Subgroups — #32991809

Order 18688.

998 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988457	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988453	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988350	C₂	1	4	yes	maximal
4	C₂²×#97×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊Dic₇₃	—	2	2	no	—
6	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
7	C₂²×C₄⋊Dic₇₃	C₂²	1	2	yes	—
8	#17988457	—	2	2	no	—
9	#17988457	C₄	1	4	yes	complemented
10	#17988453	—	2	2	no	—
11	#17988453	C₄	1	4	yes	complemented
12	C₂×#14556498	—	2	2	no	—
13	#17988350	—	2	8	no	—
14	C₂³×C₈×C₇₃	C₂²	1	2	yes	—
15	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×#97×C₇₃	—	2	2	no	—
17	C₂×#97×C₇₃	—	2	2	no	—
18	C₂×#97×C₇₃	C₄	1	4	yes	complemented
19	C₂³×Dic₇₃	—	2	2	no	—
20	C₂³×Dic₇₃	—	2	2	no	—
21	C₂×C₄⋊Dic₇₃	—	2	8	no	—
22	C₂×C₄⋊Dic₇₃	—	2	4	no	—
23	C₂×C₄⋊Dic₇₃	—	2	4	no	—
24	C₂×C₄⋊Dic₇₃	—	2	2	no	—
25	C₂×C₄⋊Dic₇₃	—	2	2	no	—
26	C₂×C₄⋊Dic₇₃	—	2	4	no	—
27	C₂×C₄⋊Dic₇₃	—	2	4	no	—
28	C₂×C₄⋊Dic₇₃	—	4	8	no	—
29	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
30	C₂×C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
31	#14556498	—	2	8	no	—
32	C₂³×C₄×C₇₃	D₄	1	2	yes	—
33	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₇₃	—	2	2	no	—
35	C₂²×C₈×C₇₃	—	2	2	no	—
36	C₂²×C₈×C₇₃	—	2	2	no	—
37	C₂²×C₈×C₇₃	—	2	2	no	—
38	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
39	C₂²×C₈×C₇₃	D₄	1	4	yes	—
40	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
41	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
42	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
43	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
44	C₂×C₄²×C₇₃	D₄	1	2	yes	—
45	C₂×C₄²×C₇₃	D₄	1	2	yes	—
46	C₂×C₄²×C₇₃	Q₈	1	2	yes	—
47	#97×C₇₃	—	2	8	no	—
48	C₂²×Dic₇₃	—	2	4	no	—
49	C₂²×Dic₇₃	—	2	4	no	—
50	C₂²×Dic₇₃	—	4	2	no	—
51	C₂²×Dic₇₃	—	4	2	no	—
52	C₂²×Dic₇₃	—	4	2	no	—
53	C₂²×Dic₇₃	—	4	2	no	—
54	C₂²×Dic₇₃	—	4	8	no	—
55	C₂²×Dic₇₃	—	4	4	no	—
56	C₂²×Dic₇₃	—	4	4	no	—
57	C₄⋊Dic₇₃	—	2	8	no	—
58	C₄⋊Dic₇₃	—	2	8	no	—
59	C₄⋊Dic₇₃	—	4	16	no	—
60	C₄⋊Dic₇₃	—	4	4	no	—
61	C₄⋊Dic₇₃	—	4	4	no	—
62	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	—	2	4	no	—
64	C₂²×C₄×C₇₃	—	2	4	no	—
65	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
66	C₂²×C₄×C₇₃	C₂²⋊C₄	1	4	yes	—
67	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
68	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
76	C₂²×C₄×C₇₃	#41	1	2	yes	—
77	C₂²×C₄×C₇₃	#41	1	2	yes	—
78	C₂×C₈×C₇₃	—	2	4	no	—
79	C₂×C₈×C₇₃	—	2	4	no	—
80	C₂×C₈×C₇₃	—	2	4	no	—
81	C₂×C₈×C₇₃	—	2	4	no	—
82	C₂×C₈×C₇₃	—	2	4	no	—
83	C₂×C₈×C₇₃	—	2	4	no	—
