Finite Group Databasebeta

← C127×#13684 #44577357

Subgroups — C127×#13684 #44577357

Order 32512.

378 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 378 conjugacy classes of subgroups — too many to draw. Only its 58 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×#3293C211yesmaximal, characteristic
2C127×#2205C212yesmaximal, complemented
3C127×#1633C211yesmaximal, characteristic, complemented
4C127×#1411C211yesmaximal, characteristic, complemented
5C127×#1405C212yesmaximal, complemented
6#578×C127—22no—
7Q82×C127C2211yescharacteristic
8#494×C127C2211yescharacteristic
9#477×C127—21no—
10#473×C127—22no—
11#456×C127—21no—
12#456×C127—22no—
13#456×C127C2212yes—
14#443×C127—21no—
15#356×C127—22no—
16#356×C127C2212yes—
17#333×C127C2211yescharacteristic
18Q8⋊C8×C127—21no—
192−1+4×C127—21no—
202−1+4×C127D412yes—
21#137×C127—22no—
22#137×C127—22no—
23#137×C127—42no—
24#137×C127—42no—
25#137×C127—42no—
26#135×C127—21no—
27#135×C127—42no—
28C2×Q16×C127—22no—
29C2×Q16×C127—42no—
30#131×C127—22no—
31C4⋊Q8×C127—22no—
32C4⋊Q8×C127—41no—
33C4⋊Q8×C127D411yescharacteristic
34C4⋊Q8×C127C2311yescharacteristic
35C4×Q8×C127—21no—
36C4×Q8×C127—41no—
37#108×C127—21no—
38#107×C127—41no—
39C4⋊C8×C127—21no—
40#104×C127—22no—
41#104×C127—42no—
42#103×C127—41no—
43#101×C127—21no—
44#101×C127—22no—
45#101×C127D412yes—
46C4×C8×C127D411yescharacteristic
47#41×C127—22no—
48#41×C127—42no—
49#41×C127—42no—
50#41×C127—42no—
51#41×C127—81no—
52C2×Q8×C127—21no—
53C2×Q8×C127—22no—
54C2×Q8×C127—21no—
55C2×Q8×C127—42no—
56C2×Q8×C127—42no—
57C2×Q8×C127C2×D412yes—
58Q16×C127—21no—
59Q16×C127—42no—
60Q16×C127—42no—
61Q16×C127—42no—
62Q16×C127—42no—
63Q16×C127—82no—
64SD16×C127—22no—
65SD16×C127—42no—
66SD16×C127—42no—
67SD16×C127—82no—
68D8×C127—21no—
69M16×C127—22no—
70M16×C127—42no—
71M16×C127—42no—
72C2×C8×C127—21no—
73C2×C8×C127—42no—
74C2×C8×C127—41no—
75C4⋊C4×C127—21no—
76C4⋊C4×C127—41no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—42no—
79C4⋊C4×C127—41no—
80C42×C127—22no—
81C42×C127—41no—
82C42×C127C2×D411yescharacteristic
83Q8×C127—22no—
84Q8×C127—22no—
85Q8×C127—42no—
86Q8×C127—41no—
87Q8×C127—42no—
88Q8×C127—42no—
89Q8×C127—81no—
90Q8×C127—82no—
91D4×C127—22no—
92D4×C127—42no—
93D4×C127—42no—
94D4×C127—81no—
95C2×C4×C127—21no—
96C2×C4×C127—22no—
97C2×C4×C127—42no—
98C2×C4×C127—41no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—82no—
102C2×C4×C127—82no—
103C2×C4×C127#12011yescharacteristic
104C8×C127—21no—
105C8×C127—21no—
106C8×C127—42no—
107C8×C127—81no—
108C8×C127—82no—
109C22×C127—42no—
110C22×C127B(4,𝔽2)11yescharacteristic
111C4×C127—21no—
112C4×C127—21no—
113C4×C127—21no—
114C4×C127—42no—
115C4×C127—42no—
116C4×C127—42no—
117C4×C127—81no—
118C4×C127—81no—
119#13684C12711yesmaximal, characteristic, direct product factor, Sylow
120C2×C127—21no—
121C2×C127—82no—
