Subgroups — #72231960

Order 1024.

745 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	#512728	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×M₆₄	C₂	1	1	yes	maximal, characteristic
3	#479914	C₂	1	1	yes	maximal, characteristic
4	#149509	C₂	1	8	yes	maximal, complemented
5	#149501	C₂	1	2	yes	maximal, complemented
6	#149496	C₂	1	2	yes	maximal, complemented
7	C₂²×M₆₄	C₄	1	2	yes	—
8	#33418	C₄	1	1	yes	characteristic
9	C₂²×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	#17323	C₄	1	1	yes	characteristic
11	#13640	C₂²	1	4	yes	—
12	#13639	C₄	1	4	yes	—
13	#13627	C₄	1	2	yes	—
14	#13553	C₄	1	2	yes	—
15	C₄×M₆₄	C₄	1	4	yes	—
16	C₄×M₆₄	C₂²	1	4	yes	—
17	C₄×M₆₄	C₂²	1	2	yes	—
18	C₂×#1397	C₂²	1	1	yes	characteristic
19	C₂×#1397	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₃₂	C₄	1	2	yes	—
21	#11408	C₂²	1	4	yes	—
22	C₁₆×M₁₆	C₄	1	4	yes	—
23	#11401	C₄	1	2	yes	—
24	#11399	C₂²	1	2	yes	—
25	#7332	C₂²	1	8	yes	complemented
26	#7330	C₂²	1	8	yes	complemented
27	C₂³×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₁₆	C₂×C₄	1	1	yes	characteristic
29	#2920	C₈	1	2	yes	—
30	C₂²×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×M₆₄	—	2	8	no	—
32	C₂×M₆₄	C₂×C₄	1	4	yes	—
33	C₂×M₆₄	C₂×C₄	1	8	yes	—
34	C₂²×C₃₂	C₂×C₄	1	2	yes	—
35	C₂²×C₃₂	C₂×C₄	1	2	yes	—
36	#2163	C₈	1	4	yes	—
37	#2162	C₂×C₄	1	4	yes	—
38	#2150	C₈	1	2	yes	—
39	#2112	C₈	1	2	yes	—
40	C₄×M₃₂	C₂×C₄	1	2	yes	—
41	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
43	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
45	C₂×C₄×C₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄×C₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₄×C₁₆	C₂³	1	2	yes	—
48	C₂×C₄×C₁₆	C₂³	1	1	yes	characteristic
49	#1452	C₈	1	2	yes	—
50	#1451	C₈	1	4	yes	—
51	#1450	C₂×C₄	1	2	yes	—
52	C₈×M₁₆	C₂×C₄	1	4	yes	—
53	C₄⋊C₃₂	C₂×C₄	1	8	yes	—
54	C₂²⋊C₃₂	C₂×C₄	1	8	yes	—
55	#1397	C₂×C₄	1	2	yes	—
56	#1397	C₂×C₄	1	2	yes	—
57	#1397	C₂³	1	4	yes	—
58	#1397	C₂³	1	2	yes	—
59	#1397	C₂³	1	2	yes	—
60	C₄×C₃₂	C₂×C₄	1	4	yes	—
61	C₄×C₃₂	C₂×C₄	1	8	yes	—
62	#1312	C₂×C₄	1	8	yes	—
63	#1311	C₂³	1	4	yes	—
64	C₈×C₁₆	C₂×C₄	1	4	yes	—
65	C₂³×C₈	C₄²	1	1	yes	characteristic
66	C₂³×C₈	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄²	C₂×C₈	1	1	yes	characteristic
68	C₂×M₃₂	C₂×C₈	1	4	yes	—
69	C₂²×C₁₆	—	2	4	no	—
70	C₂²×C₁₆	—	2	4	no	—
71	C₂²×C₁₆	C₄²	1	1	yes	characteristic
72	C₂²×C₁₆	C₄²	1	1	yes	characteristic
73	C₂²×C₁₆	C₄²	1	2	yes	—
74	C₂²×C₁₆	C₄²	1	2	yes	—
75	C₂²×C₁₆	C₂²×C₄	1	2	yes	—
76	C₂²×C₁₆	C₂²×C₄	1	2	yes	—
77	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
79	#432	—	2	2	no	—
80	#432	—	2	2	no	—
81	#431	C₂×C₈	1	4	yes	—
82	#431	C₂×C₈	1	4	yes	—
83	#423	—	2	2	no	—
84	#407	—	2	2	no	—
85	C₄×M₁₆	C₂×C₈	1	4	yes	—
86	C₂×C₄×C₈	C₄²	1	2	yes	—
