Subgroups — C₂×#22611765 #33239209

Order 18688.

662 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989005	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17989004	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988702	C₂	1	1	yes	maximal, characteristic
4	#22611765	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988407	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988370	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₇₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#350×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556611	C₄	1	1	yes	characteristic
10	C₂×#14556597	C₂²	1	1	yes	characteristic
11	#17989005	—	2	2	no	—
12	#17989005	C₂²	1	4	yes	complemented
13	#17989004	—	2	2	no	—
14	#17989004	C₂²	1	4	yes	complemented
15	C₂×#14556531	—	2	1	no	—
16	C₂×#14556531	C₂²	1	1	yes	characteristic
17	C₂×#14556530	—	2	1	no	—
18	C₂²×#11924003	—	2	1	no	—
19	C₂²×#11924003	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×D₇₃	—	2	1	no	—
21	#17988702	C₄	1	4	yes	—
22	#17988702	C₂²	1	4	yes	—
23	C₂×#14556526	C₄	1	1	yes	characteristic
24	C₂×#14556518	C₄	1	1	yes	characteristic
25	C₂×#14556517	C₄	1	1	yes	characteristic
26	C₂×C₈×Dic₇₃	C₂²	1	1	yes	characteristic
27	#17988407	C₂²	1	4	yes	complemented
28	#17988370	C₂²	1	4	yes	complemented
29	C₁₆×Dic₇₃	C₂²	1	4	yes	complemented
30	C₂×#131×C₇₃	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₇₃	C₂²	1	1	yes	characteristic
33	#350×C₇₃	C₂²	1	4	yes	complemented
34	C₂×#11924028	C₂×C₄	1	1	yes	characteristic
35	#14556611	C₈	1	4	yes	—
36	#14556611	C₂×C₄	1	4	yes	—
37	C₂×#11924018	C₈	1	1	yes	characteristic
38	C₂×#11924016	C₈	1	1	yes	characteristic
39	C₂×#11924015	C₈	1	1	yes	characteristic
40	C₂×#11924013	C₈	1	1	yes	characteristic
41	#14556597	—	2	1	no	—
42	#14556597	—	2	1	no	—
43	#14556597	C₂×C₄	1	2	yes	—
44	#14556597	C₂³	1	2	yes	—
45	C₂×#11924012	C₂×C₄	1	1	yes	characteristic
46	C₂×#11924011	—	2	1	no	—
47	C₂²×#10596831	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₇₃	—	2	1	no	—
49	C₂×C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
50	#14556531	—	2	4	no	—
51	#14556531	—	2	1	no	—
52	#14556531	C₂³	1	2	yes	—
53	#14556530	—	2	4	no	—
54	C₂×#11924003	—	2	4	no	—
55	C₂×#11924003	—	2	1	no	—
56	C₂×#11924003	D₄	1	1	yes	characteristic
57	C₂×#11924003	D₄	1	1	yes	characteristic
58	C₂×#11924003	D₄	1	2	yes	—
59	C₂×#11924003	C₂³	1	2	yes	—
60	C₁₆×D₇₃	—	2	4	no	—
61	#14556526	C₂×C₄	1	4	yes	—
62	#14556518	C₂×C₄	1	4	yes	—
63	#14556517	C₂×C₄	1	4	yes	—
64	C₈×Dic₇₃	C₂×C₄	1	2	yes	—
65	C₈×Dic₇₃	C₂³	1	2	yes	—
66	C₂×#41×C₇₃	C₂×C₄	1	1	yes	characteristic
67	#131×C₇₃	—	2	1	no	—
