Subgroups — #44408649

Order 29056.

452 conjugacy classes of subgroups.

The subgroup lattice has 282 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	#28409798	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409152	C₂	1	4	yes	maximal, complemented
4	#28409147	C₂	1	4	yes	maximal, complemented
5	C₂×#20300223	C₂	1	1	yes	maximal, characteristic, complemented
6	#28409049	C₂	1	1	yes	maximal, characteristic, complemented
7	#28409048	C₂	1	1	yes	maximal, characteristic, complemented
8	#28408990	C₂	1	1	yes	maximal, characteristic, complemented
9	#568×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16813884	C₂²	1	1	yes	characteristic
11	#20300308	C₄	1	2	yes	complemented
12	#20300308	C₄	1	2	yes	complemented
13	#20300308	C₂²	1	2	yes	—
14	#20300308	C₂²	1	2	yes	—
15	C₂×#16813874	C₄	1	1	yes	characteristic, complemented
16	C₂×#16813872	C₄	1	1	yes	characteristic, complemented
17	C₂×#16813871	C₄	1	1	yes	characteristic, complemented
18	C₂×#16813869	C₄	1	1	yes	characteristic, complemented
19	#20300296	—	2	2	no	—
20	#20300294	C₂²	1	4	yes	complemented
21	C₂×#16813868	C₂²	1	1	yes	characteristic
22	M₁₆×D₂₂₇	—	2	2	no	—
23	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
24	#20300247	C₂²	1	1	yes	characteristic
25	#20300223	C₂²	1	2	yes	complemented
26	#20300223	C₂²	1	2	yes	complemented
27	#20300215	C₂²	1	4	yes	complemented
28	#20300214	C₂²	1	4	yes	complemented
29	#20300198	C₂²	1	2	yes	complemented
30	C₈×Dic₂₂₇	C₂²	1	2	yes	complemented
31	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
32	#131×C₂₂₇	—	2	2	no	—
33	#131×C₂₂₇	C₂²	1	4	yes	complemented
34	C₂×M₁₆×C₂₂₇	—	2	2	no	—
35	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
36	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic
37	#16813884	—	2	1	no	—
38	#16813884	—	2	1	no	—
39	#16813884	C₂×C₄	1	2	yes	—
40	#16813884	C₂³	1	2	yes	—
41	C₂×Dic₄₅₄	C₂×C₄	1	1	yes	characteristic
42	C₂×D₉₀₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₂²⋊D₂₂₇	—	2	1	no	—
44	C₂×C₄×D₂₂₇	—	2	1	no	—
45	#16813874	C₂×C₄	1	2	yes	—
46	#16813874	C₂×C₄	1	2	yes	—
47	#16813872	C₂×C₄	1	2	yes	—
48	#16813872	C₂×C₄	1	2	yes	—
49	#16813871	C₂×C₄	1	2	yes	—
50	#16813871	C₂×C₄	1	2	yes	—
51	#16813869	C₂×C₄	1	2	yes	—
52	#16813869	C₂×C₄	1	2	yes	—
53	#16813868	—	2	1	no	—
54	#16813868	C₂³	1	2	yes	—
55	#16813867	—	2	4	no	—
56	C₂×C₂₂₇⋊C₈	—	2	1	no	—
57	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂₂₇⋊C₈	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
62	C₂×C₂₂₇⋊C₈	C₂³	1	1	yes	characteristic
63	C₈×D₂₂₇	—	2	4	no	—
64	C₄⋊Dic₂₂₇	—	2	1	no	—
65	C₂²⋊Dic₂₂₇	—	2	1	no	—
66	C₄×Dic₂₂₇	C₂³	1	2	yes	—
67	#41×C₂₂₇	—	2	1	no	—
68	#41×C₂₂₇	—	2	1	no	—
69	#41×C₂₂₇	C₂×C₄	1	2	yes	—
70	#41×C₂₂₇	C₂³	1	2	yes	—
71	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
72	C₂×D₄×C₂₂₇	—	2	1	no	—
73	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₂₂₇	—	2	1	no	—
75	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
76	M₁₆×C₂₂₇	—	2	1	no	—
77	M₁₆×C₂₂₇	—	2	4	no	—
78	M₁₆×C₂₂₇	D₄	1	2	yes	—
79	M₁₆×C₂₂₇	D₄	1	2	yes	—
80	M₁₆×C₂₂₇	C₂³	1	2	yes	—
81	C₂×C₈×C₂₂₇	—	2	1	no	—
82	C₂×C₈×C₂₂₇	—	2	4	no	—
83	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
84	C₂²×D₂₂₇	—	2	1	no	—
85	Dic₄₅₄	—	2	1	no	—
86	Dic₄₅₄	C₂²×C₄	1	2	yes	—
87	D₉₀₈	—	2	1	no	—
88	D₉₀₈	C₂²×C₄	1	2	yes	—
89	C₂²⋊D₂₂₇	—	2	2	no	—
90	C₂²⋊D₂₂₇	—	2	2	no	—
91	C₂×Dic₂₂₇	—	2	1	no	—
92	C₂×Dic₂₂₇	—	2	2	no	—
93	C₄×D₂₂₇	—	2	2	no	—
94	C₄×D₂₂₇	—	2	2	no	—
95	C₂₂₇⋊C₈	—	2	2	no	—
96	C₂₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
99	C₂₂₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
100	C₂³×C₂₂₇	—	2	1	no	—
101	C₂³×C₂₂₇	#41	1	1	yes	characteristic
102	Q₈×C₂₂₇	—	2	1	no	—
103	Q₈×C₂₂₇	C₂²×C₄	1	2	yes	—
104	D₄×C₂₂₇	—	2	2	no	—
105	D₄×C₂₂₇	—	2	2	no	—
106	D₄×C₂₂₇	—	2	1	no	—
107	D₄×C₂₂₇	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₂₂₇	—	2	1	no	—
109	C₂×C₄×C₂₂₇	—	2	2	no	—
110	C₂×C₄×C₂₂₇	—	2	2	no	—
111	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₂₂₇	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₂₂₇	#41	1	1	yes	characteristic
116	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
117	C₈×C₂₂₇	—	2	2	no	—
118	C₈×C₂₂₇	C₂×D₄	1	4	yes	—
119	C₂²×C₂₂₇	—	2	2	no	—
120	C₂²×C₂₂₇	—	4	1	no	—
121	C₂²×C₂₂₇	—	8	1	no	—
122	C₂²×C₂₂₇	C₄×D₄	1	1	yes	characteristic
123	C₂²×C₂₂₇	C₄×D₄	1	1	yes	characteristic
124	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
125	C₂×D₂₂₇	—	2	2	no	—
126	C₂×D₂₂₇	—	8	1	no	—
127	C₄×C₂₂₇	—	2	2	no	—
128	C₄×C₂₂₇	C₄×D₄	1	1	yes	characteristic
129	C₄×C₂₂₇	C₄×D₄	1	1	yes	characteristic
