Subgroups — #33055930

Order 18688.

678 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22619962	C₂	1	1	yes	maximal, characteristic
2	#22618644	C₂	1	1	yes	maximal, characteristic
3	#22608809	C₂	1	4	yes	maximal, complemented
4	#22608741	C₂	1	4	yes	maximal, complemented
5	#22608705	C₂	1	1	yes	maximal, characteristic, complemented
6	#22608360	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608329	C₂	1	1	yes	maximal, characteristic
8	#22608223	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₈×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556597	—	2	1	no	—
11	#17989137	C₂²	1	1	yes	characteristic
12	#17988905	C₄	1	2	yes	—
13	#17988872	C₄	1	2	yes	—
14	#17988787	C₄	1	2	yes	—
15	#17988726	C₄	1	2	yes	—
16	C₄×#11924015	C₂²	1	4	yes	—
17	#17988666	—	2	1	no	—
18	#17988666	C₂²	1	2	yes	—
19	C₈×D₂₉₂	—	2	1	no	—
20	C₈×D₂₉₂	C₂²	1	2	yes	—
21	#17988640	C₂²	1	1	yes	characteristic
22	#17988640	C₂²	1	1	yes	characteristic
23	#17988611	C₂²	1	1	yes	characteristic
24	#17988553	—	2	1	no	—
25	#17988339	C₂²	1	4	yes	—
26	Q₈⋊#10596831	C₂²	1	4	yes	—
27	#17988314	C₂²	1	4	yes	—
28	#17988309	C₂²	1	2	yes	—
29	#17988309	C₂²	1	2	yes	—
30	C₂×C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₇₃	—	2	1	no	—
32	C₈×Q₈×C₇₃	—	2	1	no	—
33	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
34	C₈×Q₈×C₇₃	C₂²	1	2	yes	—
35	C₂×C₄⋊C₈×C₇₃	—	2	1	no	—
36	C₂×C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₇₃	—	2	1	no	—
38	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×#11924028	—	2	1	no	—
40	C₂×#11924015	C₂×C₄	1	4	yes	—
41	#14556597	—	4	4	no	—
42	C₂×#11924012	—	2	1	no	—
43	C₂×#11924011	—	2	2	no	—
44	C₂×C₈×D₇₃	—	2	2	no	—
45	#14556591	C₂×C₄	1	1	yes	characteristic
46	#14556588	C₂×C₄	1	1	yes	characteristic
47	#14556579	—	2	1	no	—
48	#14556565	—	2	1	no	—
49	#14556565	C₂×C₄	1	2	yes	—
50	C₄×D₂₉₂	—	2	1	no	—
51	C₄×D₂₉₂	C₂³	1	2	yes	—
52	#14556522	C₂×C₄	1	4	yes	—
53	C₄⋊#10596831	—	2	1	no	—
54	C₄⋊#10596831	—	2	1	no	—
55	C₄⋊#10596831	C₂×C₄	1	2	yes	—
56	C₄⋊#10596831	C₂³	1	2	yes	—
57	#14556515	C₂×C₄	1	4	yes	—
58	#14556511	C₂×C₄	1	4	yes	—
59	C₂²⋊#10596831	—	2	1	no	—
60	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
61	#14556503	—	2	2	no	—
62	#14556503	—	2	2	no	—
63	C₄×#10596831	C₂³	1	2	yes	—
64	C₄×#10596831	C₂³	1	2	yes	—
65	C₂²×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₇₃	—	2	1	no	—
67	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
69	C₄×Q₈×C₇₃	—	2	1	no	—
70	C₄×Q₈×C₇₃	—	2	1	no	—
71	C₄×Q₈×C₇₃	C₂×C₄	1	2	yes	—
72	C₄×Q₈×C₇₃	C₂³	1	2	yes	—
73	C₂×C₄⋊C₄×C₇₃	—	2	1	no	—
74	C₂×C₄⋊C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄²×C₇₃	—	2	1	no	—
76	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₈×C₇₃	—	2	2	no	—
78	C₄⋊C₈×C₇₃	—	2	2	no	—
79	C₄⋊C₈×C₇₃	—	2	1	no	—
80	C₄⋊C₈×C₇₃	C₂³	1	2	yes	—
81	C₄×C₈×C₇₃	—	2	2	no	—
82	C₄×C₈×C₇₃	—	2	2	no	—
83	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
