Subgroups — #44402590

Order 29056.

532 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#20300247	C₂	1	1	yes	maximal, characteristic
2	#28408917	C₂	1	2	yes	maximal, complemented
3	#28408915	C₂	1	2	yes	maximal, complemented
4	#28408905	C₂	1	2	yes	maximal
5	#28408899	C₂	1	1	yes	maximal, characteristic, complemented
6	#28408896	C₂	1	1	yes	maximal, characteristic, complemented
7	#28408895	C₂	1	1	yes	maximal, characteristic
8	#28408883	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₂²⋊Dic₂₂₇	C₂	1	2	yes	maximal
10	#28408877	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#95×C₂₂₇	C₂	1	1	yes	maximal, characteristic
12	#20300247	C₄	1	8	yes	complemented
13	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
16	C₂×Dic₂₂₇⋊C₄	C₂²	1	2	yes	—
17	C₂×C₂²⋊Dic₂₂₇	C₂²	1	2	yes	—
18	C₂×C₂²⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊Dic₂₂₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
21	C₂×C₄×Dic₂₂₇	C₂²	1	2	yes	—
22	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
23	C₂×C₄×Dic₂₂₇	C₂²	1	1	yes	characteristic
24	C₂₂₇⋊#95	C₂²	1	1	yes	characteristic
25	C₂₂₇⋊#95	C₂²	1	2	yes	—
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	C₂³×C₄×C₂₂₇	C₂²	1	2	yes	—
32	C₂³×C₄×C₂₂₇	C₂²	1	1	yes	characteristic
33	#95×C₂₂₇	C₂²	1	1	yes	characteristic
34	#95×C₂₂₇	C₂²	1	1	yes	characteristic
35	#95×C₂₂₇	C₂²	1	2	yes	—
36	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
37	C₂²×Dic₂₂₇	C₂³	1	2	yes	—
38	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
39	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
40	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
41	C₂²×Dic₂₂₇	C₂³	1	1	yes	characteristic
42	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	complemented
43	C₄⋊Dic₂₂₇	C₂×C₄	1	4	yes	complemented
44	Dic₂₂₇⋊C₄	—	2	4	no	—
45	Dic₂₂₇⋊C₄	—	4	2	no	—
46	C₂²⋊Dic₂₂₇	—	2	4	no	—
47	C₂²⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
48	C₂²⋊Dic₂₂₇	C₂×C₄	1	4	yes	—
49	C₄×Dic₂₂₇	—	2	4	no	—
50	C₄×Dic₂₂₇	—	4	2	no	—
51	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	complemented
52	C₄×Dic₂₂₇	C₂×C₄	1	4	yes	complemented
53	C₂⁴×C₂₂₇	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₂₂₇	—	2	4	no	—
55	C₂²×C₄×C₂₂₇	—	2	4	no	—
56	C₂²×C₄×C₂₂₇	—	2	4	no	—
57	C₂²×C₄×C₂₂₇	—	4	2	no	—
58	C₂²×C₄×C₂₂₇	C₂×C₄	1	4	yes	—
59	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₂₂₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₂₂₇	C₂³	1	2	yes	—
63	C₂×Dic₂₂₇	—	2	2	no	—
64	C₂×Dic₂₂₇	—	2	2	no	—
65	C₂×Dic₂₂₇	—	2	1	no	—
66	C₂×Dic₂₂₇	—	2	1	no	—
67	C₂×Dic₂₂₇	—	2	1	no	—
68	C₂×Dic₂₂₇	—	2	1	no	—
69	C₂×Dic₂₂₇	—	2	2	no	—
70	C₂×Dic₂₂₇	—	2	2	no	—
71	C₂×Dic₂₂₇	—	2	2	no	—
72	C₂×Dic₂₂₇	—	2	2	no	—
73	C₂×Dic₂₂₇	—	2	2	no	—
74	C₂×Dic₂₂₇	—	2	2	no	—
75	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
