Subgroups — #32985331

Order 18688.

514 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	C₈×C₄⋊Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#22608679	C₂	1	1	yes	maximal, characteristic, complemented
3	#22608426	C₂	1	2	yes	maximal, complemented
4	#22608330	C₂	1	2	yes	maximal, complemented
5	C₇₃×#3147	C₂	1	1	yes	maximal, characteristic
6	#17988640	C₂²	1	2	yes	—
7	C₂×C₄⋊#10596831	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₇₃	—	2	1	no	—
9	C₄×C₄⋊Dic₇₃	C₂²	1	1	yes	characteristic
10	#17988341	—	2	1	no	—
11	#17988340	—	2	1	no	—
12	#17988338	—	2	1	no	—
13	#17988321	—	2	4	no	—
14	D₄⋊#10596831	—	2	4	no	—
15	C₂×#135×C₇₃	—	2	1	no	—
16	C₂×#135×C₇₃	—	2	1	no	—
17	#532×C₇₃	C₂²	1	2	yes	—
18	#495×C₇₃	—	2	1	no	—
19	#495×C₇₃	—	2	1	no	—
20	#494×C₇₃	C₄	1	1	yes	characteristic, complemented
21	#493×C₇₃	C₄	1	1	yes	characteristic, complemented
22	#492×C₇₃	C₄	1	2	yes	complemented
23	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
24	C₂²×#10596831	—	2	1	no	—
25	C₂×C₄⋊Dic₇₃	—	2	1	no	—
26	C₂×C₄×Dic₇₃	—	2	1	no	—
27	C₄⋊#10596831	—	2	2	no	—
28	C₄⋊#10596831	—	2	1	no	—
29	C₄⋊#10596831	D₄	1	2	yes	—
30	C₂²⋊#10596831	—	2	2	no	—
31	C₄×#10596831	—	2	2	no	—
32	C₈×Dic₇₃	—	4	2	no	—
33	#14556498	D₄	1	2	yes	—
34	C₂×#41×C₇₃	—	2	2	no	—
35	#135×C₇₃	—	4	2	no	—
36	#135×C₇₃	—	4	2	no	—
37	#135×C₇₃	—	4	2	no	—
38	#135×C₇₃	—	4	2	no	—
39	C₂×Q₁₆×C₇₃	—	2	1	no	—
40	C₂×Q₁₆×C₇₃	—	2	1	no	—
41	C₂×SD₁₆×C₇₃	—	2	2	no	—
42	C₂×SD₁₆×C₇₃	—	2	2	no	—
43	C₂×D₈×C₇₃	—	2	1	no	—
44	C₂×D₈×C₇₃	—	2	1	no	—
45	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₇₃	D₄	1	1	yes	characteristic
47	C₄⋊Q₈×C₇₃	C₂×C₄	1	2	yes	—
48	#127×C₇₃	C₂×C₄	1	2	yes	—
49	#124×C₇₃	—	2	2	no	—
50	#121×C₇₃	—	2	2	no	—
51	#121×C₇₃	—	2	2	no	—
52	C₄×D₄×C₇₃	—	2	2	no	—
53	C₄×D₄×C₇₃	—	2	2	no	—
54	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
58	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
59	C₂²×Dic₇₃	—	2	2	no	—
60	C₂×#10596831	—	2	2	no	—
61	C₂×#10596831	—	4	1	no	—
62	C₂×#10596831	—	4	1	no	—
63	C₂×#10596831	—	4	2	no	—
64	C₄⋊Dic₇₃	—	2	4	no	—
65	C₄×Dic₇₃	—	4	1	no	—
66	C₄×Dic₇₃	—	4	1	no	—
67	#41×C₇₃	—	4	2	no	—
68	#41×C₇₃	—	4	2	no	—
69	#41×C₇₃	—	4	2	no	—
70	#41×C₇₃	—	4	2	no	—
71	C₂×Q₈×C₇₃	—	2	2	no	—
72	C₂×D₄×C₇₃	—	2	2	no	—
73	C₂×D₄×C₇₃	—	2	2	no	—
74	C₂×D₄×C₇₃	—	2	2	no	—
75	C₂×D₄×C₇₃	—	4	2	no	—
76	C₂²×C₄×C₇₃	—	4	2	no	—
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
80	Q₁₆×C₇₃	—	4	2	no	—
81	Q₁₆×C₇₃	—	4	2	no	—
82	SD₁₆×C₇₃	—	4	4	no	—
83	SD₁₆×C₇₃	—	4	4	no	—
84	D₈×C₇₃	—	4	2	no	—
85	D₈×C₇₃	—	4	2	no	—
86	C₂×C₈×C₇₃	—	2	1	no	—
87	C₂×C₈×C₇₃	—	2	1	no	—
88	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₃	D₈	1	2	yes	—
93	C₂×C₈×C₇₃	SD₁₆	1	2	yes	—
94	C₄⋊C₄×C₇₃	—	2	2	no	—
95	C₄⋊C₄×C₇₃	—	2	2	no	—
96	C₂²⋊C₄×C₇₃	—	4	2	no	—
97	C₂²⋊C₄×C₇₃	—	4	2	no	—
98	C₄²×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	C₂³×C₇₃	—	4	2	no	—
103	C₂³×C₇₃	#121	1	1	yes	characteristic
104	Q₈×C₇₃	—	4	2	no	—
105	Q₈×C₇₃	—	4	2	no	—
106	D₄×C₇₃	—	4	2	no	—
107	D₄×C₇₃	—	4	2	no	—
108	D₄×C₇₃	—	4	4	no	—
109	D₄×C₇₃	—	8	2	no	—
110	D₄×C₇₃	—	8	2	no	—
111	D₄×C₇₃	—	8	2	no	—
112	C₂×C₄×C₇₃	—	2	2	no	—
113	C₂×C₄×C₇₃	—	4	1	no	—
114	C₂×C₄×C₇₃	—	4	2	no	—
115	C₂×C₄×C₇₃	—	8	2	no	—
116	C₂×C₄×C₇₃	—	8	2	no	—
