Subgroups — #44408755

Order 29056.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₄×D₂₂₇	C₂	1	1	yes	maximal, characteristic
2	C₈×D₉₀₈	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×C₂₂₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#28409097	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈×C₂²⋊D₂₂₇	C₂	1	2	yes	maximal, complemented
6	(C₂²⋊D₂₂₇)⋊C₈	C₂	1	2	yes	maximal, complemented
7	#28409084	C₂	1	1	yes	maximal, characteristic, complemented
8	#28409083	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×(C₂×D₂₂₇)⋊C₈	C₂	1	2	yes	maximal, complemented
10	C₂²⋊C₈×D₂₂₇	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×D₄×D₂₂₇	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₂₂₇	C₂²	1	2	yes	—
14	C₂²×C₂₂₇⋊C₈	C₂²	1	2	yes	complemented
15	C₂×C₈×D₂₂₇	C₂²	1	2	yes	complemented
16	C₂×C₈×D₂₂₇	C₂²	1	2	yes	complemented
17	C₄×D₉₀₈	C₂²	1	1	yes	characteristic
18	D₄×Dic₂₂₇	C₄	1	1	yes	characteristic
19	#20300250	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₂₂₇	C₂²	1	2	yes	—
21	(C₂²⋊D₂₂₇)⋊C₄	C₄	1	2	yes	—
22	C₄⋊C₄×D₂₂₇	C₄	1	1	yes	characteristic
23	C₂²⋊C₄×D₂₂₇	C₄	1	2	yes	—
24	C₄²×D₂₂₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	1	yes	characteristic, complemented
26	Dic₂₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
27	Dic₂₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
28	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	complemented
29	(C₂×D₂₂₇)⋊C₈	C₂²	1	1	yes	characteristic
30	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	—
31	(C₂×D₂₂₇)⋊C₈	C₂²	1	1	yes	characteristic
32	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	complemented
33	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	complemented
34	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
35	C₈×Dic₂₂₇	C₂²	1	2	yes	complemented
36	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	complemented
37	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
39	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	complemented
40	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
41	C₂³×D₂₂₇	C₂×C₄	1	2	yes	—
42	C₂×D₉₀₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
44	D₄×D₂₂₇	C₈	1	8	yes	complemented
45	C₂²×Dic₂₂₇	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₂₂₇	—	4	2	no	—
47	C₂×C₄×D₂₂₇	—	4	2	no	—
48	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₂₇	C₂³	1	2	yes	—
53	C₂×C₄×D₂₂₇	C₂³	1	1	yes	characteristic
54	C₂×C₂₂₇⋊C₈	—	4	2	no	—
55	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
56	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
57	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₈×D₂₂₇	—	4	2	no	—