130	C₄×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₂₂₇	C₂²×D₄	1	1	yes	characteristic
132	Dic₂₂₇	—	2	2	no	—
133	Dic₂₂₇	—	4	2	no	—
134	C₂×C₂₂₇	—	2	1	no	—
135	C₂×C₂₂₇	—	2	1	no	—
136	C₂×C₂₂₇	—	2	1	no	—
137	C₂×C₂₂₇	—	4	2	no	—
138	C₂×C₂₂₇	C₂×C₄×D₄	1	1	yes	characteristic
139	D₂₂₇	—	4	2	no	—
140	C₂₂₇	#2957	1	1	yes	characteristic, complemented, Sylow
141	#2957	—	227	1	no	maximal, Sylow
142	C₂×#135	—	227	1	no	—
143	#568	—	227	1	no	—
144	#568	D₂₂₇	1	1	yes	characteristic, complemented
145	#444	—	227	4	no	—
146	#443	—	227	4	no	—
147	C₂×#108	—	227	1	no	—
148	#421	—	227	1	no	—
149	#421	—	227	1	no	—
150	#405	—	227	1	no	—
151	C₂×#41	—	227	1	no	—
152	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
153	#135	—	227	2	no	—
154	#135	—	227	2	no	—
155	#135	—	227	2	no	—
156	#135	—	227	2	no	—
157	C₂×Q₁₆	—	227	1	no	—
158	C₂×SD₁₆	—	227	1	no	—
159	C₂×SD₁₆	—	227	1	no	—
160	C₂×D₈	—	227	1	no	—
161	#131	—	2	2	no	—
162	#131	—	454	2	no	—
163	#131	—	227	4	no	—
164	#131	C₂×D₂₂₇	1	4	yes	complemented
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	454	2	no	—
167	C₂×M₁₆	—	227	1	no	—
168	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
169	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic
170	C₂²×C₈	—	227	1	no	—
171	#117	—	227	1	no	—
172	#108	—	227	2	no	—
173	#108	—	227	2	no	—
174	#104	—	227	4	no	—
175	#104	—	227	4	no	—
176	#97	—	227	2	no	—
177	C₄×C₈	—	227	2	no	—
178	#41	—	2	1	no	—
179	#41	—	2	1	no	—
180	#41	—	454	1	no	—
181	#41	—	454	1	no	—
182	#41	—	227	2	no	—
183	#41	—	227	2	no	—
184	#41	C₄×D₂₂₇	1	2	yes	—
185	#41	C₂²×D₂₂₇	1	2	yes	—
186	C₂×Q₈	—	227	1	no	—
187	C₂×Q₈	C₄×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	C₄×D₂₂₇	1	1	yes	characteristic
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	454	1	no	—
194	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
195	Q₁₆	—	227	2	no	—
196	Q₁₆	—	227	2	no	—
197	SD₁₆	—	227	2	no	—
198	SD₁₆	—	227	2	no	—
199	SD₁₆	—	227	2	no	—
200	SD₁₆	—	227	2	no	—
201	D₈	—	227	2	no	—
202	D₈	—	227	2	no	—
203	M₁₆	—	2	1	no	—
204	M₁₆	—	2	4	no	—
205	M₁₆	—	454	1	no	—
206	M₁₆	—	454	4	no	—
207	M₁₆	—	227	2	no	—
208	M₁₆	C₂²⋊D₂₂₇	1	2	yes	—
209	M₁₆	C₂²⋊D₂₂₇	1	2	yes	—
210	M₁₆	C₂²×D₂₂₇	1	2	yes	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	454	1	no	—
214	C₂×C₈	—	454	4	no	—
215	C₂×C₈	—	227	1	no	—
216	C₂×C₈	—	227	1	no	—
217	C₂×C₈	—	227	1	no	—
218	C₂×C₈	—	227	1	no	—
219	C₂×C₈	—	227	1	no	—
220	C₂×C₈	—	227	1	no	—
221	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
222	C₄⋊C₄	—	454	1	no	—
223	C₂²⋊C₄	—	454	1	no	—
224	C₄²	—	227	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	—	454	1	no	—
227	C₂³	#16813884	1	1	yes	characteristic
228	Q₈	—	2	1	no	—
229	Q₈	—	454	1	no	—
230	Q₈	—	227	2	no	—
231	Q₈	C₂×C₄×D₂₂₇	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	2	2	no	—
234	D₄	—	2	1	no	—
235	D₄	—	454	2	no	—
236	D₄	—	454	2	no	—
237	D₄	—	454	1	no	—
238	D₄	—	227	2	no	—
239	D₄	C₂×C₄×D₂₂₇	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	454	2	no	—
244	C₂×C₄	—	454	1	no	—
245	C₂×C₄	—	454	2	no	—
246	C₂×C₄	—	454	2	no	—
247	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
248	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
249	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
250	C₂×C₄	C₂×C₂²⋊D₂₂₇	1	1	yes	characteristic
251	C₂×C₄	#16813884	1	1	yes	characteristic
252	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	454	2	no	—
255	C₈	—	227	1	no	—
256	C₈	—	227	1	no	—
257	C₈	—	227	1	no	—
258	C₈	—	227	1	no	—
259	C₈	C₂×C₂²⋊D₂₂₇	1	4	yes	—
260	C₂²	—	2	2	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	8	1	no	—
263	C₂²	—	1816	1	no	—
264	C₂²	—	454	2	no	—
265	C₂²	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
266	C₂²	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
267	C₂²	C₂×#16813884	1	1	yes	characteristic
268	C₄	—	2	2	no	—
269	C₄	—	908	2	no	—
270	C₄	—	454	2	no	—
271	C₄	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
272	C₄	C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic
273	C₄	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
274	C₄	C₂²×C₂²⋊D₂₂₇	1	1	yes	characteristic
275	C₂	—	2	1	no	—
276	C₂	—	2	1	no	—
277	C₂	—	2	1	no	—
278	C₂	—	4	2	no	—
279	C₂	—	908	2	no	—
280	C₂	C₂×C₄×C₂²⋊D₂₂₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₂₂₇
commutator	C₄×C₂₂₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₂₇
Fitting	#568×C₂₂₇	C₂
socle	C₂×C₂₂₇	C₂×C₄×D₄
prime radical	C₂×#20300308	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#568	D₂₂₇
hypocenter	C₂₂₇	#2957
LCS	G ▷ C₄×C₂₂₇ ▷ C₂×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #568

