Subgroups — C₂³×#539 #10585903

Order 512.

38914 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	C₂²×#539	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#126	C₂	1	2	yes	maximal, complemented
3	C₂³×#125	C₂	1	2	yes	maximal, complemented
4	C₂³×#123	C₂	1	2	yes	maximal, complemented
5	C₂³×#123	C₂	1	2	yes	maximal, complemented
6	C₂³×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
8	C₂³×#121	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₄×D₄	C₂	1	2	yes	maximal, complemented
10	C₂⁴×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂⁴×Q₈	C₂²	1	1	yes	characteristic
12	C₂⁴×D₄	C₂²	1	2	yes	—
13	C₂⁴×D₄	C₂²	1	1	yes	characteristic
14	C₂⁵×C₄	C₂²	1	1	yes	characteristic
15	C₂⁵×C₄	C₂²	1	1	yes	characteristic
16	C₂⁵×C₄	C₂²	1	1	yes	characteristic
17	C₂⁵×C₄	C₂²	1	2	yes	—
18	C₂×#539	C₂²	1	1792	yes	direct product factor
19	C₂²×#126	C₂²	1	112	yes	complemented
20	C₂²×#125	C₂²	1	112	yes	complemented
21	C₂²×#123	C₂²	1	112	yes	complemented
22	C₂²×#123	C₂²	1	112	yes	complemented
23	C₂²×C₂²⋊Q₈	C₂²	1	56	yes	complemented
24	C₂²×C₂²⋊Q₈	C₂²	1	112	yes	complemented
25	C₂²×#121	C₂²	1	56	yes	—
26	C₂²×C₄×D₄	C₂²	1	112	yes	complemented
27	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
28	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
29	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
30	C₂³×C₄⋊C₄	C₂²	1	1	yes	characteristic
31	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
33	C₂³×C₄⋊C₄	C₂²	1	1	yes	characteristic
34	C₂³×C₄⋊C₄	C₂²	1	2	yes	—
35	C₂³×C₄⋊C₄	C₂²	1	56	yes	complemented
36	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
37	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
38	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
39	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
40	C₂³×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂³×C₄²	C₂²	1	2	yes	—
42	C₂⁶	C₂³	1	2	yes	—
43	C₂⁶	C₂³	1	1	yes	characteristic
44	C₂³×Q₈	—	4	8	no	—
45	C₂³×Q₈	C₂³	1	28	yes	—
46	C₂³×D₄	—	2	32	no	—
47	C₂³×D₄	—	4	8	no	—
48	C₂³×D₄	C₂³	1	56	yes	—
49	C₂³×D₄	C₂³	1	28	yes	—
50	C₂⁴×C₄	—	2	16	no	—
51	C₂⁴×C₄	—	4	8	no	—
52	C₂⁴×C₄	—	4	16	no	—
53	C₂⁴×C₄	—	4	8	no	—
54	C₂⁴×C₄	C₂³	1	28	yes	—
55	C₂⁴×C₄	C₂³	1	2	yes	—
56	C₂⁴×C₄	C₂³	1	1	yes	characteristic
57	C₂⁴×C₄	C₂³	1	1	yes	characteristic
58	C₂⁴×C₄	C₂³	1	2	yes	—
59	C₂⁴×C₄	C₂³	1	2	yes	—
60	C₂⁴×C₄	C₂³	1	2	yes	—
61	C₂⁴×C₄	C₂³	1	2	yes	—
62	C₂⁴×C₄	C₂³	1	28	yes	—
63	C₂⁴×C₄	C₂³	1	28	yes	—
64	C₂⁴×C₄	C₂³	1	56	yes	—
