Subgroups — #44408797

Order 29056.

410 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409579	C₂	1	1	yes	maximal, characteristic
2	#28409111	C₂	1	1	yes	maximal, characteristic, complemented
3	#28409110	C₂	1	1	yes	maximal, characteristic, complemented
4	#28409107	C₂	1	2	yes	maximal, complemented
5	#28409097	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈×C₂²⋊D₂₂₇	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	#28408998	C₂	1	2	yes	maximal, complemented
10	#28408997	C₂	1	2	yes	maximal, complemented
11	C₈×D₄×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16813884	C₂²	1	2	yes	—
13	C₂×#16813868	C₂²	1	2	yes	—
14	C₂×#16813867	C₂²	1	2	yes	—
15	C₂×C₈×D₂₂₇	C₂²	1	2	yes	—
16	#20300287	C₄	1	1	yes	characteristic
17	#20300284	C₄	1	1	yes	characteristic
18	#20300274	C₄	1	2	yes	—
19	#20300263	C₄	1	1	yes	characteristic
20	#20300261	C₄	1	2	yes	—
21	#20300259	C₄	1	1	yes	characteristic
22	C₄×D₉₀₈	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₂₂₇	C₂²	1	2	yes	—
24	C₄²×D₂₂₇	C₂²	1	1	yes	characteristic
25	C₄⋊(C₂₂₇⋊C₈)	C₂²	1	1	yes	characteristic
26	Dic₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
27	Dic₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
28	C₂²⋊(C₂₂₇⋊C₈)	C₂²	1	2	yes	—
29	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	—
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	—
33	(C₂×D₂₂₇)⋊C₈	C₂²	1	1	yes	characteristic
34	C₄×C₂₂₇⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₂₇	C₂²	1	2	yes	—
36	C₂²×C₈×C₂₂₇	C₂²	1	2	yes	—
37	C₄×D₄×C₂₂₇	C₂²	1	1	yes	characteristic
38	C₄⋊C₈×C₂₂₇	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
40	C₄×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
41	#16813884	—	4	2	no	—
42	#16813884	—	4	2	no	—
43	C₂×Dic₄₅₄	C₂×C₄	1	2	yes	—
44	C₂×D₉₀₈	C₂×C₄	1	1	yes	characteristic
45	C₂×D₉₀₈	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
49	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₂₂₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₂₇	C₂³	1	2	yes	—
52	C₂×C₄×D₂₂₇	C₂³	1	1	yes	characteristic
53	#16813868	—	4	2	no	—
54	#16813867	—	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	C₄⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
60	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
62	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
63	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	2	yes	—
64	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
65	C₄×Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
66	C₄×Dic₂₂₇	C₂³	1	1	yes	characteristic
67	C₂×D₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
69	C₂×C₈×C₂₂₇	—	4	2	no	—
70	C₂×C₈×C₂₂₇	C₂³	1	2	yes	—
71	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₂₂₇	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₂₂₇	C₂×C₄	1	2	yes	—
75	C₄²×C₂₂₇	C₂³	1	1	yes	characteristic
76	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
