Subgroups — #33056665

Order 18688.

560 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22620517	C₂	1	1	yes	maximal, characteristic
2	#22619025	C₂	1	2	yes	maximal
3	#22608841	C₂	1	2	yes	maximal, complemented
4	#22608829	C₂	1	2	yes	maximal, complemented
5	#22608751	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₂₉₂	C₂	1	2	yes	maximal, complemented
7	#22608694	C₂	1	1	yes	maximal, characteristic
8	#22608405	C₂	1	2	yes	maximal
9	C₇₃×#1450	C₂	1	1	yes	maximal, characteristic
10	#17989144	C₂²	1	2	yes	—
11	#17988992	C₄	1	1	yes	characteristic
12	#17988967	C₄	1	1	yes	characteristic
13	#17988938	C₄	1	2	yes	—
14	#17988828	C₄	1	2	yes	—
15	#17988828	C₄	1	2	yes	—
16	#17988717	C₂²	1	4	yes	—
17	#17988716	—	2	4	no	—
18	C₈×Dic₁₄₆	C₂²	1	4	yes	—
19	C₄×Dic₂₉₂	C₂²	1	2	yes	—
20	C₄×Dic₂₉₂	C₂²	1	2	yes	—
21	#17988645	—	2	4	no	—
22	#17988644	C₂²	1	2	yes	—
23	#17988340	C₂²	1	2	yes	—
24	#17988338	C₂²	1	2	yes	—
25	#17988319	C₂²	1	4	yes	—
26	#17988319	C₂²	1	4	yes	—
27	#431×C₇₃	C₂²	1	1	yes	characteristic
28	#431×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	#322×C₇₃	C₂²	1	1	yes	characteristic
31	#322×C₇₃	C₂²	1	1	yes	characteristic
32	C₈²×C₇₃	C₂²	1	1	yes	characteristic
33	C₈²×C₇₃	C₂²	1	1	yes	characteristic
34	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
35	C₂×Dic₂₉₂	C₂×C₄	1	2	yes	—
36	#14556591	C₂×C₄	1	2	yes	—
37	#14556591	C₂×C₄	1	2	yes	—
38	#14556583	—	2	2	no	—
39	#14556570	—	2	2	no	—
40	#14556570	C₂×C₄	1	4	yes	—
41	C₄×Dic₁₄₆	—	2	2	no	—
42	C₄×Dic₁₄₆	C₂³	1	4	yes	—
43	#14556550	—	2	2	no	—
44	#14556525	C₂×C₄	1	2	yes	—
45	#14556525	C₂×C₄	1	2	yes	—
46	C₄⋊#10596831	C₂³	1	4	yes	—
47	#14556520	—	2	4	no	—
48	#14556520	—	2	4	no	—
49	#14556516	C₂×C₄	1	4	yes	—
50	#14556516	C₂×C₄	1	4	yes	—
51	C₂²⋊#10596831	—	2	2	no	—
52	#14556502	—	2	2	no	—
53	#14556501	—	2	4	no	—
54	C₈×Dic₇₃	—	2	4	no	—
55	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₇₃	—	2	1	no	—
59	C₄⋊C₈×C₇₃	D₄	1	2	yes	—
60	C₂²⋊C₈×C₇₃	—	2	1	no	—
61	C₂²⋊C₈×C₇₃	D₄	1	2	yes	—
62	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
66	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
68	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₇₃	C₂³	1	1	yes	characteristic
70	C₂×Dic₁₄₆	—	2	2	no	—
71	C₂×Dic₁₄₆	C₂²×C₄	1	4	yes	—
72	Dic₂₉₂	—	2	4	no	—
73	Dic₂₉₂	—	2	4	no	—
74	C₂×#10596831	—	2	4	no	—
75	C₄⋊Dic₇₃	—	2	2	no	—
76	C₄⋊Dic₇₃	—	2	2	no	—
77	C₄⋊Dic₇₃	—	2	2	no	—
78	C₄⋊Dic₇₃	C₂²×C₄	1	4	yes	—
79	#11924007	—	2	4	no	—
80	#11924007	—	2	4	no	—
81	C₂²⋊Dic₇₃	—	2	4	no	—
82	C₄×Dic₇₃	—	2	4	no	—
83	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
86	C₂×C₈×C₇₃	—	2	2	no	—
87	C₂×C₈×C₇₃	—	2	2	no	—
88	C₂×C₈×C₇₃	—	2	1	no	—
89	C₂×C₈×C₇₃	—	2	1	no	—
90	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
96	C₄²×C₇₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₇₃	C₂×D₄	1	1	yes	characteristic
98	C₄²×C₇₃	#41	1	1	yes	characteristic
99	C₄²×C₇₃	C₂⁴	1	1	yes	characteristic
