Subgroups — C₂×#11752364 #14414833

Order 1920.

1104 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#6758	C₂	1	2	yes	maximal, complemented
2	C₂×C₃×#4039	C₂	1	1	yes	maximal, characteristic
3	C₂×#91232	C₂	1	2	yes	maximal, complemented
4	C₂×C₄⋊(C₃⋊#170)	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄×C₃⋊#170	C₂	1	1	yes	maximal, characteristic, complemented
6	#11752364	C₂	1	8	yes	maximal, direct product factor
7	C₂×#65705	—	3	1	no	maximal, Hall
8	C₂²×C₃×D₂₀	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₄×D₅	C₂²	1	2	yes	—
10	C₂²×C₃⋊#170	C₂²	1	2	yes	—
11	C₂²×C₃⋊#170	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊#170	C₂²	1	1	yes	characteristic
13	C₂×#6758	—	2	8	no	—
14	C₂×#6758	C₂²	1	8	yes	complemented
15	C₂×C₃×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
16	C₂×C₃×#685	C₄	1	2	yes	—
17	C₂×C₃×C₄×Dic₅	C₂²	1	1	yes	characteristic
18	C₃×#4039	C₄	1	4	yes	—
19	C₃×#4039	C₂²	1	4	yes	—
20	#91232	C₂²	1	8	yes	complemented
21	C₄⋊(C₃⋊#170)	C₂²	1	4	yes	complemented
22	C₄×C₃⋊#170	C₂²	1	4	yes	complemented
23	C₂×#5012	—	5	1	no	maximal, Hall
24	C₂²×#700	—	3	2	no	—
25	C₂×#4039	D₃	1	1	yes	characteristic
26	C₂×#4003	—	3	2	no	—
27	C₂×C₄⋊#170	—	3	1	no	—
28	C₂×C₄×#170	—	3	1	no	—
29	#65705	—	3	8	no	—
30	C₂³×C₃×D₅	C₂×C₄	1	2	yes	—
31	C₂²×C₃×C₄×C₅	C₂×C₄	1	2	yes	—
32	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₃×Dic₅	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₃×Dic₅	C₂³	1	1	yes	characteristic
35	C₂×C₃×D₂₀	—	2	4	no	—
36	C₂×C₃×D₂₀	C₂×C₄	1	4	yes	—
37	C₂×C₃×C₄×D₅	—	2	4	no	—
38	C₂×C₃×C₄×D₅	—	2	4	no	—
39	C₂×C₃×C₄×D₅	C₂×C₄	1	4	yes	—
40	C₂×C₃×C₄×D₅	C₂³	1	4	yes	—
41	C₂×C₃⋊#170	—	2	4	no	—
42	C₂×C₃⋊#170	—	2	2	no	—
43	C₂×C₃⋊#170	D₄	1	4	yes	complemented
44	C₂×C₃⋊#170	C₂³	1	4	yes	—
45	C₂×C₃⋊#170	C₂³	1	2	yes	—
46	C₂×C₃⋊#170	C₂³	1	2	yes	—
47	#6758	—	2	16	no	—
48	C₃×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₃×#685	C₂×C₄	1	8	yes	—
50	C₃×C₄×Dic₅	C₂×C₄	1	2	yes	—
51	C₃×C₄×Dic₅	C₂³	1	2	yes	—
52	C₂×C₃×C₄×D₄	—	5	1	no	—
53	C₂²×#208	—	5	2	no	—
54	C₂×#830	—	5	2	no	—
55	#5012	—	5	8	no	—
56	C₂×C₄⋊(C₃⋊C₈)	—	5	1	no	—
57	C₂×C₄×C₃⋊C₈	—	5	1	no	—
58	C₂²×D₂₀	Dic₃	1	1	yes	characteristic
59	C₂²×C₄×D₅	C₂×D₃	1	2	yes	—
60	C₂²×#170	—	3	1	no	—
61	C₂²×#170	—	3	2	no	—
62	C₂²×#170	—	3	1	no	—
63	C₂×#700	—	3	8	no	—
64	C₂×#700	—	6	8	no	—
65	C₂×C₅×C₄⋊C₄	Dic₃	1	1	yes	characteristic
66	C₂×#685	Dic₃	1	2	yes	—
