Subgroups — #33003718

Order 18688.

382 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988405	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988366	C₂	1	1	yes	maximal, characteristic
4	C₈×#11924005	C₂	1	1	yes	maximal, characteristic
5	#22608009	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608008	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#346×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
9	C₂²×#11924003	C₂²	1	1	yes	characteristic
10	C₂²×#11924003	C₂²	1	1	yes	characteristic
11	C₂×#14556503	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₄×#11924005	C₄	1	1	yes	characteristic
14	#17988405	—	2	2	no	—
15	#17988366	—	2	2	no	—
16	#17988366	—	2	2	no	—
17	#17988342	C₄	1	1	yes	characteristic
18	#17988341	C₄	1	1	yes	characteristic
19	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
22	#346×C₇₃	—	2	2	no	—
23	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
26	C₂×C₈×D₇₃	—	2	1	no	—
27	C₂×C₈×D₇₃	—	2	1	no	—
28	C₂×C₈×D₇₃	—	2	1	no	—
29	C₂×C₈×D₇₃	—	2	1	no	—
30	C₂×C₈×D₇₃	—	2	1	no	—
31	C₂×C₈×D₇₃	—	2	1	no	—
32	C₂×#11924005	C₈	1	1	yes	characteristic
33	C₂×#11924005	C₈	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#11924003	—	2	1	no	—
36	C₂×#11924003	—	2	1	no	—
37	C₂×#11924003	—	2	1	no	—
38	C₂×#11924003	—	2	1	no	—
39	C₂×#11924003	—	2	1	no	—
40	C₂×#11924003	—	2	1	no	—
41	#14556503	—	2	2	no	—
42	C₈×Dic₇₃	—	2	1	no	—
43	C₈×Dic₇₃	—	2	1	no	—
44	#14556498	C₈	1	1	yes	characteristic
45	#14556498	C₈	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₁₆×C₇₃	—	2	1	no	—
50	C₂×C₁₆×C₇₃	—	2	1	no	—
51	C₂×C₁₆×C₇₃	—	2	1	no	—
52	C₂×C₁₆×C₇₃	—	2	1	no	—
53	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
54	C₂×C₁₆×C₇₃	D₄	1	2	yes	—
55	C₄×C₈×C₇₃	—	2	1	no	—
56	C₄×C₈×C₇₃	—	2	1	no	—
57	C₂³×D₇₃	C₂×C₈	1	1	yes	characteristic
58	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×Dic₇₃	C₂×C₈	1	1	yes	characteristic
61	C₂×C₄×D₇₃	—	2	1	no	—
62	C₂×C₄×D₇₃	—	2	1	no	—
63	C₂×C₄×D₇₃	—	2	1	no	—
64	C₂×C₄×D₇₃	—	2	1	no	—
65	C₂×C₄×D₇₃	—	2	1	no	—
66	C₂×C₄×D₇₃	—	2	1	no	—
67	C₂×#10596831	—	2	1	no	—
68	C₂×#10596831	—	2	1	no	—
69	C₂×#10596831	—	2	1	no	—
70	C₂×#10596831	—	2	1	no	—
71	C₂×#10596831	—	2	1	no	—
72	C₂×#10596831	—	2	1	no	—
73	C₈×D₇₃	—	4	1	no	—
74	C₈×D₇₃	—	4	1	no	—
75	C₈×D₇₃	—	4	1	no	—
76	C₈×D₇₃	—	4	1	no	—
77	#11924005	—	2	2	no	—
78	#11924005	—	2	2	no	—
79	C₄×Dic₇₃	—	2	1	no	—
80	C₄×Dic₇₃	—	2	1	no	—
81	#11924003	—	4	1	no	—
82	#11924003	—	4	1	no	—
83	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₇₃	—	2	1	no	—
87	C₂×C₈×C₇₃	—	2	1	no	—
88	C₂×C₈×C₇₃	M₁₆	1	2	yes	—
