Subgroups — C₄×M₁₆×#64408 #34019791

Order 18688.

1734 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₇₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×M₁₆×#64408	C₂	1	4	yes	maximal
3	C₂×C₄²×#64408	C₂	1	1	yes	maximal, characteristic
4	C₄×#14556645	C₂	1	2	yes	maximal, complemented
5	#97×#64408	C₂	1	4	yes	maximal, complemented
6	C₄×C₈×#64408	C₂	1	2	yes	maximal, complemented
7	C₄×M₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×D₇₃⋊M₁₆	C₂²	1	2	yes	—
9	C₂²×C₄×#64408	C₄	1	4	yes	—
10	C₂²×C₄×#64408	C₂²	1	2	yes	—
11	C₄×#11924038	C₄	1	4	yes	complemented
12	C₄×#11924037	C₄	1	4	yes	complemented
13	M₁₆×#64408	C₄	1	32	yes	direct product factor
14	#17989546	C₂²	1	2	yes	—
15	C₂×#14556645	C₂²	1	8	yes	—
16	C₄×#11924036	C₂²	1	2	yes	—
17	C₂×C₈×#64408	C₂²	1	8	yes	—
18	C₄²×#64408	C₄	1	4	yes	—
19	C₄²×#64408	C₂²	1	1	yes	characteristic
20	C₄²×#64408	C₂²	1	2	yes	—
21	C₄²×#64408	C₂²	1	1	yes	characteristic
22	C₂×M₁₆×D₇₃	C₂²	1	2	yes	—
23	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
24	C₄×#11924012	C₄	1	1	yes	characteristic, complemented
25	#17988539	C₄	1	2	yes	complemented
26	M₁₆×Dic₇₃	C₄	1	2	yes	complemented
27	C₄×#11924011	C₄	1	2	yes	complemented
28	#97×D₇₃	C₂²	1	2	yes	—
29	C₄×C₈×D₇₃	C₂²	1	2	yes	—
30	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
31	C₂³×#64408	C₂×C₄	1	4	yes	—
32	C₂×#11924038	C₂×C₄	1	4	yes	—
33	D₇₃⋊M₁₆	C₂×C₄	1	16	yes	—
34	C₂×#11924037	C₂×C₄	1	4	yes	—
35	C₂×#11924036	C₂³	1	4	yes	—
36	C₂×C₄×#64408	—	2	16	no	—
37	C₂×C₄×#64408	C₂×C₄	1	8	yes	—
38	C₂×C₄×#64408	C₂×C₄	1	4	yes	—
39	C₂×C₄×#64408	C₂×C₄	1	4	yes	—
40	C₂×C₄×#64408	C₂×C₄	1	16	yes	—
41	C₂×C₄×#64408	C₂×C₄	1	4	yes	—
42	C₂×C₄×#64408	C₂³	1	2	yes	—
43	C₂×C₄×#64408	C₂³	1	2	yes	—
44	#14556646	C₂×C₄	1	8	yes	complemented
45	#14556645	C₂×C₄	1	32	yes	complemented
46	C₄×#10596833	C₂×C₄	1	8	yes	complemented
47	C₈×#64408	C₂×C₄	1	32	yes	complemented
48	C₂²×C₄×D₇₃	C₂×C₄	1	2	yes	—
49	C₂²×C₄×D₇₃	C₂³	1	1	yes	characteristic
50	C₂×#11924012	C₂×C₄	1	2	yes	—
51	C₂×#11924011	C₂×C₄	1	4	yes	—
52	M₁₆×D₇₃	C₂×C₄	1	16	yes	—
53	C₂×C₈×D₇₃	C₂³	1	4	yes	—
54	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
55	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄²×D₇₃	C₂×C₄	1	2	yes	—
57	C₄²×D₇₃	C₂×C₄	1	2	yes	—
58	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
59	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
60	C₄²×D₇₃	C₂³	1	1	yes	characteristic
61	C₄²×D₇₃	C₂³	1	1	yes	characteristic
62	#14556502	C₂×C₄	1	2	yes	complemented
