Subgroups — #32994867

Order 18688.

614 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	C₂×#17988636	C₂	1	1	yes	maximal, characteristic
2	#22609657	C₂	1	2	yes	maximal
3	C₂×#17988345	C₂	1	1	yes	maximal, characteristic
4	#22609625	C₂	1	2	yes	maximal
5	C₇₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924012	C₂²	1	2	yes	—
7	C₂²×#11924012	C₂²	1	2	yes	—
8	#17988636	—	2	2	no	—
9	#17988636	C₄	1	4	yes	complemented
10	C₂×#14556527	—	2	2	no	—
11	#17988351	—	2	4	no	—
12	#17988345	—	2	2	no	—
13	#17988342	—	2	4	no	—
14	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
15	C₂²×M₁₆×C₇₃	C₂²	1	1	yes	characteristic
16	C₂²×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	#336×C₇₃	—	2	2	no	—
18	C₂×#11924012	—	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	—	2	2	no	—
25	C₂×#11924012	—	2	2	no	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	2	no	—
28	C₂×#11924012	—	2	2	no	—
29	C₂×#11924012	C₂×C₄	1	4	yes	—
30	C₂²×#10596831	—	2	2	no	—
31	C₂²×#10596831	D₄	1	4	yes	—
32	#14556527	—	2	8	no	—
33	C₂³×C₄×C₇₃	Q₈	1	2	yes	—
34	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
35	C₂×M₁₆×C₇₃	—	2	2	no	—
36	C₂×M₁₆×C₇₃	—	2	2	no	—
37	C₂×M₁₆×C₇₃	—	2	1	no	—
38	C₂×M₁₆×C₇₃	—	2	2	no	—
39	C₂×M₁₆×C₇₃	—	2	2	no	—
40	C₂×M₁₆×C₇₃	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₇₃	D₄	1	2	yes	—
42	C₂×M₁₆×C₇₃	Q₈	1	2	yes	—
43	C₂²×C₈×C₇₃	—	2	1	no	—
44	C₂²×C₈×C₇₃	D₄	1	2	yes	—
45	C₂×C₄²×C₇₃	—	2	1	no	—
46	C₂×C₄²×C₇₃	—	2	2	no	—
47	C₂×C₄²×C₇₃	—	2	2	no	—
48	C₂×C₄²×C₇₃	—	2	1	no	—
49	#11924012	—	2	4	no	—
50	#11924012	—	2	4	no	—
51	#11924012	—	4	2	no	—
52	#11924012	—	4	2	no	—
53	#11924012	—	4	2	no	—
54	#11924012	—	4	2	no	—
55	#11924012	—	4	2	no	—
56	#11924012	—	4	2	no	—
57	C₂×#10596831	—	2	4	no	—
58	C₂×#10596831	—	2	4	no	—
59	C₂×#10596831	—	2	4	no	—
60	C₂×#10596831	—	2	4	no	—
61	C₂×#10596831	—	4	2	no	—
62	C₂×#10596831	—	4	2	no	—
63	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
64	C₂²×C₄×C₇₃	—	2	4	no	—
65	C₂²×C₄×C₇₃	—	2	2	no	—
66	C₂²×C₄×C₇₃	—	4	4	no	—
67	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	#41	1	2	yes	—
73	C₂²×C₄×C₇₃	#41	1	2	yes	—
74	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
75	C₂²×C₄×C₇₃	#41	1	1	yes	characteristic
76	C₂²×C₄×C₇₃	#41	1	2	yes	—
77	M₁₆×C₇₃	—	2	4	no	—
78	M₁₆×C₇₃	—	2	4	no	—
79	M₁₆×C₇₃	—	2	4	no	—
80	M₁₆×C₇₃	—	4	2	no	—
81	C₂×C₈×C₇₃	—	2	2	no	—
82	C₂×C₈×C₇₃	—	2	2	no	—
83	C₂×C₈×C₇₃	—	2	2	no	—
84	C₂×C₈×C₇₃	—	4	1	no	—
85	C₂×C₈×C₇₃	—	4	1	no	—
86	C₂×C₈×C₇₃	C₂²⋊C₄	1	4	yes	—
87	C₄²×C₇₃	—	4	1	no	—
88	C₄²×C₇₃	—	4	2	no	—
89	C₄²×C₇₃	—	4	1	no	—
90	C₂³×C₇₃	—	4	2	no	—
91	C₂³×C₇₃	C₄×Q₈	1	1	yes	characteristic
92	C₂³×C₇₃	C₂²⋊Q₈	1	2	yes	—
93	C₂³×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
94	C₂³×C₇₃	#125	1	2	yes	—
95	C₂³×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
96	C₂×C₄×C₇₃	—	2	8	no	—
97	C₂×C₄×C₇₃	—	2	4	no	—
98	C₂×C₄×C₇₃	—	4	4	no	—
99	C₂×C₄×C₇₃	—	4	2	no	—
100	C₂×C₄×C₇₃	—	4	2	no	—
101	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	#117	1	2	yes	—
