Subgroups — #32976553

Order 18688.

518 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
2	C₄×#14556507	C₂	1	1	yes	maximal, characteristic
3	#22608251	C₂	1	1	yes	maximal, characteristic
4	#22608220	C₂	1	1	yes	maximal, characteristic, complemented
5	#22607782	C₂	1	1	yes	maximal, characteristic, complemented
6	#22607780	C₂	1	1	yes	maximal, characteristic, complemented
7	C₇₃×#1451	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₇₃	C₄	1	2	yes	—
9	C₂×C₄×D₂₉₂	C₂²	1	1	yes	characteristic
10	C₈×D₂₉₂	—	2	1	no	—
11	C₈×D₂₉₂	—	2	1	no	—
12	C₈×D₂₉₂	—	2	1	no	—
13	C₈×D₂₉₂	—	2	1	no	—
14	#17988640	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
16	C₂×#14556507	C₄	1	2	yes	—
17	C₂×#14556503	C₄	1	2	yes	—
18	C₄×#11924012	C₂²	1	1	yes	characteristic
19	#17988355	C₄	1	2	yes	—
20	#17988310	C₄	1	4	yes	—
21	#17988309	—	2	1	no	—
22	#17988308	—	2	1	no	—
23	#431×C₇₃	C₂²	1	1	yes	characteristic
24	C₄×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
26	#322×C₇₃	—	2	1	no	—
27	#322×C₇₃	—	2	1	no	—
28	C₂²×D₂₉₂	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₇₃	C₂×C₄	1	2	yes	—
30	C₂×#11924012	C₂×C₄	1	2	yes	—
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂×C₈×D₇₃	—	2	2	no	—
33	C₂×C₈×D₇₃	—	2	2	no	—
34	C₂×C₈×D₇₃	—	4	4	no	—
35	C₄×D₂₉₂	—	2	1	no	—
36	C₄×D₂₉₂	—	2	1	no	—
37	C₄×D₂₉₂	—	2	1	no	—
38	C₄×D₂₉₂	—	2	1	no	—
39	C₂×C₄⋊Dic₇₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#11924005	C₂×C₄	1	2	yes	—
41	C₄⋊#10596831	—	2	1	no	—
42	C₄⋊#10596831	—	2	1	no	—
43	C₄⋊#10596831	—	2	1	no	—
44	#14556507	—	2	4	no	—
45	C₂²⋊#10596831	C₂×C₄	1	2	yes	—
46	#14556503	—	2	2	no	—
47	#14556503	—	2	2	no	—
48	#14556502	—	2	1	no	—
49	C₄×#10596831	—	2	1	no	—
50	C₄×#10596831	—	2	1	no	—
51	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
52	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₇₃	—	2	1	no	—
56	C₂²⋊C₈×C₇₃	C₂×C₄	1	2	yes	—
57	#97×C₇₃	—	2	1	no	—
58	C₄×C₈×C₇₃	—	2	1	no	—
59	C₄×C₈×C₇₃	—	2	1	no	—
60	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
61	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
62	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
64	C₂³×D₇₃	C₄²	1	2	yes	—
65	C₂×D₂₉₂	—	2	1	no	—
66	C₂×D₂₉₂	—	2	1	no	—
67	C₂×D₂₉₂	—	2	2	no	—
68	C₂×D₂₉₂	M₁₆	1	4	yes	—
69	C₂²×Dic₇₃	C₄²	1	2	yes	—
70	C₂×C₄×D₇₃	—	2	2	no	—
71	C₂×C₄×D₇₃	—	2	2	no	—
72	C₂×C₄×D₇₃	—	4	2	no	—
73	C₂×C₄×D₇₃	—	4	2	no	—
74	#11924012	—	2	2	no	—
75	#11924012	—	2	2	no	—
76	C₂×#10596831	—	2	2	no	—
77	C₂×#10596831	—	2	2	no	—
78	C₂×#10596831	—	2	2	no	—
79	C₂×#10596831	—	4	2	no	—
80	C₈×D₇₃	—	4	4	no	—
81	C₈×D₇₃	—	4	4	no	—
82	C₄⋊Dic₇₃	—	2	1	no	—
83	C₄⋊Dic₇₃	—	2	1	no	—
84	#11924005	—	2	2	no	—
85	#11924005	—	2	2	no	—
86	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
89	M₁₆×C₇₃	—	2	2	no	—
90	M₁₆×C₇₃	—	2	2	no	—
91	C₂×C₈×C₇₃	—	2	2	no	—
92	C₂×C₈×C₇₃	—	2	2	no	—
93	C₂×C₈×C₇₃	—	2	2	no	—
94	C₂×C₈×C₇₃	—	4	1	no	—
95	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
99	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
101	C₄²×C₇₃	#41	1	1	yes	characteristic
102	C₄²×C₇₃	#41	1	1	yes	characteristic
103	C₂³×C₇₃	C₂×C₄²	1	1	yes	characteristic
104	C₂²×D₇₃	—	2	2	no	—
105	C₂²×D₇₃	—	2	2	no	—
106	C₂²×D₇₃	—	4	2	no	—
107	C₂²×D₇₃	—	4	2	no	—
108	C₂²×D₇₃	#97	1	4	yes	—
109	C₂×C₄×C₇₃	—	2	1	no	—
110	C₂×C₄×C₇₃	—	2	1	no	—
