Subgroups — #32994666

Order 18688.

686 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22618521	C₂	1	1	yes	maximal, characteristic
2	#22618520	C₂	1	2	yes	maximal
3	C₂×#17988345	C₂	1	1	yes	maximal, characteristic
4	#22609627	C₂	1	2	yes	maximal
5	C₇₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂×#14556550	C₂²	1	2	yes	—
8	#17988630	—	2	4	no	—
9	#17988629	—	2	4	no	—
10	#17988627	—	2	4	no	—
11	#17988625	—	2	2	no	—
12	#17988623	—	2	4	no	—
13	#17988345	C₄	1	4	yes	complemented
14	#17988341	—	2	4	no	—
15	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
18	#336×C₇₃	—	2	2	no	—
19	C₂×#11924012	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	C₂×C₄	1	4	yes	—
24	C₂×#11924012	Q₈	1	4	yes	—
25	C₂²×#10596831	D₄	1	4	yes	—
26	#14556550	—	2	2	no	—
27	#14556550	—	2	2	no	—
28	#14556550	—	2	2	no	—
29	#14556550	—	2	2	no	—
30	C₂×C₄⋊Dic₇₃	D₄	1	4	yes	—
31	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	#14556525	—	2	4	no	—
34	#14556524	—	2	8	no	—
35	#14556523	—	2	4	no	—
36	#14556522	—	2	8	no	—
37	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₇₃	—	2	2	no	—
39	C₂×M₁₆×C₇₃	—	2	2	no	—
40	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
42	C₂²×C₈×C₇₃	—	2	1	no	—
43	C₂²×C₈×C₇₃	—	2	2	no	—
44	C₂²×C₈×C₇₃	—	2	2	no	—
45	C₂²×C₈×C₇₃	—	2	1	no	—
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	#11924012	—	2	4	no	—
55	#11924012	—	2	4	no	—
56	#11924012	—	2	4	no	—
57	#11924012	—	2	4	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	C₄⋊C₄	1	8	yes	—
61	C₄⋊Dic₇₃	—	2	4	no	—
62	C₄⋊Dic₇₃	—	2	4	no	—
63	C₂²⋊Dic₇₃	—	4	2	no	—
64	C₂²⋊Dic₇₃	—	4	2	no	—
65	C₄×Dic₇₃	—	4	2	no	—
66	C₄×Dic₇₃	—	4	2	no	—
67	C₂⁴×C₇₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	—	2	2	no	—
69	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	#41	1	2	yes	—
78	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
79	M₁₆×C₇₃	—	2	4	no	—
80	M₁₆×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	—	2	2	no	—
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	—	2	4	no	—
84	C₂×C₈×C₇₃	—	2	2	no	—
85	C₂×C₈×C₇₃	—	4	1	no	—
86	C₂×C₈×C₇₃	—	4	2	no	—
87	C₂×C₈×C₇₃	—	4	1	no	—
88	C₂×C₈×C₇₃	C₄⋊C₄	1	4	yes	—
89	C₄⋊C₄×C₇₃	—	2	4	no	—
90	C₂²⋊C₄×C₇₃	—	4	2	no	—
91	C₄²×C₇₃	—	4	1	no	—
92	C₄²×C₇₃	—	4	1	no	—
93	C₂³×C₇₃	—	4	2	no	—
94	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
95	C₂³×C₇₃	#121	1	2	yes	—
96	C₂³×C₇₃	#121	1	1	yes	characteristic
97	C₂³×C₇₃	#125	1	2	yes	—
98	C₂³×C₇₃	#125	1	1	yes	characteristic
99	C₂×C₄×C₇₃	—	4	4	no	—
100	C₂×C₄×C₇₃	—	4	2	no	—
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
106	C₂×C₄×C₇₃	#120	1	2	yes	—
107	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
108	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
113	C₂×C₄×C₇₃	#123	1	2	yes	—
114	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₂×Dic₇₃	—	4	8	no	—
120	C₂×Dic₇₃	—	4	4	no	—
121	C₈×C₇₃	—	2	4	no	—
122	C₈×C₇₃	—	4	2	no	—
123	#10596831	—	2	8	no	—
124	C₂²×C₇₃	—	2	4	no	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	—	4	1	no	—
127	C₂²×C₇₃	—	4	2	no	—
128	C₂²×C₇₃	—	4	1	no	—
129	C₂²×C₇₃	#389	1	2	yes	—
130	C₂²×C₇₃	#389	1	1	yes	characteristic
131	C₂²×C₇₃	#392	1	1	yes	characteristic
132	C₂²×C₇₃	#398	1	1	yes	characteristic
133	C₂²×C₇₃	#398	1	2	yes	—
134	C₄×C₇₃	—	8	2	no	—
135	C₄×C₇₃	#384	1	1	yes	characteristic
136	C₄×C₇₃	#385	1	1	yes	characteristic
137	C₄×C₇₃	#385	1	2	yes	—
138	C₄×C₇₃	#395	1	1	yes	characteristic
139	C₄×C₇₃	#395	1	2	yes	—
140	C₄×C₇₃	#399	1	1	yes	characteristic
141	Dic₇₃	—	8	4	no	—
142	#11337	—	73	1	no	maximal, Sylow
143	C₂×C₇₃	—	2	2	no	—
144	C₂×C₇₃	—	2	1	no	—
145	C₂×C₇₃	—	2	1	no	—
146	C₂×C₇₃	—	2	2	no	—
147	C₂×C₇₃	#1444	1	1	yes	characteristic
148	C₂×C₇₃	#2017	1	2	yes	—
149	#2911	—	73	1	no	—
150	#2911	—	73	2	no	—
151	C₂×#338	—	73	1	no	—
152	#1729	—	73	2	no	—
153	#1729	D₇₃	1	1	yes	characteristic
154	C₇₃	#11337	1	1	yes	characteristic, complemented, Sylow
155	C₂²×M₁₆	—	73	2	no	—
156	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
157	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
158	C₂×#117	—	73	2	no	—
159	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
160	#428	—	146	4	no	—
161	#428	—	146	4	no	—
