Subgroups — C₂×C₅×#5734 #14483925

Order 1920.

2480 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#947	C₂	1	4	yes	maximal, complemented
3	C₅×#5734	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#895	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#892	C₂	1	4	yes	maximal, complemented
6	C₂×C₄×C₅×D₁₂	C₂	1	2	yes	maximal, complemented
7	C₂×C₅×#555	—	3	1	no	maximal, Hall
8	C₂²×C₃×C₅×Q₈	C₂²	1	2	yes	—
9	C₂²×C₅×D₁₂	C₂²	1	2	yes	—
10	C₂²×C₅×D₁₂	C₂²	1	4	yes	—
11	C₂²×C₄×C₅×D₃	C₂²	1	4	yes	—
12	C₃×C₅×C₄⋊Q₈	C₂²	1	8	yes	complemented
13	C₂×C₃×C₅×C₄⋊C₄	C₂²	1	4	yes	—
14	C₂×C₃×C₄²×C₅	C₂²	1	1	yes	characteristic
15	C₅×#947	C₂²	1	32	yes	complemented
16	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	4	yes	—
17	C₂×C₅×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
18	C₂×C₅×C₄⋊Dic₃	C₂²	1	2	yes	—
19	C₂×C₄×C₅×Dic₃	C₂²	1	4	yes	—
20	C₅×#895	C₂²	1	32	yes	complemented
21	C₅×#892	C₂²	1	32	yes	complemented
22	C₄×C₅×D₁₂	C₂²	1	16	yes	complemented
23	C₂×#5734	C₅	1	1	yes	maximal, characteristic, direct product factor, Hall
24	C₅×#555	—	3	16	no	—
25	C₂×C₅×C₄⋊Q₈	D₃	1	1	yes	characteristic, complemented
26	C₂×C₅×#124	—	3	4	no	—
27	C₂×C₅×#121	—	3	4	no	—
28	C₂×C₄×C₅×D₄	—	3	2	no	—
29	C₂×C₅×#117	—	3	4	no	—
30	C₂³×C₅×D₃	C₂³	1	4	yes	—
31	C₂×C₃×C₅×Q₈	—	4	4	no	—
32	C₂×C₃×C₅×Q₈	C₂³	1	8	yes	—
33	C₂²×C₃×C₄×C₅	C₂³	1	4	yes	—
34	C₂²×C₃×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₅×Dic₃	C₂³	1	4	yes	—
37	C₂×C₅×D₁₂	—	2	8	no	—
38	C₂×C₅×D₁₂	—	4	8	no	—
39	C₂×C₅×D₁₂	C₂³	1	8	yes	—
40	C₂×C₅×D₁₂	C₂³	1	16	yes	—
41	C₂×C₄×C₅×D₃	—	2	16	no	—
42	C₂×C₄×C₅×D₃	C₂³	1	16	yes	—
43	C₃×C₅×C₄⋊C₄	C₂³	1	16	yes	—
44	C₃×C₄²×C₅	C₂³	1	4	yes	—
45	C₅×(C₂×D₃)⋊C₄	C₂³	1	16	yes	—
46	C₅×(C₂×D₃)⋊C₄	C₂³	1	32	yes	—
47	C₅×C₄⋊Dic₃	C₂³	1	8	yes	—
48	C₄×C₅×Dic₃	C₂³	1	16	yes	—
49	C₂×C₃×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
50	C₂×#947	C₂×C₅	1	4	yes	complemented
51	#5734	C₂×C₅	1	16	yes	direct product factor
52	C₂×#895	C₂×C₅	1	4	yes	complemented
53	C₂×#892	C₂×C₅	1	4	yes	complemented
54	C₂×C₄×D₁₂	C₂×C₅	1	2	yes	complemented
55	C₂²×C₅×Q₈	C₂×D₃	1	2	yes	—
56	C₂²×C₅×D₄	—	3	2	no	—
