Subgroups — #22618663

Order 9344.

544 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×M₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₂²⋊D₇₃	C₂	1	1	yes	maximal, characteristic
4	#17988662	C₂	1	2	yes	maximal, complemented
5	#17988661	C₂	1	2	yes	maximal, complemented
6	#17988655	C₂	1	2	yes	maximal, complemented
7	#17988650	C₂	1	2	yes	maximal, complemented
8	#17988610	C₂	1	1	yes	maximal, characteristic, complemented
9	#17988550	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#407×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
13	C₂²×C₄×D₇₃	C₂²	1	1	yes	characteristic
14	C₂×#11924012	—	2	4	no	—
15	C₂×#11924012	C₂²	1	1	yes	characteristic
16	C₂×#11924012	C₂²	1	1	yes	characteristic
17	C₂×#11924012	C₂²	1	2	yes	complemented
18	C₂×#11924011	C₂²	1	2	yes	complemented
19	M₁₆×D₇₃	—	2	4	no	—
20	C₂²×#10596831	C₂²	1	2	yes	complemented
21	C₂×C₈×D₇₃	C₂²	1	2	yes	complemented
22	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
23	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
25	C₄×C₂²⋊D₇₃	C₄	1	1	yes	characteristic
26	C₄×C₂²⋊D₇₃	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₇₃	C₂²	1	1	yes	characteristic
28	C₄×C₂²⋊D₇₃	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₇₃	C₂²	1	1	yes	characteristic
30	C₂×#11924007	C₄	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₇₃	C₄	1	1	yes	characteristic
32	C₂×#11924005	C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
34	#14556520	C₂²	1	2	yes	complemented
35	#14556520	C₂²	1	2	yes	complemented
36	#14556503	C₂²	1	2	yes	complemented
37	#14556503	C₂²	1	2	yes	complemented
38	#14556501	C₂²	1	2	yes	complemented
39	C₈×Dic₇₃	C₂²	1	2	yes	complemented
40	C₂³×C₄×C₇₃	C₂²	1	1	yes	characteristic
41	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	C₂×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
43	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	complemented
44	C₂²⋊C₈×C₇₃	C₂²	1	2	yes	complemented
45	C₂³×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊D₇₃	—	2	4	no	—
47	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊D₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₇₃	—	2	2	no	—
55	C₂×C₄×D₇₃	—	2	2	no	—
56	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
59	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
60	#11924012	—	2	2	no	—
61	#11924012	—	2	4	no	—
62	#11924012	D₄	1	4	yes	complemented
63	#11924011	—	2	4	no	—
64	C₂×#10596831	—	2	4	no	—
65	C₂×#10596831	C₂³	1	2	yes	—
66	C₂×#10596831	C₂³	1	2	yes	—
67	C₈×D₇₃	—	2	4	no	—
68	#11924007	C₂×C₄	1	1	yes	characteristic
69	#11924007	C₂×C₄	1	1	yes	characteristic
70	#11924007	C₂×C₄	1	1	yes	characteristic
71	#11924007	C₂×C₄	1	1	yes	characteristic
72	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
