Finite Group Databasebeta

← C127×#12225 #44575898

Subgroups — C127×#12225 #44575898

Order 32512.

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

normal subgroups only

This group has 336 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×#2809C211yesmaximal, characteristic
2C127×#2028C211yesmaximal, characteristic
3C127×#1617C212yesmaximal
4C127×#1353C212yesmaximal
5C127×#1319C211yesmaximal, characteristic
6#544×C127C2211yescharacteristic
7#484×C127—22no—
8#419×C127—21no—
9#400×C127C2212yes—
10#398×C127—22no—
11#391×C127—21no—
12#391×C127C2211yescharacteristic
13#389×C127—21no—
14#387×C127—22no—
15#343×C127C2211yescharacteristic
16#339×C127—21no—
17#328×C127—22no—
18#324×C127C2212yes—
19C22×Q8×C127D411yescharacteristic
20C2×M16×C127—21no—
21C4⋊Q8×C127—41no—
22#125×C127—41no—
23C22⋊Q8×C127—22no—
24#117×C127—21no—
25C2×C4⋊C4×C127—22no—
26C2×C4⋊C4×C127—21no—
27C2×C4⋊C4×C127—21no—
28C2×C4⋊C4×C127—21no—
29C2×C4⋊C4×C127D412yes—
30C2×C4⋊C4×C127C2311yescharacteristic
31C2×C42×C127—21no—
32C2×C42×C127D411yescharacteristic
33#107×C127—42no—
34#103×C127—41no—
35#103×C127—41no—
36#103×C127—42no—
37C22⋊C8×C127—22no—
38#95×C127—22no—
39#95×C127—22no—
40#95×C127—22no—
41#95×C127—22no—
42#95×C127D412yes—
43C2×Q8×C127—21no—
44C2×Q8×C127—22no—
45C2×Q8×C127—41no—
46C2×Q8×C127—41no—
47C2×Q8×C127Q1612yes—
48C22×C4×C127—21no—
49C22×C4×C127—21no—
50C22×C4×C127—22no—
51C22×C4×C127—22no—
52C22×C4×C127C2×D411yescharacteristic
53C22×C4×C127C2×D412yes—
54M16×C127—41no—
55M16×C127—41no—
56C2×C8×C127—42no—
57C2×C8×C127—41no—
58C4⋊C4×C127—21no—
59C4⋊C4×C127—21no—
60C4⋊C4×C127—22no—
61C4⋊C4×C127—41no—
62C4⋊C4×C127—42no—
63C4⋊C4×C127—44no—
64C4⋊C4×C127—42no—
65C4⋊C4×C127—42no—
66C4⋊C4×C127—81no—
67C4⋊C4×C127—81no—
68C22⋊C4×C127—41no—
69C42×C127—41no—
70C42×C127—41no—
71C42×C127—81no—
72C23×C127#12011yescharacteristic
73Q8×C127—42no—
74Q8×C127—81no—
75Q8×C127—82no—
76C2×C4×C127—21no—
77C2×C4×C127—21no—
78C2×C4×C127—22no—
79C2×C4×C127—22no—
80C2×C4×C127—22no—
81C2×C4×C127—41no—
82C2×C4×C127—41no—
83C2×C4×C127—41no—
84C2×C4×C127—41no—
85C2×C4×C127—42no—
86C2×C4×C127—42no—
87C2×C4×C127—42no—
88C2×C4×C127—42no—
89C2×C4×C127—42no—
90C2×C4×C127—42no—
91C2×C4×C127—41no—
92C2×C4×C127—41no—
93C2×C4×C127—42no—
94C2×C4×C127C2×Q1611yescharacteristic
95C2×C4×C127#13711yescharacteristic
96C8×C127—81no—
97C8×C127—82no—
98C22×C127—21no—
99C22×C127—21no—
100C22×C127#45111yescharacteristic
101C22×C127#45511yescharacteristic
102C22×C127B(4,𝔽2)11yescharacteristic
103C4×C127—21no—
104C4×C127—21no—
105C4×C127—42no—
106C4×C127—42no—
107C4×C127—42no—
108C4×C127—82no—
109C4×C127—81no—
110C4×C127—81no—
111C4×C127—82no—
112#12225C12711yesmaximal, characteristic, direct product factor, Sylow
113C2×C127—21no—
114C2×C127—21no—
115C2×C127#160211yescharacteristic
116C2×C127#220111yescharacteristic
117C2×C127#220411yescharacteristic
118#2809C2×C12711yescharacteristic
119#2028C2×C12711yescharacteristic
120#1617C2×C12712yes—
121#1353C2×C12712yes—
122#1319C2×C12711yescharacteristic
123C127#1222511yescharacteristic, direct product factor, Sylow
124#544C22×C12711yescharacteristic
125#484—22no—
126#419—21no—
127#400C22×C12712yes—
128#398—22no—
129#391—21no—
130#391C22×C12711yescharacteristic
131#389—21no—
132#387—22no—
133#343C22×C12711yescharacteristic
134#339—21no—
135#328—22no—
136#324C22×C12712yes—
137C22×Q8D4×C12711yescharacteristic
138C2×M16—21no—
139C4⋊Q8—41no—
140#125—41no—
141C22⋊Q8—22no—
142#117—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4D4×C12712yes—
148C2×C4⋊C4C23×C12711yescharacteristic
149C2×C42—21no—
150C2×C42D4×C12711yescharacteristic
151#107—42no—
152#103—41no—
153#103—41no—
154#103—42no—
155C22⋊C8—22no—
156#95—22no—
157#95—22no—
158#95—22no—
159#95—22no—
160#95D4×C12712yes—
161C2×Q8—21no—
162C2×Q8—22no—
163C2×Q8—41no—
164C2×Q8—41no—
165C2×Q8Q16×C12712yes—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4C2×D4×C12711yescharacteristic
171C22×C4C2×D4×C12712yes—
172M16—41no—
173M16—41no—
174C2×C8—42no—
175C2×C8—41no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—22no—
179C4⋊C4—41no—
180C4⋊C4—42no—
181C4⋊C4—44no—
182C4⋊C4—42no—
183C4⋊C4—42no—
184C4⋊C4—81no—
185C4⋊C4—81no—
186C22⋊C4—41no—
187C42—41no—
188C42—41no—
189C42—81no—
190C23#120×C12711yescharacteristic
191Q8—42no—
192Q8—81no—
193Q8—82no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—41no—
203C2×C4—42no—
204C2×C4—42no—
205C2×C4—42no—
206C2×C4—42no—
207C2×C4—42no—
208C2×C4—42no—
209C2×C4—41no—
210C2×C4—41no—
211C2×C4—42no—
212C2×C4C2×Q16×C12711yescharacteristic
213C2×C4#137×C12711yescharacteristic
214C8—81no—
215C8—82no—
216C22—21no—
217C22—21no—
218C22#451×C12711yescharacteristic
219C22#455×C12711yescharacteristic
220C22B(4,𝔽2)×C12711yescharacteristic
221C4—21no—
222C4—21no—
223C4—42no—
224C4—42no—
225C4—42no—
226C4—82no—
227C4—81no—
228C4—82no—
229C4—81no—
230C2—21no—
231C2—21no—
232C2C127×#160211yescharacteristic
233C2C127×#220111yescharacteristic
234C2C127×#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
112. #12225C127direct (N × H)
123. C127#12225direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#28092
C127×#20282
C127×#16172
C127×#13532
C127×#13192
#12225127