63	#14556501	C₂×C₄	1	4	yes	complemented
64	C₄×#10596831	C₂×C₄	1	2	yes	complemented
65	C₈×Dic₇₃	C₂×C₄	1	4	yes	complemented
66	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
67	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
68	#97×C₇₃	C₂×C₄	1	2	yes	complemented
69	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	complemented
70	C₂²×#64408	—	2	16	no	—
71	C₂²×#64408	C₄²	1	16	yes	—
72	C₂²×#64408	C₂²×C₄	1	4	yes	—
73	C₂²×#64408	C₂²×C₄	1	4	yes	—
74	#11924038	C₄²	1	16	yes	complemented
75	#11924037	C₄²	1	16	yes	complemented
76	#11924036	C₂²×C₄	1	16	yes	—
77	C₂×#10596833	C₂²×C₄	1	8	yes	—
78	C₄×#64408	—	2	16	no	—
79	C₄×#64408	—	2	8	no	—
80	C₄×#64408	C₄²	1	16	yes	—
81	C₄×#64408	M₁₆	1	16	yes	direct product factor
82	C₄×#64408	C₂²×C₄	1	8	yes	—
83	C₄×#64408	C₂²×C₄	1	8	yes	—
84	C₂³×D₇₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×Dic₇₃	C₄²	1	2	yes	—
86	C₂²×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×D₇₃	—	2	8	no	—
88	C₂×C₄×D₇₃	C₄²	1	2	yes	—
89	C₂×C₄×D₇₃	C₄²	1	2	yes	—
90	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×D₇₃	C₂²×C₄	1	8	yes	—
95	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
96	C₂×C₄×D₇₃	C₂⁴	1	1	yes	characteristic
97	#11924012	C₄²	1	8	yes	complemented
98	#11924011	C₄²	1	16	yes	complemented
99	C₂×#10596831	C₂²×C₄	1	4	yes	—
100	C₈×D₇₃	C₂²×C₄	1	16	yes	—
101	C₄×Dic₇₃	C₄²	1	2	yes	—
102	C₄×Dic₇₃	C₄²	1	2	yes	—
103	C₄×Dic₇₃	C₄²	1	2	yes	—
104	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
107	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
109	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	M₁₆×C₇₃	C₄²	1	8	yes	complemented
111	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
112	C₄²×C₇₃	C₄²	1	2	yes	—
113	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
116	C₂²×D₇₃	—	2	4	no	—
117	C₂²×D₇₃	C₂×C₄²	1	1	yes	characteristic
118	C₂²×D₇₃	C₂×C₄²	1	4	yes	—
119	C₂²×D₇₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×#64408	—	2	8	no	—
121	C₂×#64408	—	2	32	no	—
122	C₂×#64408	C₂×C₄²	1	8	yes	—
123	C₂×#64408	C₂×C₄²	1	8	yes	—
124	C₂×#64408	C₂×M₁₆	1	16	yes	—
125	C₂×C₄×C₇₃	—	2	4	no	—
126	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
127	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂×C₄²	1	4	yes	—
129	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
131	C₂×Dic₇₃	—	2	4	no	—
132	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
133	C₂×Dic₇₃	C₂×C₄²	1	1	yes	characteristic
134	C₂×Dic₇₃	C₂×C₄²	1	4	yes	—
