Subgroups — #33056117

Order 18688.

626 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619963	C₂	1	1	yes	maximal, characteristic
2	#22619011	C₂	1	1	yes	maximal, characteristic, complemented
3	#22619006	C₂	1	1	yes	maximal, characteristic
4	#22608811	C₂	1	2	yes	maximal, complemented
5	#22608776	C₂	1	2	yes	maximal, complemented
6	#22608745	C₂	1	2	yes	maximal, complemented
7	C₈×#11924017	C₂	1	2	yes	maximal, complemented
8	#22608711	C₂	1	1	yes	maximal, characteristic, complemented
9	#22608379	C₂	1	1	yes	maximal, characteristic, complemented
10	#22608349	C₂	1	1	yes	maximal, characteristic
11	C₇₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989137	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₁₄₆	C₂²	1	1	yes	characteristic
14	#17988906	C₄	1	2	yes	—
15	#17988873	C₄	1	2	yes	—
16	#17988788	C₄	1	2	yes	—
17	#17988728	C₄	1	2	yes	—
18	C₈×Dic₁₄₆	C₂²	1	2	yes	—
19	C₈×Dic₁₄₆	C₂²	1	2	yes	—
20	#17988711	—	2	2	no	—
21	C₄×#11924017	C₂²	1	2	yes	—
22	C₄×#11924017	C₂²	1	2	yes	—
23	#17988666	C₂²	1	2	yes	—
24	#17988657	—	2	2	no	—
25	C₈×D₂₉₂	C₂²	1	2	yes	—
26	#17988644	C₂²	1	1	yes	characteristic
27	#17988640	C₂²	1	1	yes	characteristic
28	#17988637	—	2	2	no	—
29	#17988552	—	2	2	no	—
30	#17988340	C₂²	1	2	yes	—
31	#17988338	C₂²	1	2	yes	—
32	#17988319	C₂²	1	2	yes	—
33	#17988319	C₂²	1	2	yes	—
34	#17988316	C₂²	1	2	yes	—
35	#17988316	C₂²	1	2	yes	—
36	#431×C₇₃	C₂²	1	1	yes	characteristic
37	#431×C₇₃	C₂²	1	1	yes	characteristic
38	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	#322×C₇₃	C₂²	1	1	yes	characteristic
40	#322×C₇₃	C₂²	1	1	yes	characteristic
41	#322×C₇₃	C₂²	1	1	yes	characteristic
42	C₈²×C₇₃	C₂²	1	1	yes	characteristic
43	C₂×#11924028	—	2	1	no	—
44	C₂²×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
45	C₂×#11924017	C₂×C₄	1	2	yes	—
46	C₂×#11924017	C₂×C₄	1	2	yes	—
47	C₂×#11924011	—	2	2	no	—
48	C₂×C₈×D₇₃	—	2	2	no	—
49	#14556591	C₂×C₄	1	1	yes	characteristic
50	#14556588	C₂×C₄	1	1	yes	characteristic
51	#14556579	—	2	1	no	—
52	#14556565	—	2	1	no	—
53	#14556565	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₄₆	—	2	1	no	—
55	C₄×Dic₁₄₆	—	2	1	no	—
56	C₄×Dic₁₄₆	C₂×C₄	1	2	yes	—
57	C₄×Dic₁₄₆	C₂³	1	2	yes	—
58	C₄×D₂₉₂	—	2	1	no	—
59	C₄×D₂₉₂	C₂³	1	2	yes	—
60	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
61	C₂×#11924007	—	2	1	no	—
62	C₂×C₄×Dic₇₃	—	2	1	no	—
63	#14556523	C₂×C₄	1	2	yes	—
64	#14556523	C₂×C₄	1	2	yes	—
65	C₄⋊#10596831	C₂³	1	2	yes	—
66	C₄⋊#10596831	C₂³	1	2	yes	—
67	#14556520	—	2	2	no	—
68	#14556520	—	2	2	no	—
69	#14556516	C₂×C₄	1	2	yes	—
70	#14556516	C₂×C₄	1	2	yes	—
71	#14556513	C₂×C₄	1	2	yes	—
72	#14556513	C₂×C₄	1	2	yes	—
73	C₂²⋊#10596831	—	2	1	no	—
74	C₂²⋊#10596831	—	2	1	no	—
75	#14556503	—	2	2	no	—
76	#14556503	—	2	2	no	—
77	#14556502	—	2	1	no	—
78	C₄×#10596831	—	2	1	no	—
79	C₈×Dic₇₃	—	2	2	no	—
80	C₈×Dic₇₃	—	2	2	no	—
81	C₂×M₁₆×C₇₃	—	2	1	no	—
82	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
84	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
85	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
