Subgroups — #72234408

Order 8192.

1276 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72234406	C₂	1	4	yes	maximal
2	#72234404	C₂	1	2	yes	maximal
3	#72234399	C₄	1	4	yes	—
4	#72234396	C₈	1	8	yes	—
5	#72234385	C₄	1	4	yes	—
6	#72234385	C₂²	1	4	yes	—
7	#72234375	—	2	8	no	—
8	#72234375	—	2	8	no	—
9	C₄×#7511	C₂×C₄	1	8	yes	—
10	C₈×M₁₂₈	C₈	1	8	yes	—
11	#72234274	C₂	1	1	yes	maximal, characteristic
12	#72234267	C₂²	1	2	yes	—
13	#72234264	—	2	4	no	—
14	#72234255	C₄	1	2	yes	—
15	#72234254	C₂²	1	1	yes	characteristic
16	#72234249	C₈	1	4	yes	—
17	#72234245	C₈	1	4	yes	—
18	#72234239	C₂×C₄	1	2	yes	—
19	#72234239	C₂³	1	1	yes	characteristic
20	#72234233	C₂×C₄	1	2	yes	—
21	#72234233	C₂×C₄	1	1	yes	characteristic
22	#72210694	C₄	1	1	yes	characteristic
23	#72210682	—	2	2	no	—
24	#72210680	C₂×C₄	1	2	yes	—
25	#72210664	C₄	1	1	yes	characteristic
26	#72210635	—	2	1	no	—
27	#72210629	—	2	1	no	—
28	#72210619	C₂×C₄	1	2	yes	—
29	#72210602	C₂×C₄	1	1	yes	characteristic
30	C₂×#7511	—	2	8	no	—
31	C₂×#7511	—	2	8	no	—
32	C₄×M₁₂₈	C₂×C₈	1	16	yes	—
33	#151879	—	2	2	no	—
34	C₄×#1397	C₄²	1	1	yes	characteristic
35	C₄×#1397	C₂²×C₄	1	2	yes	—
36	C₄²×C₃₂	C₄²	1	1	yes	characteristic
37	#149502	—	2	2	no	—
38	C₂×#7330	M₁₆	1	8	yes	—
39	C₂×#7330	#41	1	4	yes	—
40	C₈×M₆₄	C₂×C₈	1	4	yes	—
41	C₂×#7329	C₂×C₈	1	4	yes	—
42	C₂×#7329	C₂×C₈	1	2	yes	—
43	C₂×C₈×C₃₂	C₂×C₈	1	4	yes	—
44	C₂×C₈×C₃₂	C₂×C₈	1	2	yes	—
45	#128225	—	2	2	no	—
46	#128206	M₁₆	1	2	yes	—
47	#128206	M₁₆	1	1	yes	characteristic
48	#126519	C₄²	1	2	yes	—
49	#126519	C₂²×C₄	1	1	yes	characteristic
50	#122214	—	2	2	no	—
51	#122206	C₂×C₈	1	4	yes	—
52	#96658	M₁₆	1	1	yes	characteristic
53	#94505	—	2	16	no	—
54	#94504	C₁₆	1	16	yes	complemented
55	C₈×C₆₄	C₁₆	1	16	yes	complemented
56	#93666	—	2	8	no	—
57	#93664	C₁₆	1	8	yes	—
58	C₁₆×C₃₂	C₁₆	1	8	yes	—
59	C₂×M₁₂₈	—	2	16	no	—
60	C₄×M₆₄	—	2	4	no	—
61	C₂×#1397	#97	1	4	yes	—
62	C₂×#1397	#117	1	4	yes	—
63	C₂×#1397	C₂×M₁₆	1	4	yes	—
64	C₂×C₄×C₃₂	C₄×C₈	1	2	yes	—
65	C₂×C₄×C₃₂	C₄×C₈	1	1	yes	characteristic
66	C₂×C₄×C₃₂	C₄×C₈	1	2	yes	—
67	C₂×C₄×C₃₂	C₄×C₈	1	2	yes	—
68	C₂×C₄×C₃₂	C₄×C₈	1	1	yes	characteristic
69	C₂×C₄×C₃₂	C₂²×C₈	1	4	yes	—
70	C₄×#346	#97	1	2	yes	—
71	C₄×#346	C₂×M₁₆	1	1	yes	characteristic
