Subgroups — #44408792

Order 29056.

402 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409539	C₂	1	1	yes	maximal, characteristic
2	#28409108	C₂	1	2	yes	maximal, complemented
3	#28409107	C₂	1	2	yes	maximal, complemented
4	#28409103	C₂	1	2	yes	maximal, complemented
5	#28409100	C₂	1	2	yes	maximal, complemented
6	#28409090	C₂	1	2	yes	maximal, complemented
7	#28408999	C₂	1	1	yes	maximal, characteristic, complemented
8	#28408994	C₂	1	2	yes	maximal, complemented
9	#408×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
11	C₂×#16813868	C₂²	1	1	yes	characteristic, complemented
12	C₂×#16813867	C₂²	1	2	yes	—
13	C₂×#16813867	C₂²	1	2	yes	—
14	C₂²×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
15	#20300269	C₄	1	1	yes	characteristic
16	#20300269	C₄	1	1	yes	characteristic
17	#20300264	C₄	1	2	yes	—
18	#20300259	C₄	1	2	yes	—
19	#20300256	C₄	1	1	yes	characteristic
20	#20300256	C₄	1	1	yes	characteristic
21	C₄×C₂²⋊D₂₂₇	C₂²	1	2	yes	—
22	C₄×C₂²⋊D₂₂₇	C₂²	1	2	yes	—
23	#20300247	C₂²	1	1	yes	characteristic
24	Dic₂₂₇⋊C₈	C₂²	1	2	yes	complemented
25	Dic₂₂₇⋊C₈	C₂²	1	2	yes	complemented
26	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
27	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	complemented
28	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	—
29	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	—
30	#20300198	C₂²	1	2	yes	complemented
31	C₈×Dic₂₂₇	C₂²	1	2	yes	complemented
32	C₂×#41×C₂₂₇	C₂²	1	1	yes	characteristic
33	C₂×M₁₆×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
34	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic, complemented
35	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	complemented
37	C₂³×D₂₂₇	C₂×C₄	1	1	yes	characteristic
38	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
39	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
40	C₂²×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×D₂₂₇	—	4	1	no	—
42	C₂×C₄×D₂₂₇	—	4	1	no	—
43	C₂×C₄×D₂₂₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₂₂₇	C₂³	1	2	yes	—
45	#16813868	—	4	1	no	—
46	#16813867	—	4	2	no	—
47	#16813867	—	4	2	no	—
48	C₂×C₂₂₇⋊C₈	—	4	1	no	—
49	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
50	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
51	C₄⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
52	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
53	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
54	C₂²⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
55	C₂²⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
56	C₂²⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
57	C₂²⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
58	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
59	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
60	C₄×Dic₂₂₇	C₂³	1	2	yes	—
61	#41×C₂₂₇	—	4	1	no	—
62	#41×C₂₂₇	—	4	1	no	—
63	C₂×Q₈×C₂₂₇	C₂×C₄	1	1	yes	characteristic
64	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
65	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