86	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
87	#97×C₇₃	—	2	1	no	—
88	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
89	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
94	#11924028	—	8	2	no	—
95	C₂×Dic₁₄₆	—	2	1	no	—
96	C₂×Dic₁₄₆	—	2	4	no	—
97	C₂×Dic₁₄₆	—	2	1	no	—
98	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
99	C₂×D₂₉₂	—	2	1	no	—
100	C₂×D₂₉₂	—	2	1	no	—
101	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
102	C₂×C₂²⋊D₇₃	—	2	2	no	—
103	C₂²×Dic₇₃	—	2	1	no	—
104	C₂²×Dic₇₃	—	2	1	no	—
105	C₂×C₄×D₇₃	—	2	2	no	—
106	#11924017	—	2	4	no	—
107	#11924017	—	2	4	no	—
108	#11924011	—	8	2	no	—
109	C₂×#10596831	—	2	2	no	—
110	C₂×#10596831	—	2	2	no	—
111	C₈×D₇₃	—	8	2	no	—
112	C₄⋊Dic₇₃	—	2	1	no	—
113	C₄⋊Dic₇₃	—	2	1	no	—
114	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
115	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
116	#11924007	—	2	2	no	—
117	#11924007	—	2	2	no	—
118	#11924005	—	2	2	no	—
119	#11924005	—	2	2	no	—
120	C₄×Dic₇₃	—	2	2	no	—
121	C₄×Dic₇₃	—	2	2	no	—
122	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
123	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
124	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
125	M₁₆×C₇₃	—	4	2	no	—
126	C₂×C₈×C₇₃	—	2	1	no	—
127	C₂×C₈×C₇₃	—	2	1	no	—
128	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
129	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
130	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
131	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
132	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
133	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
134	C₄²×C₇₃	#41	1	1	yes	characteristic
135	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
137	C₂²×D₇₃	—	2	2	no	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	4	1	no	—
141	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
147	C₂²⋊D₇₃	—	8	2	no	—
148	C₂×Dic₇₃	—	2	4	no	—
149	C₂×Dic₇₃	—	2	2	no	—
150	C₂×Dic₇₃	—	2	2	no	—
151	C₂×Dic₇₃	—	2	2	no	—
152	C₂×Dic₇₃	—	8	1	no	—
153	D₂₉₂	—	2	2	no	—
154	D₂₉₂	—	2	2	no	—
155	D₂₉₂	—	8	1	no	—
156	D₂₉₂	—	8	1	no	—
157	C₄×D₇₃	—	4	2	no	—
158	C₄×D₇₃	—	4	2	no	—
159	Dic₁₄₆	—	2	4	no	—
160	Dic₁₄₆	—	2	2	no	—
161	Dic₁₄₆	—	8	1	no	—
162	Dic₁₄₆	#131	1	4	yes	—
163	C₈×C₇₃	—	2	2	no	—
164	C₈×C₇₃	—	2	2	no	—
165	C₈×C₇₃	—	2	2	no	—
166	C₈×C₇₃	—	2	2	no	—
167	C₈×C₇₃	—	2	2	no	—
168	C₈×C₇₃	—	4	1	no	—
169	#10596831	—	4	2	no	—
170	#10596831	—	4	2	no	—
171	C₂²×C₇₃	—	2	1	no	—
172	C₂²×C₇₃	—	2	1	no	—
173	C₂²×C₇₃	—	2	1	no	—
174	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
175	C₂×D₇₃	—	4	2	no	—
176	C₂×D₇₃	—	4	2	no	—
177	C₂×D₇₃	—	4	2	no	—
178	C₄×C₇₃	—	2	1	no	—
179	C₄×C₇₃	—	2	1	no	—
180	C₄×C₇₃	—	2	1	no	—
181	C₄×C₇₃	—	2	1	no	—
182	C₄×C₇₃	C₂×#131	1	2	yes	—
183	C₄×C₇₃	#577	1	2	yes	—
184	Dic₇₃	—	2	4	no	—
185	Dic₇₃	—	4	2	no	—
186	Dic₇₃	—	4	2	no	—
187	#17923	—	73	1	no	maximal, Sylow
188	C₂×C₇₃	—	4	1	no	—
189	C₂×C₇₃	#2929	1	1	yes	characteristic
190	C₂×C₇₃	#2949	1	1	yes	characteristic