73	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
74	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
75	C₂²⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
76	#11924005	C₂×C₄	1	1	yes	characteristic
77	#11924005	C₂×C₄	1	1	yes	characteristic
78	#11924005	C₂×C₄	1	1	yes	characteristic
79	#11924005	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
82	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
83	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
84	C₂⁴×C₇₃	C₂×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	—	2	2	no	—
86	C₂²×C₄×C₇₃	—	2	2	no	—
87	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
91	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
93	M₁₆×C₇₃	—	2	2	no	—
94	M₁₆×C₇₃	—	2	2	no	—
95	C₂×C₈×C₇₃	C₂³	1	2	yes	—
96	C₂×C₈×C₇₃	C₂³	1	2	yes	—
97	C₂³×C₇₃	—	2	2	no	—
98	C₂³×C₇₃	—	2	2	no	—
99	C₂³×C₇₃	—	4	1	no	—
100	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×D₇₃	—	2	2	no	—
104	C₂²×D₇₃	—	2	2	no	—
105	C₂²×D₇₃	—	4	1	no	—
106	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×D₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	—	2	2	no	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	2	1	no	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	—	2	1	no	—
114	C₂×C₄×C₇₃	—	2	1	no	—
115	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₇₃	#41	1	2	yes	—
120	C₂×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₇₃	—	2	2	no	—
122	C₂²⋊D₇₃	—	2	2	no	—
123	C₂²⋊D₇₃	—	2	2	no	—
124	C₂²⋊D₇₃	M₁₆	1	4	yes	complemented
125	C₂×Dic₇₃	—	2	2	no	—
126	C₂×Dic₇₃	—	2	2	no	—
127	C₂×Dic₇₃	—	4	1	no	—
128	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
134	C₄×D₇₃	—	2	2	no	—
135	C₄×D₇₃	—	2	2	no	—
136	C₄×D₇₃	—	2	1	no	—
137	C₄×D₇₃	—	2	1	no	—
138	C₄×D₇₃	—	2	1	no	—
139	C₄×D₇₃	—	2	1	no	—
140	C₈×C₇₃	—	2	2	no	—
141	C₈×C₇₃	—	2	2	no	—
142	#10596831	—	2	2	no	—
143	#10596831	C₂×D₄	1	4	yes	complemented
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	—	2	1	no	—
150	C₂²×C₇₃	—	2	1	no	—
151	C₂²×C₇₃	—	4	4	no	—
152	C₂²×C₇₃	C₄×D₄	1	2	yes	—
153	C₂²×C₇₃	C₂×M₁₆	1	2	yes	complemented
154	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₇₃	—	2	1	no	—
156	C₂×D₇₃	—	2	1	no	—
157	C₂×D₇₃	—	2	1	no	—
158	C₂×D₇₃	—	2	1	no	—
159	C₂×D₇₃	—	2	1	no	—
160	C₂×D₇₃	—	4	4	no	—
161	C₂×D₇₃	C₂×M₁₆	1	2	yes	complemented
162	C₄×C₇₃	—	2	1	no	—
163	C₄×C₇₃	—	2	1	no	—
164	C₄×C₇₃	C₄×D₄	1	2	yes	—
165	C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
166	C₄×C₇₃	C₂×#41	1	1	yes	characteristic
167	Dic₇₃	—	2	1	no	—
168	Dic₇₃	—	2	1	no	—
169	Dic₇₃	—	2	1	no	—
170	Dic₇₃	—	2	1	no	—
171	Dic₇₃	—	2	1	no	—
172	Dic₇₃	C₂×M₁₆	1	2	yes	complemented
173	C₂×C₇₃	—	2	2	no	—
174	C₂×C₇₃	—	2	2	no	—
175	C₂×C₇₃	—	4	1	no	—
176	C₂×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
178	C₂×C₇₃	#568	1	1	yes	characteristic
