Subgroups — C₂³×#14556620 #33999780

Order 18688.

11166 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#11924027	C₂	1	1	yes	maximal, characteristic
2	C₂³×Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14556620	C₂	1	56	yes	maximal, direct product factor
4	C₂³×Dic₂₉₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×#11924018	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×#11924016	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×#11924011	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×SD₁₆×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11924027	—	2	14	no	—
10	C₂²×#11924027	C₂²	1	28	yes	—
11	C₂³×Dic₁₄₆	—	2	1	no	—
12	C₂³×Dic₁₄₆	C₂²	1	1	yes	characteristic
13	C₂³×Dic₁₄₆	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×D₇₃	—	2	14	no	—
15	C₂²×Q₈×D₇₃	C₂²	1	28	yes	complemented
16	C₂³×C₂²⋊D₇₃	—	2	1	no	—
17	C₂⁴×Dic₇₃	—	2	1	no	—
18	C₂³×C₄×D₇₃	—	2	1	no	—
19	C₂³×C₄×D₇₃	C₂²	1	1	yes	characteristic
20	C₂×#14556620	C₂²	1	448	yes	direct product factor
21	C₂²×Dic₂₉₂	C₂²	1	28	yes	complemented
22	C₂²×#11924018	C₂²	1	28	yes	complemented
23	C₂²×#11924016	C₂²	1	28	yes	complemented
24	C₂²×#11924011	C₂²	1	28	yes	complemented
25	C₂³×#10596831	C₂²	1	1	yes	characteristic
26	C₂³×Q₈×C₇₃	C₂²	1	1	yes	characteristic
27	C₂³×D₄×C₇₃	C₂²	1	1	yes	characteristic
28	C₂²×SD₁₆×C₇₃	C₂²	1	28	yes	complemented
29	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
30	C₂⁴×D₇₃	D₄	1	1	yes	characteristic
31	C₂×#11924027	—	2	168	no	—
32	C₂×#11924027	C₂³	1	112	yes	—
33	C₂²×Dic₁₄₆	—	2	28	no	—
34	C₂²×Dic₁₄₆	—	2	7	no	—
35	C₂²×Dic₁₄₆	—	2	7	no	—
36	C₂²×Dic₁₄₆	C₂³	1	14	yes	—
37	C₂²×Dic₁₄₆	C₂³	1	14	yes	—
38	C₂×Q₈×D₇₃	—	2	168	no	—
39	C₂×Q₈×D₇₃	C₂³	1	112	yes	complemented
40	C₂²×C₂²⋊D₇₃	—	2	28	no	—
41	C₂³×Dic₇₃	—	2	28	no	—
42	C₂³×Dic₇₃	—	2	1	no	—
43	C₂³×Dic₇₃	—	2	1	no	—
44	C₂³×Dic₇₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×D₇₃	—	2	28	no	—
46	C₂²×C₄×D₇₃	D₄	1	14	yes	—
47	C₂²×C₄×D₇₃	C₂³	1	14	yes	—
48	#14556620	C₂³	1	512	yes	direct product factor
49	C₂×Dic₂₉₂	C₂³	1	112	yes	complemented
50	C₂×#11924018	C₂³	1	112	yes	complemented