57	C₂²×C₅×D₄	—	3	4	no	—
58	C₂³×C₄×C₅	—	3	4	no	—
59	C₅×C₄⋊Q₈	C₂×D₃	1	8	yes	complemented
60	C₅×#124	—	3	32	no	—
61	C₅×#121	—	3	32	no	—
62	C₄×C₅×D₄	—	3	16	no	—
63	C₅×#117	—	3	32	no	—
64	C₂×C₅×C₄⋊C₄	—	3	2	no	—
65	C₂×C₅×C₄⋊C₄	C₂×D₃	1	4	yes	—
66	C₂×C₅×C₂²⋊C₄	—	3	8	no	—
67	C₂×C₅×C₂²⋊C₄	—	3	4	no	—
68	C₂×C₄²×C₅	—	3	4	no	—
69	C₂×C₄²×C₅	C₂×D₃	1	1	yes	characteristic
70	C₂×#555	—	3	1	no	Sylow
71	C₂³×C₃×C₅	C₂⁴	1	1	yes	characteristic
72	C₂²×C₅×D₃	—	2	8	no	—
73	C₂²×C₅×D₃	—	2	8	no	—
74	C₂²×C₅×D₃	—	2	8	no	—
75	C₂²×C₅×D₃	C₂⁴	1	8	yes	—
76	C₃×C₅×Q₈	—	4	8	no	—
77	C₂×C₃×C₄×C₅	—	2	8	no	—
78	C₂×C₃×C₄×C₅	—	2	2	no	—
79	C₂×C₃×C₄×C₅	#41	1	8	yes	—
80	C₂×C₃×C₄×C₅	C₂⁴	1	8	yes	—
81	C₂×C₃×C₄×C₅	C₂⁴	1	2	yes	—
82	C₂×C₃×C₄×C₅	C₂⁴	1	4	yes	—
83	C₂×C₅×Dic₃	—	2	8	no	—
84	C₂×C₅×Dic₃	C₂⁴	1	8	yes	—
85	C₅×D₁₂	—	2	16	no	—
86	C₅×D₁₂	—	4	16	no	—
87	C₄×C₅×D₃	—	2	32	no	—
88	C₂²×C₃×Q₈	C₂²×C₅	1	2	yes	—
89	C₂²×D₁₂	C₂²×C₅	1	2	yes	—
90	C₂²×D₁₂	C₂²×C₅	1	4	yes	—
91	C₂²×C₄×D₃	C₂²×C₅	1	4	yes	—
92	C₃×C₄⋊Q₈	C₂²×C₅	1	8	yes	complemented
93	C₂×C₃×C₄⋊C₄	C₂²×C₅	1	4	yes	—
94	C₂×C₃×C₄²	C₂²×C₅	1	1	yes	characteristic
95	#947	C₂²×C₅	1	32	yes	complemented
96	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	4	yes	—
97	C₂×(C₂×D₃)⋊C₄	C₂²×C₅	1	8	yes	—
98	C₂×C₄⋊Dic₃	C₂²×C₅	1	2	yes	—
99	C₂×C₄×Dic₃	C₂²×C₅	1	4	yes	—
100	#895	C₂²×C₅	1	32	yes	complemented
101	#892	C₂²×C₅	1	32	yes	complemented
102	C₄×D₁₂	C₂²×C₅	1	16	yes	complemented
103	C₂⁴×C₅	—	3	4	no	—
104	C₂×C₅×Q₈	—	4	4	no	—
105	C₂×C₅×Q₈	C₂²×D₃	1	8	yes	—
106	C₂×C₅×D₄	—	3	8	no	—
107	C₂×C₅×D₄	—	3	16	no	—
108	C₂×C₅×D₄	—	6	8	no	—
109	C₂×C₅×D₄	—	12	8	no	—
110	C₂²×C₄×C₅	—	3	4	no	—
111	C₂²×C₄×C₅	—	3	16	no	—
112	C₂²×C₄×C₅	—	6	16	no	—
113	C₂²×C₄×C₅	C₂²×D₃	1	4	yes	—
114	C₂²×C₄×C₅	C₂²×D₃	1	1	yes	characteristic
115	C₂²×C₄×C₅	C₂²×D₃	1	2	yes	—
116	C₅×C₄⋊C₄	—	3	8	no	—
117	C₅×C₄⋊C₄	C₂²×D₃	1	16	yes	—
118	C₅×C₂²⋊C₄	—	3	32	no	—
119	C₅×C₂²⋊C₄	—	3	16	no	—
120	C₄²×C₅	—	3	16	no	—
121	C₄²×C₅	C₂²×D₃	1	4	yes	—
122	#555	—	3	16	no	—
123	C₂×C₄⋊Q₈	C₅×D₃	1	1	yes	characteristic, complemented
124	C₂×#124	—	3	4	no	—
125	C₂×#121	—	3	4	no	—
126	C₂×C₄×D₄	—	3	2	no	—
127	C₂×#117	—	3	4	no	—
