Subgroups — C₂×#11751290 #14428034

Order 1920.

984 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×C₅×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#91835	C₂	1	2	yes	maximal
3	C₂²×#6705	C₂	1	1	yes	maximal, characteristic
4	C₂×#91119	C₂	1	2	yes	maximal, complemented
5	#11751290	C₂	1	8	yes	maximal, direct product factor
6	C₂×#91101	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#65478	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₄×D₁₅	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₅×Q₁₆	—	2	4	no	—
11	C₂×C₃×C₅×Q₁₆	C₂²	1	4	yes	complemented
12	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
13	#91835	—	2	4	no	—
14	#91835	C₂²	1	8	yes	—
15	C₂×#6714	—	2	2	no	—
16	C₂×C₄⋊Dic₁₅	C₂²	1	2	yes	—
17	C₂×#6712	—	2	2	no	—
18	C₂²×C₃⋊#168	C₂²	1	1	yes	characteristic
19	C₂×#6705	—	2	4	no	—
20	C₂×#6705	C₂²	1	4	yes	—
21	#91119	C₂²	1	8	yes	complemented
22	#91101	C₂²	1	4	yes	complemented
23	C₂×#5189	—	5	1	no	maximal, Hall
24	C₂²×C₅×Q₁₆	D₃	1	1	yes	characteristic, complemented
25	C₂×#4092	—	3	2	no	—
26	C₂²×#676	—	3	1	no	—
27	C₂×#3967	—	3	2	no	—
28	#65478	—	3	8	no	—
29	C₂×#3949	—	3	1	no	—
30	C₂³×D₁₅	D₄	1	1	yes	characteristic
31	C₂×C₃×C₅×Q₈	—	2	2	no	—
32	C₂×C₃×C₅×Q₈	—	2	4	no	—
33	C₂×C₃×C₅×Q₈	—	2	4	no	—
34	C₂×C₃×C₅×Q₈	C₂³	1	4	yes	—
35	C₂²×C₃×C₄×C₅	—	2	2	no	—
36	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
37	C₂²×Dic₁₅	—	2	2	no	—
38	C₂²×Dic₁₅	D₄	1	1	yes	characteristic
39	C₂×C₄×D₁₅	—	2	2	no	—
40	C₂×C₄×D₁₅	—	2	2	no	—
41	C₂×C₄×D₁₅	D₄	1	2	yes	—
42	C₂×C₄×D₁₅	C₂³	1	2	yes	—
43	C₃×C₅×Q₁₆	—	2	8	no	—
44	C₂×C₃×C₅×C₈	D₄	1	4	yes	—
45	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
46	#6714	—	2	8	no	—
47	C₄⋊Dic₁₅	—	2	2	no	—
48	C₄⋊Dic₁₅	C₂³	1	4	yes	—
49	#6712	—	2	8	no	—
50	C₂×C₃⋊#168	—	2	2	no	—
51	C₂×C₃⋊#168	C₂³	1	2	yes	—
52	#6705	—	2	8	no	—
53	C₂²×C₃×Q₁₆	D₅	1	1	yes	characteristic, complemented
54	C₂×#946	—	5	2	no	—
55	C₂²×ΓL(1,𝔽₂₅)	—	5	1	no	—
56	C₂×#835	—	5	2	no	—
57	#5189	—	5	8	no	—
58	C₂×#817	—	5	1	no	—
59	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
60	C₂²×C₄×D₅	—	3	1	no	—
61	C₂×C₅×Q₁₆	—	2	4	no	—
62	C₂×C₅×Q₁₆	C₂×D₃	1	4	yes	complemented
63	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