68	#131×C₇₃	—	2	1	no	—
69	#131×C₇₃	C₂×C₄	1	2	yes	—
70	#131×C₇₃	C₂³	1	2	yes	—
71	C₂×M₁₆×C₇₃	—	2	1	no	—
72	C₂×M₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
73	C₂²×C₈×C₇₃	—	2	1	no	—
74	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
75	M₃₂×C₇₃	—	2	1	no	—
76	M₃₂×C₇₃	C₂³	1	2	yes	—
77	C₂×C₁₆×C₇₃	—	2	1	no	—
78	C₂×C₁₆×C₇₃	—	2	1	no	—
79	C₂×C₁₆×C₇₃	—	2	1	no	—
80	C₂×C₁₆×C₇₃	C₂³	1	2	yes	—
81	#11924028	—	2	1	no	—
82	#11924028	—	2	1	no	—
83	#11924028	C₂×C₈	1	2	yes	—
84	#11924028	C₂²×C₄	1	2	yes	—
85	C₂×Dic₁₄₆	C₂×C₈	1	1	yes	characteristic
86	C₂×D₂₉₂	C₂×C₈	1	1	yes	characteristic
87	C₂×C₂²⋊D₇₃	—	2	1	no	—
88	C₂²×Dic₇₃	—	2	1	no	—
89	C₂×C₄×D₇₃	—	2	1	no	—
90	#11924018	C₂×C₈	1	4	yes	—
91	#11924016	C₂×C₈	1	4	yes	—
92	#11924015	C₂×C₈	1	4	yes	—
93	#11924013	C₂×C₈	1	4	yes	—
94	#11924012	—	2	1	no	—
95	#11924012	C₂²×C₄	1	2	yes	—
96	#11924011	—	2	2	no	—
97	#11924011	—	2	2	no	—
98	C₂×#10596831	—	2	1	no	—
99	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
100	C₂×#10596831	C₂×C₈	1	1	yes	characteristic
101	C₂×#10596831	C₂×C₈	1	2	yes	—
102	C₂×#10596831	C₂²×C₄	1	2	yes	—
103	C₈×D₇₃	—	2	2	no	—
104	C₈×D₇₃	—	2	2	no	—
105	C₄×Dic₇₃	—	2	1	no	—
106	C₄×Dic₇₃	C₂²×C₄	1	2	yes	—
107	#11924003	—	2	2	no	—
108	#11924003	C₂×D₄	1	2	yes	—
109	#11924003	C₂×D₄	1	2	yes	—
110	#41×C₇₃	—	2	1	no	—
111	#41×C₇₃	—	2	1	no	—
112	#41×C₇₃	C₂×C₈	1	2	yes	—
113	#41×C₇₃	C₂²×C₄	1	2	yes	—
114	C₂×Q₈×C₇₃	C₂×C₈	1	1	yes	characteristic
115	C₂×D₄×C₇₃	—	2	1	no	—
116	C₂×D₄×C₇₃	C₂×C₈	1	1	yes	characteristic
117	C₂²×C₄×C₇₃	—	2	1	no	—
118	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
119	M₁₆×C₇₃	—	2	1	no	—
120	M₁₆×C₇₃	—	2	2	no	—
121	M₁₆×C₇₃	—	2	2	no	—
122	M₁₆×C₇₃	C₂²×C₄	1	2	yes	—
123	C₂×C₈×C₇₃	—	2	1	no	—
124	C₂×C₈×C₇₃	—	2	2	no	—
125	C₂×C₈×C₇₃	—	2	2	no	—
126	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
130	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
131	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
132	C₁₆×C₇₃	—	2	2	no	—
133	C₁₆×C₇₃	—	2	2	no	—
134	C₂³×C₇₃	—	2	1	no	—
135	C₂³×C₇₃	#131	1	1	yes	characteristic
136	C₂²×D₇₃	—	2	1	no	—
137	Q₈×C₇₃	—	2	1	no	—
138	Q₈×C₇₃	C₂²×C₈	1	2	yes	—
139	D₄×C₇₃	—	2	2	no	—
140	D₄×C₇₃	—	2	2	no	—
141	D₄×C₇₃	—	2	1	no	—
142	D₄×C₇₃	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₇₃	—	2	1	no	—
144	C₂×C₄×C₇₃	—	2	2	no	—
145	C₂×C₄×C₇₃	—	2	2	no	—
146	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
147	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
148	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
149	C₂×C₄×C₇₃	C₂²×C₈	1	1	yes	characteristic
150	C₂×C₄×C₇₃	#131	1	1	yes	characteristic
