Finite Group Databasebeta

← C127×#12257 #44575930

Subgroups — C127×#12257 #44575930

Order 32512.

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

normal subgroups only

This group has 342 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
1C127×#2571C211yesmaximal, characteristic
2C127×#2060C211yesmaximal, characteristic
3C127×#1617C212yesmaximal
4C127×#1353C212yesmaximal
5C127×#1325C211yesmaximal, characteristic
6#544×C127C2211yescharacteristic
7#484×C127—22no—
8#428×C127—21no—
9#400×C127—21no—
10#400×C127C2212yes—
11#389×C127—22no—
12#384×C127C2211yescharacteristic
13#382×C127—22no—
14C4×C4⋊C4×C127—21no—
15#343×C127C2211yescharacteristic
16#339×C127—21no—
17#328×C127—22no—
18#324×C127C2212yes—
19C2×M16×C127—21no—
20C4⋊Q8×C127—41no—
21#125×C127—41no—
22C22⋊Q8×C127—22no—
23#117×C127—21no—
24C2×C4⋊C4×C127—21no—
25C2×C4⋊C4×C127—21no—
26C2×C4⋊C4×C127—22no—
27C2×C4⋊C4×C127—21no—
28C2×C4⋊C4×C127D411yescharacteristic
29C2×C4⋊C4×C127D412yes—
30C2×C4⋊C4×C127C2311yescharacteristic
31C2×C42×C127—22no—
32C2×C42×C127D411yescharacteristic
33#107×C127—41no—
34#107×C127—42no—
35#106×C127—41no—
36#103×C127—42no—
37C22⋊C8×C127—22no—
38#95×C127—21no—
39#95×C127—22no—
40#95×C127—22no—
41#95×C127—22no—
42#95×C127D412yes—
43C2×Q8×C127—22no—
44C22×C4×C127—21no—
45C22×C4×C127—22no—
46C22×C4×C127—22no—
47C22×C4×C127—21no—
48C22×C4×C127C2×D411yescharacteristic
49C22×C4×C127C2×D412yes—
50M16×C127—22no—
51M16×C127—41no—
52C2×C8×C127—41no—
53C2×C8×C127—42no—
54C4⋊C4×C127—21no—
55C4⋊C4×C127—21no—
56C4⋊C4×C127—22no—
57C4⋊C4×C127—41no—
58C4⋊C4×C127—42no—
59C4⋊C4×C127—41no—
60C4⋊C4×C127—42no—
61C4⋊C4×C127—44no—
62C4⋊C4×C127—42no—
63C4⋊C4×C127—42no—
64C4⋊C4×C127—81no—
65C22⋊C4×C127—41no—
66C42×C127—21no—
67C42×C127—44no—
68C42×C127—41no—
69C42×C127Q1612yes—
70C23×C127#12011yescharacteristic
71Q8×C127—82no—
72C2×C4×C127—22no—
73C2×C4×C127—22no—
74C2×C4×C127—22no—
75C2×C4×C127—21no—
76C2×C4×C127—21no—
77C2×C4×C127—22no—
78C2×C4×C127—42no—
79C2×C4×C127—42no—
80C2×C4×C127—42no—
81C2×C4×C127—42no—
82C2×C4×C127—42no—
83C2×C4×C127—42no—
84C2×C4×C127—42no—
85C2×C4×C127—41no—
86C2×C4×C127—41no—
87C2×C4×C127—41no—
88C2×C4×C127—42no—
89C2×C4×C127C2×Q1611yescharacteristic
90C2×C4×C127#13711yescharacteristic
91C8×C127—42no—
92C8×C127—82no—
93C22×C127—21no—
94C22×C127—21no—
95C22×C127#45111yescharacteristic
96C22×C127#45611yescharacteristic
97C22×C127B(4,𝔽2)11yescharacteristic
98C4×C127—24no—
99C4×C127—21no—
100C4×C127—21no—
101C4×C127—42no—
102C4×C127—82no—
103C4×C127—81no—
104C4×C127—81no—
105C4×C127—81no—
106C4×C127—82no—
107#12257C12711yesmaximal, characteristic, direct product factor, Sylow
108C2×C127—21no—
109C2×C127—21no—