67	C₂×C₄×Dic₅	C₂×D₃	1	1	yes	characteristic
68	#4039	Dic₃	1	4	yes	—
69	#4039	C₂×D₃	1	4	yes	—
70	#4003	—	3	8	no	—
71	C₄⋊#170	—	3	4	no	—
72	C₄×#170	—	3	4	no	—
73	C₂×#436	—	15	1	no	Sylow
74	C₂³×C₃×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₃×D₅	—	2	4	no	—
76	C₂²×C₃×D₅	—	2	4	no	—
77	C₂²×C₃×D₅	—	2	4	no	—
78	C₂²×C₃×D₅	C₂²×C₄	1	4	yes	—
79	C₂×C₃×C₄×C₅	—	2	4	no	—
80	C₂×C₃×C₄×C₅	M₁₆	1	4	yes	complemented
81	C₂×C₃×C₄×C₅	C₂²×C₄	1	4	yes	—
82	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂×C₃×Dic₅	—	2	2	no	—
84	C₂×C₃×Dic₅	C₂²×C₄	1	2	yes	—
85	C₂×C₃×Dic₅	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₃×Dic₅	C₂×D₄	1	2	yes	—
87	C₂×C₃×Dic₅	#41	1	2	yes	—
88	C₂×C₃×Dic₅	C₂⁴	1	1	yes	characteristic
89	C₃×D₂₀	—	2	8	no	—
90	C₃×C₄×D₅	—	2	8	no	—
91	C₃×C₄×D₅	—	2	4	no	—
92	C₃×C₄×D₅	—	2	4	no	—
93	C₃⋊#170	—	2	4	no	—
94	C₃⋊#170	—	2	2	no	—
95	C₃⋊#170	C₂×D₄	1	4	yes	complemented
96	C₂²×C₃×D₄	—	5	1	no	—
97	C₂³×C₃×C₄	—	5	2	no	—
98	C₃×C₄×D₄	—	5	4	no	—
99	C₃×C₄×D₄	—	5	4	no	—
100	C₂×C₃×C₄⋊C₄	AGL(1,𝔽₅)	1	1	yes	characteristic
101	C₂×C₃×C₂²⋊C₄	—	5	2	no	—
102	C₂×C₃×C₄²	—	5	1	no	—
103	C₂×#208	—	5	8	no	—
104	C₂×#208	—	10	8	no	—
105	C₂²×C₃⋊C₈	—	5	1	no	—
106	C₂²×C₃⋊C₈	—	5	1	no	—
107	C₂²×C₃⋊C₈	—	5	2	no	—
108	#830	—	5	8	no	—
109	C₄⋊(C₃⋊C₈)	—	5	4	no	—
110	C₄×C₃⋊C₈	—	5	4	no	—
111	C₂³×D₅	C₂×Dic₃	1	2	yes	—
112	C₂²×C₄×C₅	C₂×Dic₃	1	2	yes	—
113	C₂²×C₄×C₅	C₂×Dic₃	1	1	yes	characteristic
114	C₂²×Dic₅	C₂×Dic₃	1	1	yes	characteristic
115	C₂²×Dic₅	C₂²×D₃	1	1	yes	characteristic
116	C₂×D₂₀	—	2	4	no	—
117	C₂×D₂₀	C₂×Dic₃	1	4	yes	—
118	C₂×C₄×D₅	—	2	4	no	—
119	C₂×C₄×D₅	—	2	4	no	—
120	C₂×C₄×D₅	C₂×Dic₃	1	4	yes	—
121	C₂×C₄×D₅	C₂²×D₃	1	4	yes	—
122	C₂×#170	—	3	4	no	—
123	C₂×#170	—	3	2	no	—
124	C₂×#170	—	3	4	no	—
125	C₂×#170	—	3	2	no	—
126	C₂×#170	—	6	4	no	—
127	C₂×#170	—	6	2	no	—
128	#700	—	6	16	no	—
129	C₅×C₄⋊C₄	C₂×Dic₃	1	4	yes	—
130	#685	C₂×Dic₃	1	8	yes	—
131	C₄×Dic₅	C₂×Dic₃	1	2	yes	—
132	C₄×Dic₅	C₂²×D₃	1	2	yes	—
133	C₂²×M₁₆	—	15	2	no	—
134	C₂×C₄×D₄	—	5	1	no	—
135	#436	—	15	8	no	—
136	C₂×C₄⋊C₈	—	15	1	no	—
137	C₂×C₂²⋊C₈	—	15	2	no	—
138	C₂×C₄×C₈	—	15	1	no	—
139	C₂²×C₃×C₅	C₄×D₄	1	2	yes	—
140	C₂²×C₃×C₅	C₂×M₁₆	1	2	yes	—
141	C₂²×C₃×C₅	#131	1	2	yes	—
142	C₂²×C₃×C₅	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₃×D₅	—	2	4	no	—
144	C₂×C₃×D₅	—	2	4	no	—
145	C₂×C₃×D₅	—	2	4	no	—
146	C₂×C₃×D₅	—	4	8	no	—
147	C₃×C₄×C₅	—	2	4	no	—
148	C₃×C₄×C₅	C₂×M₁₆	1	4	yes	complemented
