Subgroups — #33241581

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	#22620572	C₂	1	2	yes	maximal, complemented
2	#22612205	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612193	C₂	1	1	yes	maximal, characteristic, complemented
4	#22612169	C₂	1	2	yes	maximal, complemented
5	#22612047	C₂	1	2	yes	maximal, complemented
6	#22612019	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612018	C₂	1	1	yes	maximal, characteristic
8	#22611793	C₂	1	2	yes	maximal, complemented
9	C₄×D₁₁₆₈	C₂	1	1	yes	maximal, characteristic, complemented
10	#22611526	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#17989209	—	2	2	no	—
13	C₂×#14556534	C₂²	1	2	yes	—
14	C₂×D₁₁₆₈	C₂²	1	1	yes	characteristic
15	C₂×#14556530	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₇₃	C₂²	1	1	yes	characteristic
17	#17988991	C₂²	1	2	yes	—
18	#17988972	C₂²	1	2	yes	—
19	#17988948	—	2	2	no	—
20	#17988836	—	2	2	no	—
21	#17988825	C₂²	1	2	yes	—
22	#17988824	C₂²	1	2	yes	—
23	#17988685	—	2	2	no	—
24	C₄×D₅₈₄	C₂²	1	2	yes	—
25	#17988541	C₂²	1	1	yes	characteristic
26	#17988413	C₂²	1	1	yes	characteristic
27	#17988412	C₂²	1	2	yes	—
28	#17988391	C₂²	1	2	yes	—
29	#17988389	C₂²	1	2	yes	—
30	#17988387	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	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×#11924011	D₄	1	1	yes	characteristic
42	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
43	#14556590	—	2	2	no	—
44	#14556585	—	4	2	no	—
45	#14556580	—	2	2	no	—
46	#14556579	—	4	2	no	—
47	#14556576	—	4	2	no	—
48	#14556569	—	4	2	no	—
49	#14556567	—	2	2	no	—
50	#14556566	—	2	2	no	—
51	C₄×D₂₉₂	—	2	2	no	—
52	#14556548	D₄	1	1	yes	characteristic
53	#14556534	—	2	4	no	—
54	D₁₁₆₈	—	2	2	no	—
55	#14556530	—	2	2	no	—
56	C₂×#11924003	C₂³	1	1	yes	characteristic
57	C₂×#11924003	C₂³	1	1	yes	characteristic
58	C₁₆×D₇₃	—	2	2	no	—
59	#14556525	C₂³	1	2	yes	—
60	#14556524	—	2	2	no	—
61	#14556515	—	2	2	no	—
62	#14556513	—	2	2	no	—
63	#14556511	—	2	2	no	—
64	#14556502	D₄	1	1	yes	characteristic
65	#14556501	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×Dic₁₄₆	—	4	2	no	—
76	C₂×D₂₉₂	—	2	2	no	—
77	C₂×D₂₉₂	—	4	2	no	—
78	C₂×C₄×D₇₃	—	4	2	no	—
79	C₂×C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
80	#11924015	—	4	4	no	—
81	D₅₈₄	—	2	2	no	—
82	D₅₈₄	—	2	2	no	—
83	#11924011	—	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	C₄⋊Dic₇₃	—	2	2	no	—
89	#11924007	—	4	2	no	—
90	#11924007	—	4	2	no	—
91	#11924007	D₈	1	1	yes	characteristic
92	#11924007	D₈	1	1	yes	characteristic
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	#11924005	D₈	1	1	yes	characteristic
99	#11924005	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	—	16	2	no	—
130	C₄×D₇₃	—	2	1	no	—
131	C₄×D₇₃	—	2	1	no	—
132	C₄×D₇₃	—	16	2	no	—
133	Dic₁₄₆	—	16	2	no	—
134	C₈×C₇₃	—	2	1	no	—
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	#135	1	2	yes	—
139	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₇₃	—	2	1	no	—
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	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	8	2	no	—
149	C₄×C₇₃	—	8	2	no	—
150	C₄×C₇₃	#532	1	1	yes	characteristic
151	C₄×C₇₃	C₂×#135	1	1	yes	characteristic
152	Dic₇₃	—	2	1	no	—
153	Dic₇₃	—	2	1	no	—
154	Dic₇₃	—	2	1	no	—
155	Dic₇₃	—	8	2	no	—
156	#33739	—	73	1	no	maximal, Sylow
157	C₂×C₇₃	#3169	1	1	yes	characteristic
158	C₂×C₇₃	#3415	1	1	yes	characteristic
