Subgroups — #32994635

Order 18688.

722 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988630	C₂	1	2	yes	maximal
2	C₂²×#14556525	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988355	C₂	1	2	yes	maximal
4	C₂×#17988352	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
8	#17988630	—	2	4	no	—
9	#17988630	—	2	4	no	—
10	C₂×#14556525	—	2	4	no	—
11	C₂×#14556525	—	2	2	no	—
12	C₂×#14556524	—	2	2	no	—
13	C₂×#14556522	—	2	2	no	—
14	#17988355	—	2	4	no	—
15	#17988352	C₄	1	4	yes	complemented
16	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
19	#336×C₇₃	—	2	2	no	—
20	C₂³×Dic₇₃	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	C₂×C₄	1	4	yes	—
25	C₂×#11924012	D₄	1	4	yes	—
26	C₂×#11924012	Q₈	1	4	yes	—
27	C₂²×#10596831	—	2	2	no	—
28	C₂×C₄⋊Dic₇₃	—	2	2	no	—
29	C₂×C₄⋊Dic₇₃	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	—	2	2	no	—
31	C₂×C₄⋊Dic₇₃	—	2	2	no	—
32	C₂×C₄⋊Dic₇₃	—	2	2	no	—
33	C₂×C₄⋊Dic₇₃	D₄	1	4	yes	—
34	#14556525	—	2	8	no	—
35	#14556524	—	2	8	no	—
36	#14556522	—	2	8	no	—
37	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	—	2	1	no	—
39	C₂×M₁₆×C₇₃	—	2	2	no	—
40	C₂×M₁₆×C₇₃	—	2	2	no	—
41	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
42	C₂²×C₈×C₇₃	—	2	2	no	—
43	C₂²×C₈×C₇₃	—	2	2	no	—
44	C₂²×C₈×C₇₃	—	2	1	no	—
45	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
46	C₂²×C₈×C₇₃	D₄	1	2	yes	—
47	C₂²×C₈×C₇₃	Q₈	1	2	yes	—
48	#117×C₇₃	—	2	2	no	—
49	#117×C₇₃	—	2	2	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	—	2	1	no	—
53	C₂×C₄²×C₇₃	—	2	1	no	—
54	C₂²×Dic₇₃	—	2	4	no	—
55	C₂²×Dic₇₃	—	4	2	no	—
56	C₂²×Dic₇₃	—	4	2	no	—
57	C₂²×Dic₇₃	—	8	4	no	—
58	#11924012	—	2	4	no	—
59	#11924012	—	2	4	no	—
60	#11924012	—	2	4	no	—
61	#11924012	C₄⋊C₄	1	8	yes	—
62	C₂×#10596831	—	2	4	no	—
63	C₂×#10596831	—	2	4	no	—
64	C₂×#10596831	—	2	4	no	—
65	C₄⋊Dic₇₃	—	2	4	no	—
66	C₄⋊Dic₇₃	—	2	4	no	—
67	C₄⋊Dic₇₃	—	4	2	no	—
68	C₄⋊Dic₇₃	—	4	2	no	—
69	C₂⁴×C₇₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	—	2	1	no	—
71	C₂²×C₄×C₇₃	—	2	1	no	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
78	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	#41	1	2	yes	—
81	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
82	M₁₆×C₇₃	—	2	4	no	—
83	M₁₆×C₇₃	—	4	2	no	—
84	C₂×C₈×C₇₃	—	2	4	no	—
85	C₂×C₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	—	2	4	no	—
87	C₂×C₈×C₇₃	—	2	2	no	—
88	C₂×C₈×C₇₃	—	2	2	no	—
89	C₂×C₈×C₇₃	—	4	1	no	—
90	C₂×C₈×C₇₃	—	4	1	no	—
91	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
92	C₄⋊C₄×C₇₃	—	2	2	no	—
93	C₄⋊C₄×C₇₃	—	2	2	no	—
94	C₂²⋊C₄×C₇₃	—	4	2	no	—
95	C₄²×C₇₃	—	4	1	no	—
96	C₄²×C₇₃	—	4	1	no	—
97	C₂³×C₇₃	—	2	2	no	—
98	C₂³×C₇₃	—	2	2	no	—
99	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
100	C₂³×C₇₃	#121	1	2	yes	—
101	C₂³×C₇₃	#121	1	1	yes	characteristic
102	C₂³×C₇₃	#125	1	2	yes	—
103	C₂³×C₇₃	#125	1	1	yes	characteristic
104	C₂×C₄×C₇₃	—	4	1	no	—
105	C₂×C₄×C₇₃	—	4	1	no	—
106	C₂×C₄×C₇₃	—	4	2	no	—
107	C₂×C₄×C₇₃	—	4	2	no	—
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
113	C₂×C₄×C₇₃	#120	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
116	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
120	C₂×C₄×C₇₃	#123	1	2	yes	—
