Finite Group Databasebeta

← C127×#13526 #44577199

Subgroups — C127×#13526 #44577199

Order 32512.

394 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 138 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
1C2×#366×C127C212yesmaximal, complemented
2C2×#357×C127C212yesmaximal
3C127×#1970C211yesmaximal, characteristic
4C127×#1845C211yesmaximal, characteristic, complemented
5C127×#1374C211yesmaximal, characteristic, complemented
6C22×D8×C127C2211yescharacteristic
7C22×C16×C127C2212yes—
8C2×#107×C127—21no—
9C2×#107×C127C2212yes—
10C2×#107×C127C2211yescharacteristic
11C2×#102×C127—21no—
12C2×C4×C8×C127C2211yescharacteristic
13#390×C127—21no—
14#384×C127—21no—
15#366×C127—24no—
16#357×C127—24no—
17C22×D4×C127—21no—
18C2×D8×C127—22no—
19C2×D8×C127—22no—
20C2×D8×C127—21no—
21C2×D8×C127Q812yes—
22C22×C8×C127D411yescharacteristic
23C22×C8×C127C2311yescharacteristic
24C2×C4⋊C4×C127—22no—
25C2×C4⋊C4×C127—21no—
26C2×C4⋊C4×C127—42no—
27C2×C22⋊C4×C127—42no—
28C2×C42×C127D411yescharacteristic
29C2×C16×C127—22no—
30C2×C16×C127—22no—
31C2×C16×C127D414yes—
32#107×C127—22no—
33#107×C127—22no—
34#107×C127—22no—
35#107×C127—42no—
36#102×C127—42no—
37C4×C8×C127—22no—
38C24×C127—41no—
39C2×D4×C127—22no—
40C2×D4×C127—41no—
41C2×D4×C127—41no—
42C2×D4×C127—41no—
43C2×D4×C127—41no—
44C2×D4×C127—41no—
45C22×C4×C127—41no—
46C22×C4×C127—42no—
47C22×C4×C127D812yes—
48C22×C4×C127C2×D411yescharacteristic
49D8×C127—24no—
50D8×C127—42no—
51C2×C8×C127—21no—
52C2×C8×C127—21no—
53C2×C8×C127D812yes—
54C2×C8×C127C2×D412yes—
55C2×C8×C127C2×Q811yescharacteristic
56C2×C8×C127#4112yes—
57C2×C8×C127#4111yescharacteristic
58C4⋊C4×C127—42no—
59C4⋊C4×C127—41no—
60C4⋊C4×C127—41no—
61C4⋊C4×C127—42no—
62C4⋊C4×C127—84no—
63C22⋊C4×C127—84no—
64C42×C127—21no—
65C42×C127—21no—
66C16×C127—24no—
67C23×C127—42no—
68C23×C127—81no—
69C23×C127—81no—
70C23×C127—82no—
71C23×C127—82no—
72C23×C127C2×D811yescharacteristic
73D4×C127—42no—
74D4×C127—42no—
75D4×C127—81no—
76D4×C127—81no—
77C2×C4×C127—22no—
78C2×C4×C127—22no—
79C2×C4×C127—81no—
80C2×C4×C127—82no—
81C2×C4×C127—82no—
82C2×C4×C127—84no—
83C2×C4×C127D1614yes—
84C2×C4×C127C22⋊Q811yescharacteristic
85C2×C4×C127#12311yescharacteristic
86C2×C4×C127#12711yescharacteristic
87C2×C4×C127C2×D811yescharacteristic
88C2×C4×C127#13711yescharacteristic
89C2×C4×C127#13711yescharacteristic
90C8×C127—22no—
91C8×C127C22⋊Q812yes—
92C8×C127#12312yes—
93C22×C127—82no—
94C22×C127—84no—
95C22×C127—162no—
96C22×C127—161no—
97C22×C127—161no—
98C22×C127#47411yescharacteristic
99C22×C127#48011yescharacteristic
100C22×C127#49311yescharacteristic
101C22×C127C2×D1612yes—
102C22×C127#51012yes—
103C4×C127—24no—
104C4×C127—161no—
105C4×C127—162no—
106C4×C127#39911yescharacteristic
107C4×C127#47411yescharacteristic
108C4×C127#48011yescharacteristic
109C4×C127#49711yescharacteristic
110#13526C12711yesmaximal, characteristic, direct product factor, Sylow
111C2×C127—162no—
112C2×C127#209311yescharacteristic
113C2×C127#222412yes—
114C2×C127#223212yes—
115C2×C127#224611yescharacteristic
116C2×C127#225111yescharacteristic
117C2×#366C2×C12712yescomplemented
118C2×#357C2×C12712yes—
