Subgroups — #44408607

Order 29056.

484 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300308	C₂	1	1	yes	maximal, characteristic
2	#28409512	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409132	C₂	1	2	yes	maximal, complemented
4	#28409130	C₂	1	2	yes	maximal, complemented
5	C₄×#16813874	C₂	1	2	yes	maximal, complemented
6	C₄×#16813869	C₂	1	2	yes	maximal, complemented
7	C₂×#20300221	C₂	1	1	yes	maximal, characteristic, complemented
8	#28409033	C₂	1	1	yes	maximal, characteristic, complemented
9	#28409031	C₂	1	1	yes	maximal, characteristic, complemented
10	#28408988	C₂	1	1	yes	maximal, characteristic, complemented
11	#520×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16813884	C₂²	1	1	yes	characteristic
13	#20300308	C₄	1	8	yes	complemented
14	C₂×#16813874	C₂²	1	1	yes	characteristic
15	C₂×#16813872	C₂²	1	1	yes	characteristic
16	C₂×#16813871	C₂²	1	1	yes	characteristic
17	C₂×#16813869	C₂²	1	1	yes	characteristic
18	C₂×#16813868	C₂²	1	1	yes	characteristic
19	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×Dic₄₅₄	C₂²	1	2	yes	—
21	C₄×D₉₀₈	C₂²	1	2	yes	—
22	(C₂²⋊D₂₂₇)⋊C₄	—	2	2	no	—
23	#20300246	—	2	1	no	—
24	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
25	C₄⋊C₄×D₂₂₇	—	2	1	no	—
26	#20300221	C₂²	1	2	yes	—
27	#20300221	C₂²	1	2	yes	—
28	#20300212	C₂²	1	2	yes	—
29	#20300211	C₂²	1	2	yes	—
30	#20300209	C₂²	1	2	yes	—
31	#20300208	C₂²	1	2	yes	—
32	#20300199	C₂²	1	2	yes	—
33	C₄×C₂₂₇⋊C₈	C₂²	1	2	yes	—
34	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
35	C₄×Q₈×C₂₂₇	C₂²	1	2	yes	—
36	C₄×D₄×C₂₂₇	—	2	2	no	—
37	C₄×D₄×C₂₂₇	C₂²	1	2	yes	—
38	#117×C₂₂₇	—	2	1	no	—
39	#117×C₂₂₇	C₂²	1	1	yes	characteristic
40	C₂×C₄⋊C₄×C₂₂₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
42	#16813884	—	2	2	no	—
43	#16813884	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₄₅₄	C₂³	1	1	yes	characteristic
45	C₂×D₉₀₈	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₂₇	—	2	1	no	—
47	C₂²×Dic₂₂₇	—	2	1	no	—
48	C₂×C₄×D₂₂₇	—	2	1	no	—
49	C₂×C₄×D₂₂₇	—	2	2	no	—
50	#16813874	C₂×C₄	1	4	yes	complemented
51	#16813872	C₂×C₄	1	4	yes	complemented
52	#16813871	C₂×C₄	1	4	yes	complemented
53	#16813869	C₂×C₄	1	4	yes	complemented
54	#16813868	—	2	2	no	—
55	C₂×C₂₂₇⋊C₈	C₂×C₄	1	4	yes	complemented
56	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
57	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
59	C₄⋊Dic₂₂₇	—	2	1	no	—
60	C₄⋊Dic₂₂₇	C₂³	1	2	yes	—
61	Dic₂₂₇⋊C₄	—	2	2	no	—
62	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
63	C₄×Dic₂₂₇	—	2	2	no	—
64	#41×C₂₂₇	—	2	2	no	—
65	#41×C₂₂₇	C₂×C₄	1	4	yes	complemented
66	C₂×Q₈×C₂₂₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₂₂₇	—	2	1	no	—
68	C₂×D₄×C₂₂₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₂₇	—	2	1	no	—
70	C₂²×C₄×C₂₂₇	—	2	1	no	—
71	C₂²×C₄×C₂₂₇	—	2	2	no	—
72	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₂₇	—	2	1	no	—
74	C₄⋊C₄×C₂₂₇	—	2	2	no	—
75	C₄⋊C₄×C₂₂₇	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₂₂₇	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₂₇	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₂₂₇	—	2	2	no	—
79	C₂²⋊C₄×C₂₂₇	D₄	1	2	yes	—
80	C₄²×C₂₂₇	—	2	2	no	—
81	C₄²×C₂₂₇	C₂³	1	2	yes	—
82	C₂²×D₂₂₇	—	2	1	no	—
83	Dic₄₅₄	—	2	1	no	—
84	Dic₄₅₄	C₂²×C₄	1	2	yes	—
85	D₉₀₈	—	2	1	no	—
86	D₉₀₈	C₂²×C₄	1	2	yes	—
87	C₂²⋊D₂₂₇	—	2	4	no	—
88	C₂×Dic₂₂₇	—	2	1	no	—
89	C₂×Dic₂₂₇	—	2	2	no	—
90	C₂×Dic₂₂₇	—	8	1	no	—
91	C₄×D₂₂₇	—	2	4	no	—
92	C₄×D₂₂₇	—	8	2	no	—
93	C₂₂₇⋊C₈	—	2	2	no	—
94	C₂₂₇⋊C₈	C₂²×C₄	1	2	yes	—
95	C₂₂₇⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂³×C₂₂₇	—	2	1	no	—
97	C₂³×C₂₂₇	C₂×D₄	1	1	yes	characteristic
98	Q₈×C₂₂₇	—	2	1	no	—
99	Q₈×C₂₂₇	C₂²×C₄	1	2	yes	—
100	D₄×C₂₂₇	—	2	4	no	—
101	D₄×C₂₂₇	—	2	1	no	—
102	D₄×C₂₂₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₂₇	—	2	1	no	—
104	C₂×C₄×C₂₂₇	—	2	4	no	—
105	C₂×C₄×C₂₂₇	—	2	2	no	—
106	C₂×C₄×C₂₂₇	—	8	2	no	—
107	C₂×C₄×C₂₂₇	—	8	1	no	—
108	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₂₂₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₂₂₇	#41	1	2	yes	—
113	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
114	C₂²×C₂₂₇	—	2	1	no	—
115	C₂²×C₂₂₇	—	2	1	no	—
116	C₂²×C₂₂₇	—	2	2	no	—
117	C₂²×C₂₂₇	—	4	1	no	—
118	C₂²×C₂₂₇	—	4	1	no	—
119	C₂²×C₂₂₇	C₄×D₄	1	2	yes	—
120	C₂²×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
121	C₂×D₂₂₇	—	2	2	no	—
122	C₂×D₂₂₇	—	4	1	no	—
123	C₂×D₂₂₇	—	4	1	no	—
124	C₄×C₂₂₇	—	2	2	no	—
125	C₄×C₂₂₇	—	2	2	no	—
126	C₄×C₂₂₇	—	2	2	no	—