66	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
70	M₁₆×C₂₂₇	—	4	1	no	—
71	C₂×C₈×C₂₂₇	—	4	1	no	—
72	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
73	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
74	C₂²×D₂₂₇	—	4	1	no	—
75	C₂²×D₂₂₇	—	4	2	no	—
76	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
77	C₂²⋊D₂₂₇	—	4	2	no	—
78	C₂²⋊D₂₂₇	—	4	2	no	—
79	C₂×Dic₂₂₇	—	4	1	no	—
80	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
81	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
82	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
83	C₄×D₂₂₇	—	2	2	no	—
84	C₄×D₂₂₇	—	2	2	no	—
85	C₄×D₂₂₇	—	4	2	no	—
86	C₂₂₇⋊C₈	—	2	2	no	—
87	C₂₂₇⋊C₈	—	2	2	no	—
88	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
89	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
90	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
91	Q₈×C₂₂₇	—	4	1	no	—
92	D₄×C₂₂₇	—	4	1	no	—
93	D₄×C₂₂₇	—	4	1	no	—
94	D₄×C₂₂₇	—	4	1	no	—
95	C₂×C₄×C₂₂₇	—	2	1	no	—
96	C₂×C₄×C₂₂₇	—	2	1	no	—
97	C₂×C₄×C₂₂₇	—	2	1	no	—
98	C₂×C₄×C₂₂₇	—	2	1	no	—
99	C₂×C₄×C₂₂₇	—	2	1	no	—
100	C₂×C₄×C₂₂₇	—	2	1	no	—
101	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
105	C₈×C₂₂₇	—	2	2	no	—
106	C₈×C₂₂₇	—	2	2	no	—
107	C₂²×C₂₂₇	—	2	1	no	—
108	C₂²×C₂₂₇	—	2	1	no	—
109	C₂²×C₂₂₇	—	2	1	no	—
110	C₂²×C₂₂₇	—	2	2	no	—
111	C₂²×C₂₂₇	—	2	2	no	—
112	C₂²×C₂₂₇	—	2	2	no	—
113	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
114	C₂×D₂₂₇	—	2	2	no	—
115	C₂×D₂₂₇	—	2	2	no	—
116	C₂×D₂₂₇	—	2	2	no	—
117	C₂×D₂₂₇	—	4	4	no	—
118	C₄×C₂₂₇	—	2	1	no	—
119	C₄×C₂₂₇	—	2	1	no	—
120	C₄×C₂₂₇	—	2	1	no	—
121	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₂₂₇	—	2	2	no	—
124	Dic₂₂₇	—	2	2	no	—
125	Dic₂₂₇	—	2	2	no	—
126	C₂×C₂₂₇	—	4	1	no	—
127	C₂×C₂₂₇	—	4	1	no	—
128	C₂×C₂₂₇	—	4	1	no	—
129	C₂×C₂₂₇	#518	1	1	yes	characteristic
130	C₂×C₂₂₇	#568	1	2	yes	—
131	D₂₂₇	—	4	2	no	—
132	C₂₂₇	#2978	1	1	yes	characteristic, complemented, Sylow
133	#2978	—	227	1	no	maximal, Sylow
134	#525	—	227	1	no	—
135	#436	—	227	2	no	—
136	#436	—	227	2	no	—
137	#435	—	227	2	no	—
138	#435	—	227	2	no	—
139	#433	—	227	2	no	—
140	#408	—	227	1	no	—
141	#408	D₂₂₇	1	1	yes	characteristic, complemented
142	#407	—	227	2	no	—
143	C₂×#41	C₂×D₂₂₇	1	1	yes	characteristic
144	C₂³×C₄	—	227	1	no	—
145	C₂×M₁₆	—	227	1	no	—
146	C₂×M₁₆	—	227	2	no	—
147	C₂×M₁₆	—	227	2	no	—
148	C₂×M₁₆	C₂×D₂₂₇	1	1	yes	characteristic, complemented
149	C₂²×C₈	—	227	1	no	—
150	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic, complemented
151	#123	—	227	1	no	—
152	#123	—	227	1	no	—
153	C₂²⋊Q₈	—	227	2	no	—
154	#121	—	227	2	no	—
155	#120	—	227	1	no	—
156	#120	—	227	1	no	—
157	C₄×D₄	—	227	2	no	—
158	C₄×D₄	—	227	2	no	—
159	#117	—	227	1	no	—
160	C₄⋊C₈	—	227	2	no	—
161	C₄⋊C₈	—	227	2	no	—
162	C₂²⋊C₈	—	227	2	no	—
163	C₂²⋊C₈	—	227	2	no	—
164	C₂²⋊C₈	—	227	2	no	—
165	C₂²⋊C₈	—	227	2	no	—
166	C₂²⋊C₈	C₂×D₂₂₇	1	2	yes	—
167	C₂²⋊C₈	C₂×D₂₂₇	1	2	yes	complemented
168	#97	—	227	2	no	—
169	C₄×C₈	—	227	2	no	—
170	C₂⁴	—	227	1	no	—
171	#41	—	4	1	no	—
172	#41	—	4	1	no	—
173	C₂×Q₈	C₄×D₂₂₇	1	1	yes	characteristic
174	C₂×D₄	—	227	2	no	—