89	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
93	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
95	C₄²×C₇₃	—	2	1	no	—
96	C₄²×C₇₃	—	2	1	no	—
97	C₁₆×C₇₃	—	2	2	no	—
98	C₁₆×C₇₃	—	2	2	no	—
99	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₇₃	—	2	1	no	—
101	C₂²×D₇₃	—	2	1	no	—
102	C₂²×D₇₃	—	2	1	no	—
103	C₂²×D₇₃	—	2	1	no	—
104	C₂²×D₇₃	—	2	1	no	—
105	C₂²×D₇₃	—	2	1	no	—
106	C₂²×D₇₃	M₃₂	1	2	yes	—
107	C₂×C₄×C₇₃	—	2	1	no	—
108	C₂×C₄×C₇₃	—	2	1	no	—
109	C₂×C₄×C₇₃	—	2	1	no	—
110	C₂×C₄×C₇₃	—	2	1	no	—
111	C₂×C₄×C₇₃	—	2	1	no	—
112	C₂×C₄×C₇₃	—	2	1	no	—
113	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
118	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
119	C₂×Dic₇₃	—	2	1	no	—
120	C₂×Dic₇₃	—	2	1	no	—
121	C₂×Dic₇₃	—	2	1	no	—
122	C₂×Dic₇₃	—	2	1	no	—
123	C₂×Dic₇₃	—	2	1	no	—
124	C₂×Dic₇₃	—	2	1	no	—
125	C₂×Dic₇₃	—	2	1	no	—
126	C₂×Dic₇₃	—	2	1	no	—
127	C₂×Dic₇₃	M₃₂	1	2	yes	—
128	C₄×D₇₃	—	4	1	no	—
129	C₄×D₇₃	—	4	1	no	—
130	C₄×D₇₃	—	4	1	no	—
131	C₄×D₇₃	—	4	1	no	—
132	C₈×C₇₃	—	2	2	no	—
133	C₈×C₇₃	#121	1	1	yes	characteristic
134	C₈×C₇₃	#123	1	1	yes	characteristic
135	C₈×C₇₃	#124	1	1	yes	characteristic
136	C₈×C₇₃	#126	1	1	yes	characteristic
137	#10596831	—	4	1	no	—
138	#10596831	—	4	1	no	—
139	C₂²×C₇₃	#431	1	1	yes	characteristic
140	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
141	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
142	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
143	C₂²×C₇₃	C₂×M₃₂	1	1	yes	characteristic
144	C₂²×C₇₃	#504	1	1	yes	characteristic
145	C₂²×C₇₃	#504	1	1	yes	characteristic
146	C₂×D₇₃	—	2	2	no	—
147	C₂×D₇₃	—	2	2	no	—
148	C₂×D₇₃	—	2	2	no	—
149	C₂×D₇₃	—	4	1	no	—
150	C₂×D₇₃	—	4	1	no	—
151	C₂×D₇₃	—	4	1	no	—
152	C₂×D₇₃	—	4	1	no	—
153	C₄×C₇₃	—	4	1	no	—
154	C₄×C₇₃	—	4	1	no	—
155	C₄×C₇₃	#388	1	1	yes	characteristic
156	C₄×C₇₃	#433	1	1	yes	characteristic
157	C₄×C₇₃	#435	1	1	yes	characteristic
158	C₄×C₇₃	#436	1	1	yes	characteristic
159	Dic₇₃	—	2	2	no	—
160	Dic₇₃	—	4	1	no	—
161	Dic₇₃	—	4	1	no	—
162	#13121	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	#1923	1	1	yes	characteristic
164	C₂×C₇₃	#2163	1	1	yes	characteristic
165	C₂×C₇₃	#2164	1	1	yes	characteristic
166	C₂×C₇₃	#2168	1	1	yes	characteristic
167	C₂×C₇₃	#2168	1	1	yes	characteristic
168	C₂×C₇₃	#2168	1	1	yes	characteristic
169	C₂×C₇₃	#2169	1	1	yes	characteristic
170	D₇₃	—	4	2	no	—
171	C₂×C₄⋊C₁₆	—	73	1	no	—
172	C₂×C₂²⋊C₁₆	—	73	1	no	—
173	C₂×C₂²⋊C₁₆	—	73	1	no	—
174	C₂×#346	D₇₃	1	1	yes	characteristic, complemented
175	C₈×C₂²⋊C₄	—	73	1	no	—
176	#1374	—	73	1	no	—
