Subgroups — #34018025

Order 18688.

712 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22626608	C₂	1	1	yes	maximal, characteristic
2	C₂×#17989609	C₂	1	1	yes	maximal, characteristic
3	#22625496	C₂	1	4	yes	maximal, complemented
4	#22625492	C₂	1	4	yes	maximal, complemented
5	#22625482	C₂	1	4	yes	maximal, complemented
6	C₄×M₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic
7	#17989609	—	2	1	no	—
8	#17989609	—	2	1	no	—
9	#17989609	C₂²	1	2	yes	—
10	#17989609	C₂²	1	2	yes	—
11	#17989609	C₂²	1	2	yes	—
12	#17989608	—	2	1	no	—
13	#17989606	—	2	1	no	—
14	#17989603	C₂²	1	2	yes	—
15	#17989591	C₂²	1	2	yes	—
16	C₂×C₄⋊#10596833	C₂²	1	2	yes	—
17	#17989558	—	2	1	no	—
18	C₂×#14556648	—	2	1	no	—
19	C₄×#11924038	—	2	1	no	—
20	C₂×#14556646	—	2	1	no	—
21	#17989450	C₂²	1	8	yes	complemented
22	#17989448	C₂²	1	8	yes	complemented
23	C₂×M₁₆×D₇₃	C₄	1	2	yes	—
24	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
25	C₄×#11924012	C₄	1	1	yes	characteristic
26	#17988539	C₂²	1	2	yes	—
27	M₁₆×Dic₇₃	C₂²	1	2	yes	—
28	C₄×#11924011	C₄	1	2	yes	—
29	#97×D₇₃	C₄	1	2	yes	—
30	C₄×C₈×D₇₃	C₂²	1	2	yes	—
31	C₄×M₁₆×C₇₃	C₄	1	1	yes	characteristic
32	C₂×#11924038	—	2	2	no	—
33	C₂²×#10596833	—	2	2	no	—
34	C₄⋊#10596833	—	2	2	no	—
35	C₄⋊#10596833	—	2	2	no	—
36	C₄⋊#10596833	C₂³	1	4	yes	—
37	C₄⋊#10596833	C₂³	1	4	yes	—
38	#14556648	—	2	2	no	—
39	#14556648	—	2	2	no	—
40	#14556648	—	2	2	no	—
41	C₂²⋊#10596833	—	2	2	no	—
42	#14556646	—	2	2	no	—
43	#14556646	—	2	2	no	—
44	#14556646	—	2	2	no	—
45	C₄×#10596833	—	2	2	no	—
46	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
48	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
49	C₂×#11924012	C₂×C₄	1	2	yes	—
50	C₂×#11924011	C₂×C₄	1	4	yes	—
51	M₁₆×D₇₃	—	2	8	no	—
52	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
53	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
54	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
55	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
56	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
58	C₄²×D₇₃	D₄	1	1	yes	characteristic
59	C₄²×D₇₃	D₄	1	1	yes	characteristic
60	C₄²×D₇₃	Q₈	1	1	yes	characteristic
61	C₄²×D₇₃	Q₈	1	1	yes	characteristic
62	C₄²×D₇₃	C₂³	1	1	yes	characteristic
63	C₄²×D₇₃	C₂³	1	1	yes	characteristic
64	#14556502	C₂×C₄	1	2	yes	—
65	#14556501	C₂×C₄	1	4	yes	—
66	C₄×#10596831	C₂×C₄	1	2	yes	—
67	C₈×Dic₇₃	C₂³	1	4	yes	—
68	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
69	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
70	#97×C₇₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
72	#11924038	—	4	4	no	—
73	C₂×#10596833	—	2	4	no	—
74	C₂×#10596833	—	2	4	no	—
75	C₂×#10596833	—	4	4	no	—
76	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₂×C₄×D₇₃	—	2	4	no	—
81	C₂×C₄×D₇₃	—	4	2	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
88	C₂×C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
