Subgroups — C₂×#11751008 #14478119

Order 1920.

736 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#91790	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃⋊#672	C₂	1	1	yes	maximal, characteristic, complemented
4	#11751008	C₂	1	8	yes	maximal, direct product factor
5	C₂×#91082	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₆×D₁₅	C₂	1	2	yes	maximal, complemented
7	C₂×#65196	—	3	1	no	maximal, Hall
8	C₂×#6817	C₄	1	1	yes	characteristic
9	C₂²×C₃×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×#6710	C₄	1	1	yes	characteristic
11	#91790	C₄	1	4	yes	—
12	#91790	C₂²	1	4	yes	—
13	C₂×#6705	C₄	1	2	yes	—
14	C₂×C₈×D₁₅	C₂²	1	2	yes	—
15	C₃×C₅×M₃₂	C₂²	1	4	yes	complemented
16	C₂×C₃×C₅×C₁₆	C₂²	1	2	yes	—
17	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊#672	C₂²	1	1	yes	characteristic
19	C₂×C₃⋊#672	C₂²	1	4	yes	complemented
20	#91082	C₂²	1	8	yes	complemented
21	C₁₆×D₁₅	C₂²	1	8	yes	complemented
22	C₂×#4907	—	5	1	no	maximal, Hall
23	C₂×#4047	—	3	1	no	—
24	C₂×C₅×M₃₂	D₃	1	1	yes	characteristic, complemented
25	C₂²×#672	—	3	1	no	—
26	#65196	—	3	8	no	—
27	C₂×#3930	—	3	2	no	—
28	C₂×D₅×C₁₆	—	3	2	no	—
29	C₂²×C₃×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂×#1220	C₈	1	2	yes	—
31	#6817	C₈	1	4	yes	—
32	#6817	C₂×C₄	1	4	yes	—
33	C₂×D₆₀	C₈	1	1	yes	characteristic
34	C₂×C₄×D₁₅	C₂×C₄	1	2	yes	—
35	C₂×Dic₃₀	C₈	1	1	yes	characteristic
36	C₂×C₃×C₅×C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃×C₅×C₈	C₂×C₄	1	2	yes	—
38	C₂×C₃×C₅×C₈	C₂³	1	1	yes	characteristic
39	C₂×C₃×C₅×C₈	C₂³	1	2	yes	—
40	#6710	C₂×C₄	1	4	yes	—
41	C₂×C₃⋊#168	C₂×C₄	1	2	yes	—
42	#6705	C₂×C₄	1	8	yes	—
43	C₈×D₁₅	C₂×C₄	1	4	yes	—
44	C₈×D₁₅	C₂³	1	4	yes	—
45	C₃×C₅×C₁₆	C₂³	1	4	yes	—
46	C₃⋊#672	C₂³	1	2	yes	—
47	C₃⋊#672	C₂³	1	2	yes	—
48	C₂×#901	—	5	1	no	—
49	C₂×C₃×M₃₂	D₅	1	1	yes	characteristic, complemented
50	C₂²×C₃⋊C₁₆	—	5	1	no	—
51	#4907	—	5	8	no	—
52	C₂×#798	—	5	2	no	—
53	C₂×D₃×C₁₆	—	5	2	no	—
54	C₂×#709	—	3	1	no	—
55	C₂²×C₅×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×#681	—	3	1	no	—
57	#4047	—	3	4	no	—
58	#4047	—	3	4	no	—
59	C₂×#676	—	3	2	no	—
60	C₂×C₈×D₅	—	3	2	no	—
61	C₅×M₃₂	C₂×D₃	1	4	yes	complemented
62	C₂×C₅×C₁₆	C₂×D₃	1	2	yes	—
63	C₂×#672	—	3	1	no	—
64	C₂×#672	—	3	1	no	—
65	C₂×#672	—	3	4	no	—
66	#3930	—	3	8	no	—
67	D₅×C₁₆	—	3	8	no	—
68	C₂×#504	—	15	1	no	Sylow
69	C₂³×C₃×C₅	C₂×C₈	1	1	yes	characteristic
70	C₂²×D₁₅	C₂×C₈	1	2	yes	—