87	C₂×C₄×C₈	C₄²	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×C₈	1	2	yes	—
90	C₂×C₄×C₈	C₂×C₈	1	2	yes	—
91	C₂×C₄×C₈	C₂×C₈	1	1	yes	characteristic
92	C₂×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₈	C₂²×C₄	1	2	yes	—
94	M₆₄	—	2	8	no	—
95	M₆₄	—	2	16	no	—
96	C₂×C₃₂	—	2	4	no	—
97	C₂×C₃₂	M₁₆	1	8	yes	complemented
98	C₂×C₃₂	C₂²×C₄	1	4	yes	—
99	C₂×C₃₂	C₂²×C₄	1	8	yes	—
100	C₂×C₃₂	C₂²×C₄	1	4	yes	—
101	C₄⋊C₁₆	C₂×C₈	1	8	yes	—
102	C₂²⋊C₁₆	C₂×C₈	1	8	yes	—
103	#346	C₂×C₈	1	4	yes	—
104	C₄×C₁₆	C₄²	1	2	yes	—
105	C₄×C₁₆	C₄²	1	2	yes	—
106	C₄×C₁₆	C₄²	1	2	yes	—
107	C₄×C₁₆	C₂²×C₄	1	2	yes	—
108	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
109	C₄×C₁₆	C₂²×C₄	1	2	yes	—
110	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
111	C₄×C₁₆	C₂²×C₄	1	2	yes	—
112	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
113	C₄×C₁₆	C₂²×C₄	1	4	yes	—
114	C₄×C₁₆	C₂⁴	1	1	yes	characteristic
115	#322	C₂×C₈	1	8	yes	—
116	#322	C₂×C₈	1	4	yes	—
117	C₈²	C₂²×C₄	1	4	yes	—
118	C₂³×C₄	C₄×C₈	1	1	yes	characteristic
119	C₂³×C₄	M₃₂	1	2	yes	—
120	C₂×M₁₆	—	2	4	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	2	2	no	—
123	C₂²×C₈	—	2	4	no	—
124	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
125	C₂²×C₈	C₄×C₈	1	2	yes	—
126	C₂²×C₈	C₄×C₈	1	1	yes	characteristic
127	C₂²×C₈	C₄×C₈	1	2	yes	—
128	C₂²×C₈	C₂×C₄²	1	2	yes	—
129	C₂²×C₈	C₂×C₄²	1	1	yes	characteristic
130	C₂²×C₈	C₂²×C₈	1	2	yes	—
131	C₂²×C₈	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄²	C₄×C₈	1	1	yes	characteristic
133	C₂×C₄²	C₄×C₈	1	2	yes	—
134	C₂×C₄²	M₃₂	1	2	yes	—
135	C₂×C₄²	M₃₂	1	2	yes	—
136	C₂×C₄²	M₃₂	1	2	yes	—
137	C₂×C₄²	C₂²×C₈	1	2	yes	—
138	C₂×C₄²	C₂²×C₈	1	1	yes	characteristic
139	M₃₂	—	2	8	no	—
140	C₂×C₁₆	—	2	2	no	—
141	C₂×C₁₆	—	2	2	no	—
142	C₂×C₁₆	—	2	4	no	—
143	C₂×C₁₆	—	2	4	no	—
144	C₂×C₁₆	—	2	4	no	—
145	C₂×C₁₆	—	2	4	no	—
146	C₂×C₁₆	C₂×C₄²	1	2	yes	—
147	C₂×C₁₆	C₂×C₄²	1	2	yes	—
148	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
149	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
150	C₂×C₁₆	C₂²×C₈	1	8	yes	—
151	C₂×C₁₆	C₂×M₁₆	1	4	yes	—
152	C₂×C₁₆	#131	1	4	yes	—
153	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₁₆	C₂³×C₄	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	C₄⋊C₈	—	2	4	no	—
157	C₂²⋊C₈	—	2	4	no	—
158	C₂²⋊C₈	—	2	4	no	—
159	#97	—	2	4	no	—
160	C₄×C₈	C₄×C₈	1	2	yes	—
161	C₄×C₈	C₄×C₈	1	2	yes	—
162	C₄×C₈	C₄×C₈	1	2	yes	—
163	C₄×C₈	C₂×C₄²	1	2	yes	—
164	C₄×C₈	C₂×C₄²	1	1	yes	characteristic
165	C₄×C₈	C₂²×C₈	1	8	yes	—
166	C₄×C₈	C₂²×C₈	1	2	yes	—
167	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
168	C₄×C₈	C₂²×C₈	1	1	yes	characteristic
169	C₄×C₈	C₂²×C₈	1	2	yes	—
170	C₄×C₈	C₂³×C₄	1	1	yes	characteristic
171	C₃₂	—	2	4	no	—
172	C₃₂	—	2	8	no	—
173	C₃₂	C₂×M₁₆	1	8	yes	—
174	C₂⁴	#346	1	1	yes	characteristic
