Subgroups — #32991800

Order 18688.

894 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612338	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988433	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988348	C₂	1	4	yes	maximal
4	C₂×#423×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14556550	C₂²	1	2	yes	—
6	C₂²×C₄×Dic₇₃	C₂²	1	2	yes	—
7	#17988457	C₄	1	2	yes	complemented
8	#17988453	C₄	1	2	yes	complemented
9	#17988447	—	2	2	no	—
10	C₄×C₄⋊Dic₇₃	—	2	2	no	—
11	#17988433	—	2	2	no	—
12	#17988433	C₄	1	4	yes	complemented
13	C₂×#14556498	—	2	2	no	—
14	#17988348	—	2	8	no	—
15	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	#423×C₇₃	—	2	2	no	—
18	#423×C₇₃	C₄	1	4	yes	complemented
19	C₂×C₄⋊C₈×C₇₃	—	2	2	no	—
20	C₂³×Dic₇₃	—	2	2	no	—
21	#14556550	—	2	4	no	—
22	#14556550	—	2	4	no	—
23	C₂×C₄⋊Dic₇₃	—	2	4	no	—
24	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
25	C₂×C₂²⋊Dic₇₃	—	2	2	no	—
26	C₂×C₄×Dic₇₃	—	2	2	no	—
27	C₂×C₄×Dic₇₃	—	2	2	no	—
28	C₂×C₄×Dic₇₃	—	2	2	no	—
29	C₂×C₄×Dic₇₃	—	2	2	no	—
30	C₂×C₄×Dic₇₃	C₂×C₄	1	4	yes	—
31	C₂×C₄×Dic₇₃	D₄	1	4	yes	—
32	C₂×C₄×Dic₇₃	Q₈	1	4	yes	—
33	#14556498	—	2	8	no	—
34	#14556498	—	2	4	no	—
35	C₂³×C₄×C₇₃	D₄	1	2	yes	—
36	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₇₃	—	2	2	no	—
38	C₂×M₁₆×C₇₃	—	2	4	no	—
39	C₂×M₁₆×C₇₃	—	2	4	no	—
40	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
41	C₂²×C₈×C₇₃	—	2	2	no	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₇₃	D₄	1	2	yes	—
47	C₂×C₄²×C₇₃	D₄	1	2	yes	—
48	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
49	C₄⋊C₈×C₇₃	—	2	8	no	—
50	C₂²×Dic₇₃	—	2	4	no	—
51	C₂²×Dic₇₃	—	2	4	no	—
52	C₂²×Dic₇₃	—	2	4	no	—
53	C₂²×Dic₇₃	—	2	4	no	—
54	C₂²×Dic₇₃	—	4	8	no	—
55	C₄⋊Dic₇₃	—	2	8	no	—
56	C₄⋊Dic₇₃	—	4	8	no	—
57	C₂²⋊Dic₇₃	—	4	4	no	—
58	C₄×Dic₇₃	—	2	4	no	—
59	C₄×Dic₇₃	—	2	4	no	—
60	C₄×Dic₇₃	—	2	4	no	—
61	C₄×Dic₇₃	—	4	2	no	—
62	C₄×Dic₇₃	—	4	2	no	—
63	C₄×Dic₇₃	C₄⋊C₄	1	8	yes	complemented
64	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₇₃	—	4	4	no	—
66	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
67	C₂²×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
68	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
69	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
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	1	yes	characteristic
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	#41	1	2	yes	—
79	C₂²×C₄×C₇₃	#41	1	2	yes	—
80	M₁₆×C₇₃	—	2	8	no	—
81	M₁₆×C₇₃	—	4	4	no	—
82	C₂×C₈×C₇₃	—	2	4	no	—
83	C₂×C₈×C₇₃	—	4	2	no	—
84	C₂×C₈×C₇₃	—	4	2	no	—
85	C₄²×C₇₃	C₄²	1	2	yes	—
86	C₄²×C₇₃	C₄²	1	2	yes	—
87	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
88	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
89	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
90	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
91	C₂³×C₇₃	—	2	4	no	—
92	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
94	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂³×C₇₃	#121	1	2	yes	—
96	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
97	C₂×C₄×C₇₃	—	2	4	no	—
98	C₂×C₄×C₇₃	—	2	4	no	—
99	C₂×C₄×C₇₃	—	4	4	no	—
100	C₂×C₄×C₇₃	—	4	4	no	—
101	C₂×C₄×C₇₃	#95	1	4	yes	—
102	C₂×C₄×C₇₃	#95	1	4	yes	—
103	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#117	1	2	yes	—