149	C₃×Dic₅	—	2	2	no	—
150	C₃×Dic₅	C₄×D₄	1	2	yes	—
151	C₃×Dic₅	C₂²×D₄	1	1	yes	characteristic
152	C₃×Dic₅	C₂×#41	1	1	yes	characteristic
153	C₂⁴×C₃	—	5	2	no	—
154	C₂×C₃×D₄	—	5	4	no	—
155	C₂×C₃×D₄	—	10	4	no	—
156	C₂²×C₃×C₄	—	5	1	no	—
157	C₂²×C₃×C₄	—	5	1	no	—
158	C₂²×C₃×C₄	—	5	4	no	—
159	C₂²×C₃×C₄	—	5	4	no	—
160	C₂²×C₃×C₄	—	10	4	no	—
161	C₂²×C₃×C₄	—	10	4	no	—
162	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
163	C₂²×C₃×C₄	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
164	C₃×C₄⋊C₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
165	C₃×C₂²⋊C₄	—	5	8	no	—
166	C₃×C₄²	—	5	2	no	—
167	C₃×C₄²	—	5	2	no	—
168	#208	—	10	16	no	—
169	C₂×C₃⋊C₈	—	5	2	no	—
170	C₂×C₃⋊C₈	—	5	4	no	—
171	C₂×C₃⋊C₈	—	5	2	no	—
172	C₂×C₃⋊C₈	—	5	4	no	—
173	C₂×C₃⋊C₈	—	10	2	no	—
174	C₂×C₃⋊C₈	—	10	4	no	—
175	C₂³×C₅	C₂²×Dic₃	1	1	yes	characteristic
176	C₂²×D₅	—	2	4	no	—
177	C₂²×D₅	—	2	4	no	—
178	C₂²×D₅	—	2	4	no	—
179	C₂²×D₅	C₂²×Dic₃	1	4	yes	—
180	C₂×C₄×C₅	—	2	4	no	—
181	C₂×C₄×C₅	#208	1	4	yes	complemented
182	C₂×C₄×C₅	C₂²×Dic₃	1	4	yes	—
183	C₂×C₄×C₅	C₂²×Dic₃	1	2	yes	—
184	C₂×Dic₅	—	2	2	no	—
185	C₂×Dic₅	D₃×D₄	1	2	yes	—
186	C₂×Dic₅	#237	1	2	yes	—
187	C₂×Dic₅	C₂²×Dic₃	1	2	yes	—
188	C₂×Dic₅	C₂²×Dic₃	1	1	yes	characteristic
189	C₂×Dic₅	C₂³×D₃	1	1	yes	characteristic
190	D₂₀	—	2	8	no	—
191	C₄×D₅	—	2	8	no	—
192	C₄×D₅	—	2	4	no	—
193	C₄×D₅	—	2	4	no	—
194	#170	—	3	4	no	—
195	#170	—	6	4	no	—
196	#170	—	6	2	no	—
197	C₂²×D₄	—	5	1	no	—
198	C₂³×C₄	—	5	2	no	—
199	C₂×M₁₆	—	15	8	no	—
200	C₂×M₁₆	—	30	8	no	—
201	C₂²×C₈	—	15	1	no	—
202	C₂²×C₈	—	15	2	no	—
203	C₂²×C₈	—	15	1	no	—
204	C₄×D₄	—	5	4	no	—
205	C₄×D₄	—	5	4	no	—
206	C₂×C₄⋊C₄	ΓL(1,𝔽₁₆)	1	1	yes	characteristic
207	C₂×C₂²⋊C₄	—	5	2	no	—
208	C₂×C₄²	—	5	1	no	—
209	C₄⋊C₈	—	15	4	no	—
210	C₂²⋊C₈	—	15	8	no	—
211	C₄×C₈	—	15	4	no	—
212	C₂×C₃×C₅	#436	1	4	yes	complemented
213	C₂×C₃×C₅	C₂×C₄×D₄	1	1	yes	characteristic
214	C₂×C₃×C₅	C₂²×M₁₆	1	1	yes	characteristic
215	C₂×C₃×C₅	C₂×#131	1	1	yes	characteristic
216	C₃×D₅	—	4	4	no	—
217	C₂³×C₃	—	5	4	no	—
218	C₂³×C₃	—	10	4	no	—
219	C₂³×C₃	—	10	4	no	—
220	C₂³×C₃	—	10	4	no	—
221	C₂³×C₃	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
222	C₃×D₄	—	10	8	no	—
223	C₂×C₃×C₄	—	2	4	no	—
224	C₂×C₃×C₄	—	5	2	no	—
225	C₂×C₃×C₄	—	5	2	no	—
226	C₂×C₃×C₄	—	5	1	no	—
227	C₂×C₃×C₄	—	5	1	no	—
228	C₂×C₃×C₄	—	5	2	no	—
229	C₂×C₃×C₄	—	10	4	no	—
230	C₂×C₃×C₄	—	10	8	no	—
