Subgroups — C₂×#11748065 #14417977

Order 1920.

928 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91793	C₂	1	1	yes	maximal, characteristic
2	C₂×#91290	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#819	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#3947	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#90981	C₂	1	2	yes	maximal, complemented
6	#11748065	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65183	—	3	1	no	maximal, Hall
8	C₂×#6817	C₂²	1	2	yes	—
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×Dic₆₀	C₄	1	1	yes	characteristic, complemented
11	#91793	C₄	1	4	yes	complemented
12	#91793	C₂²	1	4	yes	—
13	C₂×D₁₂₀	C₄	1	1	yes	characteristic, complemented
14	C₂×#6706	C₄	1	2	yes	complemented
15	C₂×C₅×#208	C₂²	1	2	yes	—
16	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
17	C₂²×C₃×#168	C₂²	1	1	yes	characteristic
18	C₂×#6651	—	2	2	no	—
19	#91290	—	2	4	no	—
20	#91290	C₂²	1	8	yes	complemented
21	C₂×#6648	—	2	2	no	—
22	C₅×#819	C₂²	1	4	yes	complemented
23	C₃×#3947	C₂²	1	4	yes	complemented
24	#90981	C₂²	1	8	yes	complemented
25	C₂×#4717	—	5	1	no	maximal, Hall
26	C₂×#4053	—	3	2	no	—
27	C₂×#4050	—	3	1	no	—
28	C₂×C₅×#108	—	3	1	no	—
29	C₂×#3957	—	3	2	no	—
30	C₂×#3947	D₃	1	1	yes	characteristic, complemented
31	#65183	—	3	8	no	—
32	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×#1220	—	2	2	no	—
34	#6817	—	2	2	no	—
35	#6817	—	2	2	no	—
36	#6817	C₂×C₄	1	4	yes	—
37	#6817	C₂³	1	4	yes	—
38	C₂×D₆₀	C₂×C₄	1	2	yes	—
39	C₂×C₄×D₁₅	—	2	2	no	—
40	C₂×Dic₃₀	C₂×C₄	1	2	yes	—
41	C₂²×C₃×Dic₅	—	2	1	no	—
42	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	Dic₆₀	C₂×C₄	1	4	yes	complemented
47	D₁₂₀	C₂×C₄	1	4	yes	complemented
48	#6706	C₂×C₄	1	8	yes	complemented
49	C₅×#208	—	2	2	no	—
50	C₅×#208	C₂³	1	4	yes	—
51	C₂×C₅×C₃⋊C₈	—	2	2	no	—
52	C₃×C₄×Dic₅	—	2	1	no	—
53	C₃×C₄×Dic₅	C₂³	1	2	yes	—
54	C₂×C₃×#168	D₄	1	2	yes	—
55	C₂×C₃×#168	D₄	1	2	yes	—
56	C₂×C₃×#168	C₂³	1	2	yes	—
57	#6651	—	2	8	no	—
58	#6648	—	2	8	no	—
59	C₂×#904	—	5	1	no	—
60	C₂×#901	—	5	2	no	—
61	C₂×C₃×#97	—	5	1	no	—
62	C₂×#819	D₅	1	1	yes	characteristic, complemented
63	C₂×#805	—	5	2	no	—
64	#4717	—	5	8	no	—
65	C₂×#709	—	3	2	no	—
66	C₂×C₅×M₁₆	—	3	2	no	—
67	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
68	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
69	C₂²×#168	C₂×D₃	1	1	yes	characteristic
70	#4053	—	3	8	no	—
71	#4053	—	6	4	no	—
72	C₂×Dic₂₀	—	3	1	no	—
73	#4050	—	3	4	no	—
74	#4050	—	3	4	no	—
75	C₂×D₄₀	—	3	1	no	—
