Subgroups — C₂²⋊C₁₆×Dic₇₃ #33001013

Order 18688.

614 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊#11924003	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×Dic₇₃	C₂	1	2	yes	maximal, complemented
4	#22608008	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×#10596831	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₇₃	C₄	1	1	yes	characteristic
8	C₂²×#11924003	C₂²	1	2	yes	—
9	C₂×C₈×Dic₇₃	—	2	4	no	—
10	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₇₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₂²⋊#11924003	C₄	1	4	yes	complemented
15	C₁₆×Dic₇₃	—	2	4	no	—
16	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂²×C₁₆×C₇₃	C₂²	1	2	yes	—
18	C₂²⋊C₁₆×C₇₃	C₄	1	4	yes	complemented
19	C₂³×Dic₇₃	C₈	1	2	yes	—
20	C₂²×#10596831	—	2	2	no	—
21	C₂²×#10596831	—	2	2	no	—
22	C₂²×#10596831	C₈	1	4	yes	—
23	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
24	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₇₃	—	2	2	no	—
26	C₂×C₄×Dic₇₃	—	2	2	no	—
27	C₂×C₄×Dic₇₃	C₈	1	2	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×#11924003	—	2	2	no	—
31	C₂×#11924003	—	2	2	no	—
32	C₂×#11924003	C₂×C₄	1	4	yes	complemented
33	C₈×Dic₇₃	—	2	2	no	—
34	C₈×Dic₇₃	—	2	2	no	—
35	C₈×Dic₇₃	—	2	2	no	—
36	C₈×Dic₇₃	D₄	1	4	yes	—
37	C₂³×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	—	2	2	no	—
39	C₂²×C₈×C₇₃	—	2	2	no	—
40	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
42	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
44	C₂×C₁₆×C₇₃	—	2	2	no	—
45	C₂×C₁₆×C₇₃	—	2	2	no	—
46	C₂×C₁₆×C₇₃	C₂×C₄	1	4	yes	complemented
47	C₂²×Dic₇₃	—	2	4	no	—
48	C₂²×Dic₇₃	—	2	4	no	—
49	C₂²×Dic₇₃	C₁₆	1	8	yes	complemented
50	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
51	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
52	C₂×#10596831	—	2	1	no	—
53	C₂×#10596831	—	2	1	no	—
54	C₂×#10596831	—	2	8	no	—
55	C₂×#10596831	C₂²⋊C₄	1	2	yes	—
56	C₂×#10596831	C₂×C₈	1	2	yes	—
57	C₂×#10596831	C₂×C₈	1	2	yes	—
58	C₂×#10596831	M₁₆	1	2	yes	—
59	C₄×Dic₇₃	—	2	1	no	—
60	C₄×Dic₇₃	—	2	1	no	—
61	C₄×Dic₇₃	—	2	4	no	—
62	C₄×Dic₇₃	C₂²⋊C₄	1	2	yes	—
63	C₄×Dic₇₃	M₁₆	1	2	yes	—
64	#11924003	—	2	4	no	—
65	C₂⁴×C₇₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₇₃	—	2	1	no	—
67	C₂²×C₄×C₇₃	—	2	1	no	—
68	C₂²×C₄×C₇₃	—	2	1	no	—
69	C₂²×C₄×C₇₃	—	2	1	no	—
70	C₂²×C₄×C₇₃	C₄²	1	2	yes	—
71	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₇₃	C₂×C₈	1	2	yes	—
73	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₇₃	—	2	1	no	—
75	C₂×C₈×C₇₃	—	2	1	no	—
76	C₂×C₈×C₇₃	—	2	4	no	—
77	C₂×C₈×C₇₃	—	2	4	no	—
78	C₂×C₈×C₇₃	C₄²	1	2	yes	—
79	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₇₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₇₃	#41	1	2	yes	—
83	C₁₆×C₇₃	—	2	4	no	—
84	C₂³×C₇₃	—	2	1	no	—
85	C₂³×C₇₃	—	2	1	no	—
86	C₂³×C₇₃	—	2	2	no	—
87	C₂³×C₇₃	C₄×C₈	1	2	yes	—
88	C₂³×C₇₃	C₂×C₁₆	1	4	yes	—
89	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₇₃	—	2	1	no	—
91	C₂×C₄×C₇₃	—	2	1	no	—
92	C₂×C₄×C₇₃	—	2	2	no	—
93	C₂×C₄×C₇₃	—	2	2	no	—
94	C₂×C₄×C₇₃	—	2	2	no	—
95	C₂×C₄×C₇₃	—	2	2	no	—