119#1970C2×C12711yescharacteristic
120#1845C2×C12711yescharacteristic, complemented
121#1374C2×C12711yescharacteristic, complemented
122C127#1352611yescharacteristic, direct product factor, Sylow
123C22×D8C22×C12711yescharacteristic
124C22×C16C22×C12712yes—
125C2×#107—21no—
126C2×#107C22×C12712yes—
127C2×#107C22×C12711yescharacteristic
128C2×#102—21no—
129C2×C4×C8C22×C12711yescharacteristic
130#390—21no—
131#384—21no—
132#366—24no—
133#357—24no—
134C22×D4—21no—
135C2×D8—22no—
136C2×D8—22no—
137C2×D8—21no—
138C2×D8Q8×C12712yes—
139C22×C8D4×C12711yescharacteristic
140C22×C8C23×C12711yescharacteristic
141C2×C4⋊C4—22no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—42no—
144C2×C22⋊C4—42no—
145C2×C42D4×C12711yescharacteristic
146C2×C16—22no—
147C2×C16—22no—
148C2×C16D4×C12714yes—
149#107—22no—
150#107—22no—
151#107—22no—
152#107—42no—
153#102—42no—
154C4×C8—22no—
155C24—41no—
156C2×D4—22no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—41no—
161C2×D4—41no—
162C22×C4—41no—
163C22×C4—42no—
164C22×C4D8×C12712yes—
165C22×C4C2×D4×C12711yescharacteristic
166D8—24no—
167D8—42no—
168C2×C8—21no—
169C2×C8—21no—
170C2×C8D8×C12712yes—
171C2×C8C2×D4×C12712yes—
172C2×C8C2×Q8×C12711yescharacteristic
173C2×C8#41×C12712yes—
174C2×C8#41×C12711yescharacteristic
175C4⋊C4—42no—
176C4⋊C4—41no—
177C4⋊C4—41no—
178C4⋊C4—42no—
179C4⋊C4—84no—
180C22⋊C4—84no—
181C42—21no—
182C42—21no—
183C16—24no—
184C23—42no—
185C23—82no—
186C23—82no—
187C23—81no—
188C23—81no—
189C23C2×D8×C12711yescharacteristic
190D4—42no—
191D4—42no—
192D4—81no—
193D4—81no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—82no—
197C2×C4—84no—
198C2×C4—81no—
199C2×C4—82no—
200C2×C4D16×C12714yes—
201C2×C4C22⋊Q8×C12711yescharacteristic
202C2×C4#123×C12711yescharacteristic
203C2×C4#127×C12711yescharacteristic
204C2×C4C2×D8×C12711yescharacteristic
205C2×C4#137×C12711yescharacteristic
206C2×C4#137×C12711yescharacteristic
207C8—22no—
208C8C22⋊Q8×C12712yes—
209C8#123×C12712yes—
210C22—82no—
211C22—84no—
212C22—162no—
213C22—161no—
214C22—161no—
215C22#474×C12711yescharacteristic
216C22#480×C12711yescharacteristic
217C22#493×C12711yescharacteristic
218C22C2×D16×C12712yes—
219C22#510×C12712yes—
220C4—24no—
221C4—161no—
222C4—162no—
223C4#399×C12711yescharacteristic
224C4#474×C12711yescharacteristic
225C4#480×C12711yescharacteristic
226C4#497×C12711yescharacteristic
227C2—162no—
228C2C127×#209311yescharacteristic
229C2C127×#222412yes—
230C2C127×#223212yes—
231C2C127×#224611yescharacteristic
232C2C127×#225111yescharacteristic
subgroupG/H
center, socleC23×C127C2×D8
commutator, Frattini, cosocleC22×C8C23×C127
prime radicalC22×D8×C127C22
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C127 ◁ C2×C42×C127 ◁ C2×C4×C8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×#366×C127C2semidirect
4. C127×#1845C2semidirect
5. C127×#1374C2semidirect
110. #13526C127direct (N × H)
117. C2×#366C2×C127semidirect
120. #1845C2×C127semidirect
121. #1374C2×C127semidirect
122. C127#13526direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#366×C1272
C2×#357×C1272
C127×#19702
C127×#18452
C127×#13742
#13526127