Subgroups — #44402618

Order 29056.

504 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300246	C₂	1	1	yes	maximal, characteristic
2	#28408931	C₂	1	1	yes	maximal, characteristic, complemented
3	#28408924	C₂	1	2	yes	maximal
4	#28408922	C₂	1	2	yes	maximal
5	#28408897	C₂	1	2	yes	maximal, complemented
6	#28408889	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₄×Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₄⋊Dic₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×(C₂×D₂₂₇)⋊C₄	C₂	1	1	yes	maximal, characteristic
10	#28408877	C₂	1	2	yes	maximal, complemented
11	#389×C₂₂₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₄×D₂₂₇	C₂²	1	1	yes	characteristic
13	#20300246	C₄	1	8	yes	complemented
14	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₂₂₇	C₂²	1	2	yes	—
16	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
17	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	2	yes	—
20	C₂×(C₂×D₂₂₇)⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
23	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
24	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
25	C₂₂₇⋊#95	C₂²	1	1	yes	characteristic
26	C₂₂₇⋊#95	C₂²	1	2	yes	—
27	C₂₂₇⋊#95	C₂²	1	2	yes	—
28	C₂₂₇⋊#95	C₂²	1	1	yes	characteristic
29	C₂₂₇⋊#95	C₂²	1	2	yes	—
30	C₂₂₇⋊#95	C₂²	1	2	yes	—
31	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	2	yes	—
33	C₂×C₄⋊C₄×C₂₂₇	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₂₂₇	C₂²	1	1	yes	characteristic
35	#95×C₂₂₇	C₂²	1	2	yes	—
36	C₂³×D₂₂₇	C₂³	1	1	yes	characteristic
37	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
38	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
39	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
41	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
42	C₂×C₄×D₂₂₇	—	2	2	no	—
43	C₂×C₄×D₂₂₇	—	2	2	no	—
44	C₂×C₄×D₂₂₇	C₂×C₄	1	4	yes	—
45	C₄⋊Dic₂₂₇	—	2	2	no	—
46	C₄⋊Dic₂₂₇	—	4	2	no	—
47	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
48	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
49	(C₂×D₂₂₇)⋊C₄	—	2	2	no	—
50	(C₂×D₂₂₇)⋊C₄	—	4	2	no	—
51	(C₂×D₂₂₇)⋊C₄	C₂×C₄	1	8	yes	—
52	C₄×Dic₂₂₇	—	2	4	no	—
53	C₄×Dic₂₂₇	—	2	4	no	—
54	C₄×Dic₂₂₇	—	2	2	no	—
55	C₄×Dic₂₂₇	C₂×C₄	1	8	yes	—
56	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₂₂₇	—	2	2	no	—
62	C₄⋊C₄×C₂₂₇	—	4	2	no	—
63	C₄⋊C₄×C₂₂₇	C₂×C₄	1	4	yes	—
64	C₄²×C₂₂₇	—	2	2	no	—
65	C₂²×D₂₂₇	—	2	1	no	—
66	C₂²×D₂₂₇	—	2	1	no	—
67	C₂²×D₂₂₇	—	2	1	no	—
68	C₂²×D₂₂₇	—	2	1	no	—
69	C₂²×D₂₂₇	—	2	1	no	—
70	C₂²×D₂₂₇	—	2	1	no	—
71	C₂²×D₂₂₇	C₂²×C₄	1	2	yes	—
72	C₂×Dic₂₂₇	—	2	2	no	—
73	C₂×Dic₂₂₇	—	2	2	no	—
74	C₂×Dic₂₂₇	—	2	1	no	—
75	C₂×Dic₂₂₇	—	2	1	no	—
76	C₂×Dic₂₂₇	—	2	1	no	—
77	C₂×Dic₂₂₇	—	2	1	no	—
78	C₂×Dic₂₂₇	—	2	1	no	—
79	C₂×Dic₂₂₇	—	2	2	no	—
80	C₂×Dic₂₂₇	—	2	1	no	—
81	C₂×Dic₂₂₇	—	2	2	no	—
82	C₂×Dic₂₂₇	—	2	2	no	—
83	C₂×Dic₂₂₇	—	2	1	no	—
84	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
85	C₂×Dic₂₂₇	C₂²×C₄	1	4	yes	—
86	C₂×Dic₂₂₇	#41	1	2	yes	—