96	C₂×C₄×C₇₃	C₄×C₈	1	2	yes	—
97	C₂×C₄×C₇₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₇₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₇₃	—	2	2	no	—
104	C₂×Dic₇₃	—	2	2	no	—
105	C₂×Dic₇₃	—	2	16	no	—
106	C₂×Dic₇₃	C₂²⋊C₈	1	4	yes	—
107	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	—
108	C₂×Dic₇₃	C₂×C₁₆	1	4	yes	—
109	C₂×Dic₇₃	M₃₂	1	4	yes	—
110	C₈×C₇₃	—	2	2	no	—
111	C₈×C₇₃	C₄×D₄	1	2	yes	—
112	C₈×C₇₃	C₄×D₄	1	2	yes	—
113	#10596831	—	2	2	no	—
114	#10596831	C₂²⋊C₈	1	4	yes	—
115	C₂²×C₇₃	—	2	2	no	—
116	C₂²×C₇₃	—	2	4	no	—
117	C₂²×C₇₃	—	2	4	no	—
118	C₂²×C₇₃	—	2	2	no	—
119	C₂²×C₇₃	C₄×C₁₆	1	4	yes	complemented
120	C₂²×C₇₃	C₂×C₄×C₈	1	1	yes	characteristic
121	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
122	C₂²×C₇₃	#431	1	1	yes	characteristic
123	C₂²×C₇₃	C₂²×C₁₆	1	2	yes	—
124	C₂²×C₇₃	C₂×M₃₂	1	2	yes	—
125	C₄×C₇₃	—	2	2	no	—
126	C₄×C₇₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
127	C₄×C₇₃	C₄×M₁₆	1	1	yes	characteristic
128	C₄×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
129	C₄×C₇₃	#431	1	1	yes	characteristic
130	Dic₇₃	—	2	4	no	—
131	Dic₇₃	C₂²⋊C₁₆	1	8	yes	direct product factor
132	C₄×C₂²⋊C₁₆	—	73	1	no	maximal, Sylow
133	C₂×C₇₃	—	2	4	no	—
134	C₂×C₇₃	C₄×C₂²⋊C₈	1	1	yes	characteristic
135	C₂×C₇₃	C₂×C₄×C₁₆	1	1	yes	characteristic
136	C₂×C₇₃	C₄×M₃₂	1	1	yes	characteristic
137	C₂×C₇₃	C₂×C₂²⋊C₁₆	1	2	yes	—
138	C₂×C₇₃	#2162	1	2	yes	—
139	C₂²×C₄×C₈	—	73	1	no	—
140	C₂×C₂²⋊C₁₆	—	73	1	no	—
141	C₂×C₂²⋊C₁₆	D₇₃	1	1	yes	characteristic
142	C₂×C₄×C₁₆	—	73	2	no	—
143	#1374	—	73	2	no	—
144	C₇₃	C₄×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
145	C₂³×C₈	—	73	1	no	—
146	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
147	C₂²×C₄²	—	73	1	no	—
148	C₂²×C₁₆	—	73	2	no	—
149	C₂²×C₁₆	C₂×D₇₃	1	2	yes	—
150	C₂×C₄×C₈	—	73	1	no	—
151	C₂×C₄×C₈	—	73	1	no	—
152	C₂×C₄×C₈	—	73	1	no	—
153	C₂×C₄×C₈	—	73	1	no	—
154	C₂×C₄×C₈	—	146	4	no	—
155	C₂²⋊C₁₆	—	73	4	no	—
156	C₂²⋊C₁₆	Dic₇₃	1	4	yes	direct product factor
157	C₄×C₁₆	—	146	4	no	—
158	C₂³×C₄	—	73	2	no	—
159	C₂³×C₄	C₄×D₇₃	1	1	yes	characteristic
160	C₂²×C₈	—	2	2	no	—
161	C₂²×C₈	—	2	2	no	—
162	C₂²×C₈	—	73	1	no	—
163	C₂²×C₈	—	73	4	no	—
164	C₂²×C₈	—	73	1	no	—
165	C₂²×C₈	—	146	2	no	—
166	C₂²×C₈	—	146	2	no	—
167	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
168	C₂²×C₈	C₄×D₇₃	1	2	yes	—
169	C₂²×C₈	C₂×Dic₇₃	1	2	yes	—
170	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
171	C₂×C₄²	—	73	1	no	—
172	C₂×C₄²	—	73	2	no	—
173	C₂×C₄²	—	73	1	no	—
174	C₂×C₄²	—	146	2	no	—
175	C₂×C₄²	—	146	2	no	—
176	C₂×C₁₆	—	2	2	no	—
177	C₂×C₁₆	—	2	2	no	—
178	C₂×C₁₆	—	73	4	no	—
179	C₂×C₁₆	—	146	2	no	—
180	C₂×C₁₆	—	146	2	no	—
181	C₂×C₁₆	C₂×Dic₇₃	1	4	yes	complemented
182	C₄×C₈	—	73	4	no	—
183	C₄×C₈	—	146	2	no	—
184	C₄×C₈	—	146	2	no	—
185	C₄×C₈	—	146	2	no	—
186	C₂⁴	C₈×D₇₃	1	1	yes	characteristic
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	2	1	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	73	2	no	—
192	C₂²×C₄	—	73	8	no	—
193	C₂²×C₄	—	73	2	no	—
194	C₂²×C₄	—	146	4	no	—
195	C₂²×C₄	—	146	4	no	—
196	C₂²×C₄	C₄×Dic₇₃	1	2	yes	—