64	#4092	—	3	8	no	—
65	#4092	—	6	4	no	—
66	C₂×#685	—	6	2	no	—
67	C₂×C₄⋊Dic₅	—	3	2	no	—
68	C₂×#683	—	6	2	no	—
69	C₂²×#168	—	3	1	no	—
70	C₂×#676	—	3	4	no	—
71	C₂×#676	—	6	4	no	—
72	#3967	—	3	8	no	—
73	#3949	—	3	4	no	—
74	C₂×#470	—	15	1	no	Sylow
75	C₂³×C₃×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×D₁₅	—	2	2	no	—
77	C₂²×D₁₅	—	2	4	no	—
78	C₂²×D₁₅	C₂×D₄	1	2	yes	—
79	C₃×C₅×Q₈	—	2	4	no	—
80	C₃×C₅×Q₈	—	2	4	no	—
81	C₃×C₅×Q₈	—	4	4	no	—
82	C₂×C₃×C₄×C₅	—	2	4	no	—
83	C₂×C₃×C₄×C₅	—	4	4	no	—
84	C₂×C₃×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂×C₃×C₄×C₅	#41	1	2	yes	—
87	C₂×C₃×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₂×Dic₁₅	—	2	2	no	—
89	C₂×Dic₁₅	—	2	4	no	—
90	C₂×Dic₁₅	—	4	4	no	—
91	C₂×Dic₁₅	C₂×D₄	1	2	yes	—
92	C₄×D₁₅	—	2	4	no	—
93	C₄×D₁₅	—	2	2	no	—
94	C₄×D₁₅	—	2	2	no	—
95	C₃×C₅×C₈	C₂×D₄	1	4	yes	—
96	C₃⋊#168	—	2	2	no	—
97	C₂²×C₃×Q₈	C₂×D₅	1	2	yes	—
98	C₂²×C₄×D₃	—	5	1	no	—
99	C₂×C₃×Q₁₆	—	2	4	no	—
100	C₂×C₃×Q₁₆	C₂×D₅	1	4	yes	complemented
101	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
102	#946	—	5	8	no	—
103	#946	—	10	4	no	—
104	C₂×(C₂×D₃)⋊C₄	—	10	2	no	—
105	C₂×C₄⋊Dic₃	—	5	2	no	—
106	C₂×Dic₃⋊C₄	—	10	2	no	—
107	C₂²×C₃⋊C₈	—	5	1	no	—
108	C₂×ΓL(1,𝔽₂₅)	—	5	4	no	—
109	C₂×ΓL(1,𝔽₂₅)	—	10	4	no	—
110	#835	—	5	8	no	—
111	#817	—	5	4	no	—
112	C₂³×D₅	—	3	1	no	—
113	C₂×C₅×Q₈	—	2	2	no	—
114	C₂×C₅×Q₈	—	2	4	no	—
115	C₂×C₅×Q₈	—	2	4	no	—
116	C₂×C₅×Q₈	C₂²×D₃	1	4	yes	—
117	C₂²×C₄×C₅	—	2	2	no	—
118	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
119	C₂²×Dic₅	—	3	1	no	—
120	C₂²×Dic₅	—	6	2	no	—
121	C₂×C₄×D₅	—	3	2	no	—
122	C₂×C₄×D₅	—	3	2	no	—
123	C₂×C₄×D₅	—	6	2	no	—
124	C₂×C₄×D₅	—	6	2	no	—
125	C₅×Q₁₆	—	2	8	no	—
126	C₂×C₅×C₈	#67	1	4	yes	—
127	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
128	#685	—	6	8	no	—
129	C₄⋊Dic₅	—	3	4	no	—
130	C₄⋊Dic₅	—	6	2	no	—
131	#683	—	6	8	no	—
132	C₂×#168	—	3	2	no	—
133	C₂×#168	—	6	2	no	—
134	#676	—	6	8	no	—
135	C₂²×Q₁₆	D₁₅	1	1	yes	characteristic, complemented
136	C₂²×M₁₆	—	15	1	no	—
137	C₂×C₂²⋊Q₈	—	15	2	no	—
138	#470	—	15	8	no	—
139	C₂×#106	—	15	1	no	—
140	C₂×#103	—	15	2	no	—
141	C₂²×C₃×C₅	#121	1	2	yes	—
142	C₂²×C₃×C₅	#137	1	4	yes	—