103	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₄×Q₈	1	1	yes	characteristic
105	C₂×C₄×C₇₃	#121	1	2	yes	—
106	C₂×C₄×C₇₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
109	C₂×C₄×C₇₃	#123	1	1	yes	characteristic
110	C₂×C₄×C₇₃	#123	1	2	yes	—
111	C₂×C₄×C₇₃	#123	1	2	yes	—
112	C₂×C₄×C₇₃	#123	1	2	yes	—
113	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₇₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₇₃	#125	1	2	yes	—
116	C₂×C₄×C₇₃	#126	1	2	yes	—
117	C₂×C₄×C₇₃	#126	1	2	yes	—
118	C₂×C₄×C₇₃	#127	1	1	yes	characteristic
119	C₈×C₇₃	—	2	4	no	—
120	C₈×C₇₃	—	4	2	no	—
121	#10596831	—	4	4	no	—
122	#10596831	—	4	4	no	—
123	C₂²×C₇₃	—	2	4	no	—
124	C₂²×C₇₃	—	2	2	no	—
125	C₂²×C₇₃	—	4	1	no	—
126	C₂²×C₇₃	—	4	2	no	—
127	C₂²×C₇₃	—	4	1	no	—
128	C₂²×C₇₃	#387	1	2	yes	—
129	C₂²×C₇₃	#391	1	1	yes	characteristic
130	C₂²×C₇₃	#398	1	2	yes	—
131	C₂²×C₇₃	#399	1	1	yes	characteristic
132	C₂²×C₇₃	#400	1	1	yes	characteristic
133	C₄×C₇₃	—	4	4	no	—
134	C₄×C₇₃	#381	1	1	yes	characteristic
135	C₄×C₇₃	#387	1	2	yes	—
136	C₄×C₇₃	#390	1	1	yes	characteristic
137	C₄×C₇₃	#397	1	2	yes	—
138	C₄×C₇₃	#399	1	1	yes	characteristic
139	C₄×C₇₃	#400	1	1	yes	characteristic
140	#11375	—	73	1	no	maximal, Sylow
141	C₂×C₇₃	—	2	2	no	—
142	C₂×C₇₃	—	2	1	no	—
143	C₂×C₇₃	—	2	2	no	—
144	C₂×C₇₃	—	2	1	no	—
145	C₂×C₇₃	#1445	1	1	yes	characteristic
146	C₂×C₇₃	#2091	1	2	yes	—
147	C₂×#430	—	73	1	no	—
148	#1738	—	73	2	no	—
149	C₂×#338	—	73	1	no	—
150	#1728	—	73	2	no	—
151	#1728	D₇₃	1	1	yes	characteristic
152	C₇₃	#11375	1	1	yes	characteristic, complemented, Sylow
153	C₂²×M₁₆	—	73	2	no	—
154	C₂²×M₁₆	—	73	2	no	—
155	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
156	C₂²×M₁₆	C₂×D₇₃	1	1	yes	characteristic
157	C₂²×C₄²	C₂×D₇₃	1	1	yes	characteristic
158	#430	—	73	4	no	—
159	#430	—	146	2	no	—
160	C₂×#108	—	146	2	no	—
161	#341	—	146	4	no	—
162	#338	—	146	2	no	—
163	#336	—	2	2	no	—
164	#336	—	146	4	no	—
165	C₂³×C₄	Dic₁₄₆	1	2	yes	—
166	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
167	C₂×M₁₆	—	2	2	no	—
168	C₂×M₁₆	—	2	2	no	—
169	C₂×M₁₆	—	2	1	no	—
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	73	4	no	—
173	C₂×M₁₆	—	146	2	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	—	146	2	no	—
180	C₂×M₁₆	—	146	2	no	—
181	C₂×M₁₆	—	146	2	no	—
182	C₂×M₁₆	—	146	2	no	—
183	C₂×M₁₆	—	146	2	no	—
184	C₂×M₁₆	Dic₁₄₆	1	2	yes	—
185	C₂×M₁₆	C₄×D₇₃	1	2	yes	—
186	C₂×M₁₆	D₂₉₂	1	2	yes	—
187	C₂²×C₈	—	2	1	no	—
188	C₂²×C₈	—	73	4	no	—
189	C₂²×C₈	—	146	2	no	—
190	C₂²×C₈	C₂²⋊D₇₃	1	2	yes	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	2	1	no	—
193	C₂×C₄²	—	2	2	no	—
194	C₂×C₄²	—	2	1	no	—
195	#108	—	146	8	no	—
196	C₂⁴	C₂×Dic₁₄₆	1	1	yes	characteristic
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
201	C₂²×C₄	D₄×D₇₃	1	2	yes	—
202	C₂²×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₂₉₂	1	1	yes	characteristic
204	C₂²×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