76	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
77	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
78	C₂×Dic₂₂₇	C₂²×C₄	1	2	yes	—
79	C₂³×C₂₂₇	—	2	2	no	—
80	C₂³×C₂₂₇	—	2	1	no	—
81	C₂³×C₂₂₇	—	2	1	no	—
82	C₂³×C₂₂₇	C₂²×C₄	1	2	yes	—
83	C₂³×C₂₂₇	#41	1	4	yes	—
84	C₂³×C₂₂₇	C₂⁴	1	1	yes	characteristic
85	C₂×C₄×C₂₂₇	—	2	8	no	—
86	C₂×C₄×C₂₂₇	—	2	2	no	—
87	C₂×C₄×C₂₂₇	—	2	2	no	—
88	C₂×C₄×C₂₂₇	—	2	2	no	—
89	C₂×C₄×C₂₂₇	—	2	2	no	—
90	C₂×C₄×C₂₂₇	—	2	2	no	—
91	C₂×C₄×C₂₂₇	—	2	2	no	—
92	C₂×C₄×C₂₂₇	—	2	2	no	—
93	C₂×C₄×C₂₂₇	—	4	8	no	—
94	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
95	C₂×C₄×C₂₂₇	C₂²×C₄	1	2	yes	—
96	C₂×C₄×C₂₂₇	#41	1	4	yes	—
97	C₂²×C₂₂₇	—	2	2	no	—
98	C₂²×C₂₂₇	—	2	2	no	—
99	C₂²×C₂₂₇	—	2	4	no	—
100	C₂²×C₂₂₇	—	2	4	no	—
101	C₂²×C₂₂₇	#117	1	4	yes	—
102	C₂²×C₂₂₇	C₂³×C₄	1	1	yes	characteristic
103	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
104	C₂²×C₂₂₇	C₂×#41	1	2	yes	—
105	C₂²×C₂₂₇	H(5,𝔽₂)	1	1	yes	characteristic
106	C₂²×C₂₂₇	2₋¹⁺⁴	1	1	yes	characteristic
107	C₄×C₂₂₇	—	2	2	no	—
108	C₄×C₂₂₇	—	2	2	no	—
109	C₄×C₂₂₇	—	2	4	no	—
110	C₄×C₂₂₇	—	4	2	no	—
111	Dic₂₂₇	—	2	2	no	—
112	Dic₂₂₇	—	2	2	no	—
113	Dic₂₂₇	—	2	2	no	—
114	Dic₂₂₇	—	2	2	no	—
115	Dic₂₂₇	—	4	2	no	—
116	Dic₂₂₇	—	4	2	no	—
117	C₂×C₂₂₇	—	2	4	no	—
118	C₂×C₂₂₇	C₂×#117	1	1	yes	characteristic
119	C₂×C₂₂₇	C₄×#41	1	1	yes	characteristic
120	C₂×C₂₂₇	#520	1	1	yes	characteristic
121	C₂×C₂₂₇	#551	1	2	yes	—
122	C₂×C₂₂₇	#552	1	2	yes	—
123	C₂₂₇	#2343	1	1	yes	characteristic, complemented, Sylow
124	#2343	—	227	1	no	maximal, Sylow
125	C₂×#117	—	227	1	no	—
126	#387	—	227	2	no	—
127	#387	—	227	2	no	—
128	#385	—	227	2	no	—
129	#383	—	227	1	no	—
130	#382	—	227	1	no	—
131	#381	—	227	1	no	—
132	C₄×C₄⋊C₄	—	227	1	no	—
133	C₄×C₂²⋊C₄	—	227	2	no	—
134	#376	—	227	1	no	—
135	C₂×#95	D₂₂₇	1	1	yes	characteristic
136	C₂³×C₄	C₂×D₂₂₇	1	2	yes	—
137	C₂³×C₄	C₂×D₂₂₇	1	1	yes	characteristic
138	#117	—	227	8	no	—
139	C₂×C₄⋊C₄	—	227	2	no	—
140	C₂×C₄⋊C₄	—	227	2	no	—
141	C₂×C₄⋊C₄	—	227	1	no	—
142	C₂×C₄⋊C₄	—	227	1	no	—
143	C₂×C₂²⋊C₄	—	227	2	no	—
144	C₂×C₂²⋊C₄	—	227	1	no	—
145	C₂×C₂²⋊C₄	—	227	1	no	—
146	C₂×C₄²	—	227	2	no	—
147	C₂×C₄²	—	227	2	no	—
148	C₂×C₄²	—	227	1	no	—
149	C₂×C₄²	—	227	1	no	—
150	#95	—	227	1	no	—
151	#95	—	227	2	no	—
152	#95	—	227	2	no	—
153	#95	—	227	2	no	—
154	#95	—	227	1	no	—
155	#95	—	227	2	no	—
156	#95	—	227	2	no	—
157	#95	C₂×D₂₂₇	1	1	yes	characteristic
158	#95	C₂×D₂₂₇	1	1	yes	characteristic
159	#95	C₂×D₂₂₇	1	2	yes	—
160	C₂⁴	C₂²×D₂₂₇	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	4	2	no	—
165	C₂²×C₄	—	227	2	no	—
166	C₂²×C₄	—	227	2	no	—
167	C₂²×C₄	—	227	1	no	—
168	C₂²×C₄	—	227	1	no	—
169	C₂²×C₄	—	227	1	no	—
170	C₂²×C₄	—	227	1	no	—
171	C₂²×C₄	C₄×D₂₂₇	1	4	yes	—
172	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
