Subgroups — #32994659

Order 18688.

790 conjugacy classes of subgroups.

The subgroup lattice has 312 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	#22612414	C₂	1	2	yes	maximal
3	#22609642	C₂	1	2	yes	maximal
4	C₂×#17988344	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556550	C₂²	1	2	yes	—
7	C₂×#14556550	C₂²	1	2	yes	—
8	#17988629	—	2	4	no	—
9	#17988625	—	2	2	no	—
10	#17988485	—	2	4	no	—
11	#17988460	—	2	4	no	—
12	C₄×#11924007	—	2	4	no	—
13	#17988348	—	2	4	no	—
14	#17988344	C₄	1	4	yes	complemented
15	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	#336×C₇₃	—	2	2	no	—
19	#14556550	—	2	2	no	—
20	#14556550	—	2	2	no	—
21	#14556550	—	2	2	no	—
22	#14556550	—	2	2	no	—
23	#14556550	—	2	2	no	—
24	#14556550	—	2	2	no	—
25	#14556550	C₂×C₄	1	4	yes	—
26	#14556550	Q₈	1	4	yes	—
27	C₂×C₄⋊Dic₇₃	—	2	2	no	—
28	C₂×C₄⋊Dic₇₃	D₄	1	4	yes	—
29	C₂×#11924007	—	2	8	no	—
30	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	C₂×C₄×Dic₇₃	D₄	1	4	yes	—
34	#14556525	—	2	4	no	—
35	#14556523	—	2	4	no	—
36	#14556498	—	2	8	no	—
37	C₂³×C₄×C₇₃	D₄	1	2	yes	—
38	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₇₃	—	2	2	no	—
40	C₂×M₁₆×C₇₃	—	2	2	no	—
41	C₂×M₁₆×C₇₃	—	2	1	no	—
42	C₂×M₁₆×C₇₃	—	2	2	no	—
43	C₂×M₁₆×C₇₃	—	2	2	no	—
44	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
45	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
46	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
47	C₂²×C₈×C₇₃	—	2	1	no	—
48	C₂²×C₈×C₇₃	D₄	1	2	yes	—
49	C₂×C₄²×C₇₃	—	2	2	no	—
50	C₂×C₄²×C₇₃	—	2	2	no	—
51	C₂×C₄²×C₇₃	—	2	1	no	—
52	C₂×C₄²×C₇₃	—	2	1	no	—
53	C₂²×Dic₇₃	—	2	4	no	—
54	C₂²×Dic₇₃	—	2	4	no	—
55	C₄⋊Dic₇₃	—	2	4	no	—
56	C₄⋊Dic₇₃	—	2	4	no	—
57	C₄⋊Dic₇₃	—	2	4	no	—
58	C₄⋊Dic₇₃	—	2	4	no	—
59	#11924007	—	4	16	no	—
60	C₂²⋊Dic₇₃	—	2	4	no	—
61	C₂²⋊Dic₇₃	—	2	4	no	—
62	C₂²⋊Dic₇₃	—	4	2	no	—
63	C₂²⋊Dic₇₃	—	4	2	no	—
64	C₄×Dic₇₃	—	2	4	no	—
65	C₄×Dic₇₃	—	4	2	no	—
66	C₄×Dic₇₃	—	4	2	no	—
67	C₄×Dic₇₃	C₄⋊C₄	1	8	yes	complemented
68	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	—	2	4	no	—
70	C₂²×C₄×C₇₃	—	2	2	no	—
71	C₂²×C₄×C₇₃	—	4	4	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×Q₈	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	#41	1	2	yes	—
80	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
82	M₁₆×C₇₃	—	2	4	no	—
83	M₁₆×C₇₃	—	2	4	no	—
84	M₁₆×C₇₃	—	2	4	no	—
85	M₁₆×C₇₃	—	4	2	no	—
86	C₂×C₈×C₇₃	—	2	2	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	complemented
92	C₄²×C₇₃	—	4	1	no	—
93	C₄²×C₇₃	—	4	2	no	—
94	C₄²×C₇₃	—	4	1	no	—
95	C₂³×C₇₃	—	4	2	no	—
96	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
97	C₂³×C₇₃	#120	1	2	yes	—
98	C₂³×C₇₃	#121	1	1	yes	characteristic
99	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	—	2	8	no	—
103	C₂×C₄×C₇₃	—	4	4	no	—
104	C₂×C₄×C₇₃	—	4	2	no	—
105	C₂×C₄×C₇₃	—	4	2	no	—
106	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#120	1	1	yes	characteristic
111	C₂×C₄×C₇₃	#121	1	2	yes	—
112	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#123	1	2	yes	—
118	C₂×C₄×C₇₃	#125	1	2	yes	—
119	C₂×C₄×C₇₃	#125	1	1	yes	characteristic
