Finite Group Databasebeta

← #127×D27⋊C3 #72276183

Subgroups — #127×D27⋊C3 #72276183

Order 5184.

2100 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 2100 conjugacy classes of subgroups — too many to draw. Only its 340 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
1#72276094C211yesmaximal, characteristic, complemented
2#72276045C216yesmaximal, complemented
3#127×C27⋊C3C211yesmaximal, characteristic, complemented
4C42×D27⋊C3C211yesmaximal, characteristic, complemented
5C2×D4×D27⋊C3C216yesmaximal, complemented
6C3×D9×#127—31nomaximal
7#127×D27C311yesmaximal, characteristic, complemented
8C23×D27⋊C3C2214yes—
9C2×D4×C27⋊C3C2216yescomplemented
10C2×#6171⋊C3C22112yescomplemented
11D4×D27⋊C3—224no—
12C2×C4⋊(D27⋊C3)C2216yescomplemented
13C2×C4×D27⋊C3C2213yes—
14C42×C27⋊C3C2211yescharacteristic, complemented
15C4×(C27⋊C3)⋊C4C2213yescomplemented
16C2×C3×D4×D9—36no—
17C3×C9×#127D311yescharacteristic
18C3×#63502—36no—
19C3×#63436—31no—
20C3×C42×D9—31no—
21C2×D4×D27C2×C316yescomplemented
22C27×#127—21no—
23C27×#127C2×C311yescharacteristic, complemented
24#11720189C2×C316yescomplemented
25#11720123C2×C311yescharacteristic, complemented
26C42×D27C2×C311yescharacteristic, complemented
27C23×C27⋊C3C2314yes—
28C22×D27⋊C3—224no—
29C22×D27⋊C3C2314yes—
30C22×D27⋊C3C2311yescharacteristic
31D4×C27⋊C3—212no—
32C2×C4×C27⋊C3C2313yes—
33#6171⋊C3—224no—
34C2×(C27⋊C3)⋊C4C2313yes—
35C4⋊(D27⋊C3)—212no—
36C4×D27⋊C3D4112yescomplemented
37C3×D3×#127—91no—
38D9×#127—31no—
39C23×C3×D9—34no—
40C2×C3×D4×C9C2×D316yes—
41C2×C3×#611—312no—
42C3×D4×D9—624no—
43C2×C3×D36—36no—
44C2×C3×C4×D9—33no—
45C23×D27C22×C314yes—
46C3×C42×C9C2×D311yescharacteristic
47C3×C4×Dic9—33no—
48C2×D4×C27—26no—
49C2×D4×C27C22×C316yescomplemented
50C2×#6171C22×C3112yescomplemented
51D4×D27—224no—
52C2×D108C22×C316yescomplemented
53C2×C4×D27C22×C313yes—
54C42×C27—21no—
55C42×C27C22×C311yescharacteristic, complemented
56C4×Dic27C22×C313yescomplemented
57C22×C27⋊C3—212no—
58C22×C27⋊C3C2411yescharacteristic
59C2×D27⋊C3—212no—
60C2×D27⋊C3—416no—
61C2×D27⋊C3C2×D416yes—
62C4×C27⋊C3C2×D416yescomplemented
63(C27⋊C3)⋊C4C2×D416yescomplemented
64C2×C3×D3×D4—96no—
65C32×#127D911yescharacteristic
66C3×#940—96no—
67C3×#874—91no—
68C3×C42×D3—91no—
69C2×D4×D9—36no—
70C9×#127—21no—
71C9×#127C3×D311yescharacteristic
72#63502—36no—
73#63436—31no—
74C42×D9—31no—
75C23×C3×C9C22×D314yes—
76C22×C3×D9—31no—
77C22×C3×D9—34no—
78C22×C3×D9—624no—
79C3×D4×C9—212no—
80C2×C3×C4×C9C22×D313yes—
81C3×#611—624no—
82C2×C3×Dic9—33no—
83C3×D36—612no—
84C3×C4×D9—312no—
85C23×C27—24no—
86C23×C27C23×C314yes—
87C22×D27—224no—
88C22×D27C23×C314yes—
89C22×D27C23×C311yescharacteristic
90D4×C27—212no—
91D4×C27—412no—
92C2×C4×C27—23no—
93C2×C4×C27C23×C313yes—
94#6171—224no—
95C2×Dic27C23×C313yes—
96D108—212no—
97C4×D27C3×D4112yescomplemented
98C2×C3×#127—271no—
99D3×#127—91no—
100C2×C27⋊C3—44no—
101C2×C27⋊C3C22×D413yes—
102D27⋊C3—44no—
103D27⋊C3#12714yesdirect product factor
104C23×C3×D3—94no—
105C2×C32×D4C2×D916yes—
106C2×C3×#67—912no—
107C3×D3×D4—1824no—
108C2×C3×D12—96no—
109C2×C3×C4×D3—93no—
110C23×D9—34no—
111C32×C42C2×D911yescharacteristic
112C3×C4×Dic3—93no—
113C2×D4×C9—26no—
114C2×D4×C9C2×C3×D316yes—
115C2×#611—312no—
116D4×D9—624no—
117C2×D36—36no—
118C2×C4×D9—33no—
119C42×C9—21no—
120C42×C9C2×C3×D311yescharacteristic
121C4×Dic9—33no—
122C22×C3×C9—212no—
123C22×C3×C9C23×D311yescharacteristic
124C2×C3×D9—36no—
125C2×C3×D9—612no—
126C2×C3×D9—1216no—
127C3×C4×C9D3×D416yes—
128C3×Dic9—36no—
129C22×C27—21no—
130C22×C27—212no—
131C22×C27—412no—
132C22×C27C24×C311yescharacteristic
133C2×D27—212no—
134C2×D27—416no—
135C2×D27C2×C3×D416yes—
136C4×C27—26no—
