2D4×C16×C127C211yesmaximal, characteristic, complemented
3C127×#1669C211yesmaximal, characteristic, complemented
4C127×#1667C211yesmaximal, characteristic
5C127×#1372C211yesmaximal, characteristic, complemented
6C127×#1362C211yesmaximal, characteristic, complemented
7C127×#1361C211yesmaximal, characteristic, complemented
8C2×SD32×C127—21no—
9C2×SD32×C127—21no—
10C22×C16×C127—21no—
11#494×C127C2211yescharacteristic
12#493×C127C2211yescharacteristic
13#474×C127—21no—
14#459×C127—21no—
15C8×D4×C127C2211yescharacteristic
16C4⋊C16×C127C2211yescharacteristic
17C22⋊C16×C127—21no—
18C4×C16×C127C2211yescharacteristic
19#335×C127C2211yescharacteristic
20#335×C127C2211yescharacteristic
21#332×C127—21no—
22#331×C127—21no—
23C2×Q16×C127—21no—
24C2×Q16×C127—21no—
25C2×D8×C127—21no—
26C2×D8×C127—21no—
27C2×D8×C127—41no—
28C22×C8×C127—21no—
29C4⋊Q8×C127—21no—
30#127×C127—21no—
31C4×D4×C127D411yescharacteristic
32SD32×C127—42no—
33SD32×C127—42no—
34C2×C16×C127—21no—
35C2×C16×C127—42no—
36C2×C16×C127D411yescharacteristic
37C2×C16×C127D411yescharacteristic
38#107×C127—41no—
39C4⋊C8×C127—21no—
40C4⋊C8×C127—21no—
41C4⋊C8×C127D411yescharacteristic
42#102×C127—41no—
43#102×C127—41no—
44C22⋊C8×C127—21no—
45C4×C8×C127C2311yescharacteristic
46C2×Q8×C127—41no—
47C2×D4×C127—41no—
48C2×D4×C127—41no—
49C2×D4×C127—41no—
50C2×D4×C127D811yescharacteristic
51C22×C4×C127—21no—
52Q16×C127—41no—
53Q16×C127—41no—
54Q16×C127—81no—
55D8×C127—41no—
56D8×C127—41no—
57D8×C127—81no—
58D8×C127—82no—
59C2×C8×C127—21no—
60C2×C8×C127—41no—
61C2×C8×C127—41no—
62C2×C8×C127—41no—
63C2×C8×C127—41no—
64C2×C8×C127C2×D411yescharacteristic
65C2×C8×C127#4111yescharacteristic
66C4⋊C4×C127—41no—
67C4⋊C4×C127—41no—
68C4⋊C4×C127D811yescharacteristic
69C22⋊C4×C127—21no—
70C42×C127C2×D411yescharacteristic
71C16×C127—21no—
72C16×C127—41no—
73C16×C127D812yescomplemented
74C23×C127—21no—
75C23×C127—81no—
76Q8×C127—81no—
77Q8×C127—81no—
78D4×C127—21no—
79D4×C127—81no—
80D4×C127—81no—
81D4×C127—81no—
82D4×C127—81no—
83D4×C127—161no—
84D4×C127SD3212yescomplemented
85C2×C4×C127—21no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—81no—
89C2×C4×C127#12111yescharacteristic
90C2×C4×C127C2×D811yescharacteristic
91C2×C4×C127#13511yescharacteristic
92C8×C127—21no—
93C8×C127—41no—
94C8×C127—81no—
95C8×C127—81no—
96C8×C127C2×D811yescharacteristic
97C8×C127AGL(1,ℤ/8)11yescharacteristic
98C22×C127—22no—
99C22×C127—41no—
100C22×C127—41no—
101C22×C127—161no—
102C22×C127—161no—
103C22×C127—161no—
104C22×C127#46611yescharacteristic
105C4×C127—21no—
106C4×C127—41no—
107C4×C127—161no—
108C4×C127#45911yescharacteristic
109C4×C127#47211yescharacteristic
110C4×C127C2×SD3211yescharacteristic
111C4×C127#50811yescharacteristic
112#12432C12711yesmaximal, characteristic, direct product factor, Sylow
113C2×C127—42no—
114C2×C127—321no—
115C2×C127#166711yescharacteristic
116C2×C127#221711yescharacteristic
117C2×C127#222311yescharacteristic
118#2248C2×C12711yescharacteristic, complemented
119D4×C16C2×C12711yescharacteristic, complemented
120#1669C2×C12711yescharacteristic, complemented
121#1667C2×C12711yescharacteristic
122#1372C2×C12711yescharacteristic, complemented
