Subgroups — #33241388

Order 18688.

834 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989017	C₂	1	4	yes	maximal, complemented
2	#22620359	C₂	1	1	yes	maximal, characteristic, complemented
3	#22612190	C₂	1	4	yes	maximal, complemented
4	#22612188	C₂	1	1	yes	maximal, characteristic, complemented
5	#22612186	C₂	1	1	yes	maximal, characteristic, complemented
6	#22612183	C₂	1	1	yes	maximal, characteristic, complemented
7	#22612180	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×#11924003	C₂	1	1	yes	maximal, characteristic, complemented
9	C₇₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14556613	—	2	1	no	—
11	C₂×#14556613	C₂²	1	2	yes	—
12	#17989137	—	2	1	no	—
13	#17989017	—	2	16	no	—
14	C₂×#14556537	C₂²	1	4	yes	—
15	C₂×#14556535	C₂²	1	4	yes	—
16	C₂×#14556534	C₂²	1	4	yes	—
17	C₂×#14556532	C₂²	1	4	yes	—
18	C₂²×#11924003	C₂²	1	2	yes	—
19	#17988971	—	2	1	no	—
20	#17988971	C₂²	1	2	yes	—
21	#17988967	C₂²	1	1	yes	characteristic
22	#17988964	C₂²	1	1	yes	characteristic
23	#17988960	—	2	1	no	—
24	C₄×#11924003	C₂²	1	1	yes	characteristic
25	C₄×#11924003	C₂²	1	1	yes	characteristic
26	C₄×#11924003	C₂²	1	2	yes	—
27	C₂×#135×C₇₃	—	2	1	no	—
28	C₂×#135×C₇₃	C₂²	1	2	yes	—
29	#532×C₇₃	—	2	1	no	—
30	#495×C₇₃	—	2	1	no	—
31	#495×C₇₃	C₂²	1	2	yes	—
32	#494×C₇₃	C₂²	1	1	yes	characteristic
33	#493×C₇₃	C₂²	1	1	yes	characteristic
34	#492×C₇₃	—	2	1	no	—
35	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
36	C₂×#11924028	—	2	2	no	—
37	#14556613	—	2	8	no	—
38	#14556613	—	4	4	no	—
39	C₂×Dic₂₉₂	—	2	1	no	—
40	C₂×Dic₂₉₂	C₂³	1	2	yes	—
41	C₂×#11924017	—	2	2	no	—
42	C₂×#11924017	—	2	2	no	—
43	C₂×D₅₈₄	—	2	1	no	—
44	C₂×D₅₈₄	C₂³	1	2	yes	—
45	#14556591	—	2	1	no	—
46	#14556588	—	2	1	no	—
47	#14556579	—	2	2	no	—
48	#14556565	—	4	2	no	—
49	C₄×D₂₉₂	—	4	2	no	—
50	#14556537	—	2	8	no	—
51	#14556535	—	2	8	no	—
52	#14556534	—	2	8	no	—
53	#14556532	—	2	8	no	—
54	C₂×#11924003	D₄	1	8	yes	complemented
55	C₂×#11924003	C₂³	1	2	yes	—
56	C₂×#11924003	C₂³	1	2	yes	—
57	C₂×#41×C₇₃	—	2	2	no	—
58	#135×C₇₃	—	2	8	no	—
59	#135×C₇₃	—	4	4	no	—
60	C₂×Q₁₆×C₇₃	—	2	1	no	—
61	C₂×Q₁₆×C₇₃	C₂³	1	2	yes	—
62	C₂×SD₁₆×C₇₃	—	2	2	no	—
63	C₂×SD₁₆×C₇₃	—	2	2	no	—
64	C₂×D₈×C₇₃	—	2	1	no	—
65	C₂×D₈×C₇₃	C₂³	1	2	yes	—
66	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₄⋊Q₈×C₇₃	—	2	1	no	—
69	#127×C₇₃	—	2	1	no	—
70	#124×C₇₃	—	2	2	no	—
71	#121×C₇₃	—	4	2	no	—
72	C₄×D₄×C₇₃	—	4	2	no	—
73	C₂×C₄²×C₇₃	D₄	1	1	yes	characteristic
74	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
78	#11924028	—	4	4	no	—
79	#11924028	—	4	4	no	—
80	C₂×Dic₁₄₆	—	2	2	no	—
81	C₂×D₂₉₂	—	2	2	no	—
82	C₂×D₂₉₂	—	4	2	no	—