151	C₂×C₄×C₇₃	C₂³×C₄	1	1	yes	characteristic
152	C₂²⋊D₇₃	—	2	2	no	—
153	C₂²⋊D₇₃	—	2	2	no	—
154	C₂×Dic₇₃	—	2	4	no	—
155	C₂×Dic₇₃	—	2	2	no	—
156	C₂×Dic₇₃	—	2	1	no	—
157	D₂₉₂	—	2	1	no	—
158	D₂₉₂	C₂²×C₈	1	2	yes	—
159	C₄×D₇₃	—	2	2	no	—
160	C₄×D₇₃	—	2	2	no	—
161	Dic₁₄₆	—	2	1	no	—
162	Dic₁₄₆	C₂²×C₈	1	2	yes	—
163	C₈×C₇₃	—	2	2	no	—
164	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
165	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
166	C₈×C₇₃	C₂²×D₄	1	1	yes	characteristic
167	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
168	#10596831	—	2	2	no	—
169	#10596831	C₂²×C₈	1	2	yes	—
170	#10596831	C₂²×C₈	1	2	yes	—
171	C₂²×C₇₃	—	2	2	no	—
172	C₂²×C₇₃	—	2	2	no	—
173	C₂²×C₇₃	—	4	2	no	—
174	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
175	C₂²×C₇₃	C₈×D₄	1	1	yes	characteristic
176	C₂²×C₇₃	C₂×#131	1	1	yes	characteristic
177	C₂×D₇₃	—	2	2	no	—
178	C₂×D₇₃	—	4	2	no	—
179	C₄×C₇₃	—	2	2	no	—
180	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
181	C₄×C₇₃	C₈×D₄	1	1	yes	characteristic
182	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
183	C₄×C₇₃	C₂³×C₈	1	1	yes	characteristic
184	Dic₇₃	—	2	4	no	—
185	Dic₇₃	—	2	2	no	—
186	C₂×#2170	—	73	1	no	maximal, Sylow
187	C₂×C₇₃	—	2	1	no	—
188	C₂×C₇₃	—	2	1	no	—
189	C₂×C₇₃	—	4	2	no	—
190	C₂×C₇₃	#2170	1	2	yes	complemented
191	C₂×C₇₃	C₂×C₈×D₄	1	1	yes	characteristic
192	D₇₃	—	4	2	no	—
193	C₂×#504	—	73	1	no	—
194	C₂×#504	—	73	1	no	—
195	C₂×#443	—	73	1	no	—
196	#2170	—	73	8	no	—
197	C₂×#364	—	73	1	no	—
198	C₂×#350	—	73	1	no	—
199	C₂×#350	D₇₃	1	1	yes	characteristic, complemented
200	C₂×C₄×C₁₆	—	73	1	no	—
201	C₇₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
202	C₂×#135	—	73	1	no	—
203	C₂×#131	—	73	1	no	—
204	C₂×#131	C₂×D₇₃	1	1	yes	characteristic
205	#504	—	73	4	no	—
206	#504	—	73	4	no	—
207	#504	—	146	2	no	—
208	#504	—	146	2	no	—
209	C₂×M₃₂	—	73	1	no	—
210	C₂×M₃₂	—	146	1	no	—
211	C₂×M₃₂	—	146	1	no	—
212	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
213	C₂²×C₁₆	—	73	1	no	—
214	C₂²×C₁₆	—	146	1	no	—
215	C₂²×C₁₆	—	146	1	no	—
216	C₂²×C₁₆	C₂×D₇₃	1	1	yes	characteristic
217	#443	—	73	4	no	—
218	#443	—	73	4	no	—
219	C₂×#108	—	73	1	no	—
220	C₂×#104	—	73	1	no	—
221	C₂×#104	—	73	1	no	—
222	C₂×C₄×C₈	—	73	1	no	—
223	#364	—	73	4	no	—
224	#350	—	73	4	no	—
225	#350	C₂×D₇₃	1	4	yes	complemented
226	C₄×C₁₆	—	73	4	no	—
227	C₂×#41	—	73	1	no	—
228	C₂×#41	C₄×D₇₃	1	1	yes	characteristic
229	#135	—	73	4	no	—
230	#135	—	73	4	no	—
231	C₂×Q₁₆	—	73	1	no	—
232	C₂×SD₁₆	—	73	1	no	—
233	C₂×SD₁₆	—	73	1	no	—
234	C₂×D₈	—	73	1	no	—
235	#131	—	2	1	no	—
236	#131	—	2	1	no	—
237	#131	—	73	2	no	—
