Finite Group Databasebeta

← C127×#12344 #44576017

Subgroups — C127×#12344 #44576017

Order 32512.

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

normal subgroups only

This group has 330 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×#3334C211yesmaximal, characteristic
2C127×#2241C211yesmaximal, characteristic, complemented
3C127×#2208C211yesmaximal, characteristic, complemented
4C127×#1669C211yesmaximal, characteristic, complemented
5C127×#1364C211yesmaximal, characteristic, complemented
6C127×#1331C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8#547×C127—21no—
9C2×SD32×C127—21no—
10#500×C127C2211yescharacteristic
11#494×C127C2211yescharacteristic
12#480×C127—22no—
13#474×C127—21no—
14#474×C127—21no—
15#466×C127—21no—
16C4×D8×C127C2211yescharacteristic
17C8×D4×C127C2211yescharacteristic
18C2×#107×C127—21no—
19C4⋊C16×C127C2211yescharacteristic
20#358×C127—21no—
21#358×C127—22no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#332×C127—21no—
25C2×#41×C127—41no—
26C2×Q16×C127—21no—
27C2×Q16×C127—21no—
28C2×D8×C127D411yescharacteristic
29C22×C8×C127—21no—
30C4⋊Q8×C127—21no—
31C4⋊Q8×C127—21no—
32#123×C127—42no—
33C22⋊Q8×C127—41no—
34#121×C127—22no—
35C4×D4×C127—21no—
36C4×D4×C127D411yescharacteristic
37C2×C4⋊C4×C127—22no—
38SD32×C127—42no—
39C2×C16×C127—21no—
40C2×C16×C127—21no—
41#107×C127—22no—
42#107×C127—21no—
43#107×C127—41no—
44#107×C127D412yes—
45#107×C127D411yescharacteristic
46C4⋊C8×C127—21no—
47C4⋊C8×C127D411yescharacteristic
48#102×C127—22no—
49#102×C127—21no—
50#102×C127—41no—
51C22⋊C8×C127—21no—
52C4×C8×C127C2311yescharacteristic
53#41×C127—82no—
54#41×C127—82no—
55C2×Q8×C127—41no—
56C2×Q8×C127—41no—
57C2×D4×C127—21no—
58C2×D4×C127—42no—
59C2×D4×C127D811yescharacteristic
60C22×C4×C127—21no—
61C22×C4×C127—42no—
62C22×C4×C127—41no—
63Q16×C127—42no—
64Q16×C127—41no—
65Q16×C127—41no—
66D8×C127—21no—
67D8×C127—21no—
68C2×C8×C127—21no—
69C2×C8×C127—41no—
70C2×C8×C127—41no—
71C2×C8×C127—41no—
72C2×C8×C127C2×D411yescharacteristic
73C2×C8×C127C2×D411yescharacteristic
74C4⋊C4×C127—21no—
75C4⋊C4×C127—22no—
76C4⋊C4×C127—42no—
77C4⋊C4×C127—41no—
78C4⋊C4×C127—41no—
79C4⋊C4×C127—41no—
80C4⋊C4×C127D811yescharacteristic
81C22⋊C4×C127—21no—
82C22⋊C4×C127—42no—
83C22⋊C4×C127—41no—
84C42×C127C2×D411yescharacteristic
85C16×C127—41no—
86C16×C127—41no—
87C23×C127—21no—
88C23×C127—41no—
89Q8×C127—81no—
90Q8×C127—81no—
91Q8×C127—81no—
92Q8×C127—81no—
93D4×C127—21no—
94D4×C127—41no—
95D4×C127—41no—
96D4×C127—84no—
97D4×C127SD3212yescomplemented
98C2×C4×C127—21no—
99C2×C4×C127—41no—
100C2×C4×C127—41no—
101C2×C4×C127—41no—
102C2×C4×C127—42no—
103C2×C4×C127—84no—
104C2×C4×C127—81no—
105C2×C4×C127—81no—
106C2×C4×C127—81no—
107C2×C4×C127#12011yescharacteristic
108C2×C4×C127C2×D811yescharacteristic
109C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
110C8×C127—21no—
111C8×C127—41no—
112C8×C127—81no—
113C8×C127#13511yescharacteristic
114C8×C127AGL(1,ℤ/8)11yescharacteristic
115C22×C127—22no—
116C22×C127—41no—
117C22×C127—41no—
118C22×C127—81no—
119C22×C127—81no—
120C22×C127—81no—
121C22×C127#44711yescharacteristic
122C4×C127—21no—
123C4×C127—41no—
124C4×C127—81no—
125C4×C127—82no—
126C4×C127—161no—
127C4×C127#44911yescharacteristic
128C4×C127#45411yescharacteristic
129C4×C127C2×SD3211yescharacteristic
130C4×C127#50911yescharacteristic
131#12344C12711yesmaximal, characteristic, direct product factor, Sylow
132C2×C127—42no—
133C2×C127—161no—
134C2×C127#162511yescharacteristic
135C2×C127#218711yescharacteristic
136C2×C127#219511yescharacteristic
137#3334C2×C12711yescharacteristic
138#2241C2×C12711yescharacteristic, complemented
