Subgroups — #33104889

Order 18688.

876 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×Dic₁₄₆	C₂	1	2	yes	maximal
2	C₄²×Dic₁₄₆	C₂	1	1	yes	maximal, characteristic
3	#22610649	C₂	1	2	yes	maximal
4	#22610646	C₂	1	4	yes	maximal
5	#22610048	C₂	1	1	yes	maximal, characteristic
6	#22610046	C₂	1	2	yes	maximal
7	#22609779	C₂	1	2	yes	maximal
8	C₄×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
10	C₂×C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
11	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
12	C₈×Dic₁₄₆	—	2	8	no	—
13	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
14	C₂×#14556520	C₂²	1	4	yes	—
15	C₂×#14556520	C₂²	1	4	yes	—
16	C₂×C₈×Dic₇₃	C₂²	1	4	yes	—
17	C₄×#11924007	C₄	1	2	yes	—
18	C₄×#11924007	C₄	1	2	yes	—
19	C₄×#11924007	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
21	C₄×C₄⋊Dic₇₃	C₄	1	1	yes	characteristic
22	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
23	C₄²×Dic₇₃	C₂²	1	2	yes	—
24	#17988342	C₂²	1	2	yes	—
25	#17988342	C₂²	1	2	yes	—
26	#17988341	C₂²	1	4	yes	—
27	C₂×C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	2	yes	—
29	C₄³×C₇₃	C₂²	1	1	yes	characteristic
30	#336×C₇₃	C₂²	1	1	yes	characteristic
31	#336×C₇₃	C₂²	1	1	yes	characteristic
32	C₂²×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
33	C₂²×#10596831	C₂³	1	4	yes	—
34	C₄×Dic₁₄₆	—	2	4	no	—
35	C₄×Dic₁₄₆	C₈	1	8	yes	complemented
36	C₄×Dic₁₄₆	D₄	1	4	yes	—
37	C₄×Dic₁₄₆	Q₈	1	4	yes	—
38	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂×#11924007	C₂×C₄	1	2	yes	—
42	C₂×#11924007	C₂×C₄	1	2	yes	—
43	C₂×#11924007	C₂×C₄	1	2	yes	—
44	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
45	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
47	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
49	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
50	C₄⋊#10596831	—	2	4	no	—
51	#14556520	—	2	8	no	—
52	#14556520	—	2	8	no	—
53	C₈×Dic₇₃	—	2	8	no	—
54	#14556498	C₂×C₄	1	2	yes	—
55	#14556498	C₂×C₄	1	2	yes	—
56	#14556498	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
58	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
65	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₇₃	—	2	4	no	—
67	C₄×C₈×C₇₃	—	2	4	no	—
68	C₂×Dic₁₄₆	C₄⋊C₄	1	4	yes	—
69	C₂×Dic₁₄₆	C₂×C₈	1	4	yes	—
70	C₂×Dic₁₄₆	M₁₆	1	4	yes	—
71	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
72	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
73	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
74	C₂²×Dic₇₃	C₂²×C₄	1	2	yes	—
75	C₂×#10596831	—	2	4	no	—
76	C₂×#10596831	—	2	4	no	—
77	C₂×#10596831	—	2	4	no	—
78	C₄⋊Dic₇₃	—	2	2	no	—
79	C₄⋊Dic₇₃	C₄⋊C₄	1	4	yes	—
80	C₄⋊Dic₇₃	C₂×C₈	1	4	yes	—
81	#11924007	—	2	4	no	—
82	#11924007	C₄⋊C₄	1	8	yes	—
83	#11924007	C₂×C₈	1	8	yes	—
