Subgroups — #33241539

Order 18688.

526 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	#22620504	C₂	1	2	yes	maximal, complemented
2	#22612208	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612193	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612164	C₂	1	2	yes	maximal, complemented
5	#22612030	C₂	1	2	yes	maximal, complemented
6	#22612019	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612018	C₂	1	1	yes	maximal, characteristic
8	C₄×#14556534	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₆×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989124	—	2	2	no	—
13	C₂×#14556534	C₂²	1	2	yes	—
14	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
17	#17988988	C₂²	1	2	yes	—
18	#17988972	C₂²	1	2	yes	—
19	#17988943	—	2	2	no	—
20	#17988825	C₂²	1	2	yes	—
21	#17988824	C₂²	1	2	yes	—
22	#17988819	—	2	2	no	—
23	C₄×#11924015	—	2	2	no	—
24	C₄×D₅₈₄	C₂²	1	2	yes	—
25	C₄×C₈×D₇₃	C₂²	1	1	yes	characteristic
26	#17988414	C₂²	1	2	yes	—
27	#17988413	C₂²	1	1	yes	characteristic
28	#17988391	C₂²	1	2	yes	—
29	#17988389	C₂²	1	2	yes	—
30	#17988387	C₂²	1	2	yes	—
31	C₄×#11924003	C₂²	1	1	yes	characteristic
32	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₇₃	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₇₃	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₇₃	C₂²	1	2	yes	—
36	#366×C₇₃	C₂²	1	1	yes	characteristic
37	#358×C₇₃	C₂²	1	2	yes	—
38	C₄×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
39	C₂×#11924015	—	2	2	no	—
40	C₂×D₅₈₄	C₂³	1	2	yes	—
41	C₂×C₈×D₇₃	D₄	1	1	yes	characteristic
42	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
43	#14556586	—	4	2	no	—
44	#14556582	—	2	2	no	—
45	#14556580	—	2	2	no	—
46	#14556576	—	4	2	no	—
47	#14556567	—	2	2	no	—
48	#14556566	—	2	2	no	—
49	C₄×D₂₉₂	—	2	2	no	—
50	C₄×D₂₉₂	—	4	2	no	—
51	#14556548	—	4	2	no	—
52	C₄²×D₇₃	D₄	1	1	yes	characteristic
53	#14556534	—	2	4	no	—
54	D₁₁₆₈	—	2	2	no	—
55	C₂×#11924003	C₂³	1	1	yes	characteristic
56	C₂×#11924003	C₂³	1	1	yes	characteristic
57	C₁₆×D₇₃	—	2	2	no	—
58	C₁₆×D₇₃	—	2	2	no	—
59	#14556525	C₂³	1	2	yes	—
60	#14556522	—	2	2	no	—
61	#14556515	—	2	2	no	—
62	#14556513	—	2	2	no	—
63	#14556511	—	2	2	no	—
64	C₄×#10596831	D₄	1	1	yes	characteristic
65	C₈×Dic₇₃	D₄	1	1	yes	characteristic
66	C₈×Dic₇₃	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
68	C₄×Q₈×C₇₃	—	2	2	no	—
69	Q₃₂×C₇₃	—	2	2	no	—
70	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
71	C₂×C₁₆×C₇₃	C₂³	1	1	yes	characteristic
72	#107×C₇₃	C₂³	1	2	yes	—
73	#103×C₇₃	—	2	2	no	—
74	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	C₂×D₂₉₂	—	2	2	no	—
76	C₂×D₂₉₂	—	4	2	no	—
77	C₂×C₄×D₇₃	—	4	2	no	—
78	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
79	C₂×C₄×D₇₃	D₈	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
81	#11924015	—	4	4	no	—
82	D₅₈₄	—	2	2	no	—
83	D₅₈₄	—	2	2	no	—
84	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
85	C₂×#10596831	C₂×D₄	1	1	yes	characteristic
86	C₈×D₇₃	—	2	1	no	—
87	C₈×D₇₃	—	2	1	no	—
88	C₈×D₇₃	—	2	2	no	—
89	C₄⋊Dic₇₃	—	2	2	no	—
90	C₄⋊Dic₇₃	—	4	2	no	—
91	#11924007	—	4	2	no	—
92	#11924007	—	4	2	no	—
93	#11924005	—	4	2	no	—
94	#11924005	—	4	2	no	—
95	#11924005	—	4	2	no	—
96	#11924005	—	4	2	no	—
97	#11924005	—	4	2	no	—
98	C₄×Dic₇₃	D₈	1	1	yes	characteristic
99	C₄×Dic₇₃	D₈	1	1	yes	characteristic
