Subgroups — C₂×C₅×#1338 #11934329

Order 1280.

346 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#346	C₂	1	1	yes	maximal, characteristic
4	C₅×#1338	C₂	1	4	yes	maximal, direct product factor
5	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
6	C₂²×C₅×C₁₆	—	2	1	no	—
7	C₂²×C₅×C₁₆	—	2	1	no	—
8	C₅×#446	—	2	1	no	—
9	C₅×#446	—	2	1	no	—
10	C₅×#446	C₄	1	2	yes	—
11	C₅×#446	C₂²	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	—	2	1	no	—
13	C₂×C₅×C₄⋊C₈	C₄	1	1	yes	characteristic
14	C₂×C₅×#97	—	2	1	no	—
15	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
16	C₅×C₄⋊C₁₆	—	2	1	no	—
17	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
18	C₅×#346	—	2	1	no	—
19	C₅×#346	C₂²	1	2	yes	—
20	C₂×#1338	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₅×Q₈	C₈	1	1	yes	characteristic
22	C₂²×C₅×C₈	—	2	1	no	—
23	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
25	C₄×C₅×Q₈	—	2	1	no	—
26	C₄×C₅×Q₈	—	2	1	no	—
27	C₄×C₅×Q₈	C₈	1	2	yes	—
28	C₄×C₅×Q₈	C₂×C₄	1	2	yes	—
29	C₂×C₅×C₄⋊C₄	—	2	1	no	—
30	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
31	C₂×C₄²×C₅	—	2	1	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
33	C₂×C₅×C₁₆	—	2	4	no	—
34	C₂×C₅×C₁₆	—	2	2	no	—
35	C₂×C₅×C₁₆	—	2	2	no	—
36	C₂×C₅×C₁₆	—	4	2	no	—
37	C₅×C₄⋊C₈	—	2	2	no	—
38	C₅×C₄⋊C₈	—	2	2	no	—
39	C₅×C₄⋊C₈	—	2	1	no	—
40	C₅×C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₅×#97	—	2	2	no	—
42	C₅×#97	—	2	2	no	—
43	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₄×C₅×C₈	D₄	1	1	yes	characteristic
45	C₄×C₅×C₈	D₄	1	1	yes	characteristic
46	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
47	C₂×#446	C₂×C₅	1	1	yes	characteristic
48	C₂×C₄⋊C₁₆	C₂×C₅	1	1	yes	characteristic
49	C₂×#346	C₂×C₅	1	1	yes	characteristic
50	#1338	C₂×C₅	1	4	yes	direct product factor
51	C₂×C₅×Q₈	—	2	1	no	—
52	C₂×C₅×Q₈	—	4	2	no	—
53	C₂×C₅×Q₈	C₂×C₈	1	2	yes	—
54	C₂²×C₄×C₅	—	2	1	no	—
55	C₂²×C₄×C₅	—	2	1	no	—
56	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
58	C₂²×C₄×C₅	M₁₆	1	1	yes	characteristic
59	C₂×C₅×C₈	—	2	2	no	—
60	C₂×C₅×C₈	—	2	2	no	—
61	C₂×C₅×C₈	—	4	2	no	—
62	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
63	C₂×C₅×C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₅×C₈	SD₁₆	1	2	yes	—
65	C₂×C₅×C₈	Q₁₆	1	2	yes	—
66	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
67	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
68	C₅×C₄⋊C₄	—	2	2	no	—
69	C₅×C₄⋊C₄	—	2	1	no	—
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
72	C₄²×C₅	—	2	2	no	—
73	C₄²×C₅	—	2	2	no	—
74	C₄²×C₅	C₂²⋊C₄	1	1	yes	characteristic
75	C₄²×C₅	C₂×C₈	1	1	yes	characteristic
76	C₄²×C₅	M₁₆	1	1	yes	characteristic