65	#539	C₂³	1	4096	yes	direct product factor
66	C₂×#126	C₂³	1	896	yes	complemented
67	C₂×#125	C₂³	1	896	yes	complemented
68	C₂×#123	C₂³	1	896	yes	complemented
69	C₂×#123	C₂³	1	896	yes	complemented
70	C₂×C₂²⋊Q₈	C₂³	1	448	yes	complemented
71	C₂×C₂²⋊Q₈	C₂³	1	896	yes	complemented
72	C₂×#121	C₂³	1	448	yes	—
73	C₂×C₄×D₄	C₂³	1	896	yes	complemented
74	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
75	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
76	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
77	C₂²×C₄⋊C₄	C₂³	1	28	yes	—
78	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
79	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
80	C₂²×C₄⋊C₄	C₂³	1	28	yes	—
81	C₂²×C₄⋊C₄	C₂³	1	56	yes	—
82	C₂²×C₄⋊C₄	C₂³	1	448	yes	complemented
83	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
84	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
85	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
86	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
87	C₂²×C₂²⋊C₄	C₂³	1	56	yes	—
88	C₂²×C₄²	C₂³	1	56	yes	—
89	C₂⁵	—	2	8	no	—
90	C₂⁵	—	2	8	no	—
91	C₂⁵	—	2	16	no	—
92	C₂⁵	—	2	16	no	—
93	C₂⁵	—	2	16	no	—
94	C₂⁵	#41	1	16	yes	—
95	C₂⁵	C₂⁴	1	28	yes	—
96	C₂⁵	C₂⁴	1	14	yes	—
97	C₂⁵	C₂⁴	1	1	yes	characteristic
98	C₂²×Q₈	—	4	112	no	—
99	C₂²×Q₈	C₂⁴	1	112	yes	—
100	C₂²×D₄	—	2	448	no	—
101	C₂²×D₄	—	4	112	no	—
102	C₂²×D₄	C₂⁴	1	224	yes	—
103	C₂²×D₄	C₂⁴	1	112	yes	—
104	C₂³×C₄	—	2	224	no	—
105	C₂³×C₄	—	2	16	no	—
106	C₂³×C₄	—	2	16	no	—
107	C₂³×C₄	—	2	16	no	—
108	C₂³×C₄	—	2	8	no	—
109	C₂³×C₄	—	2	16	no	—
110	C₂³×C₄	—	2	8	no	—
111	C₂³×C₄	—	2	16	no	—
112	C₂³×C₄	—	4	112	no	—
113	C₂³×C₄	—	4	224	no	—
114	C₂³×C₄	—	4	112	no	—
115	C₂³×C₄	C₂⁴	1	28	yes	—
116	C₂³×C₄	C₂⁴	1	28	yes	—
117	C₂³×C₄	C₂⁴	1	28	yes	—
118	C₂³×C₄	C₂⁴	1	14	yes	—
119	C₂³×C₄	C₂⁴	1	28	yes	—
120	C₂³×C₄	C₂⁴	1	28	yes	—
121	C₂³×C₄	C₂⁴	1	14	yes	—
122	C₂³×C₄	C₂⁴	1	112	yes	—
123	C₂³×C₄	C₂⁴	1	112	yes	—
124	C₂³×C₄	C₂⁴	1	112	yes	—
125	C₂³×C₄	C₂⁴	1	224	yes	—
126	#126	C₂⁴	1	1024	yes	complemented
127	#125	C₂⁴	1	1024	yes	complemented
128	#123	C₂⁴	1	1024	yes	complemented
129	#123	C₂⁴	1	1024	yes	complemented
130	C₂²⋊Q₈	C₂⁴	1	512	yes	complemented
131	C₂²⋊Q₈	C₂⁴	1	1024	yes	complemented
132	#121	C₂⁴	1	512	yes	—
133	C₄×D₄	C₂⁴	1	1024	yes	complemented
134	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
135	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
136	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