72	C₄²×C₁₆	C₂×C₄²	1	1	yes	characteristic
73	#11410	—	2	4	no	—
74	C₂×#1311	#97	1	4	yes	—
75	C₂×#1311	#117	1	2	yes	—
76	C₂×C₈×C₁₆	C₄×C₈	1	4	yes	—
77	C₂×C₈×C₁₆	C₂²×C₈	1	2	yes	—
78	#7841	#97	1	2	yes	—
79	#7841	C₂×M₁₆	1	1	yes	characteristic
80	#7511	—	4	16	no	—
81	#7511	—	4	16	no	—
82	#7510	—	2	16	no	—
83	C₄×C₆₄	C₂×C₁₆	1	32	yes	—
84	#7466	—	4	4	no	—
85	#7332	—	4	4	no	—
86	#7330	—	4	8	no	—
87	#7330	—	4	4	no	—
88	#7329	—	2	4	no	—
89	#7329	M₃₂	1	16	yes	—
90	#7329	M₃₂	1	8	yes	—
91	C₈×C₃₂	C₂×C₁₆	1	8	yes	—
92	C₈×C₃₂	C₂×C₁₆	1	8	yes	—
93	C₈×C₃₂	C₂×C₁₆	1	8	yes	—
94	C₈×C₃₂	C₂×C₁₆	1	4	yes	—
95	C₈×C₃₂	C₂×C₁₆	1	4	yes	—
96	#7145	—	4	2	no	—
97	#7142	—	4	2	no	—
98	#7137	—	4	2	no	—
99	#7137	—	4	1	no	—
100	#7053	—	4	4	no	—
101	#7052	—	2	4	no	—
102	C₁₆²	C₂×C₁₆	1	8	yes	—
103	#7014	—	4	1	no	—
104	C₂×M₆₄	—	4	4	no	—
105	C₂²×C₃₂	C₈²	1	1	yes	characteristic
106	C₂²×C₃₂	C₈²	1	2	yes	—
107	C₂²×C₃₂	C₈²	1	1	yes	characteristic
108	C₂²×C₃₂	#322	1	4	yes	—
109	C₂²×C₃₂	#431	1	8	yes	—
110	#2162	—	4	4	no	—
111	C₂×#346	—	2	2	no	—
112	C₂×#346	—	2	2	no	—
113	C₂×#346	—	2	2	no	—
114	C₂×C₄×C₁₆	#322	1	4	yes	—
115	C₂×C₄×C₁₆	#376	1	1	yes	characteristic
116	C₂×C₄×C₁₆	C₂×C₄×C₈	1	2	yes	—
117	C₂×C₄×C₁₆	C₂×C₄×C₈	1	1	yes	characteristic
118	C₂×C₄×C₁₆	C₂×#97	1	2	yes	—
119	C₂×C₄×C₁₆	#431	1	2	yes	—
120	#1744	—	2	2	no	—
121	#1744	—	2	1	no	—
122	C₄²×C₈	C₂×#97	1	1	yes	characteristic
123	C₂×#322	—	2	2	no	—
124	C₂×#322	—	2	1	no	—
125	C₂×C₈²	#322	1	4	yes	—
126	C₂×C₈²	#431	1	2	yes	—
127	M₁₂₈	—	4	16	no	—
128	M₁₂₈	—	4	16	no	—
129	C₂×C₆₄	—	2	32	no	—
130	#1397	—	2	8	no	—
131	#1397	—	2	8	no	—
132	#1397	—	2	8	no	—
133	#1397	—	4	4	no	—
134	C₄×C₃₂	—	2	8	no	—
135	C₄×C₃₂	C₄×C₁₆	1	2	yes	—
136	C₄×C₃₂	C₄×C₁₆	1	2	yes	—
137	C₄×C₃₂	C₄×C₁₆	1	4	yes	—
138	C₄×C₃₂	C₄×C₁₆	1	4	yes	—
139	C₄×C₃₂	C₄×C₁₆	1	4	yes	—
140	C₄×C₃₂	#346	1	4	yes	—
141	C₄×C₃₂	#346	1	8	yes	—
142	C₄×C₃₂	#346	1	4	yes	—
143	C₄×C₃₂	C₂²×C₁₆	1	8	yes	—
144	C₄×C₃₂	C₂×M₃₂	1	8	yes	—
145	#1312	—	4	4	no	—
146	#1311	—	2	8	no	—
147	#1311	—	2	4	no	—
148	C₈×C₁₆	—	2	8	no	—
149	C₈×C₁₆	C₄×C₁₆	1	8	yes	—
150	C₈×C₁₆	#346	1	8	yes	—
151	C₈×C₁₆	C₂²×C₁₆	1	4	yes	—
152	C₈×C₁₆	C₂×M₃₂	1	4	yes	—
153	C₂²×C₁₆	—	2	2	no	—
154	C₂²×C₁₆	—	2	4	no	—
