Subgroups — #72322779

Order 2048.

638 conjugacy classes of subgroups.

The subgroup lattice has 357 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#72321329	C₂	1	1	yes	maximal, characteristic
2	#72321153	C₂	1	2	yes	maximal, complemented
3	#72320649	C₂	1	1	yes	maximal, characteristic
4	#72286496	C₂	1	2	yes	maximal, complemented
5	#72232024	C₂	1	1	yes	maximal, characteristic
6	#513308	C₄	1	1	yes	characteristic
7	#513308	C₄	1	1	yes	characteristic
8	#512545	C₂²	1	1	yes	characteristic
9	#483413	—	2	1	no	—
10	#483409	—	2	1	no	—
11	#153425	—	2	2	no	—
12	#153425	C₄	1	2	yes	—
13	C₂×#7511	—	2	2	no	—
14	C₂×#7511	C₂²	1	2	yes	—
15	#151058	—	2	2	no	—
16	#151057	C₄	1	4	yes	—
17	#151056	C₄	1	2	yes	—
18	#151055	C₄	1	2	yes	—
19	#151054	—	2	2	no	—
20	C₂×#7464	C₂²	1	2	yes	—
21	#151038	—	2	1	no	—
22	#151037	C₂²	1	1	yes	characteristic
23	C₂×#7463	C₂²	1	1	yes	characteristic
24	#151035	—	2	1	no	—
25	C₂×#2258	C₂×C₄	1	1	yes	characteristic
26	C₂×#2258	Q₈	1	1	yes	characteristic
27	#33428	—	4	1	no	—
28	C₄×#504	D₄	1	1	yes	characteristic
29	#17799	—	4	1	no	—
30	#13737	—	2	4	no	—
31	#13737	—	2	4	no	—
32	C₂×M₁₂₈	—	2	2	no	—
33	C₂×M₁₂₈	—	2	2	no	—
34	C₂×M₁₂₈	C₂×C₄	1	2	yes	—
35	C₂×M₁₂₈	C₂×C₄	1	2	yes	—
36	#13549	C₂×C₄	1	2	yes	—
37	#13549	C₂×C₄	1	1	yes	characteristic
38	#13549	D₄	1	2	yes	—
39	#13549	Q₈	1	1	yes	characteristic
40	C₄×M₆₄	C₂×C₄	1	2	yes	—
41	C₄×M₆₄	D₄	1	1	yes	characteristic
42	C₂×#1397	D₄	1	2	yes	—
43	C₂×#1397	C₂³	1	1	yes	characteristic
44	#12027	—	2	4	no	—
45	#12026	—	4	2	no	—
46	#12025	—	4	2	no	—
47	#12024	—	2	2	no	—
48	#12023	—	2	2	no	—
49	C₂×#1373	—	2	2	no	—
50	#12009	—	4	1	no	—
51	#12008	—	2	1	no	—
52	C₂×#1372	—	2	1	no	—
53	#12002	—	4	1	no	—
54	#7511	—	2	4	no	—
55	#7511	—	2	2	no	—
56	#7511	—	2	2	no	—
57	#7464	—	2	2	no	—
58	#7464	C₂×C₄	1	4	yes	—
59	#7463	—	2	2	no	—
60	#7463	—	2	2	no	—
61	#7463	—	2	1	no	—
62	#7463	—	2	1	no	—
63	#7463	C₂×C₄	1	1	yes	characteristic
64	#7463	C₂×C₄	1	2	yes	—
65	#7463	C₂×C₄	1	1	yes	characteristic
66	#7462	—	2	4	no	—
67	C₂×#504	C₄⋊C₄	1	1	yes	characteristic
68	C₂×#504	Q₁₆	1	1	yes	characteristic
69	#2989	—	8	1	no	—
70	C₄×#131	D₈	1	1	yes	characteristic
71	#2258	—	2	4	no	—
72	#2258	—	2	2	no	—
73	#2258	—	2	2	no	—
74	C₂×M₆₄	C₄²	1	2	yes	—
75	C₂×M₆₄	C₄⋊C₄	1	1	yes	characteristic
76	C₂×M₆₄	C₄⋊C₄	1	1	yes	characteristic
77	C₂×M₆₄	C₄⋊C₄	1	2	yes	—
78	C₂²×C₃₂	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₃₂	C₄⋊C₄	1	2	yes	—
80	C₂²×C₃₂	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₃₂	C₂×Q₈	1	1	yes	characteristic
82	#2166	—	4	4	no	—
83	#2165	—	8	2	no	—
84	C₂×#364	—	4	2	no	—
85	#2115	—	8	1	no	—
