Subgroups — #32993889

Order 18688.

670 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	#22618495	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988357	C₂	1	2	yes	maximal
3	#22609670	C₂	1	1	yes	maximal, characteristic
4	C₂×#17988356	C₂	1	2	yes	maximal
5	C₇₃×#2317	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂²×#11924012	C₂²	1	2	yes	—
8	#17988614	—	2	4	no	—
9	#17988611	—	2	2	no	—
10	#17988357	—	2	4	no	—
11	#17988356	—	2	4	no	—
12	#17988356	—	2	1	no	—
13	#17988356	—	2	1	no	—
14	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄⋊C₄×C₇₃	C₂²	1	1	yes	characteristic
17	#389×C₇₃	—	2	2	no	—
18	#385×C₇₃	C₄	1	4	yes	—
19	#384×C₇₃	—	2	1	no	—
20	#382×C₇₃	—	2	1	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	—	2	2	no	—
25	C₂×#11924012	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	2	no	—
28	C₂×#11924012	—	2	2	no	—
29	C₂×#11924012	—	2	2	no	—
30	C₂×#11924012	—	2	2	no	—
31	C₂×#11924012	D₄	1	4	yes	—
32	C₂×#11924012	Q₈	1	4	yes	—
33	C₂²×#10596831	—	2	2	no	—
34	C₂²×#10596831	—	2	2	no	—
35	C₄⋊#10596831	—	2	8	no	—
36	C₂³×C₄×C₇₃	C₂×C₄	1	2	yes	—
37	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	#117×C₇₃	—	2	2	no	—
39	#117×C₇₃	—	2	1	no	—
40	#117×C₇₃	—	2	1	no	—
41	#117×C₇₃	—	2	2	no	—
42	#117×C₇₃	D₄	1	2	yes	—
43	#117×C₇₃	Q₈	1	2	yes	—
44	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
47	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
48	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
49	C₂×C₄⋊C₄×C₇₃	—	2	4	no	—
50	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
52	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
53	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₇₃	—	2	1	no	—
55	C₂×C₄²×C₇₃	D₄	1	2	yes	—
56	#95×C₇₃	—	2	4	no	—
57	#11924012	—	2	4	no	—
58	#11924012	—	2	4	no	—
59	#11924012	—	2	4	no	—
60	#11924012	—	2	4	no	—
61	#11924012	—	4	2	no	—
62	#11924012	—	4	2	no	—
63	#11924012	—	4	2	no	—
64	#11924012	—	4	2	no	—
65	C₂×#10596831	—	2	4	no	—
66	C₂×#10596831	—	2	4	no	—
67	C₂×#10596831	—	4	2	no	—
68	C₂×#10596831	—	4	2	no	—
69	C₂×#10596831	—	4	2	no	—
70	C₂×#10596831	—	4	2	no	—
71	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	—	2	2	no	—
73	C₂²×C₄×C₇₃	—	2	2	no	—
74	C₂²×C₄×C₇₃	—	2	4	no	—
75	C₂²×C₄×C₇₃	—	8	2	no	—
76	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	#41	1	2	yes	—
85	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	#41	1	2	yes	—
88	C₄⋊C₄×C₇₃	—	2	2	no	—
89	C₄⋊C₄×C₇₃	—	2	2	no	—
90	C₄⋊C₄×C₇₃	—	2	2	no	—
91	C₄⋊C₄×C₇₃	—	2	2	no	—
92	C₄⋊C₄×C₇₃	—	4	1	no	—
93	C₄⋊C₄×C₇₃	—	4	2	no	—
94	C₄⋊C₄×C₇₃	—	4	1	no	—
95	C₄⋊C₄×C₇₃	—	8	4	no	—
96	C₂²⋊C₄×C₇₃	—	2	2	no	—
97	C₂²⋊C₄×C₇₃	—	2	4	no	—
98	C₂²⋊C₄×C₇₃	C₄⋊C₄	1	4	yes	—
99	C₄²×C₇₃	—	2	2	no	—
100	C₄²×C₇₃	—	2	2	no	—
101	C₄²×C₇₃	—	2	2	no	—
102	C₄²×C₇₃	—	2	2	no	—
103	C₂³×C₇₃	—	4	1	no	—
104	C₂³×C₇₃	—	4	1	no	—
105	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₇₃	#117	1	1	yes	characteristic
108	C₂³×C₇₃	C₄×D₄	1	2	yes	—
109	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
110	C₂×C₄×C₇₃	—	2	4	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	4	2	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	—	4	2	no	—
117	C₂×C₄×C₇₃	—	4	4	no	—
118	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	—	8	4	no	—
