Subgroups — #72320439

Order 1024.

494 conjugacy classes of subgroups.

The subgroup lattice has 283 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#512731	C₂	1	1	yes	maximal, characteristic
2	#494211	C₂	1	1	yes	maximal, characteristic
3	#483342	C₂	1	1	yes	maximal, characteristic
4	#94055	C₂	1	2	yes	maximal, complemented
5	#94053	C₂	1	2	yes	maximal, complemented
6	#33401	—	2	1	no	—
7	C₄×#504	C₂²	1	1	yes	characteristic
8	#20787	—	2	1	no	—
9	#13641	C₄	1	4	yes	complemented
10	#13640	—	2	2	no	—
11	#13628	—	2	1	no	—
12	#13554	—	2	1	no	—
13	C₄×M₆₄	C₂²	1	1	yes	characteristic
14	C₂×#1397	C₂²	1	1	yes	characteristic
15	#12911	C₄	1	2	yes	—
16	#12893	—	2	1	no	—
17	#12840	C₄	1	1	yes	characteristic
18	#12839	C₄	1	1	yes	characteristic
19	#12832	—	2	1	no	—
20	#12670	C₂²	1	2	yes	—
21	#12668	—	2	1	no	—
22	#12011	—	2	1	no	—
23	#12011	C₄	1	4	yes	complemented
24	C₂×#1373	—	2	2	no	—
25	C₂×#1373	C₂²	1	2	yes	—
26	#3407	—	4	1	no	—
27	C₂×#504	C₂×C₄	1	1	yes	characteristic
28	C₂×#504	D₄	1	1	yes	characteristic
29	C₄×#131	D₄	1	1	yes	characteristic
30	C₂×M₆₄	—	2	1	no	—
31	C₂²×C₃₂	—	2	1	no	—
32	C₂×#368	C₂×C₄	1	2	yes	—
33	#2158	C₂×C₄	1	2	yes	—
34	#2155	—	2	2	no	—
35	#2153	—	4	1	no	—
36	#2153	—	4	2	no	—
37	C₂×#364	—	2	4	no	—
38	C₂×#364	—	2	2	no	—
39	#2108	C₂×C₄	1	2	yes	—
40	#2108	C₂×C₄	1	1	yes	characteristic
41	#2108	D₄	1	1	yes	characteristic
42	#2108	Q₈	1	2	yes	—
43	C₄×M₃₂	C₂×C₄	1	2	yes	—
44	C₄×M₃₂	D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₆	D₄	1	2	yes	—
46	C₂×C₄×C₁₆	C₂³	1	1	yes	characteristic
47	#1841	—	4	1	no	—
48	#1833	—	2	2	no	—
49	#1827	—	4	1	no	—
50	#1813	—	2	1	no	—
51	#1812	—	2	1	no	—
52	#1772	—	2	2	no	—
53	#1757	—	4	1	no	—
54	#1747	—	4	1	no	—
55	C₄⋊C₃₂	—	2	4	no	—
56	C₂²⋊C₃₂	—	2	4	no	—
57	#1397	—	2	1	no	—
58	#1397	—	2	1	no	—
59	#1397	C₂×C₄	1	2	yes	—
60	C₄×C₃₂	C₂×C₄	1	2	yes	—
61	#1385	—	2	1	no	—
62	#1384	—	2	1	no	—
63	#1382	—	2	2	no	—
64	#1373	—	2	4	no	—
65	#1373	—	2	2	no	—
66	#1373	C₂×C₄	1	4	yes	—
67	C₂×#131	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×#131	D₈	1	1	yes	characteristic
69	C₄×#41	SD₁₆	1	1	yes	characteristic
70	#504	—	2	2	no	—
71	#504	—	2	2	no	—
72	#504	—	4	2	no	—
73	#504	—	8	4	no	—
74	C₂×M₃₂	—	4	4	no	—
75	C₂×M₃₂	—	4	2	no	—
76	C₂×M₃₂	C₄²	1	2	yes	—
77	C₂×M₃₂	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×M₃₂	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×M₃₂	C₄⋊C₄	1	2	yes	—
80	C₂²×C₁₆	C₂²⋊C₄	1	2	yes	—
81	C₂²×C₁₆	C₄⋊C₄	1	2	yes	—
82	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₁₆	C₂×D₄	1	1	yes	characteristic
84	C₂×#108	—	2	2	no	—
85	#427	—	2	2	no	—
86	#413	—	4	1	no	—
87	#410	—	4	1	no	—
88	#405	—	4	2	no	—
89	#405	C₂²⋊C₄	1	1	yes	characteristic
90	#405	C₄⋊C₄	1	2	yes	—
91	#405	D₈	1	1	yes	characteristic
