Subgroups — #33057405

Order 18688.

794 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×D₅₈₄	C₂	1	1	yes	maximal, characteristic
2	#22618666	C₂	1	2	yes	maximal, complemented
3	#22608815	C₂	1	2	yes	maximal, complemented
4	#22608759	C₂	1	4	yes	maximal, complemented
5	#22608725	C₂	1	2	yes	maximal, complemented
6	#22608355	C₂	1	2	yes	maximal, complemented
7	C₈×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
8	C₇₃×#1567	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×D₅₈₄	C₄	1	1	yes	characteristic
10	C₂×M₁₆×D₇₃	—	2	2	no	—
11	C₂×C₄×D₂₉₂	C₂²	1	2	yes	—
12	C₄×D₅₈₄	C₄	1	2	yes	—
13	C₄×D₅₈₄	C₄	1	1	yes	characteristic
14	C₄×D₅₈₄	C₄	1	1	yes	characteristic
15	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
16	C₄×D₅₈₄	C₂²	1	2	yes	—
17	C₄×D₅₈₄	C₂²	1	1	yes	characteristic
18	#17988664	C₂²	1	4	yes	complemented
19	#17988657	—	2	4	no	—
20	#17988652	C₂²	1	4	yes	complemented
21	#17988640	C₂²	1	1	yes	characteristic
22	C₂×#14556525	C₄	1	1	yes	characteristic
23	C₂×#14556513	C₄	1	2	yes	—
24	#17988550	—	2	2	no	—
25	C₄×#11924012	C₂²	1	1	yes	characteristic
26	#17988314	C₂²	1	4	yes	complemented
27	#17988314	C₂²	1	4	yes	complemented
28	#17988309	C₂²	1	2	yes	complemented
29	C₈×#10596831	C₂²	1	2	yes	complemented
30	#423×C₇₃	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
32	#335×C₇₃	C₂²	1	2	yes	complemented
33	#335×C₇₃	C₂²	1	2	yes	complemented
34	C₂²×D₂₉₂	C₂×C₄	1	2	yes	—
35	C₂²×C₄×D₇₃	—	2	2	no	—
36	C₂×D₅₈₄	—	2	4	no	—
37	C₂×D₅₈₄	C₂×C₄	1	2	yes	—
38	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
39	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic
40	C₂×#11924012	D₄	1	1	yes	characteristic
41	C₂×#11924012	D₄	1	1	yes	characteristic
42	C₂×#11924011	—	2	4	no	—
43	M₁₆×D₇₃	—	4	8	no	—
44	C₂×C₈×D₇₃	—	2	4	no	—
45	C₄×D₂₉₂	—	2	2	no	—
46	C₄×D₂₉₂	—	2	2	no	—
47	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
48	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
49	C₄×D₂₉₂	C₂³	1	2	yes	—
50	C₄×D₂₉₂	C₂³	1	2	yes	—
51	C₂×C₄⋊Dic₇₃	C₂×C₄	1	2	yes	—
52	C₂×#11924005	—	2	2	no	—
53	#14556525	C₂×C₄	1	2	yes	—
54	#14556525	C₂×C₄	1	1	yes	characteristic
55	#14556525	C₂×C₄	1	1	yes	characteristic
56	C₄⋊#10596831	—	2	1	no	—
57	#14556513	C₂×C₄	1	2	yes	—
58	#14556513	C₂×C₄	1	2	yes	—
59	#14556513	C₂×C₄	1	2	yes	—
60	#14556513	C₂×C₄	1	2	yes	—
61	C₂²⋊#10596831	—	2	1	no	—
62	#14556503	—	2	4	no	—
63	#14556503	—	2	4	no	—
64	#14556502	D₄	1	2	yes	—
65	C₄×#10596831	C₂³	1	2	yes	—
66	C₄×#10596831	C₂³	1	2	yes	—
67	C₂×M₁₆×C₇₃	—	2	2	no	—
68	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
71	C₄⋊C₈×C₇₃	—	2	2	no	—
72	C₄⋊C₈×C₇₃	C₂³	1	4	yes	—
73	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
75	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
76	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
77	C₂³×D₇₃	—	2	2	no	—
78	C₂×D₂₉₂	—	2	4	no	—
79	C₂×D₂₉₂	—	2	2	no	—
80	C₂×D₂₉₂	—	4	2	no	—
81	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
82	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
83	C₂²×Dic₇₃	—	2	2	no	—
