Subgroups — C₂×C₄×C₅×#107 #11943168

Order 1280.

1358 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#107	C₂	1	2	yes	maximal
2	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₅×C₄⋊C₄	C₂	1	2	yes	maximal
4	C₄×C₅×#107	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₅×#340	C₂	1	2	yes	maximal
6	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
7	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₅×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂²×C₄²×C₅	—	2	2	no	—
10	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
11	C₂×C₅×#107	C₄	1	16	yes	direct product factor
12	C₂×C₅×#107	C₂²	1	8	yes	—
13	C₂×C₄×C₅×C₈	C₄	1	8	yes	complemented
14	C₂×C₄×C₅×C₈	C₂²	1	4	yes	—
15	C₄×C₅×C₄⋊C₄	—	2	4	no	—
16	C₄×C₅×C₄⋊C₄	C₂²	1	8	yes	—
17	C₂×C₅×#95	—	2	2	no	—
18	C₅×#340	C₂²	1	8	yes	—
19	C₂×C₄×#107	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
20	C₂³×C₄×C₅	—	2	4	no	—
21	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂²×C₅×C₈	—	2	4	no	—
25	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
26	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
27	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
28	C₂²×C₅×C₈	C₂³	1	2	yes	—
29	C₂²×C₅×C₈	C₂³	1	2	yes	—
30	C₂×C₅×C₄⋊C₄	—	2	4	no	—
31	C₂×C₅×C₄⋊C₄	—	2	8	no	—
32	C₂×C₅×C₄⋊C₄	C₂×C₄	1	16	yes	—
33	C₂×C₅×C₄⋊C₄	C₂³	1	8	yes	—
34	C₂×C₄²×C₅	—	2	8	no	—
35	C₂×C₄²×C₅	—	4	8	no	—
36	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	Q₈	1	4	yes	—
39	C₂×C₄²×C₅	C₂³	1	2	yes	—
40	C₅×#107	C₂×C₄	1	32	yes	direct product factor
41	C₄×C₅×C₈	C₂×C₄	1	16	yes	complemented
42	C₅×#95	—	2	8	no	—
43	C₂²×#107	C₂×C₅	1	2	yes	—
44	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic
45	C₂×C₄×C₄⋊C₄	C₂×C₅	1	2	yes	—
46	C₄×#107	C₂×C₅	1	8	yes	direct product factor
47	C₂×#340	C₂×C₅	1	2	yes	—
48	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
49	C₂²×C₄×C₅	—	2	4	no	—
50	C₂²×C₄×C₅	—	2	16	no	—
51	C₂²×C₄×C₅	—	2	8	no	—
52	C₂²×C₄×C₅	—	4	8	no	—
53	C₂²×C₄×C₅	—	4	8	no	—
54	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
55	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
56	C₂²×C₄×C₅	D₈	1	4	yes	—
57	C₂²×C₄×C₅	Q₁₆	1	4	yes	—
58	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
59	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
60	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
61	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
64	C₂²×C₄×C₅	#41	1	4	yes	—
65	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
66	C₂×C₅×C₈	—	2	8	no	—
67	C₂×C₅×C₈	—	2	4	no	—
68	C₂×C₅×C₈	C₄²	1	16	yes	complemented
69	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
70	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
71	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
72	C₅×C₄⋊C₄	—	2	16	no	—
73	C₅×C₄⋊C₄	—	2	8	no	—
74	C₅×C₄⋊C₄	C₂²×C₄	1	16	yes	—
75	C₄²×C₅	—	4	16	no	—
76	C₄²×C₅	C₄⋊C₄	1	8	yes	—
77	C₄²×C₅	C₂²×C₄	1	4	yes	—
78	C₄²×C₅	C₂×Q₈	1	4	yes	—
79	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
80	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
81	C₂²×C₄⋊C₄	C₂²×C₅	1	4	yes	—
82	C₂²×C₄²	—	2	2	no	—
83	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
84	C₂×#107	C₄×C₅	1	16	yes	direct product factor
85	C₂×#107	C₂²×C₅	1	8	yes	—
86	C₂×C₄×C₈	C₄×C₅	1	8	yes	complemented
87	C₂×C₄×C₈	C₂²×C₅	1	4	yes	—
88	C₄×C₄⋊C₄	—	2	4	no	—
89	C₄×C₄⋊C₄	C₂²×C₅	1	8	yes	—
90	C₂×#95	—	2	2	no	—
91	#340	C₂²×C₅	1	8	yes	—
92	C₂³×C₅	C₂×C₄⋊C₄	1	2	yes	—
93	C₂³×C₅	C₄×D₄	1	4	yes	—
94	C₂³×C₅	C₂×D₈	1	2	yes	—
