Subgroups — C₂×#22611832 #33239428

Order 18688.

710 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17989023	C₂	1	1	yes	maximal, characteristic
2	C₂×#17988708	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#17988699	C₂	1	1	yes	maximal, characteristic, complemented
4	#22611832	C₂	1	8	yes	maximal, direct product factor
5	C₂×#17988411	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17988398	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#17988364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#361×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14556613	C₂²	1	1	yes	characteristic
10	C₂×#14556599	—	2	1	no	—
11	C₂×D₄×Dic₇₃	—	2	1	no	—
12	#17989023	C₄	1	4	yes	complemented
13	#17989023	C₂²	1	4	yes	—
14	C₂×#14556535	C₄	1	2	yes	complemented
15	C₂×#14556532	C₄	1	2	yes	complemented
16	C₂×#14556531	C₂²	1	1	yes	characteristic
17	#17988708	—	2	2	no	—
18	#17988708	C₂²	1	4	yes	complemented
19	#17988699	—	2	2	no	—
20	#17988699	C₂²	1	4	yes	complemented
21	C₂×#14556523	C₂²	1	1	yes	characteristic
22	C₂×#14556517	—	2	1	no	—
23	C₂×#14556512	—	2	1	no	—
24	C₂×#14556501	C₂²	1	1	yes	characteristic
25	#17988411	C₂²	1	4	yes	complemented
26	#17988398	C₂²	1	4	yes	complemented
27	#17988364	C₂²	1	4	yes	complemented
28	C₂²×D₈×C₇₃	C₂²	1	1	yes	characteristic
29	C₂×M₃₂×C₇₃	C₂²	1	1	yes	characteristic
30	C₂×#108×C₇₃	C₂²	1	1	yes	characteristic
31	#361×C₇₃	C₂²	1	4	yes	complemented
32	C₂×#11924028	—	2	1	no	—
33	C₂³×Dic₇₃	—	4	1	no	—
34	#14556613	—	2	1	no	—
35	#14556613	—	2	1	no	—
36	#14556613	C₂×C₄	1	2	yes	complemented
37	#14556613	C₂³	1	2	yes	—
38	C₂×Dic₂₉₂	C₂×C₄	1	1	yes	characteristic, complemented
39	C₂×#11924017	—	2	1	no	—
40	C₂×D₅₈₄	C₂×C₄	1	1	yes	characteristic, complemented
41	#14556599	—	2	4	no	—
42	#14556599	—	4	2	no	—
43	C₂×#11924011	—	4	1	no	—
44	C₂²×#10596831	D₄	1	1	yes	characteristic
45	C₂×C₈×D₇₃	—	4	1	no	—
46	D₄×Dic₇₃	—	2	4	no	—
47	D₄×Dic₇₃	—	4	2	no	—
48	C₂×C₄⋊Dic₇₃	—	2	1	no	—
49	C₂×C₂²⋊Dic₇₃	—	4	1	no	—
50	C₂×C₄×Dic₇₃	D₄	1	1	yes	characteristic
51	#14556535	C₂×C₄	1	8	yes	complemented
52	#14556532	C₂×C₄	1	8	yes	complemented
53	#14556531	C₂×C₄	1	2	yes	complemented
54	#14556531	C₂³	1	2	yes	—
55	C₂×#11924003	C₂×C₄	1	2	yes	complemented
56	#14556523	—	2	1	no	—
57	#14556523	C₂³	1	2	yes	—
58	#14556517	—	2	4	no	—
59	#14556512	—	2	4	no	—
60	#14556501	D₄	1	2	yes	—
61	#14556501	C₂³	1	2	yes	—
62	C₂²×D₄×C₇₃	—	2	1	no	—
63	C₂×D₈×C₇₃	—	2	1	no	—
64	C₂×D₈×C₇₃	—	2	1	no	—
65	C₂×D₈×C₇₃	—	2	1	no	—