143	C₂²×C₃×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₁₅	—	2	2	no	—
145	C₂×D₁₅	—	2	4	no	—
146	C₂×D₁₅	—	4	4	no	—
147	C₃×C₄×C₅	—	4	4	no	—
148	C₃×C₄×C₅	#121	1	2	yes	—
149	C₃×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
150	C₃×C₄×C₅	C₂×#41	1	1	yes	characteristic
151	Dic₁₅	—	2	2	no	—
152	Dic₁₅	—	4	4	no	—
153	C₂³×D₃	—	5	1	no	—
154	C₂×C₃×Q₈	—	2	2	no	—
155	C₂×C₃×Q₈	—	2	4	no	—
156	C₂×C₃×Q₈	—	2	4	no	—
157	C₂×C₃×Q₈	C₂²×D₅	1	4	yes	—
158	C₂²×C₃×C₄	—	2	2	no	—
159	C₂²×C₃×C₄	C₂²×D₅	1	1	yes	characteristic
160	C₂²×Dic₃	—	5	1	no	—
161	C₂²×Dic₃	—	10	2	no	—
162	C₂×C₄×D₃	—	5	2	no	—
163	C₂×C₄×D₃	—	5	2	no	—
164	C₂×C₄×D₃	—	10	2	no	—
165	C₂×C₄×D₃	—	10	2	no	—
166	C₃×Q₁₆	—	2	8	no	—
167	C₂×C₃×C₈	C₂²⋊D₅	1	4	yes	—
168	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
169	(C₂×D₃)⋊C₄	—	10	8	no	—
170	C₄⋊Dic₃	—	5	4	no	—
171	C₄⋊Dic₃	—	10	2	no	—
172	Dic₃⋊C₄	—	10	8	no	—
173	C₂×C₃⋊C₈	—	5	2	no	—
174	C₂×C₃⋊C₈	—	10	2	no	—
175	ΓL(1,𝔽₂₅)	—	10	8	no	—
176	C₂³×C₅	D₃×D₄	1	1	yes	characteristic
177	C₂²×D₅	—	3	2	no	—
178	C₂²×D₅	—	6	4	no	—
179	C₂²×D₅	—	6	2	no	—
180	C₅×Q₈	—	2	4	no	—
181	C₅×Q₈	—	2	4	no	—
182	C₅×Q₈	—	4	4	no	—
183	C₂×C₄×C₅	—	2	4	no	—
184	C₂×C₄×C₅	—	4	4	no	—
185	C₂×C₄×C₅	D₃×D₄	1	1	yes	characteristic
186	C₂×C₄×C₅	#237	1	2	yes	—
187	C₂×C₄×C₅	C₂×#67	1	2	yes	—
188	C₂×C₄×C₅	C₂³×D₃	1	1	yes	characteristic
189	C₂×Dic₅	—	3	2	no	—
190	C₂×Dic₅	—	6	2	no	—
191	C₂×Dic₅	—	6	4	no	—
192	C₂×Dic₅	—	12	4	no	—
193	C₄×D₅	—	6	4	no	—
194	C₄×D₅	—	6	2	no	—
195	C₄×D₅	—	6	2	no	—
196	C₅×C₈	C₂×#67	1	4	yes	—
197	#168	—	6	2	no	—
198	C₂²×Q₈	C₂×D₁₅	1	2	yes	—
199	C₂³×C₄	—	15	1	no	—
200	C₂×Q₁₆	—	2	4	no	—
201	C₂×Q₁₆	C₂×D₁₅	1	4	yes	complemented
202	C₂×M₁₆	—	15	4	no	—
203	C₂×M₁₆	—	30	4	no	—
204	C₂²×C₈	—	15	1	no	—
205	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
206	C₂²⋊Q₈	—	15	8	no	—
207	C₂²⋊Q₈	—	30	4	no	—
208	C₂×C₄⋊C₄	—	15	2	no	—
209	C₂×C₄⋊C₄	—	30	2	no	—
210	C₂×C₂²⋊C₄	—	30	2	no	—
211	#106	—	15	4	no	—
212	#103	—	15	8	no	—
213	C₂×C₃×C₅	#470	1	4	yes	complemented
214	C₂×C₃×C₅	C₂×#121	1	1	yes	characteristic
215	C₂×C₃×C₅	C₂×#137	1	2	yes	—
216	D₁₅	—	4	2	no	—
217	C₂³×C₃	D₄×D₅	1	1	yes	characteristic
218	C₂²×D₃	—	5	2	no	—
219	C₂²×D₃	—	10	4	no	—
