Subgroups — C₂×#22625473 #34017965

Order 18688.

942 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989588	C₂	1	2	yes	maximal
2	C₂×#17989449	C₂	1	2	yes	maximal, complemented
3	#22625473	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17989448	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11924036	—	2	2	no	—
7	C₂×#14556658	—	2	2	no	—
8	#17989588	—	2	4	no	—
9	#17989588	C₂²	1	8	yes	—
10	C₂×C₄⋊#10596833	C₂²	1	4	yes	—
11	#17989449	C₂²	1	8	yes	complemented
12	#17989448	C₂²	1	8	yes	complemented
13	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
14	C₂²×C₈×D₇₃	C₄	1	1	yes	characteristic
15	C₂×C₄²×D₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×#10596831	C₄	1	1	yes	characteristic
17	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
18	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₇₃	C₄	1	4	yes	—
20	C₄×C₈×D₇₃	C₂²	1	4	yes	—
21	C₂×C₄×C₈×C₇₃	C₄	1	1	yes	characteristic
22	C₂×#11924036	—	2	8	no	—
23	C₂×#11924036	—	4	8	no	—
24	C₂²×#10596833	—	2	4	no	—
25	#14556658	—	2	8	no	—
26	C₄⋊#10596833	—	2	4	no	—
27	C₄⋊#10596833	C₂³	1	8	yes	—
28	C₂²×C₄×D₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×D₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×D₇₃	Q₈	1	1	yes	characteristic
31	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
32	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×D₇₃	—	2	4	no	—
34	C₂×C₈×D₇₃	C₈	1	8	yes	complemented
35	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
36	C₂×C₈×D₇₃	C₂×C₄	1	4	yes	—
37	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
38	C₂×C₄×Dic₇₃	Q₈	1	1	yes	characteristic
39	C₂×C₄×Dic₇₃	C₂³	1	1	yes	characteristic
40	C₄²×D₇₃	C₂×C₄	1	2	yes	—
41	C₄²×D₇₃	D₄	1	2	yes	—
42	C₄²×D₇₃	Q₈	1	2	yes	—
43	C₄²×D₇₃	C₂³	1	2	yes	—
44	C₄×#10596831	C₂×C₄	1	4	yes	—
45	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
46	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
47	C₈×Dic₇₃	C₂³	1	2	yes	—
48	C₈×Dic₇₃	C₂³	1	2	yes	—
49	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₇₃	C₂×C₄	1	4	yes	—
53	#11924036	—	4	16	no	—
54	C₂×#10596833	—	2	8	no	—
55	C₂×#10596833	—	4	8	no	—
56	C₂³×D₇₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂²×Dic₇₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂²×Dic₇₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×D₇₃	—	2	4	no	—
61	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
62	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
63	C₂×C₄×D₇₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₇₃	C₂×C₈	1	4	yes	—
65	C₂×C₄×D₇₃	M₁₆	1	4	yes	—
66	C₂×C₄×D₇₃	SD₁₆	1	8	yes	—
67	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
68	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
69	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
70	C₂×#10596831	—	2	2	no	—
71	C₂×#10596831	C₂×C₈	1	4	yes	complemented
72	C₂×#10596831	C₂²×C₄	1	2	yes	—
73	C₂×#10596831	C₂²×C₄	1	2	yes	—
74	C₈×D₇₃	—	2	8	no	—
75	C₈×D₇₃	C₂×C₈	1	16	yes	complemented
76	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
77	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
79	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
81	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
82	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
83	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈×C₇₃	—	2	2	no	—
88	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	complemented
89	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
91	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
92	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
