Subgroups — #33055872

Order 18688.

706 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619940	C₂	1	1	yes	maximal, characteristic
2	#22618865	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
4	#22608769	C₂	1	4	yes	maximal, complemented
5	C₈×#11924017	C₂	1	4	yes	maximal, complemented
6	#22608702	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608357	C₂	1	1	yes	maximal, characteristic
8	#22608335	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₇₃	—	2	1	no	—
11	#17989144	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
13	#17988918	C₄	1	2	yes	—
14	#17988878	C₄	1	2	yes	—
15	#17988793	C₄	1	2	yes	—
16	#17988740	C₄	1	2	yes	—
17	#17988717	—	2	1	no	—
18	#17988717	C₂²	1	2	yes	complemented
19	C₈×Dic₁₄₆	—	2	1	no	—
20	C₈×Dic₁₄₆	C₂²	1	2	yes	complemented
21	C₄×#11924017	C₂²	1	4	yes	—
22	C₈×D₂₉₂	—	2	1	no	—
23	C₈×D₂₉₂	—	2	1	no	—
24	C₈×D₂₉₂	C₂²	1	2	yes	—
25	C₈×D₂₉₂	C₂²	1	2	yes	—
26	#17988613	—	2	1	no	—
27	#17988611	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
29	C₂×#14556503	—	2	1	no	—
30	#17988542	—	2	1	no	—
31	#17988338	C₂²	1	4	yes	complemented
32	#17988319	C₂²	1	4	yes	complemented
33	#17988316	C₂²	1	4	yes	—
34	#431×C₇₃	C₂²	1	1	yes	characteristic
35	#431×C₇₃	C₂²	1	1	yes	characteristic
36	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
37	#322×C₇₃	C₂²	1	2	yes	complemented
38	C₈²×C₇₃	C₂²	1	2	yes	complemented
39	C₂²×D₂₉₂	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×D₇₃	—	2	1	no	—
41	C₂×#11924017	C₂×C₄	1	4	yes	—
42	C₂×#11924012	—	2	1	no	—
43	C₂²×#10596831	—	2	1	no	—
44	C₂×C₈×D₇₃	—	2	2	no	—
45	C₂×C₈×D₇₃	—	2	2	no	—
46	C₂×C₈×D₇₃	—	4	4	no	—
47	#14556591	C₂×C₄	1	1	yes	characteristic
48	#14556591	C₂×C₄	1	1	yes	characteristic
49	#14556583	—	2	1	no	—
50	#14556570	—	2	1	no	—
51	#14556570	C₂×C₄	1	2	yes	—
52	C₄×Dic₁₄₆	—	2	1	no	—
53	C₄×Dic₁₄₆	C₂³	1	2	yes	—
54	C₄×D₂₉₂	—	2	1	no	—
55	C₄×D₂₉₂	—	2	1	no	—
56	C₄×D₂₉₂	C₂×C₄	1	2	yes	—
57	C₄×D₂₉₂	C₂³	1	2	yes	—
58	#14556550	—	2	1	no	—
59	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
60	C₂×#11924005	—	2	1	no	—
61	#14556523	C₂×C₄	1	4	yes	—
62	C₄⋊#10596831	—	2	1	no	—
63	C₄⋊#10596831	—	2	1	no	—
64	C₄⋊#10596831	C₂³	1	2	yes	—
65	C₄⋊#10596831	C₂³	1	2	yes	—
66	#14556520	—	2	2	no	—
67	#14556520	—	2	2	no	—
68	#14556516	C₂×C₄	1	4	yes	—
69	#14556513	C₂×C₄	1	4	yes	—
70	#14556503	—	2	2	no	—
71	#14556503	—	2	2	no	—
72	#14556501	—	2	2	no	—
73	C₈×Dic₇₃	—	2	2	no	—
74	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
75	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
76	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₇₃	—	2	1	no	—
78	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
79	C₂²⋊C₈×C₇₃	—	2	1	no	—
80	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
81	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
82	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
83	C₄×C₈×C₇₃	C₂³	1	2	yes	—
84	C₄×C₈×C₇₃	C₂³	1	2	yes	—
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₂³×D₇₃	—	2	1	no	—
88	C₂×Dic₁₄₆	—	2	1	no	—
89	C₂×Dic₁₄₆	C₂²×C₄	1	2	yes	—
