Subgroups — #32991896

Order 18688.

766 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556526	C₂	1	2	yes	maximal
2	C₂²×#14556502	C₂	1	1	yes	maximal, characteristic
3	C₂×#17988351	C₂	1	2	yes	maximal
4	#22609625	C₂	1	1	yes	maximal, characteristic
5	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂³×#10596831	C₂²	1	1	yes	characteristic
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂×#14556526	—	2	4	no	—
10	C₂×#14556526	—	2	4	no	—
11	C₂×#14556526	C₄	1	8	yes	complemented
12	C₂×#14556502	—	2	4	no	—
13	C₂×#14556502	C₄	1	4	yes	—
14	#17988351	—	2	4	no	—
15	#17988342	—	2	2	no	—
16	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
17	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
18	#336×C₇₃	—	2	2	no	—
19	C₂×#11924012	—	2	2	no	—
20	C₂×#11924012	—	2	2	no	—
21	C₂×#11924012	—	2	2	no	—
22	C₂×#11924012	—	2	2	no	—
23	C₂×#11924012	—	2	2	no	—
24	C₂×#11924012	C₂×C₄	1	4	yes	—
25	C₂²×#10596831	—	2	2	no	—
26	C₂²×#10596831	—	2	2	no	—
27	C₂²×#10596831	—	2	2	no	—
28	C₂²×#10596831	C₂×C₄	1	2	yes	—
29	C₂²×#10596831	C₂×C₄	1	4	yes	—
30	C₂²×#10596831	C₂×C₄	1	2	yes	—
31	C₂²×#10596831	D₄	1	2	yes	—
32	C₂²×#10596831	D₄	1	2	yes	—
33	C₂²×#10596831	D₄	1	4	yes	—
34	C₂²×#10596831	Q₈	1	2	yes	—
35	C₂²×#10596831	Q₈	1	2	yes	—
36	#14556526	—	2	16	no	—
37	#14556502	—	2	8	no	—
38	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
39	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
40	C₂×M₁₆×C₇₃	—	2	2	no	—
41	C₂×M₁₆×C₇₃	—	2	2	no	—
42	C₂×M₁₆×C₇₃	—	2	2	no	—
43	C₂×M₁₆×C₇₃	—	2	2	no	—
44	C₂×M₁₆×C₇₃	—	2	2	no	—
45	C₂×M₁₆×C₇₃	C₂×C₄	1	4	yes	—
46	C₂²×C₈×C₇₃	—	2	2	no	—
47	C₂×C₄²×C₇₃	—	2	2	no	—
48	C₂×C₄²×C₇₃	—	2	2	no	—
49	C₂×C₄²×C₇₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₇₃	D₄	1	2	yes	—
51	#11924012	—	2	4	no	—
52	#11924012	—	2	4	no	—
53	#11924012	—	4	2	no	—
54	#11924012	—	4	2	no	—
55	C₂×#10596831	—	2	2	no	—
56	C₂×#10596831	—	2	2	no	—
57	C₂×#10596831	—	2	4	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	—	4	2	no	—
61	C₂×#10596831	—	4	2	no	—
62	C₂×#10596831	C₄²	1	4	yes	—
63	C₂×#10596831	C₂²⋊C₄	1	4	yes	—
64	C₂×#10596831	C₂²⋊C₄	1	8	yes	—
65	C₂×#10596831	C₄⋊C₄	1	4	yes	—
66	C₂×#10596831	C₄⋊C₄	1	4	yes	—
67	C₂×#10596831	C₄⋊C₄	1	8	yes	—
68	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₇₃	—	2	4	no	—
70	C₂²×C₄×C₇₃	—	4	4	no	—
71	C₂²×C₄×C₇₃	C₄⋊C₄	1	4	yes	—
72	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₇₃	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₇₃	#41	1	2	yes	—
81	C₂²×C₄×C₇₃	#41	1	2	yes	—
82	M₁₆×C₇₃	—	2	4	no	—
83	M₁₆×C₇₃	—	2	4	no	—
84	M₁₆×C₇₃	—	4	2	no	—
85	M₁₆×C₇₃	—	4	2	no	—
86	C₂×C₈×C₇₃	—	2	4	no	—
87	C₂×C₈×C₇₃	—	4	2	no	—
88	C₂×C₈×C₇₃	—	4	2	no	—
89	C₄²×C₇₃	—	2	2	no	—
90	C₄²×C₇₃	—	2	2	no	—
91	C₄²×C₇₃	—	2	4	no	—
92	C₂³×C₇₃	—	2	4	no	—
93	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂³×C₇₃	C₄×Q₈	1	2	yes	—
95	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
96	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