117	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
125	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂×Dic₇₃	—	2	4	no	—
128	C₂×Dic₇₃	—	4	2	no	—
129	C₂×Dic₇₃	—	4	2	no	—
130	C₈×C₇₃	—	2	1	no	—
131	C₈×C₇₃	—	2	1	no	—
132	C₈×C₇₃	#102	1	2	yes	—
133	C₈×C₇₃	#102	1	2	yes	—
134	#10596831	—	4	2	no	—
135	#10596831	—	8	2	no	—
136	C₂²×C₇₃	—	2	1	no	—
137	C₂²×C₇₃	—	2	1	no	—
138	C₂²×C₇₃	—	8	2	no	—
139	C₂²×C₇₃	—	8	2	no	—
140	C₂²×C₇₃	—	8	2	no	—
141	C₂²×C₇₃	#390	1	1	yes	characteristic
142	C₂²×C₇₃	#459	1	1	yes	characteristic
143	C₂²×C₇₃	#460	1	1	yes	characteristic
144	C₄×C₇₃	—	2	1	no	—
145	C₄×C₇₃	—	2	1	no	—
146	C₄×C₇₃	—	2	1	no	—
147	C₄×C₇₃	—	8	2	no	—
148	C₄×C₇₃	C₂×#102	1	1	yes	characteristic
149	C₄×C₇₃	#418	1	1	yes	characteristic
150	C₄×C₇₃	#443	1	2	yes	—
151	C₄×C₇₃	#459	1	1	yes	characteristic
152	C₄×C₇₃	#460	1	1	yes	characteristic
153	Dic₇₃	—	4	4	no	—
154	#9750	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	1	no	—
156	C₂×C₇₃	—	2	1	no	—
157	C₂×C₇₃	—	16	2	no	—
158	C₂×C₇₃	#1974	1	1	yes	characteristic
159	C₂×C₇₃	#1976	1	1	yes	characteristic
160	C₂×C₇₃	#1982	1	1	yes	characteristic
161	#3147	D₇₃	1	1	yes	characteristic
162	C₈×C₄⋊C₄	—	73	1	no	—
163	#1564	—	73	1	no	—
164	#1502	—	73	2	no	—
165	#1480	—	73	2	no	—
166	C₇₃	#9750	1	1	yes	characteristic, complemented, Sylow
167	C₂×#135	—	2	1	no	—
168	C₂×#135	—	2	1	no	—
169	#532	C₂×D₇₃	1	2	yes	—
170	#495	—	2	1	no	—
171	#495	—	2	1	no	—
172	#494	Dic₇₃	1	1	yes	characteristic, complemented
173	#493	Dic₇₃	1	1	yes	characteristic, complemented
174	#492	Dic₇₃	1	2	yes	complemented
175	#431	—	73	2	no	—
176	C₂×C₄⋊C₈	—	73	1	no	—
177	C₂×C₄×C₈	—	146	1	no	—
178	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
179	C₄×C₄⋊C₄	—	73	1	no	—
180	#336	—	146	1	no	—
181	#335	—	146	1	no	—
182	#334	—	146	1	no	—
183	#327	—	146	4	no	—
184	D₄⋊C₈	—	146	4	no	—
185	C₂×#41	—	2	2	no	—
186	#135	—	4	2	no	—
187	#135	—	4	2	no	—
188	#135	—	4	2	no	—
189	#135	—	4	2	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×SD₁₆	—	2	2	no	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×D₈	—	2	1	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂²×C₈	—	146	1	no	—
197	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
198	C₂²×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
199	C₄⋊Q₈	C₂×Dic₇₃	1	2	yes	—
200	#127	C₂×Dic₇₃	1	2	yes	—
201	#124	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	2	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	146	1	no	—
207	C₂×C₄²	—	146	1	no	—
208	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
209	C₄⋊C₈	—	73	2	no	—
210	C₄⋊C₈	—	146	2	no	—
211	C₄⋊C₈	—	146	1	no	—
212	C₂²⋊C₈	—	146	2	no	—
213	C₄×C₈	—	292	2	no	—
214	C₄×C₈	—	146	2	no	—
215	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
216	C₄×C₈	C₂×Dic₇₃	1	1	yes	characteristic
217	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
218	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
219	#95	—	73	2	no	—
220	#41	—	4	2	no	—
221	#41	—	4	2	no	—
222	#41	—	4	2	no	—
223	#41	—	4	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	146	2	no	—
231	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
233	C₂²×C₄	#11924027	1	1	yes	characteristic
234	Q₁₆	—	4	2	no	—
235	Q₁₆	—	4	2	no	—
236	SD₁₆	—	4	4	no	—