95	C₂³×C₅	C₂×Q₁₆	1	2	yes	—
96	C₂³×C₅	#135	1	4	yes	—
97	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	8	no	—
100	C₂×C₄×C₅	—	2	16	no	—
101	C₂×C₄×C₅	—	4	8	no	—
102	C₂×C₄×C₅	—	4	8	no	—
103	C₂×C₄×C₅	—	4	32	no	—
104	C₂×C₄×C₅	#107	1	16	yes	direct product factor
105	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
106	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
107	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₅	C₂×C₄⋊C₄	1	4	yes	—
109	C₂×C₄×C₅	#117	1	4	yes	—
110	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
111	C₂×C₄×C₅	C₄×Q₈	1	8	yes	—
112	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
113	C₂×C₄×C₅	C₂×Q₁₆	1	4	yes	—
114	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
115	C₂×C₄×C₅	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅	C₂×#41	1	2	yes	—
118	C₅×C₈	—	2	4	no	—
119	C₅×C₈	C₂×C₄²	1	8	yes	complemented
120	C₂³×C₄	—	2	4	no	—
121	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
122	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
123	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
124	C₂²×C₈	—	2	4	no	—
125	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
126	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
127	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
128	C₂²×C₈	C₂³×C₅	1	2	yes	—
129	C₂²×C₈	C₂³×C₅	1	2	yes	—
130	C₂×C₄⋊C₄	—	2	4	no	—
131	C₂×C₄⋊C₄	—	2	8	no	—
132	C₂×C₄⋊C₄	C₂×C₄×C₅	1	16	yes	—
133	C₂×C₄⋊C₄	C₂³×C₅	1	8	yes	—
134	C₂×C₄²	—	2	8	no	—
135	C₂×C₄²	—	4	8	no	—
136	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
137	C₂×C₄²	C₅×D₄	1	2	yes	—
138	C₂×C₄²	C₅×Q₈	1	4	yes	—
139	C₂×C₄²	C₂³×C₅	1	2	yes	—
140	#107	C₂×C₄×C₅	1	32	yes	direct product factor
141	C₄×C₈	C₂×C₄×C₅	1	16	yes	complemented
142	#95	—	2	8	no	—
143	C₂²×C₅	C₄×C₄⋊C₄	1	4	yes	—
144	C₂²×C₅	C₂×#107	1	4	yes	—
145	C₂²×C₅	#427	1	4	yes	—
146	C₂²×C₅	C₄×D₈	1	8	yes	—
147	C₂²×C₅	C₄×Q₁₆	1	8	yes	—
148	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₂²×C₅	C₂×C₄×D₄	1	2	yes	—
150	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
151	C₂²×C₅	C₂²×Q₁₆	1	1	yes	characteristic
152	C₂²×C₅	C₂×#135	1	2	yes	—
153	C₄×C₅	—	2	4	no	—
154	C₄×C₅	—	4	16	no	—
155	C₄×C₅	C₄×C₄⋊C₄	1	4	yes	—
156	C₄×C₅	C₂×#107	1	8	yes	direct product factor
157	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
158	C₄×C₅	C₂×#117	1	1	yes	characteristic
159	C₄×C₅	C₂×C₄×Q₈	1	2	yes	—
160	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	16	no	—
163	C₂²×C₄	—	2	8	no	—
164	C₂²×C₄	—	4	8	no	—
165	C₂²×C₄	—	4	8	no	—
166	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
167	C₂²×C₄	C₅×C₄⋊C₄	1	4	yes	—
168	C₂²×C₄	C₅×D₈	1	4	yes	—
169	C₂²×C₄	C₅×Q₁₆	1	4	yes	—
170	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
171	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
172	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
173	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
174	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
175	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
176	C₂²×C₄	C₅×#41	1	4	yes	—
177	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
178	C₂×C₈	—	2	8	no	—
179	C₂×C₈	—	2	4	no	—
180	C₂×C₈	C₄²×C₅	1	16	yes	complemented
181	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
182	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
183	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
184	C₄⋊C₄	—	2	16	no	—
185	C₄⋊C₄	—	2	8	no	—
186	C₄⋊C₄	C₂²×C₄×C₅	1	16	yes	—
187	C₄²	—	4	16	no	—