76	C₂×#677	—	3	2	no	—
77	C₂×#676	—	6	2	no	—
78	C₂×C₈×D₅	—	6	2	no	—
79	C₅×#108	—	3	4	no	—
80	#3957	—	3	8	no	—
81	#3947	C₂×D₃	1	4	yes	complemented
82	C₂×#444	—	15	1	no	Sylow
83	C₂³×C₃×C₅	#41	1	1	yes	characteristic
84	C₂²×D₁₅	—	2	2	no	—
85	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂×C₃×C₄×C₅	#41	1	1	yes	characteristic
90	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	#1220	—	2	4	no	—
92	#1220	—	2	4	no	—
93	C₂×Dic₁₅	—	2	2	no	—
94	D₆₀	—	2	2	no	—
95	D₆₀	C₂²×C₄	1	4	yes	—
96	C₄×D₁₅	—	2	4	no	—
97	C₄×D₁₅	—	2	4	no	—
98	Dic₃₀	—	2	2	no	—
99	Dic₃₀	C₂²×C₄	1	4	yes	—
100	C₂×C₃×Dic₅	—	2	2	no	—
101	C₂×C₃×Dic₅	—	4	2	no	—
102	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
103	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
104	C₃×#168	C₂×D₄	1	4	yes	—
105	C₅×C₃⋊C₈	—	2	4	no	—
106	C₂×#235	—	5	2	no	—
107	C₂²×C₃×C₈	—	5	1	no	—
108	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
109	C₂×C₃×C₄²	—	5	1	no	—
110	C₂×#208	C₂×D₅	1	2	yes	—
111	C₂×Dic₁₂	—	5	1	no	—
112	#904	—	5	4	no	—
113	#904	—	5	4	no	—
114	C₂×D₂₄	—	5	1	no	—
115	C₂×#204	—	5	2	no	—
116	#901	—	5	8	no	—
117	#901	—	10	4	no	—
118	C₂×ΓL(1,𝔽₂₅)	—	10	2	no	—
119	C₂×D₃×C₈	—	10	2	no	—
120	C₃×#97	—	5	4	no	—
121	#819	C₂×D₅	1	4	yes	complemented
122	#805	—	5	8	no	—
123	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
124	C₂×C₂²⋊D₅	—	6	2	no	—
125	C₂²×Dic₅	—	2	1	no	—
126	#709	—	3	4	no	—
127	#709	—	3	4	no	—
128	#709	—	6	2	no	—
129	#709	—	6	2	no	—
130	C₂×D₂₀	—	3	2	no	—
131	C₂×C₄×D₅	—	6	2	no	—
132	C₂×Dic₁₀	—	3	2	no	—
133	C₅×M₁₆	—	3	4	no	—
134	C₅×M₁₆	—	6	2	no	—
135	C₂×C₅×C₈	—	6	2	no	—
136	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
137	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
138	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
139	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
140	C₄×Dic₅	—	2	1	no	—
141	C₄×Dic₅	C₂²×D₃	1	2	yes	—
142	C₂×#168	D₁₂	1	2	yes	—
143	C₂×#168	D₁₂	1	2	yes	—
144	C₂×#168	C₂²×D₃	1	2	yes	—
145	Dic₂₀	—	3	4	no	—
146	D₄₀	—	3	4	no	—
147	#677	—	3	8	no	—
148	#676	—	6	8	no	—
149	C₈×D₅	—	6	8	no	—
150	C₂×#135	—	15	1	no	—
151	C₂×#131	—	15	2	no	—
152	#444	—	15	8	no	—
153	C₂×#108	—	3	1	no	—
154	C₂×#104	—	15	2	no	—
155	C₂×#97	—	5	1	no	—
156	C₂²×C₃×C₅	—	2	2	no	—
157	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
158	C₂²×C₃×C₅	C₄×D₄	1	1	yes	characteristic
159	C₂²×C₃×C₅	C₂×#41	1	1	yes	characteristic
160	C₂×D₁₅	—	2	4	no	—
161	C₂×D₁₅	—	4	4	no	—
162	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
