Subgroups — #72319337

Order 1024.

422 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#1398	C₂	1	1	yes	maximal, characteristic
2	C₂×#7457	C₂	1	2	yes	maximal, complemented
3	#134662	C₂	1	1	yes	maximal, characteristic
4	#134657	C₂	1	1	yes	maximal, characteristic
5	#94030	C₂	1	2	yes	maximal, complemented
6	C₂²×M₆₄	C₄	1	1	yes	characteristic
7	C₂²×M₆₄	C₄	1	1	yes	characteristic
8	C₂²×#346	C₂²	1	1	yes	characteristic
9	C₂×#1398	—	2	4	no	—
10	C₂×#1398	C₄	1	1	yes	characteristic
11	C₂×#1398	C₄	1	1	yes	characteristic
12	C₂×#1398	C₂²	1	1	yes	characteristic
13	C₂×#1398	C₂²	1	1	yes	characteristic
14	#12788	—	2	1	no	—
15	C₂×#1369	C₂²	1	2	yes	—
16	C₂×#1368	C₂²	1	2	yes	—
17	#8595	—	2	1	no	—
18	#7457	—	2	4	no	—
19	#7161	—	2	1	no	—
20	#7156	—	2	1	no	—
21	C₂³×C₁₆	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₁₆	Q₈	1	1	yes	characteristic
23	C₂²×C₄×C₈	D₄	1	1	yes	characteristic
24	C₂×M₆₄	—	2	1	no	—
25	C₂×M₆₄	—	2	2	no	—
26	C₂×M₆₄	—	2	2	no	—
27	C₂×M₆₄	—	2	2	no	—
28	C₂×M₆₄	—	2	2	no	—
29	C₂×M₆₄	C₈	1	2	yes	complemented
30	C₂×M₆₄	C₂×C₄	1	1	yes	characteristic
31	C₂×M₆₄	C₂×C₄	1	1	yes	characteristic
32	C₂×M₆₄	C₂×C₄	1	1	yes	characteristic
33	C₂×M₆₄	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃₂	—	2	1	no	—
35	C₂²×C₃₂	C₈	1	2	yes	complemented
36	C₂×#364	—	2	2	no	—
37	#2112	—	4	1	no	—
38	C₂×#346	—	2	2	no	—
39	C₂×#346	—	2	2	no	—
40	C₂×#346	C₂×C₄	1	1	yes	characteristic
41	C₂×#346	D₄	1	1	yes	characteristic
42	C₂×#346	Q₈	1	1	yes	characteristic
43	C₂×#346	C₂³	1	1	yes	characteristic
44	#1797	—	4	1	no	—
45	C₂×#334	—	2	2	no	—
46	#1398	—	2	2	no	—
47	#1398	—	2	2	no	—
48	#1398	—	2	2	no	—
49	#1398	D₄	1	4	yes	—
50	#1376	—	4	1	no	—
51	#1369	—	2	2	no	—
52	#1369	—	2	2	no	—
53	#1368	—	2	2	no	—
54	#1368	—	2	2	no	—
55	#1277	—	4	1	no	—
56	C₂³×C₈	C₄⋊C₄	1	1	yes	characteristic
57	C₂³×C₈	SD₁₆	1	1	yes	characteristic
58	C₂²×C₄²	SD₁₆	1	1	yes	characteristic
59	C₂×M₃₂	—	4	2	no	—
60	C₂²×C₁₆	—	2	1	no	—
61	C₂²×C₁₆	—	2	1	no	—
62	C₂²×C₁₆	—	2	2	no	—
63	C₂²×C₁₆	—	2	1	no	—
64	C₂²×C₁₆	—	2	1	no	—
65	C₂²×C₁₆	—	2	2	no	—
66	C₂²×C₁₆	—	2	1	no	—
67	C₂²×C₁₆	—	2	1	no	—
68	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₆	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
71	C₂²×C₁₆	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₁₆	M₁₆	1	1	yes	characteristic
73	C₂²×C₁₆	M₁₆	1	1	yes	characteristic
74	C₂²×C₁₆	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₁₆	C₂×Q₈	1	1	yes	characteristic
76	C₂×C₄⋊C₈	—	4	2	no	—
77	C₂×C₂²⋊C₈	—	8	1	no	—
78	C₂×C₄×C₈	—	2	2	no	—
79	C₂×C₄×C₈	—	2	2	no	—
80	C₂×C₄×C₈	C₄⋊C₄	1	1	yes	characteristic
81	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₈	SD₁₆	1	1	yes	characteristic
83	C₂×C₄×C₈	C₂×D₄	1	1	yes	characteristic
84	M₆₄	—	2	4	no	—
85	M₆₄	—	2	1	no	—
86	M₆₄	—	2	1	no	—
87	M₆₄	—	2	1	no	—
88	M₆₄	—	2	1	no	—
89	M₆₄	—	2	4	no	—
90	M₆₄	C₂²⋊C₄	1	2	yes	—
91	M₆₄	C₂²⋊C₄	1	2	yes	—
92	M₆₄	M₁₆	1	2	yes	—
93	M₆₄	M₁₆	1	2	yes	—
