Subgroups — #33090212

Order 18688.

822 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22612409	C₂	1	2	yes	maximal
2	#22610793	C₂	1	2	yes	maximal
3	#22610665	C₂	1	2	yes	maximal
4	#22610609	C₂	1	4	yes	maximal
5	C₄×#14556525	C₂	1	1	yes	maximal, characteristic
6	C₄×#14556515	C₂	1	2	yes	maximal
7	C₄×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Q₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#14556525	C₂²	1	2	yes	—
10	C₂×#14556515	C₂²	1	4	yes	—
11	C₂×C₄×#10596831	C₂²	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
13	C₂×C₈×Dic₇₃	C₂²	1	2	yes	—
14	#17988494	C₂²	1	4	yes	—
15	#17988486	C₂²	1	4	yes	—
16	#17988473	—	2	4	no	—
17	C₄×C₄⋊Dic₇₃	—	2	2	no	—
18	C₄×C₄⋊Dic₇₃	C₂²	1	2	yes	—
19	#17988433	—	2	2	no	—
20	C₄²×Dic₇₃	C₂²	1	1	yes	characteristic
21	#17988350	C₂²	1	2	yes	—
22	#17988349	C₂²	1	4	yes	—
23	C₂²×Q₁₆×C₇₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₇₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₇₃	C₄	1	8	yes	complemented
26	C₂×#107×C₇₃	C₂²	1	1	yes	characteristic
27	C₂×#103×C₇₃	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂²×#10596831	C₂³	1	2	yes	—
30	C₂×C₄⋊Dic₇₃	—	2	4	no	—
31	C₂×C₄⋊Dic₇₃	C₂³	1	4	yes	—
32	C₂×C₄×Dic₇₃	—	2	2	no	—
33	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
34	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
35	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
37	#14556525	—	2	4	no	—
38	#14556515	—	2	8	no	—
39	C₄×#10596831	—	2	2	no	—
40	C₈×Dic₇₃	—	2	4	no	—
41	C₈×Dic₇₃	—	2	4	no	—
42	#14556498	—	2	4	no	—
43	#14556498	—	2	4	no	—
44	#14556498	—	2	4	no	—
45	#14556498	—	2	2	no	—
46	C₂²×Q₈×C₇₃	C₂³	1	2	yes	—
47	C₂×Q₁₆×C₇₃	—	2	4	no	—
48	C₂×Q₁₆×C₇₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₇₃	—	2	4	no	—
52	C₄×Q₈×C₇₃	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₇₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₇₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₇₃	—	2	2	no	—
56	C₂×C₄²×C₇₃	C₂³	1	1	yes	characteristic
57	#107×C₇₃	C₂×C₄	1	4	yes	—
58	#103×C₇₃	C₂×C₄	1	8	yes	—
59	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
60	C₂²×Dic₇₃	—	2	4	no	—
61	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
62	C₂²×Dic₇₃	C₂×D₄	1	2	yes	—
63	C₂×#10596831	—	2	2	no	—
64	C₂×#10596831	—	2	2	no	—
65	C₂×#10596831	#41	1	4	yes	—
66	C₄⋊Dic₇₃	—	2	4	no	—
67	C₄⋊Dic₇₃	—	2	4	no	—
68	C₄⋊Dic₇₃	—	4	8	no	—
69	C₄×Dic₇₃	—	2	2	no	—
70	C₄×Dic₇₃	—	2	2	no	—
71	C₄×Dic₇₃	—	2	2	no	—
72	C₄×Dic₇₃	—	2	4	no	—
73	C₄×Dic₇₃	—	4	4	no	—
74	C₄×Dic₇₃	Q₁₆	1	4	yes	complemented
75	C₂×Q₈×C₇₃	—	2	2	no	—
76	C₂×Q₈×C₇₃	—	2	4	no	—
77	C₂×Q₈×C₇₃	—	2	4	no	—
78	C₂×Q₈×C₇₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₇₃	—	2	2	no	—
80	C₂²×C₄×C₇₃	—	2	2	no	—
81	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
84	Q₁₆×C₇₃	—	2	8	no	—
85	C₂×C₈×C₇₃	—	2	2	no	—
86	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
88	C₂×C₈×C₇₃	#41	1	4	yes	—
89	C₄⋊C₄×C₇₃	—	2	8	no	—
90	C₄⋊C₄×C₇₃	—	2	2	no	—
91	C₄⋊C₄×C₇₃	C₂²×C₄	1	4	yes	—
92	C₄²×C₇₃	—	2	8	no	—