84	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	complemented
85	C₄²×C₇₃	C₄²	1	4	yes	—
86	C₄²×C₇₃	C₂²⋊C₄	1	4	yes	—
87	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
88	C₄²×C₇₃	C₄⋊C₄	1	4	yes	—
89	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂³×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂³×C₇₃	#124	1	1	yes	characteristic
93	C₂³×C₇₃	#124	1	1	yes	characteristic
94	C₂³×C₇₃	#127	1	1	yes	characteristic
95	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂³×C₇₃	AGL(1,ℤ/8)	1	4	yes	—
97	C₂³×C₇₃	#137	1	4	yes	—
98	C₂×C₄×C₇₃	—	2	8	no	—
99	C₂×C₄×C₇₃	—	2	4	no	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	2	8	no	—
102	C₂×C₄×C₇₃	#95	1	8	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	1	yes	characteristic
106	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
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	#121	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	#125	1	2	yes	—
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
119	C₂×Dic₇₃	—	4	4	no	—
120	C₂×Dic₇₃	—	4	4	no	—
121	C₂×Dic₇₃	—	4	8	no	—
122	C₂×Dic₇₃	—	4	8	no	—
123	C₂×Dic₇₃	—	8	4	no	—
124	C₂×Dic₇₃	—	8	4	no	—
125	C₂×Dic₇₃	—	8	2	no	—
126	C₂×Dic₇₃	—	8	4	no	—
127	C₂×Dic₇₃	—	8	2	no	—
128	C₈×C₇₃	—	2	8	no	—
129	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
130	C₂²×C₇₃	#382	1	1	yes	characteristic
131	C₂²×C₇₃	#384	1	1	yes	characteristic
132	C₂²×C₇₃	#390	1	1	yes	characteristic
133	C₂²×C₇₃	#390	1	1	yes	characteristic
134	C₂²×C₇₃	#391	1	1	yes	characteristic
135	C₂²×C₇₃	#391	1	1	yes	characteristic
136	C₂²×C₇₃	#417	1	4	yes	—
137	C₂²×C₇₃	#418	1	2	yes	—
138	C₂²×C₇₃	#419	1	2	yes	—
139	C₂²×C₇₃	#428	1	4	yes	—
140	C₂²×C₇₃	#489	1	4	yes	—
141	C₂²×C₇₃	#490	1	2	yes	—
142	C₂²×C₇₃	#491	1	2	yes	—
143	C₂²×C₇₃	#496	1	2	yes	—
144	C₂²×C₇₃	#497	1	2	yes	—
145	C₂²×C₇₃	#501	1	4	yes	—
146	C₄×C₇₃	—	2	8	no	—
147	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
148	C₄×C₇₃	#380	1	2	yes	—
149	C₄×C₇₃	#389	1	2	yes	—
150	C₄×C₇₃	#389	1	2	yes	—
151	Dic₇₃	—	8	4	no	—
152	Dic₇₃	—	8	4	no	—
153	#10843	—	73	1	no	maximal, Sylow
154	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
155	C₂×C₇₃	#1736	1	2	yes	—
156	C₂×C₇₃	#1836	1	2	yes	—
157	C₂×C₇₃	#1847	1	2	yes	—
158	C₂×C₇₃	#1959	1	2	yes	—
159	C₂×C₇₃	#1960	1	1	yes	characteristic
160	C₂×C₇₃	#1961	1	1	yes	characteristic
161	C₂×C₇₃	#1992	1	2	yes	—
162	C₂×C₇₃	#1993	1	2	yes	—
163	C₂²×#97	D₇₃	1	1	yes	characteristic
164	C₂×#384	—	73	1	no	—
165	C₂×#382	—	73	1	no	—
166	C₂×#340	—	73	4	no	—
167	C₇₃	#10843	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₈	C₂×D₇₃	1	2	yes	—
169	C₂²×C₄⋊C₄	—	73	2	no	—
170	C₂²×C₄⋊C₄	—	73	2	no	—
171	C₂²×C₄⋊C₄	—	146	2	no	—