94	C₂×C₃₂	—	2	4	no	—
95	C₂×C₃₂	—	2	1	no	—
96	C₂×C₃₂	—	2	1	no	—
97	C₂×C₃₂	—	2	4	no	—
98	C₂×C₃₂	C₂×C₈	1	2	yes	—
99	C₂×C₃₂	C₂×C₈	1	2	yes	—
100	#364	—	4	2	no	—
101	#364	—	4	2	no	—
102	C₂²⋊C₁₆	—	8	1	no	—
103	C₂²⋊C₁₆	—	8	1	no	—
104	#346	—	2	1	no	—
105	#346	—	2	1	no	—
106	#346	—	2	2	no	—
107	#346	—	2	2	no	—
108	#346	C₂×D₄	1	2	yes	—
109	#346	#41	1	2	yes	—
110	#336	—	8	1	no	—
111	#334	—	4	2	no	—
112	#334	—	4	2	no	—
113	C₂³×C₄	—	2	1	no	—
114	C₂³×C₄	#106	1	1	yes	characteristic
115	C₂²×C₈	—	2	1	no	—
116	C₂²×C₈	—	2	1	no	—
117	C₂²×C₈	—	2	1	no	—
118	C₂²×C₈	—	2	1	no	—
119	C₂²×C₈	—	2	1	no	—
120	C₂²×C₈	—	2	1	no	—
121	C₂²×C₈	—	2	2	no	—
122	C₂²×C₈	—	4	1	no	—
123	C₂²×C₈	—	8	2	no	—
124	C₂²×C₈	C₄⋊C₈	1	1	yes	characteristic
125	C₂²×C₈	C₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₈	#106	1	1	yes	characteristic
127	C₂²×C₈	#106	1	1	yes	characteristic
128	C₂²×C₈	#108	1	1	yes	characteristic
129	C₂²×C₈	#108	1	1	yes	characteristic
130	C₂²×C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂²×C₈	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄²	—	2	2	no	—
133	C₂×C₄²	—	2	1	no	—
134	C₂×C₄²	—	4	1	no	—
135	C₂×C₄²	#106	1	1	yes	characteristic
136	C₂×C₄²	C₂×SD₁₆	1	1	yes	characteristic
137	M₃₂	—	8	2	no	—
138	M₃₂	—	8	2	no	—
139	C₂×C₁₆	—	2	4	no	—
140	C₂×C₁₆	—	2	4	no	—
141	C₂×C₁₆	—	2	1	no	—
142	C₂×C₁₆	—	2	1	no	—
143	C₂×C₁₆	—	2	1	no	—
144	C₂×C₁₆	—	2	1	no	—
145	C₂×C₁₆	—	2	1	no	—
146	C₂×C₁₆	—	2	1	no	—
147	C₂×C₁₆	—	2	1	no	—
148	C₂×C₁₆	—	2	1	no	—
149	C₂×C₁₆	—	2	1	no	—
150	C₂×C₁₆	—	2	1	no	—
151	C₂×C₁₆	—	2	1	no	—
152	C₂×C₁₆	—	2	1	no	—
153	C₂×C₁₆	—	2	1	no	—
154	C₂×C₁₆	—	2	1	no	—
155	C₂×C₁₆	—	8	2	no	—
156	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
157	C₂×C₁₆	C₄⋊C₈	1	1	yes	characteristic
158	C₂×C₁₆	#108	1	2	yes	—
159	C₂×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₁₆	#117	1	1	yes	characteristic
161	C₂×C₁₆	C₂²⋊Q₈	1	2	yes	—
162	C₂×C₁₆	C₂²×C₈	1	1	yes	characteristic
163	C₂×C₁₆	C₂×M₁₆	1	1	yes	characteristic
164	C₂×C₁₆	C₂×M₁₆	1	1	yes	characteristic
165	C₂×C₁₆	C₂×M₁₆	1	1	yes	characteristic
166	C₄⋊C₈	—	8	2	no	—
167	C₄⋊C₈	—	8	2	no	—
168	C₂²⋊C₈	—	16	2	no	—
169	C₄×C₈	—	2	1	no	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	#121	1	2	yes	—
174	C₄×C₈	#135	1	2	yes	—
175	C₃₂	—	2	2	no	—
176	C₃₂	—	2	2	no	—
177	C₃₂	—	2	2	no	—
178	C₃₂	C₂²⋊C₈	1	4	yes	complemented
179	C₂⁴	#365	1	1	yes	characteristic
180	C₂²×C₄	—	2	1	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	4	2	no	—
187	C₂²×C₄	—	4	2	no	—
188	C₂²×C₄	—	8	1	no	—
189	C₂²×C₄	#334	1	1	yes	characteristic
190	C₂²×C₄	#334	1	1	yes	characteristic
191	C₂²×C₄	#365	1	1	yes	characteristic
192	C₂²×C₄	#365	1	1	yes	characteristic
193	C₂²×C₄	#365	1	1	yes	characteristic
194	C₂²×C₄	C₂×#106	1	1	yes	characteristic
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	2	1	no	—
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	2	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	4	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	16	2	no	—
