Subgroups — #33051686

Order 18688.

734 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#10596831	C₂	1	1	yes	maximal, characteristic
2	#22618575	C₂	1	1	yes	maximal, characteristic
3	#22608231	C₂	1	2	yes	maximal, complemented
4	#22608228	C₂	1	2	yes	maximal, complemented
5	#22608224	C₂	1	2	yes	maximal, complemented
6	#22608221	C₂	1	1	yes	maximal, characteristic, complemented
7	#22608215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈×#11924012	C₂	1	1	yes	maximal, characteristic, complemented
9	M₁₆×#10596831	C₂	1	2	yes	maximal, complemented
10	C₂×#17988308	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2965	C₂	1	1	yes	maximal, characteristic
12	Q₈×#10596831	C₄	1	2	yes	—
13	D₄×#10596831	C₄	1	2	yes	—
14	D₄×#10596831	C₄	1	4	yes	—
15	#17988648	C₄	1	4	yes	—
16	#17988640	C₂²	1	1	yes	characteristic
17	#17988640	C₂²	1	2	yes	—
18	#17988640	C₂²	1	2	yes	—
19	#17988640	C₂²	1	2	yes	—
20	#17988614	C₄	1	2	yes	—
21	#17988554	C₄	1	2	yes	—
22	C₄×#11924012	C₂²	1	1	yes	characteristic
23	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
24	C₂×C₄×#10596831	C₂²	1	2	yes	—
25	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
26	#17988309	C₂²	1	2	yes	—
27	#17988309	C₂²	1	2	yes	—
28	#17988309	C₂²	1	2	yes	—
29	#17988308	C₂²	1	1	yes	characteristic
30	#17988308	C₂²	1	1	yes	characteristic
31	#17988308	C₂²	1	1	yes	characteristic
32	#17988308	C₂²	1	1	yes	characteristic
33	#17988308	C₂²	1	2	yes	—
34	C₈×#10596831	C₂²	1	2	yes	—
35	C₈×#10596831	C₂²	1	2	yes	—
36	C₄×#41×C₇₃	C₂²	1	1	yes	characteristic
37	#446×C₇₃	C₄	1	2	yes	—
38	#436×C₇₃	C₄	1	2	yes	—
39	#435×C₇₃	C₄	1	4	yes	—
40	#431×C₇₃	C₂²	1	1	yes	characteristic
41	#431×C₇₃	C₂²	1	2	yes	—
42	C₄×M₁₆×C₇₃	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
44	C₂×#11924012	C₂×C₄	1	2	yes	—
45	C₂²×#10596831	C₂×C₄	1	2	yes	—
46	C₂²×#10596831	C₂×C₄	1	2	yes	—
47	C₂²×#10596831	C₂×C₄	1	4	yes	—
48	C₄⋊#10596831	C₂×C₄	1	4	yes	—
49	C₄⋊#10596831	C₂×C₄	1	4	yes	—
50	C₄⋊#10596831	C₂×C₄	1	2	yes	—
51	C₄⋊#10596831	C₂×C₄	1	4	yes	—
52	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
53	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
54	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
55	C₂²⋊#10596831	C₂×C₄	1	4	yes	—
56	#14556502	C₂×C₄	1	2	yes	—
57	C₄×#10596831	C₂×C₄	1	2	yes	—
58	C₄×#10596831	C₂×C₄	1	2	yes	—
59	C₄×#10596831	C₂×C₄	1	4	yes	—
60	C₄×#10596831	C₂³	1	1	yes	characteristic
61	C₄×#10596831	C₂³	1	2	yes	—
62	C₄×#10596831	C₂³	1	1	yes	characteristic
63	C₄×#10596831	C₂³	1	1	yes	characteristic
64	C₄×#10596831	C₂³	1	2	yes	—
65	C₄×#10596831	C₂³	1	1	yes	characteristic
66	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
67	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
68	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
69	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
70	C₄×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
71	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
72	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₄×D₄×C₇₃	C₂×C₄	1	2	yes	—
