Subgroups — #33239357

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
2	#22618794	C₂	1	2	yes	maximal
3	#22611800	C₂	1	2	yes	maximal, complemented
4	#22611799	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₅₈₄	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#22611789	C₂	1	1	yes	maximal, characteristic, complemented
8	#22611712	C₂	1	1	yes	maximal, characteristic
9	#22611649	C₂	1	2	yes	maximal
10	#22611526	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₇₃	C₂²	1	2	yes	—
13	#17989063	—	2	2	no	—
14	C₂×Dic₅₈₄	C₂²	1	1	yes	characteristic
15	C₂×#14556537	C₂²	1	2	yes	—
16	Q₃₂×D₇₃	C₄	1	8	yes	complemented
17	C₂×#14556530	C₂²	1	1	yes	characteristic
18	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
19	#17988692	—	2	2	no	—
20	#17988691	—	2	2	no	—
21	C₄×Dic₂₉₂	C₂²	1	2	yes	—
22	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
23	#17988681	C₂²	1	2	yes	—
24	#17988624	C₂²	1	2	yes	—
25	#17988594	—	2	2	no	—
26	#17988541	C₂²	1	1	yes	characteristic
27	#17988414	C₂²	1	2	yes	—
28	#17988413	C₂²	1	1	yes	characteristic
29	#17988392	C₂²	1	2	yes	—
30	#17988391	C₂²	1	2	yes	—
31	#17988390	C₂²	1	2	yes	—
32	#17988363	C₂²	1	1	yes	characteristic
33	#17988362	C₂²	1	1	yes	characteristic
34	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
35	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
36	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
37	#366×C₇₃	C₂²	1	1	yes	characteristic
38	#358×C₇₃	C₂²	1	2	yes	—
39	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
40	C₂×Q₈×D₇₃	—	2	2	no	—
41	C₂×Dic₂₉₂	C₂³	1	2	yes	—
42	C₂×#11924018	—	2	2	no	—
43	Q₁₆×D₇₃	—	2	4	no	—
44	Q₁₆×D₇₃	C₂×C₄	1	8	yes	—
45	C₂×#11924011	D₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
47	Q₈×Dic₇₃	—	2	2	no	—
48	C₄×Dic₁₄₆	—	2	2	no	—
49	C₄×Dic₁₄₆	—	4	2	no	—
50	#14556556	—	2	2	no	—
51	#14556556	—	4	2	no	—
52	#14556549	—	2	2	no	—
53	#14556549	—	4	2	no	—
54	#14556548	—	4	2	no	—
55	#14556548	D₄	1	1	yes	characteristic
56	Dic₅₈₄	C₂×C₄	1	4	yes	—
57	#14556537	C₂×C₄	1	8	yes	—
58	#14556530	—	2	2	no	—
59	C₂×#11924003	C₂³	1	1	yes	characteristic
60	C₂×#11924003	C₂³	1	1	yes	characteristic
61	C₁₆×D₇₃	C₂×C₄	1	4	yes	—
62	#14556525	C₂³	1	2	yes	—
63	#14556522	—	2	2	no	—
64	#14556516	—	2	2	no	—
65	#14556515	—	2	2	no	—
66	#14556514	—	2	2	no	—
67	#14556502	D₄	1	1	yes	characteristic
68	#14556501	D₄	1	1	yes	characteristic
69	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
70	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
71	C₄×Q₈×C₇₃	—	2	2	no	—
72	Q₃₂×C₇₃	C₂×C₄	1	4	yes	—
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
75	#107×C₇₃	C₂³	1	2	yes	—
76	#103×C₇₃	—	2	2	no	—
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₄₆	—	2	2	no	—
79	C₂×Dic₁₄₆	—	4	2	no	—
80	Q₈×D₇₃	—	2	8	no	—
81	Q₈×D₇₃	—	4	4	no	—
82	C₂×C₄×D₇₃	—	4	2	no	—
83	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
84	Dic₂₉₂	—	2	2	no	—
85	Dic₂₉₂	C₂²×C₄	1	4	yes	—
86	#11924018	—	2	8	no	—
87	#11924011	—	2	2	no	—
88	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
89	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
90	C₈×D₇₃	C₂²×C₄	1	2	yes	—
91	C₈×D₇₃	#41	1	2	yes	—
92	C₄⋊Dic₇₃	—	2	2	no	—
93	C₄⋊Dic₇₃	—	4	2	no	—
94	#11924007	—	4	2	no	—
95	#11924007	—	4	2	no	—
96	#11924007	D₈	1	1	yes	characteristic
97	#11924007	D₈	1	1	yes	characteristic
98	#11924005	—	4	2	no	—
99	#11924005	D₈	1	1	yes	characteristic
