Subgroups — #33241576

Order 18688.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620567	C₂	1	2	yes	maximal, complemented
2	#22612206	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612167	C₂	1	2	yes	maximal, complemented
4	#22612041	C₂	1	2	yes	maximal, complemented
5	#22612019	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612014	C₂	1	1	yes	maximal, characteristic
7	#22611796	C₂	1	2	yes	maximal, complemented
8	Q₃₂×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	D₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₇₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989206	—	2	2	no	—
13	C₂×#14556530	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
15	#17988992	C₂²	1	2	yes	—
16	#17988946	—	2	2	no	—
17	#17988830	—	2	2	no	—
18	#17988825	C₂²	1	2	yes	—
19	#17988820	C₂²	1	2	yes	—
20	#17988688	—	2	2	no	—
21	Q₁₆×Dic₇₃	C₂²	1	2	yes	—
22	D₈×Dic₇₃	C₂²	1	2	yes	—
23	#17988541	C₂²	1	1	yes	characteristic
24	#17988413	C₂²	1	1	yes	characteristic
25	#17988413	C₂²	1	1	yes	characteristic
26	#17988413	C₂²	1	2	yes	—
27	#17988391	C₂²	1	2	yes	—
28	#17988391	C₂²	1	2	yes	—
29	#17988388	C₂²	1	2	yes	—
30	#17988388	C₂²	1	2	yes	—
31	#17988363	C₂²	1	1	yes	characteristic
32	#17988362	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
35	C₂×SD₃₂×C₇₃	C₂²	1	2	yes	—
36	C₂×D₁₆×C₇₃	C₂²	1	1	yes	characteristic
37	#495×C₇₃	C₂²	1	2	yes	—
38	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×#11924011	D₄	1	1	yes	characteristic
40	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
41	#14556591	—	2	2	no	—
42	#14556584	—	4	2	no	—
43	#14556577	—	4	2	no	—
44	#14556575	—	4	2	no	—
45	#14556568	—	4	2	no	—
46	#14556567	—	2	2	no	—
47	#14556562	—	2	2	no	—
48	Q₈×Dic₇₃	—	2	2	no	—
49	D₄×Dic₇₃	—	2	2	no	—
50	#14556548	D₄	1	1	yes	characteristic
51	#14556530	—	2	2	no	—
52	C₂×#11924003	C₂³	1	1	yes	characteristic
53	C₂×#11924003	C₂³	1	1	yes	characteristic
54	C₁₆×D₇₃	—	2	2	no	—
55	#14556525	—	2	2	no	—
56	#14556525	C₂³	1	2	yes	—
57	#14556525	C₂³	1	2	yes	—
58	#14556515	—	2	2	no	—
59	#14556515	—	2	2	no	—
60	#14556512	—	2	2	no	—
61	#14556512	—	2	2	no	—
62	#14556502	D₄	1	1	yes	characteristic
63	#14556501	D₄	1	1	yes	characteristic
64	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
65	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
66	C₂×SD₁₆×C₇₃	—	2	2	no	—
67	C₂×D₈×C₇₃	C₂³	1	2	yes	—
68	#124×C₇₃	—	2	2	no	—
69	Q₃₂×C₇₃	—	2	2	no	—
70	SD₃₂×C₇₃	—	2	4	no	—
71	D₁₆×C₇₃	—	2	2	no	—
72	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₄₆	—	4	2	no	—
76	C₂×C₂²⋊D₇₃	—	4	2	no	—
77	C₂²×Dic₇₃	—	4	2	no	—
78	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
79	#11924011	—	2	2	no	—
80	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
81	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
82	C₈×D₇₃	—	2	1	no	—
83	C₈×D₇₃	—	2	1	no	—
84	C₄⋊Dic₇₃	—	2	2	no	—
85	C₄⋊Dic₇₃	—	2	2	no	—
86	C₄⋊Dic₇₃	—	4	2	no	—
87	C₄⋊Dic₇₃	—	4	2	no	—
88	#11924007	—	4	2	no	—
89	#11924007	D₈	1	1	yes	characteristic
90	#11924007	D₈	1	1	yes	characteristic
91	C₂²⋊Dic₇₃	—	4	2	no	—
92	C₂²⋊Dic₇₃	—	4	2	no	—
93	#11924005	—	4	2	no	—
94	#11924005	D₈	1	1	yes	characteristic
95	#11924005	D₈	1	1	yes	characteristic
96	C₄×Dic₇₃	—	4	2	no	—
97	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
98	#11924003	—	2	1	no	—
99	#11924003	—	2	1	no	—
100	C₂×Q₈×C₇₃	—	2	2	no	—
101	C₂×D₄×C₇₃	—	2	2	no	—
102	Q₁₆×C₇₃	—	2	2	no	—
103	Q₁₆×C₇₃	—	2	2	no	—