100	C₂³×C₇₃	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₇₃	—	2	1	no	—
102	C₂×C₄×C₇₃	—	2	1	no	—
103	C₂×C₄×C₇₃	—	2	2	no	—
104	C₂×C₄×C₇₃	—	4	1	no	—
105	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
110	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
111	C₂×Dic₇₃	—	2	4	no	—
112	C₂×Dic₇₃	—	2	4	no	—
113	Dic₁₄₆	—	2	4	no	—
114	Dic₁₄₆	—	2	4	no	—
115	Dic₁₄₆	—	8	2	no	—
116	C₈×C₇₃	—	2	1	no	—
117	C₈×C₇₃	—	2	1	no	—
118	C₈×C₇₃	—	2	2	no	—
119	C₈×C₇₃	—	2	2	no	—
120	C₈×C₇₃	—	4	1	no	—
121	C₈×C₇₃	—	4	1	no	—
122	C₈×C₇₃	#131	1	2	yes	—
123	C₈×C₇₃	#131	1	2	yes	—
124	#10596831	—	4	4	no	—
125	C₂²×C₇₃	—	2	1	no	—
126	C₂²×C₇₃	—	2	1	no	—
127	C₂²×C₇₃	—	2	1	no	—
128	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₇₃	—	2	1	no	—
130	C₄×C₇₃	—	2	1	no	—
131	C₄×C₇₃	—	2	1	no	—
132	C₄×C₇₃	—	2	1	no	—
133	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
134	C₄×C₇₃	C₂×#131	1	1	yes	characteristic
135	C₄×C₇₃	#578	1	2	yes	—
136	Dic₇₃	—	4	4	no	—
137	Dic₇₃	—	4	4	no	—
138	#17972	—	73	1	no	maximal, Sylow
139	C₂×C₇₃	—	4	1	no	—
140	C₂×C₇₃	#2931	1	1	yes	characteristic
141	C₂×C₇₃	#2950	1	1	yes	characteristic
142	C₂×C₇₃	#2957	1	1	yes	characteristic
143	#3170	—	73	1	no	—
144	#2985	—	73	2	no	—
145	#1593	—	73	2	no	—
146	#1591	—	73	2	no	—
147	#1579	—	73	2	no	—
148	C₈×Q₁₆	—	73	2	no	—
149	#1568	—	73	1	no	—
150	#1497	—	73	2	no	—
151	#1450	D₇₃	1	1	yes	characteristic
152	C₇₃	#17972	1	1	yes	characteristic, complemented, Sylow
153	#533	—	73	2	no	—
154	#500	—	73	1	no	—
155	#494	—	73	1	no	—
156	#487	—	73	2	no	—
157	#467	—	73	2	no	—
158	#467	—	73	2	no	—
159	#446	—	73	4	no	—
160	#446	—	146	4	no	—
161	C₈×Q₈	—	73	4	no	—
162	C₄×Q₁₆	—	73	2	no	—
163	C₄×Q₁₆	—	73	2	no	—
164	#433	—	146	4	no	—
165	#432	—	73	2	no	—
166	#431	C₂×D₇₃	1	1	yes	characteristic
167	#431	C₂×D₇₃	1	1	yes	characteristic
168	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
169	#335	—	73	2	no	—
170	#334	—	73	2	no	—
171	Q₈⋊C₈	—	73	4	no	—
172	Q₈⋊C₈	—	73	4	no	—
173	#322	C₂×D₇₃	1	1	yes	characteristic
174	#322	C₂×D₇₃	1	1	yes	characteristic
175	C₈²	C₂×D₇₃	1	1	yes	characteristic
176	C₈²	C₂×D₇₃	1	1	yes	characteristic
177	C₂×Q₁₆	—	73	2	no	—
178	C₂×Q₁₆	—	73	2	no	—
179	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
180	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
181	C₄⋊Q₈	—	73	2	no	—
182	C₄⋊Q₈	—	73	2	no	—
183	#125	—	146	2	no	—
184	C₂²⋊Q₈	—	73	4	no	—
185	C₂²⋊Q₈	—	146	2	no	—
186	C₄×Q₈	—	73	4	no	—
187	C₄×Q₈	—	146	2	no	—
188	#117	—	146	2	no	—
189	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
190	#107	—	73	2	no	—
191	#107	—	73	2	no	—
192	C₄⋊C₈	—	2	1	no	—
193	C₄⋊C₈	—	73	4	no	—
194	C₄⋊C₈	—	146	4	no	—
195	C₄⋊C₈	—	146	4	no	—
196	C₄⋊C₈	D₂₉₂	1	2	yes	—
197	#103	—	73	4	no	—
198	#103	—	73	4	no	—
199	C₂²⋊C₈	—	2	1	no	—
200	C₂²⋊C₈	—	146	2	no	—
201	C₂²⋊C₈	D₂₉₂	1	2	yes	—
202	#97	—	146	4	no	—
203	#97	—	146	2	no	—
204	C₄×C₈	—	146	4	no	—