59	C₈×D₂₂₇	—	4	2	no	—
60	C₄⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
61	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
62	C₂²⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
63	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
64	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
65	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
67	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
68	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
70	C₂×C₈×C₂₂₇	—	4	2	no	—
71	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
73	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
76	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
77	C₂²×D₂₂₇	—	4	2	no	—
78	C₂²×D₂₂₇	—	4	2	no	—
79	C₂²×D₂₂₇	C₂×C₈	1	8	yes	complemented
80	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
81	C₂²×D₂₂₇	C₂²×C₄	1	1	yes	characteristic
82	D₉₀₈	C₂×C₈	1	4	yes	complemented
83	C₂²⋊D₂₂₇	C₂×C₈	1	8	yes	complemented
84	C₂×Dic₂₂₇	—	4	2	no	—
85	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
87	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
88	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₄×D₂₂₇	—	2	2	no	—
90	C₄×D₂₂₇	—	2	2	no	—
91	C₄×D₂₂₇	—	2	1	no	—
92	C₄×D₂₂₇	—	2	1	no	—
93	C₄×D₂₂₇	—	4	2	no	—
94	C₄×D₂₂₇	—	4	2	no	—
95	C₄×D₂₂₇	—	4	1	no	—
96	C₄×D₂₂₇	C₂×C₈	1	4	yes	complemented
97	C₂₂₇⋊C₈	—	2	1	no	—
98	C₂₂₇⋊C₈	—	2	2	no	—
99	C₂₂₇⋊C₈	—	2	1	no	—
100	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
101	D₄×C₂₂₇	C₂×C₈	1	4	yes	complemented
102	C₂×C₄×C₂₂₇	—	2	2	no	—
103	C₂×C₄×C₂₂₇	—	2	2	no	—
104	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
106	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
108	C₈×C₂₂₇	—	2	1	no	—
109	C₈×C₂₂₇	—	2	2	no	—
110	C₈×C₂₂₇	—	2	1	no	—
111	C₂²×C₂₂₇	—	2	2	no	—
112	C₂²×C₂₂₇	—	2	2	no	—
113	C₂²×C₂₂₇	C₂²×C₈	1	4	yes	—
114	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
115	C₂×D₂₂₇	—	2	1	no	—
116	C₂×D₂₂₇	—	2	2	no	—
117	C₂×D₂₂₇	—	2	1	no	—
118	C₂×D₂₂₇	—	2	2	no	—
119	C₂×D₂₂₇	—	4	8	no	—
120	C₂×D₂₂₇	C₂²×C₈	1	2	yes	—
121	C₂×D₂₂₇	C₂²×C₈	1	4	yes	—
122	C₄×C₂₂₇	—	2	2	no	—
123	C₄×C₂₂₇	—	2	1	no	—
124	C₄×C₂₂₇	C₂²×C₈	1	2	yes	—
125	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₂₂₇	—	2	1	no	—
128	Dic₂₂₇	—	2	2	no	—
129	Dic₂₂₇	—	2	1	no	—
130	Dic₂₂₇	C₂²×C₈	1	2	yes	—
131	C₂×C₂₂₇	—	2	4	no	—
132	C₂×C₂₂₇	#518	1	1	yes	characteristic
133	C₂×C₂₂₇	C₂³×C₈	1	1	yes	characteristic
134	C₂×C₂₂₇	#568	1	1	yes	characteristic
135	D₂₂₇	—	2	2	no	—
136	D₂₂₇	—	2	4	no	—
137	C₂₂₇	#2972	1	1	yes	characteristic, complemented, Sylow
138	#2972	—	227	1	no	maximal, Sylow
