Subgroups — C₂×#10707076 #72245987

Order 1536.

940 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#69085	C₂	1	1	yes	maximal, characteristic
2	C₂×#69301	C₂	1	2	yes	maximal, complemented
3	#10707076	C₂	1	8	yes	maximal, direct product factor
4	C₂×#67569	C₂	1	2	yes	maximal, complemented
5	C₂×#67511	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#1369	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#12068	—	3	1	no	maximal, Sylow
8	C₂×#4907	—	2	2	no	—
9	C₂×#4904	C₄	1	1	yes	characteristic
10	C₂×#4700	C₂²	1	2	yes	—
11	C₂×C₃×#364	C₂²	1	2	yes	—
12	C₂×C₃×#346	C₂²	1	1	yes	characteristic
13	C₂×#4515	—	2	2	no	—
14	C₂×#4510	C₄	1	2	yes	—
15	C₂×#4506	C₄	1	1	yes	characteristic
16	C₂×#4461	C₂²	1	1	yes	characteristic
17	C₂×#4455	C₂²	1	1	yes	characteristic
18	#69301	—	2	4	no	—
19	#69301	C₂²	1	8	yes	complemented
20	#69085	C₄	1	4	yes	—
21	#69085	C₂²	1	4	yes	—
22	C₃×#1369	C₂²	1	4	yes	complemented
23	#67569	C₂²	1	8	yes	complemented
24	#67511	C₂²	1	4	yes	complemented
25	C₂×#2179	—	3	1	no	—
26	C₂×#2171	—	3	2	no	—
27	#12068	—	3	8	no	—
28	C₂×#1369	D₃	1	1	yes	characteristic, complemented
29	C₂×#1336	—	3	2	no	—
30	C₂×#1313	—	3	1	no	—
31	C₂×#904	C₂×C₄	1	2	yes	—
32	C₂×#901	—	2	2	no	—
33	C₂×C₃×M₃₂	—	2	2	no	—
34	C₂²×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₈	C₂³	1	1	yes	characteristic
37	C₂×#819	C₂×C₄	1	2	yes	—
38	C₂²×C₃⋊C₁₆	D₄	1	2	yes	—
39	C₂×#805	—	2	2	no	—
40	C₂×C₄×C₃⋊C₈	—	2	1	no	—
41	#4907	—	2	8	no	—
42	#4907	—	4	4	no	—
43	C₂×Dic₂₄	C₈	1	1	yes	characteristic, complemented
44	#4904	C₈	1	4	yes	complemented
45	#4904	C₂×C₄	1	4	yes	—
46	C₂×#800	C₈	1	2	yes	complemented
47	C₂×D₄₈	C₈	1	1	yes	characteristic, complemented
48	C₂×#798	—	4	2	no	—
49	C₂×D₃×C₁₆	—	4	2	no	—
50	#4700	—	2	2	no	—
51	#4700	—	2	2	no	—
52	#4700	C₂×C₄	1	4	yes	—
53	#4700	C₂³	1	4	yes	—
54	C₃×#364	—	2	2	no	—
55	C₃×#364	C₂³	1	4	yes	—
56	C₃×#346	C₂×C₄	1	2	yes	—
57	C₃×#346	C₂³	1	2	yes	—
58	#4515	—	2	8	no	—
59	#4510	C₂×C₄	1	8	yes	—
60	#4506	C₂×C₄	1	4	yes	—
61	#4461	D₄	1	2	yes	—
62	#4461	C₂³	1	2	yes	—