104	SD₁₆×C₇₃	—	4	4	no	—
105	D₈×C₇₃	—	2	2	no	—
106	D₈×C₇₃	—	2	2	no	—
107	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
109	C₂²⋊C₄×C₇₃	—	4	2	no	—
110	C₂²⋊C₄×C₇₃	—	4	2	no	—
111	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₇₃	#41	1	2	yes	—
113	C₁₆×C₇₃	#41	1	2	yes	—
114	C₂³×C₇₃	—	4	2	no	—
115	C₂²×D₇₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₇₃	—	4	2	no	—
117	Q₈×C₇₃	—	4	2	no	—
118	D₄×C₇₃	—	4	2	no	—
119	D₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	—	4	2	no	—
121	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂²⋊D₇₃	—	16	2	no	—
125	C₂×Dic₇₃	—	4	2	no	—
126	C₂×Dic₇₃	—	4	2	no	—
127	C₂×Dic₇₃	—	16	2	no	—
128	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
129	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
130	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
131	C₄×D₇₃	—	2	1	no	—
132	C₄×D₇₃	—	2	1	no	—
133	Dic₁₄₆	—	16	2	no	—
134	C₈×C₇₃	#135	1	2	yes	—
135	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
136	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
137	#10596831	—	2	1	no	—
138	#10596831	—	2	1	no	—
139	C₂²×C₇₃	—	8	2	no	—
140	C₂²×C₇₃	—	8	2	no	—
141	C₂²×C₇₃	—	8	2	no	—
142	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
143	C₂×D₇₃	—	2	1	no	—
144	C₂×D₇₃	—	2	1	no	—
145	C₂×D₇₃	—	2	1	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	8	2	no	—
149	C₄×C₇₃	#532	1	1	yes	characteristic
150	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
151	Dic₇₃	—	2	1	no	—
152	Dic₇₃	—	2	1	no	—
153	Dic₇₃	—	2	1	no	—
154	Dic₇₃	—	8	2	no	—
155	Dic₇₃	—	8	2	no	—
156	#33739	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	—	16	2	no	—
158	C₂×C₇₃	#3169	1	1	yes	characteristic
159	C₂×C₇₃	#3415	1	1	yes	characteristic
160	C₂×C₇₃	#3417	1	1	yes	characteristic
161	D₇₃	—	4	1	no	—
162	#3175	—	73	2	no	—
163	#2252	—	73	1	no	—
164	#2249	D₇₃	1	1	yes	characteristic, complemented
165	#2243	—	73	2	no	—
166	#2220	—	73	2	no	—
167	#2216	—	73	1	no	—
168	#2215	—	73	1	no	—
169	#2175	—	73	2	no	—
170	C₄×Q₃₂	—	73	1	no	—
171	C₄×D₁₆	—	73	1	no	—
172	#2108	—	73	1	no	—
173	C₇₃	#33739	1	1	yes	characteristic, complemented, Sylow
174	#542	—	146	2	no	—
175	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
176	C₂×SD₃₂	C₂×D₇₃	1	2	yes	—
177	C₂×D₁₆	C₂×D₇₃	1	1	yes	characteristic
178	C₂×M₃₂	—	73	1	no	—
179	C₂²×C₁₆	—	73	1	no	—
180	#500	—	73	2	no	—
181	#495	C₂×D₇₃	1	2	yes	—
182	#489	—	146	2	no	—
183	#468	—	146	2	no	—
184	#467	—	73	2	no	—
185	#466	—	73	2	no	—
186	#440	—	146	2	no	—
187	C₄×Q₁₆	—	73	2	no	—
188	C₄×D₈	—	73	2	no	—
189	#405	—	73	1	no	—
190	#366	—	73	1	no	—
191	#366	—	73	1	no	—
192	#366	—	73	2	no	—
193	#358	—	73	2	no	—
194	#358	—	73	2	no	—
195	#357	—	73	2	no	—
196	#357	—	73	2	no	—
197	#346	—	73	1	no	—
198	#346	—	73	1	no	—
199	C₄×C₁₆	—	73	1	no	—
200	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
201	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
202	C₂×SD₁₆	—	2	2	no	—
203	C₂×D₈	C₂²×D₇₃	1	2	yes	—
204	C₂×M₁₆	—	73	1	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₄⋊Q₈	—	146	2	no	—
207	#126	—	292	2	no	—
208	#124	—	2	2	no	—
209	#124	—	292	2	no	—
210	#123	—	292	2	no	—
211	C₂²⋊Q₈	—	292	2	no	—
212	C₂²⋊Q₈	—	146	2	no	—
213	#121	—	146	2	no	—
214	C₄×Q₈	—	146	2	no	—
215	C₄×D₄	—	146	2	no	—
216	#117	—	73	1	no	—
217	Q₃₂	—	2	2	no	—
218	SD₃₂	—	2	4	no	—
219	D₁₆	—	2	2	no	—
220	M₃₂	—	146	2	no	—
221	C₂×C₁₆	—	73	1	no	—
222	C₂×C₁₆	—	73	1	no	—
223	C₂×C₁₆	—	146	2	no	—