110C2×C127#160511yescharacteristic
111C2×C127#219811yescharacteristic
112C2×C127#220411yescharacteristic
113#2571C2×C12711yescharacteristic
114#2060C2×C12711yescharacteristic
115#1617C2×C12712yes—
116#1353C2×C12712yes—
117#1325C2×C12711yescharacteristic
118C127#1225711yescharacteristic, direct product factor, Sylow
119#544C22×C12711yescharacteristic
120#484—22no—
121#428—21no—
122#400—21no—
123#400C22×C12712yes—
124#389—22no—
125#384C22×C12711yescharacteristic
126#382—22no—
127C4×C4⋊C4—21no—
128#343C22×C12711yescharacteristic
129#339—21no—
130#328—22no—
131#324C22×C12712yes—
132C2×M16—21no—
133C4⋊Q8—41no—
134#125—41no—
135C22⋊Q8—22no—
136#117—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4D4×C12711yescharacteristic
142C2×C4⋊C4D4×C12712yes—
143C2×C4⋊C4C23×C12711yescharacteristic
144C2×C42—22no—
145C2×C42D4×C12711yescharacteristic
146#107—41no—
147#107—42no—
148#106—41no—
149#103—42no—
150C22⋊C8—22no—
151#95—21no—
152#95—22no—
153#95—22no—
154#95—22no—
155#95D4×C12712yes—
156C2×Q8—22no—
157C22×C4—21no—
158C22×C4—22no—
159C22×C4—22no—
160C22×C4—21no—
161C22×C4C2×D4×C12711yescharacteristic
162C22×C4C2×D4×C12712yes—
163M16—22no—
164M16—41no—
165C2×C8—41no—
166C2×C8—42no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—22no—
170C4⋊C4—41no—
171C4⋊C4—42no—
172C4⋊C4—41no—
173C4⋊C4—42no—
174C4⋊C4—44no—
175C4⋊C4—42no—
176C4⋊C4—42no—
177C4⋊C4—81no—
178C22⋊C4—41no—
179C42—21no—
180C42—44no—
181C42—41no—
182C42Q16×C12712yes—
183C23#120×C12711yescharacteristic
184Q8—82no—
185C2×C4—22no—
186C2×C4—22no—
187C2×C4—22no—
188C2×C4—21no—
189C2×C4—21no—
190C2×C4—22no—
191C2×C4—42no—
192C2×C4—42no—
193C2×C4—42no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—42no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—42no—
202C2×C4C2×Q16×C12711yescharacteristic
203C2×C4#137×C12711yescharacteristic
204C8—42no—
205C8—82no—
206C22—21no—
207C22—21no—
208C22#451×C12711yescharacteristic
209C22#456×C12711yescharacteristic
210C22B(4,𝔽2)×C12711yescharacteristic
211C4—24no—
212C4—21no—
213C4—21no—
214C4—42no—
215C4—82no—
216C4—81no—
217C4—81no—
218C4—82no—
219C4—81no—
220C2—21no—
221C2—21no—
222C2C127×#160511yescharacteristic
223C2C127×#219811yescharacteristic
224C2C127×#220411yescharacteristic
subgroupG/H
center, socleC22×C127B(4,𝔽2)
commutator, Frattini, cosocleC2×C4⋊C4C23×C127
prime radicalC23×C127#120
LCSG ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C127 ◁ C23×C127 ◁ C2×C4⋊C4×C127 ◁ G

Complements

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

NH ≅ G/Ntype
107. #12257C127direct (N × H)
118. C127#12257direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#25712
C127×#20602
C127×#16172
C127×#13532
C127×#13252
#12257127