63	#4455	—	2	1	no	—
64	#4455	C₂³	1	2	yes	—
65	C₂×#508	—	3	1	no	—
66	C₂×#504	—	6	2	no	—
67	C₂×#443	—	3	2	no	—
68	#2179	—	3	4	no	—
69	#2179	—	3	4	no	—
70	#2171	—	3	8	no	—
71	#2171	—	6	4	no	—
72	C₂×#368	—	3	1	no	—
73	C₂×#364	C₂×D₃	1	2	yes	—
74	C₂×#359	—	3	2	no	—
75	C₂×#350	—	6	2	no	—
76	C₂×#346	—	3	1	no	—
77	C₂×#346	C₂×D₃	1	1	yes	characteristic
78	C₂×#322	—	3	1	no	—
79	#1369	C₂×D₃	1	4	yes	complemented
80	#1336	—	3	8	no	—
81	#1313	—	3	4	no	—
82	C₂×#235	—	2	2	no	—
83	C₂²×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₃×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄²	#41	1	1	yes	characteristic
86	C₂×#208	—	2	2	no	—
87	C₂²×C₃⋊C₈	—	4	1	no	—
88	C₂×Dic₁₂	C₂×C₈	1	2	yes	—
89	#904	—	2	2	no	—
90	#904	—	2	2	no	—
91	#904	C₂×C₈	1	4	yes	—
92	#904	C₂²×C₄	1	4	yes	—
93	C₂×D₂₄	C₂×C₈	1	2	yes	—
94	C₂×#204	—	2	2	no	—
95	#901	—	2	4	no	—
96	#901	—	2	4	no	—
97	#901	—	4	2	no	—
98	#901	—	4	2	no	—
99	C₂×ΓL(1,𝔽₂₅)	—	4	2	no	—
100	C₂×D₃×C₈	—	4	2	no	—
101	C₃×M₃₂	—	2	4	no	—
102	C₃×M₃₂	—	4	2	no	—
103	C₂×C₃×C₁₆	—	2	1	no	—
104	C₂×C₃×C₁₆	—	2	1	no	—
105	C₂×C₃×C₁₆	—	4	2	no	—
106	C₂×C₃×C₁₆	C₂×C₈	1	1	yes	characteristic
107	C₂×C₃×C₁₆	C₂×C₈	1	1	yes	characteristic
108	C₂×C₃×C₁₆	C₂×C₈	1	2	yes	—
109	C₂×C₃×C₁₆	C₂²×C₄	1	2	yes	—
110	C₂×C₃×C₁₆	C₂²×C₄	1	2	yes	—
111	C₃×C₄×C₈	C₂²×C₄	1	1	yes	characteristic
112	C₃×C₄×C₈	C₂×D₄	1	1	yes	characteristic
113	C₃×C₄×C₈	#41	1	1	yes	characteristic
114	C₃×C₄×C₈	C₂⁴	1	1	yes	characteristic
115	#819	—	2	2	no	—
116	#819	C₂²×C₄	1	4	yes	—
117	C₂×C₃⋊C₁₆	—	2	2	no	—
118	C₂×C₃⋊C₁₆	—	2	2	no	—
119	C₂×C₃⋊C₁₆	C₂×D₄	1	4	yes	—
120	#805	—	2	4	no	—
121	#805	—	2	4	no	—
122	C₄×C₃⋊C₈	—	2	2	no	—
123	C₄×C₃⋊C₈	—	4	1	no	—
124	Dic₂₄	C₂×C₈	1	4	yes	complemented
125	#800	C₂×C₈	1	8	yes	complemented
126	D₄₈	C₂×C₈	1	4	yes	complemented
127	#798	—	4	8	no	—
128	D₃×C₁₆	—	4	8	no	—
129	C₂×#135	—	3	2	no	—
130	C₂×#131	—	6	2	no	—
131	#508	—	3	4	no	—
132	#508	—	3	4	no	—
133	C₂×Q₃₂	—	3	1	no	—
134	C₂×SD₃₂	—	3	2	no	—
135	C₂×D₁₆	—	3	1	no	—
136	#504	—	6	8	no	—
137	#504	—	12	4	no	—
138	C₂×M₃₂	—	2	2	no	—
139	C₂×M₃₂	—	12	2	no	—