155	C₂²×C₁₆	C₂×C₈²	1	1	yes	characteristic
156	C₂²×C₁₆	C₂×#322	1	1	yes	characteristic
157	C₂²×C₁₆	#1744	1	2	yes	—
158	C₂×C₄×C₈	—	2	2	no	—
159	C₂×C₄×C₈	—	2	4	no	—
160	C₂×C₄×C₈	C₂×#322	1	2	yes	—
161	C₂×C₄×C₈	#1744	1	1	yes	characteristic
162	C₂×C₄×C₈	#2110	1	1	yes	characteristic
163	C₂×C₄×C₈	#2110	1	2	yes	—
164	C₄³	#2110	1	1	yes	characteristic
165	M₆₄	—	8	8	no	—
166	C₂×C₃₂	—	2	8	no	—
167	C₂×C₃₂	—	2	8	no	—
168	C₂×C₃₂	—	2	8	no	—
169	C₂×C₃₂	—	4	8	no	—
170	C₂×C₃₂	C₈×C₁₆	1	2	yes	—
171	C₂×C₃₂	C₈×C₁₆	1	1	yes	characteristic
172	C₂×C₃₂	C₈×C₁₆	1	2	yes	—
173	C₂×C₃₂	C₈×C₁₆	1	1	yes	characteristic
174	C₂×C₃₂	C₈×C₁₆	1	4	yes	—
175	C₂×C₃₂	C₈×C₁₆	1	2	yes	—
176	C₂×C₃₂	#1311	1	4	yes	—
177	C₂×C₃₂	#1311	1	4	yes	—
178	C₂×C₃₂	#1311	1	4	yes	—
179	C₂×C₃₂	#1312	1	8	yes	—
180	C₂×C₃₂	#2162	1	16	yes	—
181	C₄⋊C₁₆	—	8	4	no	—
182	C₂²⋊C₁₆	—	8	4	no	—
183	#346	—	8	2	no	—
184	#346	—	8	2	no	—
185	#346	—	8	2	no	—
186	C₄×C₁₆	—	2	4	no	—
187	C₄×C₁₆	—	2	2	no	—
188	C₄×C₁₆	—	4	8	no	—
189	C₄×C₁₆	#1312	1	8	yes	—
190	C₄×C₁₆	#1747	1	4	yes	—
191	C₄×C₁₆	C₂×C₄×C₁₆	1	2	yes	—
192	C₄×C₁₆	C₂×C₄×C₁₆	1	4	yes	—
193	C₄×C₁₆	C₂×#346	1	2	yes	—
194	C₄×C₁₆	C₂×#346	1	4	yes	—
195	C₄×C₁₆	#2110	1	8	yes	—
196	C₄×C₁₆	#2162	1	4	yes	—
197	#336	—	8	1	no	—
198	#336	—	8	2	no	—
199	#322	—	8	1	no	—
200	#322	—	8	2	no	—
201	C₈²	—	2	4	no	—
202	C₈²	—	2	2	no	—
203	C₈²	#1312	1	8	yes	—
204	C₈²	#2162	1	4	yes	—
205	C₆₄	—	4	32	no	—
206	C₂²×C₈	—	4	2	no	—
207	C₂²×C₈	—	4	4	no	—
208	C₂²×C₈	#8131	1	1	yes	characteristic
209	C₂²×C₈	#11405	1	2	yes	—
210	C₂²×C₈	#12593	1	1	yes	characteristic
211	C₂×C₄²	—	4	1	no	—
212	C₂×C₄²	#11405	1	2	yes	—
213	C₂×C₄²	#12593	1	1	yes	characteristic
214	C₂×C₁₆	—	2	8	no	—
215	C₂×C₁₆	—	2	4	no	—
216	C₂×C₁₆	—	4	4	no	—
217	C₂×C₁₆	—	4	4	no	—
218	C₂×C₁₆	—	4	4	no	—
219	C₂×C₁₆	—	8	8	no	—
220	C₂×C₁₆	#8137	1	4	yes	—
221	C₂×C₁₆	C₂×C₈×C₁₆	1	2	yes	—
222	C₂×C₁₆	C₂×C₈×C₁₆	1	1	yes	characteristic
223	C₂×C₁₆	C₂×#1311	1	2	yes	—
224	C₂×C₁₆	C₂×#1311	1	1	yes	characteristic
225	C₂×C₁₆	C₂×#1312	1	2	yes	—
226	C₂×C₁₆	#11405	1	4	yes	—
227	C₂×C₁₆	#12590	1	4	yes	—
228	C₂×C₁₆	#12593	1	4	yes	—
229	C₄×C₈	—	2	8	no	—
230	C₄×C₈	—	2	4	no	—
231	C₄×C₈	—	4	1	no	—
232	C₄×C₈	—	4	2	no	—
233	C₄×C₈	—	4	2	no	—
234	C₄×C₈	—	4	4	no	—