89	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
94	C₂×C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
95	#11924012	—	2	4	no	—
96	#11924011	—	2	8	no	—
97	C₂×#10596831	C₂²×C₄	1	4	yes	—
98	C₈×D₇₃	—	2	8	no	—
99	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
102	C₄×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
103	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
104	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
105	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
106	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
107	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
108	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
109	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
110	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
111	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
114	M₁₆×C₇₃	M₁₆	1	8	yes	complemented
115	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
116	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
117	C₄²×C₇₃	C₄⋊C₄	1	1	yes	characteristic
118	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
120	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×D₇₃	—	2	2	no	—
122	C₂²×D₇₃	—	2	2	no	—
123	C₂²×D₇₃	—	4	1	no	—
124	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂²×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	—	4	1	no	—
127	C₂×C₄×C₇₃	—	4	1	no	—
128	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×M₁₆	1	4	yes	—
134	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×Dic₇₃	—	2	2	no	—
136	C₂×Dic₇₃	—	2	2	no	—
137	C₂×Dic₇₃	—	4	1	no	—
138	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
140	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
141	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
142	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
144	C₄×D₇₃	—	2	2	no	—
145	C₄×D₇₃	—	2	2	no	—
146	C₄×D₇₃	—	2	2	no	—
147	C₄×D₇₃	—	2	2	no	—
148	C₄×D₇₃	—	2	2	no	—
149	C₄×D₇₃	—	2	2	no	—
150	C₄×D₇₃	—	2	2	no	—
151	C₄×D₇₃	—	2	2	no	—
152	C₄×D₇₃	—	4	1	no	—
153	C₄×D₇₃	—	4	1	no	—
154	C₄×D₇₃	—	4	1	no	—
155	C₄×D₇₃	—	4	1	no	—
156	#10596833	—	4	4	no	—
157	#10596833	—	4	4	no	—
158	C₈×C₇₃	C₂×M₁₆	1	8	yes	—
159	#10596831	—	2	4	no	—
160	C₂²×C₇₃	—	2	1	no	—
161	C₂²×C₇₃	—	2	1	no	—
162	C₂²×C₇₃	#423	1	2	yes	—
163	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
164	C₂×D₇₃	—	2	1	no	—
165	C₂×D₇₃	—	2	1	no	—
166	C₂×D₇₃	—	2	1	no	—
167	C₂×D₇₃	—	2	1	no	—
168	C₂×D₇₃	—	2	1	no	—
169	C₂×D₇₃	—	4	4	no	—
170	C₂×D₇₃	#430	1	2	yes	—
171	C₄×C₇₃	—	2	1	no	—
172	C₄×C₇₃	—	2	1	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	#423	1	2	yes	—
176	C₄×C₇₃	C₂²×M₁₆	1	2	yes	—
177	Dic₇₃	—	2	1	no	—
178	Dic₇₃	—	2	1	no	—
179	Dic₇₃	—	2	1	no	—
180	Dic₇₃	—	2	1	no	—
181	Dic₇₃	—	2	1	no	—
182	Dic₇₃	#575	1	2	yes	—
183	#17797	—	73	1	no	maximal, Sylow
184	C₂×C₇₃	—	2	2	no	—
185	C₂×C₇₃	C₂×#423	1	1	yes	characteristic
