Subgroups — #33239206

Order 18688.

682 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989005	C₂	1	2	yes	maximal, complemented
2	#22618638	C₂	1	1	yes	maximal, characteristic
3	#22611762	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611754	C₂	1	2	yes	maximal, complemented
5	#22611753	C₂	1	2	yes	maximal, complemented
6	C₁₆×D₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄⋊C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	#22611543	C₂	1	2	yes	maximal, complemented
10	C₄×#14556530	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14556597	C₂²	1	2	yes	—
13	#17989134	C₄	1	1	yes	characteristic
14	#17989005	—	2	8	no	—
15	C₂×#14556530	C₂²	1	2	yes	—
16	C₂×#14556530	C₂²	1	1	yes	characteristic
17	C₂×#14556530	C₂²	1	1	yes	characteristic
18	C₂²×#11924003	C₂²	1	2	yes	—
19	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
20	C₂×C₁₆×D₇₃	C₂²	1	2	yes	—
21	#17988665	C₄	1	1	yes	characteristic
22	#17988664	C₄	1	1	yes	characteristic
23	#17988662	C₄	1	2	yes	—
24	C₈×D₂₉₂	C₂²	1	1	yes	characteristic
25	C₈×C₂²⋊D₇₃	C₂²	1	2	yes	—
26	#17988609	C₄	1	1	yes	characteristic
27	#17988552	C₄	1	2	yes	—
28	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
29	C₄⋊#11924003	C₂²	1	1	yes	characteristic
30	#17988405	C₂²	1	2	yes	—
31	C₂²⋊#11924003	C₂²	1	2	yes	—
32	#17988366	C₂²	1	2	yes	—
33	#17988366	C₂²	1	2	yes	—
34	#17988362	C₂²	1	1	yes	characteristic
35	#17988362	C₂²	1	1	yes	characteristic
36	C₄×#11924003	C₂²	1	1	yes	characteristic
37	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
38	C₈×D₄×C₇₃	C₂²	1	1	yes	characteristic
39	C₄⋊C₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	C₂²⋊C₁₆×C₇₃	C₂²	1	2	yes	—
41	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
42	C₂×#11924028	C₂×C₄	1	2	yes	—
43	#14556597	—	2	4	no	—
44	#14556597	—	2	4	no	—
45	C₂×#11924012	C₂×C₄	1	2	yes	—
46	C₂×#11924011	C₂×C₄	1	2	yes	—
47	C₂×#11924011	C₂×C₄	1	2	yes	—
48	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₈×D₇₃	C₂³	1	2	yes	—
50	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
51	#14556590	C₈	1	1	yes	characteristic
52	#14556587	C₈	1	1	yes	characteristic
53	#14556577	C₈	1	2	yes	—
54	#14556566	C₈	1	1	yes	characteristic
55	#14556564	C₈	1	2	yes	—
56	#14556562	C₈	1	1	yes	characteristic
57	C₄×D₂₉₂	C₂×C₄	1	1	yes	characteristic
58	C₄×C₂²⋊D₇₃	C₂×C₄	1	2	yes	—
59	C₄²×D₇₃	C₂×C₄	1	1	yes	characteristic
60	#14556530	—	2	2	no	—
61	#14556530	—	2	4	no	—
62	#14556530	D₄	1	4	yes	complemented
63	C₂×#11924003	—	2	4	no	—
64	C₂×#11924003	C₂³	1	1	yes	characteristic
65	C₂×#11924003	C₂³	1	2	yes	—
66	C₂×#11924003	C₂³	1	1	yes	characteristic
67	C₁₆×D₇₃	—	2	4	no	—
68	C₁₆×D₇₃	—	2	4	no	—
69	C₄⋊#10596831	C₂×C₄	1	1	yes	characteristic
70	#14556520	C₂×C₄	1	2	yes	—
71	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
72	#14556503	C₂×C₄	1	2	yes	—
73	#14556503	C₂×C₄	1	2	yes	—
74	C₄×#10596831	C₂×C₄	1	1	yes	characteristic
75	C₈×Dic₇₃	C₂×C₄	1	1	yes	characteristic
76	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
78	C₄×D₄×C₇₃	C₂×C₄	1	1	yes	characteristic
79	M₃₂×C₇₃	—	2	4	no	—
80	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
81	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
82	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
