Subgroups — #33055285

Order 18688.

616 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619052	C₂	1	2	yes	maximal
2	C₈×#11924028	C₂	1	1	yes	maximal, characteristic
3	#22608719	C₂	1	2	yes	maximal, complemented
4	#22608713	C₂	1	4	yes	maximal, complemented
5	#22608693	C₂	1	2	yes	maximal, complemented
6	#22608690	C₂	1	1	yes	maximal, characteristic, complemented
7	#335×D₇₃	C₂	1	2	yes	maximal, complemented
8	C₇₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11924028	C₂²	1	1	yes	characteristic
10	#17988717	—	2	2	no	—
11	C₈×Dic₁₄₆	C₄	1	1	yes	characteristic
12	C₈×Dic₁₄₆	C₄	1	1	yes	characteristic
13	#17988661	—	2	4	no	—
14	C₈×D₂₉₂	—	2	2	no	—
15	C₈×D₂₉₂	C₄	1	1	yes	characteristic
16	C₈×D₂₉₂	C₄	1	1	yes	characteristic
17	C₈×C₂²⋊D₇₃	C₄	1	2	yes	—
18	C₈×C₂²⋊D₇₃	C₄	1	2	yes	—
19	#17988643	C₂²	1	4	yes	—
20	#17988640	C₂²	1	1	yes	characteristic
21	#17988638	C₂²	1	2	yes	—
22	#17988611	C₂²	1	2	yes	—
23	C₄⋊C₈×D₇₃	C₂²	1	4	yes	—
24	C₄×C₈×D₇₃	C₂²	1	2	yes	—
25	#17988340	C₂²	1	2	yes	—
26	#17988340	C₂²	1	2	yes	—
27	#17988339	C₂²	1	4	yes	—
28	#17988337	C₂²	1	4	yes	—
29	#431×C₇₃	C₂²	1	2	yes	—
30	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
31	#335×C₇₃	C₂²	1	1	yes	characteristic
32	#335×C₇₃	C₂²	1	1	yes	characteristic
33	#334×C₇₃	C₂²	1	2	yes	—
34	C₂×#11924028	Q₈	1	1	yes	characteristic
35	C₂×#11924012	—	2	2	no	—
36	C₂×C₈×D₇₃	—	2	4	no	—
37	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
39	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
40	C₄×Dic₁₄₆	D₄	1	1	yes	characteristic
41	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
42	C₄×D₂₉₂	D₄	1	1	yes	characteristic
43	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
44	C₄×C₂²⋊D₇₃	D₄	1	2	yes	—
45	#14556550	Q₈	1	1	yes	characteristic
46	#14556548	Q₈	1	2	yes	—
47	C₄²×D₇₃	C₂³	1	2	yes	—
48	C₄⋊#10596831	—	2	2	no	—
49	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
51	C₄⋊#10596831	C₂³	1	4	yes	—
52	#14556520	—	2	4	no	—
53	#14556520	C₂×C₄	1	2	yes	—
54	#14556520	C₂×C₄	1	2	yes	—
55	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
56	#14556503	—	2	4	no	—
57	#14556503	C₂×C₄	1	2	yes	—
58	#14556503	C₂×C₄	1	2	yes	—
59	#14556501	—	2	4	no	—
60	C₄×#10596831	C₂³	1	2	yes	—
61	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
62	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
67	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
68	C₂²⋊C₈×C₇₃	—	2	2	no	—
69	C₄×C₈×C₇₃	—	2	2	no	—
70	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	#11924028	—	2	4	no	—
75	C₂×Dic₁₄₆	C₄⋊C₄	1	1	yes	characteristic
76	C₂×D₂₉₂	C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₇₃	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
79	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
81	#11924012	—	8	2	no	—
82	C₂×#10596831	—	2	4	no	—
83	C₂×#10596831	C₂²×C₄	1	2	yes	—
84	C₂×#10596831	C₂²×C₄	1	2	yes	—
85	C₈×D₇₃	—	2	4	no	—
86	C₈×D₇₃	—	4	2	no	—
87	C₈×D₇₃	—	8	4	no	—
88	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
89	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
90	#11924007	C₄⋊C₄	1	2	yes	—
91	#11924007	C₄⋊C₄	1	2	yes	—
92	C₂²⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
93	#11924005	C₄⋊C₄	1	2	yes	—
94	#11924005	C₄⋊C₄	1	2	yes	—
95	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
96	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