77	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₅×C₁₆	—	2	4	no	—
79	C₅×C₁₆	—	4	2	no	—
80	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
81	C₂²×C₁₆	—	2	1	no	—
82	C₂²×C₁₆	—	2	1	no	—
83	#446	—	2	1	no	—
84	#446	—	2	1	no	—
85	#446	C₄×C₅	1	2	yes	—
86	#446	C₂²×C₅	1	2	yes	—
87	C₂×C₄⋊C₈	—	2	1	no	—
88	C₂×C₄⋊C₈	C₄×C₅	1	1	yes	characteristic
89	C₂×#97	—	2	1	no	—
90	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
91	C₄⋊C₁₆	—	2	1	no	—
92	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
93	#346	—	2	1	no	—
94	#346	C₂²×C₅	1	2	yes	—
95	C₂³×C₅	C₂²⋊C₈	1	1	yes	characteristic
96	C₅×Q₈	—	4	2	no	—
97	C₅×Q₈	—	4	2	no	—
98	C₂×C₄×C₅	—	2	2	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	2	no	—
101	C₂×C₄×C₅	—	2	2	no	—
102	C₂×C₄×C₅	—	4	2	no	—
103	C₂×C₄×C₅	—	4	2	no	—
104	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
105	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₄×C₅	C₂²⋊C₈	1	1	yes	characteristic
107	C₂×C₄×C₅	#103	1	2	yes	—
108	C₂×C₄×C₅	#104	1	2	yes	—
109	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅	C₂×M₁₆	1	1	yes	characteristic
112	C₅×C₈	—	2	1	no	—
113	C₅×C₈	—	2	1	no	—
114	C₅×C₈	—	4	2	no	—
115	C₅×C₈	#103	1	2	yes	—
116	C₅×C₈	C₂×SD₁₆	1	1	yes	characteristic
117	C₅×C₈	C₂×Q₁₆	1	1	yes	characteristic
118	C₂²×Q₈	C₅×C₈	1	1	yes	characteristic
119	C₂²×C₈	—	2	1	no	—
120	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
121	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
122	C₄×Q₈	—	2	1	no	—
123	C₄×Q₈	—	2	1	no	—
124	C₄×Q₈	C₅×C₈	1	2	yes	—
125	C₄×Q₈	C₂×C₄×C₅	1	2	yes	—
126	C₂×C₄⋊C₄	—	2	1	no	—
127	C₂×C₄⋊C₄	C₅×C₈	1	1	yes	characteristic
128	C₂×C₄²	—	2	1	no	—
129	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
130	C₂×C₁₆	—	2	4	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	2	2	no	—
133	C₂×C₁₆	—	4	2	no	—
134	C₄⋊C₈	—	2	2	no	—
135	C₄⋊C₈	—	2	2	no	—
136	C₄⋊C₈	—	2	1	no	—
137	C₄⋊C₈	C₂×C₄×C₅	1	2	yes	—
138	#97	—	2	2	no	—
139	#97	—	2	2	no	—
140	C₄×C₈	C₂×C₄×C₅	1	1	yes	characteristic
141	C₄×C₈	C₅×D₄	1	1	yes	characteristic
142	C₄×C₈	C₅×D₄	1	1	yes	characteristic
143	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
144	C₂²×C₅	Q₈⋊C₈	1	2	yes	—
145	C₂²×C₅	#349	1	2	yes	—
146	C₂²×C₅	#350	1	2	yes	—
147	C₂²×C₅	C₂×C₂²⋊C₈	1	1	yes	characteristic
148	C₄×C₅	—	2	1	no	—
149	C₄×C₅	—	2	1	no	—
150	C₄×C₅	—	2	1	no	—
151	C₄×C₅	—	2	1	no	—
152	C₄×C₅	—	4	2	no	—
153	C₄×C₅	—	4	2	no	—
154	C₄×C₅	Q₈⋊C₈	1	2	yes	—
155	C₄×C₅	C₂×#103	1	1	yes	characteristic
156	C₄×C₅	C₂×#104	1	1	yes	characteristic
157	C₂×Q₈	—	2	1	no	—
158	C₂×Q₈	—	4	2	no	—
159	C₂×Q₈	C₂×C₅×C₈	1	2	yes	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
164	C₂²×C₄	C₅×M₁₆	1	1	yes	characteristic