179	D₇₃	—	2	2	no	—
180	D₇₃	—	4	1	no	—
181	D₄×M₁₆	—	73	1	no	maximal, Sylow
182	C₇₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
183	C₂²×M₁₆	—	73	1	no	—
184	C₂²×M₁₆	—	73	1	no	—
185	C₂×C₄×D₄	—	73	1	no	—
186	#436	—	73	2	no	—
187	#435	—	73	2	no	—
188	#435	—	73	2	no	—
189	C₈×D₄	—	73	2	no	—
190	#423	—	73	1	no	—
191	#407	—	73	1	no	—
192	#407	D₇₃	1	1	yes	characteristic, complemented
193	C₄×M₁₆	—	73	1	no	—
194	C₂²×D₄	—	73	1	no	—
195	C₂³×C₄	—	73	1	no	—
196	C₂³×C₄	C₂×D₇₃	1	1	yes	characteristic
197	C₂×M₁₆	—	73	1	no	—
198	C₂×M₁₆	—	73	1	no	—
199	C₂×M₁₆	—	73	2	no	—
200	C₂×M₁₆	—	73	2	no	—
201	C₂×M₁₆	—	146	4	no	—
202	C₂×M₁₆	—	146	4	no	—
203	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
204	C₂×M₁₆	C₂×D₇₃	1	1	yes	characteristic
205	C₂²×C₈	—	73	2	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₄×D₄	—	73	1	no	—
208	C₄×D₄	—	73	1	no	—
209	C₄×D₄	—	73	1	no	—
210	C₄×D₄	—	73	1	no	—
211	C₄×D₄	—	73	1	no	—
212	C₄×D₄	—	73	1	no	—
213	C₄×D₄	—	73	1	no	—
214	C₄×D₄	—	73	1	no	—
215	C₂×C₄⋊C₄	—	73	1	no	—
216	C₂×C₂²⋊C₄	—	73	1	no	—
217	C₂×C₂²⋊C₄	—	73	1	no	—
218	C₂×C₄²	—	73	1	no	—
219	C₄⋊C₈	—	73	2	no	—
220	C₄⋊C₈	—	73	2	no	—
221	C₂²⋊C₈	—	73	2	no	—
222	C₂²⋊C₈	—	73	2	no	—
223	C₂²⋊C₈	C₂×D₇₃	1	2	yes	complemented
224	C₂²⋊C₈	C₂×D₇₃	1	2	yes	complemented
225	#97	—	73	2	no	—
226	C₄×C₈	—	73	2	no	—
227	C₂⁴	—	73	1	no	—
228	C₂⁴	C₄×D₇₃	1	1	yes	characteristic
229	C₂×D₄	—	73	1	no	—
230	C₂×D₄	—	73	1	no	—
231	C₂×D₄	—	73	1	no	—
232	C₂×D₄	—	73	1	no	—
233	C₂×D₄	—	146	4	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	73	1	no	—
237	C₂²×C₄	—	73	1	no	—
238	C₂²×C₄	—	73	1	no	—
239	C₂²×C₄	—	73	1	no	—
240	C₂²×C₄	—	73	1	no	—
241	C₂²×C₄	—	73	1	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	146	2	no	—
244	C₂²×C₄	—	146	2	no	—
245	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
246	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
247	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
248	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
249	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	73	4	no	—
254	M₁₆	—	146	4	no	—
255	M₁₆	—	146	4	no	—
256	M₁₆	—	146	2	no	—
257	C₂×C₈	—	73	2	no	—
258	C₂×C₈	—	73	2	no	—
259	C₂×C₈	—	146	4	no	—
260	C₂×C₈	—	146	4	no	—
261	C₂×C₈	C₂²×D₇₃	1	2	yes	—
262	C₂×C₈	C₂²×D₇₃	1	2	yes	—
263	C₄⋊C₄	—	73	1	no	—
264	C₄⋊C₄	—	73	1	no	—
265	C₄⋊C₄	—	73	1	no	—
266	C₄⋊C₄	—	73	1	no	—
267	C₂²⋊C₄	—	73	1	no	—
268	C₂²⋊C₄	—	73	1	no	—
269	C₂²⋊C₄	—	73	1	no	—
270	C₂²⋊C₄	—	73	1	no	—
271	C₂²⋊C₄	—	73	1	no	—
272	C₂²⋊C₄	—	73	1	no	—
273	C₂²⋊C₄	—	73	1	no	—
274	C₂²⋊C₄	—	73	1	no	—
275	C₄²	—	73	1	no	—
276	C₄²	—	73	1	no	—
277	C₄²	—	73	1	no	—
278	C₄²	—	73	1	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	292	1	no	—
283	C₂³	—	73	1	no	—
284	C₂³	—	73	1	no	—
285	C₂³	—	146	2	no	—
286	C₂³	—	146	2	no	—