127	C₄×C₂₂₇	—	4	2	no	—
128	C₄×C₂₂₇	C₄×D₄	1	2	yes	—
129	C₄×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
130	C₄×C₂₂₇	C₂×#41	1	1	yes	characteristic
131	Dic₂₂₇	—	2	2	no	—
132	Dic₂₂₇	—	4	2	no	—
133	C₂×C₂₂₇	—	2	2	no	—
134	C₂×C₂₂₇	—	4	2	no	—
135	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₂₂₇	#576	1	1	yes	characteristic
137	C₂×C₂₂₇	#578	1	1	yes	characteristic
138	D₂₂₇	—	4	2	no	—
139	C₂₂₇	#2951	1	1	yes	characteristic, complemented, Sylow
140	#2951	—	227	1	no	maximal, Sylow
141	C₂×#135	—	227	1	no	—
142	#520	—	227	1	no	—
143	#520	D₂₂₇	1	1	yes	characteristic, complemented
144	#440	—	227	2	no	—
145	#440	—	227	2	no	—
146	C₄×Q₁₆	—	227	2	no	—
147	C₄×D₈	—	227	2	no	—
148	C₂×#107	—	227	1	no	—
149	#417	—	227	1	no	—
150	#417	—	227	1	no	—
151	#405	—	227	1	no	—
152	C₂×#41	—	227	1	no	—
153	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
154	#135	—	227	8	no	—
155	C₂×Q₁₆	—	227	1	no	—
156	C₂×SD₁₆	—	227	1	no	—
157	C₂×SD₁₆	—	227	1	no	—
158	C₂×D₈	—	227	1	no	—
159	C₂×M₁₆	—	227	1	no	—
160	C₂²×C₈	—	227	1	no	—
161	C₄×Q₈	—	227	2	no	—
162	C₄×Q₈	C₂×D₂₂₇	1	2	yes	—
163	C₄×D₄	—	2	2	no	—
164	C₄×D₄	—	454	2	no	—
165	C₄×D₄	—	227	2	no	—
166	C₄×D₄	C₂×D₂₂₇	1	2	yes	—
167	#117	—	2	1	no	—
168	#117	—	454	1	no	—
169	#117	C₂×D₂₂₇	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	—	454	1	no	—
172	C₂×C₄⋊C₄	—	227	1	no	—
173	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
174	#107	—	227	2	no	—
175	#107	—	227	2	no	—
176	#103	—	227	2	no	—
177	#103	—	227	2	no	—
178	#102	—	227	2	no	—
179	#102	—	227	2	no	—
180	#97	—	227	2	no	—
181	C₄×C₈	—	227	2	no	—
182	#41	—	2	2	no	—
183	#41	—	454	2	no	—
184	#41	—	227	4	no	—
185	#41	C₄×D₂₂₇	1	4	yes	complemented
186	C₂×Q₈	—	227	1	no	—
187	C₂×Q₈	C₂²×D₂₂₇	1	1	yes	characteristic
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	454	1	no	—
190	C₂×D₄	—	227	1	no	—
191	C₂×D₄	C₂²×D₂₂₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	454	1	no	—
196	C₂²×C₄	—	454	2	no	—
197	C₂²×C₄	—	454	1	no	—
198	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
199	Q₁₆	—	227	4	no	—
200	SD₁₆	—	227	4	no	—
201	SD₁₆	—	227	4	no	—
202	D₈	—	227	4	no	—
203	M₁₆	—	454	2	no	—
204	C₂×C₈	—	227	2	no	—
205	C₂×C₈	—	227	4	no	—
206	C₂×C₈	—	227	1	no	—
207	C₂×C₈	—	227	1	no	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	454	2	no	—
211	C₄⋊C₄	—	454	1	no	—
212	C₄⋊C₄	—	227	2	no	—
213	C₄⋊C₄	C₂²⋊D₂₂₇	1	1	yes	characteristic
214	C₄⋊C₄	C₂²⋊D₂₂₇	1	1	yes	characteristic
215	C₄⋊C₄	C₂²×D₂₂₇	1	2	yes	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	454	2	no	—
218	C₂²⋊C₄	C₂²⋊D₂₂₇	1	2	yes	—
219	C₄²	—	2	2	no	—
220	C₄²	—	454	2	no	—
221	C₄²	C₂²×D₂₂₇	1	2	yes	—
222	C₂³	—	2	1	no	—
223	C₂³	—	454	1	no	—
224	C₂³	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	—	454	1	no	—
227	Q₈	—	227	2	no	—
228	Q₈	C₂×C₄×D₂₂₇	1	2	yes	—
229	D₄	—	2	4	no	—
230	D₄	—	2	1	no	—
231	D₄	—	454	4	no	—
232	D₄	—	454	1	no	—
233	D₄	—	227	2	no	—
234	D₄	C₂×C₄×D₂₂₇	1	2	yes	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	8	2	no	—
239	C₂×C₄	—	8	1	no	—
240	C₂×C₄	—	1816	2	no	—
241	C₂×C₄	—	1816	1	no	—
242	C₂×C₄	—	454	2	no	—
243	C₂×C₄	—	454	1	no	—
244	C₂×C₄	—	454	4	no	—
245	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
246	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
247	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	2	yes	—
248	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
249	C₂×C₄	#16813884	1	2	yes	—
250	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
251	C₈	—	454	2	no	—
252	C₈	—	227	2	no	—
253	C₈	—	227	2	no	—
254	C₂²	—	2	1	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	908	1	no	—
260	C₂²	—	908	1	no	—
261	C₂²	—	454	2	no	—
262	C₂²	C₄×C₂²⋊D₂₂₇	1	2	yes	—
263	C₂²	C₂²×C₂²⋊D₂₂₇	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	4	2	no	—
268	C₄	—	908	2	no	—
269	C₄	—	454	2	no	—
270	C₄	C₄×C₂²⋊D₂₂₇	1	2	yes	—
271	C₄	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
272	C₄	C₂×#16813884	1	1	yes	characteristic
273	C₂	—	2	2	no	—
274	C₂	—	4	2	no	—
275	C₂	—	908	2	no	—
276	C₂	C₂×C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
277	C₂	#28409835	1	1	yes	characteristic
278	C₂	#28409847	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₂₂₇
commutator	C₄×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	#520×C₂₂₇	C₂
socle	C₂²×C₂₂₇	C₂²×D₄
prime radical	C₂×#20300308	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#520	D₂₂₇
hypocenter	C₂₂₇	#2951
LCS	G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂² ◁ #117 ◁ #520