238	#131	—	73	2	no	—
239	#131	—	146	1	no	—
240	#131	—	146	1	no	—
241	#131	C₄×D₇₃	1	2	yes	—
242	#131	C₂²×D₇₃	1	2	yes	—
243	C₂×M₁₆	—	2	1	no	—
244	C₂×M₁₆	—	73	1	no	—
245	C₂×M₁₆	—	146	1	no	—
246	C₂×M₁₆	C₄×D₇₃	1	1	yes	characteristic
247	C₂²×C₈	—	2	1	no	—
248	C₂²×C₈	—	73	1	no	—
249	C₂²×C₈	—	146	1	no	—
250	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
251	C₂×C₄²	—	73	1	no	—
252	M₃₂	—	2	1	no	—
253	M₃₂	—	73	2	no	—
254	M₃₂	—	146	4	no	—
255	M₃₂	—	146	4	no	—
256	M₃₂	—	146	1	no	—
257	M₃₂	C₂²×D₇₃	1	2	yes	—
258	C₂×C₁₆	—	2	1	no	—
259	C₂×C₁₆	—	2	1	no	—
260	C₂×C₁₆	—	2	1	no	—
261	C₂×C₁₆	—	73	2	no	—
262	C₂×C₁₆	—	73	1	no	—
263	C₂×C₁₆	—	73	1	no	—
264	C₂×C₁₆	—	73	2	no	—
265	C₂×C₁₆	—	146	4	no	—
266	C₂×C₁₆	—	146	4	no	—
267	C₂×C₁₆	—	146	1	no	—
268	C₂×C₁₆	C₂²×D₇₃	1	2	yes	—
269	#108	—	73	4	no	—
270	#104	—	73	4	no	—
271	#104	—	73	4	no	—
272	C₄×C₈	—	73	2	no	—
273	C₄×C₈	—	73	2	no	—
274	#41	—	2	1	no	—
275	#41	—	2	1	no	—
276	#41	—	73	2	no	—
277	#41	—	73	2	no	—
278	#41	—	146	1	no	—
279	#41	—	146	1	no	—
280	#41	C₈×D₇₃	1	2	yes	—
281	#41	C₂×C₄×D₇₃	1	2	yes	—
282	C₂×Q₈	—	73	1	no	—
283	C₂×Q₈	C₈×D₇₃	1	1	yes	characteristic
284	C₂×D₄	—	2	1	no	—
285	C₂×D₄	—	73	1	no	—
286	C₂×D₄	—	146	1	no	—
287	C₂×D₄	C₈×D₇₃	1	1	yes	characteristic
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	146	1	no	—
290	C₂²×C₄	—	146	1	no	—
291	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
292	Q₁₆	—	73	4	no	—
293	SD₁₆	—	73	4	no	—
294	SD₁₆	—	73	4	no	—
295	D₈	—	73	4	no	—
296	M₁₆	—	2	1	no	—
297	M₁₆	—	2	2	no	—
298	M₁₆	—	2	2	no	—
299	M₁₆	—	73	2	no	—
300	M₁₆	—	146	1	no	—
301	M₁₆	—	146	2	no	—
302	M₁₆	—	146	2	no	—
303	M₁₆	C₂×C₄×D₇₃	1	2	yes	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	2	2	no	—
307	C₂×C₈	—	73	1	no	—
308	C₂×C₈	—	73	1	no	—
309	C₂×C₈	—	73	2	no	—
310	C₂×C₈	—	73	2	no	—
311	C₂×C₈	—	146	1	no	—
312	C₂×C₈	—	146	2	no	—
313	C₂×C₈	—	146	2	no	—
314	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
315	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
316	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
317	C₂×C₈	#11924027	1	1	yes	characteristic
318	C₂×C₈	#11924027	1	1	yes	characteristic
319	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
320	C₄²	—	73	2	no	—
321	C₄²	—	146	1	no	—
322	C₁₆	—	2	2	no	—
323	C₁₆	—	2	2	no	—
324	C₁₆	—	73	2	no	—
325	C₁₆	—	73	2	no	—
326	C₁₆	—	146	2	no	—
327	C₂³	—	2	1	no	—
328	C₂³	—	146	1	no	—
329	C₂³	#14556599	1	1	yes	characteristic
330	Q₈	—	2	1	no	—
331	Q₈	—	73	2	no	—
332	Q₈	—	146	1	no	—
333	Q₈	C₂×C₈×D₇₃	1	2	yes	—
334	D₄	—	2	2	no	—
335	D₄	—	2	2	no	—
336	D₄	—	2	1	no	—
337	D₄	—	73	2	no	—
338	D₄	—	146	2	no	—
339	D₄	—	146	2	no	—