111	C₂×C₄×C₇₃	—	4	1	no	—
112	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
119	C₂×C₄×C₇₃	#131	1	2	yes	—
120	C₂×Dic₇₃	—	2	2	no	—
121	C₂×Dic₇₃	—	4	2	no	—
122	D₂₉₂	—	2	4	no	—
123	D₂₉₂	—	2	4	no	—
124	C₄×D₇₃	—	4	4	no	—
125	C₄×D₇₃	—	4	2	no	—
126	C₄×D₇₃	—	4	2	no	—
127	C₈×C₇₃	—	2	1	no	—
128	C₈×C₇₃	—	2	1	no	—
129	C₈×C₇₃	—	2	1	no	—
130	C₈×C₇₃	—	4	2	no	—
131	C₈×C₇₃	—	4	2	no	—
132	C₈×C₇₃	#100	1	2	yes	complemented
133	#10596831	—	4	2	no	—
134	#10596831	—	4	2	no	—
135	#10596831	—	4	2	no	—
136	C₂²×C₇₃	—	2	1	no	—
137	C₂²×C₇₃	—	2	1	no	—
138	C₂²×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₇₃	C₂×#97	1	1	yes	characteristic
140	C₂²×C₇₃	#405	1	1	yes	characteristic
141	C₂×D₇₃	—	2	4	no	—
142	C₂×D₇₃	—	4	4	no	—
143	C₂×D₇₃	—	4	2	no	—
144	C₂×D₇₃	—	4	2	no	—
145	C₂×D₇₃	—	4	4	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	2	1	no	—
148	C₄×C₇₃	—	2	1	no	—
149	C₄×C₇₃	C₂×#100	1	1	yes	characteristic
150	C₄×C₇₃	#413	1	1	yes	characteristic
151	C₄×C₇₃	#435	1	2	yes	—
152	C₄×C₇₃	#435	1	2	yes	—
153	Dic₇₃	—	4	2	no	—
154	#8164	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	1	no	—
156	C₂×C₇₃	—	2	1	no	—
157	C₂×C₇₃	#1752	1	1	yes	characteristic
158	C₂×C₇₃	C₄×#100	1	1	yes	characteristic
159	C₂×C₇₃	#1757	1	1	yes	characteristic
160	D₇₃	—	4	4	no	—
161	C₂×C₈×D₄	—	73	1	no	—
162	C₄×#100	—	73	1	no	—
163	#1460	—	73	1	no	—
164	#1451	—	73	1	no	—
165	#1451	D₇₃	1	1	yes	characteristic, complemented
166	#1283	—	73	1	no	—
167	#1283	—	73	1	no	—
168	C₇₃	#8164	1	1	yes	characteristic, complemented, Sylow
169	C₂³×C₈	—	73	2	no	—
170	C₂×C₄×D₄	—	73	1	no	—
171	C₈×D₄	—	146	1	no	—
172	C₈×D₄	—	146	1	no	—
173	C₈×D₄	—	146	1	no	—
174	C₈×D₄	—	146	1	no	—
175	#431	—	73	1	no	—
176	#431	C₂×D₇₃	1	1	yes	characteristic
177	C₂×C₄⋊C₈	—	73	1	no	—
178	C₂×#100	—	73	2	no	—
179	C₂×C₂²⋊C₈	—	73	2	no	—
180	C₄×M₁₆	—	73	1	no	—
181	C₄×M₁₆	C₂×D₇₃	1	1	yes	characteristic
182	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
183	#343	—	73	2	no	—
184	#323	—	73	4	no	—
185	#322	—	2	1	no	—
186	#322	—	2	1	no	—
187	#322	—	146	1	no	—
188	#322	—	146	1	no	—
189	C₂²×D₄	—	73	1	no	—
190	C₂³×C₄	—	73	2	no	—
191	C₂×M₁₆	—	73	2	no	—
192	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
193	C₂²×C₈	—	292	4	no	—
194	C₂²×C₈	—	73	2	no	—
195	C₂²×C₈	—	146	2	no	—
196	C₂²×C₈	—	146	2	no	—
197	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
199	C₄×D₄	—	146	1	no	—
200	C₄×D₄	—	146	1	no	—
201	C₄×D₄	—	146	1	no	—
202	C₄×D₄	—	146	1	no	—
203	C₂×C₄⋊C₄	—	73	1	no	—
204	C₂×C₂²⋊C₄	—	73	2	no	—
205	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
206	C₄⋊C₈	—	2	1	no	—
207	C₄⋊C₈	—	146	1	no	—
208	C₄⋊C₈	—	146	1	no	—
209	C₄⋊C₈	—	146	1	no	—
210	#100	—	146	4	no	—
211	C₂²⋊C₈	—	73	2	no	—
212	C₂²⋊C₈	—	146	2	no	—
213	C₂²⋊C₈	—	146	2	no	—
214	C₂²⋊C₈	C₄×D₇₃	1	2	yes	—
215	#97	—	2	1	no	—
216	#97	—	146	1	no	—
217	C₄×C₈	—	2	1	no	—
218	C₄×C₈	—	2	1	no	—
219	C₄×C₈	—	146	1	no	—
220	C₄×C₈	—	146	1	no	—
221	C₄×C₈	D₂₉₂	1	1	yes	characteristic
222	C₄×C₈	D₂₉₂	1	1	yes	characteristic
223	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
224	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
225	C₂⁴	—	73	2	no	—
226	C₂×D₄	—	73	4	no	—
227	C₂×D₄	—	146	1	no	—
228	C₂×D₄	—	146	1	no	—
229	C₂×D₄	—	146	2	no	—
230	C₂²×C₄	—	292	2	no	—
231	C₂²×C₄	—	292	2	no	—