163	C₃×C₄×C₅	C₄×D₄	1	1	yes	characteristic
164	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
165	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
166	Dic₁₅	—	2	4	no	—
167	C₃×Dic₅	—	4	2	no	—
168	C₂²×C₃×C₄	—	10	1	no	—
169	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
170	C₂×#67	—	10	2	no	—
171	#235	—	5	4	no	—
172	#235	—	5	4	no	—
173	#235	—	10	2	no	—
174	#235	—	10	2	no	—
175	C₂×D₁₂	—	5	2	no	—
176	C₂×C₄×D₃	—	10	2	no	—
177	C₂×Dic₆	—	5	2	no	—
178	C₂×C₃×C₈	—	5	2	no	—
179	C₂×C₃×C₈	—	5	2	no	—
180	C₂×C₃×C₈	—	5	2	no	—
181	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
182	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
183	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
184	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
185	C₃×C₄²	—	5	2	no	—
186	C₃×C₄²	—	10	1	no	—
187	#208	—	2	2	no	—
188	#208	C₂²×D₅	1	4	yes	—
189	C₂×C₃⋊C₈	—	2	2	no	—
190	Dic₁₂	—	5	4	no	—
191	D₂₄	—	5	4	no	—
192	#204	—	5	8	no	—
193	ΓL(1,𝔽₂₅)	—	10	8	no	—
194	D₃×C₈	—	10	8	no	—
195	C₂³×C₅	#235	1	1	yes	characteristic
196	C₂²×D₅	—	6	2	no	—
197	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂×D₁₂	1	1	yes	characteristic
201	C₂×C₄×C₅	#235	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
203	C₂²⋊D₅	—	6	4	no	—
204	C₂²⋊D₅	—	6	4	no	—
205	C₂×Dic₅	—	2	2	no	—
206	C₂×Dic₅	—	4	2	no	—
207	C₂×Dic₅	—	6	2	no	—
208	D₂₀	—	3	4	no	—
209	D₂₀	—	6	2	no	—
210	C₄×D₅	—	6	4	no	—
211	C₄×D₅	—	6	4	no	—
212	Dic₁₀	—	3	4	no	—
213	Dic₁₀	—	6	2	no	—
214	C₅×C₈	—	6	4	no	—
215	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
216	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
217	#168	C₂×D₁₂	1	4	yes	—
218	C₂×#41	—	15	2	no	—
219	#135	—	15	4	no	—
220	#135	—	15	4	no	—
221	C₂×Q₁₆	—	15	1	no	—
222	C₂×SD₁₆	—	15	2	no	—
223	C₂×D₈	—	15	1	no	—
224	#131	—	15	8	no	—
225	#131	—	30	4	no	—
226	C₂×M₁₆	—	3	2	no	—
227	C₂×M₁₆	—	30	2	no	—
228	C₂²×C₈	—	5	1	no	—
229	C₂²×C₈	—	30	2	no	—
230	C₂²×C₈	D₃×D₅	1	1	yes	characteristic
231	C₂×C₄²	—	5	1	no	—
232	#108	—	3	4	no	—
233	#104	—	15	8	no	—
234	#97	—	5	4	no	—
235	C₂×C₃×C₅	—	2	1	no	—
236	C₂×C₃×C₅	—	2	1	no	—
237	C₂×C₃×C₅	#444	1	2	yes	complemented
238	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
239	D₁₅	—	4	4	no	—
240	C₂³×C₃	#713	1	1	yes	characteristic
241	C₂²×D₃	—	10	2	no	—
242	C₂×C₃×C₄	—	10	2	no	—
243	C₂×C₃×C₄	—	20	2	no	—
244	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
245	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
246	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
