Subgroups — #2318×C₂₂₇ #44396274

Order 29056.

568 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic
2	#388×C₂₂₇	C₂	1	2	yes	maximal, complemented
3	#387×C₂₂₇	C₂	1	2	yes	maximal, complemented
4	#386×C₂₂₇	C₂	1	2	yes	maximal
5	#385×C₂₂₇	C₂	1	2	yes	maximal, complemented
6	#383×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#382×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#381×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	#380×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₂²⋊C₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#95×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₂₂₇	C₂²	1	2	yes	—
14	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
16	C₄×D₄×C₂₂₇	C₄	1	8	yes	complemented
17	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
19	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	2	yes	—
21	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	2	yes	—
23	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
25	C₂×C₄²×C₂₂₇	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
27	#95×C₂₂₇	C₂²	1	1	yes	characteristic
28	#95×C₂₂₇	C₂²	1	1	yes	characteristic
29	#95×C₂₂₇	C₂²	1	2	yes	—
30	#95×C₂₂₇	C₂²	1	1	yes	characteristic
31	#95×C₂₂₇	C₂²	1	1	yes	characteristic
32	#95×C₂₂₇	C₂²	1	2	yes	—
33	#95×C₂₂₇	C₂²	1	2	yes	—
34	#95×C₂₂₇	C₂²	1	2	yes	—
35	#95×C₂₂₇	C₂²	1	2	yes	—
36	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
37	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
38	C₂×D₄×C₂₂₇	—	2	4	no	—
39	C₂×D₄×C₂₂₇	C₂×C₄	1	4	yes	—
40	C₂²×C₄×C₂₂₇	—	2	4	no	—
41	C₂²×C₄×C₂₂₇	—	4	2	no	—
42	C₂²×C₄×C₂₂₇	—	4	2	no	—
43	C₂²×C₄×C₂₂₇	—	4	2	no	—
44	C₂²×C₄×C₂₂₇	—	4	2	no	—
45	C₂²×C₄×C₂₂₇	C₂×C₄	1	4	yes	—
46	C₂²×C₄×C₂₂₇	C₂×C₄	1	4	yes	complemented
47	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
48	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
49	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
51	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
56	C₄⋊C₄×C₂₂₇	—	4	2	no	—
57	C₄⋊C₄×C₂₂₇	—	4	2	no	—
58	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	complemented
59	C₂²⋊C₄×C₂₂₇	—	2	4	no	—
60	C₂²⋊C₄×C₂₂₇	—	4	2	no	—
61	C₂²⋊C₄×C₂₂₇	—	4	2	no	—
62	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
63	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	complemented
64	C₄²×C₂₂₇	—	4	2	no	—
65	C₄²×C₂₂₇	C₂×C₄	1	4	yes	complemented
66	C₂³×C₂₂₇	—	2	2	no	—
67	C₂³×C₂₂₇	—	2	1	no	—
68	C₂³×C₂₂₇	—	2	1	no	—
69	C₂³×C₂₂₇	—	2	2	no	—
70	C₂³×C₂₂₇	—	2	1	no	—
71	C₂³×C₂₂₇	—	2	2	no	—
72	C₂³×C₂₂₇	—	2	1	no	—
73	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
74	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
75	C₂³×C₂₂₇	#41	1	4	yes	—
76	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
77	D₄×C₂₂₇	—	2	8	no	—
78	C₂×C₄×C₂₂₇	—	2	2	no	—
79	C₂×C₄×C₂₂₇	—	2	2	no	—
80	C₂×C₄×C₂₂₇	—	2	1	no	—
81	C₂×C₄×C₂₂₇	—	2	2	no	—
82	C₂×C₄×C₂₂₇	—	2	2	no	—
83	C₂×C₄×C₂₂₇	—	2	1	no	—
84	C₂×C₄×C₂₂₇	—	2	2	no	—
85	C₂×C₄×C₂₂₇	—	2	2	no	—
86	C₂×C₄×C₂₂₇	—	2	2	no	—
87	C₂×C₄×C₂₂₇	—	2	2	no	—
88	C₂×C₄×C₂₂₇	—	2	2	no	—
89	C₂×C₄×C₂₂₇	—	2	2	no	—
90	C₂×C₄×C₂₂₇	—	2	2	no	—
91	C₂×C₄×C₂₂₇	—	2	2	no	—
92	C₂×C₄×C₂₂₇	—	2	2	no	—
93	C₂×C₄×C₂₂₇	—	2	2	no	—
94	C₂×C₄×C₂₂₇	—	2	2	no	—
95	C₂×C₄×C₂₂₇	—	2	2	no	—
96	C₂×C₄×C₂₂₇	—	4	4	no	—
97	C₂×C₄×C₂₂₇	—	4	4	no	—
98	C₂×C₄×C₂₂₇	—	4	8	no	—