177	#1374	—	73	1	no	—
178	C₇₃	#13121	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₈	—	73	1	no	—
180	C₂²×C₁₆	—	73	1	no	—
181	C₂²×C₁₆	—	73	1	no	—
182	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
183	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
184	C₂×C₂²⋊C₈	—	73	1	no	—
185	C₂×C₄×C₈	—	73	1	no	—
186	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
187	C₄×C₂²⋊C₄	—	73	1	no	—
188	C₄⋊C₁₆	—	146	2	no	—
189	C₂²⋊C₁₆	—	146	2	no	—
190	C₂²⋊C₁₆	—	146	2	no	—
191	#346	—	2	2	no	—
192	#336	—	73	1	no	—
193	#336	—	73	1	no	—
194	C₂³×C₄	—	73	1	no	—
195	C₂²×C₈	—	73	1	no	—
196	C₂²×C₈	—	73	1	no	—
197	C₂²×C₈	—	146	1	no	—
198	C₂²×C₈	—	146	1	no	—
199	C₂²×C₈	—	146	1	no	—
200	C₂²×C₈	—	146	1	no	—
201	C₂²×C₈	—	146	1	no	—
202	C₂²×C₈	—	146	1	no	—
203	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
205	C₂×C₂²⋊C₄	—	73	1	no	—
206	C₂×C₂²⋊C₄	—	73	1	no	—
207	C₂×C₄²	—	73	1	no	—
208	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
209	C₂×C₁₆	—	2	1	no	—
210	C₂×C₁₆	—	2	1	no	—
211	C₂×C₁₆	—	2	1	no	—
212	C₂×C₁₆	—	2	1	no	—
213	C₂×C₁₆	—	146	1	no	—
214	C₂×C₁₆	—	146	1	no	—
215	C₂×C₁₆	—	146	1	no	—
216	C₂×C₁₆	—	146	1	no	—
217	C₂×C₁₆	—	146	1	no	—
218	C₂×C₁₆	—	146	1	no	—
219	C₂×C₁₆	D₂₉₂	1	2	yes	—
220	C₂×C₁₆	C₂²⋊D₇₃	1	2	yes	—
221	C₂²⋊C₈	—	146	2	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	—	2	1	no	—
224	C₄×C₈	—	146	1	no	—
225	C₄×C₈	—	146	1	no	—
226	#95	—	73	1	no	—
227	#95	—	73	1	no	—
228	C₂⁴	—	73	1	no	—
229	C₂²×C₄	—	73	1	no	—
230	C₂²×C₄	—	73	1	no	—
231	C₂²×C₄	—	73	1	no	—
232	C₂²×C₄	—	146	1	no	—
233	C₂²×C₄	—	146	1	no	—
234	C₂²×C₄	—	146	1	no	—
235	C₂²×C₄	—	146	1	no	—
236	C₂²×C₄	—	146	1	no	—
237	C₂²×C₄	—	146	1	no	—
238	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
239	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	292	1	no	—
244	C₂×C₈	—	292	1	no	—
245	C₂×C₈	—	292	1	no	—
246	C₂×C₈	—	292	1	no	—
247	C₂×C₈	—	146	1	no	—
248	C₂×C₈	—	146	1	no	—
249	C₂×C₈	—	146	1	no	—
250	C₂×C₈	—	146	1	no	—
251	C₂×C₈	—	146	1	no	—
252	C₂×C₈	—	146	1	no	—
253	C₂×C₈	#11924011	1	2	yes	—
254	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
255	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
256	C₂×C₈	#11924028	1	1	yes	characteristic
257	C₂×C₈	#11924028	1	1	yes	characteristic
258	C₂×C₈	#11924028	1	1	yes	characteristic
259	C₂×C₈	#11924028	1	1	yes	characteristic
260	C₂²⋊C₄	—	146	2	no	—
261	C₂²⋊C₄	—	146	2	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	146	1	no	—
265	C₄²	—	146	1	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	292	1	no	—
269	C₁₆	—	292	1	no	—
270	C₂³	—	73	2	no	—