120	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
121	C₂×C₄×C₇₃	#121	1	2	yes	—
122	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
123	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
124	C₂×C₄×C₇₃	#121	1	2	yes	—
125	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
131	C₂×C₄×C₇₃	#123	1	2	yes	—
132	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
133	C₂×C₄×C₇₃	#125	1	2	yes	—
134	C₂×C₄×C₇₃	#125	1	2	yes	—
135	C₂×C₄×C₇₃	#126	1	2	yes	—
136	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
138	#10596831	—	4	4	no	—
139	#10596831	—	4	4	no	—
140	C₂²×C₇₃	—	2	1	no	—
141	C₂²×C₇₃	—	2	2	no	—
142	C₂²×C₇₃	—	2	1	no	—
143	C₂²×C₇₃	—	2	2	no	—
144	C₂²×C₇₃	—	4	1	no	—
145	C₂²×C₇₃	—	4	1	no	—
146	C₂²×C₇₃	—	4	2	no	—
147	C₂²×C₇₃	#380	1	1	yes	characteristic
148	C₂²×C₇₃	#386	1	1	yes	characteristic
149	C₂²×C₇₃	#387	1	2	yes	—
150	C₂²×C₇₃	#389	1	2	yes	—
151	C₂²×C₇₃	#391	1	1	yes	characteristic
152	C₄×C₇₃	—	4	4	no	—
153	C₄×C₇₃	—	4	2	no	—
154	C₄×C₇₃	—	4	2	no	—
155	C₄×C₇₃	—	8	2	no	—
156	C₄×C₇₃	#394	1	1	yes	characteristic
157	C₄×C₇₃	#395	1	1	yes	characteristic
158	C₄×C₇₃	#396	1	2	yes	—
159	C₄×C₇₃	#398	1	1	yes	characteristic
160	C₄×C₇₃	#398	1	2	yes	—
161	C₄×C₇₃	#400	1	1	yes	characteristic
162	#11206	—	73	1	no	maximal, Sylow
163	C₂×C₇₃	—	2	1	no	—
164	C₂×C₇₃	—	2	2	no	—
165	C₂×C₇₃	—	2	2	no	—
166	C₂×C₇₃	—	2	1	no	—
167	C₂×C₇₃	#1441	1	1	yes	characteristic
168	C₂×C₇₃	#1914	1	1	yes	characteristic
169	C₂×C₇₃	#1915	1	1	yes	characteristic
170	#2905	—	73	1	no	—
171	#2317	D₇₃	1	1	yes	characteristic
172	C₂×#344	—	73	2	no	—
173	#1742	—	73	1	no	—
174	C₂×#343	—	73	2	no	—
175	C₇₃	#11206	1	1	yes	characteristic, complemented, Sylow
176	C₂²×M₁₆	—	73	2	no	—
177	C₂²×M₁₆	—	73	2	no	—
178	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
179	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
180	C₂²×C₄⋊C₄	C₂×D₇₃	1	1	yes	characteristic
181	#424	—	146	4	no	—
182	#423	—	146	2	no	—
183	#389	—	2	2	no	—
184	#385	Dic₇₃	1	4	yes	—
185	#384	—	2	1	no	—
186	#382	—	2	1	no	—
187	#344	—	146	4	no	—
188	#343	—	146	4	no	—
189	#343	—	146	1	no	—
190	#343	—	146	1	no	—
191	C₂³×C₄	C₂×Dic₇₃	1	2	yes	—
192	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
193	C₂×M₁₆	—	73	4	no	—
194	C₂×M₁₆	—	73	4	no	—
195	C₂×M₁₆	—	146	2	no	—
196	C₂×M₁₆	—	146	2	no	—
197	C₂×M₁₆	—	146	2	no	—
198	C₂×M₁₆	—	146	2	no	—
199	C₂×M₁₆	—	146	2	no	—
200	C₂×M₁₆	—	146	2	no	—
201	C₂×M₁₆	—	146	2	no	—
202	C₂×M₁₆	—	146	2	no	—
203	C₂×M₁₆	—	146	2	no	—
204	C₂×M₁₆	—	146	2	no	—
205	C₂²×C₈	—	146	2	no	—
206	C₂²×C₈	—	146	2	no	—
207	#117	—	2	2	no	—
208	#117	—	2	2	no	—
209	#117	—	2	1	no	—
210	#117	—	2	1	no	—
211	#117	Dic₁₄₆	1	2	yes	—
212	#117	D₂₉₂	1	2	yes	—
213	C₂×C₄⋊C₄	—	2	1	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	2	no	—
218	C₂×C₄⋊C₄	—	2	4	no	—
219	C₂×C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
220	C₂×C₄⋊C₄	C₂²⋊D₇₃	1	1	yes	characteristic
221	C₂×C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
222	C₂×C₂²⋊C₄	C₂×Dic₇₃	1	2	yes	—
223	C₂×C₄²	—	2	1	no	—
224	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
225	C₄⋊C₈	—	146	8	no	—
226	#95	—	2	4	no	—
227	C₂⁴	C₂²×Dic₇₃	1	1	yes	characteristic
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	8	2	no	—