137C4×C27C2×C3×D416yescomplemented
138Dic27C2×C3×D416yescomplemented
139C22×C3×D4—276no—
140C2×D3×D4—96no—
141C3×#127—31no—
142C3×#127—271no—
143C3×#127—276no—
144C3×#127C3×D911yescharacteristic
145C2×C3×C42—271no—
146#940—96no—
147#874—91no—
148C42×D3—91no—
149C27⋊C3C2×#12711yescharacteristic, complemented, Sylow
150C23×C32C22×D914yes—
151C22×C3×D3—91no—
152C22×C3×D3—94no—
153C22×C3×D3—1824no—
154C32×D4—212no—
155C2×C32×C4C22×D913yes—
156C3×#67—1824no—
157C2×C3×Dic3—93no—
158C3×D12—1812no—
159C3×C4×D3—912no—
160C23×C9—24no—
161C23×C9C22×C3×D314yes—
162C22×D9—31no—
163C22×D9—34no—
164C22×D9—624no—
165D4×C9—212no—
166D4×C9—412no—
167C2×C4×C9—23no—
168C2×C4×C9C22×C3×D313yes—
169#611—624no—
170C2×Dic9—33no—
171D36—612no—
172C4×D9—312no—
173C2×#127—271noSylow
174C2×C3×C9—44no—
175C2×C3×C9C2×D3×D413yes—
176C3×D9—34no—
177C3×D9—124no—
178C2×C27—23no—
179C2×C27—44no—
180C2×C27—84no—
181C2×C27C22×C3×D413yes—
182D27—44no—
183D27C3×#12714yescomplemented
184C24×C3—274no—
185C23×D3—94no—
186C2×C3×D4—36no—
187C2×C3×D4—276no—
188C2×C3×D4—2712no—
189C2×C3×D4—5424no—
190C2×C3×D4C2×C3×D916yes—
191C22×C3×C4—273no—
192C2×#67—912no—
193D3×D4—1824no—
194C2×D12—96no—
195C2×C4×D3—93no—
196C3×C42—31no—
197C3×C42—273no—
198C3×C42C2×C3×D911yescharacteristic
199C4×Dic3—93no—
200C22×C32—212no—
201C22×C32C23×D911yescharacteristic
202C2×C3×D3—96no—
203C2×C3×D3—1812no—
204C2×C3×D3—3616no—
205C32×C4D4×D916yes—
206C3×Dic3—96no—
207C22×C9—212no—
208C22×C9—21no—
209C22×C9—412no—
210C22×C9C23×C3×D311yescharacteristic
211C2×D9—36no—
212C2×D9—612no—
213C2×D9—1216no—
214C4×C9—26no—
215C4×C9C3×D3×D416yes—
216Dic9—36no—
217C22×D4—276no—
218#127—276no—
219#127—271no—
220#127D27⋊C311yescharacteristic, direct product factor
221C2×C42—271no—
222C3×C9D3×#12711yescharacteristic
223C27—21no—
224C27C2×C3×#12711yescharacteristic, complemented
225C23×C3—34no—
226C23×C3—271no—
227C23×C3—274no—
228C23×C3—5424no—
229C23×C3C22×C3×D914yes—
230C22×D3—91no—
231C22×D3—94no—
232C22×D3—1824no—
233C3×D4—212no—
234C3×D4—612no—
235C3×D4—5412no—
236C3×D4—5424no—
237C2×C3×C4—33no—
238C2×C3×C4—2712no—
239C2×C3×C4—273no—
240C2×C3×C4C22×C3×D913yes—
241#67—1824no—
242C2×Dic3—93no—
243D12—1812no—
244C4×D3—912no—
245C2×C32—44no—
246C2×C32C2×D4×D913yes—
247C3×D3—94no—
248C3×D3—364no—
249C2×C9—23no—
250C2×C9—44no—
251C2×C9—84no—
252C2×C9C2×C3×D3×D413yes—
253D9—34no—
254D9—124no—
255C24—274no—
256C2×D4—2712no—
257C2×D4—276no—
258C2×D4—5424no—
259C2×D4C2×D27⋊C316yescomplemented
260C22×C4—273no—
261C42—273no—
262C42C2×D27⋊C311yescharacteristic, complemented
263C22×C3—212no—
264C22×C3—31no—
265C22×C3—612no—
266C22×C3—276no—
267C22×C3—5412no—
268C22×C3—10816no—
269C22×C3C23×C3×D911yescharacteristic
270C2×D3—96no—
271C2×D3—1812no—
272C2×D3—3616no—
273C3×C4—36no—
274C3×C4—276no—
275C3×C4C3×D4×D916yes—
276Dic3—96no—
277C32D9×#12711yescharacteristic
278C9—21no—
279C9C3×D3×#12711yescharacteristic
280C23—271no—
281C23—274no—
282C23—5424no—
283C23C22×D27⋊C314yes—
284D4—212no—
285D4—5424no—
286D4—5412no—
287C2×C4—273no—
288C2×C4—2712no—
289C2×C4C22×D27⋊C313yes—
290C2×C3—33no—
291C2×C3—44no—
292C2×C3—124no—
293C2×C3—274no—
294C2×C3—1084no—
295C2×C3C2×C3×D4×D913yes—
296D3—94no—
297D3—364no—
298C22—212no—
299C22—276no—
300C22—5412no—
301C22—10816no—
302C22C23×D27⋊C311yescharacteristic
303C4—276no—
304C4D4×D27⋊C316yescomplemented
305C3—31no—
306C3C3×D9×#12711yescharacteristic
307C2—44no—
308C2—274no—
309C2—1084no—
310C2C2×D4×D27⋊C313yes—
subgroupG/H
centerC22C23×D27⋊C3
commutator, cosocleC22×C27C24×C3
FrattiniC22×C9C23×C3×D3
Fitting#127×C27⋊C3C2
socleC22×C3C23×C3×D9
hypercenter#127D27⋊C3
hypocenterC27C2×C3×#127
LCSG ▷ C22×C27 ▷ C27
UCSC1 ◁ C22 ◁ #127