135	C₂×Dic₇₃	C₂×C₄²	1	2	yes	—
136	C₂×Dic₇₃	C₂³×C₄	1	1	yes	characteristic
137	C₄×D₇₃	—	2	4	no	—
138	C₄×D₇₃	—	2	4	no	—
139	C₄×D₇₃	—	2	4	no	—
140	C₄×D₇₃	C₂×C₄²	1	4	yes	—
141	C₄×D₇₃	C₂×C₄²	1	4	yes	—
142	C₄×D₇₃	C₂×C₄²	1	4	yes	—
143	C₄×D₇₃	C₂×M₁₆	1	8	yes	—
144	C₄×D₇₃	C₂³×C₄	1	4	yes	—
145	#10596833	C₂×C₄²	1	16	yes	complemented
146	C₈×C₇₃	C₂×C₄²	1	8	yes	complemented
147	#10596831	C₂×C₄²	1	8	yes	complemented
148	C₂²×C₇₃	—	2	2	no	—
149	C₂²×C₇₃	C₄³	1	2	yes	—
150	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	—	2	8	no	—
153	C₂×D₇₃	C₄³	1	2	yes	—
154	C₂×D₇₃	C₂²×C₄²	1	2	yes	—
155	C₂×D₇₃	C₂²×M₁₆	1	4	yes	—
156	#64408	—	2	8	no	—
157	#64408	C₄×M₁₆	1	16	yes	direct product factor
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	C₄³	1	2	yes	—
160	C₄×C₇₃	C₄×M₁₆	1	4	yes	complemented
161	C₄×C₇₃	C₂²×C₄²	1	2	yes	—
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	C₄³	1	2	yes	—
164	Dic₇₃	C₄×M₁₆	1	4	yes	complemented
165	Dic₇₃	C₂²×C₄²	1	2	yes	—
166	C₄²×M₁₆	—	73	1	no	maximal, Sylow
167	C₂×C₇₃	—	2	2	no	—
168	C₂×C₇₃	C₂×C₄³	1	1	yes	characteristic
169	C₂×C₇₃	C₂×C₄×M₁₆	1	2	yes	—
170	D₇₃	—	2	2	no	—
171	D₇₃	C₂×C₄×M₁₆	1	4	yes	—
172	C₂×C₄×M₁₆	—	73	4	no	—
173	C₂×C₄×M₁₆	—	73	1	no	—
174	C₂×C₄×M₁₆	—	73	1	no	—
175	C₂×C₄³	—	73	1	no	—
176	C₄×#97	—	73	4	no	—
177	C₄×#97	—	73	2	no	—
178	C₄²×C₈	—	73	2	no	—
179	C₇₃	C₄²×M₁₆	1	1	yes	characteristic, complemented, Sylow
180	C₂²×M₁₆	—	73	2	no	—
181	C₂²×M₁₆	—	73	2	no	—
182	C₂²×C₄²	—	73	2	no	—
183	C₂²×C₄²	—	73	1	no	—
184	C₂²×C₄²	—	73	4	no	—
185	C₄×M₁₆	—	73	32	no	—
186	C₄×M₁₆	—	73	4	no	—
187	C₄×M₁₆	—	73	4	no	—
188	C₄×M₁₆	—	73	2	no	—
189	C₄×M₁₆	—	73	1	no	—
190	C₄×M₁₆	—	73	2	no	—
191	C₄×M₁₆	—	73	2	no	—
192	C₄×M₁₆	#64408	1	1	yes	characteristic, direct product factor
193	C₂×#97	—	73	8	no	—
194	C₂×#97	—	73	2	no	—
195	C₂×#97	—	73	2	no	—
196	C₂×C₄×C₈	—	73	8	no	—
197	C₂×C₄×C₈	—	73	2	no	—
198	C₂×C₄×C₈	—	73	2	no	—
199	C₄³	—	73	4	no	—
200	C₄³	—	73	1	no	—
201	C₄³	—	73	2	no	—
202	C₄³	—	73	1	no	—
203	C₂³×C₄	—	73	1	no	—
204	C₂³×C₄	—	73	4	no	—
205	C₂³×C₄	—	73	2	no	—
206	C₂×M₁₆	—	73	4	no	—
207	C₂×M₁₆	—	73	16	no	—
208	C₂×M₁₆	—	73	4	no	—
209	C₂×M₁₆	—	73	2	no	—
210	C₂×M₁₆	—	73	16	no	—
211	C₂×M₁₆	—	73	4	no	—
212	C₂×M₁₆	C₂×#64408	1	2	yes	—
213	C₂²×C₈	—	73	4	no	—
214	C₂²×C₈	—	73	4	no	—
215	C₂×C₄²	—	73	2	no	—
216	C₂×C₄²	—	73	2	no	—
217	C₂×C₄²	—	73	4	no	—
218	C₂×C₄²	—	73	4	no	—
219	C₂×C₄²	—	73	16	no	—
220	C₂×C₄²	—	73	2	no	—
221	C₂×C₄²	—	73	1	no	—
222	C₂×C₄²	—	73	2	no	—
223	C₂×C₄²	—	73	2	no	—
224	C₂×C₄²	—	73	1	no	—