235	C₄×C₈	#8137	1	2	yes	—
236	C₄×C₈	C₂×#1312	1	4	yes	—
237	C₄×C₈	#11405	1	4	yes	—
238	C₄×C₈	#12590	1	2	yes	—
239	C₃₂	—	2	8	no	—
240	C₃₂	—	2	16	no	—
241	C₃₂	—	8	8	no	—
242	C₃₂	C₁₆²	1	1	yes	characteristic
243	C₃₂	C₁₆²	1	2	yes	—
244	C₃₂	C₁₆²	1	1	yes	characteristic
245	C₃₂	#7052	1	8	yes	—
246	C₃₂	#7052	1	4	yes	—
247	C₂²×C₄	—	2	1	no	—
248	C₂²×C₄	—	2	2	no	—
249	C₂²×C₄	#128206	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	4	no	—
254	C₂×C₈	—	4	8	no	—
255	C₂×C₈	—	16	2	no	—
256	C₂×C₈	—	16	4	no	—
257	C₂×C₈	#96658	1	1	yes	characteristic
258	C₂×C₈	#122214	1	4	yes	—
259	C₂×C₈	#128203	1	2	yes	—
260	C₂×C₈	#128203	1	1	yes	characteristic
261	C₂×C₈	#128206	1	2	yes	—
262	C₂×C₈	#128225	1	2	yes	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	16	1	no	—
266	C₄²	#122214	1	4	yes	—
267	C₄²	#128225	1	2	yes	—
268	C₁₆	—	4	4	no	—
269	C₁₆	—	4	8	no	—
270	C₁₆	—	16	8	no	—
271	C₁₆	C₂×C₁₆²	1	1	yes	characteristic
272	C₁₆	C₂×#7052	1	2	yes	—
273	C₁₆	C₂×#7052	1	1	yes	characteristic
274	C₁₆	#122214	1	4	yes	—
275	C₁₆	#128225	1	8	yes	—
276	C₂³	#72210687	1	1	yes	characteristic
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	8	1	no	—
280	C₂×C₄	—	8	2	no	—
281	C₂×C₄	#72210645	1	1	yes	characteristic
282	C₂×C₄	#72210687	1	2	yes	—
283	C₂×C₄	#72210687	1	1	yes	characteristic
284	C₂×C₄	#72234227	1	2	yes	—
285	C₈	—	2	2	no	—
286	C₈	—	2	4	no	—
287	C₈	—	8	4	no	—
288	C₈	—	8	8	no	—
289	C₈	#72210645	1	2	yes	—
290	C₈	#72234227	1	1	yes	characteristic
291	C₈	#72234260	1	1	yes	characteristic
292	C₂²	—	4	1	no	—
293	C₂²	#72210694	1	1	yes	characteristic
294	C₂²	#72234267	1	2	yes	—
295	C₄	—	4	2	no	—
296	C₄	—	4	4	no	—
297	C₄	#72210694	1	1	yes	characteristic
298	C₄	#72234261	1	1	yes	characteristic
299	C₄	#72234267	1	2	yes	—
300	C₂	—	2	1	no	—
301	C₂	—	2	2	no	—
302	C₂	#72234269	1	1	yes	characteristic

	subgroup	G/H
center	C₃₂	C₁₆²
commutator	C₁₆	C₂×C₁₆²
Frattini, cosocle	#72234239	C₂³
socle	C₂	#72234269
prime radical	C₂³	#72210687
LCS	G ▷ C₁₆ ▷ C₁
UCS	C₁ ◁ C₃₂ ◁ G
LECS	G ▷ #72234239 ▷ C₂×C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
54. #94504	C₁₆	semidirect
55. C₈×C₆₄	C₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72234406	2
#72234404	2
#72234274	2