Finite Group Databasebeta

← C127×#12323 #44575996

Subgroups — C127×#12323 #44575996

Order 32512.

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

normal subgroups only

This group has 374 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×#3283C211yesmaximal, characteristic
2C127×#2247C211yesmaximal, characteristic, complemented
3C127×#2208C211yesmaximal, characteristic, complemented
4C127×#1669C211yesmaximal, characteristic, complemented
5C127×#1362C211yesmaximal, characteristic, complemented
6C127×#1331C211yesmaximal, characteristic, complemented
7C127×D4⋊C16C211yesmaximal, characteristic, complemented
8C2×#135×C127—21no—
9#547×C127—21no—
10C2×Q32×C127—22no—
11C2×SD32×C127—21no—
12#494×C127C2211yescharacteristic
13#494×C127C2211yescharacteristic
14#474×C127—21no—
15#467×C127—21no—
16#461×C127—21no—
17#452×C127—22no—
18C4×D8×C127C2211yescharacteristic
19C8×D4×C127C2211yescharacteristic
20C4⋊C16×C127C2211yescharacteristic
21#358×C127—21no—
22C4×C16×C127C2211yescharacteristic
23#335×C127C2211yescharacteristic
24#332×C127—21no—
25C2×#41×C127—22no—
26#135×C127—42no—
27#135×C127—42no—
28C2×Q16×C127—21no—
29C2×Q16×C127—21no—
30C2×Q16×C127—21no—
31C2×Q16×C127D412yes—
32C2×SD16×C127—22no—
33C2×D8×C127D411yescharacteristic
34C22×C8×C127—21no—
35C4⋊Q8×C127—21no—
36C4⋊Q8×C127—21no—
37#123×C127—42no—
38C22⋊Q8×C127—22no—
39#121×C127—41no—
40C4×D4×C127—21no—
41C4×D4×C127D411yescharacteristic
42C2×C4⋊C4×C127—41no—
43Q32×C127—44no—
44SD32×C127—42no—
45C2×C16×C127—21no—
46C2×C16×C127—21no—
47#107×C127—41no—
48#107×C127D411yescharacteristic
49C4⋊C8×C127—21no—
50C4⋊C8×C127D411yescharacteristic
51#103×C127—22no—
52#102×C127—21no—
53#102×C127—41no—
54C22⋊C8×C127—21no—
55C4×C8×C127C2311yescharacteristic
56#41×C127—44no—
57#41×C127—82no—
58C2×Q8×C127—22no—
59C2×Q8×C127—41no—
60C2×D4×C127—21no—
61C2×D4×C127—42no—
62C2×D4×C127D811yescharacteristic
63C22×C4×C127—21no—
64C22×C4×C127—42no—
65C22×C4×C127—41no—
66Q16×C127—42no—
67Q16×C127—42no—
68Q16×C127—41no—
69Q16×C127—41no—
70Q16×C127—41no—
71Q16×C127—41no—
72SD16×C127—44no—
73D8×C127—21no—
74D8×C127D812yes—
75C2×C8×C127—21no—
76C2×C8×C127—41no—
77C2×C8×C127—41no—
78C2×C8×C127—41no—
79C2×C8×C127C2×D411yescharacteristic
80C2×C8×C127C2×D411yescharacteristic
81C4⋊C4×C127—21no—
82C4⋊C4×C127—42no—
83C4⋊C4×C127—41no—
84C4⋊C4×C127—41no—
85C4⋊C4×C127—41no—
86C4⋊C4×C127—41no—
87C4⋊C4×C127D811yescharacteristic
88C22⋊C4×C127—21no—
89C22⋊C4×C127—42no—
90C22⋊C4×C127—41no—
91C42×C127C2×D411yescharacteristic
92C16×C127—22no—
93C16×C127—41no—
94C23×C127—21no—
95C23×C127—41no—
96Q8×C127—42no—
97Q8×C127—42no—
98Q8×C127—81no—
99Q8×C127—81no—
100D4×C127—22no—
101D4×C127—21no—
102D4×C127—21no—
103D4×C127—41no—
104D4×C127—84no—
105C2×C4×C127—21no—
106C2×C4×C127—41no—
107C2×C4×C127—41no—
108C2×C4×C127—41no—
109C2×C4×C127—42no—
110C2×C4×C127—84no—
111C2×C4×C127—81no—
112C2×C4×C127—81no—
113C2×C4×C127—81no—
114C2×C4×C127#12011yescharacteristic
115C2×C4×C127C2×D811yescharacteristic
116C2×C4×C127AGL(1,ℤ/8)11yescharacteristic
117C8×C127—21no—
118C8×C127—41no—
119C8×C127—81no—
120C8×C127C2×D811yescharacteristic
121C8×C127AGL(1,ℤ/8)11yescharacteristic
122C22×C127—41no—
123C22×C127—42no—
124C22×C127—41no—
125C22×C127—41no—
126C22×C127—81no—
127C22×C127—81no—
128C22×C127#44711yescharacteristic
129C4×C127—21no—
130C4×C127—41no—
131C4×C127—82no—
132C4×C127—81no—
133C4×C127—161no—
134C4×C127#44711yescharacteristic
135C4×C127#45311yescharacteristic
136C4×C127#50811yescharacteristic
137C4×C127#51011yescharacteristic
138#12323C12711yesmaximal, characteristic, direct product factor, Sylow
139C2×C127—81no—
140C2×C127—82no—
141C2×C127#161911yescharacteristic
142C2×C127#218911yescharacteristic
143C2×C127#219511yescharacteristic
