Subgroups — #34012118

Order 18688.

630 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989643	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556645	C₂	1	1	yes	maximal, characteristic
3	#22624971	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11924036	C₄	1	1	yes	characteristic
6	C₂²×C₄×#64408	C₄	1	1	yes	characteristic
7	C₂×#14556666	C₂²	1	2	yes	—
8	#17989643	—	2	2	no	—
9	#17989643	—	2	2	no	—
10	C₂×#14556665	C₄	1	2	yes	complemented
11	C₂×#14556664	C₄	1	2	yes	complemented
12	C₂×#14556645	—	2	2	no	—
13	C₂×#14556645	—	2	2	no	—
14	C₂×#14556645	C₄	1	2	yes	—
15	C₂×#14556645	C₂²	1	2	yes	—
16	C₂²×C₈×D₇₃	C₂²	1	1	yes	characteristic
17	C₂×#14556531	C₄	1	1	yes	characteristic, complemented
18	C₂×#14556530	C₂²	1	2	yes	—
19	M₃₂×D₇₃	—	2	2	no	—
20	M₃₂×D₇₃	—	2	2	no	—
21	C₂×C₁₆×D₇₃	C₄	1	2	yes	complemented
22	C₂×M₃₂×C₇₃	C₄	1	1	yes	characteristic, complemented
23	C₂³×#64408	—	2	1	no	—
24	C₂×#11924036	—	2	2	no	—
25	C₂×#11924036	—	2	1	no	—
26	C₂×#11924036	C₈	1	4	yes	—
27	C₂×#11924036	C₂×C₄	1	2	yes	—
28	C₂²×#10596833	—	2	1	no	—
29	C₂×C₄×#64408	—	2	2	no	—
30	C₂×C₄×#64408	—	2	1	no	—
31	C₂×C₄×#64408	C₈	1	4	yes	—
32	C₂×C₄×#64408	C₂×C₄	1	2	yes	—
33	#14556666	—	2	2	no	—
34	#14556666	—	2	2	no	—
35	#14556665	—	2	4	no	—
36	#14556664	—	2	4	no	—
37	C₂×#11924031	C₂×C₄	1	4	yes	complemented
38	#14556645	—	2	4	no	—
39	#14556645	—	2	2	no	—
40	#14556645	—	2	2	no	—
41	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₇₃	D₄	1	2	yes	—
47	C₂×C₈×D₇₃	D₄	1	2	yes	—
48	C₂×C₈×D₇₃	D₄	1	2	yes	—
49	C₂×C₈×D₇₃	Q₈	1	2	yes	—
50	C₂×C₈×D₇₃	C₂³	1	1	yes	characteristic
51	#14556531	—	2	2	no	—
52	#14556530	—	2	2	no	—
53	#14556530	—	2	2	no	—
54	C₂×#11924003	C₂×C₄	1	2	yes	complemented
55	C₁₆×D₇₃	—	2	4	no	—
56	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	M₃₂×C₇₃	—	2	2	no	—
58	C₂×C₁₆×C₇₃	C₂×C₄	1	2	yes	complemented
59	C₂²×#64408	—	2	4	no	—
60	C₂²×#64408	—	2	2	no	—
61	C₂²×#64408	—	4	4	no	—
62	#11924036	—	2	4	no	—
63	#11924036	—	2	2	no	—
64	#11924036	C₂×C₈	1	4	yes	—
65	C₂×#10596833	—	2	4	no	—
66	C₂×#10596833	—	2	2	no	—
67	C₄×#64408	—	2	4	no	—
68	C₄×#64408	—	2	2	no	—
69	C₄×#64408	C₂×C₈	1	4	yes	—
70	#11924031	—	2	4	no	—
71	C₂³×D₇₃	M₁₆	1	1	yes	characteristic
72	C₂²×Dic₇₃	M₁₆	1	1	yes	characteristic
73	C₂×C₄×D₇₃	C₄²	1	1	yes	characteristic
74	C₂×C₄×D₇₃	C₂²⋊C₄	1	2	yes	—
75	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
77	C₂×C₄×D₇₃	C₂×C₈	1	2	yes	—
78	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
79	C₂×C₄×D₇₃	M₁₆	1	1	yes	characteristic
80	C₂×C₄×D₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×#10596831	C₄²	1	1	yes	characteristic
82	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