84	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	#11924028	—	4	2	no	—
88	#11924028	—	4	2	no	—
89	C₂×Dic₁₄₆	—	2	1	no	—
90	C₂×D₂₉₂	—	2	1	no	—
91	C₂×D₂₉₂	—	2	1	no	—
92	C₂×D₂₉₂	C₂²×C₄	1	2	yes	—
93	C₂×C₂²⋊D₇₃	—	2	2	no	—
94	C₂×C₄×D₇₃	—	2	2	no	—
95	#11924015	—	2	8	no	—
96	#11924012	—	4	2	no	—
97	#11924011	—	4	4	no	—
98	C₂×#10596831	—	2	2	no	—
99	C₂×#10596831	—	2	2	no	—
100	C₂×#10596831	C₂²×C₄	1	4	yes	—
101	C₈×D₇₃	—	4	4	no	—
102	C₄⋊Dic₇₃	—	2	1	no	—
103	C₄⋊Dic₇₃	C₂²×C₄	1	2	yes	—
104	#11924005	—	2	2	no	—
105	#11924005	—	2	2	no	—
106	C₂×Q₈×C₇₃	—	2	4	no	—
107	C₂×Q₈×C₇₃	—	2	1	no	—
108	C₂×Q₈×C₇₃	C₂²×C₄	1	2	yes	—
109	C₂²×C₄×C₇₃	—	2	1	no	—
110	C₂²×C₄×C₇₃	—	2	1	no	—
111	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₇₃	—	2	2	no	—
115	C₂×C₈×C₇₃	—	2	2	no	—
116	C₂×C₈×C₇₃	—	2	2	no	—
117	C₂×C₈×C₇₃	—	4	2	no	—
118	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
122	C₄⋊C₄×C₇₃	—	2	2	no	—
123	C₄⋊C₄×C₇₃	—	2	1	no	—
124	C₄⋊C₄×C₇₃	—	2	2	no	—
125	C₄⋊C₄×C₇₃	C₂²×C₄	1	2	yes	—
126	C₄²×C₇₃	—	2	2	no	—
127	C₄²×C₇₃	—	2	2	no	—
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	2	no	—
134	Q₈×C₇₃	—	2	4	no	—
135	Q₈×C₇₃	—	2	2	no	—
136	Q₈×C₇₃	#131	1	4	yes	complemented
137	C₂×C₄×C₇₃	—	2	4	no	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	—	2	2	no	—
140	C₂×C₄×C₇₃	—	2	2	no	—
141	C₂×C₄×C₇₃	—	4	1	no	—
142	C₂×C₄×C₇₃	—	4	2	no	—
143	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
144	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₇₃	#137	1	2	yes	—
148	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
151	C₂²⋊D₇₃	—	4	4	no	—
152	C₂×Dic₇₃	—	2	2	no	—
153	D₂₉₂	—	2	4	no	—
154	D₂₉₂	—	4	2	no	—
155	D₂₉₂	—	4	2	no	—
156	C₄×D₇₃	—	4	2	no	—
157	C₄×D₇₃	—	4	2	no	—
158	Dic₁₄₆	—	4	2	no	—
159	C₈×C₇₃	—	2	1	no	—
160	C₈×C₇₃	—	2	1	no	—
161	C₈×C₇₃	—	2	1	no	—
162	C₈×C₇₃	—	2	1	no	—
163	C₈×C₇₃	—	4	2	no	—
164	#10596831	—	2	4	no	—
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	#440	1	2	yes	—
170	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
171	C₂×D₇₃	—	4	2	no	—
172	C₂×D₇₃	—	4	2	no	—
173	C₂×D₇₃	—	4	2	no	—
174	C₄×C₇₃	—	2	4	no	—
175	C₄×C₇₃	—	2	1	no	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	2	1	no	—
178	C₄×C₇₃	—	4	2	no	—
179	C₄×C₇₃	#440	1	2	yes	—
180	C₄×C₇₃	C₂×#131	1	2	yes	—
181	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₄×C₇₃	C₂×#137	1	1	yes	characteristic
183	Dic₇₃	—	4	2	no	—
184	#17908	—	73	1	no	maximal, Sylow
185	C₂×C₇₃	—	2	2	no	—
186	C₂×C₇₃	#2929	1	1	yes	characteristic
187	C₂×C₇₃	C₂×#440	1	1	yes	characteristic
188	C₂×C₇₃	#2955	1	1	yes	characteristic
189	D₇₃	—	8	2	no	—
190	#3103	—	73	1	no	—
191	C₂×C₈×Q₈	D₇₃	1	1	yes	characteristic, complemented
192	#2931	—	73	1	no	—
193	#1588	—	73	4	no	—
194	SD₁₆⋊C₈	—	73	4	no	—
195	#1571	—	73	1	no	—
196	#1487	—	73	1	no	—
197	#1480	—	73	1	no	—
198	#1452	—	73	1	no	—
199	C₇₃	#17908	1	1	yes	characteristic, complemented, Sylow