162	#428	—	146	4	no	—
163	#427	—	146	4	no	—
164	#427	—	146	2	no	—
165	#338	—	73	4	no	—
166	#336	—	2	2	no	—
167	#336	—	146	4	no	—
168	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	73	4	no	—
172	C₂×M₁₆	—	73	4	no	—
173	C₂×M₁₆	—	146	2	no	—
174	C₂×M₁₆	—	146	2	no	—
175	C₂×M₁₆	—	146	2	no	—
176	C₂×M₁₆	—	146	2	no	—
177	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
178	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	1	no	—
183	C₂²×C₈	—	73	4	no	—
184	C₂²×C₈	Dic₁₄₆	1	2	yes	—
185	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
186	#117	—	2	2	no	—
187	#117	—	2	2	no	—
188	#117	—	146	2	no	—
189	#117	—	146	2	no	—
190	#117	—	146	2	no	—
191	#117	—	146	2	no	—
192	C₂×C₄⋊C₄	—	73	4	no	—
193	C₂×C₄⋊C₄	D₂₉₂	1	2	yes	—
194	C₂×C₂²⋊C₄	—	2	1	no	—
195	C₂×C₂²⋊C₄	—	146	2	no	—
196	C₂×C₄²	—	2	1	no	—
197	C₂×C₄²	—	146	2	no	—
198	#107	—	146	8	no	—
199	#107	—	146	4	no	—
200	#106	—	146	8	no	—
201	#106	—	146	4	no	—
202	C₂⁴	#11924028	1	1	yes	characteristic
203	C₂²×C₄	—	2	2	no	—
204	C₂²×C₄	—	146	4	no	—
205	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₇₃	1	2	yes	—
207	C₂²×C₄	D₄×D₇₃	1	2	yes	—
208	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
210	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
211	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
213	C₂²×C₄	#11924027	1	2	yes	—
214	C₂²×C₄	#11924028	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	146	4	no	—
218	M₁₆	—	146	4	no	—
219	M₁₆	—	146	4	no	—
220	M₁₆	—	146	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	73	8	no	—
229	C₂×C₈	—	146	4	no	—
230	C₂×C₈	—	146	4	no	—
231	C₂×C₈	#11924007	1	4	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	146	4	no	—
234	C₄⋊C₄	—	146	4	no	—
235	C₂²⋊C₄	—	4	2	no	—
236	C₂²⋊C₄	—	292	2	no	—
237	C₂²⋊C₄	—	292	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₄²	—	292	2	no	—
241	C₄²	—	292	2	no	—
242	C₂³	—	4	2	no	—
243	C₂³	C₄×Dic₁₄₆	1	1	yes	characteristic
244	C₂³	#14556564	1	2	yes	—
245	C₂³	#14556565	1	1	yes	characteristic
246	C₂³	#14556581	1	2	yes	—
247	C₂³	#14556583	1	1	yes	characteristic
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	292	8	no	—
251	C₂×C₄	—	292	4	no	—
252	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
253	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
254	C₂×C₄	#14556551	1	2	yes	—
255	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
256	C₂×C₄	#14556559	1	1	yes	characteristic
257	C₂×C₄	#14556561	1	2	yes	—
258	C₂×C₄	#14556568	1	2	yes	—
259	C₂×C₄	#14556568	1	2	yes	—
260	C₂×C₄	#14556569	1	2	yes	—
261	C₂×C₄	#14556570	1	1	yes	characteristic
262	C₂×C₄	#14556571	1	2	yes	—
263	C₂×C₄	#14556572	1	1	yes	characteristic
264	C₂×C₄	#14556573	1	2	yes	—
265	C₂×C₄	#14556587	1	1	yes	characteristic
266	C₂×C₄	#14556589	1	1	yes	characteristic
267	C₂×C₄	#14556590	1	2	yes	—
268	C₂×C₄	#14556590	1	2	yes	—
269	C₂×C₄	#14556591	1	1	yes	characteristic
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	146	8	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	1	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	1	no	—
278	C₂²	#17988484	1	2	yes	—
279	C₂²	#17988485	1	1	yes	characteristic
280	C₂²	#17988496	1	1	yes	characteristic
281	C₂²	#17988517	1	1	yes	characteristic
282	C₂²	#17988521	1	2	yes	—
283	C₄	—	8	2	no	—
284	C₄	—	584	4	no	—
285	C₄	#17988456	1	1	yes	characteristic
286	C₄	#17988459	1	1	yes	characteristic
287	C₄	#17988461	1	2	yes	—
288	C₄	#17988507	1	1	yes	characteristic
289	C₄	#17988508	1	2	yes	—
290	C₄	#17988523	1	1	yes	characteristic
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	#22608195	1	1	yes	characteristic
296	C₂	#22611035	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556587
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#1729	C₂
socle	C₂²×C₇₃	#392
prime radical	C₂⁴×C₇₃	#41
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#1729	D₇₃
hypocenter	C₇₃	#11337
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
13. #17988345	C₄	semidirect
154. C₇₃	#11337	semidirect

Maximal subgroups

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

subgroup	index
#22618521	2
#22618520	2
C₂×#17988345	2
#22609627	2
C₇₃×#1729	2
#11337	73