Subgroups — #44408779

Order 29056.

412 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#28409543	C₂	1	1	yes	maximal, characteristic
2	#28409107	C₂	1	2	yes	maximal, complemented
3	#28409106	C₂	1	2	yes	maximal, complemented
4	#28409103	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₂₂₇	C₂	1	2	yes	maximal, complemented
6	#28409090	C₂	1	2	yes	maximal, complemented
7	#28408997	C₂	1	2	yes	maximal, complemented
8	#28408996	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₂²⋊C₈×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#16813884	C₂²	1	1	yes	characteristic
11	C₂×#16813868	C₂²	1	1	yes	characteristic
12	C₂×#16813868	C₂²	1	1	yes	characteristic
13	C₂×#16813867	C₂²	1	2	yes	complemented
14	C₂×C₈×D₂₂₇	C₂²	1	2	yes	complemented
15	#20300271	C₄	1	2	yes	—
16	#20300267	C₄	1	1	yes	characteristic
17	#20300267	C₄	1	1	yes	characteristic
18	#20300262	C₄	1	1	yes	characteristic
19	#20300262	C₄	1	1	yes	characteristic
20	#20300257	C₄	1	2	yes	—
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	—
28	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	complemented
29	(C₂×D₂₂₇)⋊C₈	C₂²	1	2	yes	complemented
30	#20300198	C₂²	1	2	yes	complemented
31	C₈×Dic₂₂₇	C₂²	1	2	yes	complemented
32	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
33	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
34	C₂²×C₈×C₂₂₇	C₂²	1	1	yes	characteristic
35	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
36	C₂²⋊C₈×C₂₂₇	C₂²	1	2	yes	—
37	#16813884	—	4	1	no	—
38	#16813884	—	4	1	no	—
39	C₂×Dic₄₅₄	C₂×C₄	1	1	yes	characteristic
40	C₂×D₉₀₈	C₂×C₄	1	1	yes	characteristic
41	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
42	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
43	C₂×C₂²⋊D₂₂₇	C₂×C₄	1	2	yes	—
44	C₂×C₄×D₂₂₇	C₂³	1	2	yes	—
45	#16813868	—	4	1	no	—
46	#16813868	—	4	1	no	—
47	#16813867	—	4	2	no	—
48	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
49	C₂×C₂₂₇⋊C₈	C₂³	1	2	yes	—
50	C₈×D₂₂₇	—	4	2	no	—
51	C₄⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
52	C₄⋊Dic₂₂₇	C₂×C₄	1	1	yes	characteristic
53	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
54	Dic₂₂₇⋊C₄	C₂×C₄	1	2	yes	—
55	C₂²⋊Dic₂₂₇	C₂×C₄	1	2	yes	—
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	C₂⁴×C₂₂₇	C₂×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₂₂₇	—	4	1	no	—
63	C₂²×C₄×C₂₂₇	—	4	1	no	—
64	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₂₂₇	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₄×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	C₂×C₈×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	C₂²×C₄	1	2	yes	—
75	Dic₄₅₄	—	4	1	no	—
76	D₉₀₈	—	4	1	no	—
77	C₂²⋊D₂₂₇	—	2	4	no	—
78	C₂²⋊D₂₂₇	—	4	2	no	—
79	C₂²⋊D₂₂₇	—	4	2	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	C₂₂₇⋊C₈	—	2	2	no	—
86	C₂₂₇⋊C₈	—	2	2	no	—
87	C₂³×C₂₂₇	—	2	2	no	—
88	C₂³×C₂₂₇	—	4	1	no	—
89	C₂³×C₂₂₇	—	4	1	no	—
90	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
91	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
92	C₂³×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₂₂₇	—	2	1	no	—
94	C₂×C₄×C₂₂₇	—	2	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	—	4	1	no	—
100	C₂×C₄×C₂₂₇	—	4	1	no	—
101	C₂×C₄×C₂₂₇	—	4	1	no	—
102	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₄×C₂₂₇	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₄×C₂₂₇	C₂⁴	1	1	yes	characteristic
106	C₈×C₂₂₇	—	2	2	no	—
107	C₈×C₂₂₇	—	2	2	no	—
108	C₂²×C₂₂₇	—	2	1	no	—
109	C₂²×C₂₂₇	—	2	1	no	—
110	C₂²×C₂₂₇	—	2	1	no	—
111	C₂²×C₂₂₇	—	2	1	no	—
112	C₂²×C₂₂₇	—	2	1	no	—
113	C₂²×C₂₂₇	—	2	1	no	—
114	C₂²×C₂₂₇	—	2	1	no	—
115	C₂²×C₂₂₇	—	2	1	no	—
116	C₂²×C₂₂₇	—	4	4	no	—
117	C₂²×C₂₂₇	#131	1	2	yes	—
118	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
119	C₂×D₂₂₇	—	2	2	no	—
120	C₂×D₂₂₇	—	2	2	no	—
121	C₂×D₂₂₇	—	2	2	no	—
122	C₄×C₂₂₇	—	2	1	no	—
123	C₄×C₂₂₇	—	2	1	no	—
124	C₄×C₂₂₇	—	2	1	no	—
125	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
126	C₄×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
127	Dic₂₂₇	—	2	2	no	—
128	Dic₂₂₇	—	2	2	no	—
129	Dic₂₂₇	—	2	2	no	—
