Subgroups — #33000825

Order 18688.

614 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556530	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988541	C₂	1	1	yes	maximal, characteristic
4	#22608009	C₂	1	1	yes	maximal, characteristic, complemented
5	#22608008	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608008	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11924011	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₂×#14556548	C₄	1	1	yes	characteristic
11	C₂×#14556530	—	2	2	no	—
12	C₂×#14556530	—	2	2	no	—
13	C₂×#14556530	C₄	1	4	yes	complemented
14	C₂²×#11924003	C₂²	1	1	yes	characteristic
15	C₂²×#11924003	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	—	2	2	no	—
17	C₂×C₁₆×D₇₃	—	2	2	no	—
18	C₂×C₁₆×D₇₃	C₄	1	4	yes	complemented
19	#17988541	—	2	2	no	—
20	#17988541	—	2	2	no	—
21	C₂×#14556502	C₄	1	1	yes	characteristic
22	C₂×#14556501	C₄	1	1	yes	characteristic
23	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
24	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
27	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#11924011	—	2	2	no	—
29	C₂×#11924011	—	2	2	no	—
30	C₂×#11924011	C₈	1	4	yes	—
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
35	C₂×C₈×D₇₃	D₄	1	2	yes	—
36	C₂×C₈×D₇₃	D₄	1	2	yes	—
37	C₂×C₈×D₇₃	D₄	1	2	yes	—
38	C₂×C₈×D₇₃	Q₈	1	2	yes	—
39	#14556548	—	2	2	no	—
40	#14556548	—	2	2	no	—
41	C₂×#11924007	C₈	1	2	yes	—
42	C₂×#11924005	C₈	1	2	yes	—
43	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
44	#14556530	—	2	8	no	—
45	C₂×#11924003	—	2	1	no	—
46	C₂×#11924003	—	2	1	no	—
47	C₂×#11924003	—	2	1	no	—
48	C₂×#11924003	—	2	1	no	—
49	C₂×#11924003	C₂×C₄	1	2	yes	—
50	C₂×#11924003	C₂×C₄	1	2	yes	—
51	C₁₆×D₇₃	—	2	8	no	—
52	#14556502	—	2	2	no	—
53	#14556501	—	2	2	no	—
54	C₈×Dic₇₃	—	2	1	no	—
55	C₈×Dic₇₃	—	2	1	no	—
56	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
59	C₂×C₁₆×C₇₃	—	2	1	no	—
60	C₂×C₁₆×C₇₃	—	2	1	no	—
61	C₂×C₁₆×C₇₃	—	2	1	no	—
62	C₂×C₁₆×C₇₃	—	2	1	no	—
63	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
64	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	—
65	C₄×C₈×C₇₃	—	2	1	no	—
66	C₄×C₈×C₇₃	—	2	1	no	—
67	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
68	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
69	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₇₃	C₄²	1	2	yes	—
71	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
72	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
74	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
75	C₂×C₄×D₇₃	M₁₆	1	2	yes	—
76	#11924011	—	2	8	no	—
77	C₂×#10596831	—	2	1	no	—
78	C₂×#10596831	—	2	1	no	—
79	C₂×#10596831	C₄²	1	2	yes	—
80	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
81	C₂×#10596831	C₄⋊C₄	1	2	yes	—
82	C₂×#10596831	C₂×C₈	1	2	yes	—
83	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
84	C₈×D₇₃	C₂²⋊C₄	1	4	yes	—
85	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
86	C₈×D₇₃	C₄⋊C₄	1	4	yes	—
87	#11924007	—	2	4	no	—
88	#11924005	—	2	4	no	—
89	C₄×Dic₇₃	—	2	1	no	—
90	C₄×Dic₇₃	—	2	1	no	—
91	#11924003	—	2	2	no	—
92	#11924003	—	2	2	no	—
93	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
94	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₇₃	—	2	1	no	—
96	C₂×C₈×C₇₃	—	2	1	no	—
97	C₂×C₈×C₇₃	C₂×C₈	1	2	yes	—
98	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₇₃	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₇₃	—	2	1	no	—
105	C₄²×C₇₃	—	2	1	no	—
106	C₁₆×C₇₃	—	2	2	no	—
107	C₁₆×C₇₃	—	2	2	no	—
108	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