128	C₂²×C₃×C₅	C₂×#41	1	4	yes	—
129	C₂²×C₃×C₅	H(5,𝔽₂)	1	2	yes	—
130	C₂²×C₃×C₅	C₂⁵	1	1	yes	characteristic
131	C₂×C₅×D₃	—	2	8	no	—
132	C₂×C₅×D₃	—	2	8	no	—
133	C₂×C₅×D₃	—	2	8	no	—
134	C₂×C₅×D₃	—	4	16	no	—
135	C₃×C₄×C₅	—	2	8	no	—
136	C₃×C₄×C₅	—	2	2	no	—
137	C₃×C₄×C₅	C₂×#41	1	8	yes	—
138	C₅×Dic₃	—	2	8	no	—
139	C₂³×D₃	C₂³×C₅	1	4	yes	—
140	C₂×C₃×Q₈	—	4	4	no	—
141	C₂×C₃×Q₈	C₂³×C₅	1	8	yes	—
142	C₂²×C₃×C₄	C₂³×C₅	1	4	yes	—
143	C₂²×C₃×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂²×C₃×C₄	C₂³×C₅	1	2	yes	—
145	C₂²×Dic₃	C₂³×C₅	1	4	yes	—
146	C₂×D₁₂	—	2	8	no	—
147	C₂×D₁₂	—	4	8	no	—
148	C₂×D₁₂	C₂³×C₅	1	8	yes	—
149	C₂×D₁₂	C₂³×C₅	1	16	yes	—
150	C₂×C₄×D₃	—	2	16	no	—
151	C₂×C₄×D₃	C₂³×C₅	1	16	yes	—
152	C₃×C₄⋊C₄	C₂³×C₅	1	16	yes	—
153	C₃×C₄²	C₂³×C₅	1	4	yes	—
154	(C₂×D₃)⋊C₄	C₂³×C₅	1	16	yes	—
155	(C₂×D₃)⋊C₄	C₂³×C₅	1	32	yes	—
156	C₄⋊Dic₃	C₂³×C₅	1	8	yes	—
157	C₄×Dic₃	C₂³×C₅	1	16	yes	—
158	C₂³×C₅	—	3	8	no	—
159	C₂³×C₅	—	6	8	no	—
160	C₂³×C₅	—	6	8	no	—
161	C₂³×C₅	—	6	8	no	—
162	C₂³×C₅	C₂³×D₃	1	1	yes	characteristic
163	C₅×Q₈	—	4	8	no	—
164	C₅×D₄	—	6	16	no	—
165	C₅×D₄	—	12	16	no	—
166	C₂×C₄×C₅	—	2	8	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	—	3	8	no	—
169	C₂×C₄×C₅	—	6	8	no	—
170	C₂×C₄×C₅	—	6	32	no	—
171	C₂×C₄×C₅	#239	1	8	yes	—
172	C₂×C₄×C₅	C₂³×D₃	1	8	yes	—
173	C₂×C₄×C₅	C₂³×D₃	1	2	yes	—
174	C₂×C₄×C₅	C₂³×D₃	1	4	yes	—
175	C₂²×Q₈	C₂×C₅×D₃	1	2	yes	—
176	C₂²×D₄	—	3	2	no	—
177	C₂²×D₄	—	3	4	no	—
178	C₂³×C₄	—	3	4	no	—
179	C₄⋊Q₈	C₂×C₅×D₃	1	8	yes	complemented
180	#124	—	3	32	no	—
181	#121	—	3	32	no	—
182	C₄×D₄	—	3	16	no	—
183	#117	—	3	32	no	—
184	C₂×C₄⋊C₄	—	3	2	no	—
185	C₂×C₄⋊C₄	C₂×C₅×D₃	1	4	yes	—
186	C₂×C₂²⋊C₄	—	3	8	no	—
187	C₂×C₂²⋊C₄	—	3	4	no	—
188	C₂×C₄²	—	3	4	no	—
189	C₂×C₄²	C₂×C₅×D₃	1	1	yes	characteristic
190	C₂×C₃×C₅	#555	1	4	yes	complemented
191	C₂×C₃×C₅	C₂²×#41	1	2	yes	—
192	C₂×C₃×C₅	C₂×H(5,𝔽₂)	1	1	yes	characteristic
193	C₅×D₃	—	4	8	no	—
194	C₂³×C₃	C₂⁴×C₅	1	1	yes	characteristic
195	C₂²×D₃	—	2	8	no	—
196	C₂²×D₃	—	2	8	no	—
197	C₂²×D₃	—	2	8	no	—
198	C₂²×D₃	C₂⁴×C₅	1	8	yes	—
199	C₃×Q₈	—	4	8	no	—
200	C₂×C₃×C₄	—	2	8	no	—
201	C₂×C₃×C₄	—	2	2	no	—