144#3283C2×C12711yescharacteristic
145#2247C2×C12711yescharacteristic, complemented
146#2208C2×C12711yescharacteristic, complemented
147#1669C2×C12711yescharacteristic, complemented
148#1362C2×C12711yescharacteristic, complemented
149#1331C2×C12711yescharacteristic, complemented
150D4⋊C16C2×C12711yescharacteristic, complemented
151C127#1232311yescharacteristic, direct product factor, Sylow
152C2×#135—21no—
153#547—21no—
154C2×Q32—22no—
155C2×SD32—21no—
156#494C22×C12711yescharacteristic
157#494C22×C12711yescharacteristic
158#474—21no—
159#467—21no—
160#461—21no—
161#452—22no—
162C4×D8C22×C12711yescharacteristic
163C8×D4C22×C12711yescharacteristic
164C4⋊C16C22×C12711yescharacteristic
165#358—21no—
166C4×C16C22×C12711yescharacteristic
167#335C22×C12711yescharacteristic
168#332—21no—
169C2×#41—22no—
170#135—42no—
171#135—42no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16D4×C12712yes—
176C2×SD16—22no—
177C2×D8D4×C12711yescharacteristic
178C22×C8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181#123—42no—
182C22⋊Q8—22no—
183#121—41no—
184C4×D4—21no—
185C4×D4D4×C12711yescharacteristic
186C2×C4⋊C4—41no—
187Q32—44no—
188SD32—42no—
189C2×C16—21no—
190C2×C16—21no—
191#107—41no—
192#107D4×C12711yescharacteristic
193C4⋊C8—21no—
194C4⋊C8D4×C12711yescharacteristic
195#103—22no—
196#102—21no—
197#102—41no—
198C22⋊C8—21no—
199C4×C8C23×C12711yescharacteristic
200#41—44no—
201#41—82no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×D4—21no—
205C2×D4—42no—
206C2×D4D8×C12711yescharacteristic
207C22×C4—21no—
208C22×C4—42no—
209C22×C4—41no—
210Q16—42no—
211Q16—42no—
212Q16—41no—
213Q16—41no—
214Q16—41no—
215Q16—41no—
216SD16—44no—
217D8—21no—
218D8D8×C12712yes—
219C2×C8—21no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8C2×D4×C12711yescharacteristic
224C2×C8C2×D4×C12711yescharacteristic
225C4⋊C4—21no—
226C4⋊C4—42no—
227C4⋊C4—41no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—41no—
231C4⋊C4D8×C12711yescharacteristic
232C22⋊C4—21no—
233C22⋊C4—42no—
234C22⋊C4—41no—
235C42C2×D4×C12711yescharacteristic
236C16—22no—
237C16—41no—
238C23—21no—
239C23—41no—
240Q8—42no—
241Q8—42no—
242Q8—81no—
243Q8—81no—
244D4—21no—
245D4—21no—
246D4—22no—
247D4—41no—
248D4—84no—
249C2×C4—21no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—84no—
255C2×C4—81no—
256C2×C4—81no—
257C2×C4—81no—
258C2×C4#120×C12711yescharacteristic
259C2×C4C2×D8×C12711yescharacteristic
260C2×C4AGL(1,ℤ/8)×C12711yescharacteristic
261C8—21no—
262C8—41no—
263C8—81no—
264C8C2×D8×C12711yescharacteristic
265C8AGL(1,ℤ/8)×C12711yescharacteristic
266C22—41no—
267C22—42no—
268C22—41no—
269C22—41no—
270C22—81no—
271C22—81no—
272C22#447×C12711yescharacteristic
273C4—21no—
274C4—41no—
275C4—82no—
276C4—81no—
277C4—161no—
278C4#447×C12711yescharacteristic
279C4#453×C12711yescharacteristic
280C4#508×C12711yescharacteristic
281C4#510×C12711yescharacteristic
282C2—81no—
283C2—82no—
284C2C127×#161911yescharacteristic
285C2C127×#218911yescharacteristic
286C2C127×#219511yescharacteristic
subgroupG/H
center, socleC22×C127#447
commutator, Frattini, cosocleC4×C8C23×C127
prime radicalC127×#3283C2
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×#2247C2semidirect
3. C127×#2208C2semidirect
4. C127×#1669C2semidirect
5. C127×#1362C2semidirect
6. C127×#1331C2semidirect
7. C127×D4⋊C16C2semidirect
138. #12323C127direct (N × H)
145. #2247C2×C127semidirect
146. #2208C2×C127semidirect
147. #1669C2×C127semidirect
148. #1362C2×C127semidirect
149. #1331C2×C127semidirect
150. D4⋊C16C2×C127semidirect
151. C127#12323direct (N × H)

Maximal subgroups

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

subgroupindex
C127×#32832
C127×#22472
C127×#22082
C127×#16692
C127×#13622
C127×#13312
C127×D4⋊C162
#12323127