200	C₂×#131	—	146	1	no	—
201	#532	—	73	1	no	—
202	C₂×C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
203	#481	—	73	2	no	—
204	#475	—	73	2	no	—
205	#460	—	73	2	no	—
206	#448	—	73	2	no	—
207	C₈×Q₈	—	2	1	no	—
208	C₈×Q₈	—	2	1	no	—
209	C₈×Q₈	C₂×D₇₃	1	2	yes	—
210	C₈×Q₈	C₂×D₇₃	1	2	yes	—
211	C₄×SD₁₆	—	73	4	no	—
212	#436	—	73	2	no	—
213	#436	—	146	1	no	—
214	C₈×D₄	—	73	2	no	—
215	C₈×D₄	—	146	1	no	—
216	#431	—	73	1	no	—
217	#431	—	73	1	no	—
218	#423	—	73	1	no	—
219	C₂×C₄⋊C₈	—	2	1	no	—
220	C₂×C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
221	#408	—	146	1	no	—
222	C₂×C₄×C₈	—	2	1	no	—
223	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
224	#335	—	73	4	no	—
225	Q₈⋊C₈	—	73	4	no	—
226	D₄⋊C₈	—	73	4	no	—
227	#322	—	73	2	no	—
228	#322	—	73	2	no	—
229	C₂×#41	—	146	1	no	—
230	C₂²×Q₈	C₄×D₇₃	1	1	yes	characteristic
231	C₂×SD₁₆	—	73	4	no	—
232	#131	—	292	4	no	—
233	C₂×M₁₆	—	146	1	no	—
234	C₂×M₁₆	—	146	2	no	—
235	C₂²×C₈	—	2	1	no	—
236	C₂²×C₈	—	146	2	no	—
237	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
238	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
239	C₄⋊Q₈	—	73	1	no	—
240	#127	—	73	1	no	—
241	#124	—	146	1	no	—
242	#121	—	73	2	no	—
243	#121	—	146	1	no	—
244	C₄×Q₈	—	2	1	no	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₄×D₇₃	1	2	yes	—
247	C₄×Q₈	C₂²×D₇₃	1	2	yes	—
248	C₄×D₄	—	73	2	no	—
249	C₄×D₄	—	146	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	C₄×D₇₃	1	1	yes	characteristic
252	C₂×C₄²	—	2	1	no	—
253	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
254	#106	—	73	4	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	2	2	no	—
257	C₄⋊C₈	—	2	1	no	—
258	C₄⋊C₈	—	73	2	no	—
259	C₄⋊C₈	—	73	2	no	—
260	C₄⋊C₈	—	146	1	no	—
261	C₄⋊C₈	—	146	1	no	—
262	C₄⋊C₈	C₂²×D₇₃	1	2	yes	—
263	#103	—	73	4	no	—
264	#102	—	73	4	no	—
265	C₂²⋊C₈	—	73	2	no	—
266	C₂²⋊C₈	—	146	1	no	—
267	C₂²⋊C₈	—	146	2	no	—
268	C₂²⋊C₈	—	146	2	no	—
269	C₄×C₈	—	2	2	no	—
270	C₄×C₈	—	2	2	no	—
271	C₄×C₈	—	73	2	no	—
272	C₄×C₈	—	73	2	no	—
273	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
274	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
275	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
276	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
277	#41	—	292	2	no	—
278	#41	—	292	2	no	—
279	C₂×Q₈	—	2	4	no	—
280	C₂×Q₈	—	2	1	no	—
281	C₂×Q₈	—	146	1	no	—
282	C₂×Q₈	C₂×C₄×D₇₃	1	2	yes	—
283	C₂×D₄	—	73	2	no	—
284	C₂×D₄	—	146	2	no	—
285	C₂×D₄	—	146	1	no	—
286	C₂×D₄	—	146	1	no	—
287	C₂²×C₄	—	2	1	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	146	2	no	—
290	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
292	C₂²×C₄	#11924028	1	1	yes	characteristic
293	SD₁₆	—	146	8	no	—
294	M₁₆	—	292	4	no	—
295	M₁₆	—	292	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	2	2	no	—
299	C₂×C₈	—	4	2	no	—
300	C₂×C₈	—	292	4	no	—
301	C₂×C₈	—	73	4	no	—