121	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
124	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
125	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
126	C₂×Dic₇₃	—	4	4	no	—
127	C₂×Dic₇₃	—	8	4	no	—
128	C₂×Dic₇₃	—	8	4	no	—
129	C₂×Dic₇₃	—	8	4	no	—
130	C₈×C₇₃	—	2	4	no	—
131	C₈×C₇₃	—	4	2	no	—
132	#10596831	—	2	8	no	—
133	C₂²×C₇₃	—	2	4	no	—
134	C₂²×C₇₃	—	2	2	no	—
135	C₂²×C₇₃	—	2	2	no	—
136	C₂²×C₇₃	—	2	2	no	—
137	C₂²×C₇₃	—	2	2	no	—
138	C₂²×C₇₃	#389	1	2	yes	—
139	C₂²×C₇₃	#389	1	1	yes	characteristic
140	C₂²×C₇₃	#392	1	1	yes	characteristic
141	C₂²×C₇₃	#398	1	1	yes	characteristic
142	C₂²×C₇₃	#398	1	2	yes	—
143	C₂²×C₇₃	#472	1	2	yes	—
144	C₂²×C₇₃	#473	1	2	yes	—
145	C₄×C₇₃	—	8	1	no	—
146	C₄×C₇₃	—	8	1	no	—
147	C₄×C₇₃	#384	1	1	yes	characteristic
148	C₄×C₇₃	#385	1	1	yes	characteristic
149	C₄×C₇₃	#385	1	2	yes	—
150	C₄×C₇₃	#395	1	1	yes	characteristic
151	C₄×C₇₃	#395	1	2	yes	—
152	C₄×C₇₃	#399	1	1	yes	characteristic
153	Dic₇₃	—	8	4	no	—
154	#11331	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	2	no	—
156	C₂×C₇₃	—	2	2	no	—
157	C₂×C₇₃	#1444	1	1	yes	characteristic
158	C₂×C₇₃	#1952	1	2	yes	—
159	C₂×C₇₃	#2018	1	1	yes	characteristic
160	C₂×C₇₃	#2020	1	1	yes	characteristic
161	C₂×C₇₃	#2089	1	2	yes	—
162	C₂×#428	—	73	2	no	—
163	C₂²×#107	—	73	1	no	—
164	C₂×#343	—	73	2	no	—
165	C₂×#341	—	73	1	no	—
166	#1729	D₇₃	1	1	yes	characteristic
167	C₇₃	#11331	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	73	2	no	—
169	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
170	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
171	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
172	C₂²×C₄⋊C₄	—	73	2	no	—
173	#428	—	146	4	no	—
174	#428	—	146	4	no	—
175	C₂×#107	—	146	4	no	—
176	C₂×#107	—	146	2	no	—
177	C₂×#107	—	146	2	no	—
178	C₂×#106	—	146	2	no	—
179	#343	—	146	4	no	—
180	#341	—	73	4	no	—
181	#336	—	2	2	no	—
182	C₂³×C₄	—	146	2	no	—
183	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	73	4	no	—
188	C₂×M₁₆	—	73	4	no	—
189	C₂×M₁₆	—	73	4	no	—
190	C₂×M₁₆	—	146	2	no	—
191	C₂×M₁₆	—	146	2	no	—
192	C₂×M₁₆	—	146	2	no	—
193	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
194	C₂²×C₈	—	2	2	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	—	146	2	no	—
198	C₂²×C₈	Dic₁₄₆	1	2	yes	—
199	C₂²×C₈	C₄×D₇₃	1	2	yes	—
200	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
201	#117	—	2	2	no	—
202	#117	—	2	2	no	—
203	C₂×C₄⋊C₄	—	73	4	no	—
204	C₂×C₄⋊C₄	—	146	2	no	—
205	C₂×C₄⋊C₄	—	146	2	no	—
206	C₂×C₄⋊C₄	—	146	2	no	—
207	C₂×C₄⋊C₄	—	146	2	no	—
208	C₂×C₄⋊C₄	—	146	2	no	—
209	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
210	C₂×C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	—	2	1	no	—
212	C₂×C₄²	—	2	1	no	—
213	#107	—	146	8	no	—
214	#107	—	146	8	no	—
215	#106	—	146	8	no	—
216	C₂⁴	#11924028	1	1	yes	characteristic
217	C₂²×C₄	—	2	1	no	—
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	292	2	no	—
220	C₂²×C₄	—	292	2	no	—
221	C₂²×C₄	—	584	4	no	—
222	C₂²×C₄	—	146	4	no	—
223	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
224	C₂²×C₄	D₄×D₇₃	1	2	yes	—
225	C₂²×C₄	D₄×D₇₃	1	2	yes	—
226	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
228	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