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	4	2	no	—
168	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
169	C₂×C₈	C₅×C₂²⋊C₄	1	1	yes	characteristic
170	C₂×C₈	C₅×SD₁₆	1	2	yes	—
171	C₂×C₈	C₅×Q₁₆	1	2	yes	—
172	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
173	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	2	no	—
176	C₄⋊C₄	—	2	1	no	—
177	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
178	C₄²	—	2	2	no	—
179	C₄²	—	2	2	no	—
180	C₄²	C₅×C₂²⋊C₄	1	1	yes	characteristic
181	C₄²	C₂×C₅×C₈	1	1	yes	characteristic
182	C₄²	C₅×M₁₆	1	1	yes	characteristic
183	C₄²	C₂²×C₄×C₅	1	1	yes	characteristic
184	C₁₆	—	2	4	no	—
185	C₁₆	—	4	2	no	—
186	C₂×C₅	#1338	1	4	yes	direct product factor
187	C₂×C₅	C₂×Q₈⋊C₈	1	1	yes	characteristic
188	C₂×C₅	C₂×#349	1	1	yes	characteristic
189	C₂×C₅	C₂×#350	1	1	yes	characteristic
190	C₂³	C₅×C₂²⋊C₈	1	1	yes	characteristic
191	Q₈	—	4	2	no	—
192	Q₈	—	4	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	—	4	2	no	—
199	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
200	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
201	C₂×C₄	C₅×C₂²⋊C₈	1	1	yes	characteristic
202	C₂×C₄	C₅×#103	1	2	yes	—
203	C₂×C₄	C₅×#104	1	2	yes	—
204	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	1	yes	characteristic
205	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
206	C₂×C₄	C₂×C₅×M₁₆	1	1	yes	characteristic
207	C₈	—	2	1	no	—
208	C₈	—	2	1	no	—
209	C₈	—	4	2	no	—
210	C₈	C₅×#103	1	2	yes	—
211	C₈	C₂×C₅×SD₁₆	1	1	yes	characteristic
212	C₈	C₂×C₅×Q₁₆	1	1	yes	characteristic
213	C₅	C₂×#1338	1	1	yes	characteristic, direct product factor, Sylow
214	C₂²	C₅×Q₈⋊C₈	1	2	yes	—
215	C₂²	C₅×#349	1	2	yes	—
216	C₂²	C₅×#350	1	2	yes	—
217	C₂²	C₂×C₅×C₂²⋊C₈	1	1	yes	characteristic
218	C₄	—	2	1	no	—
219	C₄	—	2	1	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	C₅×Q₈⋊C₈	1	2	yes	—
225	C₄	C₂×C₅×#103	1	1	yes	characteristic
226	C₄	C₂×C₅×#104	1	1	yes	characteristic
227	C₂	C₅×#1338	1	4	yes	direct product factor
228	C₂	C₂×C₅×Q₈⋊C₈	1	1	yes	characteristic
229	C₂	C₂×C₅×#349	1	1	yes	characteristic
230	C₂	C₂×C₅×#350	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₅×C₈
Frattini, cosocle	C₄×C₈	C₂³×C₅
socle, prime radical	C₂³×C₅	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
4. C₅×#1338	C₂	direct (N × H)
20. C₂×#1338	C₅	direct (N × H)
50. #1338	C₂×C₅	direct (N × H)
186. C₂×C₅	#1338	direct (N × H)
213. C₅	C₂×#1338	direct (N × H)
227. C₂	C₅×#1338	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×C₅×#446	2
C₂×C₅×C₄⋊C₁₆	2
C₂×C₅×#346	2
C₅×#1338	2
C₂×#1338	5

Subgroups — C2×C5×#1338 #11934329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1338 #11934329