66	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₇₃	C₂×C₄	1	2	yes	—
68	C₂×D₈×C₇₃	C₂³	1	2	yes	—
69	C₂×M₁₆×C₇₃	—	2	1	no	—
70	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
71	M₃₂×C₇₃	—	2	1	no	—
72	M₃₂×C₇₃	C₂³	1	2	yes	—
73	C₂×C₁₆×C₇₃	—	2	1	no	—
74	#108×C₇₃	—	2	1	no	—
75	#108×C₇₃	C₂³	1	2	yes	—
76	#11924028	—	2	2	no	—
77	#11924028	—	2	2	no	—
78	#11924028	—	4	1	no	—
79	#11924028	—	4	1	no	—
80	C₂×Dic₁₄₆	—	2	1	no	—
81	C₂×D₂₉₂	—	2	1	no	—
82	C₂×C₂²⋊D₇₃	—	4	1	no	—
83	C₂²×Dic₇₃	—	2	1	no	—
84	C₂²×Dic₇₃	—	4	4	no	—
85	C₂²×Dic₇₃	—	4	1	no	—
86	C₂²×Dic₇₃	—	8	4	no	—
87	C₂×C₄×D₇₃	—	4	1	no	—
88	Dic₂₉₂	—	2	1	no	—
89	Dic₂₉₂	C₂²×C₄	1	2	yes	complemented
90	#11924017	—	2	2	no	—
91	#11924017	—	2	2	no	—
92	D₅₈₄	—	2	1	no	—
93	D₅₈₄	C₂²×C₄	1	2	yes	complemented
94	#11924011	—	4	4	no	—
95	C₂×#10596831	—	2	1	no	—
96	C₂×#10596831	—	2	1	no	—
97	C₂×#10596831	C₂×D₄	1	2	yes	—
98	C₈×D₇₃	—	4	4	no	—
99	C₄⋊Dic₇₃	—	2	2	no	—
100	C₄⋊Dic₇₃	—	4	1	no	—
101	C₂²⋊Dic₇₃	—	4	4	no	—
102	C₄×Dic₇₃	—	2	1	no	—
103	C₄×Dic₇₃	C₂×D₄	1	2	yes	—
104	#11924003	C₂²×C₄	1	4	yes	complemented
105	C₂⁴×C₇₃	—	4	1	no	—
106	C₂×D₄×C₇₃	—	2	2	no	—
107	C₂×D₄×C₇₃	—	2	2	no	—
108	C₂×D₄×C₇₃	—	2	2	no	—
109	C₂×D₄×C₇₃	—	4	1	no	—
110	C₂×D₄×C₇₃	—	4	1	no	—
111	C₂×D₄×C₇₃	—	4	1	no	—
112	C₂²×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
113	D₈×C₇₃	—	2	2	no	—
114	D₈×C₇₃	—	2	2	no	—
115	D₈×C₇₃	—	2	2	no	—
116	D₈×C₇₃	C₂²×C₄	1	4	yes	—
117	M₁₆×C₇₃	—	2	2	no	—
118	M₁₆×C₇₃	—	4	1	no	—
119	C₂×C₈×C₇₃	—	4	1	no	—
120	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
124	C₂×C₈×C₇₃	#41	1	1	yes	characteristic
125	C₂×C₈×C₇₃	C₂⁴	1	1	yes	characteristic
126	C₁₆×C₇₃	—	2	2	no	—
127	C₂³×C₇₃	—	4	2	no	—
128	C₂³×C₇₃	—	4	2	no	—
129	C₂³×C₇₃	—	4	2	no	—
130	C₂³×C₇₃	—	8	2	no	—
131	C₂³×C₇₃	—	8	2	no	—
132	C₂³×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
133	C₂²×D₇₃	—	4	1	no	—
134	D₄×C₇₃	—	2	4	no	—
135	D₄×C₇₃	—	4	2	no	—
136	D₄×C₇₃	—	4	2	no	—
137	D₄×C₇₃	—	4	2	no	—
138	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
142	C₂×C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
143	C₂×C₄×C₇₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²⋊D₇₃	—	4	2	no	—
145	C₂²⋊D₇₃	—	4	2	no	—
146	C₂×Dic₇₃	—	2	2	no	—
147	C₂×Dic₇₃	—	4	2	no	—
148	C₂×Dic₇₃	—	4	1	no	—
149	C₂×Dic₇₃	—	4	2	no	—
150	C₂×Dic₇₃	—	8	8	no	—
151	C₂×Dic₇₃	—	8	2	no	—
152	D₂₉₂	—	2	2	no	—
153	D₂₉₂	—	4	1	no	—
154	C₄×D₇₃	—	4	2	no	—
155	C₄×D₇₃	—	4	2	no	—
156	Dic₁₄₆	—	2	2	no	—
157	Dic₁₄₆	—	4	1	no	—
158	C₈×C₇₃	—	4	2	no	—