84	C₄×Dic₇₃	—	2	4	no	—
85	C₄×Dic₇₃	—	2	4	no	—
86	C₄×Dic₇₃	—	2	4	no	—
87	C₄×Dic₇₃	—	4	2	no	—
88	C₄×Dic₇₃	C₂×C₈	1	8	yes	—
89	C₄×Dic₇₃	C₂×D₄	1	4	yes	—
90	C₄×Dic₇₃	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
95	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₈×C₇₃	—	2	2	no	—
99	C₂×C₈×C₇₃	—	2	2	no	—
100	C₂×C₈×C₇₃	—	2	2	no	—
101	C₂×C₈×C₇₃	—	2	2	no	—
102	C₂×C₈×C₇₃	—	2	2	no	—
103	C₂×C₈×C₇₃	#41	1	4	yes	—
104	C₄²×C₇₃	—	2	2	no	—
105	C₄²×C₇₃	—	2	2	no	—
106	C₄²×C₇₃	—	2	2	no	—
107	C₄²×C₇₃	—	2	2	no	—
108	C₄²×C₇₃	—	4	1	no	—
109	C₄²×C₇₃	C₂×C₈	1	4	yes	—
110	C₄²×C₇₃	C₂×D₄	1	2	yes	—
111	C₄²×C₇₃	C₂×Q₈	1	2	yes	—
112	C₂³×C₇₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	—	2	1	no	—
114	C₂×C₄×C₇₃	—	2	1	no	—
115	C₂×C₄×C₇₃	—	2	1	no	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	2	1	no	—
119	C₂×C₄×C₇₃	—	2	1	no	—
120	C₂×C₄×C₇₃	—	2	1	no	—
121	C₂×C₄×C₇₃	—	2	1	no	—
122	C₂×C₄×C₇₃	—	2	1	no	—
123	C₂×C₄×C₇₃	—	2	1	no	—
124	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
127	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
128	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
129	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
130	C₂×C₄×C₇₃	#131	1	2	yes	—
131	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
134	C₂×C₄×C₇₃	2₋¹⁺⁴	1	2	yes	—
135	C₂×Dic₇₃	—	2	2	no	—
136	C₂×Dic₇₃	—	2	2	no	—
137	C₂×Dic₇₃	—	2	2	no	—
138	C₂×Dic₇₃	—	2	2	no	—
139	C₂×Dic₇₃	—	2	2	no	—
140	C₂×Dic₇₃	—	2	2	no	—
141	C₂×Dic₇₃	—	2	2	no	—
142	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
143	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
144	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
145	C₂×Dic₇₃	C₂²×C₈	1	4	yes	—
146	C₂×Dic₇₃	C₂×M₁₆	1	4	yes	—
147	Dic₁₄₆	C₄⋊C₈	1	16	yes	complemented
148	C₈×C₇₃	—	2	4	no	—
149	C₈×C₇₃	—	4	2	no	—
150	#10596831	—	4	4	no	—
151	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₇₃	C₄×#41	1	1	yes	characteristic
153	C₂²×C₇₃	#519	1	1	yes	characteristic
154	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
155	C₂²×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
156	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
157	C₂²×C₇₃	#568	1	1	yes	characteristic
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	2	2	no	—
160	C₄×C₇₃	—	2	2	no	—
161	C₄×C₇₃	—	2	2	no	—
162	C₄×C₇₃	—	4	1	no	—
163	C₄×C₇₃	C₂×C₄⋊C₈	1	4	yes	—
164	C₄×C₇₃	#548	1	2	yes	—
165	C₄×C₇₃	#557	1	2	yes	—
166	Dic₇₃	—	2	4	no	—
167	Dic₇₃	—	2	4	no	—
168	Dic₇₃	—	4	2	no	—
169	Dic₇₃	C₂×C₄⋊C₈	1	8	yes	—
170	#22734	—	73	1	no	maximal, Sylow
171	C₂×C₇₃	#2351	1	1	yes	characteristic
172	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
173	C₂×C₇₃	#2906	1	1	yes	characteristic
174	C₂×C₇₃	C₈×#41	1	1	yes	characteristic
175	C₂×C₇₃	#2971	1	1	yes	characteristic