99	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
100	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
102	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
104	C₂²×C₂₂₇	—	2	2	no	—
105	C₂²×C₂₂₇	—	2	2	no	—
106	C₂²×C₂₂₇	—	2	4	no	—
107	C₂²×C₂₂₇	—	2	4	no	—
108	C₂²×C₂₂₇	—	2	2	no	—
109	C₂²×C₂₂₇	—	2	2	no	—
110	C₂²×C₂₂₇	—	2	2	no	—
111	C₂²×C₂₂₇	—	4	2	no	—
112	C₂²×C₂₂₇	—	4	2	no	—
113	C₂²×C₂₂₇	#117	1	4	yes	—
114	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
115	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
116	C₂²×C₂₂₇	H(5,𝔽₂)	1	2	yes	—
117	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
118	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
119	C₄×C₂₂₇	—	2	2	no	—
120	C₄×C₂₂₇	—	2	2	no	—
121	C₄×C₂₂₇	—	2	2	no	—
122	C₄×C₂₂₇	—	2	2	no	—
123	C₄×C₂₂₇	—	2	2	no	—
124	C₄×C₂₂₇	—	4	2	no	—
125	C₄×C₂₂₇	—	4	2	no	—
126	C₄×C₂₂₇	—	4	2	no	—
127	C₄×C₂₂₇	—	4	2	no	—
128	C₂×C₂₂₇	—	2	4	no	—
129	C₂×C₂₂₇	—	4	2	no	—
130	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
131	C₂×C₂₂₇	#518	1	1	yes	characteristic
132	C₂×C₂₂₇	#520	1	1	yes	characteristic
133	C₂×C₂₂₇	#538	1	2	yes	—
134	C₂×C₂₂₇	#539	1	2	yes	—
135	C₂₂₇	#2318	1	1	yes	characteristic, direct product factor, Sylow
136	#2318	C₂₂₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
137	C₂×C₄×D₄	C₂×C₂₂₇	1	1	yes	characteristic
138	#388	C₂×C₂₂₇	1	2	yes	complemented
139	#387	C₂×C₂₂₇	1	2	yes	complemented
140	#386	C₂×C₂₂₇	1	2	yes	—
141	#385	C₂×C₂₂₇	1	2	yes	complemented
142	#383	C₂×C₂₂₇	1	1	yes	characteristic, complemented
143	#382	C₂×C₂₂₇	1	1	yes	characteristic, complemented
144	#381	C₂×C₂₂₇	1	1	yes	characteristic, complemented
145	#380	C₂×C₂₂₇	1	1	yes	characteristic, complemented
146	C₄×C₂²⋊C₄	C₂×C₂₂₇	1	1	yes	characteristic, complemented
147	C₂×#95	C₂×C₂₂₇	1	1	yes	characteristic, complemented
148	C₂²×D₄	C₂²×C₂₂₇	1	1	yes	characteristic
149	C₂³×C₄	C₂²×C₂₂₇	1	2	yes	—
150	C₂³×C₄	C₂²×C₂₂₇	1	1	yes	characteristic
151	C₂³×C₄	C₂²×C₂₂₇	1	1	yes	characteristic
152	C₄×D₄	C₄×C₂₂₇	1	8	yes	complemented
153	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	2	yes	—
155	C₂×C₄⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
156	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	2	yes	—
157	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	2	yes	—
158	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	C₂²×C₂₂₇	1	1	yes	characteristic
161	C₂×C₄²	C₂²×C₂₂₇	1	2	yes	—
162	C₂×C₄²	C₂²×C₂₂₇	1	1	yes	characteristic
163	#95	C₂²×C₂₂₇	1	1	yes	characteristic
164	#95	C₂²×C₂₂₇	1	1	yes	characteristic
165	#95	C₂²×C₂₂₇	1	2	yes	—
166	#95	C₂²×C₂₂₇	1	1	yes	characteristic
167	#95	C₂²×C₂₂₇	1	1	yes	characteristic
168	#95	C₂²×C₂₂₇	1	2	yes	—
169	#95	C₂²×C₂₂₇	1	2	yes	—
170	#95	C₂²×C₂₂₇	1	2	yes	—
171	#95	C₂²×C₂₂₇	1	2	yes	—
172	C₂⁴	C₂³×C₂₂₇	1	1	yes	characteristic
173	C₂⁴	C₂³×C₂₂₇	1	1	yes	characteristic
174	C₂×D₄	—	2	4	no	—
175	C₂×D₄	C₂×C₄×C₂₂₇	1	4	yes	—
176	C₂²×C₄	—	2	4	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	—	4	2	no	—
179	C₂²×C₄	—	4	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	C₂×C₄×C₂₂₇	1	4	yes	—
182	C₂²×C₄	C₂×C₄×C₂₂₇	1	4	yes	complemented
183	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
184	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
185	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
186	C₂²×C₄	C₂³×C₂₂₇	1	2	yes	—