159	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
160	C₈×C₇₃	C₄×D₄	1	1	yes	characteristic
161	C₈×C₇₃	C₂³×C₄	1	1	yes	characteristic
162	C₈×C₇₃	C₂×#41	1	1	yes	characteristic
163	#10596831	—	2	2	no	—
164	C₂²×C₇₃	—	2	2	no	—
165	C₂²×C₇₃	—	4	4	no	—
166	C₂²×C₇₃	—	8	2	no	—
167	C₂²×C₇₃	—	8	2	no	—
168	C₂²×C₇₃	—	8	4	no	—
169	C₂²×C₇₃	—	8	4	no	—
170	C₂²×C₇₃	#442	1	1	yes	characteristic
171	C₂²×C₇₃	#442	1	1	yes	characteristic
172	C₂²×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×D₇₃	—	4	2	no	—
174	C₂×D₇₃	—	8	2	no	—
175	C₄×C₇₃	#442	1	1	yes	characteristic
176	C₄×C₇₃	#442	1	1	yes	characteristic
177	C₄×C₇₃	C₂×C₄×D₄	1	1	yes	characteristic
178	C₄×C₇₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	Dic₇₃	—	4	2	no	—
180	Dic₇₃	—	4	2	no	—
181	Dic₇₃	—	8	2	no	—
182	C₂×AGL(1,ℤ/16)	—	73	1	no	maximal, Sylow
183	C₂×C₇₃	—	2	1	no	—
184	C₂×C₇₃	—	2	1	no	—
185	C₂×C₇₃	—	8	4	no	—
186	C₂×C₇₃	AGL(1,ℤ/16)	1	2	yes	complemented
187	C₂×C₇₃	C₂×#442	1	1	yes	characteristic
188	D₇₃	—	8	2	no	—
189	C₂×#509	—	73	1	no	—
190	C₂×#444	—	73	1	no	—
191	C₂×#442	—	73	1	no	—
192	AGL(1,ℤ/16)	—	73	8	no	—
193	C₂×#365	—	73	1	no	—
194	C₂×#361	D₇₃	1	1	yes	characteristic, complemented
195	C₂×#360	—	73	1	no	—
196	C₂×#347	—	73	1	no	—
197	C₇₃	C₂×AGL(1,ℤ/16)	1	1	yes	characteristic, complemented, Sylow
198	C₂×#135	—	73	1	no	—
199	C₂²×D₈	C₂×D₇₃	1	1	yes	characteristic
200	C₂×#131	—	146	1	no	—
201	C₂×C₄×D₄	—	146	1	no	—
202	#509	—	73	4	no	—
203	#509	—	73	4	no	—
204	C₂×SD₃₂	—	73	2	no	—
205	C₂×D₁₆	—	73	2	no	—
206	C₂×M₃₂	—	73	1	no	—
207	C₂×M₃₂	C₂×D₇₃	1	1	yes	characteristic
208	#444	—	73	4	no	—
209	#444	—	146	2	no	—
210	#442	—	73	4	no	—
211	#442	—	146	2	no	—
212	C₂×#108	C₂×D₇₃	1	1	yes	characteristic
213	C₂×#106	—	73	1	no	—
214	C₂×#104	—	146	1	no	—
215	C₂×#102	—	146	1	no	—
216	C₂×#97	—	73	1	no	—
217	#365	—	73	4	no	—
218	#361	C₂×D₇₃	1	4	yes	complemented
219	#360	—	73	4	no	—
220	#347	—	73	4	no	—
221	C₂×#41	—	146	1	no	—
222	C₂²×D₄	—	2	1	no	—
223	C₂³×C₄	—	292	1	no	—
224	#135	—	73	2	no	—
225	#135	—	73	2	no	—
226	#135	—	146	1	no	—
227	#135	—	146	1	no	—
228	C₂×Q₁₆	—	73	1	no	—
229	C₂×SD₁₆	—	146	1	no	—
230	C₂×D₈	—	2	1	no	—
231	C₂×D₈	—	2	1	no	—
232	C₂×D₈	—	2	1	no	—
233	C₂×D₈	—	73	1	no	—
234	C₂×D₈	C₄×D₇₃	1	2	yes	—
235	C₂×D₈	C₄×D₇₃	1	2	yes	—
236	C₂×D₈	C₂²×D₇₃	1	2	yes	—
237	#131	—	292	2	no	—
238	#131	—	146	4	no	—
239	C₂×M₁₆	—	2	1	no	—
240	C₂×M₁₆	—	292	1	no	—
241	C₂²×C₈	—	292	1	no	—
242	C₂²×C₈	—	73	1	no	—
243	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
244	C₄×D₄	—	292	2	no	—