120	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
123	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
124	C₂×Dic₇₃	—	2	8	no	—
125	C₂×Dic₇₃	—	4	8	no	—
126	C₂×Dic₇₃	—	4	8	no	—
127	C₂×Dic₇₃	—	4	4	no	—
128	C₈×C₇₃	—	2	4	no	—
129	C₈×C₇₃	—	4	2	no	—
130	C₂²×C₇₃	—	2	4	no	—
131	C₂²×C₇₃	—	2	2	no	—
132	C₂²×C₇₃	—	4	1	no	—
133	C₂²×C₇₃	—	4	2	no	—
134	C₂²×C₇₃	—	4	1	no	—
135	C₂²×C₇₃	#385	1	2	yes	—
136	C₂²×C₇₃	#389	1	1	yes	characteristic
137	C₂²×C₇₃	#392	1	1	yes	characteristic
138	C₂²×C₇₃	#395	1	2	yes	—
139	C₂²×C₇₃	#398	1	1	yes	characteristic
140	C₄×C₇₃	—	4	4	no	—
141	C₄×C₇₃	#384	1	1	yes	characteristic
142	C₄×C₇₃	#385	1	1	yes	characteristic
143	C₄×C₇₃	#389	1	2	yes	—
144	C₄×C₇₃	#395	1	1	yes	characteristic
145	C₄×C₇₃	#398	1	2	yes	—
146	C₄×C₇₃	#399	1	1	yes	characteristic
147	Dic₇₃	—	4	8	no	—
148	Dic₇₃	—	8	4	no	—
149	#11335	—	73	1	no	maximal, Sylow
150	C₂×C₇₃	—	2	2	no	—
151	C₂×C₇₃	—	2	1	no	—
152	C₂×C₇₃	—	2	2	no	—
153	C₂×C₇₃	—	2	1	no	—
154	C₂×C₇₃	#1444	1	1	yes	characteristic
155	C₂×C₇₃	#2010	1	2	yes	—
156	#2911	—	73	1	no	—
157	#2296	—	73	2	no	—
158	#1733	—	73	2	no	—
159	C₂×H(3,ℤ/4)	—	73	1	no	—
160	#1728	D₇₃	1	1	yes	characteristic
161	C₇₃	#11335	1	1	yes	characteristic, complemented, Sylow
162	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
163	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
164	C₂×#117	—	73	2	no	—
165	C₂×#117	—	73	2	no	—
166	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
167	#428	—	146	4	no	—
168	#427	—	146	2	no	—
169	#389	—	146	4	no	—
170	#385	—	146	4	no	—
171	C₄×C₄⋊C₄	—	146	4	no	—
172	#339	—	146	4	no	—
173	H(3,ℤ/4)	—	73	4	no	—
174	#336	—	2	2	no	—
175	C₂³×C₄	D₂₉₂	1	2	yes	—
176	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
177	C₂×M₁₆	—	2	2	no	—
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₂×M₁₆	—	2	2	no	—
181	C₂×M₁₆	—	2	2	no	—
182	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
183	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
184	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
187	#117	—	73	4	no	—
188	#117	—	73	4	no	—
189	#117	—	146	2	no	—
190	#117	—	146	2	no	—
191	#117	—	146	2	no	—
192	#117	—	146	2	no	—
193	#117	—	146	2	no	—
194	#117	—	146	2	no	—
195	C₂×C₄⋊C₄	—	73	4	no	—
196	C₂×C₄⋊C₄	—	146	8	no	—
197	C₂×C₄⋊C₄	—	146	2	no	—
198	C₂×C₂²⋊C₄	—	146	2	no	—
199	C₂×C₂²⋊C₄	—	146	2	no	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	—	2	1	no	—
203	C₂×C₄²	—	2	1	no	—
204	C₂×C₄²	—	73	4	no	—
205	C₂×C₄²	—	146	2	no	—
206	#107	—	146	4	no	—
207	#106	—	146	4	no	—
208	#95	—	146	8	no	—
209	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	4	no	—
213	C₂²×C₄	—	146	4	no	—
214	C₂²×C₄	—	146	4	no	—
215	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₇₃	1	2	yes	—
217	C₂²×C₄	D₄×D₇₃	1	2	yes	—
218	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
219	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
220	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
221	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
222	C₂²×C₄	#11924027	1	2	yes	—
223	C₂²×C₄	#11924028	1	1	yes	characteristic