130	C₂×C₂₂₇	—	2	2	no	—
131	C₂×C₂₂₇	—	4	1	no	—
132	C₂×C₂₂₇	—	4	1	no	—
133	C₂×C₂₂₇	#518	1	1	yes	characteristic
134	C₂×C₂₂₇	C₂×#131	1	1	yes	characteristic
135	C₂×C₂₂₇	#568	1	1	yes	characteristic
136	D₂₂₇	—	4	2	no	—
137	C₂₂₇	#2976	1	1	yes	characteristic, complemented, Sylow
138	#2976	—	227	1	no	maximal, Sylow
139	#525	—	227	1	no	—
140	#436	—	227	2	no	—
141	#435	—	227	2	no	—
142	#435	—	227	2	no	—
143	C₈×D₄	—	227	2	no	—
144	#433	—	227	2	no	—
145	#408	—	227	2	no	—
146	#407	—	227	1	no	—
147	C₂×C₂²⋊C₈	D₂₂₇	1	1	yes	characteristic, complemented
148	C₂×#41	—	227	1	no	—
149	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
150	C₂×M₁₆	—	227	1	no	—
151	C₂×M₁₆	—	227	1	no	—
152	C₂×M₁₆	—	227	2	no	—
153	C₂²×C₈	—	227	2	no	—
154	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
155	C₂²×C₈	C₂×D₂₂₇	1	1	yes	characteristic
156	#123	—	227	2	no	—
157	C₂²⋊Q₈	—	227	1	no	—
158	C₂²⋊Q₈	—	227	1	no	—
159	#121	—	227	1	no	—
160	#121	—	227	1	no	—
161	#120	—	227	2	no	—
162	C₄×D₄	—	227	2	no	—
163	C₄×D₄	—	227	2	no	—
164	#117	—	227	1	no	—
165	C₄⋊C₈	—	227	2	no	—
166	C₄⋊C₈	—	227	2	no	—
167	C₂²⋊C₈	—	227	2	no	—
168	C₂²⋊C₈	—	227	2	no	—
169	C₂²⋊C₈	—	227	2	no	—
170	C₂²⋊C₈	—	227	2	no	—
171	C₂²⋊C₈	C₂×D₂₂₇	1	2	yes	—
172	C₂²⋊C₈	C₂×D₂₂₇	1	2	yes	—
173	#97	—	227	2	no	—
174	C₄×C₈	—	227	2	no	—
175	C₂⁴	C₄×D₂₂₇	1	1	yes	characteristic
176	#41	—	908	1	no	—
177	#41	—	908	1	no	—
178	C₂×Q₈	—	227	1	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	C₂²×C₄	—	4	1	no	—
184	C₂²×C₄	—	4	1	no	—
185	C₂²×C₄	—	227	2	no	—
186	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
187	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
188	C₂²×C₄	C₄×D₂₂₇	1	1	yes	characteristic
189	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
190	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
191	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
192	M₁₆	—	908	2	no	—
193	M₁₆	—	908	1	no	—
194	M₁₆	—	908	1	no	—
195	C₂×C₈	—	4	1	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	908	2	no	—
198	C₂×C₈	—	227	2	no	—
199	C₂×C₈	—	227	2	no	—
200	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
201	C₂×C₈	C₂²×D₂₂₇	1	2	yes	—
202	C₄⋊C₄	—	227	2	no	—
203	C₄⋊C₄	—	227	2	no	—
204	C₄⋊C₄	—	227	1	no	—
205	C₄⋊C₄	—	227	1	no	—
206	C₂²⋊C₄	—	227	2	no	—
207	C₂²⋊C₄	—	227	2	no	—
208	C₂²⋊C₄	—	227	2	no	—
209	C₂²⋊C₄	—	227	2	no	—
210	C₂²⋊C₄	—	227	2	no	—
211	C₄²	—	227	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	4	1	no	—
214	C₂³	—	4	1	no	—
215	C₂³	—	227	2	no	—
216	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
217	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
218	C₂³	C₂×C₄×D₂₂₇	1	1	yes	characteristic
219	Q₈	—	908	1	no	—
220	D₄	—	908	1	no	—
221	D₄	—	908	2	no	—
222	D₄	—	908	2	no	—
223	D₄	—	454	4	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	4	1	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	454	2	no	—
234	C₂×C₄	—	454	2	no	—
235	C₂×C₄	—	227	2	no	—
236	C₂×C₄	—	227	2	no	—
237	C₂×C₄	—	227	2	no	—
238	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
240	C₂×C₄	C₂×C₄×D₂₂₇	1	1	yes	characteristic
241	C₂×C₄	C₂³×D₂₂₇	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	2	2	no	—
244	C₈	—	454	2	no	—
245	C₈	—	454	2	no	—
246	C₂²	—	2	1	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	1	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	454	2	no	—
256	C₂²	—	454	2	no	—
257	C₂²	—	454	2	no	—
258	C₂²	#20300294	1	2	yes	—
259	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	454	2	no	—
264	C₄	—	454	2	no	—
265	C₄	—	454	2	no	—
266	C₄	#20300333	1	1	yes	characteristic
267	C₄	#20300333	1	1	yes	characteristic
268	C₂	—	2	2	no	—
269	C₂	—	4	1	no	—
270	C₂	—	4	1	no	—
271	C₂	—	908	2	no	—
272	C₂	#28409505	1	1	yes	characteristic
273	C₂	C₂×#20300294	1	1	yes	characteristic
274	C₂	#28409798	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₂₇
commutator, socle	C₂²×C₂₂₇	C₂³×C₄
Fitting	C₂×C₂²⋊C₈×C₂₂₇	C₂
prime radical	#28409543	C₂
cosocle	C₂×C₄×C₂₂₇	C₂⁴
hypercenter	C₂×C₂²⋊C₈	D₂₂₇
hypocenter	C₂₂₇	#2976
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₂²⋊C₈

