Subgroups — #32991845

Order 18688.

794 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988611	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊#10596831	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988351	C₂	1	4	yes	maximal
4	C₂×C₄×M₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924012	C₂²	1	2	yes	—
6	C₂³×#10596831	C₂²	1	2	yes	—
7	#17988611	—	2	2	no	—
8	#17988611	C₄	1	4	yes	—
9	C₂×C₄⋊#10596831	—	2	2	no	—
10	C₂×C₄⋊#10596831	—	2	2	no	—
11	C₂×C₄⋊#10596831	—	2	1	no	—
12	C₂×C₄⋊#10596831	—	2	1	no	—
13	C₂×C₄⋊#10596831	C₄	1	2	yes	—
14	C₂×C₄⋊#10596831	C₄	1	2	yes	—
15	#17988351	—	2	8	no	—
16	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	C₄×M₁₆×C₇₃	—	2	2	no	—
19	C₄×M₁₆×C₇₃	C₄	1	4	yes	—
20	C₂×#97×C₇₃	—	2	1	no	—
21	C₂×C₄×C₈×C₇₃	—	2	1	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	4	no	—
24	C₂×#11924012	—	2	4	no	—
25	C₂×#11924012	C₂×C₄	1	4	yes	—
26	C₂²×#10596831	—	2	2	no	—
27	C₂²×#10596831	—	2	2	no	—
28	C₂²×#10596831	—	2	2	no	—
29	C₂²×#10596831	—	2	4	no	—
30	C₂²×#10596831	—	2	4	no	—
31	C₂²×#10596831	C₂×C₄	1	4	yes	—
32	C₄⋊#10596831	—	2	4	no	—
33	C₄⋊#10596831	—	2	4	no	—
34	C₄⋊#10596831	—	2	4	no	—
35	C₄⋊#10596831	—	2	4	no	—
36	C₂³×C₄×C₇₃	Q₈	1	1	yes	characteristic
37	C₂³×C₄×C₇₃	Q₈	1	1	yes	characteristic
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	2	no	—
42	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
43	C₂×M₁₆×C₇₃	D₄	1	4	yes	—
44	C₂×M₁₆×C₇₃	Q₈	1	4	yes	—
45	C₂²×C₈×C₇₃	—	2	2	no	—
46	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇₃	D₄	1	2	yes	—
51	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₇₃	D₄	1	2	yes	—
54	#97×C₇₃	—	2	4	no	—
55	C₄×C₈×C₇₃	—	2	4	no	—
56	#11924012	—	2	8	no	—
57	#11924012	—	4	2	no	—
58	#11924012	—	4	2	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	—	2	8	no	—
62	C₂×#10596831	—	4	4	no	—
63	C₂×#10596831	—	4	4	no	—
64	C₂×#10596831	—	4	2	no	—
65	C₂×#10596831	—	4	2	no	—
66	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
67	C₂²×C₄×C₇₃	—	2	4	no	—
68	C₂²×C₄×C₇₃	—	4	2	no	—
69	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
73	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
76	C₂²×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
77	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₇₃	#41	1	2	yes	—
86	C₂²×C₄×C₇₃	#41	1	2	yes	—
87	M₁₆×C₇₃	—	2	4	no	—
88	M₁₆×C₇₃	—	2	4	no	—
89	M₁₆×C₇₃	—	2	4	no	—
90	M₁₆×C₇₃	C₄⋊C₄	1	8	yes	—
91	C₂×C₈×C₇₃	—	2	4	no	—
92	C₂×C₈×C₇₃	—	2	4	no	—
93	C₂×C₈×C₇₃	—	2	4	no	—
94	C₄²×C₇₃	C₄²	1	2	yes	—
95	C₄²×C₇₃	C₄²	1	2	yes	—
96	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
97	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
98	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
99	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
100	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
101	C₄²×C₇₃	C₂²⋊C₄	1	2	yes	—
102	C₂³×C₇₃	—	2	2	no	—
103	C₂³×C₇₃	—	2	2	no	—
104	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂³×C₇₃	#125	1	2	yes	—
108	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
109	C₂×C₄×C₇₃	—	2	2	no	—
110	C₂×C₄×C₇₃	—	2	2	no	—
111	C₂×C₄×C₇₃	—	2	2	no	—
112	C₂×C₄×C₇₃	—	2	4	no	—
113	C₂×C₄×C₇₃	—	2	4	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	4	2	no	—
116	C₂×C₄×C₇₃	#95	1	2	yes	—
117	C₂×C₄×C₇₃	#95	1	2	yes	—
118	C₂×C₄×C₇₃	#95	1	2	yes	—
119	C₂×C₄×C₇₃	#95	1	2	yes	—
120	C₂×C₄×C₇₃	#108	1	4	yes	—
121	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₇₃	#117	1	2	yes	—
127	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
