Subgroups — C₂×#22625476 #34017968

Order 18688.

942 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#17989451	C₂	1	2	yes	maximal, complemented
3	#22625476	C₂	1	8	yes	maximal, direct product factor
4	C₂×#17989450	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	#17989451	C₂²	1	8	yes	complemented
12	#17989450	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	D₈	1	4	yes	—
67	C₂×C₄×D₇₃	Q₁₆	1	4	yes	—
68	C₂×C₄×D₇₃	C₂²×C₄	1	2	yes	—
69	C₂×C₄×D₇₃	C₂×D₄	1	2	yes	—
70	C₂×C₄×D₇₃	C₂×Q₈	1	2	yes	—
71	C₂×#10596831	—	2	2	no	—
72	C₂×#10596831	C₂×C₈	1	4	yes	complemented
73	C₂×#10596831	C₂²×C₄	1	2	yes	—
74	C₂×#10596831	C₂²×C₄	1	2	yes	—
75	C₈×D₇₃	—	2	8	no	—
76	C₈×D₇₃	C₂×C₈	1	16	yes	complemented
77	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
79	C₄×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
80	C₄×Dic₇₃	C₂×D₄	1	1	yes	characteristic
81	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
82	C₄×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
83	C₄×Dic₇₃	C₂×Q₈	1	2	yes	—
84	C₄×Dic₇₃	C₂⁴	1	1	yes	characteristic
85	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈×C₇₃	—	2	2	no	—
89	C₂×C₈×C₇₃	C₂×C₈	1	4	yes	complemented
90	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₇₃	C₂²×C₄	1	2	yes	—
92	C₄²×C₇₃	C₄⋊C₄	1	2	yes	—
93	C₄²×C₇₃	C₂²×C₄	1	2	yes	—
94	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂²×D₇₃	C₄⋊C₈	1	4	yes	—
96	C₂²×D₇₃	#107	1	4	yes	—
97	C₂²×D₇₃	#108	1	4	yes	—
98	C₂²×D₇₃	C₂×C₄⋊C₄	1	2	yes	—
99	C₂×C₄×C₇₃	—	2	2	no	—
100	C₂×C₄×C₇₃	#107	1	4	yes	—
101	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
102	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₇₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₇₃	C₂×M₁₆	1	2	yes	—
106	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
107	C₂×Dic₇₃	—	2	2	no	—
108	C₂×Dic₇₃	C₄⋊C₈	1	4	yes	—
109	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×Dic₇₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×Dic₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×Dic₇₃	C₂×D₈	1	2	yes	—
113	C₂×Dic₇₃	C₂×Q₁₆	1	2	yes	—
114	C₂×Dic₇₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
116	C₄×D₇₃	—	2	4	no	—
117	C₄×D₇₃	—	2	4	no	—
118	C₄×D₇₃	C₄⋊C₈	1	8	yes	—
119	C₄×D₇₃	#107	1	8	yes	—
120	C₄×D₇₃	C₂²×C₈	1	4	yes	—
121	C₄×D₇₃	C₂×M₁₆	1	4	yes	—
122	C₄×D₇₃	C₂×D₈	1	4	yes	—
123	C₄×D₇₃	C₂×Q₁₆	1	4	yes	—
124	#10596833	—	4	8	no	—
125	C₈×C₇₃	—	2	2	no	—
126	C₈×C₇₃	C₂²×C₈	1	4	yes	complemented
127	#10596831	—	2	2	no	—
128	#10596831	C₂²×C₈	1	4	yes	complemented
129	C₂²×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
130	C₂²×C₇₃	C₂×#107	1	2	yes	—
131	C₂²×C₇₃	C₂×#108	1	2	yes	—
132	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
133	C₂×D₇₃	#335	1	16	yes	complemented
134	C₂×D₇₃	C₂×C₄⋊C₈	1	4	yes	—
135	C₂×D₇₃	C₂×#107	1	4	yes	—
136	C₂×D₇₃	C₂×#108	1	4	yes	—