139	C₂×C₄×D₄	—	227	1	no	—
140	C₈×D₄	—	227	2	no	—
141	C₈×D₄	—	227	1	no	—
142	C₈×D₄	—	227	1	no	—
143	C₈×D₄	—	227	2	no	—
144	C₈×D₄	—	227	1	no	—
145	C₈×D₄	D₂₂₇	1	1	yes	characteristic, complemented
146	#431	—	227	1	no	—
147	#431	—	227	1	no	—
148	C₂×C₂²⋊C₈	—	227	2	no	—
149	C₂×C₂²⋊C₈	—	227	2	no	—
150	C₂²×D₄	—	227	1	no	—
151	C₂³×C₄	—	227	2	no	—
152	C₂²×C₈	—	227	2	no	—
153	C₂²×C₈	—	227	2	no	—
154	C₂²×C₈	—	227	2	no	—
155	C₂²×C₈	C₂×D₂₂₇	1	2	yes	complemented
156	C₄×D₄	—	227	1	no	—
157	C₄×D₄	—	227	2	no	—
158	C₄×D₄	—	227	1	no	—
159	C₄×D₄	—	227	2	no	—
160	C₄×D₄	—	227	1	no	—
161	C₄×D₄	C₂×D₂₂₇	1	1	yes	characteristic
162	C₂×C₄⋊C₄	—	227	1	no	—
163	C₂×C₂²⋊C₄	—	227	2	no	—
164	C₂×C₄²	—	227	1	no	—
165	C₄⋊C₈	—	227	1	no	—
166	C₄⋊C₈	—	227	1	no	—
167	C₄⋊C₈	—	227	1	no	—
168	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic, complemented
169	C₂²⋊C₈	—	227	2	no	—
170	C₂²⋊C₈	—	227	2	no	—
171	C₂²⋊C₈	—	227	1	no	—
172	C₂²⋊C₈	—	227	1	no	—
173	C₂²⋊C₈	—	227	2	no	—
174	C₂²⋊C₈	—	227	2	no	—
175	C₂²⋊C₈	C₂×D₂₂₇	1	2	yes	complemented
176	C₄×C₈	—	227	2	no	—
177	C₄×C₈	—	227	1	no	—
178	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic, complemented
179	C₂⁴	—	227	2	no	—
180	C₂×D₄	—	227	2	no	—
181	C₂×D₄	—	227	8	no	—
182	C₂×D₄	—	227	1	no	—
183	C₂×D₄	C₄×D₂₂₇	1	1	yes	characteristic
184	C₂²×C₄	—	908	2	no	—
185	C₂²×C₄	—	908	2	no	—
186	C₂²×C₄	—	227	2	no	—
187	C₂²×C₄	—	227	1	no	—
188	C₂²×C₄	—	227	2	no	—
189	C₂²×C₄	—	227	2	no	—
190	C₂²×C₄	—	227	1	no	—
191	C₂²×C₄	—	227	2	no	—
192	C₂²×C₄	—	227	1	no	—
193	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
194	C₂×C₈	—	4	2	no	—
195	C₂×C₈	—	908	2	no	—
196	C₂×C₈	—	908	2	no	—
197	C₂×C₈	—	908	2	no	—
198	C₂×C₈	—	227	1	no	—
199	C₂×C₈	—	227	2	no	—
200	C₂×C₈	—	227	1	no	—
201	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
202	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
204	C₄⋊C₄	—	227	2	no	—
205	C₄⋊C₄	—	227	1	no	—
206	C₄⋊C₄	C₄×D₂₂₇	1	1	yes	characteristic
207	C₂²⋊C₄	—	227	2	no	—
208	C₂²⋊C₄	—	227	2	no	—
209	C₂²⋊C₄	—	227	2	no	—
210	C₂²⋊C₄	C₄×D₂₂₇	1	2	yes	—
211	C₄²	—	227	1	no	—
212	C₄²	—	227	1	no	—
213	C₄²	—	227	1	no	—
214	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
215	C₂³	—	908	2	no	—
216	C₂³	—	908	2	no	—
217	C₂³	—	227	8	no	—
218	C₂³	—	227	2	no	—
219	C₂³	—	227	1	no	—
220	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
221	D₄	—	227	8	no	—
222	D₄	—	227	4	no	—
223	D₄	C₈×D₂₂₇	1	4	yes	complemented
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	908	2	no	—
227	C₂×C₄	—	908	2	no	—
228	C₂×C₄	—	908	1	no	—
229	C₂×C₄	—	908	2	no	—
230	C₂×C₄	—	454	2	no	—
231	C₂×C₄	—	454	1	no	—
232	C₂×C₄	—	454	2	no	—
