Subgroups — C₂×#22610101 #33086438

Order 18688.

1046 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14556527	C₂	1	1	yes	maximal, characteristic
2	C₂²×#14556526	C₂	1	2	yes	maximal
3	C₂×#17988610	C₂	1	2	yes	maximal
4	C₂²×C₈×Dic₇₃	C₂	1	1	yes	maximal, characteristic
5	#22610101	C₂	1	8	yes	maximal, direct product factor
6	C₂²×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11924012	C₂²	1	2	yes	—
8	C₂³×#10596831	C₂²	1	1	yes	characteristic
9	C₂²×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
10	C₂×#14556527	—	2	4	no	—
11	C₂×#14556527	C₂²	1	4	yes	—
12	C₂×#14556526	—	2	8	no	—
13	C₂×#14556526	C₂²	1	8	yes	—
14	#17988610	—	2	4	no	—
15	#17988610	C₂²	1	8	yes	—
16	C₂×#14556520	—	2	4	no	—
17	C₂×C₈×Dic₇₃	C₄	1	4	yes	—
18	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
19	C₂×C₈×Dic₇₃	C₄	1	2	yes	—
20	C₂×C₈×Dic₇₃	C₂²	1	4	yes	—
21	C₂²×M₁₆×C₇₃	C₂²	1	2	yes	—
22	C₂³×C₈×C₇₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₇₃	—	2	4	no	—
24	C₂×#108×C₇₃	C₂²	1	4	yes	—
25	C₂³×Dic₇₃	Q₈	1	2	yes	—
26	C₂×#11924012	—	2	2	no	—
27	C₂×#11924012	—	2	4	no	—
28	C₂×#11924012	—	2	4	no	—
29	C₂×#11924012	C₂³	1	4	yes	—
30	C₂²×#10596831	—	2	2	no	—
31	C₂²×#10596831	C₂×C₄	1	2	yes	—
32	C₂²×#10596831	C₂×C₄	1	2	yes	—
33	C₂²×#10596831	D₄	1	4	yes	—
34	C₂²×#10596831	Q₈	1	4	yes	—
35	C₂²×#10596831	C₂³	1	2	yes	—
36	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
37	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
40	C₂×C₄×Dic₇₃	D₄	1	2	yes	—
41	C₂×C₄×Dic₇₃	Q₈	1	2	yes	—
42	C₂×C₄×Dic₇₃	C₂³	1	2	yes	—
43	#14556527	—	2	8	no	—
44	#14556526	—	2	16	no	—
45	#14556520	—	2	16	no	—
46	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
47	C₈×Dic₇₃	C₂×C₄	1	8	yes	—
48	C₂³×C₄×C₇₃	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₇₃	—	2	2	no	—
50	C₂×M₁₆×C₇₃	—	2	4	no	—
51	C₂×M₁₆×C₇₃	—	2	4	no	—
52	C₂×M₁₆×C₇₃	C₂³	1	4	yes	—
53	C₂²×C₈×C₇₃	—	2	2	no	—
54	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₇₃	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₇₃	D₄	1	4	yes	—
57	C₂²×C₈×C₇₃	Q₈	1	4	yes	—
58	C₂²×C₈×C₇₃	C₂³	1	2	yes	—
59	#108×C₇₃	—	2	8	no	—
60	C₂²×Dic₇₃	—	2	8	no	—
61	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
62	C₂²×Dic₇₃	C₄⋊C₄	1	4	yes	—
63	C₂²×Dic₇₃	C₄⋊C₄	1	2	yes	—
64	C₂²×Dic₇₃	C₂×Q₈	1	4	yes	—
65	#11924012	—	2	4	no	—
66	#11924012	—	2	4	no	—
67	#11924012	—	4	4	no	—
68	C₂×#10596831	—	2	4	no	—
69	C₂×#10596831	—	4	4	no	—
70	C₂×#10596831	C₄⋊C₄	1	8	yes	—
71	C₂×#10596831	C₄⋊C₄	1	8	yes	—
72	C₂×#10596831	C₂²×C₄	1	4	yes	—
73	C₂×#10596831	C₂×D₄	1	4	yes	—
74	C₂×#10596831	C₂×Q₈	1	4	yes	—
75	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
76	C₄×Dic₇₃	C₄⋊C₄	1	4	yes	—
77	C₄×Dic₇₃	C₄⋊C₄	1	2	yes	—