176	C₂×C₇₃	#2982	1	1	yes	characteristic
177	C₂×C₇₃	#2984	1	1	yes	characteristic
178	C₂×C₈×Q₈	—	73	2	no	—
179	C₄²×Q₈	—	73	1	no	—
180	#1939	—	73	4	no	—
181	#1939	—	73	2	no	—
182	#1831	—	73	1	no	—
183	#1831	—	73	2	no	—
184	C₄×C₄⋊C₈	—	73	2	no	—
185	C₄×C₄⋊C₈	D₇₃	1	1	yes	characteristic
186	C₇₃	#22734	1	1	yes	characteristic, complemented, Sylow
187	C₂×C₄×Q₈	—	73	1	no	—
188	C₂×C₄×Q₈	—	73	1	no	—
189	C₂×C₄×Q₈	—	73	1	no	—
190	C₈×Q₈	—	146	8	no	—
191	C₂×C₄⋊C₈	—	73	4	no	—
192	C₂×C₄⋊C₈	—	73	4	no	—
193	C₂×C₄⋊C₈	—	73	2	no	—
194	C₂×C₄⋊C₈	C₂×D₇₃	1	2	yes	—
195	C₂×C₄×C₈	—	73	4	no	—
196	C₂×C₄×C₈	C₂×D₇₃	1	2	yes	—
197	C₄×C₄⋊C₄	—	73	1	no	—
198	C₄×C₄⋊C₄	—	73	1	no	—
199	C₄×C₄⋊C₄	—	73	2	no	—
200	C₄×C₄⋊C₄	—	73	2	no	—
201	C₄×C₄⋊C₄	—	73	1	no	—
202	C₄×C₄⋊C₄	—	73	2	no	—
203	C₄³	—	73	2	no	—
204	C₄³	C₂×D₇₃	1	1	yes	characteristic
205	#336	—	73	4	no	—
206	#336	—	73	2	no	—
207	#336	—	73	2	no	—
208	#336	C₂×D₇₃	1	1	yes	characteristic
209	#336	C₂×D₇₃	1	1	yes	characteristic
210	C₂²×Q₈	—	73	1	no	—
211	C₂²×C₈	—	73	4	no	—
212	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
213	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
214	C₄×Q₈	—	73	8	no	—
215	C₄×Q₈	—	73	4	no	—
216	C₄×Q₈	—	73	4	no	—
217	C₄×Q₈	—	146	4	no	—
218	C₂×C₄⋊C₄	—	73	2	no	—
219	C₂×C₄⋊C₄	—	73	2	no	—
220	C₂×C₄⋊C₄	—	73	1	no	—
221	C₂×C₄⋊C₄	—	73	1	no	—
222	C₂×C₄⋊C₄	—	73	2	no	—
223	C₂×C₄⋊C₄	—	73	1	no	—
224	C₂×C₄²	—	73	2	no	—
225	C₂×C₄²	—	73	2	no	—
226	C₂×C₄²	—	73	2	no	—
227	C₂×C₄²	—	73	2	no	—
228	C₂×C₄²	—	73	2	no	—
229	C₂×C₄²	—	73	2	no	—
230	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
231	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
232	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
233	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
234	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
235	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
236	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
237	C₄⋊C₈	—	2	4	no	—
238	C₄⋊C₈	—	146	8	no	—
239	C₄⋊C₈	—	146	8	no	—
240	C₄⋊C₈	—	146	4	no	—
241	C₄×C₈	—	2	4	no	—
242	C₄×C₈	—	146	8	no	—
243	#95	—	73	2	no	—
244	#95	—	73	2	no	—
245	#95	—	73	2	no	—
246	C₂×Q₈	—	73	4	no	—
247	C₂×Q₈	—	73	4	no	—
248	C₂×Q₈	—	73	4	no	—
249	C₂²×C₄	—	73	2	no	—
250	C₂²×C₄	—	73	2	no	—
251	C₂²×C₄	—	73	2	no	—
252	C₂²×C₄	—	73	2	no	—
253	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
257	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
259	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	146	4	no	—
266	C₂×C₈	—	146	4	no	—
267	C₂×C₈	—	146	4	no	—
268	C₂×C₈	#11924028	1	4	yes	—
269	C₄⋊C₄	—	73	8	no	—
270	C₄⋊C₄	—	73	4	no	—
271	C₄⋊C₄	—	73	8	no	—
272	C₄⋊C₄	—	73	4	no	—
273	C₄⋊C₄	—	146	4	no	—
274	C₄⋊C₄	—	146	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	2	2	no	—