237	SD₁₆	—	4	4	no	—
238	D₈	—	4	2	no	—
239	D₈	—	4	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	292	1	no	—
243	C₂×C₈	—	292	1	no	—
244	C₂×C₈	—	292	2	no	—
245	C₂×C₈	—	146	2	no	—
246	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
247	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
248	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
249	C₂×C₈	C₂²⋊Dic₇₃	1	1	yes	characteristic
250	C₂×C₈	#11924013	1	2	yes	—
251	C₂×C₈	#11924016	1	2	yes	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	146	4	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₄²	—	292	1	no	—
258	C₄²	—	292	1	no	—
259	C₄²	C₂²×Dic₇₃	1	1	yes	characteristic
260	C₄²	D₄×D₇₃	1	1	yes	characteristic
261	C₄²	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
262	C₄²	#11924027	1	1	yes	characteristic
263	C₂³	—	4	2	no	—
264	C₂³	#14556562	1	1	yes	characteristic
265	Q₈	—	4	2	no	—
266	Q₈	—	4	2	no	—
267	D₄	—	4	4	no	—
268	D₄	—	4	2	no	—
269	D₄	—	4	2	no	—
270	D₄	—	8	2	no	—
271	D₄	—	8	2	no	—
272	D₄	—	8	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	4	1	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	8	2	no	—
277	C₂×C₄	—	8	2	no	—
278	C₂×C₄	—	292	2	no	—
279	C₂×C₄	—	292	2	no	—
280	C₂×C₄	—	146	4	no	—
281	C₂×C₄	C₂×C₂²⋊Dic₇₃	1	1	yes	characteristic
282	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
283	C₂×C₄	D₄×Dic₇₃	1	1	yes	characteristic
284	C₂×C₄	#14556562	1	1	yes	characteristic
285	C₂×C₄	#14556578	1	1	yes	characteristic
286	C₂×C₄	#14556587	1	1	yes	characteristic
287	C₂×C₄	C₂×#11924013	1	1	yes	characteristic
288	C₂×C₄	C₂×#11924016	1	1	yes	characteristic
289	C₂×C₄	#14556617	1	1	yes	characteristic
290	C₂×C₄	#14556617	1	1	yes	characteristic
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₈	—	292	2	no	—
294	C₈	—	584	2	no	—
295	C₈	#14556512	1	2	yes	—
296	C₈	#14556512	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	8	2	no	—
300	C₂²	—	8	2	no	—
301	C₂²	—	8	2	no	—
302	C₂²	#17988490	1	1	yes	characteristic
303	C₂²	#17988779	1	1	yes	characteristic
304	C₂²	#17988784	1	1	yes	characteristic
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	2	1	no	—
308	C₄	—	8	2	no	—
309	C₄	—	292	4	no	—
310	C₄	C₂×#14556512	1	1	yes	characteristic
311	C₄	#17988592	1	1	yes	characteristic
312	C₄	#17988704	1	2	yes	—
313	C₄	#17988779	1	1	yes	characteristic
314	C₄	#17988784	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	16	2	no	—
318	C₂	#22610820	1	1	yes	characteristic
319	C₂	#22610830	1	1	yes	characteristic
320	C₂	#22610859	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14556562
commutator	C₄²×C₇₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₇₃
Fitting, prime radical	C₇₃×#3147	C₂
socle	C₂²×C₇₃	#390
cosocle	C₂×C₄²×C₇₃	C₂³
hypercenter	#3147	D₇₃
hypocenter	C₇₃	#9750
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₈×C₄⋊Dic₇₃	C₂	semidirect
2. #22608679	C₂	semidirect
3. #22608426	C₂	semidirect
4. #22608330	C₂	semidirect
20. #494×C₇₃	C₄	semidirect
21. #493×C₇₃	C₄	semidirect
22. #492×C₇₃	C₄	semidirect
166. C₇₃	#9750	semidirect
172. #494	Dic₇₃	semidirect
173. #493	Dic₇₃	semidirect
174. #492	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₈×C₄⋊Dic₇₃	2
#22608679	2
#22608426	2
#22608330	2
C₇₃×#3147	2
#9750	73