85	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
86	#11924028	—	2	4	no	—
87	#11924028	—	2	4	no	—
88	C₂×Dic₁₄₆	C₂×C₈	1	2	yes	—
89	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
90	C₂×D₂₉₂	C₂×C₈	1	2	yes	—
91	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
92	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
93	C₂×C₂²⋊D₇₃	C₂×C₈	1	2	yes	—
94	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
95	C₂×C₄×D₇₃	C₂×C₈	1	1	yes	characteristic
96	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
98	#11924012	—	2	4	no	—
99	#11924011	—	2	4	no	—
100	#11924011	—	2	4	no	—
101	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
102	C₂×#10596831	C₂²×C₄	1	2	yes	—
103	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
104	C₈×D₇₃	—	2	2	no	—
105	C₈×D₇₃	—	2	2	no	—
106	C₈×D₇₃	—	2	2	no	—
107	C₈×D₇₃	C₂×D₄	1	2	yes	—
108	C₈×D₇₃	#41	1	2	yes	—
109	C₄⋊Dic₇₃	C₂×C₈	1	1	yes	characteristic
110	#11924007	C₂×C₈	1	2	yes	—
111	C₂²⋊Dic₇₃	C₂×C₈	1	2	yes	—
112	#11924005	C₂×C₈	1	2	yes	—
113	#11924005	C₂×C₈	1	2	yes	—
114	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
115	C₄×Dic₇₃	C₂×C₈	1	1	yes	characteristic
116	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
117	#11924003	—	2	2	no	—
118	#11924003	—	2	1	no	—
119	#11924003	C₂×D₄	1	2	yes	—
120	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
121	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
122	C₂×C₈×C₇₃	—	2	2	no	—
123	C₂×C₈×C₇₃	—	2	2	no	—
124	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
126	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
128	C₂²⋊C₄×C₇₃	C₂×C₈	1	2	yes	—
129	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
130	C₁₆×C₇₃	—	2	2	no	—
131	C₁₆×C₇₃	—	2	1	no	—
132	C₁₆×C₇₃	C₂×D₄	1	2	yes	—
133	C₂³×C₇₃	C₂²×C₈	1	2	yes	—
134	C₂²×D₇₃	C₂²×C₈	1	2	yes	—
135	C₂²×D₇₃	C₂²×C₈	1	1	yes	characteristic
136	D₄×C₇₃	—	2	2	no	—
137	C₂×C₄×C₇₃	—	2	2	no	—
138	C₂×C₄×C₇₃	—	2	2	no	—
139	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
140	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
142	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
143	C₂²⋊D₇₃	—	2	4	no	—
144	C₂²⋊D₇₃	—	2	4	no	—
145	C₂²⋊D₇₃	—	4	2	no	—
146	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
147	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
148	C₂×Dic₇₃	C₂²×C₈	1	2	yes	—
149	C₂×Dic₇₃	C₂²×C₈	1	1	yes	characteristic
150	D₂₉₂	—	2	4	no	—
151	D₂₉₂	—	2	2	no	—
152	C₄×D₇₃	—	2	2	no	—
153	C₄×D₇₃	—	2	2	no	—
154	C₄×D₇₃	—	2	2	no	—
155	C₄×D₇₃	—	4	1	no	—
156	C₄×D₇₃	C₄×D₄	1	2	yes	—
157	C₄×D₇₃	#131	1	2	yes	—
158	Dic₁₄₆	—	2	4	no	—
159	C₈×C₇₃	—	2	2	no	—
160	C₈×C₇₃	—	2	1	no	—
161	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
163	#10596831	—	2	2	no	—
164	#10596831	—	2	1	no	—
165	#10596831	C₄×D₄	1	2	yes	—
166	C₂²×C₇₃	—	2	2	no	—
167	C₂²×C₇₃	—	2	2	no	—
168	C₂²×C₇₃	—	2	2	no	—
169	C₂²×C₇₃	C₂³×C₈	1	1	yes	characteristic
170	C₂×D₇₃	—	2	1	no	—
171	C₂×D₇₃	—	2	1	no	—
172	C₂×D₇₃	—	2	2	no	—
173	C₂×D₇₃	—	2	2	no	—
174	C₂×D₇₃	—	2	2	no	—
175	C₂×D₇₃	C₈×D₄	1	2	yes	—
176	C₄×C₇₃	—	2	1	no	—
177	C₄×C₇₃	—	2	2	no	—
178	C₄×C₇₃	#504	1	2	yes	complemented
179	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
180	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