78	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
79	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
80	C₄×Dic₇₃	C₂×D₄	1	4	yes	—
81	C₂⁴×C₇₃	C₂×Q₈	1	1	yes	characteristic
82	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
85	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₇₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
89	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
90	C₂²×C₄×C₇₃	C₂×Q₈	1	2	yes	—
91	C₂²×C₄×C₇₃	C₂⁴	1	1	yes	characteristic
92	M₁₆×C₇₃	—	2	4	no	—
93	M₁₆×C₇₃	—	2	4	no	—
94	M₁₆×C₇₃	—	4	4	no	—
95	C₂×C₈×C₇₃	—	2	4	no	—
96	C₂×C₈×C₇₃	—	4	4	no	—
97	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
98	C₂×C₈×C₇₃	C₄⋊C₄	1	8	yes	—
99	C₂×C₈×C₇₃	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₇₃	C₂×D₄	1	4	yes	—
101	C₂×C₈×C₇₃	C₂×Q₈	1	4	yes	—
102	C₂³×C₇₃	—	2	4	no	—
103	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
104	C₂³×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
106	C₂³×C₇₃	C₄⋊Q₈	1	2	yes	—
107	C₂³×C₇₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
112	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
113	C₂×C₄×C₇₃	C₂×C₄⋊C₄	1	2	yes	—
114	C₂×C₄×C₇₃	#127	1	2	yes	—
115	C₂×C₄×C₇₃	#127	1	2	yes	—
116	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
117	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
118	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
119	C₂×C₄×C₇₃	C₄⋊Q₈	1	2	yes	—
120	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₇₃	C₂²×Q₈	1	1	yes	characteristic
126	C₂×Dic₇₃	—	2	4	no	—
127	C₂×Dic₇₃	—	2	8	no	—
128	C₂×Dic₇₃	—	2	8	no	—
129	C₂×Dic₇₃	#108	1	8	yes	complemented
130	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×Dic₇₃	C₂×C₄⋊C₄	1	4	yes	—
132	C₈×C₇₃	—	4	4	no	—
133	C₈×C₇₃	C₂×C₄⋊C₄	1	8	yes	—
134	#10596831	—	4	4	no	—
135	#10596831	C₂×C₄⋊C₄	1	8	yes	—
136	C₂²×C₇₃	—	2	4	no	—
137	C₂²×C₇₃	—	2	4	no	—
138	C₂²×C₇₃	—	2	2	no	—
139	C₂²×C₇₃	—	2	2	no	—
140	C₂²×C₇₃	#384	1	2	yes	—
141	C₂²×C₇₃	#384	1	2	yes	—
142	C₂²×C₇₃	C₂×#108	1	4	yes	—
143	C₂²×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
144	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
145	C₂²×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
146	C₄×C₇₃	#384	1	2	yes	—
147	C₄×C₇₃	#384	1	2	yes	—
148	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₇₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
150	C₄×C₇₃	C₂×#127	1	1	yes	characteristic
151	C₄×C₇₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
152	Dic₇₃	—	2	4	no	—
153	Dic₇₃	C₂×#108	1	8	yes	complemented
154	C₂×#1848	—	73	1	no	maximal, Sylow
155	C₂×C₇₃	—	2	1	no	—
156	C₂×C₇₃	—	2	1	no	—
157	C₂×C₇₃	—	2	2	no	—
158	C₂×C₇₃	#1848	1	4	yes	complemented
159	C₂×C₇₃	C₂×#384	1	1	yes	characteristic