202	C₂×C₃×C₄	C₅×#41	1	8	yes	—
203	C₂×C₃×C₄	C₂⁴×C₅	1	8	yes	—
204	C₂×C₃×C₄	C₂⁴×C₅	1	2	yes	—
205	C₂×C₃×C₄	C₂⁴×C₅	1	4	yes	—
206	C₂×Dic₃	—	2	8	no	—
207	C₂×Dic₃	C₂⁴×C₅	1	8	yes	—
208	D₁₂	—	2	16	no	—
209	D₁₂	—	4	16	no	—
210	C₄×D₃	—	2	32	no	—
211	C₂²×C₅	—	6	8	no	—
212	C₂²×C₅	—	6	8	no	—
213	C₂²×C₅	—	6	8	no	—
214	C₂²×C₅	—	12	16	no	—
215	C₂²×C₅	#1004	1	2	yes	—
216	C₂²×C₅	C₂×#239	1	4	yes	—
217	C₂²×C₅	C₂⁴×D₃	1	1	yes	characteristic
218	C₄×C₅	—	2	8	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	6	8	no	—
221	C₄×C₅	C₂×#239	1	8	yes	—
222	C₂⁴	—	3	4	no	—
223	C₂×Q₈	—	4	4	no	—
224	C₂×Q₈	C₂²×C₅×D₃	1	8	yes	—
225	C₂×D₄	—	3	8	no	—
226	C₂×D₄	—	3	16	no	—
227	C₂×D₄	—	6	8	no	—
228	C₂×D₄	—	12	8	no	—
229	C₂²×C₄	—	3	4	no	—
230	C₂²×C₄	—	3	16	no	—
231	C₂²×C₄	—	6	16	no	—
232	C₂²×C₄	C₂²×C₅×D₃	1	4	yes	—
233	C₂²×C₄	C₂²×C₅×D₃	1	1	yes	characteristic
234	C₂²×C₄	C₂²×C₅×D₃	1	2	yes	—
235	C₄⋊C₄	—	3	8	no	—
236	C₄⋊C₄	C₂²×C₅×D₃	1	16	yes	—
237	C₂²⋊C₄	—	3	32	no	—
238	C₂²⋊C₄	—	3	16	no	—
239	C₄²	—	3	16	no	—
240	C₄²	C₂²×C₅×D₃	1	4	yes	—
241	C₃×C₅	C₂×#555	1	1	yes	characteristic, complemented, Hall
242	C₂²×C₃	C₂×C₅×#41	1	4	yes	—
243	C₂²×C₃	C₅×H(5,𝔽₂)	1	2	yes	—
244	C₂²×C₃	C₂⁵×C₅	1	1	yes	characteristic
245	C₂×D₃	—	2	8	no	—
246	C₂×D₃	—	2	8	no	—
247	C₂×D₃	—	2	8	no	—
248	C₂×D₃	—	4	16	no	—
249	C₃×C₄	—	2	8	no	—
250	C₃×C₄	—	2	2	no	—
251	C₃×C₄	C₂×C₅×#41	1	8	yes	—
252	Dic₃	—	2	8	no	—
253	C₂×C₅	—	12	8	no	—
254	C₂×C₅	#5734	1	4	yes	direct product factor
255	C₂×C₅	C₂×#1004	1	1	yes	characteristic
256	C₂×C₅	C₂²×#239	1	2	yes	—
257	C₂³	—	3	8	no	—
258	C₂³	—	6	8	no	—
259	C₂³	—	6	8	no	—
260	C₂³	—	6	8	no	—
261	C₂³	C₂³×C₅×D₃	1	1	yes	characteristic
262	Q₈	—	4	8	no	—
263	D₄	—	6	16	no	—
264	D₄	—	12	16	no	—
265	C₂×C₄	—	2	8	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	3	8	no	—
268	C₂×C₄	—	6	8	no	—
269	C₂×C₄	—	6	32	no	—
270	C₂×C₄	C₅×#239	1	8	yes	—
271	C₂×C₄	C₂³×C₅×D₃	1	8	yes	—
272	C₂×C₄	C₂³×C₅×D₃	1	2	yes	—
273	C₂×C₄	C₂³×C₅×D₃	1	4	yes	—
274	C₂×C₃	C₅×#555	1	4	yes	complemented
275	C₂×C₃	C₂²×C₅×#41	1	2	yes	—
276	C₂×C₃	C₂×C₅×H(5,𝔽₂)	1	1	yes	characteristic
277	D₃	—	4	8	no	—