175	C₂×D₄	—	227	2	no	—
176	C₂×D₄	C₄×D₂₂₇	1	1	yes	characteristic
177	C₂×D₄	C₄×D₂₂₇	1	1	yes	characteristic
178	C₂×D₄	C₄×D₂₂₇	1	1	yes	characteristic
179	C₂²×C₄	—	908	1	no	—
180	C₂²×C₄	—	908	1	no	—
181	C₂²×C₄	—	227	1	no	—
182	C₂²×C₄	—	227	2	no	—
183	C₂²×C₄	—	227	2	no	—
184	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
187	M₁₆	—	4	1	no	—
188	M₁₆	—	908	2	no	—
189	M₁₆	—	908	2	no	—
190	M₁₆	—	908	1	no	—
191	C₂×C₈	—	4	1	no	—
192	C₂×C₈	—	908	1	no	—
193	C₂×C₈	—	227	2	no	—
194	C₂×C₈	—	227	2	no	—
195	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
196	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
197	C₄⋊C₄	—	227	2	no	—
198	C₄⋊C₄	—	227	2	no	—
199	C₄⋊C₄	—	227	2	no	—
200	C₂²⋊C₄	—	227	1	no	—
201	C₂²⋊C₄	—	227	1	no	—
202	C₂²⋊C₄	—	227	2	no	—
203	C₂²⋊C₄	—	227	2	no	—
204	C₂²⋊C₄	—	227	2	no	—
205	C₂²⋊C₄	—	227	2	no	—
206	C₄²	—	227	2	no	—
207	C₂³	—	908	1	no	—
208	C₂³	—	908	2	no	—
209	C₂³	—	227	2	no	—
210	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
211	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
212	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
213	Q₈	—	4	1	no	—
214	D₄	—	4	1	no	—
215	D₄	—	4	1	no	—
216	D₄	—	4	1	no	—
217	D₄	—	908	2	no	—
218	D₄	—	908	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	908	2	no	—
226	C₂×C₄	—	908	1	no	—
227	C₂×C₄	—	454	2	no	—
228	C₂×C₄	—	454	2	no	—
229	C₂×C₄	—	227	2	no	—
230	C₂×C₄	—	227	2	no	—
231	C₂×C₄	—	227	2	no	—
232	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	454	2	no	—
239	C₈	—	454	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	908	4	no	—
247	C₂²	—	454	2	no	—
248	C₂²	—	454	2	no	—
249	C₂²	—	454	2	no	—
250	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	454	2	no	—
255	C₄	—	454	2	no	—
256	C₄	—	454	2	no	—
257	C₄	#20300333	1	1	yes	characteristic
258	C₄	#20300333	1	1	yes	characteristic
259	C₂	—	4	1	no	—
260	C₂	—	4	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	908	2	no	—
263	C₂	#28409505	1	1	yes	characteristic
264	C₂	#28409797	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting	#408×C₂₂₇	C₂
prime radical	#28409539	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	#408	D₂₂₇
hypocenter	C₂₂₇	#2978
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. #28409108	C₂	semidirect
3. #28409107	C₂	semidirect
4. #28409103	C₂	semidirect
5. #28409100	C₂	semidirect
6. #28409090	C₂	semidirect
7. #28408999	C₂	semidirect
8. #28408994	C₂	semidirect
9. #408×C₂₂₇	C₂	semidirect
11. C₂×#16813868	C₂²	semidirect
14. C₂²×C₂₂₇⋊C₈	C₂²	semidirect
24. Dic₂₂₇⋊C₈	C₂²	semidirect
25. Dic₂₂₇⋊C₈	C₂²	semidirect
27. C₂²⋊(C₂₂₇⋊C₈)	C₂²	semidirect
30. #20300198	C₂²	semidirect
31. C₈×Dic₂₂₇	C₂²	semidirect
33. C₂×M₁₆×C₂₂₇	C₂²	semidirect
34. C₂²×C₈×C₂₂₇	C₂²	semidirect
36. C₂²⋊C₈×C₂₂₇	C₂²	semidirect
132. C₂₂₇	#2978	semidirect
141. #408	D₂₂₇	semidirect
148. C₂×M₁₆	C₂×D₂₂₇	semidirect
150. C₂²×C₈	C₂×D₂₂₇	semidirect
167. C₂²⋊C₈	C₂×D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#28409539	2
#28409108	2
#28409107	2
#28409103	2
#28409100	2
#28409090	2
#28408999	2
#28408994	2
#408×C₂₂₇	2
#2978	227