225	C₂×C₄²	—	73	1	no	—
226	C₂×C₄²	—	73	1	no	—
227	C₂×C₄²	—	73	1	no	—
228	C₂×C₄²	—	73	8	no	—
229	C₂×C₄²	—	73	4	no	—
230	C₂×C₄²	—	146	16	no	—
231	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
232	#97	—	73	32	no	—
233	#97	—	73	8	no	—
234	#97	—	73	4	no	—
235	#97	—	73	2	no	—
236	#97	C₂×#64408	1	2	yes	complemented
237	C₄×C₈	—	73	32	no	—
238	C₄×C₈	—	73	8	no	—
239	C₄×C₈	—	73	4	no	—
240	C₄×C₈	—	73	2	no	—
241	C₄×C₈	C₂×#64408	1	2	yes	complemented
242	C₂⁴	—	73	1	no	—
243	C₂²×C₄	—	73	4	no	—
244	C₂²×C₄	—	73	1	no	—
245	C₂²×C₄	—	73	1	no	—
246	C₂²×C₄	—	73	1	no	—
247	C₂²×C₄	—	73	1	no	—
248	C₂²×C₄	—	73	2	no	—
249	C₂²×C₄	—	73	1	no	—
250	C₂²×C₄	—	73	8	no	—
251	C₂²×C₄	—	73	16	no	—
252	C₂²×C₄	—	73	4	no	—
253	C₂²×C₄	—	73	2	no	—
254	C₂²×C₄	—	73	2	no	—
255	C₂²×C₄	—	73	2	no	—
256	C₂²×C₄	—	73	2	no	—
257	C₂²×C₄	—	146	8	no	—
258	C₂²×C₄	—	146	16	no	—
259	C₂²×C₄	C₄×#64408	1	2	yes	—
260	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
261	M₁₆	—	73	16	no	—
262	M₁₆	—	73	16	no	—
263	M₁₆	—	73	8	no	—
264	M₁₆	—	73	16	no	—
265	M₁₆	C₄×#64408	1	8	yes	direct product factor
266	C₂×C₈	—	73	8	no	—
267	C₂×C₈	—	73	16	no	—
268	C₂×C₈	—	73	4	no	—
269	C₂×C₈	—	73	16	no	—
270	C₂×C₈	C₂²×#64408	1	4	yes	—
271	C₄²	—	73	16	no	—
272	C₄²	—	73	16	no	—
273	C₄²	—	73	8	no	—
274	C₄²	—	73	8	no	—
275	C₄²	—	73	1	no	—
276	C₄²	—	73	1	no	—
277	C₄²	—	73	2	no	—
278	C₄²	—	73	2	no	—
279	C₄²	—	73	2	no	—
280	C₄²	—	73	2	no	—
281	C₄²	—	73	2	no	—
282	C₄²	—	146	16	no	—
283	C₄²	—	146	8	no	—
284	C₄²	C₄×#64408	1	2	yes	—
285	C₄²	C₂²×#64408	1	1	yes	characteristic
286	C₄²	C₂²×#64408	1	1	yes	characteristic
287	C₂³	—	73	1	no	—
288	C₂³	—	73	1	no	—
289	C₂³	—	73	4	no	—
290	C₂³	—	146	4	no	—
291	C₂³	C₂×C₄×#64408	1	1	yes	characteristic
292	C₂×C₄	—	2	4	no	—
293	C₂×C₄	—	73	8	no	—
294	C₂×C₄	—	73	16	no	—
295	C₂×C₄	—	73	1	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	2	no	—
298	C₂×C₄	—	73	4	no	—
299	C₂×C₄	—	73	4	no	—
300	C₂×C₄	—	73	4	no	—
301	C₂×C₄	—	73	4	no	—
302	C₂×C₄	—	73	4	no	—
303	C₂×C₄	—	73	8	no	—
304	C₂×C₄	—	73	8	no	—
305	C₂×C₄	—	73	2	no	—
306	C₂×C₄	—	146	4	no	—
307	C₂×C₄	—	146	4	no	—
308	C₂×C₄	—	146	4	no	—
309	C₂×C₄	—	146	32	no	—
310	C₂×C₄	—	146	8	no	—
311	C₂×C₄	—	146	4	no	—
312	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
313	C₂×C₄	C₂×C₄×#64408	1	4	yes	—
314	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×#64408	1	2	yes	—
316	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
317	C₈	—	73	16	no	—
318	C₈	—	73	8	no	—