87	C₄×D₂₂₇	—	2	8	no	—
88	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
89	C₂×C₄×C₂₂₇	—	2	1	no	—
90	C₂×C₄×C₂₂₇	—	2	2	no	—
91	C₂×C₄×C₂₂₇	—	2	2	no	—
92	C₂×C₄×C₂₂₇	—	2	1	no	—
93	C₂×C₄×C₂₂₇	—	2	1	no	—
94	C₂×C₄×C₂₂₇	—	2	2	no	—
95	C₂×C₄×C₂₂₇	—	2	2	no	—
96	C₂×C₄×C₂₂₇	—	2	1	no	—
97	C₂×C₄×C₂₂₇	—	2	1	no	—
98	C₂×C₄×C₂₂₇	—	2	2	no	—
99	C₂×C₄×C₂₂₇	C₂²×C₄	1	4	yes	—
100	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₂₂₇	#41	1	2	yes	—
102	C₂×C₄×C₂₂₇	#41	1	2	yes	—
103	C₂×C₄×C₂₂₇	#41	1	2	yes	—
104	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
105	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
106	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
107	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
108	C₂²×C₂₂₇	C₂×#41	1	1	yes	characteristic
109	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×D₂₂₇	—	2	2	no	—
112	C₂×D₂₂₇	—	2	2	no	—
113	C₂×D₂₂₇	—	2	2	no	—
114	C₂×D₂₂₇	—	4	1	no	—
115	C₂×D₂₂₇	—	4	1	no	—
116	C₂×D₂₂₇	—	4	1	no	—
117	C₂×D₂₂₇	—	4	1	no	—
118	C₄×C₂₂₇	—	2	4	no	—
119	C₄×C₂₂₇	—	2	2	no	—
120	C₄×C₂₂₇	—	4	1	no	—
121	C₄×C₂₂₇	—	4	2	no	—
122	C₄×C₂₂₇	#117	1	4	yes	—
123	Dic₂₂₇	—	2	2	no	—
124	Dic₂₂₇	—	2	4	no	—
125	Dic₂₂₇	—	2	2	no	—
126	Dic₂₂₇	—	4	1	no	—
127	Dic₂₂₇	—	4	2	no	—
128	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
129	C₂×C₂₂₇	C₄×#41	1	1	yes	characteristic
130	C₂×C₂₂₇	#520	1	1	yes	characteristic
131	C₂×C₂₂₇	#552	1	1	yes	characteristic
132	C₂×C₂₂₇	#552	1	1	yes	characteristic
133	C₂×C₂₂₇	#555	1	1	yes	characteristic
134	C₂×C₂₂₇	#555	1	1	yes	characteristic
135	D₂₂₇	—	4	2	no	—
136	C₂₂₇	#2346	1	1	yes	characteristic, complemented, Sylow
137	#2346	—	227	1	no	maximal, Sylow
138	C₂×#117	—	227	1	no	—
139	#389	—	227	1	no	—
140	#389	D₂₂₇	1	1	yes	characteristic, complemented
141	#388	—	227	2	no	—
142	#388	—	227	2	no	—
143	#382	—	227	2	no	—
144	#380	—	227	1	no	—
145	C₄×C₄⋊C₄	—	227	1	no	—
146	C₄×C₄⋊C₄	—	227	1	no	—
147	C₄×C₂²⋊C₄	—	227	1	no	—
148	#376	—	227	2	no	—
149	C₂³×C₄	—	227	1	no	—
150	#117	—	227	8	no	—
151	C₂×C₄⋊C₄	—	227	2	no	—
152	C₂×C₄⋊C₄	—	227	1	no	—
153	C₂×C₄⋊C₄	—	227	1	no	—
154	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂×D₂₂₇	1	2	yes	—
156	C₂×C₄⋊C₄	C₂×D₂₂₇	1	1	yes	characteristic
157	C₂×C₂²⋊C₄	—	227	2	no	—
158	C₂×C₂²⋊C₄	—	227	1	no	—
159	C₂×C₂²⋊C₄	—	227	1	no	—
160	C₂×C₂²⋊C₄	—	227	2	no	—
161	C₂×C₄²	—	227	2	no	—
162	C₂×C₄²	—	227	2	no	—
163	C₂×C₄²	—	227	1	no	—
164	C₂×C₄²	—	227	2	no	—
165	C₂×C₄²	C₂×D₂₂₇	1	1	yes	characteristic
166	#95	—	227	1	no	—
167	#95	—	227	2	no	—
168	#95	—	227	2	no	—
169	#95	—	227	1	no	—
170	#95	—	227	2	no	—
171	#95	—	227	2	no	—
172	#95	C₂×D₂₂₇	1	2	yes	—
173	C₂⁴	—	227	1	no	—
174	C₂²×C₄	—	454	2	no	—
175	C₂²×C₄	—	454	2	no	—
176	C₂²×C₄	—	227	1	no	—
177	C₂²×C₄	—	227	2	no	—
178	C₂²×C₄	—	227	1	no	—
179	C₂²×C₄	—	227	1	no	—
180	C₂²×C₄	—	227	2	no	—
181	C₂²×C₄	—	227	4	no	—
182	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
183	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