181	Dic₇₃	—	2	1	no	—
182	Dic₇₃	—	2	1	no	—
183	Dic₇₃	—	2	2	no	—
184	Dic₇₃	C₈×D₄	1	2	yes	—
185	#33438	—	73	1	no	maximal, Sylow
186	C₂×C₇₃	—	4	2	no	—
187	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
188	C₂×C₇₃	C₂×#504	1	1	yes	characteristic
189	C₂×C₇₃	#3407	1	1	yes	characteristic
190	D₇₃	—	2	2	no	—
191	D₇₃	—	4	2	no	—
192	C₂×#504	—	73	2	no	—
193	#2931	—	73	1	no	—
194	#2169	—	73	1	no	—
195	#2169	D₇₃	1	1	yes	characteristic, complemented
196	#2168	—	73	2	no	—
197	#2168	—	73	2	no	—
198	D₄×C₁₆	—	73	1	no	—
199	D₄×C₁₆	—	73	1	no	—
200	C₂×C₄⋊C₁₆	—	73	1	no	—
201	#2113	—	73	2	no	—
202	C₄×M₃₂	—	73	1	no	—
203	C₇₃	#33438	1	1	yes	characteristic, complemented, Sylow
204	C₂×#131	—	73	2	no	—
205	#532	—	73	1	no	—
206	#504	—	146	8	no	—
207	C₂×M₃₂	—	73	1	no	—
208	C₂×M₃₂	—	73	1	no	—
209	C₂×M₃₂	—	73	2	no	—
210	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
211	C₂²×C₁₆	—	73	2	no	—
212	C₂²×C₁₆	—	73	2	no	—
213	C₂²×C₁₆	—	73	2	no	—
214	#436	—	73	1	no	—
215	#436	—	73	1	no	—
216	#436	—	73	2	no	—
217	C₈×D₄	—	73	2	no	—
218	C₈×D₄	—	73	1	no	—
219	C₈×D₄	C₂×D₇₃	1	1	yes	characteristic
220	#423	—	73	1	no	—
221	#408	—	73	2	no	—
222	C₂×C₄×C₈	—	73	1	no	—
223	C₄⋊C₁₆	—	73	2	no	—
224	C₄⋊C₁₆	—	73	1	no	—
225	C₄⋊C₁₆	C₂×D₇₃	1	1	yes	characteristic
226	C₂²⋊C₁₆	—	73	2	no	—
227	C₂²⋊C₁₆	—	73	2	no	—
228	C₂²⋊C₁₆	—	73	2	no	—
229	C₂²⋊C₁₆	C₂×D₇₃	1	2	yes	—
230	#346	—	73	1	no	—
231	#346	—	73	1	no	—
232	C₄×C₁₆	—	73	1	no	—
233	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
234	C₂×#41	—	73	2	no	—
235	#131	—	146	4	no	—
236	#131	—	146	4	no	—
237	C₂×M₁₆	—	73	2	no	—
238	C₂×M₁₆	—	73	2	no	—
239	C₂×M₁₆	—	73	2	no	—
240	C₂²×C₈	—	73	1	no	—
241	C₂²×C₈	—	73	1	no	—
242	C₂²×C₈	—	73	2	no	—
243	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
244	C₄⋊Q₈	—	73	1	no	—
245	#127	—	73	1	no	—
246	#124	—	73	2	no	—
247	#121	—	73	1	no	—
248	#121	—	73	1	no	—
249	#121	—	73	2	no	—
250	C₄×D₄	—	73	2	no	—
251	C₄×D₄	—	73	1	no	—
252	C₄×D₄	C₄×D₇₃	1	1	yes	characteristic
253	C₂×C₄²	—	73	1	no	—
254	M₃₂	—	2	4	no	—
255	M₃₂	—	73	4	no	—
256	M₃₂	—	146	4	no	—
257	M₃₂	—	146	2	no	—
258	C₂×C₁₆	—	73	1	no	—
259	C₂×C₁₆	—	73	1	no	—
260	C₂×C₁₆	—	73	2	no	—
261	C₂×C₁₆	—	146	4	no	—
262	C₂×C₁₆	—	146	4	no	—
263	C₂×C₁₆	—	146	4	no	—
264	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
265	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
266	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
267	C₄⋊C₈	—	73	2	no	—
268	C₄⋊C₈	—	73	1	no	—
269	C₄⋊C₈	C₄×D₇₃	1	1	yes	characteristic
270	C₂²⋊C₈	—	73	2	no	—
271	C₂²⋊C₈	—	73	2	no	—
272	C₂²⋊C₈	—	73	2	no	—
273	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
274	C₄×C₈	—	73	1	no	—
275	C₄×C₈	—	73	1	no	—
276	C₄×C₈	—	73	1	no	—
277	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
278	#41	—	146	4	no	—
279	#41	—	146	4	no	—
280	C₂×Q₈	—	73	2	no	—
281	C₂×D₄	—	73	2	no	—
282	C₂×D₄	—	73	2	no	—
283	C₂×D₄	—	73	2	no	—
284	C₂×D₄	—	73	1	no	—
285	C₂×D₄	—	73	2	no	—
286	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
287	C₂²×C₄	—	73	1	no	—
288	C₂²×C₄	—	73	1	no	—
289	C₂²×C₄	—	73	1	no	—
290	C₂²×C₄	—	73	2	no	—
291	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
292	M₁₆	—	146	4	no	—
293	M₁₆	—	146	4	no	—
294	M₁₆	—	146	4	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	2	2	no	—