86	#2110	—	8	1	no	—
87	#2108	C₂²⋊C₄	1	2	yes	—
88	#2108	C₄⋊C₄	1	1	yes	characteristic
89	#2108	SD₁₆	1	2	yes	—
90	#2108	Q₁₆	1	1	yes	characteristic
91	C₄×M₃₂	D₈	1	1	yes	characteristic
92	C₄×M₃₂	SD₁₆	1	2	yes	—
93	C₂×C₄×C₁₆	C₂²⋊C₄	1	2	yes	—
94	C₂×C₄×C₁₆	C₂×D₄	1	1	yes	characteristic
95	#1569	—	8	2	no	—
96	#1568	—	4	2	no	—
97	#1567	—	4	2	no	—
98	#1565	—	8	1	no	—
99	#1564	—	4	1	no	—
100	C₂×#335	—	4	1	no	—
101	M₁₂₈	—	2	4	no	—
102	M₁₂₈	—	2	4	no	—
103	M₁₂₈	—	2	2	no	—
104	M₁₂₈	—	2	2	no	—
105	M₁₂₈	—	2	2	no	—
106	M₁₂₈	M₁₆	1	4	yes	complemented
107	C₂×C₆₄	—	2	4	no	—
108	C₂×C₆₄	—	2	4	no	—
109	C₂×C₆₄	—	2	2	no	—
110	C₂×C₆₄	M₁₆	1	4	yes	complemented
111	#1397	C₂²⋊C₄	1	2	yes	—
112	#1397	C₂²⋊C₄	1	2	yes	—
113	#1397	C₄⋊C₄	1	2	yes	—
114	#1397	C₂²×C₄	1	1	yes	characteristic
115	#1397	C₂²×C₄	1	1	yes	characteristic
116	#1397	C₂×D₄	1	1	yes	characteristic
117	#1397	C₂×D₄	1	1	yes	characteristic
118	C₄×C₃₂	C₄²	1	2	yes	—
119	C₄×C₃₂	C₂²⋊C₄	1	2	yes	—
120	C₄×C₃₂	C₄⋊C₄	1	2	yes	—
121	#1373	—	2	4	no	—
122	#1373	—	4	2	no	—
123	#1372	—	2	2	no	—
124	#1372	—	2	1	no	—
125	#1372	—	2	1	no	—
126	#1372	—	4	2	no	—
127	#1372	—	4	1	no	—
128	#1371	—	4	2	no	—
129	#1371	—	4	1	no	—
130	#1369	—	4	4	no	—
131	C₂×#131	#107	1	1	yes	characteristic
132	C₂×#131	SD₃₂	1	1	yes	characteristic
133	C₄×#41	SD₃₂	1	1	yes	characteristic
134	#504	—	2	4	no	—
135	#504	—	2	2	no	—
136	#504	—	2	2	no	—
137	C₂×M₃₂	—	8	2	no	—
138	C₂×M₃₂	#103	1	2	yes	—
139	C₂×M₃₂	#106	1	2	yes	—
140	C₂×M₃₂	#107	1	1	yes	characteristic
141	C₂×M₃₂	#107	1	1	yes	characteristic
142	C₂²×C₁₆	#95	1	2	yes	—
143	C₂²×C₁₆	#103	1	2	yes	—
144	C₂²×C₁₆	C₂×C₄⋊C₄	1	1	yes	characteristic
145	C₂²×C₁₆	C₂×Q₁₆	1	1	yes	characteristic
146	#446	—	16	1	no	—
147	#436	—	16	1	no	—
148	#435	—	16	2	no	—
149	#432	—	8	2	no	—
150	#431	—	8	1	no	—
151	#423	—	8	2	no	—
152	C₂×C₄⋊C₈	—	8	1	no	—
153	#405	#102	1	2	yes	—
154	#405	#107	1	1	yes	characteristic
155	#405	SD₃₂	1	1	yes	characteristic
156	#405	Q₃₂	1	2	yes	—
157	C₄×M₁₆	D₁₆	1	2	yes	—
158	C₄×M₁₆	SD₃₂	1	1	yes	characteristic
159	C₂×C₄×C₈	#102	1	2	yes	—
160	C₂×C₄×C₈	C₂×D₈	1	1	yes	characteristic
161	M₆₄	—	2	2	no	—
162	M₆₄	—	2	4	no	—
163	M₆₄	—	2	4	no	—
164	M₆₄	#108	1	4	yes	—
165	C₂×C₃₂	—	2	2	no	—
166	C₂×C₃₂	—	2	4	no	—
167	C₂×C₃₂	—	2	2	no	—
168	C₂×C₃₂	—	2	2	no	—
169	C₂×C₃₂	#95	1	2	yes	—
170	C₂×C₃₂	#95	1	2	yes	—
171	C₂×C₃₂	C₂×C₄²	1	1	yes	characteristic
172	C₂×C₃₂	C₂×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₃₂	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₃₂	C₂×M₁₆	1	2	yes	—
176	C₂×C₃₂	C₂×M₁₆	1	2	yes	—
177	#364	—	4	4	no	—
178	#364	—	8	2	no	—
179	#349	—	8	4	no	—
180	#347	—	8	4	no	—