109	C₂²×D₇₃	—	2	2	no	—
110	C₂²×D₇₃	—	2	2	no	—
111	C₂²×D₇₃	C₄×C₈	1	2	yes	—
112	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
113	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
114	C₂×C₄×C₇₃	—	2	2	no	—
115	C₂×C₄×C₇₃	—	2	2	no	—
116	C₂×C₄×C₇₃	—	2	2	no	—
117	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
123	C₂×Dic₇₃	—	2	2	no	—
124	C₂×Dic₇₃	—	2	2	no	—
125	C₂×Dic₇₃	—	2	2	no	—
126	C₂×Dic₇₃	C₄×C₈	1	2	yes	—
127	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
128	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
129	C₄×D₇₃	#95	1	4	yes	—
130	C₄×D₇₃	#97	1	4	yes	—
131	C₄×D₇₃	C₂²⋊C₈	1	4	yes	—
132	C₄×D₇₃	C₄⋊C₈	1	4	yes	—
133	C₈×C₇₃	—	2	2	no	—
134	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
136	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
138	#10596831	—	2	2	no	—
139	#10596831	#95	1	4	yes	—
140	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
141	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
142	C₂²×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
143	C₂²×C₇₃	#504	1	2	yes	—
144	C₂²×C₇₃	#504	1	2	yes	—
145	C₂×D₇₃	—	2	2	no	—
146	C₂×D₇₃	—	2	2	no	—
147	C₂×D₇₃	—	2	4	no	—
148	C₂×D₇₃	—	2	4	no	—
149	C₂×D₇₃	#336	1	4	yes	—
150	C₄×C₇₃	—	4	2	no	—
151	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#97	1	1	yes	characteristic
153	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
154	C₄×C₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
155	Dic₇₃	—	4	2	no	—
156	Dic₇₃	#336	1	4	yes	—
157	#12545	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
159	C₂×C₇₃	#2108	1	1	yes	characteristic
160	C₂×C₇₃	#2108	1	1	yes	characteristic
161	C₂×C₇₃	#2113	1	2	yes	—
162	C₂×C₇₃	#2151	1	2	yes	—
163	D₇₃	—	2	4	no	—
164	C₂²×M₃₂	—	73	1	no	—
165	C₂³×C₁₆	—	73	1	no	—
166	C₂×#405	—	73	1	no	—
167	#1374	—	73	1	no	—
168	#1374	—	73	1	no	—
169	#1374	—	73	1	no	—
170	#1374	D₇₃	1	1	yes	characteristic, complemented
171	C₇₃	#12545	1	1	yes	characteristic, complemented, Sylow
172	C₂²×M₁₆	—	73	1	no	—
173	C₂³×C₈	—	73	1	no	—
174	C₂×#117	—	73	1	no	—
175	C₂×M₃₂	—	73	4	no	—
176	C₂×M₃₂	—	146	2	no	—
177	C₂×M₃₂	—	146	2	no	—
178	C₂²×C₁₆	—	73	1	no	—
179	C₂²×C₁₆	—	73	1	no	—
180	C₂²×C₁₆	—	73	4	no	—
181	C₂²×C₁₆	—	146	2	no	—
182	C₂²×C₁₆	—	146	2	no	—
183	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
184	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
185	#405	—	146	2	no	—
186	#405	—	146	2	no	—
187	C₂×#97	—	73	1	no	—
188	C₂×#97	—	73	1	no	—
189	C₂×C₄×C₈	—	73	1	no	—
190	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
191	C₂³×C₄	—	73	1	no	—
192	C₂×M₁₆	—	73	4	no	—
193	C₂×M₁₆	—	146	2	no	—
194	C₂×M₁₆	—	146	2	no	—
195	C₂²×C₈	—	73	1	no	—
196	C₂²×C₈	—	73	2	no	—
197	C₂²×C₈	—	73	2	no	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	—	73	2	no	—
200	C₂²×C₈	—	73	2	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	—	73	1	no	—
203	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
205	#117	—	146	2	no	—
206	#117	—	146	2	no	—
207	C₂×C₄⋊C₄	—	73	2	no	—
208	C₂×C₂²⋊C₄	—	73	2	no	—
209	C₂×C₄²	—	73	1	no	—
210	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
211	M₃₂	—	146	8	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	2	1	no	—
214	C₂×C₁₆	—	2	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	73	2	no	—
217	C₂×C₁₆	—	73	2	no	—
218	C₂×C₁₆	—	146	8	no	—
219	C₂×C₁₆	—	146	1	no	—
220	C₂×C₁₆	—	146	1	no	—
221	C₂×C₁₆	—	146	1	no	—
222	C₂×C₁₆	—	146	1	no	—
223	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
224	C₂×C₁₆	C₄×D₇₃	1	2	yes	—
225	#97	—	146	2	no	—
226	#97	—	146	2	no	—
