Subgroups — #33083942

Order 18688.

1110 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×#17988614	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988542	C₂	1	1	yes	maximal, characteristic
3	C₂²×#14556501	C₂	1	1	yes	maximal, characteristic
4	#22612415	C₂	1	1	yes	maximal, characteristic
5	#22609803	C₂	1	4	yes	maximal, complemented
6	#22609798	C₂	1	4	yes	maximal, complemented
7	#22609629	C₂	1	1	yes	maximal, characteristic
8	#22609624	C₂	1	1	yes	maximal, characteristic
9	C₂×#405×C₇₃	C₂	1	1	yes	maximal, characteristic
10	C₂²×#11924012	C₂²	1	1	yes	characteristic
11	C₂³×#10596831	C₂²	1	1	yes	characteristic
12	C₂×#14556550	C₂²	1	1	yes	characteristic
13	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
14	#17988614	C₄	1	8	yes	complemented
15	C₂×C₄⋊#10596831	C₂²	1	4	yes	—
16	#17988542	—	2	4	no	—
17	C₂×#14556501	—	2	4	no	—
18	C₂×#14556501	C₂²	1	2	yes	—
19	C₂×#14556501	C₂²	1	2	yes	—
20	C₂×#14556501	C₂²	1	2	yes	—
21	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
22	#17988486	C₄	1	4	yes	—
23	#17988462	C₄	1	4	yes	—
24	C₄×C₄⋊Dic₇₃	C₂²	1	4	yes	—
25	#17988341	C₂²	1	4	yes	—
26	#17988341	C₂²	1	4	yes	—
27	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
28	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×#117×C₇₃	C₂²	1	1	yes	characteristic
30	#405×C₇₃	C₄	1	8	yes	complemented
31	C₂×#97×C₇₃	C₂²	1	2	yes	—
32	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
33	C₂³×Dic₇₃	C₂×C₄	1	2	yes	—
34	C₂×#11924012	—	2	2	no	—
35	C₂×#11924012	—	2	2	no	—
36	C₂×#11924012	C₂×C₄	1	4	yes	—
37	C₂²×#10596831	—	2	2	no	—
38	C₂²×#10596831	—	2	2	no	—
39	C₂²×#10596831	C₂×C₄	1	4	yes	—
40	C₂²×#10596831	C₂³	1	2	yes	—
41	C₂²×#10596831	C₂³	1	2	yes	—
42	#14556550	—	2	2	no	—
43	#14556550	—	2	2	no	—
44	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊Dic₇₃	C₂×C₄	1	8	yes	—
46	C₂×C₂²⋊Dic₇₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₇₃	—	2	2	no	—
48	C₂×C₄×Dic₇₃	—	2	2	no	—
49	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
50	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
51	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
52	C₄⋊#10596831	C₂×C₄	1	16	yes	complemented
53	#14556501	—	2	4	no	—
54	#14556501	—	2	4	no	—
55	#14556501	—	2	4	no	—
56	C₈×Dic₇₃	—	2	4	no	—
57	#14556498	C₂×C₄	1	8	yes	—
58	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
59	C₂×M₁₆×C₇₃	—	2	2	no	—
60	C₂×M₁₆×C₇₃	—	2	2	no	—
61	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
62	C₂²×C₈×C₇₃	C₂×C₄	1	4	yes	—
63	C₂²×C₈×C₇₃	D₄	1	4	yes	—
64	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
65	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	#117×C₇₃	C₂×C₄	1	4	yes	—
69	#117×C₇₃	C₂×C₄	1	4	yes	—
70	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
71	C₂×C₂²⋊C₄×C₇₃	C₂×C₄	1	2	yes	—
72	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
73	#97×C₇₃	C₂×C₄	1	8	yes	complemented
74	C₄×C₈×C₇₃	C₂×C₄	1	8	yes	complemented
75	C₂²×Dic₇₃	—	2	4	no	—
76	C₂²×Dic₇₃	—	4	4	no	—
77	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
78	C₂²×Dic₇₃	C₂²×C₄	1	4	yes	—
79	#11924012	—	2	8	no	—
80	C₂×#10596831	—	2	8	no	—
81	C₂×#10596831	—	2	2	no	—
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	—	2	2	no	—