319	C₈	C₂×C₄×#64408	1	8	yes	complemented
320	C₂²	—	2	2	no	—
321	C₂²	—	73	2	no	—
322	C₂²	—	73	4	no	—
323	C₂²	—	73	2	no	—
324	C₂²	—	146	8	no	—
325	C₂²	—	146	2	no	—
326	C₂²	C₄²×#64408	1	2	yes	—
327	C₂²	C₂²×C₄×#64408	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	73	16	no	—
330	C₄	—	73	2	no	—
331	C₄	—	73	2	no	—
332	C₄	—	73	4	no	—
333	C₄	—	146	8	no	—
334	C₄	—	146	2	no	—
335	C₄	C₄²×#64408	1	2	yes	—
336	C₄	M₁₆×#64408	1	4	yes	direct product factor
337	C₄	C₂²×C₄×#64408	1	2	yes	—
338	C₂	—	2	2	no	—
339	C₂	—	73	4	no	—
340	C₂	—	146	2	no	—
341	C₂	C₂×C₄²×#64408	1	1	yes	characteristic
342	C₂	C₂×M₁₆×#64408	1	2	yes	—

	subgroup	G/H
center	C₄²	C₂²×#64408
commutator	C₂×C₇₃	C₂×C₄³
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×M₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄²
prime radical	C₂³×D₇₃	C₂²×C₄
cosocle	C₂×C₄×D₇₃	C₂⁴
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	C₄²×M₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
4. C₄×#14556645	C₂	semidirect
5. #97×#64408	C₂	semidirect
6. C₄×C₈×#64408	C₂	semidirect
11. C₄×#11924038	C₄	semidirect
12. C₄×#11924037	C₄	semidirect
13. M₁₆×#64408	C₄	direct (N × H)
24. C₄×#11924012	C₄	semidirect
25. #17988539	C₄	semidirect
26. M₁₆×Dic₇₃	C₄	semidirect
27. C₄×#11924011	C₄	semidirect
30. C₄×M₁₆×C₇₃	C₄	semidirect
44. #14556646	C₂×C₄	semidirect
45. #14556645	C₂×C₄	semidirect
46. C₄×#10596833	C₂×C₄	semidirect
47. C₈×#64408	C₂×C₄	semidirect
62. #14556502	C₂×C₄	semidirect
63. #14556501	C₂×C₄	semidirect
64. C₄×#10596831	C₂×C₄	semidirect
65. C₈×Dic₇₃	C₂×C₄	semidirect
68. #97×C₇₃	C₂×C₄	semidirect
69. C₄×C₈×C₇₃	C₂×C₄	semidirect
74. #11924038	C₄²	semidirect
75. #11924037	C₄²	semidirect
81. C₄×#64408	M₁₆	direct (N × H)
97. #11924012	C₄²	semidirect
98. #11924011	C₄²	semidirect
110. M₁₆×C₇₃	C₄²	semidirect
145. #10596833	C₂×C₄²	semidirect
146. C₈×C₇₃	C₂×C₄²	semidirect
147. #10596831	C₂×C₄²	semidirect
157. #64408	C₄×M₁₆	direct (N × H)
160. C₄×C₇₃	C₄×M₁₆	semidirect
164. Dic₇₃	C₄×M₁₆	semidirect
179. C₇₃	C₄²×M₁₆	semidirect
192. C₄×M₁₆	#64408	direct (N × H)
236. #97	C₂×#64408	semidirect
241. C₄×C₈	C₂×#64408	semidirect
265. M₁₆	C₄×#64408	direct (N × H)
319. C₈	C₂×C₄×#64408	semidirect
336. C₄	M₁₆×#64408	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×D₇₃⋊M₁₆	2
C₂×M₁₆×#64408	2
C₂×C₄²×#64408	2
C₄×#14556645	2
#97×#64408	2
C₄×C₈×#64408	2
C₄×M₁₆×D₇₃	2
C₄²×M₁₆	73

Subgroups — C4×M16×#64408 #34019791

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×M₁₆×#64408 #34019791