139#2208C2×C12711yescharacteristic, complemented
140#1669C2×C12711yescharacteristic, complemented
141#1364C2×C12711yescharacteristic, complemented
142#1331C2×C12711yescharacteristic, complemented
143D4⋊C16C2×C12711yescharacteristic, complemented
144C127#1234411yescharacteristic, direct product factor, Sylow
145#547—21no—
146C2×SD32—21no—
147#500C22×C12711yescharacteristic
148#494C22×C12711yescharacteristic
149#480—22no—
150#474—21no—
151#474—21no—
152#466—21no—
153C4×D8C22×C12711yescharacteristic
154C8×D4C22×C12711yescharacteristic
155C2×#107—21no—
156C4⋊C16C22×C12711yescharacteristic
157#358—21no—
158#358—22no—
159C4×C16C22×C12711yescharacteristic
160#335C22×C12711yescharacteristic
161#332—21no—
162C2×#41—41no—
163C2×Q16—21no—
164C2×Q16—21no—
165C2×D8D4×C12711yescharacteristic
166C22×C8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169#123—42no—
170C22⋊Q8—41no—
171#121—22no—
172C4×D4—21no—
173C4×D4D4×C12711yescharacteristic
174C2×C4⋊C4—22no—
175SD32—42no—
176C2×C16—21no—
177C2×C16—21no—
178#107—22no—
179#107—21no—
180#107—41no—
181#107D4×C12711yescharacteristic
182#107D4×C12712yes—
183C4⋊C8—21no—
184C4⋊C8D4×C12711yescharacteristic
185#102—22no—
186#102—21no—
187#102—41no—
188C22⋊C8—21no—
189C4×C8C23×C12711yescharacteristic
190#41—82no—
191#41—82no—
192C2×Q8—41no—
193C2×Q8—41no—
194C2×D4—21no—
195C2×D4—42no—
196C2×D4D8×C12711yescharacteristic
197C22×C4—21no—
198C22×C4—42no—
199C22×C4—41no—
200Q16—42no—
201Q16—41no—
202Q16—41no—
203D8—21no—
204D8—21no—
205C2×C8—21no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8C2×D4×C12711yescharacteristic
210C2×C8C2×D4×C12711yescharacteristic
211C4⋊C4—21no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—41no—
215C4⋊C4—41no—
216C4⋊C4—41no—
217C4⋊C4D8×C12711yescharacteristic
218C22⋊C4—21no—
219C22⋊C4—42no—
220C22⋊C4—41no—
221C42C2×D4×C12711yescharacteristic
222C16—41no—
223C16—41no—
224C23—21no—
225C23—41no—
226Q8—81no—
227Q8—81no—
228Q8—81no—
229Q8—81no—
230D4—21no—
231D4—41no—
232D4—41no—
233D4—84no—
234D4SD32×C12712yescomplemented
235C2×C4—21no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—42no—
240C2×C4—84no—
241C2×C4—81no—
242C2×C4—81no—
243C2×C4—81no—
244C2×C4#120×C12711yescharacteristic
245C2×C4C2×D8×C12711yescharacteristic
246C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
247C8—21no—
248C8—41no—
249C8—81no—
250C8#135×C12711yescharacteristic
251C8AGL(1,ℤ/8)×C12711yescharacteristic
252C22—22no—
253C22—41no—
254C22—41no—
255C22—81no—
256C22—81no—
257C22—81no—
258C22#447×C12711yescharacteristic
259C4—21no—
260C4—41no—
261C4—81no—
262C4—82no—
263C4—161no—
264C4#449×C12711yescharacteristic
265C4#454×C12711yescharacteristic
266C4C2×SD32×C12711yescharacteristic
267C4#509×C12711yescharacteristic
268C2—42no—
269C2—161no—
270C2C127×#162511yescharacteristic
271C2C127×#218711yescharacteristic
272C2C127×#219511yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3334C2
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
2. C127×#2241C2semidirect
3. C127×#2208C2semidirect
4. C127×#1669C2semidirect
5. C127×#1364C2semidirect
6. C127×#1331C2semidirect
7. C127×D4⋊C16C2semidirect
97. D4×C127SD32semidirect
131. #12344C127direct (N × H)
138. #2241C2×C127semidirect
139. #2208C2×C127semidirect
140. #1669C2×C127semidirect
141. #1364C2×C127semidirect
142. #1331C2×C127semidirect
143. D4⋊C16C2×C127semidirect
144. C127#12344direct (N × H)
234. D4SD32×C127semidirect

Maximal subgroups

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

subgroupindex
C127×#33342
C127×#22412
C127×#22082
C127×#16692
C127×#13642
C127×#13312
C127×D4⋊C162
#12344127