140	C₂²×C₁₆	—	3	2	no	—
141	C₂²×C₁₆	—	12	2	no	—
142	C₂²×C₁₆	C₄×D₃	1	1	yes	characteristic
143	C₂²×C₁₆	C₄×D₃	1	1	yes	characteristic
144	#443	—	3	4	no	—
145	#443	—	3	4	no	—
146	#443	—	6	2	no	—
147	#443	—	6	2	no	—
148	C₂×#108	—	3	2	no	—
149	C₂×#104	—	6	2	no	—
150	C₂×C₄×C₈	—	6	1	no	—
151	C₂×C₄×C₈	C₂²×D₃	1	1	yes	characteristic
152	#368	—	3	4	no	—
153	#364	—	2	2	no	—
154	#364	C₂²×D₃	1	4	yes	—
155	#359	—	3	8	no	—
156	#350	—	6	8	no	—
157	#346	—	3	2	no	—
158	#346	—	3	2	no	—
159	#346	C₄×D₃	1	2	yes	—
160	#346	C₂²×D₃	1	2	yes	—
161	#322	—	3	2	no	—
162	#322	—	6	1	no	—
163	C₂²×C₃×C₄	—	2	1	no	—
164	C₂²×C₃×C₄	C₄×D₄	1	1	yes	characteristic
165	C₂×#67	—	4	2	no	—
166	#235	—	2	4	no	—
167	#235	—	2	4	no	—
168	#235	—	4	2	no	—
169	#235	—	4	2	no	—
170	C₂×D₁₂	—	2	2	no	—
171	C₂×C₄×D₃	—	4	2	no	—
172	C₂×Dic₆	—	2	2	no	—
173	C₂×C₃×C₈	C₄×D₄	1	1	yes	characteristic
174	C₂×C₃×C₈	C₄×D₄	1	1	yes	characteristic
175	C₂×C₃×C₈	C₂²×C₈	1	1	yes	characteristic
176	C₂×C₃×C₈	C₂²×C₈	1	1	yes	characteristic
177	C₂×C₃×C₈	#131	1	1	yes	characteristic
178	C₂×C₃×C₈	#131	1	1	yes	characteristic
179	C₂×C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
180	C₂×C₃×C₈	AGL(1,ℤ/8)	1	2	yes	—
181	C₂×C₃×C₈	C₂³×C₄	1	1	yes	characteristic
182	C₂×C₃×C₈	C₂²×D₄	1	1	yes	characteristic
183	C₃×C₄²	—	2	1	no	—
184	C₃×C₄²	C₄×D₄	1	1	yes	characteristic
185	C₃×C₄²	C₂×#41	1	1	yes	characteristic
186	#208	—	2	4	no	—
187	#208	—	4	2	no	—
188	C₂×C₃⋊C₈	—	4	2	no	—
189	C₂×C₃⋊C₈	—	4	2	no	—
190	C₂×C₃⋊C₈	—	8	2	no	—
191	Dic₁₂	—	2	2	no	—
192	Dic₁₂	C₂²×C₈	1	4	yes	—
193	D₂₄	—	2	2	no	—
194	D₂₄	C₂²×C₈	1	4	yes	—
195	#204	—	2	4	no	—
196	#204	—	2	4	no	—
197	ΓL(1,𝔽₂₅)	—	4	4	no	—
198	ΓL(1,𝔽₂₅)	—	4	4	no	—
199	D₃×C₈	—	4	4	no	—
200	D₃×C₈	—	4	4	no	—
201	C₃×C₁₆	—	2	2	no	—
202	C₃×C₁₆	—	4	4	no	—
203	C₃×C₁₆	C₂²×C₈	1	2	yes	—
204	C₃×C₁₆	C₂²×C₈	1	2	yes	—
205	C₃⋊C₁₆	—	2	4	no	—
206	C₂×#41	—	6	2	no	—
207	#135	—	3	4	no	—
208	#135	—	3	4	no	—
209	#135	—	6	2	no	—
210	#135	—	6	2	no	—
211	C₂×Q₁₆	—	3	2	no	—
212	C₂×SD₁₆	—	6	2	no	—
213	C₂×D₈	—	3	2	no	—
214	#131	—	6	4	no	—