277	C₄²	—	2	2	no	—
278	C₄²	—	2	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	—	292	2	no	—
281	C₄²	—	73	8	no	—
282	C₄²	—	73	4	no	—
283	C₄²	—	73	4	no	—
284	C₄²	—	146	4	no	—
285	C₄²	—	146	4	no	—
286	C₄²	—	146	4	no	—
287	C₄²	C₈×D₇₃	1	4	yes	—
288	C₄²	D₄×D₇₃	1	2	yes	—
289	C₄²	Q₈×D₇₃	1	2	yes	—
290	C₂³	C₂²×C₄×D₇₃	1	1	yes	characteristic
291	Q₈	—	73	16	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	73	4	no	—
304	C₂×C₄	—	73	4	no	—
305	C₂×C₄	—	73	4	no	—
306	C₂×C₄	—	73	4	no	—
307	C₂×C₄	—	73	4	no	—
308	C₂×C₄	—	146	2	no	—
309	C₂×C₄	—	146	2	no	—
310	C₂×C₄	—	146	2	no	—
311	C₂×C₄	—	146	2	no	—
312	C₂×C₄	—	146	2	no	—
313	C₂×C₄	—	146	2	no	—
314	C₂×C₄	—	146	2	no	—
315	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
316	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
317	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
318	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
319	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
320	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
321	C₂×C₄	#14556597	1	2	yes	—
322	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
323	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
324	C₂×C₄	C₂×#11924028	1	2	yes	—
325	C₂×C₄	#14556638	1	2	yes	—
326	C₈	—	2	4	no	—
327	C₈	—	4	2	no	—
328	C₈	—	292	4	no	—
329	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
330	C₂²	C₄×#11924028	1	1	yes	characteristic
331	C₂²	#17989064	1	1	yes	characteristic
332	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
333	C₂²	C₂×M₁₆×D₇₃	1	1	yes	characteristic
334	C₂²	C₂×#14556597	1	1	yes	characteristic
335	C₂²	#17989353	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	2	2	no	—
339	C₄	—	2	2	no	—
340	C₄	—	4	1	no	—
341	C₄	—	292	2	no	—
342	C₄	—	73	8	no	—
343	C₄	—	146	4	no	—
344	C₄	—	146	4	no	—
345	C₄	C₄⋊C₈×D₇₃	1	4	yes	—
346	C₄	#17989246	1	2	yes	—
347	C₄	#17989315	1	2	yes	—
348	C₂	#22612851	1	1	yes	characteristic
349	C₂	C₂×C₄⋊C₈×D₇₃	1	1	yes	characteristic
350	C₂	#22618496	1	1	yes	characteristic
351	C₂	C₈×#11924028	1	1	yes	characteristic
352	C₂	#22618949	1	1	yes	characteristic
353	C₂	#22619011	1	1	yes	characteristic
354	C₂	#22619018	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₃
commutator	C₂²×C₇₃	C₂³×C₈
Fitting	C₄×C₄⋊C₈×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂³×C₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₄×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#22734
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
35. C₄×Dic₁₄₆	C₈	semidirect
147. Dic₁₄₆	C₄⋊C₈	semidirect
186. C₇₃	#22734	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×Dic₁₄₆	2
C₄²×Dic₁₄₆	2
#22610649	2
#22610646	2
#22610048	2
#22610046	2
#22609779	2
C₄×C₄⋊C₈×C₇₃	2
#22734	73