224	C₂²×C₄	#11924028	1	1	yes	characteristic
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	2	4	no	—
228	M₁₆	—	4	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	4	1	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	#11924007	1	4	yes	complemented
235	C₄⋊C₄	—	292	16	no	—
236	C₄⋊C₄	—	146	4	no	—
237	C₄⋊C₄	—	146	4	no	—
238	C₄⋊C₄	—	146	4	no	—
239	C₄⋊C₄	—	146	4	no	—
240	C₂²⋊C₄	—	292	2	no	—
241	C₂²⋊C₄	—	292	2	no	—
242	C₂²⋊C₄	—	146	4	no	—
243	C₂²⋊C₄	—	146	4	no	—
244	C₄²	—	4	1	no	—
245	C₄²	—	4	2	no	—
246	C₄²	—	4	1	no	—
247	C₄²	—	292	2	no	—
248	C₄²	—	292	2	no	—
249	C₄²	—	73	8	no	—
250	C₄²	—	146	4	no	—
251	C₂³	—	4	2	no	—
252	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
253	C₂³	#14556561	1	2	yes	—
254	C₂³	#14556565	1	1	yes	characteristic
255	C₂³	#14556571	1	2	yes	—
256	C₂³	#14556591	1	1	yes	characteristic
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	292	8	no	—
263	C₂×C₄	—	292	8	no	—
264	C₂×C₄	—	292	4	no	—
265	C₂×C₄	—	146	8	no	—
266	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
267	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
268	C₂×C₄	#14556551	1	2	yes	—
269	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
270	C₂×C₄	#14556559	1	1	yes	characteristic
271	C₂×C₄	#14556564	1	2	yes	—
272	C₂×C₄	#14556568	1	2	yes	—
273	C₂×C₄	#14556568	1	2	yes	—
274	C₂×C₄	#14556569	1	2	yes	—
275	C₂×C₄	#14556570	1	1	yes	characteristic
276	C₂×C₄	#14556572	1	1	yes	characteristic
277	C₂×C₄	#14556573	1	2	yes	—
278	C₂×C₄	#14556581	1	2	yes	—
279	C₂×C₄	#14556583	1	1	yes	characteristic
280	C₂×C₄	#14556587	1	1	yes	characteristic
281	C₂×C₄	#14556589	1	1	yes	characteristic
282	C₂×C₄	#14556590	1	2	yes	—
283	C₂×C₄	#14556590	1	2	yes	—
284	C₈	—	2	4	no	—
285	C₈	—	4	2	no	—
286	C₂²	—	2	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	4	1	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	1	no	—
291	C₂²	#17988461	1	2	yes	—
292	C₂²	#17988485	1	1	yes	characteristic
293	C₂²	#17988496	1	1	yes	characteristic
294	C₂²	#17988508	1	2	yes	—
295	C₂²	#17988517	1	1	yes	characteristic
296	C₄	—	4	4	no	—
297	C₄	—	292	8	no	—
298	C₄	—	584	4	no	—
299	C₄	#17988456	1	1	yes	characteristic
300	C₄	#17988459	1	1	yes	characteristic
301	C₄	#17988484	1	2	yes	—
302	C₄	#17988507	1	1	yes	characteristic
303	C₄	#17988521	1	2	yes	—
304	C₄	#17988523	1	1	yes	characteristic
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	#22608195	1	1	yes	characteristic
310	C₂	#22610998	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556587
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#1728	C₂
socle	C₂²×C₇₃	#392
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#1728	D₇₃
hypocenter	C₇₃	#11335
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
14. #17988344	C₄	semidirect
67. C₄×Dic₇₃	C₄⋊C₄	semidirect
91. C₂×C₈×C₇₃	C₄⋊C₄	semidirect
161. C₇₃	#11335	semidirect
234. C₂×C₈	#11924007	semidirect

Maximal subgroups

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

subgroup	index
#22618521	2
#22612414	2
#22609642	2
C₂×#17988344	2
C₇₃×#1728	2
#11335	73