215	#131	—	6	4	no	—
216	#131	—	12	2	no	—
217	#131	—	12	2	no	—
218	C₂×M₁₆	—	6	2	no	—
219	C₂×M₁₆	—	12	2	no	—
220	C₂²×C₈	—	12	2	no	—
221	C₂²×C₈	—	12	1	no	—
222	C₂²×C₈	C₂×C₄×D₃	1	1	yes	characteristic
223	C₂²×C₈	D₃×D₄	1	1	yes	characteristic
224	C₂×C₄²	#239	1	1	yes	characteristic
225	Q₃₂	—	3	4	no	—
226	SD₃₂	—	3	8	no	—
227	D₁₆	—	3	4	no	—
228	M₃₂	—	2	4	no	—
229	M₃₂	—	4	2	no	—
230	M₃₂	—	12	8	no	—
231	C₂×C₁₆	—	2	1	no	—
232	C₂×C₁₆	—	2	1	no	—
233	C₂×C₁₆	—	3	4	no	—
234	C₂×C₁₆	—	4	2	no	—
235	C₂×C₁₆	—	6	2	no	—
236	C₂×C₁₆	—	6	2	no	—
237	C₂×C₁₆	—	12	8	no	—
238	C₂×C₁₆	D₃×C₈	1	1	yes	characteristic
239	C₂×C₁₆	D₃×C₈	1	1	yes	characteristic
240	C₂×C₁₆	D₃×C₈	1	2	yes	—
241	C₂×C₁₆	C₂×C₄×D₃	1	2	yes	—
242	C₂×C₁₆	C₂×C₄×D₃	1	2	yes	—
243	#108	—	3	4	no	—
244	#108	—	6	2	no	—
245	#104	—	6	4	no	—
246	#104	—	6	4	no	—
247	C₄×C₈	—	6	2	no	—
248	C₄×C₈	—	12	1	no	—
249	C₄×C₈	C₂×C₄×D₃	1	1	yes	characteristic
250	C₄×C₈	D₃×D₄	1	1	yes	characteristic
251	C₄×C₈	#239	1	1	yes	characteristic
252	C₄×C₈	C₂³×D₃	1	1	yes	characteristic
253	C₂³×C₃	#444	1	1	yes	characteristic
254	C₂²×D₃	—	4	2	no	—
255	C₂×C₃×C₄	—	2	1	no	—
256	C₂×C₃×C₄	—	2	1	no	—
257	C₂×C₃×C₄	—	4	2	no	—
258	C₂×C₃×C₄	C₈×D₄	1	1	yes	characteristic
259	C₂×C₃×C₄	C₈×D₄	1	1	yes	characteristic
260	C₂×C₃×C₄	#442	1	1	yes	characteristic
261	C₂×C₃×C₄	#442	1	1	yes	characteristic
262	C₂×C₃×C₄	#444	1	1	yes	characteristic
263	C₂×C₃×C₄	C₂×C₄×D₄	1	1	yes	characteristic
264	#67	—	4	4	no	—
265	#67	—	4	4	no	—
266	C₂×Dic₃	—	4	2	no	—
267	D₁₂	—	2	4	no	—
268	D₁₂	—	4	2	no	—
269	C₄×D₃	—	4	4	no	—
270	C₄×D₃	—	4	4	no	—
271	Dic₆	—	2	4	no	—
272	Dic₆	—	4	2	no	—
273	C₃×C₈	C₈×D₄	1	1	yes	characteristic
274	C₃×C₈	C₈×D₄	1	1	yes	characteristic
275	C₃×C₈	#442	1	2	yes	—
276	C₃×C₈	C₂³×C₈	1	1	yes	characteristic
277	C₃×C₈	C₂×#131	1	1	yes	characteristic
278	C₃×C₈	C₂×AGL(1,ℤ/8)	1	2	yes	—
279	C₃⋊C₈	—	4	4	no	—
280	C₃⋊C₈	—	8	2	no	—
281	#41	—	6	4	no	—
282	#41	—	6	4	no	—
283	#41	—	12	2	no	—
284	#41	—	12	2	no	—
285	C₂×Q₈	—	6	2	no	—
286	C₂×D₄	—	6	2	no	—
287	C₂×D₄	—	12	2	no	—
288	C₂²×C₄	—	2	1	no	—
289	C₂²×C₄	—	12	2	no	—