247	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
248	C₂×C₃×C₄	#713	1	1	yes	characteristic
249	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
250	#67	—	10	4	no	—
251	#67	—	10	4	no	—
252	C₂×Dic₃	—	10	2	no	—
253	D₁₂	—	5	4	no	—
254	D₁₂	—	10	2	no	—
255	C₄×D₃	—	10	4	no	—
256	C₄×D₃	—	10	4	no	—
257	Dic₆	—	5	4	no	—
258	Dic₆	—	10	2	no	—
259	C₃×C₈	—	5	4	no	—
260	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
261	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
262	C₃⋊C₈	—	2	4	no	—
263	C₂²×C₅	—	2	2	no	—
264	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
265	C₂²×C₅	C₄×D₁₂	1	1	yes	characteristic
266	C₂²×C₅	C₂×#235	1	1	yes	characteristic
267	C₂×D₅	—	6	4	no	—
268	C₂×D₅	—	12	4	no	—
269	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
270	C₄×C₅	C₄×D₁₂	1	1	yes	characteristic
271	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
272	C₄×C₅	C₂²×D₁₂	1	1	yes	characteristic
273	Dic₅	—	4	2	no	—
274	Dic₅	—	6	4	no	—
275	#41	—	15	4	no	—
276	#41	—	15	4	no	—
277	#41	—	30	2	no	—
278	#41	—	30	2	no	—
279	C₂×Q₈	—	15	2	no	—
280	C₂×D₄	—	15	2	no	—
281	C₂×D₄	—	30	2	no	—
282	C₂²×C₄	—	10	1	no	—
283	C₂²×C₄	—	30	2	no	—
284	C₂²×C₄	C₂×D₃×D₅	1	1	yes	characteristic
285	Q₁₆	—	15	4	no	—
286	SD₁₆	—	15	8	no	—
287	D₈	—	15	4	no	—
288	M₁₆	—	3	4	no	—
289	M₁₆	—	6	2	no	—
290	M₁₆	—	30	8	no	—
291	C₂×C₈	—	5	2	no	—
292	C₂×C₈	—	5	2	no	—
293	C₂×C₈	—	5	2	no	—
294	C₂×C₈	—	6	2	no	—
295	C₂×C₈	—	30	8	no	—
296	C₂×C₈	#1200	1	1	yes	characteristic
297	C₂×C₈	#1200	1	1	yes	characteristic
298	C₂×C₈	#1200	1	2	yes	—
299	C₂×C₈	C₂×D₃×D₅	1	2	yes	—
300	C₄²	—	5	2	no	—
301	C₄²	—	10	1	no	—
302	C₃×C₅	C₂×#444	1	1	yes	characteristic, complemented, Hall
303	C₂²×C₃	—	2	2	no	—
304	C₂²×C₃	#4039	1	1	yes	characteristic
305	C₂²×C₃	#4039	1	1	yes	characteristic
306	C₂²×C₃	C₂×#713	1	1	yes	characteristic
307	C₂×D₃	—	10	4	no	—
308	C₂×D₃	—	20	4	no	—
309	C₃×C₄	—	20	2	no	—
310	C₃×C₄	#4039	1	1	yes	characteristic
311	C₃×C₄	#4039	1	1	yes	characteristic
312	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
313	C₃×C₄	C₂×D₄×D₅	1	1	yes	characteristic
314	Dic₃	—	10	4	no	—
315	C₂×C₅	—	2	1	no	—
316	C₂×C₅	—	2	1	no	—
317	C₂×C₅	#4717	1	2	yes	complemented
318	C₂×C₅	C₂×C₄×D₁₂	1	1	yes	characteristic
319	D₅	—	12	4	no	—
320	C₂³	—	30	2	no	—
321	C₂³	#6773	1	1	yes	characteristic
322	Q₈	—	15	4	no	—
323	Q₈	—	30	2	no	—
324	D₄	—	15	4	no	—
325	D₄	—	30	4	no	—
326	D₄	—	30	4	no	—
327	D₄	—	30	2	no	—
328	C₂×C₄	—	10	2	no	—
329	C₂×C₄	—	20	2	no	—
330	C₂×C₄	—	30	2	no	—
331	C₂×C₄	—	30	4	no	—