84	C₂×#10596831	—	2	2	no	—
85	C₂×#10596831	C₂²×C₄	1	4	yes	—
86	C₂×#10596831	C₂²×C₄	1	4	yes	—
87	C₄⋊Dic₇₃	—	2	4	no	—
88	C₄⋊Dic₇₃	—	4	8	no	—
89	C₂²⋊Dic₇₃	—	2	4	no	—
90	C₄×Dic₇₃	—	2	4	no	—
91	C₄×Dic₇₃	—	2	2	no	—
92	C₄×Dic₇₃	—	2	2	no	—
93	C₄×Dic₇₃	—	2	2	no	—
94	C₄×Dic₇₃	—	2	2	no	—
95	C₂⁴×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂²×C₄	1	4	yes	—
97	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
98	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
99	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
100	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
102	C₂²×C₄×C₇₃	#41	1	2	yes	—
103	C₂²×C₄×C₇₃	#41	1	2	yes	—
104	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
105	M₁₆×C₇₃	—	2	8	no	—
106	C₂×C₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	—	2	2	no	—
108	C₂×C₈×C₇₃	C₄⋊C₄	1	16	yes	complemented
109	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
110	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
112	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
113	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
114	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
115	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
116	C₄⋊C₄×C₇₃	C₄²	1	8	yes	—
117	C₂²⋊C₄×C₇₃	C₄²	1	8	yes	—
118	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
119	C₄²×C₇₃	C₂²×C₄	1	4	yes	—
120	C₂³×C₇₃	—	2	4	no	—
121	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
122	C₂³×C₇₃	C₄×D₄	1	2	yes	—
123	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
124	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	—	2	8	no	—
126	C₂×C₄×C₇₃	C₂×C₄²	1	8	yes	—
127	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
128	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×C₄×C₇₃	#117	1	4	yes	—
130	C₂×C₄×C₇₃	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₇₃	C₄×Q₈	1	4	yes	—
133	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
134	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
140	C₂×Dic₇₃	—	2	4	no	—
141	C₂×Dic₇₃	—	2	4	no	—
142	C₂×Dic₇₃	—	2	4	no	—
143	C₂×Dic₇₃	—	2	8	no	—
144	C₂×Dic₇₃	—	2	4	no	—
145	C₂×Dic₇₃	—	4	8	no	—
146	C₈×C₇₃	—	2	4	no	—
147	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
148	C₈×C₇₃	C₂×C₄⋊C₄	1	4	yes	—
149	#10596831	—	2	4	no	—
150	#10596831	—	2	4	no	—
151	C₂²×C₇₃	—	2	4	no	—
152	C₂²×C₇₃	—	2	4	no	—
153	C₂²×C₇₃	—	2	4	no	—
154	C₂²×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
155	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
156	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
157	C₂²×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₇₃	#568	1	4	yes	—
159	C₄×C₇₃	—	2	8	no	—
160	C₄×C₇₃	C₄×C₄⋊C₄	1	4	yes	—
161	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
162	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
163	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₄×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
165	Dic₇₃	—	4	4	no	—
166	Dic₇₃	—	4	4	no	—
167	#20436	—	73	1	no	maximal, Sylow
168	C₂×C₇₃	—	2	4	no	—
169	C₂×C₇₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
170	C₂×C₇₃	#2875	1	2	yes	—
171	C₂×C₇₃	#2930	1	2	yes	—
172	C₂×C₇₃	#2961	1	2	yes	—