188	C₄²	C₅×C₄⋊C₄	1	8	yes	—
189	C₄²	C₂²×C₄×C₅	1	4	yes	—
190	C₄²	C₂×C₅×Q₈	1	4	yes	—
191	C₂×C₅	C₄×#107	1	8	yes	direct product factor
192	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
193	C₂×C₅	C₂²×#107	1	1	yes	characteristic
194	C₂×C₅	C₂×#427	1	1	yes	characteristic
195	C₂×C₅	C₂×C₄×D₈	1	2	yes	—
196	C₂×C₅	C₂×C₄×Q₁₆	1	2	yes	—
197	C₂³	C₂×C₅×C₄⋊C₄	1	2	yes	—
198	C₂³	C₄×C₅×D₄	1	4	yes	—
199	C₂³	C₂×C₅×D₈	1	2	yes	—
200	C₂³	C₂×C₅×Q₁₆	1	2	yes	—
201	C₂³	C₅×#135	1	4	yes	—
202	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
203	C₂×C₄	—	2	8	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	4	32	no	—
207	C₂×C₄	—	4	8	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	C₅×#107	1	16	yes	direct product factor
210	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
211	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
212	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
213	C₂×C₄	C₂×C₅×C₄⋊C₄	1	4	yes	—
214	C₂×C₄	C₅×#117	1	4	yes	—
215	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
216	C₂×C₄	C₄×C₅×Q₈	1	8	yes	—
217	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
218	C₂×C₄	C₂×C₅×Q₁₆	1	4	yes	—
219	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
220	C₂×C₄	C₂³×C₄×C₅	1	2	yes	—
221	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
222	C₂×C₄	C₂×C₅×#41	1	2	yes	—
223	C₈	—	2	4	no	—
224	C₈	C₂×C₄²×C₅	1	8	yes	complemented
225	C₅	C₂×C₄×#107	1	1	yes	characteristic, direct product factor, Sylow
226	C₂²	C₄×C₅×C₄⋊C₄	1	4	yes	—
227	C₂²	C₂×C₅×#107	1	4	yes	—
228	C₂²	C₅×#427	1	4	yes	—
229	C₂²	C₄×C₅×D₈	1	8	yes	—
230	C₂²	C₄×C₅×Q₁₆	1	8	yes	—
231	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
232	C₂²	C₂×C₄×C₅×D₄	1	2	yes	—
233	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
234	C₂²	C₂²×C₅×Q₁₆	1	1	yes	characteristic
235	C₂²	C₂×C₅×#135	1	2	yes	—
236	C₄	—	2	4	no	—
237	C₄	—	4	16	no	—
238	C₄	C₄×C₅×C₄⋊C₄	1	4	yes	—
239	C₄	C₂×C₅×#107	1	8	yes	direct product factor
240	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
241	C₄	C₂×C₅×#117	1	1	yes	characteristic
242	C₄	C₂×C₄×C₅×Q₈	1	2	yes	—
243	C₂	C₄×C₅×#107	1	8	yes	direct product factor
244	C₂	C₂×C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
245	C₂	C₂²×C₅×#107	1	1	yes	characteristic
246	C₂	C₂×C₅×#427	1	1	yes	characteristic
247	C₂	C₂×C₄×C₅×D₈	1	2	yes	—
248	C₂	C₂×C₄×C₅×Q₁₆	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	D₄
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂⁴×C₅	C₂×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₄×C₅×#107	C₂	direct (N × H)
11. C₂×C₅×#107	C₄	direct (N × H)
13. C₂×C₄×C₅×C₈	C₄	semidirect
19. C₂×C₄×#107	C₅	direct (N × H)
40. C₅×#107	C₂×C₄	direct (N × H)
41. C₄×C₅×C₈	C₂×C₄	semidirect
46. C₄×#107	C₂×C₅	direct (N × H)
68. C₂×C₅×C₈	C₄²	semidirect
84. C₂×#107	C₄×C₅	direct (N × H)
86. C₂×C₄×C₈	C₄×C₅	semidirect
104. C₂×C₄×C₅	#107	direct (N × H)
119. C₅×C₈	C₂×C₄²	semidirect
140. #107	C₂×C₄×C₅	direct (N × H)
141. C₄×C₈	C₂×C₄×C₅	semidirect
156. C₄×C₅	C₂×#107	direct (N × H)
180. C₂×C₈	C₄²×C₅	semidirect
191. C₂×C₅	C₄×#107	direct (N × H)
209. C₂×C₄	C₅×#107	direct (N × H)
224. C₈	C₂×C₄²×C₅	semidirect
225. C₅	C₂×C₄×#107	direct (N × H)
239. C₄	C₂×C₅×#107	direct (N × H)
243. C₂	C₄×C₅×#107	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×C₅×#107	2
C₂²×C₄×C₅×C₈	2
C₂×C₄×C₅×C₄⋊C₄	2
C₄×C₅×#107	2
C₂×C₅×#340	2
C₂×C₄×#107	5

Subgroups — C2×C4×C5×#107 #11943168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄×C₅×#107 #11943168