220	C₂²×D₃	—	10	2	no	—
221	C₃×Q₈	—	2	4	no	—
222	C₃×Q₈	—	2	4	no	—
223	C₃×Q₈	—	4	4	no	—
224	C₂×C₃×C₄	—	2	4	no	—
225	C₂×C₃×C₄	—	4	4	no	—
226	C₂×C₃×C₄	D₄×D₅	1	1	yes	characteristic
227	C₂×C₃×C₄	#711	1	2	yes	—
228	C₂×C₃×C₄	C₂×C₂²⋊D₅	1	2	yes	—
229	C₂×C₃×C₄	C₂³×D₅	1	1	yes	characteristic
230	C₂×Dic₃	—	5	2	no	—
231	C₂×Dic₃	—	10	2	no	—
232	C₂×Dic₃	—	10	4	no	—
233	C₂×Dic₃	—	20	4	no	—
234	C₄×D₃	—	10	2	no	—
235	C₄×D₃	—	10	2	no	—
236	C₄×D₃	—	10	4	no	—
237	C₃×C₈	C₂×C₂²⋊D₅	1	4	yes	—
238	C₃⋊C₈	—	10	2	no	—
239	C₂²×C₅	#918	1	4	yes	—
240	C₂²×C₅	#938	1	2	yes	—
241	C₂²×C₅	C₂×D₃×D₄	1	1	yes	characteristic
242	C₂×D₅	—	6	4	no	—
243	C₂×D₅	—	6	2	no	—
244	C₂×D₅	—	12	4	no	—
245	C₄×C₅	—	4	4	no	—
246	C₄×C₅	#938	1	2	yes	—
247	C₄×C₅	C₂×#237	1	1	yes	characteristic
248	C₄×C₅	C₂²×#67	1	1	yes	characteristic
249	Dic₅	—	6	2	no	—
250	Dic₅	—	12	4	no	—
251	C₂⁴	—	15	1	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	2	4	no	—
254	C₂×Q₈	—	2	4	no	—
255	C₂×Q₈	C₂²×D₁₅	1	4	yes	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	15	1	no	—
258	C₂²×C₄	—	15	2	no	—
259	C₂²×C₄	—	15	2	no	—
260	C₂²×C₄	—	30	2	no	—
261	C₂²×C₄	—	30	2	no	—
262	C₂²×C₄	—	30	2	no	—
263	C₂²×C₄	C₂²×D₁₅	1	1	yes	characteristic
264	Q₁₆	—	2	8	no	—
265	M₁₆	—	30	8	no	—
266	C₂×C₈	—	15	2	no	—
267	C₂×C₈	—	30	2	no	—
268	C₂×C₈	#1220	1	4	yes	—
269	C₂×C₈	C₂²×D₁₅	1	2	yes	—
270	C₄⋊C₄	—	15	4	no	—
271	C₄⋊C₄	—	30	8	no	—
272	C₄⋊C₄	—	30	2	no	—
273	C₂²⋊C₄	—	30	8	no	—
274	C₃×C₅	C₂×#470	1	1	yes	characteristic, complemented, Hall
275	C₂²×C₃	#4064	1	4	yes	—
276	C₂²×C₃	#4084	1	2	yes	—
277	C₂²×C₃	C₂×D₄×D₅	1	1	yes	characteristic
278	C₂×D₃	—	10	4	no	—
279	C₂×D₃	—	10	2	no	—
280	C₂×D₃	—	20	4	no	—
281	C₃×C₄	—	4	4	no	—
282	C₃×C₄	#4084	1	2	yes	—
283	C₃×C₄	C₂×#711	1	1	yes	characteristic
284	C₃×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
285	Dic₃	—	10	2	no	—
286	Dic₃	—	20	4	no	—
287	C₂×C₅	#5189	1	4	yes	complemented
288	C₂×C₅	C₂×#918	1	2	yes	—
289	C₂×C₅	C₂×#938	1	1	yes	characteristic
290	D₅	—	12	2	no	—
291	C₂³	—	15	2	no	—
292	C₂³	—	30	4	no	—
293	C₂³	—	30	2	no	—
294	C₂³	D₄×D₁₅	1	1	yes	characteristic
295	Q₈	—	2	4	no	—
296	Q₈	—	2	4	no	—
297	Q₈	—	4	4	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	15	2	no	—