290	C₂²×C₄	#893	1	1	yes	characteristic
291	Q₁₆	—	3	4	no	—
292	Q₁₆	—	6	2	no	—
293	SD₁₆	—	6	4	no	—
294	SD₁₆	—	6	4	no	—
295	D₈	—	3	4	no	—
296	D₈	—	6	2	no	—
297	M₁₆	—	6	4	no	—
298	M₁₆	—	12	2	no	—
299	M₁₆	—	12	4	no	—
300	M₁₆	—	12	4	no	—
301	C₂×C₈	—	12	2	no	—
302	C₂×C₈	—	12	4	no	—
303	C₂×C₈	—	12	4	no	—
304	C₂×C₈	—	12	2	no	—
305	C₂×C₈	—	24	2	no	—
306	C₂×C₈	#893	1	1	yes	characteristic
307	C₂×C₈	#893	1	1	yes	characteristic
308	C₂×C₈	C₂×D₃×C₈	1	1	yes	characteristic
309	C₂×C₈	C₂×D₃×C₈	1	1	yes	characteristic
310	C₂×C₈	#907	1	1	yes	characteristic
311	C₂×C₈	#907	1	1	yes	characteristic
312	C₂×C₈	#914	1	2	yes	—
313	C₂×C₈	#914	1	2	yes	—
314	C₂×C₈	C₂²×C₄×D₃	1	1	yes	characteristic
315	C₂×C₈	C₂×D₃×D₄	1	1	yes	characteristic
316	C₄²	—	2	1	no	—
317	C₄²	#893	1	1	yes	characteristic
318	C₄²	C₂×#239	1	1	yes	characteristic
319	C₁₆	—	2	2	no	—
320	C₁₆	—	4	4	no	—
321	C₁₆	—	6	4	no	—
322	C₁₆	C₂×D₃×C₈	1	2	yes	—
323	C₁₆	C₂×D₃×C₈	1	2	yes	—
324	C₂²×C₃	—	2	2	no	—
325	C₂²×C₃	#1580	1	1	yes	characteristic
326	C₂²×C₃	#1580	1	1	yes	characteristic
327	C₂²×C₃	C₂×#444	1	1	yes	characteristic
328	C₂×D₃	—	4	4	no	—
329	C₂×D₃	—	8	4	no	—
330	C₃×C₄	—	4	1	no	—
331	C₃×C₄	—	4	1	no	—
332	C₃×C₄	#1580	1	1	yes	characteristic
333	C₃×C₄	#1580	1	1	yes	characteristic
334	C₃×C₄	C₂×C₈×D₄	1	1	yes	characteristic
335	C₃×C₄	C₂×#442	1	1	yes	characteristic
336	Dic₃	—	4	4	no	—
337	C₂³	—	12	2	no	—
338	C₂³	#4894	1	1	yes	characteristic
339	Q₈	—	6	4	no	—
340	Q₈	—	12	2	no	—
341	D₄	—	6	4	no	—
342	D₄	—	12	4	no	—
343	D₄	—	12	4	no	—
344	D₄	—	12	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	2	no	—
348	C₂×C₄	—	12	2	no	—
349	C₂×C₄	—	12	4	no	—
350	C₂×C₄	—	12	4	no	—
351	C₂×C₄	#4834	1	1	yes	characteristic
352	C₂×C₄	#4834	1	1	yes	characteristic
353	C₂×C₄	#4869	1	1	yes	characteristic
354	C₂×C₄	#4869	1	1	yes	characteristic
355	C₂×C₄	#4894	1	1	yes	characteristic
356	C₂×C₄	C₂×#893	1	1	yes	characteristic
357	C₈	—	12	4	no	—
358	C₈	—	24	2	no	—
359	C₈	#4834	1	1	yes	characteristic
360	C₈	#4834	1	1	yes	characteristic
361	C₈	#4869	1	2	yes	—