160	C₂×C₇₃	C₂²×#108	1	2	yes	—
161	C₂²×#108	—	73	2	no	—
162	C₂²×#108	—	73	1	no	—
163	C₂²×#108	D₇₃	1	1	yes	characteristic
164	C₂×#423	—	73	2	no	—
165	C₂²×C₄×C₈	—	73	1	no	—
166	#1848	—	73	8	no	—
167	C₇₃	C₂×#1848	1	1	yes	characteristic, complemented, Sylow
168	C₂²×M₁₆	—	73	2	no	—
169	C₂²×M₁₆	C₂×D₇₃	1	2	yes	—
170	C₂³×C₈	—	73	1	no	—
171	C₂³×C₈	C₂×D₇₃	1	1	yes	characteristic
172	C₂²×C₄²	—	73	1	no	—
173	C₂×#108	—	2	4	no	—
174	C₂×#108	—	73	8	no	—
175	C₂×#108	—	73	4	no	—
176	C₂×#108	—	146	8	no	—
177	C₂×#108	—	146	4	no	—
178	C₂×#108	C₂×D₇₃	1	4	yes	—
179	#423	—	73	8	no	—
180	#423	—	146	4	no	—
181	C₂×C₄⋊C₈	—	146	4	no	—
182	C₂×C₄×C₈	—	73	2	no	—
183	C₂×C₄×C₈	—	73	4	no	—
184	C₂×C₄×C₈	—	73	2	no	—
185	C₂×C₄×C₈	—	73	4	no	—
186	C₂³×C₄	—	73	2	no	—
187	C₂³×C₄	C₂²×D₇₃	1	1	yes	characteristic
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	2	4	no	—
191	C₂×M₁₆	—	73	4	no	—
192	C₂×M₁₆	—	146	4	no	—
193	C₂×M₁₆	—	146	4	no	—
194	C₂×M₁₆	—	146	2	no	—
195	C₂×M₁₆	C₂²×D₇₃	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	73	2	no	—
198	C₂²×C₈	—	73	2	no	—
199	C₂²×C₈	—	73	4	no	—
200	C₂²×C₈	—	73	4	no	—
201	C₂²×C₈	—	73	2	no	—
202	C₂²×C₈	—	146	2	no	—
203	C₂²×C₈	Dic₁₄₆	1	4	yes	—
204	C₂²×C₈	C₄×D₇₃	1	2	yes	—
205	C₂²×C₈	C₄×D₇₃	1	2	yes	—
206	C₂²×C₈	C₂²⋊D₇₃	1	4	yes	—
207	C₂²×C₈	C₂²×D₇₃	1	2	yes	—
208	C₂×C₄²	—	73	1	no	—
209	C₂×C₄²	—	73	2	no	—
210	C₂×C₄²	—	73	1	no	—
211	C₂×C₄²	—	73	2	no	—
212	C₂×C₄²	—	73	2	no	—
213	C₂×C₄²	—	73	2	no	—
214	C₂×C₄²	—	73	2	no	—
215	#108	—	2	8	no	—
216	#108	—	146	16	no	—
217	#108	—	146	8	no	—
218	C₄⋊C₈	—	146	16	no	—
219	C₄×C₈	—	73	8	no	—
220	C₄×C₈	—	73	8	no	—
221	C₂⁴	Q₈×D₇₃	1	1	yes	characteristic
222	C₂²×C₄	—	73	2	no	—
223	C₂²×C₄	—	73	4	no	—
224	C₂²×C₄	—	73	2	no	—
225	C₂²×C₄	—	73	4	no	—
226	C₂²×C₄	—	146	8	no	—
227	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
228	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₇₃	1	2	yes	—
230	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₂²⋊D₇₃	1	2	yes	—
233	C₂²×C₄	Q₈×D₇₃	1	1	yes	characteristic
234	C₂²×C₄	Q₈×D₇₃	1	2	yes	—
235	C₂²×C₄	C₂×Dic₁₄₆	1	2	yes	—
236	C₂²×C₄	C₂³×D₇₃	1	1	yes	characteristic
237	M₁₆	—	2	4	no	—
238	M₁₆	—	2	4	no	—
239	M₁₆	—	4	4	no	—
240	M₁₆	—	292	4	no	—
241	M₁₆	—	146	4	no	—
242	M₁₆	—	146	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	4	4	no	—
245	C₂×C₈	—	292	4	no	—
246	C₂×C₈	—	73	8	no	—
247	C₂×C₈	—	73	8	no	—
248	C₂×C₈	—	73	4	no	—
249	C₂×C₈	—	73	4	no	—
250	C₂×C₈	—	73	4	no	—
251	C₂×C₈	—	146	4	no	—
252	C₂×C₈	#11924007	1	8	yes	—
253	C₂×C₈	#11924007	1	8	yes	—
254	C₂×C₈	C₂×C₄×D₇₃	1	4	yes	—
255	C₂×C₈	C₂×C₂²⋊D₇₃	1	4	yes	—
256	C₂×C₈	C₂×Dic₁₄₆	1	4	yes	—
257	C₄²	—	73	2	no	—
258	C₄²	—	73	4	no	—
259	C₄²	—	73	2	no	—
260	C₄²	—	73	2	no	—