186	C₂×C₇₃	#2911	1	1	yes	characteristic
187	C₂×C₇₃	C₂×#430	1	1	yes	characteristic
188	D₇₃	—	2	2	no	—
189	D₇₃	—	4	1	no	—
190	#2920	—	73	1	no	—
191	C₂×#432	—	73	1	no	—
192	C₂×C₄×M₁₆	—	73	1	no	—
193	#1569	—	73	4	no	—
194	#1568	—	73	4	no	—
195	#1565	—	73	4	no	—
196	C₇₃	#17797	1	1	yes	characteristic, complemented, Sylow
197	C₂²×M₁₆	—	73	2	no	—
198	C₂²×C₄²	—	73	1	no	—
199	#432	—	73	2	no	—
200	#432	—	73	2	no	—
201	#432	—	73	2	no	—
202	#432	—	146	1	no	—
203	#432	—	146	1	no	—
204	#432	—	146	1	no	—
205	#431	—	73	2	no	—
206	#431	—	146	1	no	—
207	#423	—	73	2	no	—
208	C₂×C₄⋊C₈	—	73	2	no	—
209	#407	—	146	1	no	—
210	C₂×C₂²⋊C₈	—	146	1	no	—
211	C₄×M₁₆	—	73	2	no	—
212	C₄×M₁₆	—	73	1	no	—
213	C₄×M₁₆	—	73	2	no	—
214	C₄×M₁₆	—	73	2	no	—
215	C₄×M₁₆	—	146	1	no	—
216	C₄×M₁₆	#64408	1	1	yes	characteristic
217	C₂×#97	—	73	2	no	—
218	C₂×#97	—	146	1	no	—
219	C₂×C₄×C₈	—	73	2	no	—
220	#335	—	73	8	no	—
221	#334	—	73	8	no	—
222	C₂³×C₄	—	73	1	no	—
223	C₂³×C₄	—	73	1	no	—
224	C₂³×C₄	—	73	1	no	—
225	C₂×M₁₆	—	73	2	no	—
226	C₂×M₁₆	—	73	4	no	—
227	C₂×M₁₆	—	146	2	no	—
228	C₂×M₁₆	—	146	8	no	—
229	C₂×M₁₆	C₂×#64408	1	2	yes	—
230	C₂²×C₈	—	73	4	no	—
231	C₂²×C₈	—	146	2	no	—
232	C₂×C₄²	—	73	1	no	—
233	C₂×C₄²	—	73	1	no	—
234	C₂×C₄²	—	73	1	no	—
235	C₂×C₄²	—	73	1	no	—
236	C₂×C₄²	—	73	1	no	—
237	C₂×C₄²	—	73	1	no	—
238	C₂×C₄²	—	73	1	no	—
239	C₂×C₄²	—	73	1	no	—
240	C₂×C₄²	—	73	1	no	—
241	C₂×C₄²	—	73	1	no	—
242	C₂×C₄²	—	73	1	no	—
243	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
244	C₄⋊C₈	—	73	4	no	—
245	C₄⋊C₈	—	73	4	no	—
246	C₄⋊C₈	—	146	2	no	—
247	C₄⋊C₈	—	146	2	no	—
248	C₂²⋊C₈	—	146	2	no	—
249	C₂²⋊C₈	—	146	2	no	—
250	C₂²⋊C₈	—	146	2	no	—
251	C₂²⋊C₈	—	146	2	no	—
252	#97	—	73	4	no	—
253	#97	—	73	2	no	—
254	#97	—	146	2	no	—
255	#97	—	146	2	no	—
256	#97	—	146	2	no	—
257	#97	C₂×#64408	1	2	yes	—
258	C₄×C₈	—	73	4	no	—
259	C₄×C₈	—	73	2	no	—
260	C₄×C₈	—	146	2	no	—
261	C₄×C₈	C₂×#64408	1	2	yes	—
262	C₂⁴	—	73	1	no	—
263	C₂²×C₄	—	292	2	no	—
264	C₂²×C₄	—	292	2	no	—
265	C₂²×C₄	—	73	1	no	—
266	C₂²×C₄	—	73	1	no	—
267	C₂²×C₄	—	73	1	no	—
268	C₂²×C₄	—	73	1	no	—
269	C₂²×C₄	—	73	1	no	—
270	C₂²×C₄	—	73	1	no	—
271	C₂²×C₄	—	73	1	no	—
272	C₂²×C₄	—	73	1	no	—
273	C₂²×C₄	—	73	1	no	—
274	C₂²×C₄	—	73	1	no	—
275	C₂²×C₄	—	73	1	no	—
276	C₂²×C₄	—	73	1	no	—
277	C₂²×C₄	—	73	1	no	—
278	C₂²×C₄	—	73	1	no	—
279	C₂²×C₄	—	73	1	no	—
280	C₂²×C₄	—	146	4	no	—
281	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
282	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
283	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
284	M₁₆	—	292	4	no	—
285	M₁₆	—	146	4	no	—
286	M₁₆	—	146	8	no	—
287	M₁₆	#11924038	1	8	yes	complemented
288	C₂×C₈	—	292	4	no	—
289	C₂×C₈	—	73	4	no	—
290	C₂×C₈	—	146	4	no	—
291	C₂×C₈	—	146	4	no	—
292	C₂×C₈	—	146	8	no	—
293	C₂×C₈	C₂²×#64408	1	4	yes	—
294	C₄²	—	73	1	no	—
295	C₄²	—	73	1	no	—
296	C₄²	—	73	1	no	—