173	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
174	C₂²×C₄	C₂²×D₂₂₇	1	1	yes	characteristic
175	C₂²×C₄	C₂²×D₂₂₇	1	2	yes	—
176	C₄⋊C₄	—	908	2	no	—
177	C₄⋊C₄	—	454	4	no	—
178	C₄⋊C₄	—	227	4	no	—
179	C₄⋊C₄	—	227	4	no	—
180	C₂²⋊C₄	—	454	4	no	—
181	C₂²⋊C₄	—	227	4	no	—
182	C₂²⋊C₄	—	227	4	no	—
183	C₄²	—	908	2	no	—
184	C₄²	—	454	4	no	—
185	C₄²	—	227	4	no	—
186	C₄²	—	227	4	no	—
187	C₂³	—	2	2	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	2	1	no	—
190	C₂³	C₂×C₄×D₂₂₇	1	2	yes	—
191	C₂³	#16813883	1	4	yes	—
192	C₂³	C₂³×D₂₂₇	1	1	yes	characteristic
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	8	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	4	8	no	—
202	C₂×C₄	—	454	2	no	—
203	C₂×C₄	—	454	2	no	—
204	C₂×C₄	—	454	2	no	—
205	C₂×C₄	—	454	2	no	—
206	C₂×C₄	—	454	2	no	—
207	C₂×C₄	—	454	2	no	—
208	C₂×C₄	—	454	2	no	—
209	C₂×C₄	—	454	2	no	—
210	C₂×C₄	—	454	1	no	—
211	C₂×C₄	—	454	1	no	—
212	C₂×C₄	—	454	1	no	—
213	C₂×C₄	—	454	1	no	—
214	C₂×C₄	—	227	2	no	—
215	C₂×C₄	—	227	2	no	—
216	C₂×C₄	—	227	2	no	—
217	C₂×C₄	—	227	2	no	—
218	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
219	C₂×C₄	C₂×C₄×D₂₂₇	1	2	yes	—
220	C₂×C₄	#16813884	1	4	yes	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	#20300244	1	4	yes	—
226	C₂²	C₂²×C₄×D₂₂₇	1	1	yes	characteristic
227	C₂²	C₂×#16813883	1	2	yes	—
228	C₂²	C₂×#16813884	1	2	yes	—
229	C₂²	#20300333	1	1	yes	characteristic
230	C₂²	#20300336	1	1	yes	characteristic
231	C₄	—	2	4	no	—
232	C₄	—	2	2	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	908	2	no	—
236	C₄	—	908	2	no	—
237	C₄	—	454	2	no	—
238	C₄	—	454	2	no	—
239	C₄	—	454	2	no	—
240	C₄	—	454	2	no	—
241	C₂	—	2	4	no	—
242	C₂	C₂×#20300244	1	1	yes	characteristic
243	C₂	C₄×#16813884	1	1	yes	characteristic
244	C₂	#28409513	1	1	yes	characteristic
245	C₂	#28409708	1	2	yes	—
246	C₂	#28409717	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₂₇
commutator	C₂²×C₂₂₇	C₂³×C₄
Fitting	C₂×#95×C₂₂₇	C₂
socle, cosocle	C₂³×C₂₂₇	C₂⁴
prime radical	C₂⁴×C₂₂₇	C₂³
hypercenter	C₂×#95	D₂₂₇
hypocenter	C₂₂₇	#2343
LCS	G ▷ C₂²×C₂₂₇ ▷ C₂₂₇
UCS	C₁ ◁ C₂³ ◁ C₂×#95

Complements

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

N	H ≅ G/N	type
2. #28408917	C₂	semidirect
3. #28408915	C₂	semidirect
5. #28408899	C₂	semidirect
6. #28408896	C₂	semidirect
8. #28408883	C₂	semidirect
10. #28408877	C₂	semidirect
12. #20300247	C₄	semidirect
42. C₄⋊Dic₂₂₇	C₂×C₄	semidirect
43. C₄⋊Dic₂₂₇	C₂×C₄	semidirect
51. C₄×Dic₂₂₇	C₂×C₄	semidirect
52. C₄×Dic₂₂₇	C₂×C₄	semidirect
123. C₂₂₇	#2343	semidirect

Maximal subgroups

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

subgroup	index
C₂×#20300247	2
#28408917	2
#28408915	2
#28408905	2
#28408899	2
#28408896	2
#28408895	2
#28408883	2
C₄×C₂²⋊Dic₂₂₇	2
#28408877	2
C₂×#95×C₂₂₇	2
#2343	227