74	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
75	#117×C₇₃	C₂×C₄	1	1	yes	characteristic
76	#117×C₇₃	C₂×C₄	1	2	yes	—
77	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
78	C₂×C₄²×C₇₃	C₂³	1	2	yes	—
79	C₄⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
80	C₄⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
81	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
82	C₂²⋊C₈×C₇₃	C₂×C₄	1	4	yes	—
83	#97×C₇₃	C₂×C₄	1	4	yes	—
84	C₄×C₈×C₇₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
87	C₄×C₈×C₇₃	C₂³	1	2	yes	—
88	#11924012	—	2	4	no	—
89	C₂×#10596831	—	2	4	no	—
90	C₂×#10596831	—	4	2	no	—
91	C₂×#10596831	—	4	4	no	—
92	C₂×#10596831	C₂²×C₄	1	2	yes	—
93	C₂×#10596831	C₂²×C₄	1	2	yes	—
94	C₂×#10596831	C₂²×C₄	1	2	yes	—
95	C₂×#10596831	C₂²×C₄	1	4	yes	—
96	C₂×#10596831	C₂²×C₄	1	2	yes	—
97	C₂×#10596831	C₂²×C₄	1	4	yes	—
98	#41×C₇₃	—	2	4	no	—
99	C₂×Q₈×C₇₃	C₄²	1	1	yes	characteristic
100	C₂×D₄×C₇₃	C₄²	1	2	yes	—
101	C₂×D₄×C₇₃	C₄²	1	1	yes	characteristic
102	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
103	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
104	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
108	M₁₆×C₇₃	—	2	8	no	—
109	C₂×C₈×C₇₃	—	2	4	no	—
110	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
112	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
113	C₄⋊C₄×C₇₃	C₄²	1	2	yes	—
114	C₄⋊C₄×C₇₃	C₄²	1	1	yes	characteristic
115	C₄⋊C₄×C₇₃	C₄²	1	2	yes	—
116	C₄⋊C₄×C₇₃	C₄²	1	1	yes	characteristic
117	C₂²⋊C₄×C₇₃	C₄²	1	2	yes	—
118	C₂²⋊C₄×C₇₃	C₄²	1	2	yes	—
119	C₂²⋊C₄×C₇₃	C₄²	1	2	yes	—
120	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
122	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
123	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
126	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
127	C₂³×C₇₃	C₂×C₄²	1	2	yes	—
128	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
129	Q₈×C₇₃	—	2	2	no	—
130	D₄×C₇₃	—	2	4	no	—
131	D₄×C₇₃	—	2	2	no	—
132	C₂×C₄×C₇₃	—	2	2	no	—
133	C₂×C₄×C₇₃	—	2	4	no	—
134	C₂×C₄×C₇₃	—	2	4	no	—
135	C₂×C₄×C₇₃	—	2	2	no	—
136	C₂×C₄×C₇₃	—	2	4	no	—
137	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
141	C₂×C₄×C₇₃	C₂×C₄²	1	2	yes	—
142	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
143	C₂×C₄×C₇₃	#131	1	4	yes	—
144	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
145	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
147	C₈×C₇₃	—	2	2	no	—
148	C₈×C₇₃	—	2	4	no	—
149	C₈×C₇₃	#131	1	4	yes	—
150	#10596831	—	2	2	no	—
151	#10596831	—	2	2	no	—
152	#10596831	—	2	4	no	—
153	#10596831	—	2	2	no	—
154	#10596831	—	2	4	no	—
155	#10596831	—	2	2	no	—
156	C₂²×C₇₃	—	2	1	no	—
157	C₂²×C₇₃	—	2	1	no	—
158	C₂²×C₇₃	—	2	2	no	—
159	C₂²×C₇₃	—	2	2	no	—
160	C₂²×C₇₃	—	2	2	no	—
161	C₂²×C₇₃	#405	1	2	yes	—
162	C₂²×C₇₃	C₂²×C₄²	1	1	yes	characteristic
163	C₄×C₇₃	—	2	2	no	—
164	C₄×C₇₃	—	2	1	no	—
165	C₄×C₇₃	—	2	2	no	—
166	C₄×C₇₃	—	2	1	no	—
167	C₄×C₇₃	—	2	2	no	—
168	C₄×C₇₃	#405	1	2	yes	—
169	C₄×C₇₃	C₂×#131	1	2	yes	—
170	C₄×C₇₃	C₂×#131	1	2	yes	—
171	C₄×C₇₃	#568	1	2	yes	—
172	#17331	—	73	1	no	maximal, Sylow
173	C₂×C₇₃	—	2	2	no	—