93	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₇₃	#41	1	2	yes	—
95	C₂³×C₇₃	C₂²×D₄	1	1	yes	characteristic
96	Q₈×C₇₃	—	2	4	no	—
97	Q₈×C₇₃	—	2	4	no	—
98	Q₈×C₇₃	—	4	4	no	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	—	2	4	no	—
101	C₂×C₄×C₇₃	—	2	4	no	—
102	C₂×C₄×C₇₃	—	4	4	no	—
103	C₂×C₄×C₇₃	—	4	4	no	—
104	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₇₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₇₃	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₇₃	#135	1	2	yes	—
108	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
109	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×#41	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×#41	1	2	yes	—
112	C₂×Dic₇₃	—	2	2	no	—
113	C₂×Dic₇₃	—	2	2	no	—
114	C₂×Dic₇₃	—	2	2	no	—
115	C₂×Dic₇₃	—	2	2	no	—
116	C₂×Dic₇₃	—	4	4	no	—
117	C₂×Dic₇₃	—	4	4	no	—
118	C₂×Dic₇₃	—	4	4	no	—
119	C₂×Dic₇₃	C₂×Q₁₆	1	4	yes	—
120	C₂×Dic₇₃	#135	1	4	yes	—
121	C₈×C₇₃	—	2	2	no	—
122	C₈×C₇₃	#117	1	4	yes	—
123	#10596831	—	2	4	no	—
124	C₂²×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
125	C₂²×C₇₃	#532	1	2	yes	—
126	C₂²×C₇₃	C₂²×Q₁₆	1	1	yes	characteristic
127	C₂²×C₇₃	C₂×#135	1	2	yes	—
128	C₂²×C₇₃	C₂×#135	1	1	yes	characteristic
129	C₄×C₇₃	—	2	2	no	—
130	C₄×C₇₃	—	4	4	no	—
131	C₄×C₇₃	—	4	4	no	—
132	C₄×C₇₃	C₄×Q₁₆	1	4	yes	—
133	C₄×C₇₃	C₂×#117	1	1	yes	characteristic
134	C₄×C₇₃	C₄×#41	1	1	yes	characteristic
135	C₄×C₇₃	#529	1	2	yes	—
136	Dic₇₃	—	2	4	no	—
137	Dic₇₃	—	2	4	no	—
138	Dic₇₃	—	8	4	no	—
139	#21276	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	#2293	1	1	yes	characteristic
141	C₂×C₇₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
142	C₂×C₇₃	C₄×#135	1	1	yes	characteristic
143	C₂×C₇₃	#3168	1	2	yes	—
144	C₂×C₇₃	#3170	1	2	yes	—
145	C₂×C₄×Q₁₆	D₇₃	1	1	yes	characteristic
146	#2295	—	73	2	no	—
147	#1969	—	73	2	no	—
148	#1942	—	73	2	no	—
149	#1933	—	73	4	no	—
150	C₄×#107	—	73	1	no	—
151	C₄×#103	—	73	2	no	—
152	C₄²×C₈	—	73	1	no	—
153	C₇₃	#21276	1	1	yes	characteristic, complemented, Sylow
154	C₂²×Q₁₆	C₂×D₇₃	1	1	yes	characteristic
155	C₂×C₄×Q₈	C₂×D₇₃	1	2	yes	—
156	C₄×Q₁₆	Dic₇₃	1	8	yes	complemented
157	C₂×#107	—	73	2	no	—
158	C₂×#107	C₂×D₇₃	1	1	yes	characteristic
159	C₂×#103	—	73	4	no	—
160	C₂×#103	C₂×D₇₃	1	2	yes	—
161	C₂×C₄×C₈	—	73	2	no	—
162	C₂×C₄×C₈	—	73	2	no	—
163	C₂×C₄×C₈	—	73	1	no	—
164	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
165	#391	—	73	4	no	—
166	#389	—	73	4	no	—
167	#387	—	146	4	no	—
168	C₄×C₄⋊C₄	—	73	2	no	—
169	C₄×C₄⋊C₄	—	146	2	no	—
170	#376	—	146	2	no	—
171	C₄³	—	73	1	no	—
172	#340	—	73	2	no	—
173	#340	—	73	4	no	—
174	C₂²×Q₈	C₂²×D₇₃	1	2	yes	—
175	C₂×Q₁₆	—	2	4	no	—
176	C₂×Q₁₆	C₂×Dic₇₃	1	4	yes	—
177	C₂²×C₈	—	73	2	no	—
178	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	C₂×Dic₇₃	1	8	yes	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	73	4	no	—
184	C₂×C₄⋊C₄	—	146	4	no	—
185	C₂×C₄⋊C₄	C₂²×D₇₃	1	2	yes	—
186	C₂×C₄²	—	2	2	no	—
187	C₂×C₄²	—	73	1	no	—
188	C₂×C₄²	—	73	2	no	—
189	C₂×C₄²	—	73	1	no	—
190	C₂×C₄²	—	73	2	no	—
191	C₂×C₄²	—	146	2	no	—