122C2×C127#219611yescharacteristic
123#3293C2×C12711yescharacteristic
124#2205C2×C12712yescomplemented
125#1633C2×C12711yescharacteristic, complemented
126#1411C2×C12711yescharacteristic, complemented
127#1405C2×C12712yescomplemented
128C127#1368411yescharacteristic, direct product factor, Sylow
129#578—22no—
130Q82C22×C12711yescharacteristic
131#494C22×C12711yescharacteristic
132#477—21no—
133#473—22no—
134#456—21no—
135#456—22no—
136#456C22×C12712yes—
137#443—21no—
138#356—22no—
139#356C22×C12712yes—
140#333C22×C12711yescharacteristic
141Q8⋊C8—21no—
1422−1+4—21no—
1432−1+4D4×C12712yes—
144#137—22no—
145#137—22no—
146#137—42no—
147#137—42no—
148#137—42no—
149#135—21no—
150#135—42no—
151C2×Q16—22no—
152C2×Q16—42no—
153#131—22no—
154C4⋊Q8—22no—
155C4⋊Q8—41no—
156C4⋊Q8D4×C12711yescharacteristic
157C4⋊Q8C23×C12711yescharacteristic
158C4×Q8—21no—
159C4×Q8—41no—
160#108—21no—
161#107—41no—
162C4⋊C8—21no—
163#104—22no—
164#104—42no—
165#103—41no—
166#101—21no—
167#101—22no—
168#101D4×C12712yes—
169C4×C8D4×C12711yescharacteristic
170#41—22no—
171#41—42no—
172#41—42no—
173#41—42no—
174#41—81no—
175C2×Q8—21no—
176C2×Q8—22no—
177C2×Q8—21no—
178C2×Q8—42no—
179C2×Q8—42no—
180C2×Q8C2×D4×C12712yes—
181Q16—21no—
182Q16—42no—
183Q16—42no—
184Q16—42no—
185Q16—42no—
186Q16—82no—
187SD16—22no—
188SD16—42no—
189SD16—42no—
190SD16—82no—
191D8—21no—
192M16—22no—
193M16—42no—
194M16—42no—
195C2×C8—21no—
196C2×C8—42no—
197C2×C8—41no—
198C4⋊C4—21no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C4⋊C4—42no—
202C4⋊C4—41no—
203C42—22no—
204C42—41no—
205C42C2×D4×C12711yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—42no—
209Q8—41no—
210Q8—42no—
211Q8—42no—
212Q8—81no—
213Q8—82no—
214D4—22no—
215D4—42no—
216D4—42no—
217D4—81no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—41no—
222C2×C4—41no—
223C2×C4—41no—
224C2×C4—82no—
225C2×C4—82no—
226C2×C4#120×C12711yescharacteristic
227C8—21no—
228C8—21no—
229C8—42no—
230C8—81no—
231C8—82no—
232C22—42no—
233C22B(4,𝔽2)×C12711yescharacteristic
234C4—21no—
235C4—21no—
236C4—21no—
237C4—42no—
238C4—42no—
239C4—42no—
240C4—81no—
241C4—81no—
242C2—21no—
243C2—82no—
244C2C127×#219611yescharacteristic
subgroupG/H
center, socleC2×C127#2196
commutator, Frattini, cosocleC4⋊Q8C23×C127
prime radicalC127×#3293C2
LCSG ▷ C4⋊Q8 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C127 ◁ C22×C127 ◁ C2×C4×C127 ◁ C4⋊Q8×C127 ◁ G

Complements

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

NH ≅ G/Ntype
2. C127×#2205C2semidirect
3. C127×#1633C2semidirect
4. C127×#1411C2semidirect
5. C127×#1405C2semidirect
119. #13684C127direct (N × H)
124. #2205C2×C127semidirect
125. #1633C2×C127semidirect
126. #1411C2×C127semidirect
127. #1405C2×C127semidirect
128. C127#13684direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32932
C127×#22052
C127×#16332
C127×#14112
C127×#14052
#13684127