84	C₂×C₄×D₇₃	—	2	2	no	—
85	C₂×C₄×D₇₃	—	2	2	no	—
86	C₂×C₄×D₇₃	—	2	2	no	—
87	C₂×C₄×D₇₃	—	2	2	no	—
88	C₂×C₄×D₇₃	—	4	4	no	—
89	C₂×C₄×D₇₃	—	4	4	no	—
90	D₅₈₄	—	2	4	no	—
91	D₅₈₄	—	2	2	no	—
92	D₅₈₄	M₁₆	1	4	yes	complemented
93	#11924012	—	2	2	no	—
94	#11924012	D₈	1	4	yes	complemented
95	#11924011	—	4	8	no	—
96	C₂×#10596831	—	2	2	no	—
97	C₂×#10596831	C₂×D₄	1	2	yes	—
98	C₂×#10596831	C₂×D₄	1	2	yes	—
99	C₈×D₇₃	—	4	8	no	—
100	C₄⋊Dic₇₃	—	2	2	no	—
101	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
103	#11924005	—	2	2	no	—
104	#11924005	—	2	2	no	—
105	#11924005	—	2	2	no	—
106	#11924005	—	2	2	no	—
107	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
108	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
110	M₁₆×C₇₃	—	4	4	no	—
111	C₂×C₈×C₇₃	—	2	2	no	—
112	C₂×C₈×C₇₃	—	2	4	no	—
113	C₂×C₈×C₇₃	—	4	1	no	—
114	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₃	#41	1	1	yes	characteristic
121	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
122	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
123	C₂²×D₇₃	—	2	2	no	—
124	C₂²×D₇₃	—	2	2	no	—
125	C₂²×D₇₃	—	4	4	no	—
126	C₂²×D₇₃	—	4	4	no	—
127	C₂²×D₇₃	—	8	2	no	—
128	C₂×C₄×C₇₃	—	2	1	no	—
129	C₂×C₄×C₇₃	—	2	1	no	—
130	C₂×C₄×C₇₃	—	4	1	no	—
131	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₇₃	#135	1	2	yes	—
136	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
139	C₂×Dic₇₃	—	2	2	no	—
140	C₂×Dic₇₃	—	2	2	no	—
141	C₂×Dic₇₃	—	4	4	no	—
142	D₂₉₂	—	2	2	no	—
143	D₂₉₂	—	2	2	no	—
144	D₂₉₂	—	2	2	no	—
145	D₂₉₂	—	4	4	no	—
146	D₂₉₂	C₂×M₁₆	1	4	yes	complemented
147	C₄×D₇₃	—	4	2	no	—
148	C₄×D₇₃	—	4	2	no	—
149	C₄×D₇₃	—	4	2	no	—
150	C₄×D₇₃	—	4	2	no	—
151	C₄×D₇₃	—	4	4	no	—
152	C₄×D₇₃	—	4	4	no	—
153	C₈×C₇₃	—	2	1	no	—
154	C₈×C₇₃	—	2	1	no	—
155	C₈×C₇₃	—	2	1	no	—
156	C₈×C₇₃	—	4	4	no	—
157	C₈×C₇₃	C₂×M₁₆	1	2	yes	complemented
158	#10596831	—	2	2	no	—
159	#10596831	—	4	2	no	—
160	#10596831	C₂×D₈	1	4	yes	complemented
161	C₂²×C₇₃	—	2	1	no	—
162	C₂²×C₇₃	—	2	1	no	—
163	C₂²×C₇₃	C₄×D₈	1	2	yes	—
164	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
165	C₂×D₇₃	—	2	4	no	—
166	C₂×D₇₃	—	4	2	no	—
167	C₂×D₇₃	—	4	2	no	—
168	C₂×D₇₃	—	4	2	no	—
169	C₂×D₇₃	—	4	2	no	—
170	C₂×D₇₃	—	4	2	no	—
171	C₂×D₇₃	—	8	8	no	—
172	C₄×C₇₃	—	2	1	no	—
173	C₄×C₇₃	—	2	1	no	—
174	C₄×C₇₃	—	2	1	no	—
175	C₄×C₇₃	C₄×D₈	1	2	yes	—
176	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
177	C₄×C₇₃	#568	1	1	yes	characteristic
178	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
179	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
180	Dic₇₃	—	4	2	no	—
181	Dic₇₃	—	4	2	no	—
182	M₁₆×D₈	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	—	2	2	no	—
184	C₂×C₇₃	D₄×M₁₆	1	1	yes	characteristic
185	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
186	C₂×C₇₃	#2957	1	1	yes	characteristic
187	D₇₃	—	4	4	no	—
188	D₇₃	—	8	2	no	—
189	C₂×C₄×D₈	—	73	1	no	—
190	D₄×M₁₆	—	73	2	no	—
191	#1589	—	73	2	no	—
192	#1581	—	73	4	no	—
193	C₈×D₈	—	73	2	no	—
194	#1567	D₇₃	1	1	yes	characteristic, complemented
195	#1486	—	73	2	no	—
196	C₈×M₁₆	—	73	1	no	—