51	C₂×#11924016	C₂³	1	112	yes	complemented
52	C₂×#11924011	C₂³	1	112	yes	complemented
53	C₂²×#10596831	C₂³	1	14	yes	—
54	C₂⁵×C₇₃	—	2	1	no	—
55	C₂²×Q₈×C₇₃	—	2	7	no	—
56	C₂²×Q₈×C₇₃	C₂³	1	14	yes	—
57	C₂²×D₄×C₇₃	—	2	7	no	—
58	C₂²×D₄×C₇₃	C₂³	1	14	yes	—
59	C₂³×C₄×C₇₃	—	2	1	no	—
60	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
61	C₂×SD₁₆×C₇₃	C₂³	1	112	yes	complemented
62	C₂²×C₈×C₇₃	C₂³	1	14	yes	—
63	C₂³×D₇₃	—	2	8	no	—
64	C₂³×D₇₃	C₂×D₄	1	14	yes	—
65	#11924027	—	2	224	no	—
66	#11924027	C₂⁴	1	64	yes	—
67	C₂×Dic₁₄₆	—	2	42	no	—
68	C₂×Dic₁₄₆	—	2	42	no	—
69	C₂×Dic₁₄₆	—	2	112	no	—
70	C₂×Dic₁₄₆	C₂⁴	1	28	yes	—
71	C₂×Dic₁₄₆	C₂⁴	1	28	yes	—
72	Q₈×D₇₃	—	2	224	no	—
73	Q₈×D₇₃	C₂⁴	1	64	yes	complemented
74	C₂×C₂²⋊D₇₃	—	2	112	no	—
75	C₂²×Dic₇₃	—	2	14	no	—
76	C₂²×Dic₇₃	—	2	14	no	—
77	C₂²×Dic₇₃	—	2	112	no	—
78	C₂²×Dic₇₃	C₂×D₄	1	14	yes	—
79	C₂×C₄×D₇₃	—	2	112	no	—
80	C₂×C₄×D₇₃	C₂×D₄	1	84	yes	—
81	C₂×C₄×D₇₃	C₂⁴	1	28	yes	—
82	Dic₂₉₂	C₂⁴	1	64	yes	complemented
83	#11924018	C₂⁴	1	64	yes	complemented
84	#11924016	C₂⁴	1	64	yes	complemented
85	#11924011	C₂⁴	1	64	yes	complemented
86	C₂×#10596831	C₂⁴	1	28	yes	—
87	C₂⁴×C₇₃	—	2	14	no	—
88	C₂⁴×C₇₃	—	4	8	no	—
89	C₂⁴×C₇₃	C₂×D₄	1	1	yes	characteristic
90	C₂×Q₈×C₇₃	—	2	42	no	—
91	C₂×Q₈×C₇₃	C₂⁴	1	28	yes	—
92	C₂×D₄×C₇₃	—	2	42	no	—
93	C₂×D₄×C₇₃	C₂⁴	1	28	yes	—
94	C₂²×C₄×C₇₃	—	2	14	no	—
95	C₂²×C₄×C₇₃	C₂×D₄	1	7	yes	—
96	C₂²×C₄×C₇₃	C₂⁴	1	7	yes	—
97	SD₁₆×C₇₃	C₂⁴	1	64	yes	complemented
98	C₂×C₈×C₇₃	C₂⁴	1	28	yes	—
99	C₂³×C₇₃	—	2	28	no	—
100	C₂³×C₇₃	—	4	56	no	—
101	C₂³×C₇₃	#137	1	8	yes	complemented
102	C₂³×C₇₃	C₂²×D₄	1	7	yes	—
103	C₂²×D₇₃	—	2	56	no	—
104	C₂²×D₇₃	C₂²×D₄	1	28	yes	—
105	Q₈×C₇₃	—	2	28	no	—
106	Q₈×C₇₃	C₂⁵	1	8	yes	—
107	D₄×C₇₃	—	2	28	no	—
108	D₄×C₇₃	C₂⁵	1	8	yes	—
109	C₂×C₄×C₇₃	—	2	28	no	—
110	C₂×C₄×C₇₃	C₂²×D₄	1	21	yes	—
111	C₂×C₄×C₇₃	C₂⁵	1	7	yes	—
112	C₂²⋊D₇₃	—	2	64	no	—
113	C₂×Dic₇₃	—	2	28	no	—
114	C₂×Dic₇₃	—	2	28	no	—
115	C₂×Dic₇₃	—	2	64	no	—
116	C₂×Dic₇₃	C₂²×D₄	1	28	yes	—
117	C₄×D₇₃	—	2	64	no	—
118	C₄×D₇₃	C₂²×D₄	1	56	yes	—
119	C₄×D₇₃	C₂⁵	1	8	yes	—
120	Dic₁₄₆	—	2	28	no	—
121	Dic₁₄₆	—	2	28	no	—