192	C₂×C₄²	C₂²×D₇₃	1	1	yes	characteristic
193	#107	—	146	4	no	—
194	#107	C₂×Dic₇₃	1	4	yes	—
195	#103	—	146	8	no	—
196	#103	C₂×Dic₇₃	1	8	yes	—
197	C₄×C₈	—	146	4	no	—
198	C₄×C₈	—	146	4	no	—
199	C₄×C₈	—	146	2	no	—
200	C₄×C₈	C₂×Dic₇₃	1	4	yes	—
201	#95	—	146	4	no	—
202	#95	—	146	4	no	—
203	#95	—	146	4	no	—
204	#95	—	146	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂×Q₈	C₂²×Dic₇₃	1	4	yes	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	73	2	no	—
212	C₂²×C₄	—	73	2	no	—
213	C₂²×C₄	—	146	4	no	—
214	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
216	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
217	Q₁₆	—	2	8	no	—
218	C₂×C₈	—	2	2	no	—
219	C₂×C₈	—	73	4	no	—
220	C₂×C₈	—	146	2	no	—
221	C₂×C₈	—	146	2	no	—
222	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
223	C₂×C₈	C₂²×Dic₇₃	1	2	yes	—
224	C₂×C₈	#11924028	1	4	yes	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	292	8	no	—
228	C₄⋊C₄	—	146	4	no	—
229	C₄⋊C₄	—	146	4	no	—
230	C₄⋊C₄	C₂²×Dic₇₃	1	4	yes	—
231	C₄²	—	2	8	no	—
232	C₄²	—	292	4	no	—
233	C₄²	—	73	4	no	—
234	C₄²	—	146	4	no	—
235	C₄²	—	146	2	no	—
236	C₄²	—	146	2	no	—
237	C₄²	—	146	2	no	—
238	C₄²	C₂²×Dic₇₃	1	2	yes	—
239	C₄²	#11924027	1	2	yes	—
240	C₂³	C₂×D₄×D₇₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	4	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	292	4	no	—
250	C₂×C₄	—	292	4	no	—
251	C₂×C₄	—	292	4	no	—
252	C₂×C₄	—	73	4	no	—
253	C₂×C₄	—	73	4	no	—
254	C₂×C₄	—	146	2	no	—
255	C₂×C₄	—	146	2	no	—
256	C₂×C₄	—	146	2	no	—
257	C₂×C₄	—	146	2	no	—
258	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
259	C₂×C₄	D₄×Dic₇₃	1	2	yes	—
260	C₂×C₄	Q₁₆×D₇₃	1	2	yes	—
261	C₂×C₄	#14556614	1	2	yes	—
262	C₂×C₄	C₂³×Dic₇₃	1	1	yes	characteristic
263	C₂×C₄	#41×D₇₃	1	2	yes	—
264	C₂×C₄	C₂×#11924027	1	1	yes	characteristic
265	C₂×C₄	C₂×#11924028	1	2	yes	—
266	C₈	—	2	2	no	—
267	C₈	—	146	4	no	—
268	C₈	#14556550	1	4	yes	—
269	C₂²	C₂×D₄×Dic₇₃	1	1	yes	characteristic
270	C₂²	#17989134	1	2	yes	—
271	C₂²	C₂×Q₁₆×D₇₃	1	1	yes	characteristic
272	C₂²	#135×D₇₃	1	2	yes	—
273	C₂²	C₂×#14556614	1	1	yes	characteristic
274	C₄	—	2	2	no	—
275	C₄	—	4	4	no	—
276	C₄	—	4	4	no	—
277	C₄	—	584	4	no	—
278	C₄	—	146	4	no	—
279	C₄	—	146	4	no	—
280	C₄	Q₁₆×Dic₇₃	1	4	yes	—
281	C₄	C₂×#14556550	1	1	yes	characteristic
282	C₄	#41×Dic₇₃	1	1	yes	characteristic
283	C₄	#17989119	1	2	yes	—
284	C₂	#22612396	1	1	yes	characteristic
285	C₂	C₂×Q₁₆×Dic₇₃	1	1	yes	characteristic
286	C₂	#135×Dic₇₃	1	1	yes	characteristic
287	C₂	#22620504	1	2	yes	—
288	C₂	#22620514	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂×C₄×Q₁₆×C₇₃	C₂
socle, prime radical	C₂³×C₇₃	C₂²×D₄
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂×C₄×Q₁₆	D₇₃
hypocenter	C₇₃	#21276
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
25. C₄×Q₁₆×C₇₃	C₄	semidirect
74. C₄×Dic₇₃	Q₁₆	semidirect
153. C₇₃	#21276	semidirect
156. C₄×Q₁₆	Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
#22612409	2
#22610793	2
#22610665	2
#22610609	2
C₄×#14556525	2
C₄×#14556515	2
C₄×C₈×Dic₇₃	2
C₂×C₄×Q₁₆×C₇₃	2
#21276	73