97	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
98	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₇₃	—	2	2	no	—
102	C₂×C₈×C₇₃	—	2	2	no	—
103	C₂×C₈×C₇₃	—	2	2	no	—
104	C₂×C₈×C₇₃	—	4	1	no	—
105	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
113	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	2	no	—
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂²⋊D₇₃	—	2	4	no	—
127	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
130	D₂₉₂	#108	1	4	yes	—
131	C₄×D₇₃	—	2	4	no	—
132	C₄×D₇₃	—	2	2	no	—
133	C₄×D₇₃	—	2	2	no	—
134	C₄×D₇₃	—	2	2	no	—
135	C₄×D₇₃	—	2	2	no	—
136	Dic₁₄₆	—	2	2	no	—
137	C₈×C₇₃	—	2	1	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	—	4	2	no	—
141	C₈×C₇₃	—	4	2	no	—
142	C₈×C₇₃	#131	1	2	yes	—
143	#10596831	—	2	2	no	—
144	#10596831	—	2	2	no	—
145	#10596831	—	4	4	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₂×D₇₃	—	2	2	no	—
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	C₂×#108	1	4	yes	—
153	C₄×C₇₃	—	2	1	no	—
154	C₄×C₇₃	—	2	1	no	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	C₂×#108	1	2	yes	—
157	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
158	C₄×C₇₃	#568	1	1	yes	characteristic
159	C₄×C₇₃	#576	1	1	yes	characteristic
160	C₄×C₇₃	#578	1	1	yes	characteristic
161	Dic₇₃	—	2	2	no	—
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	—	2	2	no	—
164	#17836	—	73	1	no	maximal, Sylow
165	C₂×C₇₃	—	4	1	no	—
166	C₂×C₇₃	#2906	1	1	yes	characteristic
167	C₂×C₇₃	#2913	1	1	yes	characteristic
168	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
169	D₇₃	—	2	4	no	—
170	#2990	—	73	2	no	—
171	C₈×#41	—	73	1	no	—
172	#1574	—	73	2	no	—
173	#1573	—	73	4	no	—
174	#1568	—	73	2	no	—
175	#1567	—	73	1	no	—
176	#1564	D₇₃	1	1	yes	characteristic, complemented
177	C₂×#335	—	73	2	no	—
178	C₇₃	#17836	1	1	yes	characteristic, complemented, Sylow
179	C₄×#41	—	73	1	no	—
180	#446	—	146	2	no	—
181	C₈×Q₈	—	73	1	no	—
182	C₈×Q₈	—	73	1	no	—
183	#435	—	146	4	no	—
184	C₈×D₄	—	73	2	no	—
185	C₈×D₄	—	73	2	no	—
186	C₈×D₄	—	73	1	no	—
187	C₈×D₄	—	73	1	no	—
188	C₈×D₄	—	146	2	no	—
189	#432	—	73	4	no	—
190	#431	—	73	1	no	—
191	#431	—	73	2	no	—
192	#431	C₂×D₇₃	1	2	yes	—
193	#423	—	73	2	no	—
194	C₂×C₄⋊C₈	—	73	4	no	—
195	C₂×C₄×C₈	—	73	2	no	—
196	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
197	#335	—	73	4	no	—
198	#335	—	73	2	no	—
199	#335	—	73	2	no	—
200	#335	C₂×D₇₃	1	1	yes	characteristic
201	#335	C₂×D₇₃	1	1	yes	characteristic
202	#334	—	73	4	no	—
203	#334	C₂×D₇₃	1	2	yes	—
204	C₂×#41	—	73	1	no	—
205	C₂×M₁₆	—	146	2	no	—
206	C₂²×C₈	—	73	2	no	—
207	C₂²×C₈	—	73	2	no	—
208	C₂²×C₈	—	146	4	no	—
209	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
210	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
211	C₄×Q₈	—	73	1	no	—
212	C₄×Q₈	—	73	1	no	—
213	C₄×D₄	—	73	2	no	—
214	C₄×D₄	—	73	2	no	—
215	C₄×D₄	—	73	1	no	—
216	C₄×D₄	—	73	1	no	—
217	#117	—	73	1	no	—
218	#117	—	73	2	no	—
219	C₂×C₄²	—	73	2	no	—
220	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
221	C₄⋊C₈	—	73	2	no	—
222	C₄⋊C₈	—	73	2	no	—
223	C₄⋊C₈	—	73	1	no	—
224	C₄⋊C₈	—	73	1	no	—
225	C₄⋊C₈	—	73	4	no	—
226	C₄⋊C₈	—	146	4	no	—
227	C₄⋊C₈	—	146	2	no	—
228	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
229	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
230	C₂²⋊C₈	—	2	2	no	—
231	C₂²⋊C₈	—	73	2	no	—
232	C₂²⋊C₈	—	73	2	no	—
233	C₂²⋊C₈	—	73	2	no	—