205	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
206	C₄×C₈	C₄×D₇₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
208	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
209	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
210	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
211	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
212	C₄×C₈	C₂²×D₇₃	1	1	yes	characteristic
213	C₂×Q₈	—	73	4	no	—
214	C₂×Q₈	—	146	2	no	—
215	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
217	C₂²×C₄	#11924028	1	1	yes	characteristic
218	Q₁₆	—	146	4	no	—
219	Q₁₆	—	146	4	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	146	4	no	—
225	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
228	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
229	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
230	C₂×C₈	C₂×D₂₉₂	1	2	yes	—
231	C₄⋊C₄	—	73	4	no	—
232	C₄⋊C₄	—	146	4	no	—
233	C₄⋊C₄	—	146	2	no	—
234	C₄⋊C₄	—	146	2	no	—
235	C₄⋊C₄	—	146	2	no	—
236	C₄⋊C₄	—	146	4	no	—
237	C₂²⋊C₄	—	146	4	no	—
238	C₄²	—	146	4	no	—
239	C₄²	C₂×C₄×D₇₃	1	1	yes	characteristic
240	C₄²	C₂×D₂₉₂	1	1	yes	characteristic
241	C₄²	#11924028	1	1	yes	characteristic
242	C₄²	C₂³×D₇₃	1	1	yes	characteristic
243	C₂³	C₄×D₂₉₂	1	1	yes	characteristic
244	Q₈	—	584	2	no	—
245	Q₈	—	146	4	no	—
246	Q₈	—	146	4	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	146	4	no	—
252	C₂×C₄	—	146	4	no	—
253	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
254	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
255	C₂×C₄	C₄×D₂₉₂	1	1	yes	characteristic
256	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₂₉₂	1	1	yes	characteristic
258	C₂×C₄	C₂×#11924028	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	—	2	2	no	—
261	C₈	—	2	1	no	—
262	C₈	—	2	1	no	—
263	C₈	—	4	1	no	—
264	C₈	—	4	1	no	—
265	C₈	—	292	4	no	—
266	C₈	#14556597	1	2	yes	—
267	C₈	#14556597	1	2	yes	—
268	C₂²	—	2	1	no	—
269	C₂²	—	2	1	no	—
270	C₂²	—	2	1	no	—
271	C₂²	C₂×C₄×D₂₉₂	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	292	4	no	—
277	C₄	—	292	4	no	—
278	C₄	C₂×#14556597	1	1	yes	characteristic
279	C₄	C₂×#14556597	1	1	yes	characteristic
280	C₄	#17989404	1	2	yes	—
281	C₂	—	4	1	no	—
282	C₂	#22618644	1	1	yes	characteristic
283	C₂	#22618800	1	1	yes	characteristic
284	C₂	#22618849	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₉₂
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₄²	C₂³×D₇₃
Fitting	C₇₃×#1450	C₂
socle	C₂²×C₇₃	C₂×C₄×D₄
prime radical	C₂³×C₇₃	C₄×D₄
cosocle	C₄²×C₇₃	C₂⁴
hypercenter	#1450	D₇₃
hypocenter	C₇₃	#17972
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #22608841	C₂	semidirect
4. #22608829	C₂	semidirect
5. #22608751	C₂	semidirect
6. C₈×Dic₂₉₂	C₂	semidirect
152. C₇₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#22620517	2
#22619025	2
#22608841	2
#22608829	2
#22608751	2
C₈×Dic₂₉₂	2
#22608694	2
#22608405	2
C₇₃×#1450	2
#17972	73