332	C₂×C₄	—	30	4	no	—
333	C₂×C₄	#6773	1	1	yes	characteristic
334	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
335	C₂×C₄	D₅×D₁₂	1	1	yes	characteristic
336	C₂×C₄	C₂×#1200	1	1	yes	characteristic
337	C₂×C₄	C₂×#1200	1	1	yes	characteristic
338	C₂×C₄	C₂²×D₃×D₅	1	1	yes	characteristic
339	C₈	—	5	4	no	—
340	C₈	—	6	4	no	—
341	C₈	C₂×#1200	1	2	yes	—
342	C₈	C₂×#1200	1	2	yes	—
343	C₂×C₃	—	2	1	no	—
344	C₂×C₃	—	2	1	no	—
345	C₂×C₃	#65183	1	2	yes	complemented
346	C₂×C₃	C₂×#4039	1	1	yes	characteristic
347	D₃	—	20	4	no	—
348	C₅	C₂×#4717	1	1	yes	characteristic, complemented, Sylow
349	C₂²	—	2	2	no	—
350	C₂²	—	30	4	no	—
351	C₂²	—	60	4	no	—
352	C₂²	#91418	1	1	yes	characteristic
353	C₂²	#91418	1	1	yes	characteristic
354	C₂²	C₂×#6773	1	1	yes	characteristic
355	C₄	—	20	2	no	—
356	C₄	—	30	4	no	—
357	C₄	#91418	1	1	yes	characteristic
358	C₄	#91418	1	1	yes	characteristic
359	C₄	C₂×D₅×D₁₂	1	1	yes	characteristic
360	C₄	C₂²×#1200	1	1	yes	characteristic
361	C₃	C₂×#65183	1	1	yes	characteristic, complemented, Sylow
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	60	4	no	—
365	C₂	#11748065	1	2	yes	direct product factor
366	C₂	C₂×#91418	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₅×D₁₂
commutator	C₃×C₄×C₅	C₂³×C₄
Frattini	C₂×C₄	C₂²×D₃×D₅
Fitting	C₂²×C₃×C₅×C₈	C₂²
socle	C₂²×C₃×C₅	C₄×D₄
prime radical	C₂×#91793	C₂
cosocle	C₂×C₃×C₄×C₅	C₂⁴
hypercenter	C₂²×C₈	D₃×D₅
hypocenter	C₃×C₅	C₂×#444
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
2. C₂×#91290	C₂	semidirect
3. C₂×C₅×#819	C₂	semidirect
4. C₂×C₃×#3947	C₂	semidirect
5. C₂×#90981	C₂	semidirect
6. #11748065	C₂	direct (N × H)
10. C₂×Dic₆₀	C₄	semidirect
11. #91793	C₄	semidirect
13. C₂×D₁₂₀	C₄	semidirect
14. C₂×#6706	C₄	semidirect
20. #91290	C₂²	semidirect
22. C₅×#819	C₂²	semidirect
23. C₃×#3947	C₂²	semidirect
24. #90981	C₂²	semidirect
30. C₂×#3947	D₃	semidirect
46. Dic₆₀	C₂×C₄	semidirect
47. D₁₂₀	C₂×C₄	semidirect
48. #6706	C₂×C₄	semidirect
62. C₂×#819	D₅	semidirect
81. #3947	C₂×D₃	semidirect
121. #819	C₂×D₅	semidirect
237. C₂×C₃×C₅	#444	semidirect
302. C₃×C₅	C₂×#444	semidirect
317. C₂×C₅	#4717	semidirect
345. C₂×C₃	#65183	semidirect
348. C₅	C₂×#4717	semidirect
361. C₃	C₂×#65183	semidirect
365. C₂	#11748065	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×#91793	2
C₂×#91290	2
C₂×C₅×#819	2
C₂×C₃×#3947	2
C₂×#90981	2
#11748065	2
C₂×#65183	3
C₂×#4717	5

Subgroups — C2×#11748065 #14417977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11748065 #14417977