229	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
230	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
231	C₂²×C₄	#11924027	1	2	yes	—
232	C₂²×C₄	#11924028	1	1	yes	characteristic
233	M₁₆	—	2	4	no	—
234	M₁₆	—	4	2	no	—
235	M₁₆	—	73	8	no	—
236	M₁₆	—	146	4	no	—
237	M₁₆	—	146	4	no	—
238	M₁₆	—	146	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	4	1	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	146	4	no	—
247	C₂×C₈	—	146	4	no	—
248	C₂×C₈	—	146	4	no	—
249	C₂×C₈	#11924007	1	4	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	292	2	no	—
253	C₄⋊C₄	—	292	2	no	—
254	C₄⋊C₄	—	146	4	no	—
255	C₄⋊C₄	—	146	4	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	4	1	no	—
258	C₄²	—	4	1	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₄×Dic₁₄₆	1	1	yes	characteristic
262	C₂³	#14556564	1	2	yes	—
263	C₂³	#14556565	1	1	yes	characteristic
264	C₂³	#14556581	1	2	yes	—
265	C₂³	#14556583	1	1	yes	characteristic
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	4	1	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	4	2	no	—
270	C₂×C₄	—	292	4	no	—
271	C₂×C₄	—	584	4	no	—
272	C₂×C₄	—	584	4	no	—
273	C₂×C₄	—	584	4	no	—
274	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
275	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
276	C₂×C₄	#14556551	1	2	yes	—
277	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
278	C₂×C₄	#14556559	1	1	yes	characteristic
279	C₂×C₄	#14556561	1	2	yes	—
280	C₂×C₄	#14556568	1	2	yes	—
281	C₂×C₄	#14556568	1	2	yes	—
282	C₂×C₄	#14556569	1	2	yes	—
283	C₂×C₄	#14556570	1	1	yes	characteristic
284	C₂×C₄	#14556571	1	2	yes	—
285	C₂×C₄	#14556572	1	1	yes	characteristic
286	C₂×C₄	#14556573	1	2	yes	—
287	C₂×C₄	#14556587	1	1	yes	characteristic
288	C₂×C₄	#14556589	1	1	yes	characteristic
289	C₂×C₄	#14556590	1	2	yes	—
290	C₂×C₄	#14556590	1	2	yes	—
291	C₂×C₄	#14556591	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	4	2	no	—
294	C₈	—	146	8	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	#17988484	1	2	yes	—
301	C₂²	#17988485	1	1	yes	characteristic
302	C₂²	#17988496	1	1	yes	characteristic
303	C₂²	#17988517	1	1	yes	characteristic
304	C₂²	#17988521	1	2	yes	—
305	C₂²	#17988857	1	2	yes	—
306	C₂²	#17988862	1	2	yes	—
307	C₄	—	8	1	no	—
308	C₄	—	8	1	no	—
309	C₄	—	584	4	no	—
310	C₄	#17988456	1	1	yes	characteristic
311	C₄	#17988459	1	1	yes	characteristic
312	C₄	#17988461	1	2	yes	—
313	C₄	#17988507	1	1	yes	characteristic
314	C₄	#17988508	1	2	yes	—
315	C₄	#17988523	1	1	yes	characteristic
316	C₂	—	2	2	no	—
317	C₂	—	2	2	no	—
318	C₂	#22608195	1	1	yes	characteristic
319	C₂	#22610716	1	2	yes	—
320	C₂	#22611040	1	1	yes	characteristic
321	C₂	#22611052	1	1	yes	characteristic
322	C₂	#22611436	1	2	yes	—

	subgroup	G/H
center	C₂³	#14556565
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#1729	C₂
socle	C₂³×C₇₃	#121
prime radical	C₂⁴×C₇₃	#41
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#1729	D₇₃
hypocenter	C₇₃	#11331
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
15. #17988352	C₄	semidirect
167. C₇₃	#11331	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988630	2
C₂²×#14556525	2
C₂×#17988355	2
C₂×#17988352	2
C₇₃×#1729	2
#11331	73