83	C₂×#10596831	C₄⋊C₄	1	2	yes	—
84	C₂×#10596831	C₂²×C₄	1	1	yes	characteristic
85	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
86	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
87	C₈×D₇₃	C₂²⋊C₄	1	2	yes	—
88	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
89	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
90	C₈×D₇₃	C₄⋊C₄	1	2	yes	—
91	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
92	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
93	C₈×D₇₃	C₂×D₄	1	1	yes	characteristic
94	C₈×D₇₃	C₂×Q₈	1	1	yes	characteristic
95	#11924003	—	2	2	no	—
96	C₂²×C₄×C₇₃	C₄²	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₄²	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²⋊C₄	1	2	yes	—
99	C₂×C₈×C₇₃	C₄⋊C₄	1	2	yes	—
100	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
101	C₁₆×C₇₃	—	2	2	no	—
102	C₂³×C₇₃	#97	1	1	yes	characteristic
103	C₂²×D₇₃	—	2	4	no	—
104	C₂²×D₇₃	#97	1	1	yes	characteristic
105	C₂²×D₇₃	C₂²⋊C₈	1	2	yes	—
106	C₂²×D₇₃	C₄⋊C₈	1	2	yes	—
107	C₂²×D₇₃	C₂×M₁₆	1	1	yes	characteristic
108	C₂×#64408	—	2	4	no	—
109	C₂×#64408	—	4	4	no	—
110	C₂×#64408	—	4	8	no	—
111	C₂×C₄×C₇₃	#95	1	2	yes	—
112	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
113	C₂×C₄×C₇₃	#97	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×Dic₇₃	#97	1	1	yes	characteristic
116	C₂×Dic₇₃	C₂²⋊C₈	1	2	yes	—
117	C₂×Dic₇₃	C₄⋊C₈	1	2	yes	—
118	C₂×Dic₇₃	C₂×M₁₆	1	1	yes	characteristic
119	C₄×D₇₃	#95	1	2	yes	—
120	C₄×D₇₃	C₄×C₈	1	2	yes	—
121	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
122	C₄×D₇₃	C₂²⋊C₈	1	2	yes	—
123	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
124	C₄×D₇₃	C₄⋊C₈	1	2	yes	—
125	C₄×D₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
126	C₄×D₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
128	C₄×D₇₃	C₂²×C₈	1	1	yes	characteristic
129	#10596833	—	2	4	no	—
130	C₈×C₇₃	#95	1	2	yes	—
131	C₈×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₈×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	#10596831	#95	1	2	yes	—
134	#10596831	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	#10596831	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₇₃	—	2	2	no	—
137	C₂²×C₇₃	#336	1	2	yes	—
138	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
139	C₂×D₇₃	—	2	2	no	—
140	C₂×D₇₃	—	2	4	no	—
141	C₂×D₇₃	—	2	4	no	—
142	C₂×D₇₃	#336	1	2	yes	—
143	C₂×D₇₃	#347	1	4	yes	—
144	C₂×D₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₂×D₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
146	#64408	—	4	4	no	—
147	C₄×C₇₃	#336	1	2	yes	—
148	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
