Subgroups — #33055230

Order 18688.

610 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11924028	C₂	1	1	yes	maximal, characteristic
2	#22618599	C₂	1	2	yes	maximal
3	#22608710	C₂	1	4	yes	maximal, complemented
4	#22608703	C₂	1	2	yes	maximal, complemented
5	#22608693	C₂	1	2	yes	maximal, complemented
6	#22608678	C₂	1	2	yes	maximal, complemented
7	C₂×#17988340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11924028	C₂²	1	1	yes	characteristic
10	C₈×Dic₁₄₆	C₄	1	1	yes	characteristic
11	C₈×Dic₁₄₆	C₄	1	1	yes	characteristic
12	C₈×D₂₉₂	C₄	1	1	yes	characteristic
13	C₈×D₂₉₂	C₄	1	1	yes	characteristic
14	C₈×C₂²⋊D₇₃	C₄	1	2	yes	—
15	C₈×C₂²⋊D₇₃	C₄	1	2	yes	—
16	#17988647	—	2	4	no	—
17	#17988643	C₂²	1	4	yes	—
18	#17988640	C₂²	1	1	yes	characteristic
19	#17988639	C₂²	1	4	yes	—
20	#17988638	C₂²	1	2	yes	—
21	C₂×C₄⋊#10596831	C₂²	1	2	yes	—
22	#17988555	—	2	2	no	—
23	#17988545	—	2	2	no	—
24	C₄×C₈×D₇₃	C₂²	1	2	yes	—
25	#17988340	C₂²	1	1	yes	characteristic
26	#17988340	C₂²	1	1	yes	characteristic
27	#17988340	C₂²	1	2	yes	—
28	#17988337	C₂²	1	4	yes	—
29	#17988337	C₂²	1	4	yes	—
30	#423×C₇₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#335×C₇₃	C₂²	1	2	yes	—
33	#335×C₇₃	C₂²	1	2	yes	—
34	C₂×#11924028	D₄	1	1	yes	characteristic
35	C₂²×#10596831	—	2	2	no	—
36	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₇₃	C₂×C₄	1	2	yes	—
38	C₄×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
39	C₄×Dic₁₄₆	Q₈	1	1	yes	characteristic
40	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
41	C₄×D₂₉₂	Q₈	1	1	yes	characteristic
42	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
43	C₄×C₂²⋊D₇₃	Q₈	1	2	yes	—
44	#14556550	D₄	1	1	yes	characteristic
45	#14556548	D₄	1	2	yes	—
46	C₄²×D₇₃	C₂³	1	2	yes	—
47	C₄⋊#10596831	—	2	2	no	—
48	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
49	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#10596831	C₂³	1	2	yes	—
51	C₄⋊#10596831	C₂³	1	2	yes	—
52	#14556520	C₂×C₄	1	2	yes	—
53	#14556520	C₂×C₄	1	2	yes	—
54	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
55	#14556503	—	2	4	no	—
56	#14556503	—	2	4	no	—
57	#14556503	C₂×C₄	1	2	yes	—
58	#14556503	C₂×C₄	1	2	yes	—
59	#14556501	—	2	4	no	—
60	C₄×#10596831	C₂³	1	2	yes	—
61	C₈×Dic₇₃	—	2	4	no	—
62	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
63	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
64	C₂×M₁₆×C₇₃	—	2	2	no	—
65	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
68	C₄⋊C₈×C₇₃	—	2	2	no	—
69	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
70	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
73	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
74	#11924028	—	2	2	no	—
75	#11924028	—	2	2	no	—
76	C₂×Dic₁₄₆	C₄⋊C₄	1	1	yes	characteristic
77	C₂×D₂₉₂	C₄⋊C₄	1	1	yes	characteristic
78	C₂×C₂²⋊D₇₃	C₄⋊C₄	1	2	yes	—
79	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
82	C₂×#10596831	—	2	2	no	—
83	C₂×#10596831	—	2	2	no	—
84	C₂×#10596831	—	4	4	no	—
85	C₂×#10596831	C₂²×C₄	1	2	yes	—
86	C₂×#10596831	C₂²×C₄	1	2	yes	—
87	C₈×D₇₃	—	2	4	no	—
88	C₈×D₇₃	—	4	2	no	—
89	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
90	C₄⋊Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
91	#11924007	C₄⋊C₄	1	2	yes	—
92	#11924007	C₄⋊C₄	1	2	yes	—
93	C₂²⋊Dic₇₃	C₄⋊C₄	1	2	yes	—
94	#11924005	C₄⋊C₄	1	2	yes	—
95	#11924005	C₄⋊C₄	1	2	yes	—
96	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
98	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
99	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