93	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
94	C₂²×D₇₃	C₄⋊C₈	1	4	yes	—
95	C₂²×D₇₃	#106	1	4	yes	—
96	C₂²×D₇₃	#108	1	4	yes	—
97	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
98	C₂×C₄×C₇₃	—	2	2	no	—
99	C₂×C₄×C₇₃	#106	1	4	yes	—
100	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
103	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₇₃	—	2	2	no	—
107	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
108	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
109	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×Dic₇₃	C₂×SD₁₆	1	4	yes	—
112	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
114	C₄×D₇₃	—	2	4	no	—
115	C₄×D₇₃	—	2	4	no	—
116	C₄×D₇₃	C₄⋊C₈	1	8	yes	—
117	C₄×D₇₃	#106	1	8	yes	—
118	C₄×D₇₃	C₂²×C₈	1	4	yes	—
119	C₄×D₇₃	C₂×M₁₆	1	4	yes	—
120	C₄×D₇₃	C₂×SD₁₆	1	8	yes	—
121	#10596833	—	4	8	no	—
122	C₈×C₇₃	—	2	2	no	—
123	C₈×C₇₃	C₂²×C₈	1	4	yes	complemented
124	#10596831	—	2	2	no	—
125	#10596831	C₂²×C₈	1	4	yes	complemented
126	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
127	C₂²×C₇₃	C₂×#106	1	2	yes	—
128	C₂²×C₇₃	C₂×#108	1	2	yes	—
129	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
130	C₂×D₇₃	#334	1	16	yes	complemented
131	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
132	C₂×D₇₃	C₂×#106	1	4	yes	—
133	C₂×D₇₃	C₂×#108	1	4	yes	—
134	C₄×C₇₃	—	2	2	no	—
135	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
136	C₄×C₇₃	C₂×#106	1	4	yes	—
137	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
138	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
139	Dic₇₃	—	2	1	no	—
140	Dic₇₃	—	2	1	no	—
141	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
142	Dic₇₃	C₂×#106	1	2	yes	—
143	Dic₇₃	C₂²×SD₁₆	1	2	yes	—
144	C₂²×#334	—	73	1	no	maximal, Sylow
145	C₂×C₇₃	C₂×#334	1	4	yes	complemented
146	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
147	C₂×C₇₃	C₂²×#106	1	1	yes	characteristic
148	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
149	D₇₃	C₂×#334	1	8	yes	complemented
150	C₂²×C₄⋊C₈	—	73	2	no	—
151	C₂²×C₄×C₈	—	73	1	no	—
152	C₂×#334	—	73	2	no	—
153	C₂×#334	—	73	8	no	—
154	C₂×#334	—	73	2	no	—
155	C₇₃	C₂²×#334	1	1	yes	characteristic, complemented, Sylow
156	C₂³×C₈	—	73	1	no	—
157	C₂³×C₈	—	73	1	no	—
158	C₂³×C₈	—	146	2	no	—
159	C₂²×C₄²	—	73	1	no	—
160	C₂×C₄⋊C₈	—	73	4	no	—
161	C₂×C₄⋊C₈	—	73	8	no	—
162	C₂×C₄⋊C₈	—	146	4	no	—
163	C₂×C₄⋊C₈	—	146	2	no	—
164	C₂×C₄×C₈	—	73	1	no	—
165	C₂×C₄×C₈	—	73	1	no	—
166	C₂×C₄×C₈	—	73	1	no	—
167	C₂×C₄×C₈	—	73	4	no	—
168	C₂×C₄×C₈	—	73	4	no	—
169	C₂×C₄×C₈	#64408	1	1	yes	characteristic
170	#334	—	73	8	no	—
171	#334	—	73	8	no	—
172	C₂³×C₄	—	73	1	no	—
173	C₂³×C₄	—	73	1	no	—
174	C₂³×C₄	—	73	1	no	—
175	C₂²×C₈	—	292	8	no	—
176	C₂²×C₈	—	73	1	no	—
177	C₂²×C₈	—	73	8	no	—
178	C₂²×C₈	—	73	4	no	—
179	C₂²×C₈	—	73	1	no	—
180	C₂²×C₈	—	73	4	no	—
181	C₂²×C₈	—	146	4	no	—
182	C₂²×C₈	—	146	4	no	—
183	C₂²×C₈	—	146	8	no	—
184	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
185	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
186	C₂×C₄²	—	73	1	no	—
187	C₂×C₄²	—	73	1	no	—
188	C₂×C₄²	—	73	1	no	—
189	C₂×C₄²	—	73	2	no	—
190	C₂×C₄²	—	73	2	no	—
191	C₂×C₄²	—	73	2	no	—
192	C₂×C₄²	—	73	2	no	—
193	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
194	C₄⋊C₈	—	73	8	no	—
195	C₄⋊C₈	—	146	4	no	—
196	C₄⋊C₈	—	146	8	no	—
197	C₄×C₈	—	73	2	no	—
198	C₄×C₈	—	73	2	no	—
199	C₄×C₈	—	73	2	no	—
200	C₄×C₈	—	73	2	no	—
201	C₄×C₈	—	73	4	no	—
202	C₄×C₈	C₂×#64408	1	4	yes	—
203	C₂⁴	—	73	1	no	—
204	C₂²×C₄	—	73	1	no	—
205	C₂²×C₄	—	73	1	no	—
206	C₂²×C₄	—	73	4	no	—
207	C₂²×C₄	—	73	4	no	—
208	C₂²×C₄	—	73	2	no	—
209	C₂²×C₄	—	73	8	no	—