213	C₂×C₈	—	16	2	no	—
214	C₂×C₈	—	16	2	no	—
215	C₂×C₈	#334	1	1	yes	characteristic
216	C₂×C₈	#334	1	1	yes	characteristic
217	C₂×C₈	#380	1	1	yes	characteristic
218	C₂×C₈	C₂×C₄⋊C₈	1	1	yes	characteristic
219	C₂×C₈	#423	1	1	yes	characteristic
220	C₂×C₈	#427	1	1	yes	characteristic
221	C₂×C₈	C₂×#108	1	1	yes	characteristic
222	C₂×C₈	C₂×#108	1	1	yes	characteristic
223	C₂×C₈	#465	1	2	yes	—
224	C₄²	—	2	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	2	1	no	—
227	C₄²	—	4	2	no	—
228	C₄²	#465	1	2	yes	—
229	C₁₆	—	2	1	no	—
230	C₁₆	—	2	1	no	—
231	C₁₆	—	2	2	no	—
232	C₁₆	—	2	2	no	—
233	C₁₆	—	16	2	no	—
234	C₁₆	C₂×C₂²⋊C₈	1	1	yes	characteristic
235	C₁₆	#407	1	1	yes	characteristic
236	C₁₆	#431	1	1	yes	characteristic
237	C₁₆	#432	1	1	yes	characteristic
238	C₂³	—	2	1	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	1	no	—
241	C₂³	—	4	1	no	—
242	C₂³	#1368	1	1	yes	characteristic
243	C₂³	#1368	1	1	yes	characteristic
244	C₂³	C₂×#365	1	1	yes	characteristic
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	1	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	4	1	no	—
257	C₂×C₄	—	4	1	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	4	1	no	—
261	C₂×C₄	—	8	4	no	—
262	C₂×C₄	#1368	1	1	yes	characteristic
263	C₂×C₄	#1368	1	1	yes	characteristic
264	C₂×C₄	C₂×#334	1	1	yes	characteristic
265	C₂×C₄	#1566	1	1	yes	characteristic
266	C₂×C₄	#1807	1	1	yes	characteristic
267	C₂×C₄	C₂×#365	1	1	yes	characteristic
268	C₂×C₄	#2155	1	1	yes	characteristic
269	C₂×C₄	#2155	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	—	2	1	no	—
272	C₈	—	2	1	no	—
273	C₈	—	4	1	no	—
274	C₈	—	32	2	no	—
275	C₈	#1564	1	1	yes	characteristic
276	C₈	#1568	1	1	yes	characteristic
277	C₈	#1797	1	1	yes	characteristic
278	C₈	#1810	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	1	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	#7457	1	2	yes	complemented
289	C₂²	C₂×#1368	1	1	yes	characteristic
290	C₂²	#12019	1	1	yes	characteristic
291	C₂²	#12831	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	2	no	—
295	C₄	—	8	1	no	—
296	C₄	#8595	1	1	yes	characteristic
297	C₄	#11999	1	1	yes	characteristic
298	C₄	#12019	1	1	yes	characteristic
299	C₄	#12831	1	1	yes	characteristic
300	C₂	—	2	1	no	—
301	C₂	—	2	2	no	—
302	C₂	—	2	1	no	—
303	C₂	—	4	1	no	—
304	C₂	#134657	1	1	yes	characteristic
305	C₂	C₂×#7457	1	1	yes	characteristic
306	C₂	#151024	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#1368
commutator	C₂×C₁₆	C₂²×C₈
Frattini, cosocle	C₂×#346	C₂³
prime radical	C₂⁴	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ C₂²×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₂×#7457	C₂	semidirect
5. #94030	C₂	semidirect
29. C₂×M₆₄	C₈	semidirect
35. C₂²×C₃₂	C₈	semidirect
178. C₃₂	C₂²⋊C₈	semidirect
288. C₂²	#7457	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#1398	2
C₂×#7457	2
#134662	2
#134657	2
#94030	2