174	C₂×C₇₃	—	4	2	no	—
175	C₂×C₇₃	C₂×#405	1	1	yes	characteristic
176	C₂×C₇₃	C₄×#131	1	1	yes	characteristic
177	C₂×C₇₃	#2875	1	1	yes	characteristic
178	#2965	D₇₃	1	1	yes	characteristic
179	C₈×#41	—	73	1	no	—
180	#2922	—	73	1	no	—
181	#1454	—	73	2	no	—
182	#1453	—	73	2	no	—
183	#1452	—	73	2	no	—
184	#1451	—	73	1	no	—
185	#1450	—	73	1	no	—
186	C₈×M₁₆	—	73	2	no	—
187	C₈×M₁₆	—	73	1	no	—
188	C₂×#322	—	73	1	no	—
189	C₇₃	#17331	1	1	yes	characteristic, complemented, Sylow
190	C₄×#41	C₂×D₇₃	1	1	yes	characteristic
191	#446	Dic₇₃	1	2	yes	—
192	C₈×Q₈	—	73	2	no	—
193	#436	Dic₇₃	1	2	yes	—
194	#435	Dic₇₃	1	4	yes	—
195	C₈×D₄	—	73	2	no	—
196	C₈×D₄	—	73	4	no	—
197	#433	—	73	4	no	—
198	#431	—	73	2	no	—
199	#431	—	73	2	no	—
200	#431	—	73	1	no	—
201	#431	—	73	2	no	—
202	#431	C₂×D₇₃	1	1	yes	characteristic
203	#431	C₂×D₇₃	1	2	yes	—
204	#424	—	73	2	no	—
205	#408	—	73	2	no	—
206	C₄×M₁₆	—	73	1	no	—
207	C₄×M₁₆	C₂×D₇₃	1	2	yes	—
208	C₂×C₄×C₈	—	73	1	no	—
209	C₂×C₄×C₈	—	73	1	no	—
210	C₂×C₄×C₈	—	73	2	no	—
211	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
212	#322	—	73	1	no	—
213	#322	—	73	1	no	—
214	#322	—	73	1	no	—
215	#322	—	73	1	no	—
216	#322	—	73	2	no	—
217	#322	—	73	2	no	—
218	#322	—	73	2	no	—
219	#322	—	73	2	no	—
220	C₈²	—	73	2	no	—
221	C₈²	—	73	2	no	—
222	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
223	C₂×M₁₆	—	73	2	no	—
224	C₂×M₁₆	C₂×Dic₇₃	1	4	yes	—
225	C₂²×C₈	—	73	2	no	—
226	C₂²×C₈	—	73	4	no	—
227	C₂²×C₈	—	73	2	no	—
228	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
229	C₄×Q₈	C₄×D₇₃	1	1	yes	characteristic
230	C₄×Q₈	C₂×Dic₇₃	1	1	yes	characteristic
231	C₄×D₄	C₄×D₇₃	1	2	yes	—
232	C₄×D₄	C₄×D₇₃	1	1	yes	characteristic
233	C₄×D₄	C₂×Dic₇₃	1	2	yes	—
234	C₄×D₄	C₂×Dic₇₃	1	1	yes	characteristic
235	#117	C₄×D₇₃	1	1	yes	characteristic
236	#117	C₄×D₇₃	1	2	yes	—
237	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₇₃	1	2	yes	—
239	C₄⋊C₈	—	73	4	no	—
240	C₄⋊C₈	—	73	4	no	—
241	C₄⋊C₈	—	73	2	no	—
242	C₄⋊C₈	—	73	4	no	—
243	C₄⋊C₈	C₂×Dic₇₃	1	4	yes	—
244	C₄⋊C₈	C₂×Dic₇₃	1	2	yes	—
245	C₂²⋊C₈	—	73	4	no	—
246	C₂²⋊C₈	—	73	4	no	—
247	C₂²⋊C₈	—	73	2	no	—
248	C₂²⋊C₈	—	73	4	no	—
249	C₂²⋊C₈	C₂×Dic₇₃	1	2	yes	—
250	C₂²⋊C₈	C₂×Dic₇₃	1	4	yes	—
251	#97	—	73	2	no	—
252	#97	C₂×Dic₇₃	1	4	yes	—
253	C₄×C₈	—	73	2	no	—
254	C₄×C₈	—	73	2	no	—
255	C₄×C₈	—	73	4	no	—
256	C₄×C₈	—	73	2	no	—
257	C₄×C₈	—	73	1	no	—
258	C₄×C₈	—	73	2	no	—
259	C₄×C₈	—	73	1	no	—
260	C₄×C₈	—	73	1	no	—
261	C₄×C₈	—	73	1	no	—
262	C₄×C₈	C₂×Dic₇₃	1	2	yes	—
263	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₇₃	1	2	yes	—
265	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
266	#41	—	2	4	no	—
267	C₂×Q₈	C₄×Dic₇₃	1	1	yes	characteristic
268	C₂×D₄	C₄×Dic₇₃	1	2	yes	—
269	C₂×D₄	C₄×Dic₇₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
271	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
272	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
273	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
274	C₂²×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
275	C₂²×C₄	C₂²×Dic₇₃	1	2	yes	—
276	M₁₆	—	2	8	no	—