173	C₂×#424	—	73	1	no	—
174	C₂×#405	—	73	1	no	—
175	C₂×#405	D₇₃	1	1	yes	characteristic
176	C₂²×#97	—	73	1	no	—
177	#2296	—	73	1	no	—
178	#1771	—	73	4	no	—
179	#1770	—	73	4	no	—
180	#1729	—	73	1	no	—
181	#1728	—	73	1	no	—
182	C₇₃	#20436	1	1	yes	characteristic, complemented, Sylow
183	C₂²×M₁₆	—	73	1	no	—
184	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
185	C₂³×C₈	—	73	1	no	—
186	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
187	C₂×#117	—	73	1	no	—
188	C₂×#117	C₂×D₇₃	1	1	yes	characteristic
189	C₂²×C₄²	—	73	1	no	—
190	#424	—	73	8	no	—
191	C₂×C₄⋊C₈	—	73	4	no	—
192	#405	—	146	4	no	—
193	#405	Dic₇₃	1	8	yes	complemented
194	C₂×#97	—	73	2	no	—
195	C₂×#97	—	73	2	no	—
196	C₂×#97	—	73	2	no	—
197	C₂×#97	—	146	4	no	—
198	C₂×#97	C₂×D₇₃	1	2	yes	—
199	C₂×C₄×C₈	—	73	2	no	—
200	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
201	#389	—	73	4	no	—
202	#385	—	73	4	no	—
203	C₄×C₄⋊C₄	—	73	4	no	—
204	#336	—	73	4	no	—
205	#336	—	73	4	no	—
206	C₂³×C₄	—	73	2	no	—
207	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	—	2	2	no	—
210	C₂×M₁₆	—	73	4	no	—
211	C₂×M₁₆	—	146	2	no	—
212	C₂×M₁₆	—	146	2	no	—
213	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
214	C₂²×C₈	—	73	4	no	—
215	C₂²×C₈	—	73	2	no	—
216	C₂²×C₈	—	73	2	no	—
217	C₂²×C₈	—	146	2	no	—
218	C₂²×C₈	—	146	2	no	—
219	C₂²×C₈	Dic₁₄₆	1	4	yes	—
220	C₂²×C₈	D₂₉₂	1	4	yes	—
221	C₂²×C₈	C₂×Dic₇₃	1	4	yes	—
222	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
223	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
224	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
225	#117	—	146	2	no	—
226	#117	—	146	2	no	—
227	#117	C₄×D₇₃	1	4	yes	—
228	#117	C₂×Dic₇₃	1	4	yes	—
229	C₂×C₄⋊C₄	—	73	8	no	—
230	C₂×C₄⋊C₄	—	73	2	no	—
231	C₂×C₄⋊C₄	C₄×D₇₃	1	2	yes	—
232	C₂×C₂²⋊C₄	—	73	2	no	—
233	C₂×C₂²⋊C₄	C₄×D₇₃	1	2	yes	—
234	C₂×C₄²	—	73	2	no	—
235	C₂×C₄²	—	73	2	no	—
236	C₂×C₄²	—	73	2	no	—
237	C₂×C₄²	—	146	2	no	—
238	C₂×C₄²	—	146	2	no	—
239	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
240	C₄⋊C₈	—	73	16	no	—
241	#97	—	146	4	no	—
242	#97	—	146	4	no	—
243	#97	—	146	4	no	—
244	#97	C₂×Dic₇₃	1	8	yes	complemented
245	C₄×C₈	—	146	4	no	—
246	C₄×C₈	C₂×Dic₇₃	1	8	yes	complemented
247	#95	—	73	8	no	—
248	C₂⁴	C₂×C₄×D₇₃	1	1	yes	characteristic
249	C₂²×C₄	—	292	4	no	—
250	C₂²×C₄	—	73	4	no	—
251	C₂²×C₄	—	73	4	no	—
252	C₂²×C₄	—	146	4	no	—
253	C₂²×C₄	C₂×C₄×D₇₃	1	4	yes	—
254	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
256	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
257	C₂²×C₄	C₂×D₂₉₂	1	2	yes	—
258	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
259	C₂²×C₄	#11924028	1	2	yes	—
260	C₂²×C₄	#11924028	1	2	yes	—
261	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
262	M₁₆	—	2	8	no	—
263	M₁₆	—	146	8	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	73	4	no	—
267	C₂×C₈	—	73	4	no	—
268	C₂×C₈	—	146	8	no	—
269	C₂×C₈	—	146	2	no	—
270	C₂×C₈	—	146	2	no	—
271	C₂×C₈	—	146	2	no	—
272	C₂×C₈	—	146	2	no	—
273	C₂×C₈	C₄⋊Dic₇₃	1	16	yes	complemented
274	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
275	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
276	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