149	C₄×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
150	Dic₇₃	#336	1	2	yes	—
151	Dic₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
152	Dic₇₃	C₂×C₄⋊C₈	1	1	yes	characteristic
153	C₂×#1375	—	73	1	no	maximal, Sylow
154	C₂×C₇₃	—	2	2	no	—
155	C₂×C₇₃	C₂×#336	1	1	yes	characteristic
156	C₂×C₇₃	C₂×#347	1	2	yes	—
157	D₇₃	—	2	1	no	—
158	D₇₃	—	2	1	no	—
159	D₇₃	#1375	1	4	yes	complemented
160	C₂²×M₃₂	—	73	1	no	—
161	C₂²×M₃₂	—	73	1	no	—
162	C₂²×#97	—	73	1	no	—
163	#1375	—	73	4	no	—
164	C₇₃	C₂×#1375	1	1	yes	characteristic, complemented, Sylow
165	C₂³×C₈	—	73	1	no	—
166	C₂³×C₈	—	73	1	no	—
167	C₂²×C₄²	—	73	1	no	—
168	C₂×M₃₂	—	73	2	no	—
169	C₂×M₃₂	—	73	2	no	—
170	C₂×M₃₂	—	73	1	no	—
171	C₂×M₃₂	—	73	2	no	—
172	C₂×M₃₂	—	146	2	no	—
173	C₂×M₃₂	—	146	2	no	—
174	C₂×M₃₂	—	146	2	no	—
175	C₂×M₃₂	—	146	2	no	—
176	C₂×M₃₂	#64408	1	1	yes	characteristic, complemented
177	C₂²×C₁₆	—	73	2	no	—
178	C₂²×C₁₆	—	73	2	no	—
179	C₂×#97	—	73	2	no	—
180	C₂×#97	—	73	2	no	—
181	C₂×#97	—	146	2	no	—
182	C₂×#97	—	146	2	no	—
183	C₂³×C₄	—	73	1	no	—
184	C₂³×C₄	—	146	1	no	—
185	C₂²×C₈	—	73	1	no	—
186	C₂²×C₈	—	73	1	no	—
187	C₂²×C₈	—	73	1	no	—
188	C₂²×C₈	—	73	2	no	—
189	C₂²×C₈	—	73	2	no	—
190	C₂²×C₈	—	73	2	no	—
191	C₂²×C₈	—	73	2	no	—
192	C₂²×C₈	—	73	1	no	—
193	C₂²×C₈	—	73	1	no	—
194	C₂²×C₈	—	73	4	no	—
195	C₂²×C₈	—	73	2	no	—
196	C₂²×C₈	—	146	1	no	—
197	C₂²×C₈	—	146	1	no	—
198	C₂²×C₈	—	146	2	no	—
199	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
200	C₂×C₄²	—	73	2	no	—
201	C₂×C₄²	—	73	4	no	—
202	C₂×C₄²	—	146	2	no	—
203	C₂×C₄²	—	146	1	no	—
204	M₃₂	—	2	2	no	—
205	M₃₂	—	146	2	no	—
206	M₃₂	—	146	2	no	—
207	M₃₂	—	146	4	no	—
208	M₃₂	—	146	2	no	—
209	M₃₂	—	146	2	no	—
210	M₃₂	—	146	2	no	—
211	C₂×C₁₆	—	73	4	no	—
212	C₂×C₁₆	—	73	2	no	—
213	C₂×C₁₆	—	146	4	no	—
214	C₂×C₁₆	—	146	4	no	—
215	C₂×C₁₆	C₂×#64408	1	2	yes	complemented
216	#97	—	146	4	no	—
217	#97	—	146	2	no	—
218	#97	—	146	2	no	—
219	C₂⁴	—	73	1	no	—
220	C₂²×C₄	—	292	4	no	—
221	C₂²×C₄	—	73	1	no	—
222	C₂²×C₄	—	73	2	no	—
223	C₂²×C₄	—	73	2	no	—
224	C₂²×C₄	—	73	2	no	—
225	C₂²×C₄	—	73	2	no	—
226	C₂²×C₄	—	73	1	no	—
227	C₂²×C₄	—	73	1	no	—
228	C₂²×C₄	—	73	1	no	—
229	C₂²×C₄	—	73	1	no	—
230	C₂²×C₄	—	146	4	no	—
231	C₂²×C₄	—	146	2	no	—
232	C₂²×C₄	C₄×#64408	1	1	yes	characteristic
233	C₂×C₈	—	73	2	no	—
234	C₂×C₈	—	73	2	no	—
235	C₂×C₈	—	73	1	no	—
236	C₂×C₈	—	73	1	no	—
237	C₂×C₈	—	73	2	no	—
238	C₂×C₈	—	73	2	no	—
239	C₂×C₈	—	73	2	no	—
240	C₂×C₈	—	73	2	no	—
241	C₂×C₈	—	73	2	no	—
242	C₂×C₈	—	73	1	no	—
243	C₂×C₈	—	73	1	no	—
244	C₂×C₈	—	73	2	no	—
245	C₂×C₈	—	73	1	no	—
246	C₂×C₈	—	73	1	no	—
247	C₂×C₈	—	73	4	no	—
248	C₂×C₈	—	146	4	no	—
249	C₂×C₈	—	146	2	no	—
250	C₂×C₈	—	146	2	no	—
