Subgroups — #44402419

Order 29056.

556 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300264	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#20300245	C₂	1	1	yes	maximal, characteristic
4	#28408930	C₂	1	1	yes	maximal, characteristic, complemented
5	#28408929	C₂	1	2	yes	maximal, complemented
6	#28408924	C₂	1	2	yes	maximal
7	#28408921	C₂	1	2	yes	maximal
8	#28408914	C₂	1	2	yes	maximal, complemented
9	#28408889	C₂	1	1	yes	maximal, characteristic
10	C₄⋊C₄×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
11	#391×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
13	#20300264	C₄	1	8	yes	complemented
14	Q₈×Dic₂₂₇	—	2	4	no	—
15	#20300245	—	2	4	no	—
16	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
18	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₂₇⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₂₇⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
22	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
23	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	2	yes	—
26	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	2	yes	—
27	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
28	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
29	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
30	C₂₂₇⋊#95	C₂²	1	2	yes	—
31	C₂₂₇⋊#95	C₂²	1	2	yes	—
32	C₂₂₇⋊#95	C₂²	1	2	yes	—
33	C₂²×Q₈×C₂₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
35	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
36	#95×C₂₂₇	C₂²	1	2	yes	—
37	#95×C₂₂₇	C₂²	1	2	yes	—
38	C₂³×D₂₂₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
40	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
42	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
43	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
44	C₂×C₄×D₂₂₇	—	2	2	no	—
45	C₂×C₄×D₂₂₇	—	2	2	no	—
46	C₂×C₄×D₂₂₇	C₂×C₄	1	4	yes	—
47	C₄⋊Dic₂₂₇	—	2	4	no	—
48	C₄⋊Dic₂₂₇	—	2	2	no	—
49	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
50	Dic₂₂₇⋊C₄	—	2	2	no	—
51	Dic₂₂₇⋊C₄	—	2	2	no	—
52	Dic₂₂₇⋊C₄	—	4	2	no	—
53	Dic₂₂₇⋊C₄	C₂×C₄	1	8	yes	—
54	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
55	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
56	(C₂×D₂₂₇)⋊C₄	—	4	2	no	—
57	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	8	yes	—
58	C₄×Dic₂₂₇	—	2	4	no	—
59	C₄×Dic₂₂₇	—	4	2	no	—
60	C₄×Dic₂₂₇	D₄	1	4	yes	—
61	C₂×Q₈×C₂₂₇	—	2	2	no	—
62	C₂×Q₈×C₂₂₇	—	2	2	no	—
63	C₂×Q₈×C₂₂₇	C₂×C₄	1	4	yes	—
64	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
67	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
68	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₂₂₇	—	2	2	no	—
70	C₄²×C₂₂₇	—	2	2	no	—
71	C₂²×D₂₂₇	—	2	1	no	—
72	C₂²×D₂₂₇	—	2	1	no	—
73	C₂²×D₂₂₇	—	2	1	no	—
74	C₂²×D₂₂₇	—	2	1	no	—
75	C₂²×D₂₂₇	—	2	1	no	—
76	C₂²×D₂₂₇	—	2	1	no	—
77	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₂₇	—	2	1	no	—
79	C₂×Dic₂₂₇	—	2	1	no	—
80	C₂×Dic₂₂₇	—	2	1	no	—
81	C₂×Dic₂₂₇	—	2	2	no	—
82	C₂×Dic₂₂₇	—	2	2	no	—
83	C₂×Dic₂₂₇	—	2	1	no	—
84	C₂×Dic₂₂₇	—	2	1	no	—
85	C₂×Dic₂₂₇	—	2	1	no	—
86	C₂×Dic₂₂₇	—	2	2	no	—
87	C₂×Dic₂₂₇	—	2	2	no	—
88	C₂×Dic₂₂₇	—	2	2	no	—
89	C₂×Dic₂₂₇	C₂²×C₄	1	4	yes	—