301	C₂×C₄	—	30	2	no	—
302	C₂×C₄	—	30	4	no	—
303	C₂×C₄	—	30	2	no	—
304	C₂×C₄	—	30	2	no	—
305	C₂×C₄	—	30	4	no	—
306	C₂×C₄	—	60	4	no	—
307	C₂×C₄	D₄×D₁₅	1	1	yes	characteristic
308	C₂×C₄	#6819	1	2	yes	—
309	C₂×C₄	C₂×#1220	1	2	yes	—
310	C₂×C₄	C₂³×D₁₅	1	1	yes	characteristic
311	C₈	—	30	2	no	—
312	C₈	C₂×#1220	1	4	yes	—
313	C₂×C₃	#65478	1	4	yes	complemented
314	C₂×C₃	C₂×#4064	1	2	yes	—
315	C₂×C₃	C₂×#4084	1	1	yes	characteristic
316	D₃	—	20	2	no	—
317	C₅	C₂×#5189	1	1	yes	characteristic, complemented, Sylow
318	C₂²	—	30	2	no	—
319	C₂²	—	30	4	no	—
320	C₂²	—	60	4	no	—
321	C₂²	#91807	1	4	yes	—
322	C₂²	#91827	1	2	yes	—
323	C₂²	C₂×D₄×D₁₅	1	1	yes	characteristic
324	C₄	—	4	4	no	—
325	C₄	—	30	2	no	—
326	C₄	—	60	4	no	—
327	C₄	#91827	1	2	yes	—
328	C₄	C₂×#6819	1	1	yes	characteristic
329	C₄	C₂²×#1220	1	1	yes	characteristic
330	C₃	C₂×#65478	1	1	yes	characteristic, complemented, Sylow
331	C₂	—	60	2	no	—
332	C₂	#11751290	1	4	yes	direct product factor
333	C₂	C₂×#91807	1	2	yes	—
334	C₂	C₂×#91827	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₄×D₁₅
commutator, cosocle	C₂×C₃×C₄×C₅	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅
Fitting	C₂²×C₃×C₅×Q₁₆	C₂
socle	C₂³×C₃×C₅	C₂×D₄
prime radical	C₂³×D₁₅	D₄
hypercenter	C₂²×Q₁₆	D₁₅
hypocenter	C₃×C₅	C₂×#470
LCS	G ▷ C₂×C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×C₅×Q₁₆	C₂	semidirect
4. C₂×#91119	C₂	semidirect
5. #11751290	C₂	direct (N × H)
6. C₂×#91101	C₂	semidirect
11. C₂×C₃×C₅×Q₁₆	C₂²	semidirect
21. #91119	C₂²	semidirect
22. #91101	C₂²	semidirect
24. C₂²×C₅×Q₁₆	D₃	semidirect
53. C₂²×C₃×Q₁₆	D₅	semidirect
62. C₂×C₅×Q₁₆	C₂×D₃	semidirect
100. C₂×C₃×Q₁₆	C₂×D₅	semidirect
135. C₂²×Q₁₆	D₁₅	semidirect
201. C₂×Q₁₆	C₂×D₁₅	semidirect
213. C₂×C₃×C₅	#470	semidirect
274. C₃×C₅	C₂×#470	semidirect
287. C₂×C₅	#5189	semidirect
313. C₂×C₃	#65478	semidirect
317. C₅	C₂×#5189	semidirect
330. C₃	C₂×#65478	semidirect
332. C₂	#11751290	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×C₃×C₅×Q₁₆	2
C₂×#91835	2
C₂²×#6705	2
C₂×#91119	2
#11751290	2
C₂×#91101	2
C₂×#65478	3
C₂×#5189	5

Subgroups — C2×#11751290 #14428034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751290 #14428034