122	Dic₁₄₆	—	2	64	no	—
123	Dic₁₄₆	C₂⁵	1	8	yes	—
124	Dic₁₄₆	C₂⁵	1	8	yes	—
125	C₈×C₇₃	C₂⁵	1	8	yes	—
126	#10596831	C₂⁵	1	8	yes	—
127	C₂²×C₇₃	—	2	8	no	—
128	C₂²×C₇₃	—	4	56	no	—
129	C₂²×C₇₃	C₂×#137	1	28	yes	complemented
130	C₂²×C₇₃	C₂³×D₄	1	7	yes	—
131	C₂×D₇₃	—	2	56	no	—
132	C₂×D₇₃	C₂³×D₄	1	8	yes	—
133	C₄×C₇₃	—	2	8	no	—
134	C₄×C₇₃	C₂³×D₄	1	7	yes	—
135	C₄×C₇₃	C₂⁶	1	1	yes	characteristic
136	Dic₇₃	—	2	8	no	—
137	Dic₇₃	—	2	8	no	—
138	Dic₇₃	C₂³×D₄	1	8	yes	—
139	C₂³×#137	—	73	1	no	maximal, Sylow
140	C₂×C₇₃	—	4	8	no	—
141	C₂×C₇₃	C₂²×#137	1	14	yes	complemented
142	C₂×C₇₃	C₂⁴×D₄	1	1	yes	characteristic
143	D₇₃	—	2	8	no	—
144	C₂³×#41	—	73	1	no	—
145	C₂⁴×Q₈	—	73	1	no	—
146	C₂²×#137	—	73	56	no	—
147	C₂³×Q₁₆	—	73	1	no	—
148	C₂³×Q₁₆	—	73	1	no	—
149	C₂³×SD₁₆	—	73	1	no	—
150	C₂³×SD₁₆	D₇₃	1	1	yes	characteristic, complemented
151	C₂³×M₁₆	—	73	1	no	—
152	C₇₃	C₂³×#137	1	1	yes	characteristic, complemented, Sylow
153	C₂²×#41	—	73	28	no	—
154	C₂²×#41	—	146	14	no	—
155	C₂³×Q₈	—	73	28	no	—
156	C₂³×Q₈	—	73	1	no	—
157	C₂³×Q₈	—	73	1	no	—
158	C₂³×Q₈	—	146	1	no	—
159	C₂³×Q₈	—	146	14	no	—
160	C₂³×Q₈	C₂×D₇₃	1	1	yes	characteristic
161	C₂³×D₄	—	146	1	no	—
162	C₂³×D₄	C₂×D₇₃	1	1	yes	characteristic
163	C₂⁴×C₄	—	73	1	no	—
164	C₂⁴×C₄	—	146	1	no	—
165	C₂⁴×C₄	—	146	1	no	—
166	C₂×#137	—	73	448	no	—
167	C₂²×Q₁₆	—	73	28	no	—
168	C₂²×Q₁₆	—	73	28	no	—
169	C₂²×SD₁₆	—	73	28	no	—
170	C₂²×SD₁₆	C₂×D₇₃	1	28	yes	complemented
171	C₂²×M₁₆	—	73	28	no	—
172	C₂³×C₈	—	73	1	no	—
173	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
174	C₂⁵	—	2	1	no	—
175	C₂⁵	—	73	1	no	—
176	C₂×#41	—	73	112	no	—
177	C₂×#41	—	146	168	no	—
178	C₂²×Q₈	—	2	7	no	—
179	C₂²×Q₈	—	73	112	no	—
180	C₂²×Q₈	—	73	14	no	—
181	C₂²×Q₈	—	73	14	no	—
182	C₂²×Q₈	—	146	28	no	—
183	C₂²×Q₈	—	146	168	no	—
184	C₂²×Q₈	—	146	7	no	—
185	C₂²×Q₈	—	146	7	no	—
186	C₂²×Q₈	C₂²×D₇₃	1	14	yes	—
187	C₂²×D₄	—	2	7	no	—
188	C₂²×D₄	—	146	28	no	—
189	C₂²×D₄	C₂²×D₇₃	1	14	yes	—
190	C₂³×C₄	—	2	1	no	—
191	C₂³×C₄	—	73	1	no	—
192	C₂³×C₄	—	73	14	no	—
193	C₂³×C₄	—	73	14	no	—
194	C₂³×C₄	—	146	28	no	—
195	C₂³×C₄	—	146	28	no	—
196	C₂³×C₄	—	146	1	no	—