83	C₂×C₂²⋊D₇₃	—	4	2	no	—
84	C₂×C₄×D₇₃	—	4	2	no	—
85	Dic₂₉₂	—	2	4	no	—
86	Dic₂₉₂	—	4	2	no	—
87	#11924017	—	4	4	no	—
88	#11924017	—	4	4	no	—
89	D₅₈₄	—	2	4	no	—
90	D₅₈₄	—	4	2	no	—
91	C₄⋊Dic₇₃	—	4	2	no	—
92	#11924005	—	4	4	no	—
93	#11924003	C₂×D₄	1	4	yes	—
94	#11924003	C₂×D₄	1	4	yes	—
95	#41×C₇₃	—	4	4	no	—
96	#41×C₇₃	—	4	4	no	—
97	C₂×Q₈×C₇₃	—	2	2	no	—
98	C₂×D₄×C₇₃	—	2	2	no	—
99	C₂×D₄×C₇₃	—	4	2	no	—
100	C₂×D₄×C₇₃	—	4	2	no	—
101	C₂²×C₄×C₇₃	—	4	2	no	—
102	C₂²×C₄×C₇₃	D₈	1	2	yes	—
103	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
104	Q₁₆×C₇₃	—	2	4	no	—
105	Q₁₆×C₇₃	—	4	2	no	—
106	SD₁₆×C₇₃	—	4	4	no	—
107	SD₁₆×C₇₃	—	4	4	no	—
108	D₈×C₇₃	—	2	4	no	—
109	D₈×C₇₃	—	4	2	no	—
110	C₂×C₈×C₇₃	D₈	1	4	yes	—
111	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
112	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
116	C₄⋊C₄×C₇₃	—	4	2	no	—
117	C₂²⋊C₄×C₇₃	—	4	4	no	—
118	C₄²×C₇₃	D₈	1	2	yes	—
119	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
121	C₂³×C₇₃	—	4	2	no	—
122	C₂³×C₇₃	C₂×D₈	1	1	yes	characteristic
123	C₂²×D₇₃	—	4	2	no	—
124	Q₈×C₇₃	—	4	2	no	—
125	Q₈×C₇₃	—	4	2	no	—
126	D₄×C₇₃	—	4	2	no	—
127	D₄×C₇₃	—	4	2	no	—
128	D₄×C₇₃	—	8	4	no	—
129	D₄×C₇₃	—	8	4	no	—
130	C₂×C₄×C₇₃	—	2	4	no	—
131	C₂×C₄×C₇₃	—	4	2	no	—
132	C₂×C₄×C₇₃	—	8	4	no	—
133	C₂×C₄×C₇₃	#127	1	2	yes	—
134	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₇₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
139	C₂²⋊D₇₃	—	8	4	no	—
140	C₂×Dic₇₃	—	4	2	no	—
141	D₂₉₂	—	4	2	no	—
142	D₂₉₂	—	4	2	no	—
143	D₂₉₂	—	8	4	no	—
144	C₄×D₇₃	—	8	4	no	—
145	Dic₁₄₆	—	4	2	no	—
146	Dic₁₄₆	—	4	2	no	—
147	C₈×C₇₃	#127	1	2	yes	—
148	C₈×C₇₃	C₂×D₈	1	2	yes	—
149	C₈×C₇₃	C₂×D₈	1	2	yes	—
150	C₈×C₇₃	C₂²×D₄	1	2	yes	—
151	C₂²×C₇₃	—	2	2	no	—
152	C₂²×C₇₃	—	8	2	no	—
153	C₂²×C₇₃	—	8	4	no	—
154	C₂²×C₇₃	#493	1	2	yes	—
155	C₂²×C₇₃	C₂²×D₈	1	1	yes	characteristic
156	C₂×D₇₃	—	8	2	no	—
157	C₂×D₇₃	—	8	4	no	—
158	C₄×C₇₃	—	2	2	no	—
159	C₄×C₇₃	—	8	2	no	—
160	C₄×C₇₃	#493	1	2	yes	—
161	C₄×C₇₃	#508	1	4	yes	complemented
162	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
163	C₄×C₇₃	C₂²×D₈	1	1	yes	characteristic
164	Dic₇₃	—	8	2	no	—
165	#33713	—	73	1	no	maximal, Sylow
166	C₂×C₇₃	—	2	2	no	—
167	C₂×C₇₃	—	16	2	no	—
168	C₂×C₇₃	C₂×#493	1	1	yes	characteristic
169	C₂×C₇₃	C₂×#508	1	2	yes	—
170	D₇₃	—	16	2	no	—
171	C₂×#508	—	73	4	no	—
172	#3147	—	73	1	no	—
173	#3147	D₇₃	1	1	yes	characteristic, complemented
174	#2249	—	73	4	no	—
175	#2248	—	73	1	no	—
176	#2248	—	73	1	no	—
177	#2247	—	73	1	no	—
178	#2246	—	73	1	no	—
179	C₂×C₄×C₁₆	—	73	1	no	—
180	C₇₃	#33713	1	1	yes	characteristic, complemented, Sylow
181	C₂×#135	—	2	1	no	—
182	C₂×#135	—	73	2	no	—