197	C₇₃	M₁₆×D₈	1	1	yes	characteristic, complemented, Sylow
198	C₂²×D₈	—	73	1	no	—
199	C₂²×M₁₆	—	146	2	no	—
200	C₂×C₄×D₄	—	73	2	no	—
201	C₄×D₈	—	73	1	no	—
202	C₄×D₈	—	73	2	no	—
203	C₄×D₈	—	73	1	no	—
204	C₄×D₈	—	73	1	no	—
205	C₄×D₈	—	73	2	no	—
206	C₄×D₈	—	73	1	no	—
207	#436	—	73	4	no	—
208	#435	—	146	4	no	—
209	C₈×D₄	—	73	4	no	—
210	#431	—	73	1	no	—
211	C₂×#107	—	73	1	no	—
212	#423	C₂×D₇₃	1	2	yes	—
213	C₂×#102	—	73	2	no	—
214	#407	—	146	2	no	—
215	C₄×M₁₆	—	73	1	no	—
216	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
217	#335	C₂×D₇₃	1	2	yes	complemented
218	#335	C₂×D₇₃	1	2	yes	complemented
219	D₄⋊C₈	—	73	4	no	—
220	D₄⋊C₈	—	73	4	no	—
221	#322	—	73	2	no	—
222	C₈²	—	73	2	no	—
223	C₂²×D₄	—	73	2	no	—
224	C₂³×C₄	—	146	2	no	—
225	C₂×D₈	—	73	1	no	—
226	C₂×D₈	—	73	2	no	—
227	C₂×D₈	—	73	1	no	—
228	C₂×D₈	—	146	4	no	—
229	C₂×M₁₆	—	2	2	no	—
230	C₂×M₁₆	—	292	8	no	—
231	C₂×M₁₆	—	73	1	no	—
232	C₂×M₁₆	—	73	1	no	—
233	C₂×M₁₆	—	146	4	no	—
234	C₂²×C₈	—	146	4	no	—
235	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
236	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
237	C₄×D₄	—	73	2	no	—
238	C₄×D₄	—	73	2	no	—
239	C₄×D₄	—	73	2	no	—
240	C₄×D₄	—	73	2	no	—
241	C₄×D₄	—	146	2	no	—
242	C₄×D₄	—	146	2	no	—
243	C₂×C₄⋊C₄	—	73	2	no	—
244	C₂×C₂²⋊C₄	—	146	2	no	—
245	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
246	#107	—	73	2	no	—
247	#107	—	73	1	no	—
248	#107	—	73	1	no	—
249	C₄⋊C₈	—	2	2	no	—
250	C₄⋊C₈	—	146	1	no	—
251	C₄⋊C₈	C₂²×D₇₃	1	4	yes	—
252	#102	—	73	2	no	—
253	#102	—	73	2	no	—
254	#102	—	73	2	no	—
255	#102	—	73	2	no	—
256	C₂²⋊C₈	—	146	4	no	—
257	C₂²⋊C₈	—	146	4	no	—
258	C₂²⋊C₈	—	146	1	no	—
259	#97	—	73	2	no	—
260	C₄×C₈	—	73	2	no	—
261	C₄×C₈	—	73	2	no	—
262	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
263	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
265	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
266	C₂⁴	—	146	2	no	—
267	C₂×D₄	—	292	2	no	—
268	C₂×D₄	—	73	2	no	—
269	C₂×D₄	—	73	2	no	—
270	C₂×D₄	—	146	4	no	—
271	C₂×D₄	—	146	2	no	—
272	C₂²×C₄	—	292	4	no	—
273	C₂²×C₄	—	292	4	no	—
274	C₂²×C₄	—	146	2	no	—
275	C₂²×C₄	—	146	2	no	—
276	C₂²×C₄	—	146	2	no	—
277	C₂²×C₄	—	146	2	no	—
278	C₂²×C₄	—	146	2	no	—
279	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
280	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
281	C₂²×C₄	#11924029	1	1	yes	characteristic
282	D₈	—	73	4	no	—
283	D₈	—	146	4	no	—
284	D₈	—	146	2	no	—
285	M₁₆	—	4	4	no	—
286	M₁₆	—	292	8	no	—
287	M₁₆	—	73	4	no	—
288	M₁₆	—	146	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	4	no	—
291	C₂×C₈	—	4	1	no	—
292	C₂×C₈	—	292	8	no	—
293	C₂×C₈	—	73	2	no	—
294	C₂×C₈	—	73	2	no	—
295	C₂×C₈	—	146	2	no	—
296	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
298	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
299	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
300	C₄⋊C₄	—	73	2	no	—
301	C₄⋊C₄	—	73	2	no	—
302	C₄⋊C₄	—	146	2	no	—
303	C₂²⋊C₄	—	146	2	no	—
304	C₂²⋊C₄	—	146	2	no	—
305	C₂²⋊C₄	—	146	2	no	—
306	C₂²⋊C₄	—	146	2	no	—