109	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
113	C₂×C₄×C₇₃	#121	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
116	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₇₃	#125	1	2	yes	—
118	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
121	C₂×Dic₇₃	—	2	8	no	—
122	C₂×Dic₇₃	—	4	8	no	—
123	C₂×Dic₇₃	—	4	8	no	—
124	C₂×Dic₇₃	—	4	8	no	—
125	C₂×Dic₇₃	—	4	4	no	—
126	C₂×Dic₇₃	—	4	8	no	—
127	C₈×C₇₃	—	4	4	no	—
128	C₂²×C₇₃	—	2	4	no	—
129	C₂²×C₇₃	—	2	2	no	—
130	C₂²×C₇₃	—	2	2	no	—
131	C₂²×C₇₃	—	2	4	no	—
132	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
133	C₂²×C₇₃	#380	1	2	yes	—
134	C₂²×C₇₃	#389	1	2	yes	—
135	C₂²×C₇₃	#390	1	1	yes	characteristic
136	C₂²×C₇₃	#391	1	1	yes	characteristic
137	C₂²×C₇₃	#421	1	4	yes	—
138	C₄×C₇₃	—	2	4	no	—
139	C₄×C₇₃	—	2	4	no	—
140	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
141	C₄×C₇₃	#382	1	1	yes	characteristic
142	C₄×C₇₃	#384	1	1	yes	characteristic
143	C₄×C₇₃	#389	1	2	yes	—
144	C₄×C₇₃	#390	1	1	yes	characteristic
145	C₄×C₇₃	#391	1	1	yes	characteristic
146	Dic₇₃	—	4	8	no	—
147	Dic₇₃	—	8	4	no	—
148	#10840	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	—	2	2	no	—
150	C₂×C₇₃	—	2	2	no	—
151	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
152	C₂×C₇₃	#1733	1	2	yes	—
153	C₂×C₇₃	#1963	1	2	yes	—
154	C₂×C₇₃	#1995	1	2	yes	—
155	C₂×#423	D₇₃	1	1	yes	characteristic
156	#2285	—	73	1	no	—
157	C₂×#376	—	73	1	no	—
158	C₂×#339	—	73	4	no	—
159	C₇₃	#10840	1	1	yes	characteristic, complemented, Sylow
160	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
161	C₂×#117	—	73	2	no	—
162	C₂²×C₄²	—	73	2	no	—
163	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
164	#423	—	2	2	no	—
165	#423	Dic₇₃	1	4	yes	complemented
166	C₂×C₄⋊C₈	—	2	2	no	—
167	#384	—	73	2	no	—
168	#382	—	73	2	no	—
169	#380	—	146	2	no	—
170	C₄×C₄⋊C₄	—	146	2	no	—
171	#376	—	73	4	no	—
172	#376	—	146	2	no	—
173	C₂×#95	—	146	2	no	—
174	#339	—	146	8	no	—
175	C₂³×C₄	—	146	2	no	—
176	C₂³×C₄	D₂₉₂	1	2	yes	—
177	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
178	C₂×M₁₆	—	2	2	no	—
179	C₂×M₁₆	—	2	4	no	—
180	C₂×M₁₆	—	2	4	no	—
181	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
182	C₂²×C₈	—	2	2	no	—
183	#117	—	146	4	no	—
184	#117	—	146	4	no	—
185	C₂×C₄⋊C₄	—	73	4	no	—
186	C₂×C₄⋊C₄	—	146	4	no	—
187	C₂×C₂²⋊C₄	—	146	2	no	—
188	C₂×C₄²	—	73	4	no	—
189	C₂×C₄²	—	73	4	no	—
190	C₂×C₄²	—	73	4	no	—
191	C₂×C₄²	—	146	2	no	—
192	C₂×C₄²	—	146	2	no	—
193	C₂×C₄²	—	146	2	no	—
194	C₂×C₄²	—	146	2	no	—
195	C₂×C₄²	Dic₁₄₆	1	2	yes	—
196	C₂×C₄²	C₄×D₇₃	1	2	yes	—
197	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
199	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
200	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
201	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
202	C₄⋊C₈	—	2	8	no	—
203	#95	—	146	8	no	—
204	#95	—	146	4	no	—
205	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
206	C₂²×C₄	—	4	4	no	—
207	C₂²×C₄	—	292	8	no	—
208	C₂²×C₄	—	146	4	no	—
209	C₂²×C₄	—	146	4	no	—
210	C₂²×C₄	—	146	4	no	—
211	C₂²×C₄	—	146	4	no	—
212	C₂²×C₄	#11924005	1	2	yes	—
213	C₂²×C₄	#11924005	1	2	yes	—
214	C₂²×C₄	#11924005	1	4	yes	—
215	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
216	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