123#1362C2×C12711yescharacteristic, complemented
124#1361C2×C12711yescharacteristic, complemented
125C127#1243211yescharacteristic, direct product factor, Sylow
126C2×SD32—21no—
127C2×SD32—21no—
128C22×C16—21no—
129#494C22×C12711yescharacteristic
130#493C22×C12711yescharacteristic
131#474—21no—
132#459—21no—
133C8×D4C22×C12711yescharacteristic
134C4⋊C16C22×C12711yescharacteristic
135C22⋊C16—21no—
136C4×C16C22×C12711yescharacteristic
137#335C22×C12711yescharacteristic
138#335C22×C12711yescharacteristic
139#332—21no—
140#331—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—41no—
146C22×C8—21no—
147C4⋊Q8—21no—
148#127—21no—
149C4×D4D4×C12711yescharacteristic
150SD32—42no—
151SD32—42no—
152C2×C16—21no—
153C2×C16—42no—
154C2×C16D4×C12711yescharacteristic
155C2×C16D4×C12711yescharacteristic
156#107—41no—
157C4⋊C8—21no—
158C4⋊C8—21no—
159C4⋊C8D4×C12711yescharacteristic
160#102—41no—
161#102—41no—
162C22⋊C8—21no—
163C4×C8C23×C12711yescharacteristic
164C2×Q8—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4D8×C12711yescharacteristic
169C22×C4—21no—
170Q16—41no—
171Q16—41no—
172Q16—81no—
173D8—41no—
174D8—41no—
175D8—81no—
176D8—82no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8C2×D4×C12711yescharacteristic
183C2×C8#41×C12711yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4D8×C12711yescharacteristic
187C22⋊C4—21no—
188C42C2×D4×C12711yescharacteristic
189C16—21no—
190C16—41no—
191C16D8×C12712yescomplemented
192C23—21no—
193C23—81no—
194Q8—81no—
195Q8—81no—
196D4—21no—
197D4—81no—
198D4—81no—
199D4—81no—
200D4—81no—
201D4—161no—
202D4SD32×C12712yescomplemented
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—81no—
207C2×C4#121×C12711yescharacteristic
208C2×C4C2×D8×C12711yescharacteristic
209C2×C4#135×C12711yescharacteristic
210C8—21no—
211C8—41no—
212C8—81no—
213C8—81no—
214C8C2×D8×C12711yescharacteristic
215C8AGL(1,ℤ/8)×C12711yescharacteristic
216C22—22no—
217C22—41no—
218C22—41no—
219C22—161no—
220C22—161no—
221C22—161no—
222C22#466×C12711yescharacteristic
223C4—21no—
224C4—41no—
225C4—161no—
226C4#459×C12711yescharacteristic
227C4#472×C12711yescharacteristic
228C4C2×SD32×C12711yescharacteristic
229C4#508×C12711yescharacteristic
230C2—42no—
231C2—321no—
232C2C127×#166711yescharacteristic
233C2C127×#221711yescharacteristic
234C2C127×#222311yescharacteristic
subgroupG/H
center, socleC22×C127#466
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#1667C2
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C42×C127 ◁ C8×D4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C127×#2248C2semidirect
2. D4×C16×C127C2semidirect
3. C127×#1669C2semidirect
5. C127×#1372C2semidirect
6. C127×#1362C2semidirect
7. C127×#1361C2semidirect
73. C16×C127D8semidirect
84. D4×C127SD32semidirect
112. #12432C127direct (N × H)
118. #2248C2×C127semidirect
119. D4×C16C2×C127semidirect
120. #1669C2×C127semidirect
122. #1372C2×C127semidirect
123. #1362C2×C127semidirect
124. #1361C2×C127semidirect
125. C127#12432direct (N × H)
191. C16D8×C127semidirect
202. D4SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#22482
D4×C16×C1272
C127×#16692
C127×#16672
C127×#13722
C127×#13622
C127×#13612
#12432127