205	C₂²×C₄	#11924027	1	2	yes	—
206	C₂²×C₄	#11924027	1	2	yes	—
207	C₂²×C₄	#11924028	1	1	yes	characteristic
208	C₂²×C₄	#11924028	1	1	yes	characteristic
209	C₂²×C₄	#11924029	1	2	yes	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	292	2	no	—
215	M₁₆	—	292	2	no	—
216	M₁₆	—	292	2	no	—
217	M₁₆	—	292	2	no	—
218	M₁₆	—	292	2	no	—
219	M₁₆	—	292	2	no	—
220	M₁₆	—	146	4	no	—
221	M₁₆	—	146	4	no	—
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	4	1	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	292	2	no	—
228	C₂×C₈	—	292	2	no	—
229	C₂×C₈	—	146	4	no	—
230	C₂×C₈	—	146	4	no	—
231	C₂×C₈	—	146	4	no	—
232	C₂×C₈	—	146	4	no	—
233	C₂×C₈	#11924005	1	4	yes	—
234	C₄²	—	4	1	no	—
235	C₄²	—	4	2	no	—
236	C₄²	—	4	1	no	—
237	C₂³	—	4	2	no	—
238	C₂³	C₄×Dic₁₄₆	1	1	yes	characteristic
239	C₂³	#14556568	1	2	yes	—
240	C₂³	#14556570	1	1	yes	characteristic
241	C₂³	#14556582	1	2	yes	—
242	C₂³	#14556591	1	1	yes	characteristic
243	C₂×C₄	—	2	8	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	4	4	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
249	C₂×C₄	#14556547	1	2	yes	—
250	C₂×C₄	#14556553	1	2	yes	—
251	C₂×C₄	C₄×Dic₁₄₆	1	1	yes	characteristic
252	C₂×C₄	#14556566	1	2	yes	—
253	C₂×C₄	#14556568	1	2	yes	—
254	C₂×C₄	#14556570	1	1	yes	characteristic
255	C₂×C₄	#14556572	1	1	yes	characteristic
256	C₂×C₄	#14556572	1	1	yes	characteristic
257	C₂×C₄	#14556573	1	2	yes	—
258	C₂×C₄	#14556573	1	2	yes	—
259	C₂×C₄	#14556575	1	2	yes	—
260	C₂×C₄	#14556579	1	1	yes	characteristic
261	C₂×C₄	#14556580	1	1	yes	characteristic
262	C₂×C₄	#14556582	1	2	yes	—
263	C₂×C₄	#14556586	1	2	yes	—
264	C₂×C₄	#14556586	1	2	yes	—
265	C₂×C₄	#14556587	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	2	no	—
268	C₈	—	292	4	no	—
269	C₈	—	292	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	—	4	2	no	—
274	C₂²	—	4	1	no	—
275	C₂²	#17988470	1	2	yes	—
276	C₂²	#17988493	1	1	yes	characteristic
277	C₂²	#17988519	1	2	yes	—
278	C₂²	#17988523	1	1	yes	characteristic
279	C₂²	#17988526	1	1	yes	characteristic
280	C₄	—	4	4	no	—
281	C₄	#17988448	1	1	yes	characteristic
282	C₄	#17988470	1	2	yes	—
283	C₄	#17988488	1	1	yes	characteristic
284	C₄	#17988514	1	2	yes	—
285	C₄	#17988523	1	1	yes	characteristic
286	C₄	#17988526	1	1	yes	characteristic
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	2	no	—
290	C₂	—	2	1	no	—
291	C₂	#22608200	1	1	yes	characteristic
292	C₂	#22611447	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14556587
commutator	C₂²×C₄×C₇₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₇₃
Fitting	C₇₃×#1728	C₂
socle	C₂²×C₇₃	#399
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂³×C₄×C₇₃	C₂³
hypercenter	#1728	D₇₃
hypocenter	C₇₃	#11375
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
9. #17988636	C₄	semidirect
152. C₇₃	#11375	semidirect

Maximal subgroups

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

subgroup	index
C₂×#17988636	2
#22609657	2
C₂×#17988345	2
#22609625	2
C₇₃×#1728	2
#11375	73