340	D₄	—	146	1	no	—
341	D₄	C₂×C₈×D₇₃	1	2	yes	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	146	4	no	—
346	C₂×C₄	—	146	1	no	—
347	C₂×C₄	—	146	2	no	—
348	C₂×C₄	—	146	2	no	—
349	C₂×C₄	—	146	2	no	—
350	C₂×C₄	#14556551	1	1	yes	characteristic
351	C₂×C₄	#14556551	1	1	yes	characteristic
352	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
353	C₂×C₄	C₂×C₈×D₇₃	1	1	yes	characteristic
354	C₂×C₄	#14556599	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₄×D₇₃	1	1	yes	characteristic
356	C₈	—	2	2	no	—
357	C₈	—	73	2	no	—
358	C₈	—	73	2	no	—
359	C₈	—	146	2	no	—
360	C₈	#14556551	1	1	yes	characteristic
361	C₈	#14556551	1	1	yes	characteristic
362	C₈	C₂×D₄×D₇₃	1	1	yes	characteristic
363	C₈	C₂×#11924027	1	1	yes	characteristic
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	4	2	no	—
367	C₂²	—	292	2	no	—
368	C₂²	—	146	2	no	—
369	C₂²	#17988650	1	1	yes	characteristic
370	C₂²	#17988650	1	1	yes	characteristic
371	C₂²	C₂×#14556599	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	146	4	no	—
374	C₄	—	146	2	no	—
375	C₄	#17988650	1	1	yes	characteristic
376	C₄	#17988650	1	1	yes	characteristic
377	C₄	C₂×#14556551	1	1	yes	characteristic
378	C₄	C₂²×C₈×D₇₃	1	1	yes	characteristic
379	C₂	—	2	1	no	—
380	C₂	—	2	1	no	—
381	C₂	—	4	2	no	—
382	C₂	—	292	2	no	—
383	C₂	#22611765	1	2	yes	direct product factor
384	C₂	C₂×#17988650	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₇₃
commutator	C₄×C₇₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#350×C₇₃	C₂
socle	C₂²×C₇₃	C₈×D₄
prime radical	C₂×#14556611	C₄
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#350	D₇₃
hypocenter	C₇₃	C₂×#2170
LCS	G ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#350

Complements

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

N	H ≅ G/N	type
1. C₂×#17989005	C₂	semidirect
2. C₂×#17989004	C₂	semidirect
4. #22611765	C₂	direct (N × H)
5. C₂×#17988407	C₂	semidirect
6. C₂×#17988370	C₂	semidirect
7. C₂×C₁₆×Dic₇₃	C₂	semidirect
8. C₂×#350×C₇₃	C₂	semidirect
12. #17989005	C₂²	semidirect
14. #17989004	C₂²	semidirect
27. #17988407	C₂²	semidirect
28. #17988370	C₂²	semidirect
29. C₁₆×Dic₇₃	C₂²	semidirect
33. #350×C₇₃	C₂²	semidirect
190. C₂×C₇₃	#2170	semidirect
199. C₂×#350	D₇₃	semidirect
201. C₇₃	C₂×#2170	semidirect
225. #350	C₂×D₇₃	semidirect
383. C₂	#22611765	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×#17989005	2
C₂×#17989004	2
C₂×#17988702	2
#22611765	2
C₂×#17988407	2
C₂×#17988370	2
C₂×C₁₆×Dic₇₃	2
C₂×#350×C₇₃	2
C₂×#2170	73

Subgroups — C2×#22611765 #33239209

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611765 #33239209