277	M₁₆	—	146	4	no	—
278	C₂×C₈	—	2	4	no	—
279	C₂×C₈	—	292	2	no	—
280	C₂×C₈	—	292	4	no	—
281	C₂×C₈	—	73	2	no	—
282	C₂×C₈	—	73	2	no	—
283	C₂×C₈	—	73	2	no	—
284	C₂×C₈	—	73	4	no	—
285	C₂×C₈	—	73	2	no	—
286	C₂×C₈	—	73	4	no	—
287	C₂×C₈	—	146	4	no	—
288	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
289	C₂×C₈	C₂²×Dic₇₃	1	4	yes	—
290	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
291	C₄⋊C₄	C₄×Dic₇₃	1	2	yes	—
292	C₄⋊C₄	C₄×Dic₇₃	1	1	yes	characteristic
293	C₄⋊C₄	C₄×Dic₇₃	1	2	yes	—
294	C₄⋊C₄	C₄×Dic₇₃	1	1	yes	characteristic
295	C₂²⋊C₄	C₄×Dic₇₃	1	2	yes	—
296	C₂²⋊C₄	C₄×Dic₇₃	1	2	yes	—
297	C₂²⋊C₄	C₄×Dic₇₃	1	2	yes	—
298	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
299	C₄²	C₂×C₄×D₇₃	1	2	yes	—
300	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
301	C₄²	C₂×C₄×D₇₃	1	2	yes	—
302	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
303	C₄²	C₂²×Dic₇₃	1	2	yes	—
304	C₄²	C₂³×D₇₃	1	1	yes	characteristic
305	C₂³	C₂×C₄×Dic₇₃	1	2	yes	—
306	C₂³	C₂×C₄×Dic₇₃	1	1	yes	characteristic
307	Q₈	—	2	2	no	—
308	D₄	—	2	4	no	—
309	D₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	4	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
317	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
318	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
319	C₂×C₄	C₂×C₄×Dic₇₃	1	2	yes	—
320	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
321	C₂×C₄	#14556599	1	4	yes	—
322	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
324	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
325	C₈	—	2	2	no	—
326	C₈	—	2	4	no	—
327	C₈	—	146	2	no	—
328	C₈	—	146	2	no	—
329	C₈	—	146	4	no	—
330	C₈	—	146	2	no	—
331	C₈	—	146	4	no	—
332	C₈	—	146	2	no	—
333	C₈	#14556598	1	4	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	#17988545	1	2	yes	—
340	C₂²	C₂²×C₄×Dic₇₃	1	1	yes	characteristic
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	2	no	—
344	C₄	—	2	1	no	—
345	C₄	—	2	2	no	—
346	C₄	#17988545	1	2	yes	—
347	C₄	C₂×#14556598	1	2	yes	—
348	C₄	C₂×#14556599	1	2	yes	—
349	C₄	#17989352	1	2	yes	—
350	C₂	—	2	2	no	—
351	C₂	—	4	2	no	—
352	C₂	C₂×#17988545	1	1	yes	characteristic
353	C₂	C₄×#14556598	1	1	yes	characteristic
354	C₂	#22618357	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂²×C₄²
Fitting	C₇₃×#2965	C₂
prime radical	C₂×#41×C₇₃	C₂×C₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#2965	D₇₃
hypocenter	C₇₃	#17331
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄² ◁ #2965

Complements

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

N	H ≅ G/N	type
3. #22608231	C₂	semidirect
4. #22608228	C₂	semidirect
5. #22608224	C₂	semidirect
6. #22608221	C₂	semidirect
7. #22608215	C₂	semidirect
8. C₈×#11924012	C₂	semidirect
9. M₁₆×#10596831	C₂	semidirect
10. C₂×#17988308	C₂	semidirect
189. C₇₃	#17331	semidirect

Maximal subgroups

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

subgroup	index
#41×#10596831	2
#22618575	2
#22608231	2
#22608228	2
#22608224	2
#22608221	2
#22608215	2
C₈×#11924012	2
M₁₆×#10596831	2
C₂×#17988308	2
C₇₃×#2965	2
#17331	73