137	C₄×C₇₃	—	2	2	no	—
138	C₄×C₇₃	C₂×C₄⋊C₈	1	2	yes	—
139	C₄×C₇₃	C₂×#107	1	4	yes	—
140	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
141	C₄×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
142	Dic₇₃	—	2	1	no	—
143	Dic₇₃	—	2	1	no	—
144	Dic₇₃	C₂×C₄⋊C₈	1	4	yes	—
145	Dic₇₃	C₂×#107	1	2	yes	—
146	Dic₇₃	C₂²×D₈	1	1	yes	characteristic
147	Dic₇₃	C₂²×Q₁₆	1	1	yes	characteristic
148	C₂²×#335	—	73	1	no	maximal, Sylow
149	C₂×C₇₃	C₂×#335	1	4	yes	complemented
150	C₂×C₇₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
151	C₂×C₇₃	C₂²×#107	1	1	yes	characteristic
152	C₂×C₇₃	C₂²×#108	1	1	yes	characteristic
153	D₇₃	C₂×#335	1	8	yes	complemented
154	C₂²×C₄⋊C₈	—	73	2	no	—
155	C₂²×C₄×C₈	—	73	1	no	—
156	C₂×#335	—	73	2	no	—
157	C₂×#335	—	73	8	no	—
158	C₂×#335	—	73	2	no	—
159	C₇₃	C₂²×#335	1	1	yes	characteristic, complemented, Sylow
160	C₂³×C₈	—	73	1	no	—
161	C₂³×C₈	—	73	1	no	—
162	C₂³×C₈	—	146	2	no	—
163	C₂²×C₄²	—	73	1	no	—
164	C₂×C₄⋊C₈	—	73	4	no	—
165	C₂×C₄⋊C₈	—	73	8	no	—
166	C₂×C₄⋊C₈	—	146	4	no	—
167	C₂×C₄⋊C₈	—	146	2	no	—
168	C₂×C₄×C₈	—	73	1	no	—
169	C₂×C₄×C₈	—	73	1	no	—
170	C₂×C₄×C₈	—	73	1	no	—
171	C₂×C₄×C₈	—	73	4	no	—
172	C₂×C₄×C₈	—	73	4	no	—
173	C₂×C₄×C₈	#64408	1	1	yes	characteristic
174	#335	—	73	8	no	—
175	#335	—	73	8	no	—
176	C₂³×C₄	—	73	1	no	—
177	C₂³×C₄	—	73	1	no	—
178	C₂³×C₄	—	73	1	no	—
179	C₂²×C₈	—	292	8	no	—
180	C₂²×C₈	—	73	1	no	—
181	C₂²×C₈	—	73	8	no	—
182	C₂²×C₈	—	73	4	no	—
183	C₂²×C₈	—	73	1	no	—
184	C₂²×C₈	—	73	4	no	—
185	C₂²×C₈	—	146	4	no	—
186	C₂²×C₈	—	146	4	no	—
187	C₂²×C₈	—	146	8	no	—
188	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
189	C₂²×C₈	C₂×#64408	1	1	yes	characteristic
190	C₂×C₄²	—	73	1	no	—
191	C₂×C₄²	—	73	1	no	—
192	C₂×C₄²	—	73	1	no	—
193	C₂×C₄²	—	73	2	no	—
194	C₂×C₄²	—	73	2	no	—
195	C₂×C₄²	—	73	2	no	—
196	C₂×C₄²	—	73	2	no	—
197	C₂×C₄²	C₂×#64408	1	1	yes	characteristic
198	C₄⋊C₈	—	73	8	no	—
199	C₄⋊C₈	—	146	4	no	—
200	C₄⋊C₈	—	146	8	no	—
201	C₄×C₈	—	73	2	no	—
202	C₄×C₈	—	73	2	no	—
203	C₄×C₈	—	73	2	no	—
204	C₄×C₈	—	73	2	no	—
205	C₄×C₈	—	73	4	no	—
206	C₄×C₈	C₂×#64408	1	4	yes	—
207	C₂⁴	—	73	1	no	—
208	C₂²×C₄	—	73	1	no	—
209	C₂²×C₄	—	73	1	no	—
210	C₂²×C₄	—	73	4	no	—
211	C₂²×C₄	—	73	4	no	—
212	C₂²×C₄	—	73	2	no	—
213	C₂²×C₄	—	73	4	no	—
214	C₂²×C₄	—	73	4	no	—
215	C₂²×C₄	—	73	2	no	—
216	C₂²×C₄	—	73	2	no	—
217	C₂²×C₄	—	73	2	no	—
218	C₂²×C₄	—	73	1	no	—
219	C₂²×C₄	—	73	2	no	—
220	C₂²×C₄	—	73	2	no	—
221	C₂²×C₄	—	146	4	no	—
222	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
223	C₂²×C₄	C₄⋊#64408	1	1	yes	characteristic
224	C₂²×C₄	C₂²×#64408	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	292	8	no	—
227	C₂×C₈	—	292	16	no	—
228	C₂×C₈	—	73	4	no	—
229	C₂×C₈	—	73	2	no	—
230	C₂×C₈	—	73	16	no	—
231	C₂×C₈	—	73	2	no	—
232	C₂×C₈	—	146	8	no	—
233	C₂×C₈	—	146	8	no	—
234	C₂×C₈	—	146	2	no	—