100	#11924005	D₈	1	1	yes	characteristic
101	C₄×Dic₇₃	—	4	2	no	—
102	C₄×Dic₇₃	—	4	2	no	—
103	C₄×Dic₇₃	—	4	2	no	—
104	C₄×Dic₇₃	—	4	2	no	—
105	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
106	#11924003	—	2	1	no	—
107	#11924003	C₂²×C₄	1	2	yes	—
108	C₂×Q₈×C₇₃	—	2	2	no	—
109	Q₁₆×C₇₃	—	2	2	no	—
110	Q₁₆×C₇₃	C₂²×C₄	1	4	yes	—
111	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₇₃	—	2	2	no	—
114	C₄⋊C₄×C₇₃	—	4	2	no	—
115	C₄²×C₇₃	—	4	2	no	—
116	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
117	C₁₆×C₇₃	—	2	1	no	—
118	C₁₆×C₇₃	C₂²×C₄	1	2	yes	—
119	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
120	Q₈×C₇₃	—	2	4	no	—
121	Q₈×C₇₃	—	4	2	no	—
122	C₂×C₄×C₇₃	—	4	2	no	—
123	C₂×C₄×C₇₃	—	4	2	no	—
124	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×Dic₇₃	—	4	2	no	—
128	C₂×Dic₇₃	—	4	2	no	—
129	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
131	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
132	C₄×D₇₃	—	4	8	no	—
133	C₄×D₇₃	C₄×D₄	1	2	yes	—
134	C₄×D₇₃	#135	1	2	yes	—
135	Dic₁₄₆	—	2	4	no	—
136	Dic₁₄₆	—	4	2	no	—
137	Dic₁₄₆	—	4	8	no	—
138	C₈×C₇₃	—	2	1	no	—
139	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
140	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
141	#10596831	—	2	1	no	—
142	#10596831	C₄×D₄	1	2	yes	—
143	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
144	C₂×D₇₃	—	2	1	no	—
145	C₂×D₇₃	—	2	1	no	—
146	C₂×D₇₃	C₄×D₈	1	2	yes	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	—	4	4	no	—
150	C₄×C₇₃	—	8	2	no	—
151	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
153	Dic₇₃	—	2	1	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	4	4	no	—
156	Dic₇₃	—	8	2	no	—
157	Dic₇₃	C₄×D₈	1	2	yes	—
158	#33458	—	73	1	no	maximal, Sylow
159	C₂×C₇₃	C₂×C₄×D₈	1	1	yes	characteristic
160	C₂×C₇₃	#3417	1	1	yes	characteristic
161	C₂×C₇₃	#3417	1	1	yes	characteristic
162	D₇₃	—	2	2	no	—
163	C₂²×Q₃₂	—	73	1	no	—
164	#2950	—	73	2	no	—
165	#2176	—	73	2	no	—
166	#2176	—	73	2	no	—
167	C₄×Q₃₂	—	73	1	no	—
168	C₄×Q₃₂	—	73	1	no	—
169	C₄×Q₃₂	—	73	1	no	—
170	C₄×Q₃₂	D₇₃	1	1	yes	characteristic, complemented
171	#2158	—	73	1	no	—
172	#2141	—	73	2	no	—
173	#2108	—	73	1	no	—
174	C₇₃	#33458	1	1	yes	characteristic, complemented, Sylow
175	C₂²×Q₁₆	—	73	2	no	—
176	#519	—	146	2	no	—
177	C₂×Q₃₂	—	73	2	no	—
178	C₂×Q₃₂	—	73	8	no	—
179	C₂×Q₃₂	—	73	1	no	—
180	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
181	C₂×M₃₂	—	73	1	no	—
182	C₂²×C₁₆	—	73	1	no	—
183	#441	—	146	2	no	—
184	#441	—	146	2	no	—
185	C₄×Q₁₆	—	73	2	no	—
186	C₄×Q₁₆	—	73	2	no	—
187	C₄×Q₁₆	—	73	2	no	—
188	C₄×Q₁₆	C₂×D₇₃	1	2	yes	—
189	#427	—	73	2	no	—
190	#419	—	146	2	no	—
191	#405	—	73	1	no	—
192	#367	—	73	2	no	—
193	#366	—	73	1	no	—
194	#366	C₂×D₇₃	1	1	yes	characteristic
195	#358	—	73	2	no	—
196	#358	—	73	2	no	—
197	#358	—	73	2	no	—
198	#358	C₂×D₇₃	1	2	yes	—
199	#346	—	73	1	no	—
200	#346	—	73	1	no	—
201	C₄×C₁₆	—	73	1	no	—
202	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
203	C₂²×Q₈	—	146	2	no	—
204	C₂×Q₁₆	—	73	8	no	—
205	C₂×Q₁₆	—	73	2	no	—
206	C₂×Q₁₆	—	146	4	no	—
207	C₂×Q₁₆	—	146	2	no	—
208	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
209	C₂×M₁₆	—	73	1	no	—
210	C₂²×C₈	—	73	1	no	—
211	C₄×Q₈	—	2	2	no	—
212	C₄×Q₈	—	292	2	no	—
213	C₄×Q₈	—	292	2	no	—
214	C₄×Q₈	—	146	2	no	—
215	C₄×Q₈	—	146	2	no	—
216	C₄×Q₈	—	146	2	no	—
217	#117	—	292	2	no	—
218	#117	—	292	2	no	—