251	C₂×C₈	—	146	4	no	—
252	C₂×C₈	C₄×#64408	1	1	yes	characteristic
253	C₂×C₈	C₂²⋊#64408	1	2	yes	—
254	C₂×C₈	C₄⋊#64408	1	2	yes	—
255	C₂×C₈	C₂²×#64408	1	1	yes	characteristic
256	C₄²	—	73	4	no	—
257	C₄²	—	146	2	no	—
258	C₄²	—	146	4	no	—
259	C₁₆	—	2	2	no	—
260	C₁₆	—	146	4	no	—
261	C₁₆	—	146	2	no	—
262	C₂³	—	73	1	no	—
263	C₂³	—	73	2	no	—
264	C₂³	—	73	2	no	—
265	C₂³	—	73	1	no	—
266	C₂³	—	146	4	no	—
267	C₂³	#14556645	1	1	yes	characteristic
268	C₂×C₄	—	292	8	no	—
269	C₂×C₄	—	292	4	no	—
270	C₂×C₄	—	73	2	no	—
271	C₂×C₄	—	73	2	no	—
272	C₂×C₄	—	73	1	no	—
273	C₂×C₄	—	73	1	no	—
274	C₂×C₄	—	73	2	no	—
275	C₂×C₄	—	73	2	no	—
276	C₂×C₄	—	73	2	no	—
277	C₂×C₄	—	73	2	no	—
278	C₂×C₄	—	73	1	no	—
279	C₂×C₄	—	73	2	no	—
280	C₂×C₄	—	73	1	no	—
281	C₂×C₄	—	73	1	no	—
282	C₂×C₄	—	73	1	no	—
283	C₂×C₄	—	73	2	no	—
284	C₂×C₄	—	146	4	no	—
285	C₂×C₄	(C₂×#64408)⋊C₄	1	2	yes	—
286	C₂×C₄	C₈×#64408	1	2	yes	—
287	C₂×C₄	#14556645	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄×#64408	1	1	yes	characteristic
289	C₈	—	73	1	no	—
290	C₈	—	73	2	no	—
291	C₈	—	73	1	no	—
292	C₈	—	146	4	no	—
293	C₈	(C₂×#64408)⋊C₄	1	2	yes	—
294	C₈	C₂×C₂²⋊#64408	1	1	yes	characteristic
295	C₈	C₂×C₄⋊#64408	1	1	yes	characteristic
296	C₂²	—	2	2	no	—
297	C₂²	—	73	2	no	—
298	C₂²	—	73	4	no	—
299	C₂²	—	73	1	no	—
300	C₂²	—	73	1	no	—
301	C₂²	—	146	4	no	—
302	C₂²	—	146	4	no	—
303	C₂²	—	146	2	no	—
304	C₂²	#17989452	1	2	yes	—
305	C₂²	C₂×#14556645	1	1	yes	characteristic
306	C₄	—	292	4	no	—
307	C₄	—	73	1	no	—
308	C₄	—	73	2	no	—
309	C₄	—	73	1	no	—
310	C₄	#17989452	1	2	yes	—
311	C₄	C₂×(C₂×#64408)⋊C₄	1	1	yes	characteristic
312	C₄	C₂×C₈×#64408	1	1	yes	characteristic
313	C₂	—	2	2	no	—
314	C₂	—	73	4	no	—
315	C₂	—	146	1	no	—
316	C₂	—	146	1	no	—
317	C₂	C₂×#17989452	1	1	yes	characteristic
318	C₂	#22626146	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×#64408
commutator	C₄×C₇₃	C₂×C₄×C₈
Frattini	C₂×C₈	C₂²×#64408
Fitting	C₂×M₃₂×C₇₃	C₄
socle	C₂²×C₇₃	C₂×#97
prime radical	C₂³×D₇₃	M₁₆
cosocle	C₂×C₈×D₇₃	C₂³
hypercenter	C₂×M₃₂	#64408
hypocenter	C₇₃	C₂×#1375
LCS	G ▷ C₄×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
3. #22624971	C₂	semidirect
10. C₂×#14556665	C₄	semidirect
11. C₂×#14556664	C₄	semidirect
17. C₂×#14556531	C₄	semidirect
21. C₂×C₁₆×D₇₃	C₄	semidirect
22. C₂×M₃₂×C₇₃	C₄	semidirect
37. C₂×#11924031	C₂×C₄	semidirect
54. C₂×#11924003	C₂×C₄	semidirect
58. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
159. D₇₃	#1375	semidirect
164. C₇₃	C₂×#1375	semidirect
176. C₂×M₃₂	#64408	semidirect
215. C₂×C₁₆	C₂×#64408	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17989643	2
C₂²×#14556645	2
#22624971	2
C₂×M₃₂×D₇₃	2
C₂×#1375	73