159	C₂×C₇₃	#3417	1	1	yes	characteristic
160	D₇₃	—	4	1	no	—
161	D₇₃	—	16	2	no	—
162	#3175	—	73	2	no	—
163	#2252	—	73	1	no	—
164	#2249	—	73	1	no	—
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	D₇₃	1	1	yes	characteristic, complemented
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	—	73	2	no	—
177	C₂×D₁₆	—	73	1	no	—
178	C₂×M₃₂	—	73	1	no	—
179	C₂²×C₁₆	—	73	1	no	—
180	#500	—	73	2	no	—
181	#495	—	73	2	no	—
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	C₂×D₇₃	1	2	yes	—
188	C₄×D₈	—	73	2	no	—
189	#405	—	73	1	no	—
190	#366	—	73	1	no	—
191	#366	—	73	2	no	—
192	#366	C₂×D₇₃	1	1	yes	characteristic
193	#358	—	73	2	no	—
194	#358	C₂×D₇₃	1	2	yes	—
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	—	146	2	no	—
203	C₂×D₈	—	73	2	no	—
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	—	292	2	no	—
209	#124	—	146	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	—	2	2	no	—
215	C₄×D₄	—	146	2	no	—
216	#117	—	73	1	no	—
217	Q₃₂	—	2	2	no	—
218	SD₃₂	—	146	4	no	—
219	D₁₆	—	146	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	—	146	2	no	—
228	#107	C₂²×D₇₃	1	2	yes	—
229	#103	—	2	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	—	292	2	no	—
240	C₂×D₄	—	146	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	—	292	4	no	—
246	D₈	—	146	2	no	—
247	D₈	—	146	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	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	292	2	no	—
258	C₄⋊C₄	—	292	2	no	—
259	C₄⋊C₄	—	73	1	no	—
260	C₄⋊C₄	—	73	1	no	—
261	C₄⋊C₄	—	146	2	no	—
262	C₂²⋊C₄	—	292	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	—	73	1	no	—
268	C₂²⋊C₄	—	73	1	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	73	1	no	—
271	C₄²	D₄×D₇₃	1	1	yes	characteristic
272	C₁₆	—	2	1	no	—
273	C₁₆	—	73	2	no	—
274	C₁₆	—	146	1	no	—
275	C₁₆	#11924028	1	2	yes	—
276	C₂³	—	292	2	no	—
277	C₂³	—	73	1	no	—
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	1168	2	no	—
281	D₄	—	292	2	no	—
282	D₄	—	292	2	no	—
283	D₄	—	1168	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	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	—	2	1	no	—
297	C₈	—	73	2	no	—
298	C₈	—	146	1	no	—
299	C₈	#41×D₇₃	1	1	yes	characteristic
300	C₈	C₂×#11924028	1	1	yes	characteristic
301	C₂²	—	584	2	no	—
302	C₂²	—	584	2	no	—
303	C₂²	—	584	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	—	8	2	no	—
312	C₄	—	584	2	no	—
313	C₄	—	146	1	no	—
314	C₄	—	146	1	no	—
315	C₄	—	146	1	no	—
316	C₄	#17989134	1	1	yes	characteristic
317	C₄	#135×D₇₃	1	1	yes	characteristic
318	C₂	—	292	1	no	—
319	C₂	—	1168	2	no	—
320	C₂	#22620507	1	1	yes	characteristic
321	C₂	#22623069	1	1	yes	characteristic
322	C₂	#22623080	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×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	#33739
LCS	G ▷ C₂×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
1. #22620572	C₂	semidirect
2. #22612205	C₂	semidirect
3. #22612193	C₂	semidirect
4. #22612169	C₂	semidirect
5. #22612047	C₂	semidirect
6. #22612019	C₂	semidirect
8. #22611793	C₂	semidirect
9. C₄×D₁₁₆₈	C₂	semidirect
10. #22611526	C₂	semidirect
11. C₄×Q₃₂×C₇₃	C₂	semidirect
170. C₄×Q₃₂	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
#22620572	2
#22612205	2
#22612193	2
#22612169	2
#22612047	2
#22612019	2
#22612018	2
#22611793	2
C₄×D₁₁₆₈	2
#22611526	2
C₄×Q₃₂×C₇₃	2
#33739	73