90	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
91	C₂×Dic₂₂₇	C₂×D₄	1	2	yes	—
92	C₂×Dic₂₂₇	C₂×D₄	1	2	yes	—
93	C₄×D₂₂₇	—	2	8	no	—
94	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
95	Q₈×C₂₂₇	—	2	8	no	—
96	C₂×C₄×C₂₂₇	—	2	2	no	—
97	C₂×C₄×C₂₂₇	—	2	2	no	—
98	C₂×C₄×C₂₂₇	—	2	1	no	—
99	C₂×C₄×C₂₂₇	—	2	1	no	—
100	C₂×C₄×C₂₂₇	—	2	2	no	—
101	C₂×C₄×C₂₂₇	—	2	2	no	—
102	C₂×C₄×C₂₂₇	—	2	1	no	—
103	C₂×C₄×C₂₂₇	—	2	1	no	—
104	C₂×C₄×C₂₂₇	—	2	1	no	—
105	C₂×C₄×C₂₂₇	—	2	1	no	—
106	C₂×C₄×C₂₂₇	—	2	2	no	—
107	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
108	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₂₂₇	#41	1	2	yes	—
111	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
112	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
113	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
114	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
115	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
118	C₂×D₂₂₇	—	2	2	no	—
119	C₂×D₂₂₇	—	2	2	no	—
120	C₂×D₂₂₇	—	2	2	no	—
121	C₂×D₂₂₇	—	4	1	no	—
122	C₂×D₂₂₇	—	4	1	no	—
123	C₂×D₂₂₇	—	4	1	no	—
124	C₂×D₂₂₇	—	4	1	no	—
125	C₄×C₂₂₇	—	2	4	no	—
126	C₄×C₂₂₇	—	4	1	no	—
127	C₄×C₂₂₇	—	4	1	no	—
128	C₄×C₂₂₇	—	4	2	no	—
129	C₄×C₂₂₇	C₄×D₄	1	4	yes	—
130	Dic₂₂₇	—	2	2	no	—
131	Dic₂₂₇	—	2	4	no	—
132	Dic₂₂₇	—	2	2	no	—
133	Dic₂₂₇	—	2	2	no	—
134	Dic₂₂₇	—	4	2	no	—
135	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₂₂₇	#519	1	1	yes	characteristic
137	C₂×C₂₂₇	#520	1	1	yes	characteristic
138	C₂×C₂₂₇	#536	1	1	yes	characteristic
139	C₂×C₂₂₇	#537	1	1	yes	characteristic
140	C₂×C₂₂₇	#541	1	1	yes	characteristic
141	C₂×C₂₂₇	#542	1	1	yes	characteristic
142	D₂₂₇	—	4	2	no	—
143	C₂₂₇	#2323	1	1	yes	characteristic, complemented, Sylow
144	#2323	—	227	1	no	maximal, Sylow
145	C₂×C₂²⋊Q₈	—	227	1	no	—
146	C₂×C₄×Q₈	—	227	1	no	—
147	C₂×#117	—	227	1	no	—
148	#391	D₂₂₇	1	1	yes	characteristic, complemented
149	#389	—	227	1	no	—
150	#389	—	227	2	no	—
151	#388	—	227	2	no	—
152	#388	—	227	2	no	—
153	#387	—	227	2	no	—
154	#380	—	227	1	no	—
155	C₄×C₄⋊C₄	—	227	1	no	—
156	C₂²×Q₈	C₂×D₂₂₇	1	1	yes	characteristic
157	C₂³×C₄	—	227	1	no	—
158	C₂²⋊Q₈	—	227	8	no	—
159	C₄×Q₈	—	454	4	no	—
160	#117	—	454	4	no	—
161	C₂×C₄⋊C₄	—	227	2	no	—
162	C₂×C₄⋊C₄	—	227	1	no	—
163	C₂×C₄⋊C₄	—	227	1	no	—
164	C₂×C₄⋊C₄	—	227	1	no	—
165	C₂×C₄⋊C₄	—	227	2	no	—
166	C₂×C₄⋊C₄	—	227	1	no	—
167	C₂×C₄⋊C₄	—	227	2	no	—
168	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	227	2	no	—
170	C₂×C₂²⋊C₄	—	227	1	no	—
171	C₂×C₂²⋊C₄	—	227	1	no	—
172	C₂×C₂²⋊C₄	—	227	2	no	—
173	C₂×C₄²	—	227	2	no	—
174	C₂×C₄²	—	227	2	no	—
175	C₂×C₄²	—	227	1	no	—
176	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
177	#95	—	227	2	no	—
178	#95	—	227	2	no	—
179	#95	—	227	2	no	—
180	#95	C₂×D₂₂₇	1	2	yes	—
181	#95	C₂×D₂₂₇	1	2	yes	—
182	C₂⁴	—	227	1	no	—
183	C₂×Q₈	—	2	2	no	—
184	C₂×Q₈	—	2	2	no	—
185	C₂×Q₈	C₄×D₂₂₇	1	4	yes	—
186	C₂²×C₄	—	454	2	no	—
187	C₂²×C₄	—	454	2	no	—
188	C₂²×C₄	—	227	2	no	—
189	C₂²×C₄	—	227	1	no	—
190	C₂²×C₄	—	227	2	no	—
191	C₂²×C₄	—	227	1	no	—
192	C₂²×C₄	—	227	1	no	—
193	C₂²×C₄	—	227	4	no	—
194	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