71	C₂×C₃×C₄×C₅	C₂×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄×C₅	C₂×C₈	1	2	yes	—
73	C₂×C₃×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄×C₅	C₂²×C₄	1	2	yes	—
75	#1220	C₂×C₈	1	8	yes	—
76	C₂×Dic₁₅	C₂×C₈	1	2	yes	—
77	D₆₀	C₂×C₈	1	4	yes	—
78	C₄×D₁₅	C₂×C₈	1	4	yes	—
79	C₄×D₁₅	C₂²×C₄	1	4	yes	—
80	Dic₃₀	C₂×C₈	1	4	yes	—
81	C₃×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
82	C₃×C₅×C₈	C₂²×C₄	1	2	yes	—
83	C₃×C₅×C₈	C₂⁴	1	1	yes	characteristic
84	C₃⋊#168	C₂²×C₄	1	4	yes	—
85	C₂×#235	—	5	1	no	—
86	C₂²×C₃×C₈	C₂×D₅	1	1	yes	characteristic
87	C₂×#208	—	5	1	no	—
88	#901	—	5	4	no	—
89	#901	—	5	4	no	—
90	C₂×ΓL(1,𝔽₂₅)	—	5	2	no	—
91	C₂×D₃×C₈	—	5	2	no	—
92	C₃×M₃₂	C₂×D₅	1	4	yes	complemented
93	C₂×C₃×C₁₆	C₂×D₅	1	2	yes	—
94	C₂×C₃⋊C₁₆	—	5	1	no	—
95	C₂×C₃⋊C₁₆	—	5	1	no	—
96	C₂×C₃⋊C₁₆	—	5	4	no	—
97	#798	—	5	8	no	—
98	D₃×C₁₆	—	5	8	no	—
99	C₂²×C₄×C₅	C₄×D₃	1	1	yes	characteristic
100	C₂×C₂²⋊D₅	—	3	2	no	—
101	#709	—	3	4	no	—
102	#709	—	3	4	no	—
103	C₂×D₂₀	—	3	1	no	—
104	C₂×C₄×D₅	—	3	2	no	—
105	C₂×Dic₁₀	—	3	1	no	—
106	C₂×C₅×C₈	C₄×D₃	1	1	yes	characteristic
107	C₂×C₅×C₈	C₄×D₃	1	2	yes	—
108	C₂×C₅×C₈	C₂²×D₃	1	2	yes	—
109	C₂×C₅×C₈	C₂²×D₃	1	1	yes	characteristic
110	#681	—	3	4	no	—
111	C₂×#168	—	3	2	no	—
112	#676	—	3	8	no	—
113	C₈×D₅	—	3	4	no	—
114	C₈×D₅	—	3	4	no	—
115	C₅×C₁₆	C₂²×D₃	1	4	yes	—
116	#672	—	3	2	no	—
117	#672	—	3	2	no	—
118	C₂×#131	—	15	1	no	—
119	#504	—	15	8	no	—
120	C₂×M₃₂	—	15	2	no	—
121	C₂×M₃₂	D₁₅	1	1	yes	characteristic, complemented
122	C₂²×C₁₆	—	15	1	no	—
123	C₂²×C₁₆	—	15	2	no	—
124	C₂²×C₃×C₅	—	2	2	no	—
125	C₂²×C₃×C₅	C₂²×C₈	1	2	yes	—
126	C₂²×C₃×C₅	C₂²×C₈	1	1	yes	characteristic
127	C₂×D₁₅	—	2	4	no	—
128	C₂×D₁₅	C₂²×C₈	1	4	yes	—
129	C₃×C₄×C₅	C₂²×C₈	1	2	yes	—
130	C₃×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
131	C₃×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
132	Dic₁₅	C₂²×C₈	1	4	yes	—
133	C₂²×C₃×C₄	C₄×D₅	1	1	yes	characteristic
134	C₂×#67	—	5	2	no	—
135	#235	—	5	4	no	—
136	#235	—	5	4	no	—
137	C₂×D₁₂	—	5	1	no	—
138	C₂×C₄×D₃	—	5	2	no	—
139	C₂×Dic₆	—	5	1	no	—
140	C₂×C₃×C₈	C₄×D₅	1	1	yes	characteristic
141	C₂×C₃×C₈	C₄×D₅	1	2	yes	—
142	C₂×C₃×C₈	C₂²×D₅	1	2	yes	—
143	C₂×C₃×C₈	C₂²×D₅	1	1	yes	characteristic
144	#208	—	5	4	no	—
145	C₂×C₃⋊C₈	—	5	2	no	—
146	ΓL(1,𝔽₂₅)	—	5	8	no	—
147	D₃×C₈	—	5	4	no	—
148	D₃×C₈	—	5	4	no	—
149	C₃×C₁₆	C₂²×D₅	1	4	yes	—
150	C₃⋊C₁₆	—	5	2	no	—
151	C₃⋊C₁₆	—	5	2	no	—
152	C₂³×C₅	D₃×C₈	1	1	yes	characteristic
153	C₂²×D₅	—	3	2	no	—
154	C₂×C₄×C₅	D₃×C₈	1	1	yes	characteristic