197	C₂³×C₄	—	146	1	no	—
198	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
199	#137	—	73	512	no	—
200	C₂×Q₁₆	—	73	112	no	—
201	C₂×Q₁₆	—	73	112	no	—
202	C₂×SD₁₆	—	73	112	no	—
203	C₂×SD₁₆	C₂²×D₇₃	1	112	yes	complemented
204	C₂×M₁₆	—	73	112	no	—
205	C₂²×C₈	—	73	14	no	—
206	C₂²×C₈	C₂²×D₇₃	1	14	yes	—
207	C₂⁴	—	2	14	no	—
208	C₂⁴	—	4	8	no	—
209	C₂⁴	—	73	14	no	—
210	C₂⁴	—	146	8	no	—
211	C₂⁴	D₄×D₇₃	1	1	yes	characteristic
212	#41	—	73	64	no	—
213	#41	—	146	224	no	—
214	C₂×Q₈	—	2	42	no	—
215	C₂×Q₈	—	73	64	no	—
216	C₂×Q₈	—	73	28	no	—
217	C₂×Q₈	—	73	28	no	—
218	C₂×Q₈	—	146	112	no	—
219	C₂×Q₈	—	146	224	no	—
220	C₂×Q₈	—	146	42	no	—
221	C₂×Q₈	—	146	42	no	—
222	C₂×Q₈	C₂³×D₇₃	1	28	yes	—
223	C₂×D₄	—	2	42	no	—
224	C₂×D₄	—	146	112	no	—
225	C₂×D₄	C₂³×D₇₃	1	28	yes	—
226	C₂²×C₄	—	2	14	no	—
227	C₂²×C₄	—	73	14	no	—
228	C₂²×C₄	—	73	84	no	—
229	C₂²×C₄	—	73	28	no	—
230	C₂²×C₄	—	146	112	no	—
231	C₂²×C₄	—	146	112	no	—
232	C₂²×C₄	—	146	14	no	—
233	C₂²×C₄	—	146	14	no	—
234	C₂²×C₄	D₄×D₇₃	1	7	yes	—
235	C₂²×C₄	C₂³×D₇₃	1	7	yes	—
236	Q₁₆	—	73	64	no	—
237	Q₁₆	—	73	64	no	—
238	SD₁₆	—	73	64	no	—
239	SD₁₆	C₂³×D₇₃	1	64	yes	complemented
240	M₁₆	—	73	64	no	—
241	C₂×C₈	—	73	28	no	—
242	C₂×C₈	C₂³×D₇₃	1	28	yes	—
243	C₂³	—	2	28	no	—
244	C₂³	—	4	56	no	—
245	C₂³	—	73	28	no	—
246	C₂³	—	146	56	no	—
247	C₂³	#14556620	1	8	yes	direct product factor
248	C₂³	C₂×D₄×D₇₃	1	7	yes	—
249	Q₈	—	2	28	no	—
250	Q₈	—	73	8	no	—
251	Q₈	—	73	8	no	—
252	Q₈	—	146	64	no	—
253	Q₈	—	146	28	no	—
254	Q₈	—	146	28	no	—
255	Q₈	C₂⁴×D₇₃	1	8	yes	—
256	D₄	—	2	28	no	—
257	D₄	—	146	64	no	—
258	D₄	C₂⁴×D₇₃	1	8	yes	—
259	C₂×C₄	—	2	28	no	—
260	C₂×C₄	—	73	28	no	—
261	C₂×C₄	—	73	56	no	—
262	C₂×C₄	—	73	8	no	—
263	C₂×C₄	—	146	64	no	—
264	C₂×C₄	—	146	64	no	—
265	C₂×C₄	—	146	28	no	—
266	C₂×C₄	—	146	28	no	—
267	C₂×C₄	C₂×D₄×D₇₃	1	21	yes	—
268	C₂×C₄	C₂⁴×D₇₃	1	7	yes	—
269	C₈	—	73	8	no	—
270	C₈	C₂⁴×D₇₃	1	8	yes	—
271	C₂²	—	2	8	no	—
272	C₂²	—	4	56	no	—
273	C₂²	—	73	8	no	—
274	C₂²	—	146	56	no	—
275	C₂²	C₂×#14556620	1	28	yes	direct product factor
276	C₂²	C₂²×D₄×D₇₃	1	7	yes	—
277	C₄	—	2	8	no	—
278	C₄	—	73	8	no	—