Complements

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

N	H ≅ G/N	type
2. #28409798	C₂	semidirect
3. #28409152	C₂	semidirect
4. #28409147	C₂	semidirect
5. C₂×#20300223	C₂	semidirect
6. #28409049	C₂	semidirect
7. #28409048	C₂	semidirect
8. #28408990	C₂	semidirect
9. #568×C₂₂₇	C₂	semidirect
11. #20300308	C₄	semidirect
12. #20300308	C₄	semidirect
15. C₂×#16813874	C₄	semidirect
16. C₂×#16813872	C₄	semidirect
17. C₂×#16813871	C₄	semidirect
18. C₂×#16813869	C₄	semidirect
20. #20300294	C₂²	semidirect
25. #20300223	C₂²	semidirect
26. #20300223	C₂²	semidirect
27. #20300215	C₂²	semidirect
28. #20300214	C₂²	semidirect
29. #20300198	C₂²	semidirect
30. C₈×Dic₂₂₇	C₂²	semidirect
33. #131×C₂₂₇	C₂²	semidirect
140. C₂₂₇	#2957	semidirect
144. #568	D₂₂₇	semidirect
164. #131	C₂×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
#28409798	2
#28409152	2
#28409147	2
C₂×#20300223	2
#28409049	2
#28409048	2
#28408990	2
#568×C₂₂₇	2
#2957	227