Complements

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

N	H ≅ G/N	type
2. #28409512	C₂	semidirect
3. #28409132	C₂	semidirect
4. #28409130	C₂	semidirect
5. C₄×#16813874	C₂	semidirect
6. C₄×#16813869	C₂	semidirect
7. C₂×#20300221	C₂	semidirect
8. #28409033	C₂	semidirect
9. #28409031	C₂	semidirect
10. #28408988	C₂	semidirect
11. #520×C₂₂₇	C₂	semidirect
13. #20300308	C₄	semidirect
43. #16813884	C₂×C₄	semidirect
50. #16813874	C₂×C₄	semidirect
51. #16813872	C₂×C₄	semidirect
52. #16813871	C₂×C₄	semidirect
53. #16813869	C₂×C₄	semidirect
55. C₂×C₂₂₇⋊C₈	C₂×C₄	semidirect
65. #41×C₂₂₇	C₂×C₄	semidirect
139. C₂₂₇	#2951	semidirect
143. #520	D₂₂₇	semidirect
185. #41	C₄×D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300308	2
#28409512	2
#28409132	2
#28409130	2
C₄×#16813874	2
C₄×#16813869	2
C₂×#20300221	2
#28409033	2
#28409031	2
#28408988	2
#520×C₂₂₇	2
#2951	227