183	C₂×#135	—	146	1	no	—
184	C₂×#135	C₂×D₇₃	1	2	yes	—
185	#532	—	2	1	no	—
186	#532	—	146	1	no	—
187	#508	—	146	16	no	—
188	C₂×Q₃₂	—	73	4	no	—
189	C₂×SD₃₂	—	73	4	no	—
190	C₂×SD₃₂	—	73	4	no	—
191	C₂×D₁₆	—	73	4	no	—
192	C₂²×C₁₆	—	73	2	no	—
193	#495	—	2	1	no	—
194	#495	—	73	2	no	—
195	#495	—	146	1	no	—
196	#495	C₂×D₇₃	1	2	yes	—
197	#494	—	73	1	no	—
198	#494	C₂×D₇₃	1	1	yes	characteristic
199	#493	—	73	1	no	—
200	#493	C₂×D₇₃	1	1	yes	characteristic
201	#492	—	2	1	no	—
202	#492	—	146	1	no	—
203	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
204	C₄×C₁₆	—	73	1	no	—
205	C₄×C₁₆	—	73	1	no	—
206	C₄×C₁₆	—	73	2	no	—
207	C₂×#41	—	2	2	no	—
208	C₂×#41	—	146	2	no	—
209	#135	—	2	8	no	—
210	#135	—	4	4	no	—
211	#135	—	292	4	no	—
212	#135	—	146	8	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	73	2	no	—
215	C₂×Q₁₆	—	146	1	no	—
216	C₂×Q₁₆	C₂²×D₇₃	1	2	yes	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×SD₁₆	—	2	2	no	—
219	C₂×SD₁₆	—	146	2	no	—
220	C₂×SD₁₆	—	146	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	—	73	2	no	—
223	C₂×D₈	—	146	1	no	—
224	C₂×D₈	C₂²×D₇₃	1	2	yes	—
225	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
226	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	1	no	—
228	C₄⋊Q₈	—	146	1	no	—
229	#127	—	2	1	no	—
230	#127	—	146	1	no	—
231	#124	—	2	2	no	—
232	#124	—	146	2	no	—
233	#121	—	4	2	no	—
234	#121	—	292	2	no	—
235	C₄×D₄	—	4	2	no	—
236	C₄×D₄	—	292	2	no	—
237	C₂×C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
238	Q₃₂	—	146	8	no	—
239	SD₃₂	—	146	8	no	—
240	SD₃₂	—	146	8	no	—
241	D₁₆	—	146	8	no	—
242	C₂×C₁₆	—	73	8	no	—
243	C₂×C₁₆	—	73	2	no	—
244	C₂×C₁₆	—	73	2	no	—
245	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
246	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
247	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
248	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
249	#41	—	4	4	no	—
250	#41	—	4	4	no	—
251	#41	—	292	4	no	—
252	#41	—	292	4	no	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	146	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	—	4	2	no	—
258	C₂×D₄	—	292	2	no	—
259	C₂×D₄	—	292	2	no	—
260	C₂×D₄	—	146	2	no	—
261	C₂²×C₄	—	4	2	no	—
262	C₂²×C₄	—	292	2	no	—
263	C₂²×C₄	#11924013	1	2	yes	—
264	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	4	2	no	—
267	Q₁₆	—	292	2	no	—
268	Q₁₆	—	146	4	no	—
269	SD₁₆	—	4	4	no	—
270	SD₁₆	—	4	4	no	—
271	SD₁₆	—	292	4	no	—
272	SD₁₆	—	292	4	no	—
273	D₈	—	2	4	no	—
274	D₈	—	4	2	no	—
275	D₈	—	292	2	no	—
276	D₈	—	146	4	no	—
277	C₂×C₈	#11924013	1	4	yes	—
278	C₂×C₈	D₄×D₇₃	1	4	yes	—
279	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