234	C₂²⋊C₈	—	146	4	no	—
235	#97	—	146	4	no	—
236	C₄×C₈	—	2	2	no	—
237	C₄×C₈	—	73	2	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	—	73	2	no	—
240	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
241	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
242	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
244	#41	—	146	4	no	—
245	C₂×Q₈	—	73	1	no	—
246	C₂×D₄	—	73	2	no	—
247	C₂×D₄	—	73	1	no	—
248	C₂²×C₄	—	73	2	no	—
249	C₂²×C₄	—	73	2	no	—
250	C₂²×C₄	—	73	2	no	—
251	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
253	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
254	M₁₆	—	584	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	4	1	no	—
259	C₂×C₈	—	292	2	no	—
260	C₂×C₈	—	584	4	no	—
261	C₂×C₈	—	73	2	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	146	4	no	—
264	C₂×C₈	—	146	4	no	—
265	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
266	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
267	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
269	C₄⋊C₄	—	73	2	no	—
270	C₄⋊C₄	—	73	2	no	—
271	C₄⋊C₄	—	73	1	no	—
272	C₄⋊C₄	—	73	1	no	—
273	C₂²⋊C₄	—	73	2	no	—
274	C₂²⋊C₄	—	73	2	no	—
275	C₂²⋊C₄	—	73	2	no	—
276	C₄²	—	73	2	no	—
277	C₄²	—	73	2	no	—
278	C₄²	—	73	2	no	—
279	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
280	C₄²	D₄×D₇₃	1	1	yes	characteristic
281	C₄²	Q₈×D₇₃	1	1	yes	characteristic
282	C₄²	C₂³×D₇₃	1	1	yes	characteristic
283	C₂³	—	73	2	no	—
284	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
285	Q₈	—	146	2	no	—
286	D₄	—	73	4	no	—
287	D₄	—	146	4	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	73	2	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	73	2	no	—
296	C₂×C₄	—	146	4	no	—
297	C₂×C₄	—	146	2	no	—
298	C₂×C₄	—	146	2	no	—
299	C₂×C₄	—	146	2	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
304	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
305	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
306	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	4	2	no	—
311	C₈	—	4	2	no	—
312	C₈	—	292	4	no	—
313	C₈	—	146	2	no	—
314	C₈	—	146	2	no	—
315	C₈	#14556597	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	73	4	no	—
320	C₂²	—	146	2	no	—
321	C₂²	—	146	2	no	—
322	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	146	2	no	—
327	C₄	—	146	2	no	—
328	C₄	—	146	2	no	—
329	C₄	#108×D₇₃	1	2	yes	—
330	C₄	C₂×#14556597	1	1	yes	characteristic
331	C₄	#17989353	1	1	yes	characteristic
332	C₄	#17989389	1	1	yes	characteristic
333	C₄	#17989401	1	1	yes	characteristic
334	C₂	—	4	1	no	—
335	C₂	—	146	4	no	—
336	C₂	#22618496	1	1	yes	characteristic
337	C₂	#22618528	1	1	yes	characteristic
338	C₂	C₂×#108×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1564	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#11924028	Q₈
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1564	D₇₃
hypocenter	C₇₃	#17836
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
3. #22608719	C₂	semidirect
4. #22608713	C₂	semidirect
5. #22608693	C₂	semidirect
6. #22608690	C₂	semidirect
7. #335×D₇₃	C₂	semidirect
8. C₇₃×#1564	C₂	semidirect
176. #1564	D₇₃	semidirect
178. C₇₃	#17836	semidirect

Maximal subgroups

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

subgroup	index
#22619052	2
C₈×#11924028	2
#22608719	2
#22608713	2
#22608693	2
#22608690	2
#335×D₇₃	2
C₇₃×#1564	2
#17836	73