261	C₄²	—	73	2	no	—
262	C₄²	—	73	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
265	C₂³	C₄⋊C₄×D₇₃	1	1	yes	characteristic
266	C₂³	#14556589	1	2	yes	—
267	C₂³	#14556590	1	2	yes	—
268	C₂³	C₂×Q₈×D₇₃	1	1	yes	characteristic
269	C₂×C₄	—	73	8	no	—
270	C₂×C₄	—	73	4	no	—
271	C₂×C₄	—	73	4	no	—
272	C₂×C₄	—	146	8	no	—
273	C₂×C₄	—	146	8	no	—
274	C₂×C₄	—	146	4	no	—
275	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
276	C₂×C₄	C₄⋊C₄×D₇₃	1	1	yes	characteristic
277	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
278	C₂×C₄	C₄⋊C₄×D₇₃	1	2	yes	—
279	C₂×C₄	C₂×#11924007	1	2	yes	—
280	C₂×C₄	C₂×#11924007	1	2	yes	—
281	C₂×C₄	#14556587	1	2	yes	—
282	C₂×C₄	#14556587	1	2	yes	—
283	C₂×C₄	#14556589	1	2	yes	—
284	C₂×C₄	#14556590	1	2	yes	—
285	C₂×C₄	#14556590	1	2	yes	—
286	C₂×C₄	#14556590	1	2	yes	—
287	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
288	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₂²⋊D₇₃	1	1	yes	characteristic
291	C₂×C₄	C₂×Q₈×D₇₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×Dic₁₄₆	1	1	yes	characteristic
293	C₈	—	4	4	no	—
294	C₈	—	292	4	no	—
295	C₈	—	73	8	no	—
296	C₈	C₂×#11924007	1	8	yes	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	4	no	—
301	C₂²	#17988456	1	2	yes	—
302	C₂²	#17988456	1	2	yes	—
303	C₂²	#108×D₇₃	1	4	yes	—
304	C₂²	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
305	C₂²	C₂×#14556589	1	1	yes	characteristic
306	C₂²	C₂×#14556590	1	1	yes	characteristic
307	C₄	—	73	8	no	—
308	C₄	—	146	4	no	—
309	C₄	#17988456	1	2	yes	—
310	C₄	#17988456	1	2	yes	—
311	C₄	C₂×C₄⋊C₄×D₇₃	1	1	yes	characteristic
312	C₄	C₂²×#11924007	1	1	yes	characteristic
313	C₄	C₂×#14556587	1	1	yes	characteristic
314	C₄	C₂×#14556590	1	1	yes	characteristic
315	C₂	—	2	1	no	—
316	C₂	—	2	1	no	—
317	C₂	—	2	2	no	—
318	C₂	#22610101	1	4	yes	direct product factor
319	C₂	C₂×#17988456	1	1	yes	characteristic
320	C₂	C₂×#108×D₇₃	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₇₃
commutator	C₂×C₄×C₇₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₇₃
Fitting	C₂²×#108×C₇₃	C₂
socle	C₂³×C₇₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₇₃	C₂×Q₈
cosocle	C₂²×C₄×C₇₃	C₂⁴
hypercenter	C₂²×#108	D₇₃
hypocenter	C₇₃	C₂×#1848
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
5. #22610101	C₂	direct (N × H)
129. C₂×Dic₇₃	#108	semidirect
153. Dic₇₃	C₂×#108	semidirect
158. C₂×C₇₃	#1848	semidirect
167. C₇₃	C₂×#1848	semidirect
318. C₂	#22610101	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14556527	2
C₂²×#14556526	2
C₂×#17988610	2
C₂²×C₈×Dic₇₃	2
#22610101	2
C₂²×#108×C₇₃	2
C₂×#1848	73

Subgroups — C2×#22610101 #33086438

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22610101 #33086438