102	M₁₆×C₇₃	—	4	4	no	—
103	C₂×C₈×C₇₃	—	2	2	no	—
104	C₂×C₈×C₇₃	—	2	4	no	—
105	C₂×C₈×C₇₃	—	4	1	no	—
106	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
111	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₇₃	C₂×Q₈	1	1	yes	characteristic
113	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
114	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
115	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₇₃	—	2	1	no	—
117	C₂×C₄×C₇₃	—	2	1	no	—
118	C₂×C₄×C₇₃	—	4	1	no	—
119	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	#135	1	2	yes	—
123	C₂×C₄×C₇₃	#135	1	2	yes	—
124	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
127	C₂²⋊D₇₃	—	2	4	no	—
128	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
130	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
131	D₂₉₂	—	2	2	no	—
132	C₄×D₇₃	—	2	2	no	—
133	C₄×D₇₃	—	2	2	no	—
134	C₄×D₇₃	—	2	2	no	—
135	C₄×D₇₃	—	2	2	no	—
136	C₄×D₇₃	—	4	2	no	—
137	Dic₁₄₆	—	2	2	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	—	2	1	no	—
140	C₈×C₇₃	—	2	1	no	—
141	C₈×C₇₃	—	4	4	no	—
142	C₈×C₇₃	#131	1	2	yes	—
143	#10596831	—	2	2	no	—
144	#10596831	—	2	2	no	—
145	#10596831	—	4	2	no	—
146	#10596831	—	4	2	no	—
147	C₂²×C₇₃	—	2	1	no	—
148	C₂²×C₇₃	—	2	1	no	—
149	C₂²×C₇₃	#427	1	2	yes	—
150	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
151	C₂×D₇₃	—	2	2	no	—
152	C₂×D₇₃	—	2	2	no	—
153	C₂×D₇₃	—	2	2	no	—
154	C₄×C₇₃	—	2	1	no	—
155	C₄×C₇₃	—	2	1	no	—
156	C₄×C₇₃	—	2	1	no	—
157	C₄×C₇₃	#427	1	2	yes	—
158	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
159	C₄×C₇₃	#568	1	1	yes	characteristic
160	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
161	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
162	Dic₇₃	—	2	2	no	—
163	Dic₇₃	—	2	2	no	—
164	Dic₇₃	—	2	2	no	—
165	#17829	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	2	2	no	—
167	C₂×C₇₃	#2906	1	1	yes	characteristic
168	C₂×C₇₃	C₂×#427	1	1	yes	characteristic
169	C₂×C₇₃	#2916	1	1	yes	characteristic
170	D₇₃	—	4	2	no	—
171	C₈×#41	—	73	1	no	—
172	#2925	—	73	2	no	—
173	#1572	—	73	4	no	—
174	#1570	—	73	2	no	—
175	#1568	—	73	2	no	—
176	#1567	D₇₃	1	1	yes	characteristic, complemented
177	#1564	—	73	2	no	—
178	C₂×#335	—	73	1	no	—
179	C₇₃	#17829	1	1	yes	characteristic, complemented, Sylow
180	C₄×#41	—	73	1	no	—
181	C₈×Q₈	—	73	1	no	—
182	C₈×Q₈	—	73	1	no	—
183	C₈×D₄	—	73	2	no	—
184	C₈×D₄	—	73	2	no	—
185	C₈×D₄	—	73	1	no	—
186	C₈×D₄	—	73	1	no	—
187	#433	—	146	4	no	—
188	#432	—	73	4	no	—
189	#431	—	73	1	no	—
190	#431	—	73	2	no	—
191	#431	—	73	4	no	—
192	#423	C₂×D₇₃	1	2	yes	—
193	C₂×C₄⋊C₈	—	73	2	no	—
194	#408	—	146	2	no	—
195	#405	—	146	2	no	—
196	C₂×C₄×C₈	—	73	2	no	—
197	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
198	#335	—	73	1	no	—
199	#335	—	73	1	no	—
200	#335	—	73	2	no	—
201	#335	C₂×D₇₃	1	2	yes	—
202	#335	C₂×D₇₃	1	2	yes	—
203	#334	—	73	4	no	—
204	#334	—	73	4	no	—
205	C₂×#41	—	73	1	no	—
206	C₂×M₁₆	—	2	2	no	—
207	C₂²×C₈	—	73	2	no	—
208	C₂²×C₈	—	73	2	no	—
209	C₂²×C₈	—	146	2	no	—
210	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
211	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
212	C₄×Q₈	—	73	1	no	—
213	C₄×Q₈	—	73	1	no	—
214	C₄×D₄	—	73	2	no	—
215	C₄×D₄	—	73	2	no	—
216	C₄×D₄	—	73	1	no	—
217	C₄×D₄	—	73	1	no	—
218	#117	—	73	1	no	—
219	#117	—	73	2	no	—
220	C₂×C₄²	—	73	2	no	—
221	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
222	C₄⋊C₈	—	2	2	no	—
223	C₄⋊C₈	—	73	2	no	—
224	C₄⋊C₈	—	73	2	no	—
225	C₄⋊C₈	—	73	1	no	—
226	C₄⋊C₈	—	73	1	no	—
227	C₄⋊C₈	—	73	2	no	—
228	C₄⋊C₈	—	73	2	no	—