92	#405	SD₁₆	1	2	yes	—
93	C₄×M₁₆	SD₁₆	1	1	yes	characteristic
94	C₄×M₁₆	Q₁₆	1	2	yes	—
95	C₂×C₄×C₈	C₂²⋊C₄	1	2	yes	—
96	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
97	M₆₄	—	4	2	no	—
98	C₂×C₃₂	—	2	2	no	—
99	C₂×C₃₂	—	2	2	no	—
100	C₂×C₃₂	—	4	2	no	—
101	#368	—	2	2	no	—
102	#368	—	2	2	no	—
103	#367	—	2	2	no	—
104	#366	—	2	2	no	—
105	#365	—	4	4	no	—
106	#364	—	2	4	no	—
107	#364	—	4	4	no	—
108	#364	—	4	2	no	—
109	#346	C₄²	1	2	yes	—
110	#346	C₄⋊C₄	1	2	yes	—
111	#346	C₄⋊C₄	1	2	yes	—
112	C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
113	C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
114	C₄×C₁₆	C₄⋊C₄	1	2	yes	—
115	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
116	C₄×C₁₆	C₂²×C₄	1	1	yes	characteristic
117	C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
118	C₄×C₁₆	C₂×Q₈	1	1	yes	characteristic
119	#344	—	4	2	no	—
120	#339	—	4	2	no	—
121	#338	—	4	1	no	—
122	H(3,ℤ/4)	—	4	1	no	—
123	H(3,ℤ/4)	—	4	2	no	—
124	C₂×#41	#102	1	1	yes	characteristic
125	#131	—	2	2	no	—
126	#131	—	2	2	no	—
127	#131	—	2	2	no	—
128	#131	—	2	2	no	—
129	C₂×M₁₆	—	4	2	no	—
130	C₂×M₁₆	#102	1	1	yes	characteristic
131	C₂×M₁₆	#102	1	1	yes	characteristic
132	C₂×M₁₆	#103	1	2	yes	—
133	C₂×M₁₆	#107	1	2	yes	—
134	C₂²×C₈	#95	1	2	yes	—
135	C₂²×C₈	#102	1	2	yes	—
136	C₂²×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
137	C₂²×C₈	C₂×D₈	1	1	yes	characteristic
138	C₄×Q₈	—	2	1	no	—
139	C₄×D₄	—	2	1	no	—
140	C₄×D₄	—	2	2	no	—
141	#117	—	4	2	no	—
142	#117	#102	1	1	yes	characteristic
143	#117	#106	1	2	yes	—
144	C₂×C₄²	#103	1	2	yes	—
145	C₂×C₄²	C₂×SD₁₆	1	1	yes	characteristic
146	M₃₂	—	2	2	no	—
147	M₃₂	—	2	2	no	—
148	M₃₂	—	2	4	no	—
149	M₃₂	—	4	2	no	—
150	M₃₂	—	4	4	no	—
151	M₃₂	—	8	4	no	—
152	M₃₂	—	8	2	no	—
153	C₂×C₁₆	—	2	2	no	—
154	C₂×C₁₆	—	2	2	no	—
155	C₂×C₁₆	—	2	2	no	—
156	C₂×C₁₆	—	4	2	no	—
157	C₂×C₁₆	—	8	4	no	—
158	C₂×C₁₆	—	8	2	no	—
159	C₂×C₁₆	#95	1	2	yes	—
160	C₂×C₁₆	#95	1	2	yes	—
161	C₂×C₁₆	C₂×C₄²	1	1	yes	characteristic
162	C₂×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
165	C₂×C₁₆	#131	1	2	yes	—
166	C₂×C₁₆	#131	1	2	yes	—
167	#108	—	4	2	no	—
168	#108	—	4	2	no	—
169	#107	—	4	2	no	—
170	#106	—	4	2	no	—
171	#101	—	8	1	no	—
172	#100	—	8	1	no	—
173	#99	—	8	2	no	—
174	#97	—	8	2	no	—
175	#97	#103	1	2	yes	—
176	#97	#106	1	2	yes	—
177	#97	#107	1	2	yes	—
178	C₄×C₈	—	8	2	no	—
179	C₄×C₈	#95	1	2	yes	—
180	C₄×C₈	#102	1	2	yes	—
181	C₄×C₈	#103	1	2	yes	—
182	C₄×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
183	C₄×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
184	C₄×C₈	C₂×SD₁₆	1	1	yes	characteristic
185	C₄×C₈	C₂×Q₁₆	1	1	yes	characteristic
186	C₃₂	—	4	2	no	—
187	C₃₂	—	4	2	no	—
188	#41	—	2	2	no	—
189	#41	—	2	2	no	—
190	C₂×Q₈	#359	1	1	yes	characteristic