172	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
173	C₂×#97	—	2	2	no	—
174	C₂×#97	—	2	2	no	—
175	C₂×#97	Dic₇₃	1	4	yes	complemented
176	#384	—	73	4	no	—
177	#384	—	146	2	no	—
178	#382	—	73	4	no	—
179	#382	—	146	2	no	—
180	C₂×#95	—	146	2	no	—
181	#340	—	146	8	no	—
182	C₂³×C₄	—	146	2	no	—
183	C₂³×C₄	—	146	2	no	—
184	C₂³×C₄	D₂₉₂	1	2	yes	—
185	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	—	2	2	no	—
188	C₂²×C₈	—	2	2	no	—
189	C₂²×C₈	—	2	2	no	—
190	C₂²×C₈	Dic₁₄₆	1	4	yes	—
191	C₂²×C₈	D₂₉₂	1	4	yes	—
192	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
193	C₂×C₄⋊C₄	—	292	8	no	—
194	C₂×C₄⋊C₄	—	73	4	no	—
195	C₂×C₄⋊C₄	—	73	4	no	—
196	C₂×C₄⋊C₄	—	146	8	no	—
197	C₂×C₄⋊C₄	—	146	2	no	—
198	C₂×C₄⋊C₄	—	146	2	no	—
199	C₂×C₄⋊C₄	—	146	4	no	—
200	C₂×C₄⋊C₄	—	146	4	no	—
201	C₂×C₄⋊C₄	—	146	4	no	—
202	C₂×C₄⋊C₄	—	146	4	no	—
203	C₂×C₄²	Dic₁₄₆	1	2	yes	—
204	C₂×C₄²	C₄×D₇₃	1	2	yes	—
205	C₂×C₄²	C₄×D₇₃	1	2	yes	—
206	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
207	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
208	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
209	#97	—	2	8	no	—
210	#95	—	146	8	no	—
211	C₂⁴	C₂×D₂₉₂	1	1	yes	characteristic
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	292	2	no	—
215	C₂²×C₄	—	292	2	no	—
216	C₂²×C₄	—	292	2	no	—
217	C₂²×C₄	—	292	2	no	—
218	C₂²×C₄	—	292	8	no	—
219	C₂²×C₄	—	292	4	no	—
220	C₂²×C₄	—	292	4	no	—
221	C₂²×C₄	—	146	4	no	—
222	C₂²×C₄	—	146	4	no	—
223	C₂²×C₄	#11924005	1	4	yes	—
224	C₂²×C₄	#11924005	1	4	yes	—
225	C₂²×C₄	C₄⋊Dic₇₃	1	4	yes	—
226	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
232	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
233	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
234	C₂²×C₄	#11924028	1	2	yes	—
235	C₂²×C₄	#11924028	1	2	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	2	4	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	C₄⋊Dic₇₃	1	8	yes	complemented
243	C₄⋊C₄	—	292	16	no	—
244	C₄⋊C₄	—	292	4	no	—
245	C₄⋊C₄	—	292	4	no	—
246	C₄⋊C₄	—	146	8	no	—
247	C₄⋊C₄	—	146	8	no	—
248	C₄²	C₄×Dic₇₃	1	4	yes	—
249	C₄²	C₂²⋊Dic₇₃	1	4	yes	—
250	C₄²	#11924007	1	4	yes	—
251	C₄²	#11924007	1	4	yes	—
252	C₂³	C₂×#11924005	1	1	yes	characteristic
253	C₂³	C₂×#11924005	1	1	yes	characteristic
254	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
255	C₂³	#14556579	1	1	yes	characteristic
256	C₂³	#14556579	1	1	yes	characteristic
257	C₂³	#14556588	1	1	yes	characteristic
258	C₂³	#14556591	1	1	yes	characteristic
259	C₂³	#14556619	1	4	yes	—
260	C₂³	#14556624	1	4	yes	—
261	C₂×C₄	—	2	8	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	292	4	no	—
266	C₂×C₄	—	292	4	no	—