137	C₂×C₄⋊C₄	C₂⁴	1	112	yes	—
138	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
139	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
140	C₂×C₄⋊C₄	C₂⁴	1	112	yes	—
141	C₂×C₄⋊C₄	C₂⁴	1	224	yes	—
142	C₂×C₄⋊C₄	C₂⁴	1	512	yes	complemented
143	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
144	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
145	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
146	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
147	C₂×C₂²⋊C₄	C₂⁴	1	224	yes	—
148	C₂×C₄²	C₂⁴	1	224	yes	—
149	C₂⁴	—	2	128	no	—
150	C₂⁴	—	2	56	no	—
151	C₂⁴	—	2	56	no	—
152	C₂⁴	—	2	112	no	—
153	C₂⁴	—	2	112	no	—
154	C₂⁴	—	2	112	no	—
155	C₂⁴	—	4	128	no	—
156	C₂⁴	C₂×#41	1	112	yes	—
157	C₂⁴	C₂×#41	1	8	yes	—
158	C₂⁴	H(5,𝔽₂)	1	8	yes	—
159	C₂⁴	2₋¹⁺⁴	1	8	yes	—
160	C₂⁴	C₂⁵	1	56	yes	—
161	C₂⁴	C₂⁵	1	28	yes	—
162	C₂⁴	C₂⁵	1	7	yes	—
163	C₂×Q₈	—	4	224	no	—
164	C₂×Q₈	C₂⁵	1	64	yes	—
165	C₂×D₄	—	2	896	no	—
166	C₂×D₄	—	4	224	no	—
167	C₂×D₄	C₂⁵	1	128	yes	—
168	C₂×D₄	C₂⁵	1	64	yes	—
169	C₂²×C₄	—	2	448	no	—
170	C₂²×C₄	—	2	112	no	—
171	C₂²×C₄	—	2	112	no	—
172	C₂²×C₄	—	2	112	no	—
173	C₂²×C₄	—	2	56	no	—
174	C₂²×C₄	—	2	112	no	—
175	C₂²×C₄	—	2	56	no	—
176	C₂²×C₄	—	2	112	no	—
177	C₂²×C₄	—	4	224	no	—
178	C₂²×C₄	—	4	448	no	—
179	C₂²×C₄	—	4	224	no	—
180	C₂²×C₄	C₂⁵	1	56	yes	—
181	C₂²×C₄	C₂⁵	1	56	yes	—
182	C₂²×C₄	C₂⁵	1	56	yes	—
183	C₂²×C₄	C₂⁵	1	56	yes	—
184	C₂²×C₄	C₂⁵	1	56	yes	—
185	C₂²×C₄	C₂⁵	1	28	yes	—
186	C₂²×C₄	C₂⁵	1	28	yes	—
187	C₂²×C₄	C₂⁵	1	64	yes	—
188	C₂²×C₄	C₂⁵	1	64	yes	—
189	C₂²×C₄	C₂⁵	1	64	yes	—
190	C₂²×C₄	C₂⁵	1	128	yes	—
191	C₄⋊C₄	C₂⁵	1	128	yes	—
192	C₄⋊C₄	C₂⁵	1	128	yes	—
193	C₄⋊C₄	C₂⁵	1	128	yes	—
194	C₄⋊C₄	C₂⁵	1	64	yes	—
195	C₄⋊C₄	C₂⁵	1	128	yes	—
196	C₄⋊C₄	C₂⁵	1	128	yes	—
197	C₄⋊C₄	C₂⁵	1	64	yes	—
198	C₄⋊C₄	C₂⁵	1	128	yes	—
199	C₂²⋊C₄	C₂⁵	1	128	yes	—
200	C₂²⋊C₄	C₂⁵	1	128	yes	—
201	C₂²⋊C₄	C₂⁵	1	128	yes	—
202	C₂²⋊C₄	C₂⁵	1	128	yes	—
203	C₂²⋊C₄	C₂⁵	1	128	yes	—
204	C₄²	C₂⁵	1	128	yes	—
205	C₂³	—	2	56	no	—
206	C₂³	—	2	56	no	—
207	C₂³	—	2	448	no	—
208	C₂³	—	2	112	no	—
209	C₂³	—	2	112	no	—
210	C₂³	—	2	112	no	—
211	C₂³	—	4	448	no	—
212	C₂³	#539	1	64	yes	direct product factor
213	C₂³	C₂²×#41	1	112	yes	—
214	C₂³	C₂²×#41	1	28	yes	—
215	C₂³	C₂×H(5,𝔽₂)	1	28	yes	—
216	C₂³	C₂×2₋¹⁺⁴	1	28	yes	—
217	C₂³	C₂⁶	1	16	yes	—
218	C₂³	C₂⁶	1	8	yes	—
219	C₂³	C₂⁶	1	7	yes	—
220	Q₈	—	4	64	no	—
221	D₄	—	2	256	no	—
222	D₄	—	4	64	no	—
223	C₂×C₄	—	2	128	no	—
224	C₂×C₄	—	2	112	no	—
225	C₂×C₄	—	2	112	no	—