181	#346	#102	1	2	yes	—
182	#346	#103	1	2	yes	—
183	#346	#106	1	2	yes	—
184	C₄×C₁₆	#95	1	2	yes	—
185	C₄×C₁₆	#102	1	2	yes	—
186	C₄×C₁₆	#103	1	2	yes	—
187	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
188	C₄×C₁₆	C₂×C₂²⋊C₄	1	1	yes	characteristic
189	C₄×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
190	C₄×C₁₆	C₂×SD₁₆	1	1	yes	characteristic
191	#335	—	4	2	no	—
192	#335	—	4	1	no	—
193	#335	—	4	1	no	—
194	#335	—	8	2	no	—
195	#335	—	8	1	no	—
196	#334	—	8	2	no	—
197	#334	—	8	1	no	—
198	C₆₄	—	2	4	no	—
199	C₆₄	—	2	4	no	—
200	C₂×#41	#367	1	1	yes	characteristic
201	#131	—	2	2	no	—
202	#131	—	2	2	no	—
203	#131	—	4	2	no	—
204	C₂×M₁₆	—	16	2	no	—
205	C₂×M₁₆	#357	1	2	yes	—
206	C₂×M₁₆	#366	1	2	yes	—
207	C₂×M₁₆	#367	1	1	yes	characteristic
208	C₂×M₁₆	#367	1	1	yes	characteristic
209	C₂²×C₈	—	16	1	no	—
210	C₂²×C₈	#340	1	2	yes	—
211	C₂²×C₈	#358	1	2	yes	—
212	C₂²×C₈	C₂×#107	1	1	yes	characteristic
213	C₂²×C₈	C₂×SD₃₂	1	1	yes	characteristic
214	C₄×Q₈	—	2	1	no	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	1	no	—
217	#117	#358	1	2	yes	—
218	#117	#367	1	1	yes	characteristic
219	C₂×C₄²	#357	1	2	yes	—
220	C₂×C₄²	C₂×SD₃₂	1	1	yes	characteristic
221	M₃₂	—	2	2	no	—
222	M₃₂	—	2	4	no	—
223	M₃₂	—	2	4	no	—
224	M₃₂	—	8	4	no	—
225	M₃₂	—	16	2	no	—
226	M₃₂	#368	1	4	yes	—
227	C₂×C₁₆	—	2	2	no	—
228	C₂×C₁₆	—	2	4	no	—
229	C₂×C₁₆	—	2	2	no	—
230	C₂×C₁₆	—	2	2	no	—
231	C₂×C₁₆	—	16	2	no	—
232	C₂×C₁₆	#340	1	2	yes	—
233	C₂×C₁₆	#340	1	2	yes	—
234	C₂×C₁₆	C₂×#95	1	1	yes	characteristic
235	C₂×C₁₆	C₂×#103	1	1	yes	characteristic
236	C₂×C₁₆	C₂×#103	1	1	yes	characteristic
237	C₂×C₁₆	#423	1	2	yes	—
238	C₂×C₁₆	C₂×#106	1	1	yes	characteristic
239	C₂×C₁₆	C₂×#108	1	2	yes	—
240	C₄⋊C₈	—	8	1	no	—
241	C₄⋊C₈	—	8	2	no	—
242	C₄⋊C₈	—	8	1	no	—
243	C₄⋊C₈	—	16	1	no	—
244	C₄⋊C₈	—	16	2	no	—
245	C₂²⋊C₈	—	16	1	no	—
246	C₂²⋊C₈	—	16	2	no	—
247	#97	—	16	2	no	—
248	#97	#357	1	2	yes	—
249	#97	#358	1	2	yes	—
250	#97	#366	1	2	yes	—
251	C₄×C₈	—	16	1	no	—
252	C₄×C₈	#340	1	2	yes	—
253	C₄×C₈	#357	1	2	yes	—
254	C₄×C₈	#358	1	2	yes	—
255	C₄×C₈	C₂×#102	1	1	yes	characteristic
256	C₄×C₈	C₂×#102	1	1	yes	characteristic
257	C₄×C₈	C₂×D₁₆	1	1	yes	characteristic
258	C₄×C₈	C₂×Q₃₂	1	1	yes	characteristic
259	C₃₂	—	2	2	no	—
260	C₃₂	—	2	1	no	—
261	C₃₂	—	2	1	no	—
262	C₃₂	#341	1	4	yes	—
263	C₃₂	C₄×M₁₆	1	2	yes	—
264	C₃₂	#407	1	1	yes	characteristic
265	C₃₂	#407	1	1	yes	characteristic
266	#41	—	2	2	no	—
267	#41	—	2	2	no	—
268	C₂×Q₈	#1426	1	1	yes	characteristic
269	C₂×D₄	#1426	1	2	yes	—
270	C₂×D₄	#1426	1	1	yes	characteristic
271	C₂²×C₄	—	2	2	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	#1380	1	2	yes	—