77	C₂²×D₂₂₇	C₂²×C₄	1	1	yes	characteristic
78	Dic₄₅₄	—	4	2	no	—
79	D₉₀₈	—	2	2	no	—
80	D₉₀₈	—	4	2	no	—
81	C₂²⋊D₂₂₇	—	2	4	no	—
82	C₂²⋊D₂₂₇	—	4	2	no	—
83	C₂²⋊D₂₂₇	—	4	2	no	—
84	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
85	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
86	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
87	C₂×Dic₂₂₇	C₂²×C₄	1	1	yes	characteristic
88	C₄×D₂₂₇	—	2	2	no	—
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	—	2	2	no	—
94	C₂₂₇⋊C₈	—	2	1	no	—
95	C₂₂₇⋊C₈	—	2	2	no	—
96	C₂₂₇⋊C₈	—	2	1	no	—
97	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
98	D₄×C₂₂₇	—	2	2	no	—
99	C₂×C₄×C₂₂₇	—	2	2	no	—
100	C₂×C₄×C₂₂₇	—	2	2	no	—
101	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
103	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
105	C₈×C₂₂₇	—	2	1	no	—
106	C₈×C₂₂₇	—	2	2	no	—
107	C₈×C₂₂₇	—	2	1	no	—
108	C₂²×C₂₂₇	—	2	2	no	—
109	C₂²×C₂₂₇	—	2	2	no	—
110	C₂²×C₂₂₇	—	2	2	no	—
111	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
112	C₂×D₂₂₇	—	2	1	no	—
113	C₂×D₂₂₇	—	2	2	no	—
114	C₂×D₂₂₇	—	2	1	no	—
115	C₂×D₂₂₇	—	2	1	no	—
116	C₂×D₂₂₇	—	2	2	no	—
117	C₂×D₂₂₇	—	2	2	no	—
118	C₄×C₂₂₇	—	2	2	no	—
119	C₄×C₂₂₇	—	2	1	no	—
120	C₄×C₂₂₇	#131	1	2	yes	—
121	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
122	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
123	Dic₂₂₇	—	2	1	no	—
124	Dic₂₂₇	—	2	2	no	—
125	Dic₂₂₇	—	2	1	no	—
126	Dic₂₂₇	#131	1	2	yes	—
127	C₂×C₂₂₇	—	4	2	no	—
128	C₂×C₂₂₇	#518	1	1	yes	characteristic
129	C₂×C₂₂₇	C₂×#131	1	1	yes	characteristic
130	C₂×C₂₂₇	C₂×#131	1	1	yes	characteristic
131	D₂₂₇	—	4	1	no	—
132	D₂₂₇	—	4	2	no	—
133	C₂₂₇	#2979	1	1	yes	characteristic, complemented, Sylow
134	#2979	—	227	1	no	maximal, Sylow
135	#532	—	227	1	no	—
136	#436	—	227	1	no	—
137	#436	—	227	2	no	—
138	#436	—	227	1	no	—
139	C₈×D₄	—	227	2	no	—
140	C₈×D₄	—	227	1	no	—
141	C₈×D₄	D₂₂₇	1	1	yes	characteristic, complemented
142	#431	—	227	1	no	—
143	#431	—	227	1	no	—
144	#408	—	227	2	no	—
145	#408	—	227	2	no	—
146	C₂×#41	—	227	2	no	—
147	C₂×M₁₆	—	227	2	no	—
148	C₂×M₁₆	—	227	2	no	—
149	C₂²×C₈	—	227	2	no	—
150	C₂²×C₈	C₂×D₂₂₇	1	2	yes	—
151	C₄⋊Q₈	—	227	1	no	—
152	#127	—	227	1	no	—
153	#124	—	227	2	no	—
154	#121	—	227	1	no	—
155	#121	—	227	1	no	—
156	#121	—	227	2	no	—
157	C₄×D₄	—	227	2	no	—
158	C₄×D₄	—	227	1	no	—
159	C₄×D₄	C₂×D₂₂₇	1	1	yes	characteristic
160	C₂×C₄²	—	227	1	no	—
161	C₄⋊C₈	—	227	1	no	—
162	C₄⋊C₈	—	227	1	no	—
163	C₄⋊C₈	—	227	1	no	—
164	C₄⋊C₈	C₂×D₂₂₇	1	1	yes	characteristic
165	C₂²⋊C₈	—	227	2	no	—
166	C₂²⋊C₈	—	227	1	no	—
167	C₂²⋊C₈	—	227	2	no	—
168	C₂²⋊C₈	—	227	1	no	—
169	C₂²⋊C₈	—	227	2	no	—
170	C₂²⋊C₈	—	227	2	no	—
171	C₂²⋊C₈	C₂×D₂₂₇	1	2	yes	—
172	C₄×C₈	—	227	2	no	—
173	C₄×C₈	—	227	1	no	—
174	C₄×C₈	C₂×D₂₂₇	1	1	yes	characteristic
175	#41	—	908	2	no	—
176	#41	—	908	2	no	—
177	C₂×Q₈	—	227	2	no	—
178	C₂×D₄	—	227	2	no	—
179	C₂×D₄	—	227	2	no	—
180	C₂×D₄	—	227	2	no	—
181	C₂×D₄	—	227	2	no	—
182	C₂×D₄	—	227	1	no	—