245	C₄×D₄	—	146	4	no	—
246	C₂×C₄⋊C₄	—	146	1	no	—
247	C₂×C₂²⋊C₄	—	292	1	no	—
248	C₂×C₄²	—	73	1	no	—
249	SD₃₂	—	73	8	no	—
250	D₁₆	—	73	8	no	—
251	M₃₂	—	2	1	no	—
252	M₃₂	—	73	2	no	—
253	M₃₂	—	73	2	no	—
254	M₃₂	C₂²×D₇₃	1	2	yes	—
255	C₂×C₁₆	—	2	1	no	—
256	C₂×C₁₆	—	73	2	no	—
257	#108	—	2	1	no	—
258	#108	C₂²×D₇₃	1	2	yes	—
259	#106	—	73	2	no	—
260	#106	—	146	1	no	—
261	#104	—	146	4	no	—
262	#102	—	146	4	no	—
263	#97	—	73	2	no	—
264	#97	—	73	2	no	—
265	C₂⁴	—	4	1	no	—
266	#41	—	292	1	no	—
267	#41	—	292	1	no	—
268	#41	—	146	2	no	—
269	#41	—	146	2	no	—
270	C₂×Q₈	—	146	1	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	—	2	2	no	—
274	C₂×D₄	—	4	1	no	—
275	C₂×D₄	—	4	1	no	—
276	C₂×D₄	—	4	1	no	—
277	C₂×D₄	—	292	1	no	—
278	C₂×D₄	—	146	1	no	—
279	C₂²×C₄	—	292	1	no	—
280	C₂²×C₄	—	292	1	no	—
281	C₂²×C₄	—	292	4	no	—
282	C₂²×C₄	—	584	4	no	—
283	C₂²×C₄	—	146	1	no	—
284	C₂²×C₄	D₄×D₇₃	1	1	yes	characteristic
285	Q₁₆	—	73	2	no	—
286	Q₁₆	—	146	1	no	—
287	SD₁₆	—	146	2	no	—
288	SD₁₆	—	146	2	no	—
289	D₈	—	2	2	no	—
290	D₈	—	2	2	no	—
291	D₈	—	2	2	no	—
292	D₈	—	73	2	no	—
293	D₈	—	146	1	no	—
294	D₈	C₂×C₄×D₇₃	1	4	yes	—
295	M₁₆	—	2	2	no	—
296	M₁₆	—	4	1	no	—
297	M₁₆	—	292	4	no	—
298	C₂×C₈	—	4	1	no	—
299	C₂×C₈	—	292	4	no	—
300	C₂×C₈	—	73	2	no	—
301	C₂×C₈	—	146	1	no	—
302	C₂×C₈	—	146	1	no	—
303	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₇₃	1	1	yes	characteristic
305	C₂×C₈	D₄×D₇₃	1	1	yes	characteristic
306	C₂×C₈	#11924027	1	1	yes	characteristic
307	C₂×C₈	#11924027	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₇₃	1	1	yes	characteristic
309	C₄⋊C₄	—	292	1	no	—
310	C₄⋊C₄	—	146	2	no	—
311	C₂²⋊C₄	—	292	4	no	—
312	C₄²	—	73	2	no	—
313	C₄²	—	146	1	no	—
314	C₁₆	—	2	2	no	—
315	C₁₆	—	73	4	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	8	2	no	—
320	C₂³	—	8	2	no	—
321	C₂³	—	292	1	no	—
322	C₂³	#14556618	1	1	yes	characteristic
323	Q₈	—	292	1	no	—
324	Q₈	—	146	2	no	—
325	D₄	—	2	4	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	4	2	no	—
329	D₄	—	292	2	no	—
330	D₄	—	292	2	no	—
331	D₄	—	292	1	no	—
332	D₄	—	146	2	no	—
333	C₂×C₄	—	292	2	no	—
334	C₂×C₄	—	292	1	no	—
335	C₂×C₄	—	292	2	no	—
336	C₂×C₄	—	292	2	no	—
337	C₂×C₄	—	292	2	no	—
338	C₂×C₄	—	584	8	no	—
339	C₂×C₄	—	584	2	no	—
340	C₂×C₄	—	146	2	no	—
341	C₂×C₄	#14556551	1	1	yes	characteristic
342	C₂×C₄	#14556551	1	1	yes	characteristic
343	C₂×C₄	#14556615	1	1	yes	characteristic
344	C₂×C₄	#14556615	1	1	yes	characteristic
345	C₂×C₄	#14556618	1	1	yes	characteristic