279	C₄	—	146	8	no	—
280	C₄	—	146	8	no	—
281	C₄	C₂²×D₄×D₇₃	1	7	yes	—
282	C₄	C₂⁵×D₇₃	1	1	yes	characteristic
283	C₂	—	4	8	no	—
284	C₂	—	146	8	no	—
285	C₂	C₂²×#14556620	1	14	yes	direct product factor
286	C₂	C₂³×D₄×D₇₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	D₄×D₇₃
commutator, cosocle	C₄×C₇₃	C₂⁶
Frattini	C₄	C₂⁵×D₇₃
Fitting	C₂³×SD₁₆×C₇₃	C₂
socle	C₂⁴×C₇₃	C₂×D₄
prime radical	C₂³×#11924027	C₂
hypercenter	C₂³×SD₁₆	D₇₃
hypocenter	C₇₃	C₂³×#137
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂³×SD₁₆

Complements

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

N	H ≅ G/N	type
2. C₂³×Q₈×D₇₃	C₂	semidirect
3. C₂²×#14556620	C₂	direct (N × H)
4. C₂³×Dic₂₉₂	C₂	semidirect
5. C₂³×#11924018	C₂	semidirect
6. C₂³×#11924016	C₂	semidirect
7. C₂³×#11924011	C₂	semidirect
8. C₂³×SD₁₆×C₇₃	C₂	semidirect
15. C₂²×Q₈×D₇₃	C₂²	semidirect
20. C₂×#14556620	C₂²	direct (N × H)
21. C₂²×Dic₂₉₂	C₂²	semidirect
22. C₂²×#11924018	C₂²	semidirect
23. C₂²×#11924016	C₂²	semidirect
24. C₂²×#11924011	C₂²	semidirect
28. C₂²×SD₁₆×C₇₃	C₂²	semidirect
39. C₂×Q₈×D₇₃	C₂³	semidirect
48. #14556620	C₂³	direct (N × H)
49. C₂×Dic₂₉₂	C₂³	semidirect
50. C₂×#11924018	C₂³	semidirect
51. C₂×#11924016	C₂³	semidirect
52. C₂×#11924011	C₂³	semidirect
61. C₂×SD₁₆×C₇₃	C₂³	semidirect
73. Q₈×D₇₃	C₂⁴	semidirect
82. Dic₂₉₂	C₂⁴	semidirect
83. #11924018	C₂⁴	semidirect
84. #11924016	C₂⁴	semidirect
85. #11924011	C₂⁴	semidirect
97. SD₁₆×C₇₃	C₂⁴	semidirect
101. C₂³×C₇₃	#137	semidirect
129. C₂²×C₇₃	C₂×#137	semidirect
141. C₂×C₇₃	C₂²×#137	semidirect
150. C₂³×SD₁₆	D₇₃	semidirect
152. C₇₃	C₂³×#137	semidirect
170. C₂²×SD₁₆	C₂×D₇₃	semidirect
203. C₂×SD₁₆	C₂²×D₇₃	semidirect
239. SD₁₆	C₂³×D₇₃	semidirect
247. C₂³	#14556620	direct (N × H)
275. C₂²	C₂×#14556620	direct (N × H)
285. C₂	C₂²×#14556620	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#11924027	2
C₂³×Q₈×D₇₃	2
C₂²×#14556620	2
C₂³×Dic₂₉₂	2
C₂³×#11924018	2
C₂³×#11924016	2
C₂³×#11924011	2
C₂³×SD₁₆×C₇₃	2
C₂³×#137	73

Subgroups — C23×#14556620 #33999780

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14556620 #33999780