191	C₂×C₇₃	#2956	1	1	yes	characteristic
192	D₇₃	—	8	2	no	—
193	#3103	—	73	1	no	—
194	#2982	—	73	1	no	—
195	#2981	—	73	1	no	—
196	#1588	—	73	2	no	—
197	#1583	—	73	2	no	—
198	SD₁₆⋊C₈	—	73	2	no	—
199	C₈×SD₁₆	—	73	2	no	—
200	#1572	—	73	1	no	—
201	#1491	—	73	1	no	—
202	#1484	—	73	1	no	—
203	#1453	D₇₃	1	1	yes	characteristic, complemented
204	C₇₃	#17923	1	1	yes	characteristic, complemented, Sylow
205	#532	—	73	1	no	—
206	C₂×C₄×Q₈	—	73	1	no	—
207	#481	—	73	2	no	—
208	#475	—	73	2	no	—
209	#460	—	73	2	no	—
210	#448	—	73	2	no	—
211	C₈×Q₈	—	73	2	no	—
212	C₈×Q₈	—	73	2	no	—
213	C₈×Q₈	—	146	2	no	—
214	C₄×SD₁₆	—	73	2	no	—
215	C₄×SD₁₆	—	73	2	no	—
216	#436	—	73	2	no	—
217	#435	—	146	2	no	—
218	C₈×D₄	—	73	2	no	—
219	#432	—	73	1	no	—
220	#431	—	73	1	no	—
221	#431	—	146	2	no	—
222	#431	C₂×D₇₃	1	1	yes	characteristic
223	#431	C₂×D₇₃	1	1	yes	characteristic
224	#408	—	146	2	no	—
225	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
226	#335	—	73	2	no	—
227	#334	—	73	2	no	—
228	Q₈⋊C₈	—	73	2	no	—
229	Q₈⋊C₈	—	73	2	no	—
230	D₄⋊C₈	—	73	2	no	—
231	D₄⋊C₈	—	73	2	no	—
232	#322	C₂×D₇₃	1	1	yes	characteristic
233	#322	C₂×D₇₃	1	1	yes	characteristic
234	#322	C₂×D₇₃	1	1	yes	characteristic
235	C₈²	C₂×D₇₃	1	1	yes	characteristic
236	C₂×#41	—	146	1	no	—
237	C₂²×Q₈	—	73	1	no	—
238	C₂×SD₁₆	—	73	2	no	—
239	C₂×SD₁₆	—	73	2	no	—
240	C₂×M₁₆	—	2	1	no	—
241	C₂×M₁₆	—	146	2	no	—
242	C₂²×C₈	—	146	2	no	—
243	C₄⋊Q₈	—	73	1	no	—
244	#127	—	73	1	no	—
245	#124	—	146	1	no	—
246	#121	—	73	2	no	—
247	#121	—	146	1	no	—
248	C₄×Q₈	—	73	2	no	—
249	C₄×Q₈	—	73	2	no	—
250	C₄×Q₈	—	146	1	no	—
251	C₄×Q₈	—	146	1	no	—
252	C₄×D₄	—	73	2	no	—
253	C₄×D₄	—	146	1	no	—
254	C₂×C₄⋊C₄	—	73	1	no	—
255	C₂×C₄⋊C₄	—	146	1	no	—
256	C₂×C₄²	—	146	1	no	—
257	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
258	#106	—	73	2	no	—
259	#106	—	73	2	no	—
260	C₄⋊C₈	—	73	2	no	—
261	C₄⋊C₈	—	73	2	no	—
262	C₄⋊C₈	—	146	2	no	—
263	C₄⋊C₈	—	146	2	no	—
264	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
265	C₄⋊C₈	D₂₉₂	1	2	yes	—
266	#103	—	73	2	no	—
267	#103	—	73	2	no	—
268	#102	—	73	2	no	—
269	#102	—	73	2	no	—
270	C₂²⋊C₈	—	146	1	no	—
271	C₂²⋊C₈	—	146	1	no	—
272	C₂²⋊C₈	—	146	2	no	—
273	C₂²⋊C₈	—	146	2	no	—
274	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
275	C₂²⋊C₈	D₂₉₂	1	2	yes	—
276	#97	—	2	1	no	—
277	#97	—	146	1	no	—
278	C₄×C₈	—	146	2	no	—
279	C₄×C₈	—	146	2	no	—
280	C₄×C₈	—	146	1	no	—
281	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
286	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
287	#41	—	584	2	no	—
288	C₂×Q₈	—	73	2	no	—
289	C₂×Q₈	—	146	4	no	—
290	C₂×Q₈	—	146	1	no	—
291	C₂×Q₈	—	146	1	no	—
292	C₂×D₄	—	73	2	no	—
293	C₂×D₄	—	146	2	no	—
294	C₂×D₄	—	146	1	no	—
295	C₂×D₄	—	146	1	no	—
296	C₂²×C₄	—	146	2	no	—
297	C₂²×C₄	—	146	1	no	—
298	C₂²×C₄	—	146	1	no	—
299	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
300	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
301	C₂²×C₄	#11924028	1	1	yes	characteristic
302	SD₁₆	—	146	4	no	—
303	SD₁₆	—	146	4	no	—
304	M₁₆	—	4	2	no	—