Complements

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

NH ≅ G/Ntype
1. #72276094C2semidirect
2. #72276045C2semidirect
3. #127×C27⋊C3C2semidirect
4. C42×D27⋊C3C2semidirect
5. C2×D4×D27⋊C3C2semidirect
7. #127×D27C3semidirect
9. C2×D4×C27⋊C3C22semidirect
10. C2×#6171⋊C3C22semidirect
12. C2×C4⋊(D27⋊C3)C22semidirect
14. C42×C27⋊C3C22semidirect
15. C4×(C27⋊C3)⋊C4C22semidirect
21. C2×D4×D27C2×C3semidirect
23. C27×#127C2×C3semidirect
24. #11720189C2×C3semidirect
25. #11720123C2×C3semidirect
26. C42×D27C2×C3semidirect
36. C4×D27⋊C3D4semidirect
49. C2×D4×C27C22×C3semidirect
50. C2×#6171C22×C3semidirect
52. C2×D108C22×C3semidirect
55. C42×C27C22×C3semidirect
56. C4×Dic27C22×C3semidirect
62. C4×C27⋊C3C2×D4semidirect
63. (C27⋊C3)⋊C4C2×D4semidirect
97. C4×D27C3×D4semidirect
103. D27⋊C3#127direct (N × H)
137. C4×C27C2×C3×D4semidirect
138. Dic27C2×C3×D4semidirect
149. C27⋊C3C2×#127semidirect
183. D27C3×#127semidirect
220. #127D27⋊C3direct (N × H)
224. C27C2×C3×#127semidirect
259. C2×D4C2×D27⋊C3semidirect
262. C42C2×D27⋊C3semidirect
304. C4D4×D27⋊C3semidirect

Maximal subgroups

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

subgroupindex
#722760942
#722760452
#127×C27⋊C32
C42×D27⋊C32
C2×D4×D27⋊C32
C3×D9×#1273
#127×D273