183	C₂×D₄	C₄×D₂₂₇	1	1	yes	characteristic
184	C₂²×C₄	—	227	1	no	—
185	C₂²×C₄	—	227	1	no	—
186	C₂²×C₄	—	227	1	no	—
187	C₂²×C₄	—	227	2	no	—
188	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
189	M₁₆	—	908	2	no	—
190	M₁₆	—	908	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	908	2	no	—
193	C₂×C₈	—	227	1	no	—
194	C₂×C₈	—	227	2	no	—
195	C₂×C₈	—	227	1	no	—
196	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
197	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
198	C₂×C₈	C₂²×D₂₂₇	1	1	yes	characteristic
199	C₄⋊C₄	—	227	2	no	—
200	C₄⋊C₄	—	227	1	no	—
201	C₄⋊C₄	C₄×D₂₂₇	1	1	yes	characteristic
202	C₂²⋊C₄	—	227	2	no	—
203	C₂²⋊C₄	—	227	2	no	—
204	C₂²⋊C₄	—	227	2	no	—
205	C₂²⋊C₄	C₄×D₂₂₇	1	2	yes	—
206	C₄²	—	227	1	no	—
207	C₄²	—	227	1	no	—
208	C₄²	—	227	1	no	—
209	C₄²	C₂²×D₂₂₇	1	1	yes	characteristic
210	C₂³	—	227	2	no	—
211	C₂³	—	227	1	no	—
212	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
213	Q₈	—	908	2	no	—
214	D₄	—	2	2	no	—
215	D₄	—	908	2	no	—
216	D₄	—	908	2	no	—
217	D₄	—	908	2	no	—
218	D₄	—	454	4	no	—
219	D₄	—	454	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	454	2	no	—
223	C₂×C₄	—	454	2	no	—
224	C₂×C₄	—	454	2	no	—
225	C₂×C₄	—	454	1	no	—
226	C₂×C₄	—	454	2	no	—
227	C₂×C₄	—	454	1	no	—
228	C₂×C₄	—	227	1	no	—
229	C₂×C₄	—	227	1	no	—
230	C₂×C₄	—	227	2	no	—
231	C₂×C₄	—	227	1	no	—
232	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
234	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	2	no	—
238	C₈	—	2	1	no	—
239	C₈	—	454	1	no	—
240	C₈	—	454	2	no	—
241	C₈	—	454	1	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	454	1	no	—
246	C₂²	—	454	2	no	—
247	C₂²	—	454	1	no	—
248	C₂²	—	454	2	no	—
249	C₂²	—	454	1	no	—
250	C₂²	—	454	2	no	—
251	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	2	1	no	—
254	C₄	—	454	1	no	—
255	C₄	—	454	2	no	—
256	C₄	—	454	1	no	—
257	C₄	—	227	2	no	—
258	C₄	#20300294	1	2	yes	—
259	C₄	#20300333	1	1	yes	characteristic
260	C₄	#20300334	1	1	yes	characteristic
261	C₂	—	4	2	no	—
262	C₂	—	908	1	no	—
263	C₂	—	908	2	no	—
264	C₂	#28409504	1	1	yes	characteristic
265	C₂	#131×D₂₂₇	1	1	yes	characteristic
266	C₂	C₂×#20300294	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting	C₈×D₄×C₂₂₇	C₂
prime radical	#28409579	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₈×D₄	D₂₂₇
hypocenter	C₂₂₇	#2979
LCS	G ▷ C₂²×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. #28409111	C₂	semidirect
3. #28409110	C₂	semidirect
4. #28409107	C₂	semidirect
5. #28409097	C₂	semidirect
6. C₈×C₂²⋊D₂₂₇	C₂	semidirect
7. #28409084	C₂	semidirect
8. #28409083	C₂	semidirect
9. #28408998	C₂	semidirect
10. #28408997	C₂	semidirect
11. C₈×D₄×C₂₂₇	C₂	semidirect
133. C₂₂₇	#2979	semidirect
141. C₈×D₄	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#28409579	2
#28409111	2
#28409110	2
#28409107	2
#28409097	2
C₈×C₂²⋊D₂₂₇	2
#28409084	2
#28409083	2
#28408998	2
#28408997	2
C₈×D₄×C₂₂₇	2
#2979	227