100	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
101	#11924003	—	2	1	no	—
102	#11924003	#41	1	2	yes	—
103	C₂×Q₈×C₇₃	—	2	2	no	—
104	Q₁₆×C₇₃	—	2	2	no	—
105	Q₁₆×C₇₃	—	2	2	no	—
106	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₇₃	—	2	2	no	—
109	C₄⋊C₄×C₇₃	—	4	2	no	—
110	C₄²×C₇₃	—	4	2	no	—
111	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₇₃	—	2	1	no	—
113	C₁₆×C₇₃	#41	1	2	yes	—
114	C₂²×D₇₃	—	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	C₂×C₄×C₇₃	—	4	2	no	—
119	C₂×C₄×C₇₃	—	4	2	no	—
120	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×Dic₇₃	—	4	2	no	—
124	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₇₃	C₂×D₈	1	1	yes	characteristic
127	D₂₉₂	—	4	2	no	—
128	D₂₉₂	—	4	2	no	—
129	D₂₉₂	—	8	4	no	—
130	C₄×D₇₃	—	2	2	no	—
131	C₄×D₇₃	—	2	1	no	—
132	C₄×D₇₃	—	2	1	no	—
133	C₄×D₇₃	—	2	2	no	—
134	C₄×D₇₃	—	8	4	no	—
135	C₈×C₇₃	—	2	1	no	—
136	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
137	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
138	#10596831	—	2	1	no	—
139	#10596831	#135	1	2	yes	—
140	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₇₃	—	2	1	no	—
142	C₂×D₇₃	—	2	1	no	—
143	C₂×D₇₃	—	8	2	no	—
144	C₂×D₇₃	—	8	2	no	—
145	C₂×D₇₃	—	8	2	no	—
146	C₂×D₇₃	#508	1	2	yes	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	8	2	no	—
149	C₄×C₇₃	—	8	2	no	—
150	C₄×C₇₃	#508	1	2	yes	—
151	C₄×C₇₃	#532	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	—	2	1	no	—
156	Dic₇₃	—	8	2	no	—
157	#33735	—	73	1	no	maximal, Sylow
158	C₂×C₇₃	#3169	1	1	yes	characteristic
159	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
160	C₂×C₇₃	C₂×#508	1	1	yes	characteristic
161	D₇₃	—	2	2	no	—
162	D₇₃	—	16	2	no	—
163	#3168	—	73	2	no	—
164	#2253	—	73	1	no	—
165	#2249	—	73	1	no	—
166	#2242	—	73	2	no	—
167	#2217	—	73	2	no	—
168	#2216	—	73	1	no	—
169	#2215	—	73	1	no	—
170	C₄×Q₃₂	D₇₃	1	1	yes	characteristic, complemented
171	C₄×SD₃₂	—	73	2	no	—
172	C₄×D₁₆	—	73	1	no	—
173	C₂×C₄×C₁₆	—	73	1	no	—
174	C₇₃	#33735	1	1	yes	characteristic, complemented, Sylow
175	#529	—	146	2	no	—
176	C₂×Q₃₂	C₂×D₇₃	1	1	yes	characteristic
177	C₂×SD₃₂	—	73	2	no	—
178	C₂×D₁₆	—	73	1	no	—
179	C₂²×C₁₆	—	73	1	no	—
180	C₂²×C₁₆	—	73	1	no	—
181	#499	—	73	2	no	—
182	#495	—	73	2	no	—
183	#488	—	146	2	no	—
184	#467	—	73	2	no	—
185	#466	—	73	2	no	—
186	#465	—	146	2	no	—
187	C₄×Q₁₆	C₂×D₇₃	1	2	yes	—
188	C₄×SD₁₆	—	146	2	no	—
189	C₄×D₈	—	73	2	no	—
190	C₂×C₄×C₈	—	73	1	no	—
191	#367	—	73	2	no	—
192	#366	—	73	1	no	—
193	#366	C₂×D₇₃	1	1	yes	characteristic
194	#358	—	73	2	no	—
195	#358	C₂×D₇₃	1	2	yes	—
196	#357	—	73	2	no	—
197	#357	—	73	2	no	—
198	C₄×C₁₆	—	73	1	no	—
199	C₄×C₁₆	—	73	1	no	—
200	C₄×C₁₆	—	73	1	no	—
201	C₄×C₁₆	C₂×D₇₃	1	1	yes	characteristic
202	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
203	C₂×SD₁₆	—	146	2	no	—
204	C₂×D₈	—	73	2	no	—
205	C₂²×C₈	—	73	1	no	—
206	C₂²×C₈	—	73	1	no	—
207	#126	—	292	2	no	—
208	#125	—	146	2	no	—
209	#124	—	146	2	no	—
210	#123	—	292	2	no	—
211	C₂²⋊Q₈	—	146	2	no	—
212	#121	—	146	2	no	—
213	C₄×Q₈	—	2	2	no	—
214	C₄×D₄	—	292	2	no	—
215	C₄×D₄	—	146	2	no	—
216	#117	—	292	2	no	—
217	C₂×C₄²	—	73	1	no	—
218	Q₃₂	—	2	2	no	—
219	SD₃₂	—	146	4	no	—
220	D₁₆	—	146	2	no	—
221	C₂×C₁₆	—	73	1	no	—
222	C₂×C₁₆	—	73	1	no	—
223	C₂×C₁₆	—	146	2	no	—
224	C₂×C₁₆	—	146	2	no	—