155	C₂×C₄×C₅	D₃×C₈	1	2	yes	—
156	C₂×C₄×C₅	C₂×C₄×D₃	1	1	yes	characteristic
157	C₂×C₄×C₅	C₂×C₄×D₃	1	2	yes	—
158	C₂²⋊D₅	—	3	8	no	—
159	C₂×Dic₅	—	3	2	no	—
160	D₂₀	—	3	4	no	—
161	C₄×D₅	—	3	4	no	—
162	C₄×D₅	—	3	4	no	—
163	Dic₁₀	—	3	4	no	—
164	C₅×C₈	C₂×C₄×D₃	1	1	yes	characteristic
165	C₅×C₈	C₂×C₄×D₃	1	2	yes	—
166	C₅×C₈	C₂³×D₃	1	1	yes	characteristic
167	#168	—	3	4	no	—
168	C₂×#41	—	15	1	no	—
169	#131	—	15	4	no	—
170	#131	—	15	4	no	—
171	C₂×M₁₆	—	15	1	no	—
172	C₂×M₁₆	—	15	2	no	—
173	C₂²×C₈	—	15	2	no	—
174	C₂²×C₈	C₂×D₁₅	1	1	yes	characteristic
175	M₃₂	—	15	8	no	—
176	M₃₂	C₂×D₁₅	1	4	yes	complemented
177	C₂×C₁₆	—	15	4	no	—
178	C₂×C₁₆	—	15	1	no	—
179	C₂×C₁₆	—	15	1	no	—
180	C₂×C₁₆	—	15	8	no	—
181	C₂×C₁₆	C₂×D₁₅	1	2	yes	—
182	C₂×C₃×C₅	—	2	2	no	—
183	C₂×C₃×C₅	#504	1	2	yes	complemented
184	C₂×C₃×C₅	C₂³×C₈	1	1	yes	characteristic
185	D₁₅	—	2	4	no	—
186	C₂³×C₃	C₈×D₅	1	1	yes	characteristic
187	C₂²×D₃	—	5	2	no	—
188	C₂×C₃×C₄	C₈×D₅	1	1	yes	characteristic
189	C₂×C₃×C₄	C₈×D₅	1	2	yes	—
190	C₂×C₃×C₄	C₂×C₄×D₅	1	1	yes	characteristic
191	C₂×C₃×C₄	C₂×C₄×D₅	1	2	yes	—
192	#67	—	5	8	no	—
193	C₂×Dic₃	—	5	2	no	—
194	D₁₂	—	5	4	no	—
195	C₄×D₃	—	5	4	no	—
196	C₄×D₃	—	5	4	no	—
197	Dic₆	—	5	4	no	—
198	C₃×C₈	C₂×C₄×D₅	1	1	yes	characteristic
199	C₃×C₈	C₂×C₄×D₅	1	2	yes	—
200	C₃×C₈	C₂³×D₅	1	1	yes	characteristic
201	C₃⋊C₈	—	5	4	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	C₂×D₃×C₈	1	2	yes	—
204	C₂²×C₅	C₂×D₃×C₈	1	1	yes	characteristic
205	C₂×D₅	—	3	4	no	—
206	C₂×D₅	—	6	4	no	—
207	C₄×C₅	C₂×D₃×C₈	1	2	yes	—
208	C₄×C₅	C₂×D₃×C₈	1	1	yes	characteristic
209	C₄×C₅	C₂²×C₄×D₃	1	1	yes	characteristic
210	Dic₅	—	3	4	no	—
211	#41	—	15	4	no	—
212	#41	—	15	4	no	—
213	C₂×Q₈	—	15	1	no	—
214	C₂×D₄	—	15	2	no	—
215	C₂×D₄	—	15	1	no	—
216	C₂²×C₄	—	15	2	no	—
217	C₂²×C₄	C₄×D₁₅	1	1	yes	characteristic
218	M₁₆	—	15	4	no	—
219	M₁₆	—	15	8	no	—
220	C₂×C₈	—	15	2	no	—
221	C₂×C₈	—	15	4	no	—
222	C₂×C₈	—	15	4	no	—
223	C₂×C₈	C₄×D₁₅	1	1	yes	characteristic
224	C₂×C₈	C₄×D₁₅	1	2	yes	—
225	C₂×C₈	C₂²×D₁₅	1	2	yes	—
226	C₂×C₈	C₂²×D₁₅	1	1	yes	characteristic
227	C₁₆	—	15	2	no	—
228	C₁₆	—	15	2	no	—
229	C₁₆	C₂²×D₁₅	1	4	yes	—
230	C₃×C₅	C₂×#504	1	1	yes	characteristic, complemented, Hall
231	C₂²×C₃	—	2	2	no	—
232	C₂²×C₃	C₂×C₈×D₅	1	2	yes	—
233	C₂²×C₃	C₂×C₈×D₅	1	1	yes	characteristic
234	C₂×D₃	—	5	4	no	—
235	C₂×D₃	—	10	4	no	—
236	C₃×C₄	C₂×C₈×D₅	1	2	yes	—
237	C₃×C₄	C₂×C₈×D₅	1	1	yes	characteristic
238	C₃×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