307	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
308	C₄²	D₄×D₇₃	1	1	yes	characteristic
309	C₄²	#11924029	1	1	yes	characteristic
310	C₄²	C₂³×D₇₃	1	1	yes	characteristic
311	C₂³	—	292	4	no	—
312	C₂³	—	292	4	no	—
313	C₂³	—	584	2	no	—
314	C₂³	—	146	2	no	—
315	C₂³	—	146	2	no	—
316	C₂³	#14556553	1	1	yes	characteristic
317	D₄	—	292	4	no	—
318	D₄	—	73	4	no	—
319	D₄	—	146	2	no	—
320	D₄	—	146	2	no	—
321	D₄	—	146	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	292	4	no	—
326	C₂×C₄	—	292	4	no	—
327	C₂×C₄	—	292	4	no	—
328	C₂×C₄	—	292	2	no	—
329	C₂×C₄	—	292	2	no	—
330	C₂×C₄	—	292	2	no	—
331	C₂×C₄	—	292	2	no	—
332	C₂×C₄	—	146	2	no	—
333	C₂×C₄	—	146	2	no	—
334	C₂×C₄	#14556553	1	1	yes	characteristic
335	C₂×C₄	#14556553	1	1	yes	characteristic
336	C₂×C₄	#14556553	1	1	yes	characteristic
337	C₂×C₄	D₈×D₇₃	1	2	yes	—
338	C₂×C₄	#14556614	1	2	yes	—
339	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
340	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
341	C₂×C₄	C₂×#11924029	1	1	yes	characteristic
342	C₈	—	2	1	no	—
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	4	no	—
346	C₈	—	292	2	no	—
347	C₈	—	73	4	no	—
348	C₈	—	146	2	no	—
349	C₈	M₁₆×D₇₃	1	2	yes	complemented
350	C₂²	—	2	1	no	—
351	C₂²	—	2	1	no	—
352	C₂²	—	292	2	no	—
353	C₂²	—	292	2	no	—
354	C₂²	—	292	2	no	—
355	C₂²	—	292	2	no	—
356	C₂²	—	292	2	no	—
357	C₂²	—	584	8	no	—
358	C₂²	—	146	4	no	—
359	C₂²	#17988669	1	2	yes	—
360	C₂²	C₂×#14556553	1	1	yes	characteristic
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	292	2	no	—
365	C₄	—	292	2	no	—
366	C₄	#17988669	1	2	yes	—
367	C₄	C₂×M₁₆×D₇₃	1	1	yes	characteristic
368	C₄	#17989353	1	1	yes	characteristic
369	C₄	C₂×D₈×D₇₃	1	1	yes	characteristic
370	C₄	C₂×#14556614	1	1	yes	characteristic
371	C₂	—	2	2	no	—
372	C₂	—	292	4	no	—
373	C₂	—	584	2	no	—
374	C₂	#22618666	1	1	yes	characteristic
375	C₂	C₂×#17988669	1	1	yes	characteristic
376	C₂	#22618847	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	C₂×C₄×D₄
prime radical	C₂²×D₅₈₄	C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1567	D₇₃
hypocenter	C₇₃	M₁₆×D₈
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
2. #22618666	C₂	semidirect
3. #22608815	C₂	semidirect
4. #22608759	C₂	semidirect
5. #22608725	C₂	semidirect
6. #22608355	C₂	semidirect
7. C₈×#11924012	C₂	semidirect
8. C₇₃×#1567	C₂	semidirect
18. #17988664	C₂²	semidirect
20. #17988652	C₂²	semidirect
26. #17988314	C₂²	semidirect
27. #17988314	C₂²	semidirect
28. #17988309	C₂²	semidirect
29. C₈×#10596831	C₂²	semidirect
32. #335×C₇₃	C₂²	semidirect
33. #335×C₇₃	C₂²	semidirect
92. D₅₈₄	M₁₆	semidirect
94. #11924012	D₈	semidirect
146. D₂₉₂	C₂×M₁₆	semidirect
157. C₈×C₇₃	C₂×M₁₆	semidirect
160. #10596831	C₂×D₈	semidirect
194. #1567	D₇₃	semidirect
197. C₇₃	M₁₆×D₈	semidirect
217. #335	C₂×D₇₃	semidirect
218. #335	C₂×D₇₃	semidirect
349. C₈	M₁₆×D₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×D₅₈₄	2
#22618666	2
#22608815	2
#22608759	2
#22608725	2
#22608355	2
C₈×#11924012	2
C₇₃×#1567	2
M₁₆×D₈	73