90	C₂×D₂₉₂	—	2	1	no	—
91	C₂×D₂₉₂	—	2	4	no	—
92	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
93	C₂²×Dic₇₃	—	2	1	no	—
94	C₂×C₄×D₇₃	—	2	2	no	—
95	C₂×C₄×D₇₃	—	2	2	no	—
96	C₂×C₄×D₇₃	—	4	2	no	—
97	C₂×C₄×D₇₃	—	4	2	no	—
98	#11924017	—	2	8	no	—
99	#11924012	—	4	2	no	—
100	C₂×#10596831	—	2	2	no	—
101	C₂×#10596831	—	2	2	no	—
102	C₂×#10596831	—	4	2	no	—
103	C₈×D₇₃	—	4	4	no	—
104	C₈×D₇₃	—	4	4	no	—
105	C₄⋊Dic₇₃	—	2	1	no	—
106	C₄⋊Dic₇₃	—	2	1	no	—
107	C₄⋊Dic₇₃	—	2	1	no	—
108	C₄⋊Dic₇₃	—	2	1	no	—
109	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
110	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
111	#11924007	—	2	2	no	—
112	#11924007	—	2	2	no	—
113	C₂²⋊Dic₇₃	—	2	2	no	—
114	#11924005	—	2	2	no	—
115	#11924005	—	2	2	no	—
116	C₄×Dic₇₃	—	2	2	no	—
117	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
120	C₂×C₈×C₇₃	—	2	2	no	—
121	C₂×C₈×C₇₃	—	2	2	no	—
122	C₂×C₈×C₇₃	SD₁₆	1	4	yes	complemented
123	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
124	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₇₃	#41	1	1	yes	characteristic
131	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
132	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
133	C₂²×D₇₃	—	2	4	no	—
134	C₂²×D₇₃	—	2	2	no	—
135	C₂²×D₇₃	—	4	2	no	—
136	C₂²×D₇₃	—	4	2	no	—
137	C₂×C₄×C₇₃	—	2	2	no	—
138	C₂×C₄×C₇₃	—	4	1	no	—
139	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂×SD₁₆	1	2	yes	—
143	C₂×C₄×C₇₃	#135	1	2	yes	—
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	C₂×Dic₇₃	—	2	2	no	—
148	C₂×Dic₇₃	—	2	2	no	—
149	C₂×Dic₇₃	—	2	2	no	—
150	C₂×Dic₇₃	—	4	2	no	—
151	D₂₉₂	—	2	4	no	—
152	D₂₉₂	—	2	2	no	—
153	D₂₉₂	#131	1	4	yes	—
154	C₄×D₇₃	—	4	2	no	—
155	C₄×D₇₃	—	4	2	no	—
156	C₄×D₇₃	—	4	4	no	—
157	Dic₁₄₆	—	2	4	no	—
158	Dic₁₄₆	—	4	2	no	—
159	C₈×C₇₃	—	2	4	no	—
160	C₈×C₇₃	—	2	1	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	—	4	2	no	—
163	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
164	C₈×C₇₃	C₂×SD₁₆	1	2	yes	—
165	#10596831	—	4	2	no	—
166	#10596831	—	4	2	no	—
167	C₂²×C₇₃	—	2	1	no	—
168	C₂²×C₇₃	—	2	1	no	—
169	C₂²×C₇₃	C₄×SD₁₆	1	2	yes	—
170	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₇₃	—	2	4	no	—
172	C₂×D₇₃	—	4	8	no	—
173	C₂×D₇₃	—	4	2	no	—
174	C₂×D₇₃	—	4	2	no	—
175	C₄×C₇₃	—	2	1	no	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	C₄×SD₁₆	1	2	yes	—
179	C₄×C₇₃	C₂×#131	1	2	yes	—
180	C₄×C₇₃	C₂²×SD₁₆	1	1	yes	characteristic
181	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
182	Dic₇₃	—	4	2	no	—
183	Dic₇₃	—	4	2	no	—
184	Dic₇₃	—	4	2	no	—
185	#17902	—	73	1	no	maximal, Sylow
186	C₂×C₇₃	—	2	2	no	—
187	C₂×C₇₃	#2929	1	1	yes	characteristic
188	C₂×C₇₃	C₂×C₄×SD₁₆	1	1	yes	characteristic
189	C₂×C₇₃	#2955	1	1	yes	characteristic
190	D₇₃	—	4	4	no	—
191	#3101	—	73	1	no	—
192	#2960	—	73	1	no	—
193	C₂×C₈×D₄	—	73	1	no	—
194	#1582	—	73	4	no	—
195	C₈×SD₁₆	—	73	4	no	—
196	#1570	—	73	1	no	—
197	#1486	—	73	1	no	—
198	#1481	—	73	1	no	—
199	#1450	D₇₃	1	1	yes	characteristic, complemented
200	C₇₃	#17902	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₈	—	146	1	no	—