231	C₂×C₃×C₄	—	10	4	no	—
232	C₂×C₃×C₄	—	10	2	no	—
233	C₂×C₃×C₄	#700	1	4	yes	complemented
234	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
235	C₂×C₃×C₄	C₂²×AGL(1,𝔽₅)	1	2	yes	—
236	C₃⋊C₈	—	5	4	no	—
237	C₃⋊C₈	—	10	2	no	—
238	C₃⋊C₈	—	10	4	no	—
239	C₂²×C₅	C₂×#208	1	2	yes	—
240	C₂²×C₅	D₄×Dic₃	1	2	yes	—
241	C₂²×C₅	#948	1	2	yes	—
242	C₂²×C₅	C₂³×Dic₃	1	1	yes	characteristic
243	C₂×D₅	—	2	4	no	—
244	C₂×D₅	—	2	4	no	—
245	C₂×D₅	—	2	4	no	—
246	C₂×D₅	—	4	8	no	—
247	C₄×C₅	—	2	4	no	—
248	C₄×C₅	C₂×#208	1	4	yes	complemented
249	Dic₅	—	2	2	no	—
250	Dic₅	D₄×Dic₃	1	2	yes	—
251	Dic₅	C₂×D₃×D₄	1	1	yes	characteristic
252	Dic₅	C₂×#237	1	1	yes	characteristic
253	C₂⁴	—	5	2	no	—
254	C₂×D₄	—	5	4	no	—
255	C₂×D₄	—	10	4	no	—
256	C₂²×C₄	—	5	1	no	—
257	C₂²×C₄	—	5	1	no	—
258	C₂²×C₄	—	5	4	no	—
259	C₂²×C₄	—	5	4	no	—
260	C₂²×C₄	—	10	4	no	—
261	C₂²×C₄	—	10	4	no	—
262	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	2	yes	—
263	C₂²×C₄	C₂×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
264	M₁₆	—	30	16	no	—
265	C₂×C₈	—	15	2	no	—
266	C₂×C₈	—	15	4	no	—
267	C₂×C₈	—	15	2	no	—
268	C₂×C₈	—	15	4	no	—
269	C₂×C₈	—	30	4	no	—
270	C₂×C₈	—	30	2	no	—
271	C₄⋊C₄	C₂×ΓL(1,𝔽₁₆)	1	4	yes	—
272	C₂²⋊C₄	—	5	8	no	—
273	C₄²	—	5	2	no	—
274	C₄²	—	5	2	no	—
275	C₃×C₅	C₂×#436	1	1	yes	characteristic, complemented, Hall
276	C₂²×C₃	—	10	4	no	—
277	C₂²×C₃	—	10	4	no	—
278	C₂²×C₃	—	10	4	no	—
279	C₂²×C₃	—	20	8	no	—
280	C₂²×C₃	C₂×#700	1	2	yes	—
281	C₂²×C₃	D₄×AGL(1,𝔽₅)	1	2	yes	—
282	C₂²×C₃	#4133	1	2	yes	—
283	C₂²×C₃	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
284	C₃×C₄	—	2	4	no	—
285	C₃×C₄	—	5	2	no	—
286	C₃×C₄	—	5	1	no	—
287	C₃×C₄	—	5	1	no	—
288	C₃×C₄	—	10	2	no	—
289	C₃×C₄	C₂×#700	1	4	yes	complemented
290	C₂×C₅	#5012	1	4	yes	complemented
291	C₂×C₅	C₂²×#208	1	1	yes	characteristic
292	C₂×C₅	C₂×D₄×Dic₃	1	1	yes	characteristic
293	C₂×C₅	C₂×#948	1	1	yes	characteristic
294	D₅	—	4	4	no	—
295	C₂³	—	5	4	no	—
296	C₂³	—	10	4	no	—
297	C₂³	—	10	4	no	—
298	C₂³	—	10	4	no	—
299	C₂³	C₂²×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
300	D₄	—	10	8	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	5	2	no	—
303	C₂×C₄	—	5	2	no	—
304	C₂×C₄	—	5	1	no	—
305	C₂×C₄	—	5	1	no	—
306	C₂×C₄	—	5	2	no	—
307	C₂×C₄	—	10	4	no	—
308	C₂×C₄	—	10	4	no	—
309	C₂×C₄	—	10	8	no	—
310	C₂×C₄	—	10	2	no	—
311	C₂×C₄	#6758	1	4	yes	complemented
312	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	4	yes	—