278	C₅	C₂×#5734	1	1	yes	characteristic, direct product factor, Sylow
279	C₂²	—	6	8	no	—
280	C₂²	—	6	8	no	—
281	C₂²	—	6	8	no	—
282	C₂²	—	12	16	no	—
283	C₂²	C₅×#1004	1	2	yes	—
284	C₂²	C₂×C₅×#239	1	4	yes	—
285	C₂²	C₂⁴×C₅×D₃	1	1	yes	characteristic
286	C₄	—	2	8	no	—
287	C₄	—	2	2	no	—
288	C₄	—	6	8	no	—
289	C₄	C₂×C₅×#239	1	8	yes	—
290	C₃	C₂×C₅×#555	1	1	yes	characteristic, complemented, Sylow
291	C₂	—	12	8	no	—
292	C₂	C₅×#5734	1	4	yes	direct product factor
293	C₂	C₂×C₅×#1004	1	1	yes	characteristic
294	C₂	C₂²×C₅×#239	1	2	yes	—

	subgroup	G/H
center	C₂³×C₅	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵×C₅
Frattini	C₂²	C₂⁴×C₅×D₃
Fitting	C₂×C₃×C₅×C₄⋊Q₈	C₂
socle	C₂³×C₃×C₅	C₂⁴
hypercenter	C₂×C₅×C₄⋊Q₈	D₃
hypocenter	C₃	C₂×C₅×#555
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₅ ◁ C₂×C₅×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₅×C₄⋊Q₈	C₂	semidirect
2. C₂×C₅×#947	C₂	semidirect
3. C₅×#5734	C₂	direct (N × H)
4. C₂×C₅×#895	C₂	semidirect
5. C₂×C₅×#892	C₂	semidirect
6. C₂×C₄×C₅×D₁₂	C₂	semidirect
12. C₃×C₅×C₄⋊Q₈	C₂²	semidirect
15. C₅×#947	C₂²	semidirect
20. C₅×#895	C₂²	semidirect
21. C₅×#892	C₂²	semidirect
22. C₄×C₅×D₁₂	C₂²	semidirect
23. C₂×#5734	C₅	direct (N × H)
25. C₂×C₅×C₄⋊Q₈	D₃	semidirect
49. C₂×C₃×C₄⋊Q₈	C₂×C₅	semidirect
50. C₂×#947	C₂×C₅	semidirect
51. #5734	C₂×C₅	direct (N × H)
52. C₂×#895	C₂×C₅	semidirect
53. C₂×#892	C₂×C₅	semidirect
54. C₂×C₄×D₁₂	C₂×C₅	semidirect
59. C₅×C₄⋊Q₈	C₂×D₃	semidirect
92. C₃×C₄⋊Q₈	C₂²×C₅	semidirect
95. #947	C₂²×C₅	semidirect
100. #895	C₂²×C₅	semidirect
101. #892	C₂²×C₅	semidirect
102. C₄×D₁₂	C₂²×C₅	semidirect
123. C₂×C₄⋊Q₈	C₅×D₃	semidirect
179. C₄⋊Q₈	C₂×C₅×D₃	semidirect
190. C₂×C₃×C₅	#555	semidirect
241. C₃×C₅	C₂×#555	semidirect
254. C₂×C₅	#5734	direct (N × H)
274. C₂×C₃	C₅×#555	semidirect
278. C₅	C₂×#5734	direct (N × H)
290. C₃	C₂×C₅×#555	semidirect
292. C₂	C₅×#5734	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×C₃×C₅×C₄⋊Q₈	2
C₂×C₅×#947	2
C₅×#5734	2
C₂×C₅×#895	2
C₂×C₅×#892	2
C₂×C₄×C₅×D₁₂	2
C₂×C₅×#555	3
C₂×#5734	5

Subgroups — C2×C5×#5734 #14483925

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#5734 #14483925