131	C₂×C₄×C₇₃	#121	1	2	yes	—
132	C₂×C₄×C₇₃	#121	1	2	yes	—
133	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
134	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
135	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
136	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
137	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
139	C₈×C₇₃	—	2	8	no	—
140	#10596831	—	4	4	no	—
141	#10596831	—	4	4	no	—
142	C₂²×C₇₃	—	2	4	no	—
143	C₂²×C₇₃	—	2	4	no	—
144	C₂²×C₇₃	—	2	1	no	—
145	C₂²×C₇₃	—	2	1	no	—
146	C₂²×C₇₃	—	2	1	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	C₂×#95	1	1	yes	characteristic
149	C₂²×C₇₃	#382	1	1	yes	characteristic
150	C₂²×C₇₃	#384	1	1	yes	characteristic
151	C₂²×C₇₃	#389	1	2	yes	—
152	C₂²×C₇₃	#389	1	2	yes	—
153	C₂²×C₇₃	C₂×#108	1	2	yes	—
154	C₂²×C₇₃	#430	1	2	yes	—
155	C₄×C₇₃	—	2	2	no	—
156	C₄×C₇₃	—	2	2	no	—
157	C₄×C₇₃	—	2	2	no	—
158	C₄×C₇₃	#341	1	4	yes	complemented
159	C₄×C₇₃	C₄×C₄⋊C₄	1	2	yes	—
160	C₄×C₇₃	#380	1	2	yes	—
161	C₄×C₇₃	#390	1	1	yes	characteristic
162	C₄×C₇₃	#390	1	1	yes	characteristic
163	C₄×C₇₃	#391	1	1	yes	characteristic
164	C₄×C₇₃	#391	1	1	yes	characteristic
165	#10853	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	2	1	no	—
167	C₂×C₇₃	—	2	1	no	—
168	C₂×C₇₃	—	2	2	no	—
169	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
170	C₂×C₇₃	C₂×#341	1	1	yes	characteristic
171	C₂×C₇₃	#1738	1	1	yes	characteristic
172	C₂×C₇₃	#1950	1	4	yes	—
173	C₂×#423	—	73	1	no	—
174	C₂²×C₄⋊C₈	—	73	1	no	—
175	C₂×C₄×M₁₆	D₇₃	1	1	yes	characteristic
176	C₂×#341	—	73	4	no	—
177	C₇₃	#10853	1	1	yes	characteristic, complemented, Sylow
178	C₂²×M₁₆	—	73	2	no	—
179	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
180	C₂³×C₈	—	73	2	no	—
181	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
182	#423	—	73	4	no	—
183	#423	—	146	2	no	—
184	C₂×C₄⋊C₈	—	73	2	no	—
185	C₂×C₄⋊C₈	—	73	2	no	—
186	C₂×C₄⋊C₈	—	146	2	no	—
187	C₂×C₄⋊C₈	—	146	2	no	—
188	C₂×C₄⋊C₈	—	146	1	no	—
189	C₂×C₄⋊C₈	—	146	1	no	—
190	C₄×M₁₆	—	2	2	no	—
191	C₄×M₁₆	Dic₇₃	1	4	yes	—
192	C₂×#97	—	2	1	no	—
193	C₂×C₄×C₈	—	2	1	no	—
194	#341	—	146	8	no	—
195	C₂³×C₄	Dic₁₄₆	1	1	yes	characteristic
196	C₂³×C₄	Dic₁₄₆	1	1	yes	characteristic
197	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂×M₁₆	—	73	4	no	—
202	C₂×M₁₆	—	146	4	no	—
203	C₂×M₁₆	—	146	4	no	—
204	C₂×M₁₆	—	146	2	no	—
205	C₂×M₁₆	Dic₁₄₆	1	4	yes	—
206	C₂×M₁₆	D₂₉₂	1	4	yes	—
207	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
208	C₂²×C₈	—	2	2	no	—
209	C₂²×C₈	—	73	4	no	—
210	C₂²×C₈	—	146	2	no	—
211	C₂²×C₈	—	146	2	no	—
212	C₂²×C₈	—	146	4	no	—
213	C₂²×C₈	—	146	2	no	—
214	C₂²×C₈	—	146	4	no	—
215	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
216	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
217	C₂×C₄²	C₄×D₇₃	1	2	yes	—
218	C₂×C₄²	D₂₉₂	1	2	yes	—
219	C₂×C₄²	C₂×Dic₇₃	1	2	yes	—
220	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
221	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
222	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
223	C₄⋊C₈	—	146	4	no	—
224	C₄⋊C₈	—	146	4	no	—
225	C₄⋊C₈	—	146	4	no	—
226	C₄⋊C₈	—	146	4	no	—
227	#97	—	2	4	no	—
228	C₄×C₈	—	2	4	no	—
229	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	#11924007	1	1	yes	characteristic
233	C₂²×C₄	#11924007	1	1	yes	characteristic
234	C₂²×C₄	#11924007	1	1	yes	characteristic
235	C₂²×C₄	#11924007	1	2	yes	—
236	C₂²×C₄	#11924007	1	2	yes	—
237	C₂²×C₄	#11924007	1	1	yes	characteristic
238	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