227	C₄×C₈	—	2	1	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	146	1	no	—
230	C₄×C₈	—	146	1	no	—
231	C₂⁴	—	73	1	no	—
232	C₂²×C₄	—	73	1	no	—
233	C₂²×C₄	—	73	2	no	—
234	C₂²×C₄	—	73	2	no	—
235	C₂²×C₄	—	73	2	no	—
236	C₂²×C₄	—	73	2	no	—
237	C₂²×C₄	—	73	2	no	—
238	C₂²×C₄	—	73	2	no	—
239	C₂²×C₄	—	73	2	no	—
240	C₂²×C₄	C₈×D₇₃	1	2	yes	—
241	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
242	M₁₆	—	146	8	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	73	2	no	—
246	C₂×C₈	—	73	2	no	—
247	C₂×C₈	—	73	2	no	—
248	C₂×C₈	—	73	4	no	—
249	C₂×C₈	—	73	4	no	—
250	C₂×C₈	—	73	4	no	—
251	C₂×C₈	—	73	4	no	—
252	C₂×C₈	—	73	2	no	—
253	C₂×C₈	—	146	1	no	—
254	C₂×C₈	—	146	1	no	—
255	C₂×C₈	C₈×D₇₃	1	2	yes	—
256	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
257	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
259	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
260	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
261	C₂×C₈	Q₈×D₇₃	1	1	yes	characteristic
262	C₄⋊C₄	—	146	4	no	—
263	C₂²⋊C₄	—	146	4	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	146	1	no	—
267	C₄²	—	146	1	no	—
268	C₁₆	—	2	2	no	—
269	C₁₆	—	2	2	no	—
270	C₁₆	—	146	2	no	—
271	C₁₆	—	146	2	no	—
272	C₂³	—	73	2	no	—
273	C₂³	—	73	2	no	—
274	C₂³	—	73	2	no	—
275	C₂³	—	146	2	no	—
276	C₂³	—	146	2	no	—
277	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	73	2	no	—
282	C₂×C₄	—	73	2	no	—
283	C₂×C₄	—	73	2	no	—
284	C₂×C₄	—	73	4	no	—
285	C₂×C₄	—	73	4	no	—
286	C₂×C₄	—	73	4	no	—
287	C₂×C₄	—	73	4	no	—
288	C₂×C₄	—	146	2	no	—
289	C₂×C₄	—	146	2	no	—
290	C₂×C₄	—	146	2	no	—
291	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
292	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
293	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
294	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
295	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
296	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	73	4	no	—
299	C₈	—	146	2	no	—
300	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
301	C₈	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
302	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
303	C₈	C₄⋊C₄×D₇₃	1	1	yes	characteristic
304	C₂²	—	73	4	no	—
305	C₂²	—	146	2	no	—
306	C₂²	—	146	2	no	—
307	C₂²	—	146	4	no	—
308	C₂²	—	146	4	no	—
309	C₂²	C₄×C₈×D₇₃	1	1	yes	characteristic
310	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
311	C₂²	C₄⋊C₈×D₇₃	1	1	yes	characteristic
312	C₂²	#17989003	1	2	yes	—
313	C₂²	#17989005	1	2	yes	—
314	C₄	—	4	2	no	—
315	C₄	—	292	2	no	—
316	C₄	—	73	4	no	—
317	C₄	#95×D₇₃	1	1	yes	characteristic
318	C₄	#97×D₇₃	1	1	yes	characteristic
319	C₄	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
320	C₄	C₄⋊C₈×D₇₃	1	1	yes	characteristic
321	C₂	—	146	4	no	—
322	C₂	#336×D₇₃	1	1	yes	characteristic
323	C₂	#22611526	1	1	yes	characteristic
324	C₂	#22611529	1	1	yes	characteristic
325	C₂	#22611543	1	2	yes	—
326	C₂	#22611688	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂²×C₇₃	C₂×C₄×C₈
Fitting	C₇₃×#1374	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	#1374	D₇₃
hypocenter	C₇₃	#12545
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
4. #22608009	C₂	semidirect
5. #22608008	C₂	semidirect
6. #22608008	C₂	semidirect
7. C₇₃×#1374	C₂	semidirect
13. C₂×#14556530	C₄	semidirect
18. C₂×C₁₆×D₇₃	C₄	semidirect
170. #1374	D₇₃	semidirect
171. C₇₃	#12545	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556530	2
C₂²×C₁₆×D₇₃	2
C₂×#17988541	2
#22608009	2
#22608008	2
C₇₃×#1374	2
#12545	73