235	C₂×C₈	#11924036	1	4	yes	complemented
236	C₂×C₈	C₂²×#64408	1	2	yes	—
237	C₂×C₈	C₂²×#64408	1	2	yes	—
238	C₄²	—	73	1	no	—
239	C₄²	—	73	1	no	—
240	C₄²	—	73	2	no	—
241	C₄²	—	73	2	no	—
242	C₄²	—	73	2	no	—
243	C₄²	—	73	1	no	—
244	C₄²	—	73	1	no	—
245	C₄²	—	73	2	no	—
246	C₄²	C₄⋊#64408	1	2	yes	—
247	C₄²	C₂²×#64408	1	2	yes	—
248	C₂³	—	73	2	no	—
249	C₂³	—	73	4	no	—
250	C₂³	—	73	4	no	—
251	C₂³	—	73	4	no	—
252	C₂³	C₂×C₄⋊#64408	1	1	yes	characteristic
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	73	1	no	—
255	C₂×C₄	—	73	2	no	—
256	C₂×C₄	—	73	1	no	—
257	C₂×C₄	—	73	4	no	—
258	C₂×C₄	—	73	2	no	—
259	C₂×C₄	—	73	2	no	—
260	C₂×C₄	—	73	4	no	—
261	C₂×C₄	—	73	4	no	—
262	C₂×C₄	—	73	8	no	—
263	C₂×C₄	—	73	4	no	—
264	C₂×C₄	—	73	8	no	—
265	C₂×C₄	—	73	4	no	—
266	C₂×C₄	—	73	1	no	—
267	C₂×C₄	—	73	1	no	—
268	C₂×C₄	—	146	4	no	—
269	C₂×C₄	—	146	4	no	—
270	C₂×C₄	—	146	2	no	—
271	C₂×C₄	#14556660	1	4	yes	—
272	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
273	C₂×C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
274	C₂×C₄	C₂×C₄⋊#64408	1	2	yes	—
275	C₂×C₄	C₂×#11924036	1	2	yes	—
276	C₂×C₄	D₇₃⋊M₁₆	1	2	yes	—
277	C₂×C₄	C₂³×#64408	1	1	yes	characteristic
278	C₈	—	2	2	no	—
279	C₈	—	292	8	no	—
280	C₈	—	73	4	no	—
281	C₈	—	146	2	no	—
282	C₈	C₂×#11924036	1	4	yes	complemented
283	C₂²	—	73	4	no	—
284	C₂²	—	73	4	no	—
285	C₂²	—	73	4	no	—
286	C₂²	—	73	16	no	—
287	C₂²	#17989588	1	2	yes	—
288	C₂²	C₂×#14556660	1	2	yes	—
289	C₂²	D₇₃⋊#108	1	2	yes	—
290	C₂²	C₂²×C₄⋊#64408	1	1	yes	characteristic
291	C₄	—	2	2	no	—
292	C₄	—	73	4	no	—
293	C₄	—	73	1	no	—
294	C₄	—	73	1	no	—
295	C₄	—	73	2	no	—
296	C₄	—	146	1	no	—
297	C₄	—	146	1	no	—
298	C₄	#17989588	1	2	yes	—
299	C₄	C₂×#14556660	1	4	yes	—
300	C₄	C₂²×#11924036	1	1	yes	characteristic
301	C₄	C₂×D₇₃⋊M₁₆	1	1	yes	characteristic
302	C₂	—	73	8	no	—
303	C₂	#22625476	1	4	yes	direct product factor
304	C₂	C₂×#17989588	1	1	yes	characteristic
305	C₂	C₂²×#14556660	1	1	yes	characteristic
306	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×#335
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×#17989451	C₂	semidirect
3. #22625476	C₂	direct (N × H)
4. C₂×#17989450	C₂	semidirect
11. #17989451	C₂²	semidirect
12. #17989450	C₂²	semidirect
34. C₂×C₈×D₇₃	C₈	semidirect
72. C₂×#10596831	C₂×C₈	semidirect
76. C₈×D₇₃	C₂×C₈	semidirect
89. C₂×C₈×C₇₃	C₂×C₈	semidirect
126. C₈×C₇₃	C₂²×C₈	semidirect
128. #10596831	C₂²×C₈	semidirect
133. C₂×D₇₃	#335	semidirect
149. C₂×C₇₃	C₂×#335	semidirect
153. D₇₃	C₂×#335	semidirect
159. C₇₃	C₂²×#335	semidirect
235. C₂×C₈	#11924036	semidirect
282. C₈	C₂×#11924036	semidirect
303. C₂	#22625476	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×#17989451	2
#22625476	2
C₂×#17989450	2
C₂×C₄×C₈×D₇₃	2
C₂²×#335	73

Subgroups — C2×#22625476 #34017968

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22625476 #34017968