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	4	4	no	—
178	C₂²×C₄	#346	1	2	yes	—
179	C₂²×C₄	#346	1	2	yes	—
180	C₂²×C₄	#346	1	1	yes	characteristic
181	C₂²×C₄	#346	1	2	yes	—
182	C₂²×C₄	#346	1	2	yes	—
183	C₂²×C₄	C₂×C₄×C₈	1	2	yes	—
184	C₂²×C₄	C₂×C₄×C₈	1	1	yes	characteristic
185	C₂²×C₄	C₂×M₃₂	1	2	yes	—
186	C₂²×C₄	C₂×M₃₂	1	2	yes	—
187	C₂²×C₄	C₂×M₃₂	1	2	yes	—
188	M₁₆	—	4	8	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	8	no	—
197	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
198	C₂×C₈	C₂×C₄×C₈	1	2	yes	—
199	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
200	C₂×C₈	C₂×C₄×C₈	1	1	yes	characteristic
201	C₂×C₈	C₄×M₁₆	1	2	yes	—
202	C₂×C₈	#405	1	2	yes	—
203	C₂×C₈	#504	1	4	yes	—
204	C₂×C₈	C₂²×C₄²	1	1	yes	characteristic
205	C₂×C₈	C₂³×C₈	1	1	yes	characteristic
206	C₄²	#346	1	2	yes	—
207	C₄²	#346	1	1	yes	characteristic
208	C₄²	#346	1	2	yes	—
209	C₄²	#346	1	1	yes	characteristic
210	C₄²	C₂×C₄×C₈	1	1	yes	characteristic
211	C₄²	C₂×C₄×C₈	1	2	yes	—
212	C₄²	C₂×M₃₂	1	2	yes	—
213	C₄²	C₂×M₃₂	1	2	yes	—
214	C₄²	C₂×M₃₂	1	2	yes	—
215	C₄²	C₂³×C₈	1	1	yes	characteristic
216	C₁₆	—	2	2	no	—
217	C₁₆	—	2	2	no	—
218	C₁₆	—	2	8	no	—
219	C₁₆	C₄×M₁₆	1	2	yes	—
220	C₁₆	#405	1	2	yes	—
221	C₁₆	C₂²×M₁₆	1	2	yes	—
222	C₁₆	C₂×#131	1	2	yes	—
223	C₂³	—	2	2	no	—
224	C₂³	—	4	2	no	—
225	C₂³	C₂×#346	1	2	yes	—
226	C₂³	C₂×#346	1	1	yes	characteristic
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	4	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₈×M₁₆	1	2	yes	—
238	C₂×C₄	C₂×#346	1	2	yes	—
239	C₂×C₄	C₂×#346	1	2	yes	—
240	C₂×C₄	C₂×#346	1	2	yes	—
241	C₂×C₄	C₂×#346	1	1	yes	characteristic
242	C₂×C₄	C₂×#346	1	2	yes	—
243	C₂×C₄	#2108	1	2	yes	—
244	C₂×C₄	C₂²×C₄×C₈	1	1	yes	characteristic
245	C₂×C₄	C₂²×M₃₂	1	2	yes	—
246	C₈	—	2	2	no	—
247	C₈	—	2	2	no	—
248	C₈	—	4	8	no	—
249	C₈	C₈×M₁₆	1	2	yes	—
250	C₈	#2108	1	2	yes	—
251	C₈	C₂×C₄×M₁₆	1	1	yes	characteristic
252	C₈	C₂×#405	1	1	yes	characteristic
253	C₈	C₂×#504	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	4	4	no	—
259	C₂²	#11408	1	2	yes	—
260	C₂²	C₂²×#346	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	2	2	no	—
265	C₄	#11408	1	2	yes	—
266	C₄	#13551	1	4	yes	—
267	C₄	C₂×C₈×M₁₆	1	1	yes	characteristic
268	C₄	C₂×#2108	1	1	yes	characteristic
269	C₂	—	2	2	no	—
270	C₂	—	4	2	no	—
271	C₂	C₂×#11408	1	1	yes	characteristic
272	C₂	C₂×#13551	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄×C₈
Frattini, cosocle	C₄×C₁₆	C₂⁴
socle	C₂²	C₂²×#346
prime radical	C₂⁴	#346
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #149509	C₂	semidirect
5. #149501	C₂	semidirect
6. #149496	C₂	semidirect
25. #7332	C₂²	semidirect
26. #7330	C₂²	semidirect
97. C₂×C₃₂	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#512728	2
C₂×C₄×M₆₄	2
#479914	2
#149509	2
#149501	2
#149496	2