229	C₄⋊C₈	—	146	2	no	—
230	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
231	C₂²⋊C₈	—	73	2	no	—
232	C₂²⋊C₈	—	73	2	no	—
233	C₂²⋊C₈	—	73	2	no	—
234	C₂²⋊C₈	—	146	4	no	—
235	C₂²⋊C₈	—	146	4	no	—
236	#97	—	146	4	no	—
237	C₄×C₈	—	73	2	no	—
238	C₄×C₈	—	73	2	no	—
239	C₄×C₈	—	73	2	no	—
240	C₄×C₈	—	146	4	no	—
241	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
242	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
243	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
245	#41	—	146	2	no	—
246	#41	—	146	2	no	—
247	C₂×Q₈	—	73	1	no	—
248	C₂×D₄	—	73	2	no	—
249	C₂×D₄	—	73	1	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	D₄×D₇₃	1	1	yes	characteristic
255	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
256	M₁₆	—	4	4	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	292	2	no	—
261	C₂×C₈	—	292	4	no	—
262	C₂×C₈	—	73	2	no	—
263	C₂×C₈	—	73	2	no	—
264	C₂×C₈	—	146	2	no	—
265	C₂×C₈	—	146	2	no	—
266	C₂×C₈	—	146	4	no	—
267	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
268	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
271	C₄⋊C₄	—	73	2	no	—
272	C₄⋊C₄	—	73	2	no	—
273	C₄⋊C₄	—	73	1	no	—
274	C₄⋊C₄	—	73	1	no	—
275	C₂²⋊C₄	—	73	2	no	—
276	C₂²⋊C₄	—	73	2	no	—
277	C₂²⋊C₄	—	73	2	no	—
278	C₄²	—	73	2	no	—
279	C₄²	—	73	2	no	—
280	C₄²	—	73	2	no	—
281	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
282	C₄²	D₄×D₇₃	1	1	yes	characteristic
283	C₄²	Q₈×D₇₃	1	1	yes	characteristic
284	C₄²	C₂³×D₇₃	1	1	yes	characteristic
285	C₂³	—	73	2	no	—
286	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
287	Q₈	—	146	2	no	—
288	D₄	—	146	4	no	—
289	D₄	—	146	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	1	no	—
293	C₂×C₄	—	292	2	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	73	2	no	—
296	C₂×C₄	—	73	2	no	—
297	C₂×C₄	—	146	2	no	—
298	C₂×C₄	—	146	2	no	—
299	C₂×C₄	—	146	2	no	—
300	C₂×C₄	—	146	2	no	—
301	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
302	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
303	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
304	C₂×C₄	#14556610	1	2	yes	—
305	C₂×C₄	#14556614	1	2	yes	—
306	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	292	2	no	—
314	C₈	—	292	2	no	—
315	C₈	—	146	2	no	—
316	C₈	—	146	2	no	—
317	C₈	#14556599	1	2	yes	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	146	2	no	—
321	C₂²	—	146	2	no	—
322	C₂²	—	146	2	no	—
323	C₂²	#17988624	1	2	yes	—
324	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	146	2	no	—
329	C₄	—	146	2	no	—
330	C₄	—	146	2	no	—
331	C₄	#17988624	1	2	yes	—
332	C₄	C₂×#14556599	1	1	yes	characteristic
333	C₄	#17989353	1	1	yes	characteristic
334	C₄	C₂×#14556610	1	1	yes	characteristic
335	C₄	C₂×#14556614	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	292	2	no	—
338	C₂	#22618500	1	1	yes	characteristic
339	C₂	C₂×#17988624	1	1	yes	characteristic
340	C₂	#22618539	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1567	C₂
socle	C₂²×C₇₃	C₂²×C₄⋊C₄
prime radical	C₂×#11924028	D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1567	D₇₃
hypocenter	C₇₃	#17829
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. #22608710	C₂	semidirect
4. #22608703	C₂	semidirect
5. #22608693	C₂	semidirect
6. #22608678	C₂	semidirect
7. C₂×#17988340	C₂	semidirect
8. C₇₃×#1567	C₂	semidirect
176. #1567	D₇₃	semidirect
179. C₇₃	#17829	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11924028	2
#22618599	2
#22608710	2
#22608703	2
#22608693	2
#22608678	2
C₂×#17988340	2
C₇₃×#1567	2
#17829	73