271	C₂³	—	146	1	no	—
272	C₂³	—	146	1	no	—
273	C₂³	—	146	1	no	—
274	C₂³	—	146	1	no	—
275	C₂³	—	146	1	no	—
276	C₂³	—	146	1	no	—
277	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	292	1	no	—
285	C₂×C₄	—	292	1	no	—
286	C₂×C₄	—	292	1	no	—
287	C₂×C₄	—	292	1	no	—
288	C₂×C₄	—	73	2	no	—
289	C₂×C₄	—	146	1	no	—
290	C₂×C₄	—	146	1	no	—
291	C₂×C₄	—	146	1	no	—
292	C₂×C₄	—	146	1	no	—
293	C₂×C₄	—	146	1	no	—
294	C₂×C₄	—	146	1	no	—
295	C₂×C₄	—	146	1	no	—
296	C₂×C₄	—	146	1	no	—
297	C₂×C₄	#14556548	1	1	yes	characteristic
298	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
299	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
300	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
301	C₂×C₄	#14556597	1	1	yes	characteristic
302	C₂×C₄	#14556597	1	1	yes	characteristic
303	C₈	—	2	2	no	—
304	C₈	—	292	1	no	—
305	C₈	—	292	1	no	—
306	C₈	#14556565	1	1	yes	characteristic
307	C₈	#14556574	1	1	yes	characteristic
308	C₈	#14556579	1	1	yes	characteristic
309	C₈	#14556585	1	1	yes	characteristic
310	C₂²	—	292	1	no	—
311	C₂²	—	292	1	no	—
312	C₂²	—	292	1	no	—
313	C₂²	—	292	1	no	—
314	C₂²	—	146	2	no	—
315	C₂²	—	146	2	no	—
316	C₂²	—	146	2	no	—
317	C₂²	#17988638	1	1	yes	characteristic
318	C₂²	C₈×C₂²⋊D₇₃	1	1	yes	characteristic
319	C₂²	C₈×D₂₉₂	1	1	yes	characteristic
320	C₂²	M₃₂×D₇₃	1	1	yes	characteristic
321	C₂²	M₃₂×D₇₃	1	1	yes	characteristic
322	C₂²	#17989005	1	1	yes	characteristic
323	C₂²	#17989005	1	1	yes	characteristic
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	292	1	no	—
327	C₄	—	292	1	no	—
328	C₄	—	146	2	no	—
329	C₄	#17988478	1	1	yes	characteristic
330	C₄	#17988646	1	1	yes	characteristic
331	C₄	#17988658	1	1	yes	characteristic
332	C₄	#17988666	1	1	yes	characteristic
333	C₂	—	292	2	no	—
334	C₂	#22610537	1	1	yes	characteristic
335	C₂	#22611732	1	1	yes	characteristic
336	C₂	#22611739	1	1	yes	characteristic
337	C₂	#22611754	1	1	yes	characteristic
338	C₂	#22611758	1	1	yes	characteristic
339	C₂	#22611758	1	1	yes	characteristic
340	C₂	#22611761	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator, socle	C₂³×C₇₃	C₂²×C₈
Fitting	C₂×#346×C₇₃	C₂
prime radical	C₂³×D₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×#346	D₇₃
hypocenter	C₇₃	#13121
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#17988405	C₂	semidirect
5. #22608009	C₂	semidirect
6. #22608008	C₂	semidirect
7. C₂×#346×C₇₃	C₂	semidirect
174. C₂×#346	D₇₃	semidirect
178. C₇₃	#13121	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988405	2
C₂×#17988366	2
C₈×#11924005	2
#22608009	2
#22608008	2
C₂×#346×C₇₃	2
#13121	73