197	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
198	C₂²×C₄	C₈×D₇₃	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	2	4	no	—
204	C₂×C₈	—	73	2	no	—
205	C₂×C₈	—	73	2	no	—
206	C₂×C₈	—	73	2	no	—
207	C₂×C₈	—	73	2	no	—
208	C₂×C₈	—	146	8	no	—
209	C₂×C₈	—	146	1	no	—
210	C₂×C₈	—	146	1	no	—
211	C₂×C₈	C₄×Dic₇₃	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
213	C₂×C₈	C₂²×Dic₇₃	1	1	yes	characteristic
214	C₂×C₈	D₄×D₇₃	1	2	yes	—
215	C₂×C₈	#11924027	1	2	yes	—
216	C₄²	—	73	2	no	—
217	C₄²	—	73	2	no	—
218	C₄²	—	146	4	no	—
219	C₄²	—	146	1	no	—
220	C₄²	—	146	1	no	—
221	C₁₆	—	2	4	no	—
222	C₁₆	—	146	4	no	—
223	C₂³	—	2	1	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	1	no	—
226	C₂³	C₈×Dic₇₃	1	2	yes	—
227	C₂³	C₁₆×D₇₃	1	4	yes	—
228	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	73	4	no	—
236	C₂×C₄	—	73	4	no	—
237	C₂×C₄	—	73	4	no	—
238	C₂×C₄	—	73	4	no	—
239	C₂×C₄	—	146	16	no	—
240	C₂×C₄	—	146	2	no	—
241	C₂×C₄	—	146	2	no	—
242	C₂×C₄	C₈×Dic₇₃	1	2	yes	—
243	C₂×C₄	C₂×C₄×Dic₇₃	1	1	yes	characteristic
244	C₂×C₄	C₂²⋊C₄×D₇₃	1	1	yes	characteristic
245	C₂×C₄	#14556547	1	1	yes	characteristic
246	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
247	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
248	C₂×C₄	M₁₆×D₇₃	1	1	yes	characteristic
249	C₈	—	2	2	no	—
250	C₈	—	73	4	no	—
251	C₈	—	146	2	no	—
252	C₈	#14556551	1	2	yes	—
253	C₈	D₄×Dic₇₃	1	2	yes	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	4	no	—
258	C₂²	C₁₆×Dic₇₃	1	4	yes	complemented
259	C₂²	C₂×C₈×Dic₇₃	1	1	yes	characteristic
260	C₂²	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
261	C₂²	#17988637	1	1	yes	characteristic
262	C₂²	C₂×C₁₆×D₇₃	1	2	yes	—
263	C₂²	M₃₂×D₇₃	1	2	yes	—
264	C₄	—	2	2	no	—
265	C₄	—	73	8	no	—
266	C₄	—	146	4	no	—
267	C₄	C₂²⋊C₄×Dic₇₃	1	1	yes	characteristic
268	C₄	M₁₆×Dic₇₃	1	1	yes	characteristic
269	C₄	C₂²⋊C₈×D₇₃	1	1	yes	characteristic
270	C₄	#17988637	1	1	yes	characteristic
271	C₂	—	2	4	no	—
272	C₂	C₂²⋊C₈×Dic₇₃	1	1	yes	characteristic
273	C₂	C₂×C₁₆×Dic₇₃	1	1	yes	characteristic
274	C₂	M₃₂×Dic₇₃	1	1	yes	characteristic
275	C₂	C₂²⋊C₁₆×D₇₃	1	2	yes	—
276	C₂	#22611727	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₇₃
commutator	C₂×C₇₃	C₂×C₄×C₁₆
Fitting	C₂×C₂²⋊C₁₆×C₇₃	C₂
socle	C₂³×C₇₃	C₂²×C₈
prime radical	C₂⁴×C₇₃	C₂×C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₇₃
hypocenter	C₇₃	C₄×C₂²⋊C₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₆×Dic₇₃	C₂	semidirect
4. #22608008	C₂	semidirect
14. C₂²⋊#11924003	C₄	semidirect
18. C₂²⋊C₁₆×C₇₃	C₄	semidirect
32. C₂×#11924003	C₂×C₄	semidirect
46. C₂×C₁₆×C₇₃	C₂×C₄	semidirect
49. C₂²×Dic₇₃	C₁₆	semidirect
119. C₂²×C₇₃	C₄×C₁₆	semidirect
131. Dic₇₃	C₂²⋊C₁₆	direct (N × H)
144. C₇₃	C₄×C₂²⋊C₁₆	semidirect
156. C₂²⋊C₁₆	Dic₇₃	direct (N × H)
181. C₂×C₁₆	C₂×Dic₇₃	semidirect
258. C₂²	C₁₆×Dic₇₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₈×Dic₇₃	2
C₂×C₂²⋊#11924003	2
C₂×C₁₆×Dic₇₃	2
#22608008	2
C₂×C₂²⋊C₁₆×C₇₃	2
C₄×C₂²⋊C₁₆	73

Subgroups — C22⋊C16×Dic73 #33001013

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊C₁₆×Dic₇₃ #33001013