233	C₂×C₄	—	454	1	no	—
234	C₂×C₄	—	227	1	no	—
235	C₂×C₄	—	227	1	no	—
236	C₂×C₄	—	227	2	no	—
237	C₂×C₄	—	227	1	no	—
238	C₂×C₄	—	227	4	no	—
239	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
241	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	454	1	no	—
247	C₈	—	454	2	no	—
248	C₈	—	454	1	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	2	2	no	—
251	C₂²	—	908	8	no	—
252	C₂²	—	454	1	no	—
253	C₂²	—	454	2	no	—
254	C₂²	—	454	1	no	—
255	C₂²	—	454	2	no	—
256	C₂²	—	227	2	no	—
257	C₂²	—	227	4	no	—
258	C₂²	C₂×C₈×D₂₂₇	1	4	yes	—
259	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	454	1	no	—
263	C₄	—	454	2	no	—
264	C₄	—	454	1	no	—
265	C₄	—	227	2	no	—
266	C₄	C₂×C₈×D₂₂₇	1	2	yes	—
267	C₄	#20300333	1	1	yes	characteristic
268	C₄	#20300334	1	1	yes	characteristic
269	C₂	—	2	4	no	—
270	C₂	—	454	2	no	—
271	C₂	—	454	4	no	—
272	C₂	#28409504	1	1	yes	characteristic
273	C₂	C₂²×C₈×D₂₂₇	1	1	yes	characteristic
274	C₂	#28409797	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator	C₂×C₂₂₇	C₂³×C₈
Fitting	C₈×D₄×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂³×C₄
prime radical	C₂×D₄×D₂₂₇	C₄
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₈×D₄	D₂₂₇
hypocenter	C₂₂₇	#2972
LCS	G ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
2. C₈×D₉₀₈	C₂	semidirect
3. D₄×C₂₂₇⋊C₈	C₂	semidirect
4. #28409097	C₂	semidirect
5. C₈×C₂²⋊D₂₂₇	C₂	semidirect
6. (C₂²⋊D₂₂₇)⋊C₈	C₂	semidirect
7. #28409084	C₂	semidirect
8. #28409083	C₂	semidirect
9. C₂×(C₂×D₂₂₇)⋊C₈	C₂	semidirect
10. C₂²⋊C₈×D₂₂₇	C₂	semidirect
11. C₈×D₄×C₂₂₇	C₂	semidirect
14. C₂²×C₂₂₇⋊C₈	C₂²	semidirect
15. C₂×C₈×D₂₂₇	C₂²	semidirect
16. C₂×C₈×D₂₂₇	C₂²	semidirect
25. C₄⋊(C₂₂₇⋊C₈)	C₂²	semidirect
26. Dic₂₂₇⋊C₈	C₂²	semidirect
27. Dic₂₂₇⋊C₈	C₂²	semidirect
28. C₂²⋊(C₂₂₇⋊C₈)	C₂²	semidirect
32. (C₂×D₂₂₇)⋊C₈	C₂²	semidirect
33. (C₂×D₂₂₇)⋊C₈	C₂²	semidirect
34. C₄×C₂₂₇⋊C₈	C₂²	semidirect
35. C₈×Dic₂₂₇	C₂²	semidirect
36. C₂²×C₈×C₂₂₇	C₂²	semidirect
38. C₄⋊C₈×C₂₂₇	C₂²	semidirect
39. C₂²⋊C₈×C₂₂₇	C₂²	semidirect
40. C₄×C₈×C₂₂₇	C₂²	semidirect
44. D₄×D₂₂₇	C₈	semidirect
79. C₂²×D₂₂₇	C₂×C₈	semidirect
82. D₉₀₈	C₂×C₈	semidirect
83. C₂²⋊D₂₂₇	C₂×C₈	semidirect
96. C₄×D₂₂₇	C₂×C₈	semidirect
101. D₄×C₂₂₇	C₂×C₈	semidirect
137. C₂₂₇	#2972	semidirect
145. C₈×D₄	D₂₂₇	semidirect
155. C₂²×C₈	C₂×D₂₂₇	semidirect
168. C₄⋊C₈	C₂×D₂₂₇	semidirect
175. C₂²⋊C₈	C₂×D₂₂₇	semidirect
178. C₄×C₈	C₂×D₂₂₇	semidirect
223. D₄	C₈×D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₄×D₂₂₇	2
C₈×D₉₀₈	2
D₄×C₂₂₇⋊C₈	2
#28409097	2
C₈×C₂²⋊D₂₂₇	2
(C₂²⋊D₂₂₇)⋊C₈	2
#28409084	2
#28409083	2
C₂×(C₂×D₂₂₇)⋊C₈	2
C₂²⋊C₈×D₂₂₇	2
C₈×D₄×C₂₂₇	2
#2972	227