302	C₂×C₈	—	146	2	no	—
303	C₂×C₈	—	146	2	no	—
304	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
305	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
308	C₄⋊C₄	—	2	2	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	73	2	no	—
312	C₄⋊C₄	—	146	1	no	—
313	C₄⋊C₄	C₂×C₄×D₇₃	1	2	yes	—
314	C₂²⋊C₄	—	146	2	no	—
315	C₂²⋊C₄	—	146	2	no	—
316	C₄²	—	2	2	no	—
317	C₄²	—	2	2	no	—
318	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
319	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
320	C₄²	#11924028	1	1	yes	characteristic
321	C₄²	C₂³×D₇₃	1	1	yes	characteristic
322	C₂³	—	146	2	no	—
323	C₂³	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
324	Q₈	—	2	4	no	—
325	Q₈	—	2	2	no	—
326	Q₈	—	292	2	no	—
327	Q₈	#14556597	1	4	yes	complemented
328	D₄	—	292	2	no	—
329	D₄	—	292	4	no	—
330	D₄	—	292	2	no	—
331	D₄	—	146	4	no	—
332	C₂×C₄	—	2	4	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	4	1	no	—
337	C₂×C₄	—	4	2	no	—
338	C₂×C₄	—	292	2	no	—
339	C₂×C₄	—	292	2	no	—
340	C₂×C₄	—	146	2	no	—
341	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
342	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
343	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
344	C₂×C₄	#14556616	1	2	yes	—
345	C₂×C₄	#14556622	1	2	yes	—
346	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
347	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
348	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
349	C₈	—	2	1	no	—
350	C₈	—	2	1	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	—	146	4	no	—
357	C₂²	—	2	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	292	2	no	—
360	C₂²	—	292	2	no	—
361	C₂²	—	292	2	no	—
362	C₂²	#17988685	1	2	yes	—
363	C₂²	C₂×C₄×C₂²⋊D₇₃	1	1	yes	characteristic
364	C₄	—	2	4	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	292	2	no	—
370	C₄	#17988685	1	2	yes	—
371	C₄	C₂×#14556597	1	2	yes	—
372	C₄	C₂×#14556616	1	1	yes	characteristic
373	C₄	C₂×#14556622	1	1	yes	characteristic
374	C₂	—	2	2	no	—
375	C₂	—	584	2	no	—
376	C₂	#22618625	1	1	yes	characteristic
377	C₂	C₂×#17988685	1	1	yes	characteristic
378	C₂	#22618832	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₂×C₈×Q₈×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	#17989137	C₂²
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₃
hypocenter	C₇₃	#17908
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
3. #22608809	C₂	semidirect
4. #22608741	C₂	semidirect
5. #22608705	C₂	semidirect
6. #22608360	C₂	semidirect
8. #22608223	C₂	semidirect
9. C₂×C₈×Q₈×C₇₃	C₂	semidirect
136. Q₈×C₇₃	#131	semidirect
191. C₂×C₈×Q₈	D₇₃	semidirect
199. C₇₃	#17908	semidirect
327. Q₈	#14556597	semidirect

Maximal subgroups

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

subgroup	index
#22619962	2
#22618644	2
#22608809	2
#22608741	2
#22608705	2
#22608360	2
#22608329	2
#22608223	2
C₂×C₈×Q₈×C₇₃	2
#17908	73