362	C₈	C₂²×D₃×C₈	1	1	yes	characteristic
363	C₈	C₂×#907	1	1	yes	characteristic
364	C₈	C₂×#914	1	2	yes	—
365	C₂×C₃	—	2	1	no	—
366	C₂×C₃	—	2	1	no	—
367	C₂×C₃	#12068	1	2	yes	complemented
368	C₂×C₃	C₂×#1580	1	1	yes	characteristic
369	D₃	—	8	4	no	—
370	C₂²	—	2	2	no	—
371	C₂²	—	12	4	no	—
372	C₂²	—	24	4	no	—
373	C₂²	#68518	1	1	yes	characteristic
374	C₂²	#68518	1	1	yes	characteristic
375	C₂²	C₂×#4894	1	1	yes	characteristic
376	C₄	—	4	1	no	—
377	C₄	—	4	1	no	—
378	C₄	—	12	4	no	—
379	C₄	#68518	1	1	yes	characteristic
380	C₄	#68518	1	1	yes	characteristic
381	C₄	C₂×#4834	1	1	yes	characteristic
382	C₄	C₂×#4869	1	1	yes	characteristic
383	C₃	C₂×#12068	1	1	yes	characteristic, complemented, Sylow
384	C₂	—	2	1	no	—
385	C₂	—	2	1	no	—
386	C₂	—	24	4	no	—
387	C₂	#10707076	1	2	yes	direct product factor
388	C₂	C₂×#68518	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#4869
commutator	C₃×C₈	C₂³×C₈
Frattini	C₄×C₈	C₂³×D₃
Fitting	C₂×C₃×#1369	C₂
socle	C₂²×C₃	#1580
prime radical	C₂×#4904	C₄
cosocle	C₃×C₄×C₈	C₂⁴
hypercenter	C₂×#1369	D₃
hypocenter	C₃	C₂×#12068
LCS	G ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#1369

Complements

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

N	H ≅ G/N	type
2. C₂×#69301	C₂	semidirect
3. #10707076	C₂	direct (N × H)
4. C₂×#67569	C₂	semidirect
5. C₂×#67511	C₂	semidirect
6. C₂×C₃×#1369	C₂	semidirect
19. #69301	C₂²	semidirect
22. C₃×#1369	C₂²	semidirect
23. #67569	C₂²	semidirect
24. #67511	C₂²	semidirect
28. C₂×#1369	D₃	semidirect
43. C₂×Dic₂₄	C₈	semidirect
44. #4904	C₈	semidirect
46. C₂×#800	C₈	semidirect
47. C₂×D₄₈	C₈	semidirect
79. #1369	C₂×D₃	semidirect
124. Dic₂₄	C₂×C₈	semidirect
125. #800	C₂×C₈	semidirect
126. D₄₈	C₂×C₈	semidirect
367. C₂×C₃	#12068	semidirect
383. C₃	C₂×#12068	semidirect
387. C₂	#10707076	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×#69085	2
C₂×#69301	2
#10707076	2
C₂×#67569	2
C₂×#67511	2
C₂×C₃×#1369	2
C₂×#12068	3

Subgroups — C2×#10707076 #72245987

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10707076 #72245987