239	Dic₃	—	5	4	no	—
240	C₂×C₅	—	2	2	no	—
241	C₂×C₅	#4907	1	2	yes	complemented
242	C₂×C₅	C₂²×D₃×C₈	1	1	yes	characteristic
243	D₅	—	6	4	no	—
244	C₂³	—	15	2	no	—
245	C₂³	C₈×D₁₅	1	1	yes	characteristic
246	Q₈	—	15	4	no	—
247	D₄	—	15	8	no	—
248	D₄	—	15	4	no	—
249	C₂×C₄	—	15	2	no	—
250	C₂×C₄	—	15	4	no	—
251	C₂×C₄	—	15	4	no	—
252	C₂×C₄	C₈×D₁₅	1	1	yes	characteristic
253	C₂×C₄	C₈×D₁₅	1	2	yes	—
254	C₂×C₄	C₂×C₄×D₁₅	1	1	yes	characteristic
255	C₂×C₄	C₂×C₄×D₁₅	1	2	yes	—
256	C₈	—	15	4	no	—
257	C₈	C₂×C₄×D₁₅	1	1	yes	characteristic
258	C₈	C₂×C₄×D₁₅	1	2	yes	—
259	C₈	C₂³×D₁₅	1	1	yes	characteristic
260	C₂×C₃	—	2	2	no	—
261	C₂×C₃	#65196	1	2	yes	complemented
262	C₂×C₃	C₂²×C₈×D₅	1	1	yes	characteristic
263	D₃	—	10	4	no	—
264	C₅	C₂×#4907	1	1	yes	characteristic, complemented, Sylow
265	C₂²	—	2	2	no	—
266	C₂²	—	15	4	no	—
267	C₂²	—	30	4	no	—
268	C₂²	C₂×C₈×D₁₅	1	2	yes	—
269	C₂²	C₂×C₈×D₁₅	1	1	yes	characteristic
270	C₄	—	15	4	no	—
271	C₄	C₂×C₈×D₁₅	1	1	yes	characteristic
272	C₄	C₂×C₈×D₁₅	1	2	yes	—
273	C₄	C₂²×C₄×D₁₅	1	1	yes	characteristic
274	C₃	C₂×#65196	1	1	yes	characteristic, complemented, Sylow
275	C₂	—	2	2	no	—
276	C₂	—	30	4	no	—
277	C₂	#11751008	1	2	yes	direct product factor
278	C₂	C₂²×C₈×D₁₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₁₅
commutator	C₂×C₃×C₅	C₂³×C₈
Frattini	C₈	C₂³×D₁₅
Fitting	C₂×C₃×C₅×M₃₂	C₂
socle	C₂²×C₃×C₅	C₂²×C₈
prime radical	C₂×#6817	C₄
cosocle	C₃×C₅×C₈	C₂⁴
hypercenter	C₂×M₃₂	D₁₅
hypocenter	C₃×C₅	C₂×#504
LCS	G ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₅×M₃₂	C₂	semidirect
3. C₂²×C₃⋊#672	C₂	semidirect
4. #11751008	C₂	direct (N × H)
5. C₂×#91082	C₂	semidirect
6. C₂×C₁₆×D₁₅	C₂	semidirect
15. C₃×C₅×M₃₂	C₂²	semidirect
19. C₂×C₃⋊#672	C₂²	semidirect
20. #91082	C₂²	semidirect
21. C₁₆×D₁₅	C₂²	semidirect
24. C₂×C₅×M₃₂	D₃	semidirect
49. C₂×C₃×M₃₂	D₅	semidirect
61. C₅×M₃₂	C₂×D₃	semidirect
92. C₃×M₃₂	C₂×D₅	semidirect
121. C₂×M₃₂	D₁₅	semidirect
176. M₃₂	C₂×D₁₅	semidirect
183. C₂×C₃×C₅	#504	semidirect
230. C₃×C₅	C₂×#504	semidirect
241. C₂×C₅	#4907	semidirect
261. C₂×C₃	#65196	semidirect
264. C₅	C₂×#4907	semidirect
274. C₃	C₂×#65196	semidirect
277. C₂	#11751008	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×#91790	2
C₂×C₃×C₅×M₃₂	2
C₂²×C₃⋊#672	2
#11751008	2
C₂×#91082	2
C₂×C₁₆×D₁₅	2
C₂×#65196	3
C₂×#4907	5

Subgroups — C2×#11751008 #14478119

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11751008 #14478119