232	C₂²×C₄	—	73	2	no	—
233	C₂²×C₄	—	146	2	no	—
234	C₂²×C₄	—	146	2	no	—
235	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
238	M₁₆	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	M₁₆	—	146	2	no	—
241	M₁₆	—	146	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	4	1	no	—
246	C₂×C₈	—	292	4	no	—
247	C₂×C₈	—	292	4	no	—
248	C₂×C₈	—	292	2	no	—
249	C₂×C₈	—	146	2	no	—
250	C₂×C₈	—	146	2	no	—
251	C₂×C₈	—	146	2	no	—
252	C₂×C₈	#11924005	1	1	yes	characteristic
253	C₂×C₈	#11924005	1	1	yes	characteristic
254	C₂×C₈	#11924005	1	1	yes	characteristic
255	C₂×C₈	#11924005	1	1	yes	characteristic
256	C₄⋊C₄	—	146	1	no	—
257	C₄⋊C₄	—	146	1	no	—
258	C₂²⋊C₄	—	146	2	no	—
259	C₂²⋊C₄	—	146	2	no	—
260	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
261	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
262	C₄²	#11924028	1	1	yes	characteristic
263	C₄²	#11924028	1	1	yes	characteristic
264	C₂³	—	292	2	no	—
265	C₂³	—	292	2	no	—
266	C₂³	—	73	4	no	—
267	C₂³	—	146	2	no	—
268	C₂³	—	146	2	no	—
269	C₂³	C₄²×D₇₃	1	1	yes	characteristic
270	D₄	—	146	4	no	—
271	D₄	—	146	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	292	4	no	—
276	C₂×C₄	—	292	2	no	—
277	C₂×C₄	—	292	2	no	—
278	C₂×C₄	—	292	2	no	—
279	C₂×C₄	—	146	2	no	—
280	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
281	C₂×C₄	#14556548	1	1	yes	characteristic
282	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
283	C₂×C₄	C₄×C₂²⋊D₇₃	1	1	yes	characteristic
284	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
285	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₇₃	1	2	yes	—
287	C₂×C₄	#14556599	1	2	yes	—
288	C₈	—	2	1	no	—
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	4	2	no	—
292	C₈	—	4	2	no	—
293	C₈	—	292	2	no	—
294	C₈	—	292	2	no	—
295	C₈	—	292	2	no	—
296	C₈	#14556507	1	2	yes	complemented
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	292	4	no	—
300	C₂²	—	292	4	no	—
301	C₂²	—	292	2	no	—
302	C₂²	—	292	2	no	—
303	C₂²	—	146	4	no	—
304	C₂²	C₄×#11924005	1	1	yes	characteristic
305	C₂²	#97×D₇₃	1	1	yes	characteristic
306	C₂²	#17988544	1	1	yes	characteristic
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	292	2	no	—
311	C₄	C₂×#14556507	1	1	yes	characteristic
312	C₄	#17988569	1	1	yes	characteristic
313	C₄	#17988656	1	2	yes	—
314	C₄	#17988660	1	2	yes	—
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	292	4	no	—
318	C₂	#22609708	1	1	yes	characteristic
319	C₂	C₄×#14556507	1	1	yes	characteristic
320	C₂	#22609732	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11924005
commutator	C₂³×C₇₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting	C₇₃×#1451	C₂
socle	C₂²×C₇₃	C₄×C₂²⋊C₄
prime radical	C₂²×D₂₉₂	C₂×C₄
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#1451	D₇₃
hypocenter	C₇₃	#8164
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #1451

Complements

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

N	H ≅ G/N	type
4. #22608220	C₂	semidirect
5. #22607782	C₂	semidirect
6. #22607780	C₂	semidirect
7. C₇₃×#1451	C₂	semidirect
132. C₈×C₇₃	#100	semidirect
165. #1451	D₇₃	semidirect
168. C₇₃	#8164	semidirect
296. C₈	#14556507	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₂₉₂	2
C₄×#14556507	2
#22608251	2
#22608220	2
#22607782	2
#22607780	2
C₇₃×#1451	2
#8164	73