277	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
278	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
279	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
280	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
281	C₄⋊C₄	—	292	8	no	—
282	C₄⋊C₄	—	146	4	no	—
283	C₄⋊C₄	C₄×Dic₇₃	1	8	yes	—
284	C₂²⋊C₄	—	146	4	no	—
285	C₂²⋊C₄	C₄×Dic₇₃	1	8	yes	—
286	C₄²	—	146	4	no	—
287	C₄²	—	146	2	no	—
288	C₄²	—	146	2	no	—
289	C₄²	—	146	2	no	—
290	C₄²	—	146	2	no	—
291	C₄²	C₂×C₄×D₇₃	1	4	yes	—
292	C₄²	C₂²×Dic₇₃	1	4	yes	—
293	C₂³	—	2	4	no	—
294	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
295	C₂³	C₄×D₂₉₂	1	2	yes	—
296	C₂³	C₄×Dic₁₄₆	1	2	yes	—
297	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
298	C₂×C₄	—	2	8	no	—
299	C₂×C₄	—	292	8	no	—
300	C₂×C₄	—	146	4	no	—
301	C₂×C₄	—	146	4	no	—
302	C₂×C₄	—	146	4	no	—
303	C₂×C₄	—	146	8	no	—
304	C₂×C₄	—	146	4	no	—
305	C₂×C₄	C₂×C₄×Dic₇₃	1	8	yes	—
306	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
307	C₂×C₄	C₂×C₄⋊Dic₇₃	1	4	yes	—
308	C₂×C₄	#14556550	1	4	yes	—
309	C₂×C₄	C₄×D₂₉₂	1	2	yes	—
310	C₂×C₄	C₄×D₂₉₂	1	4	yes	—
311	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
312	C₂×C₄	C₄×Dic₁₄₆	1	4	yes	—
313	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
314	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
315	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
316	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
317	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
318	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
319	C₈	—	2	4	no	—
320	C₈	—	146	4	no	—
321	C₈	—	146	4	no	—
322	C₈	C₂×C₄⋊Dic₇₃	1	4	yes	—
323	C₈	C₂×C₄⋊Dic₇₃	1	4	yes	—
324	C₂²	—	2	4	no	—
325	C₂²	—	2	4	no	—
326	C₂²	—	2	4	no	—
327	C₂²	C₄×C₄⋊Dic₇₃	1	4	yes	—
328	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
329	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
330	C₂²	C₂×C₄×Dic₁₄₆	1	1	yes	characteristic
331	C₂²	#17989352	1	4	yes	—
332	C₄	—	2	8	no	—
333	C₄	—	292	4	no	—
334	C₄	—	292	4	no	—
335	C₄	C₄×C₄⋊Dic₇₃	1	4	yes	—
336	C₄	C₂²×C₄⋊Dic₇₃	1	1	yes	characteristic
337	C₄	C₂×#14556550	1	1	yes	characteristic
338	C₄	C₂×C₄×D₂₉₂	1	1	yes	characteristic
339	C₄	C₂×C₄×Dic₁₄₆	1	1	yes	characteristic
340	C₂	—	2	4	no	—
341	C₂	C₂×C₄×C₄⋊Dic₇₃	1	1	yes	characteristic
342	C₂	#22618357	1	2	yes	—
343	C₂	#22618636	1	2	yes	—
344	C₂	#22618869	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂²×C₄²
Fitting	C₂×#405×C₇₃	C₂
socle	C₂³×C₇₃	C₂³×C₄
prime radical	C₂⁴×C₇₃	C₂²×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×#405	D₇₃
hypocenter	C₇₃	#20436
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
5. #22609803	C₂	semidirect
6. #22609798	C₂	semidirect
14. #17988614	C₄	semidirect
30. #405×C₇₃	C₄	semidirect
52. C₄⋊#10596831	C₂×C₄	semidirect
73. #97×C₇₃	C₂×C₄	semidirect
74. C₄×C₈×C₇₃	C₂×C₄	semidirect
108. C₂×C₈×C₇₃	C₄⋊C₄	semidirect
182. C₇₃	#20436	semidirect
193. #405	Dic₇₃	semidirect
244. #97	C₂×Dic₇₃	semidirect
246. C₄×C₈	C₂×Dic₇₃	semidirect
273. 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×#17988614	2
C₂×#17988542	2
C₂²×#14556501	2
#22612415	2
#22609803	2
#22609798	2
#22609629	2
#22609624	2
C₂×#405×C₇₃	2
#20436	73