274	C₂²×C₄	C₂×#367	1	1	yes	characteristic
275	M₁₆	—	2	2	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	4	no	—
278	M₁₆	—	4	2	no	—
279	M₁₆	—	64	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	4	2	no	—
284	C₂×C₈	—	16	1	no	—
285	C₂×C₈	—	16	1	no	—
286	C₂×C₈	—	16	2	no	—
287	C₂×C₈	—	64	1	no	—
288	C₂×C₈	#1380	1	2	yes	—
289	C₂×C₈	#1380	1	2	yes	—
290	C₂×C₈	#1567	1	2	yes	—
291	C₂×C₈	C₂×#340	1	1	yes	characteristic
292	C₂×C₈	C₂×#357	1	1	yes	characteristic
293	C₂×C₈	C₂×#358	1	1	yes	characteristic
294	C₂×C₈	C₂×#366	1	1	yes	characteristic
295	C₂×C₈	C₂×#368	1	2	yes	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	1	no	—
298	C₂²⋊C₄	—	2	2	no	—
299	C₂²⋊C₄	—	2	1	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	1	no	—
302	C₄²	#1380	1	2	yes	—
303	C₄²	C₂×#357	1	1	yes	characteristic
304	C₄²	C₂×#358	1	1	yes	characteristic
305	C₁₆	—	2	1	no	—
306	C₁₆	—	2	1	no	—
307	C₁₆	—	2	2	no	—
308	C₁₆	—	16	4	no	—
309	C₁₆	#1381	1	4	yes	—
310	C₁₆	#1566	1	2	yes	—
311	C₁₆	#1728	1	1	yes	characteristic
312	C₁₆	C₂×#341	1	1	yes	characteristic
313	C₂³	#7474	1	2	yes	—
314	C₂³	C₂×#1426	1	1	yes	characteristic
315	Q₈	—	2	2	no	—
316	D₄	—	2	4	no	—
317	D₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	8	2	no	—
325	C₂×C₄	#7474	1	2	yes	—
326	C₂×C₄	#12023	1	2	yes	—
327	C₂×C₄	C₂×#1380	1	1	yes	characteristic
328	C₂×C₄	C₂×#1426	1	1	yes	characteristic
329	C₂×C₄	#13638	1	2	yes	—
330	C₈	—	2	2	no	—
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	2	2	no	—
334	C₈	—	32	1	no	—
335	C₈	—	32	2	no	—
336	C₈	—	32	1	no	—
337	C₈	#7874	1	1	yes	characteristic
338	C₈	#12022	1	2	yes	—
339	C₈	C₂×#1381	1	1	yes	characteristic
340	C₂²	—	2	2	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	2	no	—
343	C₂²	#151054	1	2	yes	—
344	C₂²	C₂×#7474	1	1	yes	characteristic
345	C₄	—	2	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	2	no	—
349	C₄	#126685	1	1	yes	characteristic
350	C₄	#151054	1	2	yes	—
351	C₄	#151774	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	2	no	—
354	C₂	—	4	2	no	—
355	C₂	#72322776	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂	#72322776
commutator	C₂×C₃₂	C₂×C₄²
Frattini, cosocle	C₂×#1397	C₂³
prime radical	C₂×#41	#367
LCS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ C₄×#504 ◁ #512545 ◁ G
LECS	G ▷ C₂×#1397 ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #72321153	C₂	semidirect
4. #72286496	C₂	semidirect
106. M₁₂₈	M₁₆	semidirect
110. C₂×C₆₄	M₁₆	semidirect

Maximal subgroups

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

subgroup	index
#72321329	2
#72321153	2
#72320649	2
#72286496	2
#72232024	2