280	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
281	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
282	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	292	2	no	—
285	C₂²⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	292	4	no	—
287	C₄²	#11924013	1	2	yes	—
288	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
289	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
290	C₁₆	—	73	4	no	—
291	C₁₆	—	73	4	no	—
292	C₂³	—	4	2	no	—
293	C₂³	—	292	2	no	—
294	C₂³	C₂×#11924013	1	1	yes	characteristic
295	Q₈	—	4	2	no	—
296	Q₈	—	4	2	no	—
297	Q₈	—	292	2	no	—
298	Q₈	—	292	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	292	2	no	—
304	D₄	—	292	2	no	—
305	D₄	—	584	4	no	—
306	D₄	—	584	4	no	—
307	C₂×C₄	—	2	4	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	4	no	—
310	C₂×C₄	—	292	2	no	—
311	C₂×C₄	—	584	4	no	—
312	C₂×C₄	#14556587	1	2	yes	—
313	C₂×C₄	C₂×#11924013	1	2	yes	—
314	C₂×C₄	C₂×#11924013	1	2	yes	—
315	C₂×C₄	C₂×#11924013	1	2	yes	—
316	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
317	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
318	C₈	#14556587	1	2	yes	—
319	C₈	C₂×#11924013	1	2	yes	—
320	C₈	C₂×#11924013	1	2	yes	—
321	C₈	C₂×D₄×D₇₃	1	2	yes	—
322	C₂²	—	2	2	no	—
323	C₂²	—	8	4	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	584	4	no	—
326	C₂²	—	584	2	no	—
327	C₂²	#17988963	1	2	yes	—
328	C₂²	C₂²×#11924013	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	8	2	no	—
331	C₄	—	584	2	no	—
332	C₄	#17988963	1	2	yes	—
333	C₄	#17989017	1	4	yes	complemented
334	C₄	C₂×#14556587	1	1	yes	characteristic
335	C₄	C₂²×#11924013	1	1	yes	characteristic
336	C₂	—	2	2	no	—
337	C₂	—	16	2	no	—
338	C₂	—	1168	2	no	—
339	C₂	C₂×#17988963	1	1	yes	characteristic
340	C₂	C₂×#17989017	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#11924013
commutator, cosocle	C₂×C₈×C₇₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₇₃×#3147	C₂
socle	C₂²×C₇₃	C₂²×D₈
hypercenter	#3147	D₇₃
hypocenter	C₇₃	#33713
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#17989017	C₂	semidirect
2. #22620359	C₂	semidirect
3. #22612190	C₂	semidirect
4. #22612188	C₂	semidirect
5. #22612186	C₂	semidirect
6. #22612183	C₂	semidirect
7. #22612180	C₂	semidirect
8. C₂×C₄×#11924003	C₂	semidirect
9. C₇₃×#3147	C₂	semidirect
54. C₂×#11924003	D₄	semidirect
161. C₄×C₇₃	#508	semidirect
173. #3147	D₇₃	semidirect
180. C₇₃	#33713	semidirect
333. C₄	#17989017	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989017	2
#22620359	2
#22612190	2
#22612188	2
#22612186	2
#22612183	2
#22612180	2
C₂×C₄×#11924003	2
C₇₃×#3147	2
#33713	73