184	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
185	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
186	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	908	2	no	—
190	C₄⋊C₄	—	454	2	no	—
191	C₄⋊C₄	—	227	4	no	—
192	C₄⋊C₄	C₄×D₂₂₇	1	4	yes	—
193	C₂²⋊C₄	—	908	2	no	—
194	C₂²⋊C₄	—	454	2	no	—
195	C₂²⋊C₄	—	454	2	no	—
196	C₂²⋊C₄	—	227	8	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	454	4	no	—
199	C₄²	—	454	4	no	—
200	C₄²	—	454	2	no	—
201	C₄²	—	227	8	no	—
202	C₂³	—	454	1	no	—
203	C₂³	—	454	1	no	—
204	C₂³	—	454	1	no	—
205	C₂³	—	454	1	no	—
206	C₂³	—	454	1	no	—
207	C₂³	—	454	1	no	—
208	C₂³	—	227	2	no	—
209	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	454	1	no	—
221	C₂×C₄	—	454	2	no	—
222	C₂×C₄	—	454	1	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	—	454	1	no	—
229	C₂×C₄	—	454	2	no	—
230	C₂×C₄	—	454	1	no	—
231	C₂×C₄	—	454	1	no	—
232	C₂×C₄	—	454	8	no	—
233	C₂×C₄	—	227	2	no	—
234	C₂×C₄	—	227	4	no	—
235	C₂×C₄	—	227	2	no	—
236	C₂×C₄	C₂×C₄×D₂₂₇	1	4	yes	—
237	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
238	C₂×C₄	#16813883	1	2	yes	—
239	C₂×C₄	#16813885	1	2	yes	—
240	C₂×C₄	#16813885	1	2	yes	—
241	C₂²	—	908	1	no	—
242	C₂²	—	908	1	no	—
243	C₂²	—	908	1	no	—
244	C₂²	—	908	1	no	—
245	C₂²	—	454	2	no	—
246	C₂²	—	454	2	no	—
247	C₂²	—	454	2	no	—
248	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
249	C₂²	#41×D₂₂₇	1	1	yes	characteristic
250	C₂²	C₂×#16813883	1	1	yes	characteristic
251	C₂²	C₂×#16813885	1	1	yes	characteristic
252	C₂²	C₂×#16813885	1	1	yes	characteristic
253	C₂²	#20300333	1	1	yes	characteristic
254	C₂²	#20300335	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	2	4	no	—
257	C₄	—	4	2	no	—
258	C₄	—	4	1	no	—
259	C₄	—	908	1	no	—
260	C₄	—	908	2	no	—
261	C₄	—	454	2	no	—
262	C₄	—	454	2	no	—
263	C₄	—	454	4	no	—
264	C₄	#20300246	1	4	yes	—
265	C₂	—	908	2	no	—
266	C₂	C₂×#20300246	1	1	yes	characteristic
267	C₂	C₄×#16813885	1	1	yes	characteristic
268	C₂	#28409511	1	1	yes	characteristic
269	C₂	#28409720	1	1	yes	characteristic
270	C₂	#28409724	1	1	yes	characteristic
271	C₂	#28409746	1	1	yes	characteristic
272	C₂	#28409748	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting	#389×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
prime radical	C₂³×D₂₂₇	C₂³
hypercenter	#389	D₂₂₇
hypocenter	C₂₂₇	#2346
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ #389

Complements

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

N	H ≅ G/N	type
2. #28408931	C₂	semidirect
5. #28408897	C₂	semidirect
7. C₄⋊C₄×Dic₂₂₇	C₂	semidirect
8. C₄×C₄⋊Dic₂₂₇	C₂	semidirect
10. #28408877	C₂	semidirect
11. #389×C₂₂₇	C₂	semidirect
13. #20300246	C₄	semidirect
136. C₂₂₇	#2346	semidirect
140. #389	D₂₂₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300246	2
#28408931	2
#28408924	2
#28408922	2
#28408897	2
#28408889	2
C₄⋊C₄×Dic₂₂₇	2
C₄×C₄⋊Dic₂₂₇	2
C₄×(C₂×D₂₂₇)⋊C₄	2
#28408877	2
#389×C₂₂₇	2
#2346	227