232	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
233	C₂²×C₄	C₂²⋊Dic₇₃	1	2	yes	—
234	C₂²×C₄	D₄×D₇₃	1	2	yes	—
235	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
238	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
239	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
240	C₂²×C₄	#11924027	1	2	yes	—
241	C₂²×C₄	#11924028	1	1	yes	characteristic
242	C₂²×C₄	#11924028	1	1	yes	characteristic
243	C₂²×C₄	#11924029	1	2	yes	—
244	M₁₆	—	292	2	no	—
245	M₁₆	—	292	2	no	—
246	M₁₆	—	292	2	no	—
247	M₁₆	—	292	2	no	—
248	M₁₆	—	146	4	no	—
249	M₁₆	—	146	4	no	—
250	M₁₆	—	146	4	no	—
251	M₁₆	—	146	4	no	—
252	C₂×C₈	—	292	2	no	—
253	C₂×C₈	—	292	2	no	—
254	C₂×C₈	—	292	2	no	—
255	C₂×C₈	—	292	2	no	—
256	C₂×C₈	—	146	4	no	—
257	C₂×C₈	—	146	4	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	8	4	no	—
266	C₂²⋊C₄	—	2	2	no	—
267	C₂²⋊C₄	—	2	4	no	—
268	C₂²⋊C₄	C₄⋊Dic₇₃	1	4	yes	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	2	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
276	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
277	C₂³	#14556550	1	1	yes	characteristic
278	C₂³	D₄×Dic₇₃	1	2	yes	—
279	C₂³	Q₈×Dic₇₃	1	2	yes	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	#14556559	1	1	yes	characteristic
291	C₂×C₄	#14556563	1	2	yes	—
292	C₂×C₄	#14556565	1	1	yes	characteristic
293	C₂×C₄	#14556565	1	1	yes	characteristic
294	C₂×C₄	#14556566	1	2	yes	—
295	C₂×C₄	#14556567	1	2	yes	—
296	C₂×C₄	#14556569	1	2	yes	—
297	C₂×C₄	#14556570	1	1	yes	characteristic
298	C₂×C₄	#14556570	1	1	yes	characteristic
299	C₂×C₄	#14556571	1	2	yes	—
300	C₂×C₄	#14556572	1	1	yes	characteristic
301	C₂×C₄	#14556576	1	2	yes	—
302	C₂×C₄	#14556580	1	1	yes	characteristic
303	C₂×C₄	#14556581	1	2	yes	—
304	C₂×C₄	#14556582	1	2	yes	—
305	C₂×C₄	#14556586	1	2	yes	—
306	C₂×C₄	#14556589	1	1	yes	characteristic
307	C₂×C₄	#14556590	1	2	yes	—
308	C₈	—	292	4	no	—
309	C₈	—	292	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	#17988447	1	1	yes	characteristic
318	C₂²	#17988467	1	1	yes	characteristic
319	C₂²	#17988473	1	2	yes	—
320	C₂²	#17988486	1	2	yes	—
321	C₂²	#17988494	1	1	yes	characteristic
322	C₄	—	4	4	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	—	8	2	no	—
326	C₄	#17988504	1	1	yes	characteristic
327	C₄	#17988507	1	1	yes	characteristic
328	C₄	#17988509	1	2	yes	—
329	C₄	#17988517	1	1	yes	characteristic
330	C₄	#17988520	1	2	yes	—
331	C₄	#17988526	1	1	yes	characteristic
332	C₂	—	2	1	no	—
333	C₂	—	2	2	no	—
334	C₂	—	2	2	no	—
335	C₂	—	2	1	no	—
336	C₂	#22608184	1	1	yes	characteristic
337	C₂	#22610479	1	1	yes	characteristic
338	C₂	#22610484	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988467
commutator, prime radical	C₂⁴×C₇₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#2317	C₂
socle	C₂²×C₇₃	#386
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#2317	D₇₃
hypocenter	C₇₃	#11206
LCS	G ▷ C₂⁴×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2317

Complements

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

N	H ≅ G/N	type
175. C₇₃	#11206	semidirect

Maximal subgroups

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

subgroup	index
#22618495	2
C₂×#17988357	2
#22609670	2
C₂×#17988356	2
C₇₃×#2317	2
#11206	73