305	M₁₆	—	584	2	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	584	2	no	—
309	C₂×C₈	—	146	2	no	—
310	C₂×C₈	—	146	2	no	—
311	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
312	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
313	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
314	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
315	C₄⋊C₄	—	73	2	no	—
316	C₄⋊C₄	—	73	2	no	—
317	C₄⋊C₄	—	146	2	no	—
318	C₄⋊C₄	—	146	1	no	—
319	C₄⋊C₄	—	146	1	no	—
320	C₄⋊C₄	—	146	2	no	—
321	C₂²⋊C₄	—	146	2	no	—
322	C₂²⋊C₄	—	146	2	no	—
323	C₄²	—	146	2	no	—
324	C₄²	—	146	2	no	—
325	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
326	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
327	C₄²	#11924028	1	1	yes	characteristic
328	C₄²	C₂³×D₇₃	1	1	yes	characteristic
329	C₂³	—	146	2	no	—
330	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
331	Q₈	—	584	1	no	—
332	Q₈	—	73	4	no	—
333	Q₈	—	146	4	no	—
334	Q₈	—	146	2	no	—
335	D₄	—	584	1	no	—
336	D₄	—	584	1	no	—
337	D₄	—	584	2	no	—
338	D₄	—	146	2	no	—
339	D₄	—	146	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	292	2	no	—
344	C₂×C₄	—	292	2	no	—
345	C₂×C₄	—	584	1	no	—
346	C₂×C₄	—	146	2	no	—
347	C₂×C₄	—	146	2	no	—
348	C₂×C₄	—	146	4	no	—
349	C₂×C₄	—	146	2	no	—
350	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
351	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
352	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
353	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
354	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
355	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	2	2	no	—
358	C₈	—	2	2	no	—
359	C₈	—	2	2	no	—
360	C₈	—	2	2	no	—
361	C₈	—	4	1	no	—
362	C₈	—	292	2	no	—
363	C₈	—	292	2	no	—
364	C₂²	—	2	1	no	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	292	2	no	—
368	C₂²	—	292	2	no	—
369	C₂²	—	292	2	no	—
370	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	292	2	no	—
376	C₄	—	292	2	no	—
377	C₄	—	146	4	no	—
378	C₄	#131×D₇₃	1	2	yes	—
379	C₄	#17989399	1	2	yes	—
380	C₂	—	4	1	no	—
381	C₂	—	584	2	no	—
382	C₂	#22618627	1	1	yes	characteristic
383	C₂	#22618792	1	1	yes	characteristic
384	C₂	#22618842	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1453	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17989137	C₂²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1453	D₇₃
hypocenter	C₇₃	#17923
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1453

Complements

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

N	H ≅ G/N	type
2. #22619011	C₂	semidirect
4. #22608811	C₂	semidirect
5. #22608776	C₂	semidirect
6. #22608745	C₂	semidirect
7. C₈×#11924017	C₂	semidirect
8. #22608711	C₂	semidirect
9. #22608379	C₂	semidirect
11. C₇₃×#1453	C₂	semidirect
203. #1453	D₇₃	semidirect
204. C₇₃	#17923	semidirect

Maximal subgroups

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

subgroup	index
#22619963	2
#22619011	2
#22619006	2
#22608811	2
#22608776	2
#22608745	2
C₈×#11924017	2
#22608711	2
#22608379	2
#22608349	2
C₇₃×#1453	2
#17923	73