97	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₇₃	—	2	4	no	—
99	C₂×C₄×C₇₃	—	2	8	no	—
100	C₂×C₄×C₇₃	—	4	8	no	—
101	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
106	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₇₃	#121	1	2	yes	—
113	C₂×C₄×C₇₃	#121	1	2	yes	—
114	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
115	C₂×C₄×C₇₃	#124	1	2	yes	—
116	C₂×C₄×C₇₃	#125	1	2	yes	—
117	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
119	C₈×C₇₃	—	4	4	no	—
120	#10596831	—	2	4	no	—
121	#10596831	—	4	4	no	—
122	#10596831	#95	1	8	yes	—
123	C₂²×C₇₃	—	2	4	no	—
124	C₂²×C₇₃	—	2	2	no	—
125	C₂²×C₇₃	—	2	2	no	—
126	C₂²×C₇₃	—	2	4	no	—
127	C₂²×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₂²×C₇₃	#380	1	1	yes	characteristic
129	C₂²×C₇₃	#384	1	1	yes	characteristic
130	C₂²×C₇₃	#389	1	2	yes	—
131	C₂²×C₇₃	#391	1	2	yes	—
132	C₂²×C₇₃	#430	1	4	yes	—
133	C₄×C₇₃	—	4	4	no	—
134	C₄×C₇₃	C₂×#95	1	1	yes	characteristic
135	C₄×C₇₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₄×C₇₃	#380	1	1	yes	characteristic
137	C₄×C₇₃	#382	1	1	yes	characteristic
138	C₄×C₇₃	#389	1	2	yes	—
139	C₄×C₇₃	#390	1	2	yes	—
140	#10865	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	2	no	—
142	C₂×C₇₃	—	2	2	no	—
143	C₂×C₇₃	C₄⋊#95	1	1	yes	characteristic
144	C₂×C₇₃	#1778	1	2	yes	—
145	C₂×C₇₃	#1811	1	2	yes	—
146	C₂×C₇₃	#1849	1	2	yes	—
147	C₂²×#108	—	73	2	no	—
148	C₂²×#97	—	73	1	no	—
149	C₂×#341	—	73	2	no	—
150	#1728	—	73	1	no	—
151	#1728	D₇₃	1	1	yes	characteristic
152	C₇₃	#10865	1	1	yes	characteristic, complemented, Sylow
153	C₂²×M₁₆	—	73	2	no	—
154	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
155	C₂³×C₈	—	73	1	no	—
156	C₂³×C₈	—	73	1	no	—
157	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
158	C₂×#108	—	73	8	no	—
159	C₂×#108	—	146	4	no	—
160	C₂×#108	—	146	4	no	—
161	C₂×#97	—	73	4	no	—
162	C₂×#97	—	146	4	no	—
163	#341	—	146	4	no	—
164	#336	—	2	2	no	—
165	#336	—	146	2	no	—
166	C₂³×C₄	Dic₁₄₆	1	2	yes	—
167	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	2	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	73	4	no	—
174	C₂×M₁₆	—	146	2	no	—
175	C₂×M₁₆	—	146	2	no	—
176	C₂×M₁₆	—	146	2	no	—
177	C₂×M₁₆	—	146	2	no	—
178	C₂×M₁₆	—	146	2	no	—
179	C₂×M₁₆	C₄×D₇₃	1	4	yes	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	73	2	no	—
182	C₂²×C₈	—	73	4	no	—
183	C₂²×C₈	—	73	4	no	—
184	C₂²×C₈	—	73	2	no	—
185	C₂²×C₈	—	73	2	no	—
186	C₂²×C₈	—	73	2	no	—
187	C₂²×C₈	—	73	2	no	—
188	C₂²×C₈	—	73	2	no	—
189	C₂²×C₈	—	146	2	no	—
190	C₂²×C₈	—	146	2	no	—
191	C₂²×C₈	—	146	2	no	—
192	C₂×C₄²	—	2	2	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	C₄×D₇₃	1	2	yes	—
195	C₂×C₄²	C₂²⋊D₇₃	1	2	yes	—
196	#108	—	146	16	no	—
197	#97	—	146	8	no	—
198	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	4	4	no	—
201	C₂²×C₄	#11924007	1	4	yes	—
202	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₄×D₇₃	1	2	yes	—
204	C₂²×C₄	D₄×D₇₃	1	2	yes	—
205	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