225	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂²×D₇₃	1	1	yes	characteristic
227	#107	—	73	2	no	—
228	#107	C₂²×D₇₃	1	2	yes	—
229	#106	—	146	2	no	—
230	#103	—	2	2	no	—
231	#103	—	146	2	no	—
232	#102	—	146	2	no	—
233	#102	—	146	2	no	—
234	C₄×C₈	—	73	1	no	—
235	C₄×C₈	—	73	1	no	—
236	C₄×C₈	—	73	1	no	—
237	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
238	C₂×Q₈	—	2	2	no	—
239	C₂×D₄	—	292	2	no	—
240	C₂×D₄	—	146	2	no	—
241	C₂²×C₄	—	292	2	no	—
242	C₂²×C₄	—	73	1	no	—
243	C₂²×C₄	—	73	1	no	—
244	C₂²×C₄	—	73	1	no	—
245	Q₁₆	—	2	2	no	—
246	Q₁₆	—	2	2	no	—
247	SD₁₆	—	292	4	no	—
248	D₈	—	146	2	no	—
249	D₈	—	146	2	no	—
250	C₂×C₈	—	73	1	no	—
251	C₂×C₈	—	73	1	no	—
252	C₂×C₈	—	146	2	no	—
253	C₂×C₈	—	146	1	no	—
254	C₂×C₈	—	146	1	no	—
255	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	292	2	no	—
260	C₄⋊C₄	—	292	2	no	—
261	C₄⋊C₄	—	292	2	no	—
262	C₄⋊C₄	—	146	2	no	—
263	C₂²⋊C₄	—	292	2	no	—
264	C₂²⋊C₄	—	292	2	no	—
265	C₂²⋊C₄	—	292	2	no	—
266	C₂²⋊C₄	—	292	2	no	—
267	C₂²⋊C₄	—	292	2	no	—
268	C₄²	—	4	2	no	—
269	C₄²	—	73	1	no	—
270	C₄²	—	73	1	no	—
271	C₄²	—	73	1	no	—
272	C₄²	D₄×D₇₃	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	73	2	no	—
275	C₁₆	—	146	1	no	—
276	C₁₆	#11924028	1	2	yes	—
277	C₂³	—	292	2	no	—
278	C₂³	—	73	1	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	4	2	no	—
281	D₄	—	292	2	no	—
282	D₄	—	292	2	no	—
283	D₄	—	584	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	292	2	no	—
287	C₂×C₄	—	584	4	no	—
288	C₂×C₄	—	73	1	no	—
289	C₂×C₄	—	73	1	no	—
290	C₂×C₄	—	73	1	no	—
291	C₂×C₄	—	146	2	no	—
292	C₂×C₄	—	146	1	no	—
293	C₂×C₄	—	146	2	no	—
294	C₂×C₄	—	146	1	no	—
295	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
296	C₂×C₄	D₈×D₇₃	1	1	yes	characteristic
297	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	73	2	no	—
300	C₈	—	146	1	no	—
301	C₈	#41×D₇₃	1	1	yes	characteristic
302	C₈	C₂×#11924028	1	1	yes	characteristic
303	C₂²	—	584	2	no	—
304	C₂²	—	584	2	no	—
305	C₂²	—	584	2	no	—
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	—	8	2	no	—
312	C₄	—	8	2	no	—
313	C₄	—	584	2	no	—
314	C₄	—	146	1	no	—
315	C₄	—	146	1	no	—
316	C₄	—	146	1	no	—
317	C₄	#17989020	1	2	yes	—
318	C₄	#17989134	1	1	yes	characteristic
319	C₄	#135×D₇₃	1	1	yes	characteristic
320	C₂	—	1168	2	no	—
321	C₂	—	146	2	no	—
322	C₂	#22620507	1	1	yes	characteristic
323	C₂	#508×D₇₃	1	1	yes	characteristic
324	C₂	C₂×#17989020	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₇₃
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₄×Q₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂²×D₈
prime radical	#22612018	C₂
hypercenter	C₄×Q₃₂	D₇₃
hypocenter	C₇₃	#33735
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
1. #22620504	C₂	semidirect
2. #22612208	C₂	semidirect
3. #22612193	C₂	semidirect
4. #22612164	C₂	semidirect
5. #22612030	C₂	semidirect
6. #22612019	C₂	semidirect
8. C₄×#14556534	C₂	semidirect
9. C₄×D₁₁₆₈	C₂	semidirect
10. C₄×C₁₆×D₇₃	C₂	semidirect
11. C₄×Q₃₂×C₇₃	C₂	semidirect
170. C₄×Q₃₂	D₇₃	semidirect
174. C₇₃	#33735	semidirect

Maximal subgroups

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

subgroup	index
#22620504	2
#22612208	2
#22612193	2
#22612164	2
#22612030	2
#22612019	2
#22612018	2
C₄×#14556534	2
C₄×D₁₁₆₈	2
C₄×C₁₆×D₇₃	2
C₄×Q₃₂×C₇₃	2
#33735	73