Subgroups — #32994630

Order 18688.

722 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#14556523	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×#14556524	—	2	2	no	—
11	C₂×#14556523	—	2	4	no	—
12	C₂×#14556523	—	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	#14556524	—	2	8	no	—
35	#14556523	—	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	2	yes	—
51	C₂×C₂²⋊C₄×C₇₃	—	2	1	no	—
52	C₂×C₄²×C₇₃	—	2	1	no	—
53	C₂²×Dic₇₃	—	2	4	no	—
54	C₂²×Dic₇₃	—	4	2	no	—
55	C₂²×Dic₇₃	—	4	2	no	—
56	C₂²×Dic₇₃	—	8	4	no	—
57	#11924012	—	2	4	no	—
58	#11924012	—	2	4	no	—
59	#11924012	—	2	4	no	—
60	#11924012	C₄⋊C₄	1	8	yes	—
61	C₂×#10596831	—	2	4	no	—
62	C₂×#10596831	—	2	4	no	—
63	C₂×#10596831	—	2	4	no	—
64	C₄⋊Dic₇₃	—	2	4	no	—
65	C₄⋊Dic₇₃	—	2	4	no	—
66	C₄⋊Dic₇₃	—	4	2	no	—
67	C₄⋊Dic₇₃	—	4	2	no	—
68	C₂⁴×C₇₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	—	2	2	no	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	#41	1	2	yes	—
79	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
80	M₁₆×C₇₃	—	2	4	no	—
81	M₁₆×C₇₃	—	4	2	no	—
82	C₂×C₈×C₇₃	—	2	4	no	—
83	C₂×C₈×C₇₃	—	2	2	no	—
84	C₂×C₈×C₇₃	—	2	4	no	—
85	C₂×C₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	—	2	2	no	—
87	C₂×C₈×C₇₃	—	4	1	no	—
88	C₂×C₈×C₇₃	—	4	1	no	—
89	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
90	C₄⋊C₄×C₇₃	—	2	2	no	—
91	C₄⋊C₄×C₇₃	—	2	2	no	—
92	C₂²⋊C₄×C₇₃	—	4	2	no	—
93	C₄²×C₇₃	—	4	1	no	—
94	C₄²×C₇₃	—	4	1	no	—
95	C₂³×C₇₃	—	2	4	no	—
96	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
97	C₂³×C₇₃	#121	1	2	yes	—
98	C₂³×C₇₃	#121	1	1	yes	characteristic
99	C₂³×C₇₃	#125	1	2	yes	—
100	C₂³×C₇₃	#125	1	1	yes	characteristic
101	C₂×C₄×C₇₃	—	4	2	no	—
102	C₂×C₄×C₇₃	—	4	4	no	—
103	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#120	1	2	yes	—
109	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₇₃	#123	1	2	yes	—
116	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
121	C₂×Dic₇₃	—	4	4	no	—
122	C₂×Dic₇₃	—	8	4	no	—
123	C₂×Dic₇₃	—	8	4	no	—
124	C₂×Dic₇₃	—	8	4	no	—
125	C₈×C₇₃	—	2	4	no	—
126	C₈×C₇₃	—	4	2	no	—
127	#10596831	—	2	8	no	—
128	C₂²×C₇₃	—	2	4	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	2	2	no	—
133	C₂²×C₇₃	#389	1	2	yes	—
134	C₂²×C₇₃	#389	1	1	yes	characteristic
135	C₂²×C₇₃	#392	1	1	yes	characteristic
136	C₂²×C₇₃	#398	1	1	yes	characteristic
137	C₂²×C₇₃	#398	1	2	yes	—
138	C₂²×C₇₃	#471	1	4	yes	—
139	C₄×C₇₃	—	8	2	no	—
140	C₄×C₇₃	#384	1	1	yes	characteristic
141	C₄×C₇₃	#385	1	1	yes	characteristic
142	C₄×C₇₃	#385	1	2	yes	—
143	C₄×C₇₃	#395	1	1	yes	characteristic
144	C₄×C₇₃	#395	1	2	yes	—
145	C₄×C₇₃	#399	1	1	yes	characteristic
146	Dic₇₃	—	8	4	no	—
147	#11330	—	73	1	no	maximal, Sylow
148	C₂×C₇₃	—	2	2	no	—
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	#1444	1	1	yes	characteristic
151	C₂×C₇₃	#1951	1	2	yes	—
152	C₂×C₇₃	#2019	1	2	yes	—
153	C₂×C₇₃	#2090	1	2	yes	—
154	C₂×#428	—	73	2	no	—
155	C₂²×#106	—	73	1	no	—
156	C₂×#343	—	73	2	no	—
157	C₂×#341	—	73	1	no	—
158	#1729	D₇₃	1	1	yes	characteristic
159	C₇₃	#11330	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	—	73	2	no	—
161	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
162	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
163	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
164	C₂²×C₄⋊C₄	—	73	2	no	—
165	#428	—	146	4	no	—
166	#428	—	146	4	no	—
167	C₂×#107	—	146	2	no	—
168	C₂×#106	—	146	4	no	—
169	C₂×#106	—	146	2	no	—
170	C₂×#106	—	146	2	no	—
171	#343	—	146	4	no	—
172	#341	—	73	4	no	—
173	#336	—	2	2	no	—
174	C₂³×C₄	—	146	2	no	—
175	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
176	C₂×M₁₆	—	2	1	no	—
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	73	4	no	—
180	C₂×M₁₆	—	73	4	no	—