346	C₂×C₄	C₂×D₄×D₇₃	1	1	yes	characteristic
347	C₈	—	4	2	no	—
348	C₈	—	146	2	no	—
349	C₈	#14556551	1	1	yes	characteristic
350	C₈	#14556551	1	1	yes	characteristic
351	C₈	C₂²×C₄×D₇₃	1	1	yes	characteristic
352	C₈	C₂×#11924027	1	1	yes	characteristic
353	C₂²	—	2	2	no	—
354	C₂²	—	4	4	no	—
355	C₂²	—	8	2	no	—
356	C₂²	—	8	2	no	—
357	C₂²	—	8	4	no	—
358	C₂²	—	8	4	no	—
359	C₂²	—	292	2	no	—
360	C₂²	—	584	2	no	—
361	C₂²	#17988696	1	1	yes	characteristic
362	C₂²	#17988696	1	1	yes	characteristic
363	C₂²	C₂×#14556618	1	1	yes	characteristic
364	C₄	—	292	2	no	—
365	C₄	—	292	2	no	—
366	C₄	—	584	2	no	—
367	C₄	#17988696	1	1	yes	characteristic
368	C₄	#17988696	1	1	yes	characteristic
369	C₄	C₂×#14556551	1	1	yes	characteristic
370	C₄	C₂×#14556615	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	1	no	—
373	C₂	—	8	4	no	—
374	C₂	—	584	2	no	—
375	C₂	#22611832	1	2	yes	direct product factor
376	C₂	C₂×#17988696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#17988696
commutator	C₈×C₇₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₇₃
Fitting	C₂×#361×C₇₃	C₂
socle	C₂²×C₇₃	#442
prime radical	C₂×#17989023	C₂
cosocle	C₂×C₈×C₇₃	C₂⁴
hypercenter	C₂×#361	D₇₃
hypocenter	C₇₃	C₂×AGL(1,ℤ/16)
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. C₂×#17988708	C₂	semidirect
3. C₂×#17988699	C₂	semidirect
4. #22611832	C₂	direct (N × H)
5. C₂×#17988411	C₂	semidirect
6. C₂×#17988398	C₂	semidirect
7. C₂×#17988364	C₂	semidirect
8. C₂×#361×C₇₃	C₂	semidirect
12. #17989023	C₄	semidirect
14. C₂×#14556535	C₄	semidirect
15. C₂×#14556532	C₄	semidirect
18. #17988708	C₂²	semidirect
20. #17988699	C₂²	semidirect
25. #17988411	C₂²	semidirect
26. #17988398	C₂²	semidirect
27. #17988364	C₂²	semidirect
31. #361×C₇₃	C₂²	semidirect
36. #14556613	C₂×C₄	semidirect
38. C₂×Dic₂₉₂	C₂×C₄	semidirect
40. C₂×D₅₈₄	C₂×C₄	semidirect
51. #14556535	C₂×C₄	semidirect
52. #14556532	C₂×C₄	semidirect
53. #14556531	C₂×C₄	semidirect
55. C₂×#11924003	C₂×C₄	semidirect
89. Dic₂₉₂	C₂²×C₄	semidirect
93. D₅₈₄	C₂²×C₄	semidirect
104. #11924003	C₂²×C₄	semidirect
186. C₂×C₇₃	AGL(1,ℤ/16)	semidirect
194. C₂×#361	D₇₃	semidirect
197. C₇₃	C₂×AGL(1,ℤ/16)	semidirect
218. #361	C₂×D₇₃	semidirect
375. C₂	#22611832	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×#17989023	2
C₂×#17988708	2
C₂×#17988699	2
#22611832	2
C₂×#17988411	2
C₂×#17988398	2
C₂×#17988364	2
C₂×#361×C₇₃	2
C₂×AGL(1,ℤ/16)	73

Subgroups — C2×#22611832 #33239428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#22611832 #33239428