224	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
226	#107	—	73	2	no	—
227	#107	—	73	2	no	—
228	#107	—	146	2	no	—
229	#103	—	146	2	no	—
230	#103	—	146	2	no	—
231	#102	—	146	2	no	—
232	#102	—	146	2	no	—
233	#97	—	73	1	no	—
234	#97	—	73	1	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	292	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	292	2	no	—
241	C₂²×C₄	—	292	2	no	—
242	C₂²×C₄	—	73	1	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	2	2	no	—
245	SD₁₆	—	4	4	no	—
246	D₈	—	2	2	no	—
247	D₈	—	2	2	no	—
248	M₁₆	—	146	2	no	—
249	C₂×C₈	—	73	1	no	—
250	C₂×C₈	—	73	1	no	—
251	C₂×C₈	—	146	1	no	—
252	C₂×C₈	—	146	1	no	—
253	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
254	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
255	C₄⋊C₄	—	292	2	no	—
256	C₄⋊C₄	—	292	2	no	—
257	C₄⋊C₄	—	292	2	no	—
258	C₄⋊C₄	—	73	1	no	—
259	C₄⋊C₄	—	73	1	no	—
260	C₄⋊C₄	—	146	2	no	—
261	C₄⋊C₄	—	146	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	292	2	no	—
265	C₂²⋊C₄	—	292	2	no	—
266	C₂²⋊C₄	—	292	2	no	—
267	C₂²⋊C₄	—	73	1	no	—
268	C₂²⋊C₄	—	73	1	no	—
269	C₄²	—	292	2	no	—
270	C₄²	—	73	1	no	—
271	C₄²	D₄×D₇₃	1	1	yes	characteristic
272	C₁₆	—	146	1	no	—
273	C₁₆	—	146	1	no	—
274	C₁₆	#11924027	1	2	yes	—
275	C₁₆	#11924027	1	2	yes	—
276	C₂³	—	4	2	no	—
277	C₂³	—	73	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	1168	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	1168	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	292	2	no	—
286	C₂×C₄	—	292	2	no	—
287	C₂×C₄	—	73	1	no	—
288	C₂×C₄	—	73	1	no	—
289	C₂×C₄	—	73	1	no	—
290	C₂×C₄	—	1168	2	no	—
291	C₂×C₄	—	146	1	no	—
292	C₂×C₄	—	146	1	no	—
293	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
294	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
295	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
296	C₈	—	146	1	no	—
297	C₈	—	146	1	no	—
298	C₈	#14556610	1	2	yes	—
299	C₈	C₂×#11924027	1	1	yes	characteristic
300	C₈	C₂×#11924027	1	1	yes	characteristic
301	C₂²	—	8	2	no	—
302	C₂²	—	8	2	no	—
303	C₂²	—	8	2	no	—
304	C₂²	—	146	1	no	—
305	C₂²	—	146	1	no	—
306	C₂²	—	146	1	no	—
307	C₂²	C₂×D₈×D₇₃	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	8	2	no	—
311	C₄	—	584	2	no	—
312	C₄	—	584	2	no	—
313	C₄	—	146	1	no	—
314	C₄	—	146	1	no	—
315	C₄	—	146	1	no	—
316	C₄	#17989135	1	1	yes	characteristic
317	C₄	C₂×#14556610	1	1	yes	characteristic
318	C₂	—	16	2	no	—
319	C₂	—	292	1	no	—
320	C₂	#22620508	1	1	yes	characteristic
321	C₂	#22623069	1	1	yes	characteristic
322	C₂	#22623081	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#2249	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22612014	C₂
hypercenter	#2249	D₇₃
hypocenter	C₇₃	#33739
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2249

Complements

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

N	H ≅ G/N	type
1. #22620567	C₂	semidirect
2. #22612206	C₂	semidirect
3. #22612167	C₂	semidirect
4. #22612041	C₂	semidirect
5. #22612019	C₂	semidirect
7. #22611796	C₂	semidirect
8. Q₃₂×Dic₇₃	C₂	semidirect
9. D₁₆×Dic₇₃	C₂	semidirect
10. #22611526	C₂	semidirect
11. C₇₃×#2249	C₂	semidirect
164. #2249	D₇₃	semidirect
173. C₇₃	#33739	semidirect

Maximal subgroups

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

subgroup	index
#22620567	2
#22612206	2
#22612167	2
#22612041	2
#22612019	2
#22612014	2
#22611796	2
Q₃₂×Dic₇₃	2
D₁₆×Dic₇₃	2
#22611526	2
C₇₃×#2249	2
#33739	73