195	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
196	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
197	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
198	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
199	C₄⋊C₄	—	2	2	no	—
200	C₄⋊C₄	—	908	2	no	—
201	C₄⋊C₄	—	454	2	no	—
202	C₄⋊C₄	—	454	2	no	—
203	C₄⋊C₄	—	454	4	no	—
204	C₄⋊C₄	—	454	2	no	—
205	C₄⋊C₄	—	227	8	no	—
206	C₄⋊C₄	—	227	4	no	—
207	C₂²⋊C₄	—	908	2	no	—
208	C₂²⋊C₄	—	454	2	no	—
209	C₂²⋊C₄	—	454	2	no	—
210	C₂²⋊C₄	—	227	8	no	—
211	C₄²	—	2	2	no	—
212	C₄²	—	908	2	no	—
213	C₄²	—	454	4	no	—
214	C₄²	—	227	4	no	—
215	C₂³	—	454	1	no	—
216	C₂³	—	454	1	no	—
217	C₂³	—	454	1	no	—
218	C₂³	—	454	1	no	—
219	C₂³	—	454	1	no	—
220	C₂³	—	454	1	no	—
221	C₂³	—	227	2	no	—
222	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
223	Q₈	—	2	8	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	454	2	no	—
236	C₂×C₄	—	454	2	no	—
237	C₂×C₄	—	454	2	no	—
238	C₂×C₄	—	454	1	no	—
239	C₂×C₄	—	454	1	no	—
240	C₂×C₄	—	454	1	no	—
241	C₂×C₄	—	454	2	no	—
242	C₂×C₄	—	454	1	no	—
243	C₂×C₄	—	454	1	no	—
244	C₂×C₄	—	454	1	no	—
245	C₂×C₄	—	454	2	no	—
246	C₂×C₄	—	454	8	no	—
247	C₂×C₄	—	227	4	no	—
248	C₂×C₄	—	227	2	no	—
249	C₂×C₄	—	227	2	no	—
250	C₂×C₄	—	227	2	no	—
251	C₂×C₄	C₂×C₄×D₂₂₇	1	4	yes	—
252	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
253	C₂×C₄	D₄×D₂₂₇	1	2	yes	—
254	C₂×C₄	#16813883	1	2	yes	—
255	C₂²	—	908	1	no	—
256	C₂²	—	908	1	no	—
257	C₂²	—	908	1	no	—
258	C₂²	—	908	1	no	—
259	C₂²	—	454	2	no	—
260	C₂²	—	454	2	no	—
261	C₂²	—	454	2	no	—
262	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
263	C₂²	C₂×D₄×D₂₂₇	1	1	yes	characteristic
264	C₂²	C₂×#16813883	1	1	yes	characteristic
265	C₂²	#20300334	1	1	yes	characteristic
266	C₂²	#20300335	1	1	yes	characteristic
267	C₂²	#20300336	1	1	yes	characteristic
268	C₂²	#20300336	1	1	yes	characteristic
269	C₄	—	2	4	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	—	4	2	no	—
273	C₄	—	908	2	no	—
274	C₄	—	454	2	no	—
275	C₄	—	454	2	no	—
276	C₄	—	454	2	no	—
277	C₄	—	454	4	no	—
278	C₄	(C₂²⋊D₂₂₇)⋊C₄	1	4	yes	—
279	C₂	—	908	2	no	—
280	C₂	C₂×(C₂²⋊D₂₂₇)⋊C₄	1	1	yes	characteristic
281	C₂	#28409508	1	1	yes	characteristic
282	C₂	#28409514	1	1	yes	characteristic
283	C₂	#28409605	1	1	yes	characteristic
284	C₂	#28409612	1	1	yes	characteristic
285	C₂	#28409643	1	1	yes	characteristic
286	C₂	#28409651	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting	#391×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
prime radical	C₂³×D₂₂₇	C₂³
hypercenter	#391	D₂₂₇
hypocenter	C₂₂₇	#2323
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ #391

Complements

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

N	H ≅ G/N	type
2. C₂×Q₈×Dic₂₂₇	C₂	semidirect
4. #28408930	C₂	semidirect
5. #28408929	C₂	semidirect
8. #28408914	C₂	semidirect
10. C₄⋊C₄×Dic₂₂₇	C₂	semidirect
11. #391×C₂₂₇	C₂	semidirect
13. #20300264	C₄	semidirect
143. C₂₂₇	#2323	semidirect
148. #391	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300264	2
C₂×Q₈×Dic₂₂₇	2
C₂×#20300245	2
#28408930	2
#28408929	2
#28408924	2
#28408921	2
#28408914	2
#28408889	2
C₄⋊C₄×Dic₂₂₇	2
#391×C₂₂₇	2
#2323	227