267	C₂×C₄	—	292	8	no	—
268	C₂×C₄	—	292	8	no	—
269	C₂×C₄	—	584	4	no	—
270	C₂×C₄	—	584	4	no	—
271	C₂×C₄	—	584	2	no	—
272	C₂×C₄	—	584	4	no	—
273	C₂×C₄	—	584	2	no	—
274	C₂×C₄	#14556498	1	8	yes	—
275	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
276	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
277	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
278	C₂×C₄	C₂×#11924007	1	1	yes	characteristic
279	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
280	C₂×C₄	#14556550	1	2	yes	—
281	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
282	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
283	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
284	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
285	C₂×C₄	#14556565	1	2	yes	—
286	C₂×C₄	#14556565	1	2	yes	—
287	C₂×C₄	#14556570	1	2	yes	—
288	C₂×C₄	#14556570	1	2	yes	—
289	C₂×C₄	#14556583	1	2	yes	—
290	C₂×C₄	#14556591	1	2	yes	—
291	C₈	—	2	8	no	—
292	C₂²	C₂×#14556498	1	1	yes	characteristic
293	C₂²	#17988453	1	1	yes	characteristic
294	C₂²	#17988457	1	1	yes	characteristic
295	C₂²	#17988489	1	1	yes	characteristic
296	C₂²	#17988489	1	1	yes	characteristic
297	C₂²	#17988493	1	1	yes	characteristic
298	C₂²	#17988493	1	1	yes	characteristic
299	C₂²	#17988589	1	4	yes	—
300	C₂²	#17988593	1	2	yes	—
301	C₂²	#17988597	1	2	yes	—
302	C₂²	#17988629	1	4	yes	—
303	C₂²	#17988947	1	4	yes	—
304	C₂²	#17988952	1	2	yes	—
305	C₂²	#17988956	1	2	yes	—
306	C₂²	#17988976	1	2	yes	—
307	C₂²	#17988981	1	2	yes	—
308	C₂²	#17988996	1	4	yes	—
309	C₄	—	2	8	no	—
310	C₄	—	584	4	no	—
311	C₄	—	584	4	no	—
312	C₄	C₄×C₄⋊Dic₇₃	1	2	yes	—
313	C₄	#17988447	1	2	yes	—
314	C₄	#17988485	1	2	yes	—
315	C₄	#17988485	1	2	yes	—
316	C₂	#22608161	1	1	yes	characteristic
317	C₂	#22609651	1	2	yes	—
318	C₂	#22610065	1	2	yes	—
319	C₂	#22610099	1	2	yes	—
320	C₂	#22610747	1	2	yes	—
321	C₂	#22610751	1	1	yes	characteristic
322	C₂	#22610755	1	1	yes	characteristic
323	C₂	#22610900	1	2	yes	—
324	C₂	#22610904	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂²×#97×C₇₃	C₂
socle, prime radical	C₂⁴×C₇₃	C₂×D₄
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂²×#97	D₇₃
hypocenter	C₇₃	#10843
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ C₂²×#97

Complements

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

N	H ≅ G/N	type
9. #17988457	C₄	semidirect
11. #17988453	C₄	semidirect
18. C₂×#97×C₇₃	C₄	semidirect
84. C₂×C₈×C₇₃	C₄⋊C₄	semidirect
167. C₇₃	#10843	semidirect
175. C₂×#97	Dic₇₃	semidirect
242. C₂×C₈	C₄⋊Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988457	2
C₂×#17988453	2
C₂×#17988350	2
C₂²×#97×C₇₃	2
#10843	73