Complements

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

N	H ≅ G/N	type
2. #28409107	C₂	semidirect
3. #28409106	C₂	semidirect
4. #28409103	C₂	semidirect
5. C₈×C₂²⋊D₂₂₇	C₂	semidirect
6. #28409090	C₂	semidirect
7. #28408997	C₂	semidirect
8. #28408996	C₂	semidirect
9. C₂×C₂²⋊C₈×C₂₂₇	C₂	semidirect
13. C₂×#16813867	C₂²	semidirect
14. C₂×C₈×D₂₂₇	C₂²	semidirect
24. Dic₂₂₇⋊C₈	C₂²	semidirect
25. Dic₂₂₇⋊C₈	C₂²	semidirect
28. (C₂×D₂₂₇)⋊C₈	C₂²	semidirect
29. (C₂×D₂₂₇)⋊C₈	C₂²	semidirect
30. #20300198	C₂²	semidirect
31. C₈×Dic₂₂₇	C₂²	semidirect
137. C₂₂₇	#2976	semidirect
147. C₂×C₂²⋊C₈	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
#28409543	2
#28409107	2
#28409106	2
#28409103	2
C₈×C₂²⋊D₂₂₇	2
#28409090	2
#28408997	2
#28408996	2
C₂×C₂²⋊C₈×C₂₂₇	2
#2976	227