187	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
188	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
189	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
190	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₂₂₇	1	1	yes	characteristic
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	C₂×C₄×C₂₂₇	1	4	yes	complemented
195	C₂²⋊C₄	—	2	4	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₂²⋊C₄	C₂×C₄×C₂₂₇	1	4	yes	—
199	C₂²⋊C₄	C₂×C₄×C₂₂₇	1	4	yes	complemented
200	C₄²	—	4	2	no	—
201	C₄²	C₂×C₄×C₂₂₇	1	4	yes	complemented
202	C₂³	—	2	2	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	1	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	2	1	no	—
209	C₂³	C₂²×C₄×C₂₂₇	1	2	yes	—
210	C₂³	C₂²×C₄×C₂₂₇	1	2	yes	—
211	C₂³	#41×C₂₂₇	1	4	yes	—
212	C₂³	C₂⁴×C₂₂₇	1	1	yes	characteristic
213	D₄	—	2	8	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	8	no	—
235	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
236	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
237	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
238	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
239	C₂×C₄	C₂²×C₄×C₂₂₇	1	2	yes	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	4	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	#117×C₂₂₇	1	4	yes	—
250	C₂²	C₂³×C₄×C₂₂₇	1	1	yes	characteristic
251	C₂²	C₂×#41×C₂₂₇	1	2	yes	—
252	C₂²	H(5,𝔽₂)×C₂₂₇	1	2	yes	—
253	C₂²	H(5,𝔽₂)×C₂₂₇	1	1	yes	characteristic
254	C₂²	2₋¹⁺⁴×C₂₂₇	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	2	no	—
257	C₄	—	2	2	no	—
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	2	no	—
264	C₂	—	2	4	no	—
265	C₂	—	4	2	no	—
266	C₂	C₂×#117×C₂₂₇	1	1	yes	characteristic
267	C₂	#518×C₂₂₇	1	1	yes	characteristic
268	C₂	#520×C₂₂₇	1	1	yes	characteristic
269	C₂	#538×C₂₂₇	1	2	yes	—
270	C₂	#539×C₂₂₇	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₂₂₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₂₂₇
Frattini, cosocle	C₂³	C₂⁴×C₂₂₇
prime radical	C₂²×D₄×C₂₂₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₂₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #388×C₂₂₇	C₂	semidirect
3. #387×C₂₂₇	C₂	semidirect
5. #385×C₂₂₇	C₂	semidirect
6. #383×C₂₂₇	C₂	semidirect
7. #382×C₂₂₇	C₂	semidirect
8. #381×C₂₂₇	C₂	semidirect
9. #380×C₂₂₇	C₂	semidirect
10. C₄×C₂²⋊C₄×C₂₂₇	C₂	semidirect
11. C₂×#95×C₂₂₇	C₂	semidirect
16. C₄×D₄×C₂₂₇	C₄	semidirect
46. C₂²×C₄×C₂₂₇	C₂×C₄	semidirect
58. C₄⋊C₄×C₂₂₇	C₂×C₄	semidirect
63. C₂²⋊C₄×C₂₂₇	C₂×C₄	semidirect
65. C₄²×C₂₂₇	C₂×C₄	semidirect
135. C₂₂₇	#2318	direct (N × H)
136. #2318	C₂₂₇	direct (N × H)
138. #388	C₂×C₂₂₇	semidirect
139. #387	C₂×C₂₂₇	semidirect
141. #385	C₂×C₂₂₇	semidirect
142. #383	C₂×C₂₂₇	semidirect
143. #382	C₂×C₂₂₇	semidirect
144. #381	C₂×C₂₂₇	semidirect
145. #380	C₂×C₂₂₇	semidirect
146. C₄×C₂²⋊C₄	C₂×C₂₂₇	semidirect
147. C₂×#95	C₂×C₂₂₇	semidirect
152. C₄×D₄	C₄×C₂₂₇	semidirect
182. C₂²×C₄	C₂×C₄×C₂₂₇	semidirect
194. C₄⋊C₄	C₂×C₄×C₂₂₇	semidirect
199. C₂²⋊C₄	C₂×C₄×C₂₂₇	semidirect
201. C₄²	C₂×C₄×C₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₄×C₂₂₇	2
#388×C₂₂₇	2
#387×C₂₂₇	2
#386×C₂₂₇	2
#385×C₂₂₇	2
#383×C₂₂₇	2
#382×C₂₂₇	2
#381×C₂₂₇	2
#380×C₂₂₇	2
C₄×C₂²⋊C₄×C₂₂₇	2
C₂×#95×C₂₂₇	2
#2318	227

Subgroups — #2318×C227 #44396274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #2318×C₂₂₇ #44396274