297	C₄²	—	73	1	no	—
298	C₄²	—	73	1	no	—
299	C₄²	—	73	1	no	—
300	C₄²	—	73	1	no	—
301	C₄²	—	73	1	no	—
302	C₄²	—	73	1	no	—
303	C₄²	—	73	1	no	—
304	C₄²	—	73	1	no	—
305	C₄²	—	73	1	no	—
306	C₄²	C₄⋊#64408	1	1	yes	characteristic
307	C₄²	C₄⋊#64408	1	1	yes	characteristic
308	C₄²	C₂²×#64408	1	1	yes	characteristic
309	C₄²	C₂²×#64408	1	1	yes	characteristic
310	C₂³	—	292	1	no	—
311	C₂³	—	73	1	no	—
312	C₂³	—	73	1	no	—
313	C₂³	—	146	2	no	—
314	C₂³	—	146	2	no	—
315	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
316	C₂×C₄	—	4	1	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	292	1	no	—
319	C₂×C₄	—	292	1	no	—
320	C₂×C₄	—	292	1	no	—
321	C₂×C₄	—	292	1	no	—
322	C₂×C₄	—	292	1	no	—
323	C₂×C₄	—	73	1	no	—
324	C₂×C₄	—	73	1	no	—
325	C₂×C₄	—	73	1	no	—
326	C₂×C₄	—	73	1	no	—
327	C₂×C₄	—	73	1	no	—
328	C₂×C₄	—	73	1	no	—
329	C₂×C₄	—	146	2	no	—
330	C₂×C₄	—	146	2	no	—
331	C₂×C₄	—	146	2	no	—
332	C₂×C₄	—	146	2	no	—
333	C₂×C₄	—	146	2	no	—
334	C₂×C₄	—	146	2	no	—
335	C₂×C₄	—	146	2	no	—
336	C₂×C₄	—	146	2	no	—
337	C₂×C₄	—	146	2	no	—
338	C₂×C₄	—	146	2	no	—
339	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
340	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
341	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
342	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
343	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
344	C₂×C₄	C₂×#11924038	1	4	yes	—
345	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
346	C₈	—	292	4	no	—
347	C₈	—	292	4	no	—
348	C₈	—	146	4	no	—
349	C₈	C₂×#11924038	1	8	yes	—
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	292	4	no	—
353	C₂²	—	73	2	no	—
354	C₂²	—	146	1	no	—
355	C₂²	—	146	1	no	—
356	C₂²	—	146	1	no	—
357	C₂²	—	146	1	no	—
358	C₂²	—	146	1	no	—
359	C₂²	#17989591	1	2	yes	—
360	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	2	1	no	—
365	C₄	—	73	2	no	—
366	C₄	—	146	1	no	—
367	C₄	—	146	1	no	—
368	C₄	—	146	1	no	—
369	C₄	—	146	1	no	—
370	C₄	—	146	1	no	—
371	C₄	#17989591	1	2	yes	—
372	C₄	C₂²×#11924038	1	2	yes	—
373	C₂	—	2	2	no	—
374	C₂	—	292	1	no	—
375	C₂	—	146	2	no	—
376	C₂	C₂×#17989591	1	1	yes	characteristic
377	C₂	#22626584	1	1	yes	characteristic
378	C₂	D₇₃⋊#430	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₄×M₁₆×C₇₃	C₄
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₄×M₁₆	#64408
hypocenter	C₇₃	#17797
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

Complements

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

N	H ≅ G/N	type
3. #22625496	C₂	semidirect
4. #22625492	C₂	semidirect
5. #22625482	C₂	semidirect
21. #17989450	C₂²	semidirect
22. #17989448	C₂²	semidirect
114. M₁₆×C₇₃	M₁₆	semidirect
196. C₇₃	#17797	semidirect
287. M₁₆	#11924038	semidirect

Maximal subgroups

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

subgroup	index
#22626608	2
C₂×#17989609	2
#22625496	2
#22625492	2
#22625482	2
C₄×M₁₆×D₇₃	2
#17797	73