287	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
288	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₇₃	1	1	yes	characteristic
290	D₄	—	73	4	no	—
291	D₄	—	146	2	no	—
292	D₄	—	146	2	no	—
293	D₄	—	146	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	292	1	no	—
302	C₂×C₄	—	73	1	no	—
303	C₂×C₄	—	73	1	no	—
304	C₂×C₄	—	73	1	no	—
305	C₂×C₄	—	73	1	no	—
306	C₂×C₄	—	73	1	no	—
307	C₂×C₄	—	73	1	no	—
308	C₂×C₄	—	146	2	no	—
309	C₂×C₄	—	146	2	no	—
310	C₂×C₄	—	146	2	no	—
311	C₂×C₄	—	146	2	no	—
312	C₂×C₄	—	146	1	no	—
313	C₂×C₄	—	146	1	no	—
314	C₂×C₄	—	146	1	no	—
315	C₂×C₄	—	146	1	no	—
316	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
319	C₂×C₄	D₄×D₇₃	1	2	yes	—
320	C₂×C₄	#11924027	1	2	yes	—
321	C₂×C₄	C₂³×D₇₃	1	1	yes	characteristic
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	73	4	no	—
325	C₈	—	146	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	292	4	no	—
335	C₂²	—	73	2	no	—
336	C₂²	—	146	1	no	—
337	C₂²	—	146	1	no	—
338	C₂²	—	146	1	no	—
339	C₂²	—	146	1	no	—
340	C₂²	—	146	1	no	—
341	C₂²	#14556551	1	2	yes	—
342	C₂²	M₁₆×D₇₃	1	2	yes	complemented
343	C₂²	C₂²×C₄×D₇₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	73	2	no	—
347	C₄	—	146	1	no	—
348	C₄	—	146	1	no	—
349	C₄	—	146	1	no	—
350	C₄	—	146	1	no	—
351	C₄	—	146	1	no	—
352	C₄	#14556551	1	2	yes	—
353	C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
354	C₄	C₂×#11924027	1	1	yes	characteristic
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	4	1	no	—
358	C₂	—	292	1	no	—
359	C₂	—	146	2	no	—
360	C₂	C₂×#14556551	1	1	yes	characteristic
361	C₂	C₂×M₁₆×D₇₃	1	1	yes	characteristic
362	C₂	#17989353	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	#407×C₇₃	C₂
hypercenter	#407	D₇₃
hypocenter	C₇₃	D₄×M₁₆
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
1. C₂²×#11924012	C₂	semidirect
2. C₂×M₁₆×D₇₃	C₂	semidirect
4. #17988662	C₂	semidirect
5. #17988661	C₂	semidirect
6. #17988655	C₂	semidirect
7. #17988650	C₂	semidirect
8. #17988610	C₂	semidirect
9. #17988550	C₂	semidirect
10. M₁₆×Dic₇₃	C₂	semidirect
11. #407×C₇₃	C₂	semidirect
17. C₂×#11924012	C₂²	semidirect
18. C₂×#11924011	C₂²	semidirect
20. C₂²×#10596831	C₂²	semidirect
21. C₂×C₈×D₇₃	C₂²	semidirect
34. #14556520	C₂²	semidirect
35. #14556520	C₂²	semidirect
36. #14556503	C₂²	semidirect
37. #14556503	C₂²	semidirect
38. #14556501	C₂²	semidirect
39. C₈×Dic₇₃	C₂²	semidirect
43. C₂²⋊C₈×C₇₃	C₂²	semidirect
44. C₂²⋊C₈×C₇₃	C₂²	semidirect
62. #11924012	D₄	semidirect
124. C₂²⋊D₇₃	M₁₆	semidirect
143. #10596831	C₂×D₄	semidirect
153. C₂²×C₇₃	C₂×M₁₆	semidirect
161. C₂×D₇₃	C₂×M₁₆	semidirect
172. Dic₇₃	C₂×M₁₆	semidirect
182. C₇₃	D₄×M₁₆	semidirect
192. #407	D₇₃	semidirect
223. C₂²⋊C₈	C₂×D₇₃	semidirect
224. C₂²⋊C₈	C₂×D₇₃	semidirect
342. C₂²	M₁₆×D₇₃	semidirect