210	C₂²×C₄	—	73	2	no	—
211	C₂²×C₄	—	73	2	no	—
212	C₂²×C₄	—	73	2	no	—
213	C₂²×C₄	—	73	1	no	—
214	C₂²×C₄	—	73	2	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	146	4	no	—
217	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
218	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
219	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	292	8	no	—
222	C₂×C₈	—	292	16	no	—
223	C₂×C₈	—	73	4	no	—
224	C₂×C₈	—	73	2	no	—
225	C₂×C₈	—	73	16	no	—
226	C₂×C₈	—	73	2	no	—
227	C₂×C₈	—	146	8	no	—
228	C₂×C₈	—	146	8	no	—
229	C₂×C₈	—	146	2	no	—
230	C₂×C₈	#11924036	1	4	yes	complemented
231	C₂×C₈	C₂²×#64408	1	2	yes	—
232	C₂×C₈	C₂²×#64408	1	2	yes	—
233	C₄²	—	73	1	no	—
234	C₄²	—	73	1	no	—
235	C₄²	—	73	2	no	—
236	C₄²	—	73	2	no	—
237	C₄²	—	73	2	no	—
238	C₄²	—	73	1	no	—
239	C₄²	—	73	1	no	—
240	C₄²	—	73	2	no	—
241	C₄²	C₄⋊#64408	1	2	yes	—
242	C₄²	C₂²×#64408	1	2	yes	—
243	C₂³	—	73	2	no	—
244	C₂³	—	73	4	no	—
245	C₂³	—	73	4	no	—
246	C₂³	—	73	4	no	—
247	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	73	1	no	—
250	C₂×C₄	—	73	2	no	—
251	C₂×C₄	—	73	1	no	—
252	C₂×C₄	—	73	4	no	—
253	C₂×C₄	—	73	4	no	—
254	C₂×C₄	—	73	4	no	—
255	C₂×C₄	—	73	4	no	—
256	C₂×C₄	—	73	8	no	—
257	C₂×C₄	—	73	8	no	—
258	C₂×C₄	—	73	8	no	—
259	C₂×C₄	—	73	1	no	—
260	C₂×C₄	—	73	1	no	—
261	C₂×C₄	—	146	4	no	—
262	C₂×C₄	—	146	4	no	—
263	C₂×C₄	—	146	2	no	—
264	C₂×C₄	#14556659	1	4	yes	—
265	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
266	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
267	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
268	C₂×C₄	C₂×#11924036	1	2	yes	—
269	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
270	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	—	292	8	no	—
273	C₈	—	73	4	no	—
274	C₈	—	146	2	no	—
275	C₈	C₂×#11924036	1	4	yes	complemented
276	C₂²	—	73	4	no	—
277	C₂²	—	73	4	no	—
278	C₂²	—	73	4	no	—
279	C₂²	—	73	16	no	—
280	C₂²	#17989588	1	2	yes	—
281	C₂²	C₂×#14556659	1	2	yes	—
282	C₂²	D₇₃⋊#108	1	2	yes	—
283	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	73	4	no	—
286	C₄	—	73	2	no	—
287	C₄	—	73	2	no	—
288	C₄	—	146	1	no	—
289	C₄	—	146	1	no	—
290	C₄	#17989588	1	2	yes	—
291	C₄	C₂×#14556659	1	4	yes	—
292	C₄	C₂²×#11924036	1	1	yes	characteristic
293	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
294	C₂	—	73	8	no	—
295	C₂	#22625473	1	4	yes	direct product factor
296	C₂	C₂×#17989588	1	1	yes	characteristic
297	C₂	C₂²×#14556659	1	1	yes	characteristic
298	C₂	C₂×D₇₃⋊#108	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₄⋊#64408
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#64408
Fitting	C₂×C₄×C₈×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂³×D₇₃	C₄⋊C₄
cosocle	C₄×Dic₇₃	C₂⁴
hypercenter	C₂×C₄×C₈	#64408
hypocenter	C₇₃	C₂²×#334
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#17989449	C₂	semidirect
3. #22625473	C₂	direct (N × H)
4. C₂×#17989448	C₂	semidirect
11. #17989449	C₂²	semidirect
12. #17989448	C₂²	semidirect
34. C₂×C₈×D₇₃	C₈	semidirect
71. C₂×#10596831	C₂×C₈	semidirect
75. C₈×D₇₃	C₂×C₈	semidirect
88. C₂×C₈×C₇₃	C₂×C₈	semidirect
123. C₈×C₇₃	C₂²×C₈	semidirect
125. #10596831	C₂²×C₈	semidirect
130. C₂×D₇₃	#334	semidirect
145. C₂×C₇₃	C₂×#334	semidirect
149. D₇₃	C₂×#334	semidirect
155. C₇₃	C₂²×#334	semidirect
230. C₂×C₈	#11924036	semidirect
275. C₈	C₂×#11924036	semidirect
295. C₂	#22625473	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#17989588	2
C₂×#17989449	2
#22625473	2
C₂×#17989448	2
C₂×C₄×C₈×D₇₃	2
C₂²×#334	73

Subgroups — C2×#22625473 #34017965

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625473 #34017965