181	C₂×M₁₆	—	73	4	no	—
182	C₂×M₁₆	—	146	2	no	—
183	C₂×M₁₆	—	146	2	no	—
184	C₂×M₁₆	—	146	2	no	—
185	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	146	2	no	—
190	C₂²×C₈	Dic₁₄₆	1	2	yes	—
191	C₂²×C₈	C₄×D₇₃	1	2	yes	—
192	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	C₂×C₄⋊C₄	—	73	4	no	—
196	C₂×C₄⋊C₄	—	146	2	no	—
197	C₂×C₄⋊C₄	—	146	2	no	—
198	C₂×C₄⋊C₄	—	146	2	no	—
199	C₂×C₄⋊C₄	—	146	2	no	—
200	C₂×C₄⋊C₄	—	146	2	no	—
201	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
202	C₂×C₂²⋊C₄	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	#107	—	146	8	no	—
205	#106	—	146	8	no	—
206	#106	—	146	8	no	—
207	C₂⁴	#11924028	1	1	yes	characteristic
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	292	2	no	—
210	C₂²×C₄	—	292	2	no	—
211	C₂²×C₄	—	584	4	no	—
212	C₂²×C₄	—	146	4	no	—
213	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
214	C₂²×C₄	D₄×D₇₃	1	2	yes	—
215	C₂²×C₄	D₄×D₇₃	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
218	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
219	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
220	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
221	C₂²×C₄	#11924027	1	2	yes	—
222	C₂²×C₄	#11924028	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	73	8	no	—
226	M₁₆	—	146	4	no	—
227	M₁₆	—	146	4	no	—
228	M₁₆	—	146	4	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	146	4	no	—
237	C₂×C₈	—	146	4	no	—
238	C₂×C₈	—	146	4	no	—
239	C₂×C₈	#11924007	1	4	yes	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	292	2	no	—
243	C₄⋊C₄	—	292	2	no	—
244	C₄⋊C₄	—	146	4	no	—
245	C₄⋊C₄	—	146	4	no	—
246	C₂²⋊C₄	—	4	2	no	—
247	C₄²	—	4	1	no	—
248	C₄²	—	4	1	no	—
249	C₂³	—	2	4	no	—
250	C₂³	C₄×Dic₁₄₆	1	1	yes	characteristic
251	C₂³	#14556564	1	2	yes	—
252	C₂³	#14556565	1	1	yes	characteristic
253	C₂³	#14556581	1	2	yes	—
254	C₂³	#14556583	1	1	yes	characteristic
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	292	4	no	—
258	C₂×C₄	—	584	4	no	—
259	C₂×C₄	—	584	4	no	—
260	C₂×C₄	—	584	4	no	—
261	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
262	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
263	C₂×C₄	#14556551	1	2	yes	—
264	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
265	C₂×C₄	#14556559	1	1	yes	characteristic
266	C₂×C₄	#14556561	1	2	yes	—
267	C₂×C₄	#14556568	1	2	yes	—
268	C₂×C₄	#14556568	1	2	yes	—
269	C₂×C₄	#14556569	1	2	yes	—
270	C₂×C₄	#14556570	1	1	yes	characteristic
271	C₂×C₄	#14556571	1	2	yes	—
272	C₂×C₄	#14556572	1	1	yes	characteristic
273	C₂×C₄	#14556573	1	2	yes	—
274	C₂×C₄	#14556587	1	1	yes	characteristic
275	C₂×C₄	#14556589	1	1	yes	characteristic
276	C₂×C₄	#14556590	1	2	yes	—
277	C₂×C₄	#14556590	1	2	yes	—
278	C₂×C₄	#14556591	1	1	yes	characteristic
279	C₈	—	2	4	no	—
280	C₈	—	4	2	no	—
281	C₈	—	146	8	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	#17988484	1	2	yes	—
288	C₂²	#17988485	1	1	yes	characteristic
289	C₂²	#17988496	1	1	yes	characteristic
290	C₂²	#17988517	1	1	yes	characteristic
291	C₂²	#17988521	1	2	yes	—
292	C₂²	#17988850	1	4	yes	—
293	C₄	—	8	2	no	—
294	C₄	—	584	4	no	—
295	C₄	#17988456	1	1	yes	characteristic
296	C₄	#17988459	1	1	yes	characteristic
297	C₄	#17988461	1	2	yes	—
298	C₄	#17988507	1	1	yes	characteristic
299	C₄	#17988508	1	2	yes	—
300	C₄	#17988523	1	1	yes	characteristic
301	C₂	—	2	2	no	—
302	C₂	—	2	2	no	—
303	C₂	#22608195	1	1	yes	characteristic
304	C₂	#22610711	1	2	yes	—
305	C₂	#22611046	1	2	yes	—
306	C₂	#22611441	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	#11330
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
159. C₇₃	#11330	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×#14556523	2
C₂×#17988355	2
C₂×#17988352	2
C₇₃×#1729	2
#11330	73