191	C₂×D₄	#359	1	1	yes	characteristic
192	C₂×D₄	#359	1	2	yes	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	#340	1	2	yes	—
196	C₂²×C₄	C₂×#102	1	1	yes	characteristic
197	M₁₆	—	2	2	no	—
198	M₁₆	—	2	2	no	—
199	M₁₆	—	2	4	no	—
200	M₁₆	—	2	4	no	—
201	M₁₆	—	8	2	no	—
202	M₁₆	—	8	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	8	2	no	—
208	C₂×C₈	#340	1	2	yes	—
209	C₂×C₈	#340	1	2	yes	—
210	C₂×C₈	C₂×#95	1	1	yes	characteristic
211	C₂×C₈	#408	1	2	yes	—
212	C₂×C₈	C₂×#102	1	1	yes	characteristic
213	C₂×C₈	C₂×#103	1	1	yes	characteristic
214	C₂×C₈	#421	1	2	yes	—
215	C₂×C₈	C₂×#107	1	1	yes	characteristic
216	C₂×C₈	#509	1	2	yes	—
217	C₂×C₈	#510	1	2	yes	—
218	C₄⋊C₄	—	2	1	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	8	2	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	2	no	—
223	C₂²⋊C₄	—	8	2	no	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	8	2	no	—
227	C₄²	#340	1	2	yes	—
228	C₄²	C₂×#103	1	1	yes	characteristic
229	C₄²	C₂×#106	1	1	yes	characteristic
230	C₁₆	—	2	2	no	—
231	C₁₆	—	2	1	no	—
232	C₁₆	—	2	1	no	—
233	C₁₆	—	2	2	no	—
234	C₁₆	—	8	4	no	—
235	C₁₆	#405	1	2	yes	—
236	C₁₆	#424	1	1	yes	characteristic
237	C₁₆	#424	1	1	yes	characteristic
238	C₂³	#1382	1	2	yes	—
239	C₂³	C₂×#359	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	D₄	—	2	4	no	—
242	D₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	8	1	no	—
249	C₂×C₄	—	8	2	no	—
250	C₂×C₄	—	16	2	no	—
251	C₂×C₄	#1382	1	2	yes	—
252	C₂×C₄	#1480	1	2	yes	—
253	C₂×C₄	C₂×#340	1	1	yes	characteristic
254	C₂×C₄	#2139	1	2	yes	—
255	C₂×C₄	C₂×#359	1	1	yes	characteristic
256	C₈	—	2	2	no	—
257	C₈	—	2	2	no	—
258	C₈	—	16	2	no	—
259	C₈	#1481	1	2	yes	—
260	C₈	#1729	1	1	yes	characteristic
261	C₈	#1733	1	1	yes	characteristic
262	C₈	#2140	1	1	yes	characteristic
263	C₈	#2141	1	1	yes	characteristic
264	C₈	#2159	1	2	yes	—
265	C₂²	—	2	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	2	no	—
268	C₂²	#11648	1	2	yes	—
269	C₂²	C₂×#1382	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	4	1	no	—
274	C₄	—	32	2	no	—
275	C₄	#7893	1	1	yes	characteristic
276	C₄	#11648	1	2	yes	—
277	C₄	#12557	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	2	no	—
280	C₂	—	4	2	no	—
281	C₂	#126832	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#12557
commutator	C₂×C₁₆	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₁₆	C₂³
socle	C₂	#126832
prime radical	C₂×#41	#102
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₄×#131 ◁ C₄×#504 ◁ G
LECS	G ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #94055	C₂	semidirect
5. #94053	C₂	semidirect
9. #13641	C₄	semidirect
23. #12011	C₄	semidirect

Maximal subgroups

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

subgroup	index
#512731	2
#494211	2
#483342	2
#94055	2
#94053	2