202	#533	—	73	1	no	—
203	C₂×C₄×D₄	—	73	1	no	—
204	#483	—	73	2	no	—
205	#476	—	73	2	no	—
206	#461	—	73	2	no	—
207	#450	—	73	2	no	—
208	#446	—	73	2	no	—
209	#446	—	146	1	no	—
210	C₈×Q₈	—	73	2	no	—
211	C₈×Q₈	—	146	1	no	—
212	C₄×SD₁₆	—	73	4	no	—
213	C₈×D₄	—	73	2	no	—
214	C₈×D₄	—	73	2	no	—
215	C₈×D₄	—	146	1	no	—
216	C₈×D₄	—	146	1	no	—
217	#431	C₂×D₇₃	1	1	yes	characteristic
218	#431	C₂×D₇₃	1	1	yes	characteristic
219	#424	—	146	1	no	—
220	#423	—	73	1	no	—
221	C₂×C₄⋊C₈	—	73	1	no	—
222	C₂×C₂²⋊C₈	—	146	1	no	—
223	#405	—	146	1	no	—
224	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
225	#334	—	73	4	no	—
226	Q₈⋊C₈	—	73	4	no	—
227	D₄⋊C₈	—	73	4	no	—
228	#322	C₂×D₇₃	1	2	yes	complemented
229	C₈²	C₂×D₇₃	1	2	yes	complemented
230	C₂²×D₄	—	73	1	no	—
231	C₂³×C₄	—	146	1	no	—
232	C₂×SD₁₆	—	73	4	no	—
233	C₂×M₁₆	—	146	1	no	—
234	C₂²×C₈	—	292	4	no	—
235	C₂²×C₈	—	146	1	no	—
236	C₂²×C₈	—	146	2	no	—
237	C₂²×C₈	—	146	2	no	—
238	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
239	C₂²×C₈	D₂₉₂	1	1	yes	characteristic
240	C₄⋊Q₈	—	73	1	no	—
241	C₄⋊Q₈	—	73	1	no	—
242	#125	—	146	1	no	—
243	C₂²⋊Q₈	—	73	2	no	—
244	C₂²⋊Q₈	—	146	1	no	—
245	C₄×Q₈	—	73	2	no	—
246	C₄×Q₈	—	146	1	no	—
247	C₄×D₄	—	73	2	no	—
248	C₄×D₄	—	73	2	no	—
249	C₄×D₄	—	146	1	no	—
250	C₄×D₄	—	146	1	no	—
251	#117	—	146	1	no	—
252	C₂×C₄⋊C₄	—	73	1	no	—
253	C₂×C₂²⋊C₄	—	146	1	no	—
254	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
255	#106	—	73	4	no	—
256	C₄⋊C₈	—	2	1	no	—
257	C₄⋊C₈	—	73	2	no	—
258	C₄⋊C₈	—	73	2	no	—
259	C₄⋊C₈	—	146	2	no	—
260	C₄⋊C₈	—	146	2	no	—
261	C₄⋊C₈	—	146	1	no	—
262	C₄⋊C₈	—	146	1	no	—
263	C₄⋊C₈	C₄×D₇₃	1	2	yes	—
264	#103	—	73	4	no	—
265	#102	—	73	4	no	—
266	C₂²⋊C₈	—	2	1	no	—
267	C₂²⋊C₈	—	146	2	no	—
268	C₂²⋊C₈	—	146	2	no	—
269	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
270	#97	—	146	2	no	—
271	C₄×C₈	—	146	2	no	—
272	C₄×C₈	D₂₉₂	1	1	yes	characteristic
273	C₄×C₈	D₂₉₂	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₇₃	1	2	yes	—
275	C₄×C₈	C₂²×D₇₃	1	2	yes	—
276	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
277	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
278	C₂⁴	—	146	1	no	—
279	C₂×Q₈	—	73	2	no	—
280	C₂×Q₈	—	146	1	no	—
281	C₂×D₄	—	73	2	no	—
282	C₂×D₄	—	146	4	no	—
283	C₂×D₄	—	146	1	no	—
284	C₂²×C₄	—	292	2	no	—
285	C₂²×C₄	—	292	2	no	—
286	C₂²×C₄	—	146	1	no	—
287	C₂²×C₄	—	146	2	no	—
288	C₂²×C₄	—	146	2	no	—
289	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
290	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
291	C₂²×C₄	#11924028	1	1	yes	characteristic
292	SD₁₆	—	146	8	no	—
293	M₁₆	—	292	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	292	2	no	—
297	C₂×C₈	—	292	4	no	—
298	C₂×C₈	—	292	4	no	—
299	C₂×C₈	—	146	2	no	—
300	C₂×C₈	—	146	2	no	—
301	C₂×C₈	#11924017	1	4	yes	complemented
302	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
303	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
306	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
307	C₄⋊C₄	—	73	2	no	—
308	C₄⋊C₄	—	73	2	no	—