297	C₂×C₈	—	73	1	no	—
298	C₂×C₈	—	73	2	no	—
299	C₂×C₈	—	73	2	no	—
300	C₂×C₈	—	73	1	no	—
301	C₂×C₈	—	73	2	no	—
302	C₂×C₈	—	146	2	no	—
303	C₂×C₈	—	146	2	no	—
304	C₂×C₈	—	146	2	no	—
305	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
306	C₂×C₈	C₂×C₄×D₇₃	1	2	yes	—
307	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
308	C₄⋊C₄	—	73	2	no	—
309	C₄⋊C₄	—	73	1	no	—
310	C₄⋊C₄	C₈×D₇₃	1	1	yes	characteristic
311	C₂²⋊C₄	—	73	2	no	—
312	C₂²⋊C₄	—	73	2	no	—
313	C₂²⋊C₄	—	73	2	no	—
314	C₂²⋊C₄	C₈×D₇₃	1	2	yes	—
315	C₄²	—	73	1	no	—
316	C₄²	—	73	1	no	—
317	C₄²	—	73	1	no	—
318	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	2	1	no	—
321	C₁₆	—	73	2	no	—
322	C₁₆	—	146	2	no	—
323	C₁₆	—	146	1	no	—
324	C₁₆	D₄×D₇₃	1	2	yes	—
325	C₂³	—	73	1	no	—
326	C₂³	—	73	2	no	—
327	C₂³	C₂×C₈×D₇₃	1	2	yes	—
328	Q₈	—	146	4	no	—
329	D₄	—	2	2	no	—
330	D₄	—	292	2	no	—
331	D₄	—	146	4	no	—
332	D₄	—	146	4	no	—
333	D₄	—	146	4	no	—
334	D₄	—	146	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	292	1	no	—
338	C₂×C₄	—	73	1	no	—
339	C₂×C₄	—	73	2	no	—
340	C₂×C₄	—	73	2	no	—
341	C₂×C₄	—	73	1	no	—
342	C₂×C₄	—	73	1	no	—
343	C₂×C₄	—	73	2	no	—
344	C₂×C₄	—	146	2	no	—
345	C₂×C₄	—	146	2	no	—
346	C₂×C₄	—	146	2	no	—
347	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
348	C₂×C₄	C₂×C₈×D₇₃	1	2	yes	—
349	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
350	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	73	2	no	—
354	C₈	—	146	2	no	—
355	C₈	—	146	1	no	—
356	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
357	C₈	#41×D₇₃	1	1	yes	characteristic
358	C₂²	—	2	2	no	—
359	C₂²	—	2	2	no	—
360	C₂²	—	2	2	no	—
361	C₂²	—	73	2	no	—
362	C₂²	—	146	2	no	—
363	C₂²	—	146	2	no	—
364	C₂²	—	146	2	no	—
365	C₂²	—	146	1	no	—
366	C₂²	—	146	1	no	—
367	C₂²	C₂²×C₈×D₇₃	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	2	no	—
370	C₄	—	73	2	no	—
371	C₄	—	146	1	no	—
372	C₄	—	146	1	no	—
373	C₄	—	146	2	no	—
374	C₄	#17989005	1	2	yes	complemented
375	C₄	C₄×D₄×D₇₃	1	1	yes	characteristic
376	C₄	#131×D₇₃	1	1	yes	characteristic
377	C₂	—	4	2	no	—
378	C₂	—	292	2	no	—
379	C₂	—	146	2	no	—
380	C₂	C₈×D₄×D₇₃	1	1	yes	characteristic
381	C₂	C₂×#17989005	1	1	yes	characteristic
382	C₂	#22623032	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₇₃
commutator, socle	C₂²×C₇₃	C₂³×C₈
Fitting	C₇₃×#2169	C₂
prime radical	#17989134	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	#2169	D₇₃
hypocenter	C₇₃	#33438
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ #2169

Complements

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

N	H ≅ G/N	type
1. C₂×#17989005	C₂	semidirect
3. #22611762	C₂	semidirect
4. #22611754	C₂	semidirect
5. #22611753	C₂	semidirect
6. C₁₆×D₂₉₂	C₂	semidirect
7. D₄×#11924003	C₂	semidirect
8. C₄⋊C₁₆×D₇₃	C₂	semidirect
9. #22611543	C₂	semidirect
10. C₄×#14556530	C₂	semidirect
11. C₇₃×#2169	C₂	semidirect
62. #14556530	D₄	semidirect
178. C₄×C₇₃	#504	semidirect
195. #2169	D₇₃	semidirect
203. C₇₃	#33438	semidirect
374. C₄	#17989005	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989005	2
#22618638	2
#22611762	2
#22611754	2
#22611753	2
C₁₆×D₂₉₂	2
D₄×#11924003	2
C₄⋊C₁₆×D₇₃	2
#22611543	2
C₄×#14556530	2
C₇₃×#2169	2
#33438	73