219	#117	—	73	1	no	—
220	#117	—	146	2	no	—
221	Q₃₂	—	73	8	no	—
222	Q₃₂	—	73	4	no	—
223	Q₃₂	C₄×D₇₃	1	4	yes	—
224	M₃₂	—	146	2	no	—
225	C₂×C₁₆	—	73	1	no	—
226	C₂×C₁₆	—	73	4	no	—
227	C₂×C₁₆	—	73	1	no	—
228	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
229	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
230	#107	—	73	2	no	—
231	#107	C₂²×D₇₃	1	2	yes	—
232	#106	—	146	2	no	—
233	#103	—	2	2	no	—
234	#103	—	146	2	no	—
235	#103	—	146	2	no	—
236	#103	—	146	2	no	—
237	#97	—	73	1	no	—
238	#97	—	73	1	no	—
239	C₄×C₈	—	73	1	no	—
240	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
241	C₂×Q₈	—	2	2	no	—
242	C₂×Q₈	—	292	2	no	—
243	C₂×Q₈	—	292	4	no	—
244	C₂×Q₈	—	146	8	no	—
245	C₂×Q₈	—	146	2	no	—
246	C₂²×C₄	—	292	2	no	—
247	C₂²×C₄	—	73	1	no	—
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	73	4	no	—
250	Q₁₆	—	146	8	no	—
251	Q₁₆	—	146	2	no	—
252	Q₁₆	C₂×C₄×D₇₃	1	4	yes	—
253	M₁₆	—	146	2	no	—
254	C₂×C₈	—	73	1	no	—
255	C₂×C₈	—	73	2	no	—
256	C₂×C₈	—	73	2	no	—
257	C₂×C₈	—	73	1	no	—
258	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
259	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	292	2	no	—
263	C₄⋊C₄	—	292	2	no	—
264	C₄⋊C₄	—	292	2	no	—
265	C₄⋊C₄	—	73	1	no	—
266	C₄⋊C₄	—	73	1	no	—
267	C₄⋊C₄	—	146	2	no	—
268	C₂²⋊C₄	—	292	2	no	—
269	C₂²⋊C₄	—	73	1	no	—
270	C₂²⋊C₄	—	73	1	no	—
271	C₄²	—	4	2	no	—
272	C₄²	—	292	2	no	—
273	C₄²	—	292	2	no	—
274	C₄²	—	292	2	no	—
275	C₄²	—	292	2	no	—
276	C₄²	—	73	1	no	—
277	C₄²	D₄×D₇₃	1	1	yes	characteristic
278	C₁₆	—	2	1	no	—
279	C₁₆	—	73	2	no	—
280	C₁₆	—	146	1	no	—
281	C₁₆	C₂×C₄×D₇₃	1	2	yes	—
282	C₂³	—	73	1	no	—
283	Q₈	—	2	4	no	—
284	Q₈	—	4	2	no	—
285	Q₈	—	292	8	no	—
286	Q₈	—	292	2	no	—
287	Q₈	—	146	4	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	292	8	no	—
291	C₂×C₄	—	292	2	no	—
292	C₂×C₄	—	292	2	no	—
293	C₂×C₄	—	73	1	no	—
294	C₂×C₄	—	73	2	no	—
295	C₂×C₄	—	73	2	no	—
296	C₂×C₄	—	73	1	no	—
297	C₂×C₄	—	73	1	no	—
298	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
299	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
300	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	73	2	no	—
303	C₈	—	146	1	no	—
304	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
305	C₈	#41×D₇₃	1	1	yes	characteristic
306	C₂²	—	73	2	no	—
307	C₂²	—	146	1	no	—
308	C₂²	—	146	1	no	—
309	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	2	no	—
314	C₄	—	292	4	no	—
315	C₄	—	584	2	no	—
316	C₄	—	73	2	no	—
317	C₄	—	146	1	no	—
318	C₄	—	146	1	no	—
319	C₄	C₄×D₄×D₇₃	1	1	yes	characteristic
320	C₄	#135×D₇₃	1	1	yes	characteristic
321	C₂	—	146	2	no	—
322	C₂	C₄×D₈×D₇₃	1	1	yes	characteristic
323	C₂	#22623080	1	1	yes	characteristic
324	C₂	#22623081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×Q₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	C₂²×D₇₃	C₂×D₈
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₄×Q₃₂	D₇₃
hypocenter	C₇₃	#33458
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
3. #22611800	C₂	semidirect
4. #22611799	C₂	semidirect
5. C₄×Dic₅₈₄	C₂	semidirect
6. Q₃₂×Dic₇₃	C₂	semidirect
7. #22611789	C₂	semidirect
11. C₄×Q₃₂×C₇₃	C₂	semidirect
16. Q₃₂×D₇₃	C₄	semidirect
170. C₄×Q₃₂	D₇₃	semidirect
174. C₇₃	#33458	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₃₂×D₇₃	2
#22618794	2
#22611800	2
#22611799	2
C₄×Dic₅₈₄	2
Q₃₂×Dic₇₃	2
#22611789	2
#22611712	2
#22611649	2
#22611526	2
C₄×Q₃₂×C₇₃	2
#33458	73