313	C₂×C₄	C₂²×ΓL(1,𝔽₁₆)	1	2	yes	—
314	C₈	—	15	4	no	—
315	C₈	—	30	4	no	—
316	C₈	—	30	2	no	—
317	C₂×C₃	—	20	4	no	—
318	C₂×C₃	#65705	1	4	yes	complemented
319	C₂×C₃	C₂²×#700	1	1	yes	characteristic
320	C₂×C₃	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
321	C₂×C₃	C₂×#4133	1	1	yes	characteristic
322	C₅	C₂×#5012	1	1	yes	characteristic, complemented, Sylow
323	C₂²	—	10	4	no	—
324	C₂²	—	10	4	no	—
325	C₂²	—	10	4	no	—
326	C₂²	—	20	8	no	—
327	C₂²	C₂×#6758	1	2	yes	—
328	C₂²	D₄×ΓL(1,𝔽₁₆)	1	2	yes	—
329	C₂²	#91992	1	2	yes	—
330	C₂²	C₂³×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
331	C₄	—	2	4	no	—
332	C₄	—	5	1	no	—
333	C₄	—	5	1	no	—
334	C₄	—	5	2	no	—
335	C₄	—	10	2	no	—
336	C₄	C₂×#6758	1	4	yes	complemented
337	C₃	C₂×#65705	1	1	yes	characteristic, complemented, Sylow
338	C₂	—	20	4	no	—
339	C₂	#11752364	1	4	yes	direct product factor
340	C₂	C₂²×#6758	1	1	yes	characteristic
341	C₂	C₂×D₄×ΓL(1,𝔽₁₆)	1	1	yes	characteristic
342	C₂	C₂×#91992	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×ΓL(1,𝔽₁₆)
commutator	C₂²×C₃×C₅	C₂³×C₄
Frattini	C₂²	C₂³×ΓL(1,𝔽₁₆)
Fitting	C₂×C₃×C₅×C₄⋊C₄	C₄
socle	C₂³×C₃×C₅	C₂²×C₄
prime radical	C₂²×C₃×D₂₀	C₄
cosocle	C₂×C₃×Dic₅	C₂⁴
hypercenter	C₂×C₄⋊C₄	ΓL(1,𝔽₁₆)
hypocenter	C₃×C₅	C₂×#436
LCS	G ▷ C₂²×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#6758	C₂	semidirect
3. C₂×#91232	C₂	semidirect
4. C₂×C₄⋊(C₃⋊#170)	C₂	semidirect
5. C₂×C₄×C₃⋊#170	C₂	semidirect
6. #11752364	C₂	direct (N × H)
14. C₂×#6758	C₂²	semidirect
20. #91232	C₂²	semidirect
21. C₄⋊(C₃⋊#170)	C₂²	semidirect
22. C₄×C₃⋊#170	C₂²	semidirect
43. C₂×C₃⋊#170	D₄	semidirect
80. C₂×C₃×C₄×C₅	M₁₆	semidirect
95. C₃⋊#170	C₂×D₄	semidirect
148. C₃×C₄×C₅	C₂×M₁₆	semidirect
181. C₂×C₄×C₅	#208	semidirect
212. C₂×C₃×C₅	#436	semidirect
233. C₂×C₃×C₄	#700	semidirect
248. C₄×C₅	C₂×#208	semidirect
275. C₃×C₅	C₂×#436	semidirect
289. C₃×C₄	C₂×#700	semidirect
290. C₂×C₅	#5012	semidirect
311. C₂×C₄	#6758	semidirect
318. C₂×C₃	#65705	semidirect
322. C₅	C₂×#5012	semidirect
336. C₄	C₂×#6758	semidirect
337. C₃	C₂×#65705	semidirect
339. C₂	#11752364	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×#6758	2
C₂×C₃×#4039	2
C₂×#91232	2
C₂×C₄⋊(C₃⋊#170)	2
C₂×C₄×C₃⋊#170	2
#11752364	2
C₂×#65705	3
C₂×#5012	5

Subgroups — C2×#11752364 #14414833

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11752364 #14414833