226	C₂×C₄	—	2	112	no	—
227	C₂×C₄	—	2	56	no	—
228	C₂×C₄	—	2	112	no	—
229	C₂×C₄	—	2	56	no	—
230	C₂×C₄	—	2	112	no	—
231	C₂×C₄	—	4	64	no	—
232	C₂×C₄	—	4	128	no	—
233	C₂×C₄	—	4	64	no	—
234	C₂×C₄	C₂⁶	1	16	yes	—
235	C₂×C₄	C₂⁶	1	16	yes	—
236	C₂×C₄	C₂⁶	1	16	yes	—
237	C₂×C₄	C₂⁶	1	16	yes	—
238	C₂×C₄	C₂⁶	1	16	yes	—
239	C₂×C₄	C₂⁶	1	8	yes	—
240	C₂×C₄	C₂⁶	1	8	yes	—
241	C₂²	—	2	8	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	224	no	—
244	C₂²	—	2	16	no	—
245	C₂²	—	2	16	no	—
246	C₂²	—	2	16	no	—
247	C₂²	—	4	224	no	—
248	C₂²	C₂×#539	1	112	yes	direct product factor
249	C₂²	C₂³×#41	1	16	yes	—
250	C₂²	C₂³×#41	1	14	yes	—
251	C₂²	C₂²×H(5,𝔽₂)	1	14	yes	—
252	C₂²	C₂²×2₋¹⁺⁴	1	14	yes	—
253	C₂²	C₂⁷	1	1	yes	characteristic
254	C₄	—	2	16	no	—
255	C₄	—	2	16	no	—
256	C₄	—	2	16	no	—
257	C₄	—	2	8	no	—
258	C₄	—	2	16	no	—
259	C₄	—	2	8	no	—
260	C₄	—	2	16	no	—
261	C₂	—	2	16	no	—
262	C₂	—	4	16	no	—
263	C₂	C₂²×#539	1	28	yes	direct product factor
264	C₂	C₂⁴×#41	1	1	yes	characteristic
265	C₂	C₂³×H(5,𝔽₂)	1	1	yes	characteristic
266	C₂	C₂³×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂³×#121	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×#539	C₂	direct (N × H)
2. C₂³×#126	C₂	semidirect
3. C₂³×#125	C₂	semidirect
4. C₂³×#123	C₂	semidirect
5. C₂³×#123	C₂	semidirect
6. C₂³×C₂²⋊Q₈	C₂	semidirect
7. C₂³×C₂²⋊Q₈	C₂	semidirect
9. C₂³×C₄×D₄	C₂	semidirect
10. C₂⁴×C₄⋊C₄	C₂	semidirect
18. C₂×#539	C₂²	direct (N × H)
19. C₂²×#126	C₂²	semidirect
20. C₂²×#125	C₂²	semidirect
21. C₂²×#123	C₂²	semidirect
22. C₂²×#123	C₂²	semidirect
23. C₂²×C₂²⋊Q₈	C₂²	semidirect
24. C₂²×C₂²⋊Q₈	C₂²	semidirect
26. C₂²×C₄×D₄	C₂²	semidirect
35. C₂³×C₄⋊C₄	C₂²	semidirect
65. #539	C₂³	direct (N × H)
66. C₂×#126	C₂³	semidirect
67. C₂×#125	C₂³	semidirect
68. C₂×#123	C₂³	semidirect
69. C₂×#123	C₂³	semidirect
70. C₂×C₂²⋊Q₈	C₂³	semidirect
71. C₂×C₂²⋊Q₈	C₂³	semidirect
73. C₂×C₄×D₄	C₂³	semidirect
82. C₂²×C₄⋊C₄	C₂³	semidirect
126. #126	C₂⁴	semidirect
127. #125	C₂⁴	semidirect
128. #123	C₂⁴	semidirect
129. #123	C₂⁴	semidirect
130. C₂²⋊Q₈	C₂⁴	semidirect
131. C₂²⋊Q₈	C₂⁴	semidirect
133. C₄×D₄	C₂⁴	semidirect
142. C₂×C₄⋊C₄	C₂⁴	semidirect
212. C₂³	#539	direct (N × H)
248. C₂²	C₂×#539	direct (N × H)
263. C₂	C₂²×#539	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#539	2
C₂³×#126	2
C₂³×#125	2
C₂³×#123	2
C₂³×C₂²⋊Q₈	2
C₂³×#121	2
C₂³×C₄×D₄	2
C₂⁴×C₄⋊C₄	2

Subgroups — C23×#539 #10585903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#539 #10585903