219	C₂²×C₄	D₄×D₇₃	1	2	yes	—
220	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
222	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
223	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
224	C₂²×C₄	#11924027	1	2	yes	—
225	C₂²×C₄	#11924029	1	2	yes	—
226	M₁₆	—	2	8	no	—
227	M₁₆	—	4	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₄⋊C₄	—	292	8	no	—
232	C₄⋊C₄	—	146	8	no	—
233	C₂²⋊C₄	—	292	4	no	—
234	C₄²	—	292	2	no	—
235	C₄²	—	292	2	no	—
236	C₄²	—	73	8	no	—
237	C₄²	—	146	4	no	—
238	C₄²	—	146	4	no	—
239	C₄²	—	146	4	no	—
240	C₄²	C₄×Dic₇₃	1	2	yes	—
241	C₄²	C₄×Dic₇₃	1	2	yes	—
242	C₄²	C₂²⋊Dic₇₃	1	4	yes	—
243	C₄²	#11924007	1	4	yes	—
244	C₄²	#11924007	1	2	yes	—
245	C₄²	#11924007	1	2	yes	—
246	C₂³	—	2	4	no	—
247	C₂³	C₂×#11924005	1	1	yes	characteristic
248	C₂³	C₂×#11924005	1	1	yes	characteristic
249	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
250	C₂³	#14556566	1	2	yes	—
251	C₂³	#14556571	1	2	yes	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	4	no	—
256	C₂×C₄	—	292	8	no	—
257	C₂×C₄	—	292	8	no	—
258	C₂×C₄	—	292	8	no	—
259	C₂×C₄	—	292	4	no	—
260	C₂×C₄	—	292	8	no	—
261	C₂×C₄	—	146	8	no	—
262	C₂×C₄	#14556498	1	4	yes	—
263	C₂×C₄	#14556498	1	4	yes	—
264	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
266	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
267	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
268	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
269	C₂×C₄	#14556549	1	2	yes	—
270	C₂×C₄	#14556553	1	2	yes	—
271	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
272	C₂×C₄	#14556556	1	2	yes	—
273	C₂×C₄	Q₈×Dic₇₃	1	2	yes	—
274	C₂×C₄	#14556562	1	2	yes	—
275	C₂×C₄	#14556567	1	2	yes	—
276	C₂×C₄	#14556578	1	1	yes	characteristic
277	C₂×C₄	#14556580	1	1	yes	characteristic
278	C₂×C₄	#14556582	1	2	yes	—
279	C₂×C₄	#14556587	1	1	yes	characteristic
280	C₂×C₄	#14556589	1	1	yes	characteristic
281	C₂×C₄	#14556590	1	2	yes	—
282	C₈	—	4	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₂×#14556498	1	1	yes	characteristic
288	C₂²	#17988444	1	2	yes	—
289	C₂²	#17988486	1	2	yes	—
290	C₂²	#17988488	1	1	yes	characteristic
291	C₂²	#17988492	1	1	yes	characteristic
292	C₂²	#17988603	1	4	yes	—
293	C₄	—	2	4	no	—
294	C₄	—	2	4	no	—
295	C₄	—	292	8	no	—
296	C₄	—	584	4	no	—
297	C₄	C₄⋊C₄×Dic₇₃	1	2	yes	—
298	C₄	#17988452	1	1	yes	characteristic
299	C₄	#17988456	1	1	yes	characteristic
300	C₄	#17988483	1	2	yes	—
301	C₄	#17988490	1	1	yes	characteristic
302	C₄	#17988494	1	1	yes	characteristic
303	C₂	—	2	2	no	—
304	C₂	—	2	2	no	—
305	C₂	#22608160	1	1	yes	characteristic
306	C₂	#22609642	1	2	yes	—
307	C₂	#22610762	1	2	yes	—
308	C₂	#22610911	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂×#423×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂×#423	D₇₃
hypocenter	C₇₃	#10840
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×#423

Complements

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

N	H ≅ G/N	type
7. #17988457	C₄	semidirect
8. #17988453	C₄	semidirect
12. #17988433	C₄	semidirect
18. #423×C₇₃	C₄	semidirect
63. C₄×Dic₇₃	C₄⋊C₄	semidirect
159. C₇₃	#10840	semidirect
165. #423	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612338	2
C₂×#17988433	2
C₂×#17988348	2
C₂×#423×C₇₃	2
#10840	73