208	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
209	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
210	C₂²×C₄	#11924027	1	2	yes	—
211	C₂²×C₄	#11924028	1	2	yes	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	4	2	no	—
216	M₁₆	—	292	2	no	—
217	M₁₆	—	292	2	no	—
218	M₁₆	—	146	4	no	—
219	M₁₆	—	146	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	4	2	no	—
222	C₂×C₈	—	4	2	no	—
223	C₂×C₈	—	292	2	no	—
224	C₂×C₈	—	292	2	no	—
225	C₂×C₈	—	73	4	no	—
226	C₂×C₈	—	73	4	no	—
227	C₂×C₈	—	73	8	no	—
228	C₂×C₈	—	73	4	no	—
229	C₂×C₈	—	73	8	no	—
230	C₂×C₈	—	73	4	no	—
231	C₂×C₈	—	146	4	no	—
232	C₂×C₈	—	146	2	no	—
233	C₂×C₈	—	146	2	no	—
234	C₂×C₈	—	146	4	no	—
235	C₂×C₈	—	146	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	4	no	—
239	C₂³	—	2	4	no	—
240	C₂³	C₂×#11924007	1	1	yes	characteristic
241	C₂³	C₄×Dic₁₄₆	1	2	yes	—
242	C₂³	#14556568	1	2	yes	—
243	C₂³	#14556590	1	1	yes	characteristic
244	C₂³	#14556590	1	1	yes	characteristic
245	C₂×C₄	—	2	8	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	8	no	—
248	C₂×C₄	C₄²×D₇₃	1	1	yes	characteristic
249	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
250	C₂×C₄	C₂²⋊C₄×D₇₃	1	2	yes	—
251	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
252	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
253	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
254	C₂×C₄	#14556547	1	1	yes	characteristic
255	C₂×C₄	#14556548	1	1	yes	characteristic
256	C₂×C₄	#14556551	1	2	yes	—
257	C₂×C₄	C₄×C₂²⋊D₇₃	1	2	yes	—
258	C₂×C₄	#14556556	1	2	yes	—
259	C₂×C₄	#14556563	1	2	yes	—
260	C₂×C₄	#14556564	1	2	yes	—
261	C₂×C₄	#14556569	1	2	yes	—
262	C₂×C₄	#14556577	1	2	yes	—
263	C₂×C₄	#14556581	1	2	yes	—
264	C₂×C₄	#14556587	1	1	yes	characteristic
265	C₂×C₄	#14556589	1	1	yes	characteristic
266	C₈	—	4	4	no	—
267	C₈	—	292	4	no	—
268	C₈	—	73	8	no	—
269	C₈	—	146	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	C₄×#11924007	1	1	yes	characteristic
275	C₂²	#17988445	1	1	yes	characteristic
276	C₂²	#17988456	1	1	yes	characteristic
277	C₂²	#17988484	1	2	yes	—
278	C₂²	#17988491	1	2	yes	—
279	C₂²	#17988635	1	4	yes	—
280	C₄	—	4	4	no	—
281	C₄	#95×D₇₃	1	1	yes	characteristic
282	C₄	#17988438	1	1	yes	characteristic
283	C₄	#17988446	1	1	yes	characteristic
284	C₄	#17988451	1	1	yes	characteristic
285	C₄	#17988482	1	2	yes	—
286	C₄	#17988487	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#22608159	1	1	yes	characteristic
290	C₂	#22609838	1	2	yes	—
291	C₂	#22609971	1	2	yes	—
292	C₂	#22610105	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₃
commutator	C₂×C₄×C₇₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#1728	C₂
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#1728	D₇₃
hypocenter	C₇₃	#10865
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
11. C₂×#14556526	C₄	semidirect
152. C₇₃	#10865	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14556526	2
C₂²×#14556502	2
C₂×#17988351	2
#22609625	2
C₇₃×#1728	2
#10865	73