309	C₄⋊C₄	—	146	2	no	—
310	C₄⋊C₄	—	146	1	no	—
311	C₄⋊C₄	—	146	1	no	—
312	C₄⋊C₄	—	146	1	no	—
313	C₄⋊C₄	—	146	1	no	—
314	C₄⋊C₄	—	146	2	no	—
315	C₂²⋊C₄	—	146	2	no	—
316	C₂²⋊C₄	—	146	2	no	—
317	C₂²⋊C₄	—	146	2	no	—
318	C₄²	—	146	2	no	—
319	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
320	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
321	C₄²	#11924028	1	1	yes	characteristic
322	C₄²	C₂³×D₇₃	1	1	yes	characteristic
323	C₂³	—	292	2	no	—
324	C₂³	—	292	2	no	—
325	C₂³	—	146	4	no	—
326	C₂³	—	146	2	no	—
327	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
328	Q₈	—	292	2	no	—
329	Q₈	—	146	4	no	—
330	D₄	—	73	4	no	—
331	D₄	—	146	4	no	—
332	D₄	—	146	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	292	2	no	—
336	C₂×C₄	—	292	4	no	—
337	C₂×C₄	—	292	2	no	—
338	C₂×C₄	—	292	2	no	—
339	C₂×C₄	—	146	2	no	—
340	C₂×C₄	—	146	2	no	—
341	C₂×C₄	—	146	2	no	—
342	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
343	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
344	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
345	C₂×C₄	C₂×#11924017	1	2	yes	—
346	C₂×C₄	#14556613	1	2	yes	—
347	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
348	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
349	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	2	1	no	—
352	C₈	—	2	1	no	—
353	C₈	—	4	2	no	—
354	C₈	—	292	2	no	—
355	C₈	—	292	2	no	—
356	C₈	C₂×#11924017	1	2	yes	—
357	C₈	C₂×#11924017	1	2	yes	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	292	8	no	—
361	C₂²	—	292	2	no	—
362	C₂²	—	292	2	no	—
363	C₂²	—	146	4	no	—
364	C₂²	C₄×#11924017	1	2	yes	—
365	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	292	2	no	—
370	C₄	—	292	2	no	—
371	C₄	—	292	2	no	—
372	C₄	C₄×#11924017	1	2	yes	—
373	C₄	#131×D₇₃	1	2	yes	—
374	C₄	C₂²×#11924017	1	1	yes	characteristic
375	C₄	C₂×#14556613	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	292	4	no	—
378	C₂	#22618627	1	1	yes	characteristic
379	C₂	C₂×C₄×#11924017	1	1	yes	characteristic
380	C₂	#22618835	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1450	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂²×D₂₉₂	C₂×C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1450	D₇₃
hypocenter	C₇₃	#17902
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
2. #22618865	C₂	semidirect
4. #22608769	C₂	semidirect
5. C₈×#11924017	C₂	semidirect
6. #22608702	C₂	semidirect
8. #22608335	C₂	semidirect
9. C₇₃×#1450	C₂	semidirect
18. #17988717	C₂²	semidirect
20. C₈×Dic₁₄₆	C₂²	semidirect
31. #17988338	C₂²	semidirect
32. #17988319	C₂²	semidirect
37. #322×C₇₃	C₂²	semidirect
38. C₈²×C₇₃	C₂²	semidirect
122. C₂×C₈×C₇₃	SD₁₆	semidirect
199. #1450	D₇₃	semidirect
200. C₇₃	#17902	semidirect
228. #322	C₂×D₇₃	semidirect
229. C₈²	C₂×D₇₃	semidirect
301. C₂×C₈	#11924017	semidirect

Maximal subgroups

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

subgroup	index
#22619940	2
#22618865	2
C₂×C₈×D₂₉₂	2
#22608769	2
C₈×#11924017	2
#22608702	2
#22608357	2
#22608335	2
C₇₃×#1450	2
#17902	73