239	C₂²×C₄	C₄⋊Dic₇₃	1	2	yes	—
240	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
246	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
247	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
248	C₂²×C₄	#11924028	1	2	yes	—
249	C₂²×C₄	#11924028	1	2	yes	—
250	M₁₆	—	2	4	no	—
251	M₁₆	—	2	4	no	—
252	M₁₆	—	2	4	no	—
253	M₁₆	—	292	2	no	—
254	M₁₆	—	292	2	no	—
255	M₁₆	—	146	8	no	—
256	M₁₆	C₄⋊Dic₇₃	1	8	yes	—
257	C₂×C₈	—	2	4	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	292	4	no	—
261	C₂×C₈	—	292	4	no	—
262	C₂×C₈	—	292	2	no	—
263	C₂×C₈	—	292	2	no	—
264	C₂×C₈	—	146	4	no	—
265	C₂×C₈	—	146	4	no	—
266	C₂×C₈	—	146	8	no	—
267	C₄²	C₄×Dic₇₃	1	2	yes	—
268	C₄²	C₄×Dic₇₃	1	2	yes	—
269	C₄²	#11924005	1	2	yes	—
270	C₄²	#11924005	1	2	yes	—
271	C₄²	#11924005	1	2	yes	—
272	C₄²	#11924005	1	2	yes	—
273	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
274	C₄²	C₂²⋊Dic₇₃	1	2	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	2	no	—
277	C₂³	C₂×#11924007	1	1	yes	characteristic
278	C₂³	C₂×#11924007	1	1	yes	characteristic
279	C₂³	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
280	C₂³	#14556583	1	2	yes	—
281	C₂³	#14556591	1	2	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	#14556498	1	2	yes	—
290	C₂×C₄	#14556498	1	2	yes	—
291	C₂×C₄	#14556498	1	2	yes	—
292	C₂×C₄	#14556498	1	2	yes	—
293	C₂×C₄	#14556526	1	4	yes	—
294	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
295	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
296	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
297	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
298	C₂×C₄	C₂×C₄⋊Dic₇₃	1	2	yes	—
299	C₂×C₄	#14556550	1	2	yes	—
300	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
301	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
302	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
303	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
304	C₂×C₄	#14556565	1	2	yes	—
305	C₂×C₄	#14556565	1	2	yes	—
306	C₂×C₄	#14556570	1	2	yes	—
307	C₂×C₄	#14556570	1	2	yes	—
308	C₂×C₄	#14556579	1	1	yes	characteristic
309	C₂×C₄	#14556579	1	1	yes	characteristic
310	C₂×C₄	#14556588	1	1	yes	characteristic
311	C₂×C₄	#14556591	1	1	yes	characteristic
312	C₈	—	2	8	no	—
313	C₈	—	292	4	no	—
314	C₈	—	292	4	no	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	C₂×#14556498	1	1	yes	characteristic
322	C₂²	#17988453	1	1	yes	characteristic
323	C₂²	#17988457	1	1	yes	characteristic
324	C₂²	#17988485	1	2	yes	—
325	C₂²	#17988485	1	2	yes	—
326	C₂²	C₂×#14556526	1	2	yes	—
327	C₂²	#17988635	1	2	yes	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	#17988351	1	4	yes	complemented
332	C₄	C₄×C₄⋊Dic₇₃	1	2	yes	—
333	C₄	#17988447	1	2	yes	—
334	C₄	#17988489	1	1	yes	characteristic
335	C₄	#17988489	1	1	yes	characteristic
336	C₄	#17988493	1	1	yes	characteristic
337	C₄	#17988493	1	1	yes	characteristic
338	C₂	—	2	1	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	2	no	—
341	C₂	#22608161	1	1	yes	characteristic
342	C₂	C₂×#17988351	1	1	yes	characteristic
343	C₂	#22609657	1	1	yes	characteristic
344	C₂	#22610706	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₂×C₄×M₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	C₂×C₄×M₁₆	D₇₃
hypocenter	C₇₃	#10853
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
158. C₄×C₇₃	#341	semidirect
177. C₇₃	#10853	semidirect
331. C₄	#17988351	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988611	2
C₂²×C₄⋊#10596831	2
C₂×#17988351	2
C₂×C₄×M₁₆×C₇₃	2
#10853	73