Subgroups — C₂×D₄×#674 #13043499

Order 1280.

806 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#674	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×#168	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#674	C₂	1	2	yes	maximal, complemented
4	D₄×#674	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊#674	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#674	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₅×D₄	C₄	1	1	yes	characteristic
8	C₂³×#168	C₂²	1	2	yes	—
9	C₂²×#674	—	2	8	no	—
10	C₂²×#674	C₂²	1	1	yes	characteristic
11	C₂²×#674	C₂²	1	2	yes	—
12	C₂²×#674	C₂²	1	1	yes	characteristic
13	C₂²×#674	C₂²	1	8	yes	complemented
14	C₂×C₂²⋊#168	C₄	1	2	yes	—
15	D₄×#168	C₄	1	4	yes	—
16	D₄×#168	C₂²	1	4	yes	—
17	C₂×C₄⋊#168	C₄	1	1	yes	characteristic
18	C₂×C₄×#168	C₂²	1	1	yes	characteristic
19	C₂²⋊#674	C₂²	1	8	yes	complemented
20	C₄⋊#674	C₂²	1	4	yes	complemented
21	C₄×#674	C₂²	1	4	yes	complemented
22	C₂×D₄×C₁₆	—	5	1	no	maximal, Sylow
23	C₂²×C₅×D₄	C₈	1	1	yes	characteristic
24	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
25	C₄×C₅×D₄	C₈	1	4	yes	—
26	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
27	C₂×C₅×C₄⋊C₄	C₈	1	1	yes	characteristic
28	C₂×C₅×C₂²⋊C₄	C₈	1	2	yes	—
29	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂²×#168	—	2	4	no	—
31	C₂²×#168	—	2	4	no	—
32	C₂²×#168	C₂×C₄	1	4	yes	—
33	C₂²×#168	C₂×C₄	1	2	yes	—
34	C₂²×#168	C₂×C₄	1	1	yes	characteristic
35	C₂²×#168	C₂³	1	4	yes	—
36	C₂²×#168	C₂³	1	1	yes	characteristic
37	C₂×#674	—	2	4	no	—
38	C₂×#674	—	2	2	no	—
39	C₂×#674	—	2	16	no	—
40	C₂×#674	D₄	1	4	yes	direct product factor
41	C₂×#674	C₂³	1	2	yes	—
42	C₂×#674	C₂³	1	4	yes	—
43	C₂×#674	C₂³	1	2	yes	—
44	C₂²⋊#168	C₂×C₄	1	8	yes	—
45	C₄⋊#168	C₂×C₄	1	4	yes	—
46	C₄×#168	C₂×C₄	1	2	yes	—
47	C₄×#168	C₂³	1	2	yes	—
48	C₂³×C₁₆	—	5	2	no	—
49	C₂×C₈×D₄	—	5	1	no	—
50	D₄×C₁₆	—	5	8	no	—
51	C₂×C₄⋊C₁₆	—	5	1	no	—
52	C₂×C₂²⋊C₁₆	—	5	2	no	—
53	C₂×C₄×C₁₆	—	5	1	no	—
54	C₂⁴×C₅	C₂×C₈	1	2	yes	—
55	C₂×C₅×D₄	C₁₆	1	8	yes	complemented
56	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
57	C₂²×C₄×C₅	—	2	4	no	—
58	C₂²×C₄×C₅	—	2	4	no	—
59	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
61	C₂²×C₄×C₅	C₂×C₈	1	4	yes	—
62	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
64	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
65	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
66	C₅×C₂²⋊C₄	C₂×C₈	1	8	yes	—
67	C₄²×C₅	C₂×C₈	1	2	yes	—
68	C₄²×C₅	C₂²×C₄	1	2	yes	—
69	C₂×#168	—	2	4	no	—
70	C₂×#168	—	2	2	no	—
71	C₂×#168	—	2	8	no	—
72	C₂×#168	—	2	4	no	—
73	C₂×#168	—	2	4	no	—
74	C₂×#168	C₂²×C₄	1	2	yes	—
75	C₂×#168	C₂²×C₄	1	4	yes	—
76	C₂×#168	C₂²×C₄	1	1	yes	characteristic
77	C₂×#168	C₂×D₄	1	2	yes	—
78	C₂×#168	#41	1	2	yes	—
79	C₂×#168	C₂⁴	1	1	yes	characteristic
80	#674	—	2	4	no	—
81	#674	—	2	2	no	—
82	#674	C₂×D₄	1	4	yes	direct product factor
83	C₂³×C₈	—	5	2	no	—
84	C₂×C₄×D₄	AGL(1,𝔽₅)	1	1	yes	characteristic
85	C₂²×C₁₆	—	5	1	no	—
86	C₂²×C₁₆	—	5	1	no	—
87	C₂²×C₁₆	—	5	2	no	—
88	C₂²×C₁₆	—	5	8	no	—
89	C₂²×C₁₆	—	10	8	no	—
90	C₈×D₄	—	5	4	no	—
91	C₈×D₄	—	5	4	no	—
92	C₂×C₄⋊C₈	—	5	1	no	—
93	C₂×C₂²⋊C₈	—	5	2	no	—
94	C₂×C₄×C₈	—	5	1	no	—
95	C₄⋊C₁₆	—	5	4	no	—
96	C₂²⋊C₁₆	—	5	8	no	—
97	C₄×C₁₆	—	5	4	no	—
98	C₂³×C₅	—	2	4	no	—
99	C₂³×C₅	—	2	4	no	—
100	C₂³×C₅	C₂×C₁₆	1	8	yes	complemented
101	C₂³×C₅	C₂²×C₈	1	4	yes	—
102	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
103	C₅×D₄	C₂×C₁₆	1	16	yes	complemented
104	C₂×C₄×C₅	—	2	4	no	—
105	C₂×C₄×C₅	—	2	2	no	—
106	C₂×C₄×C₅	—	2	4	no	—
107	C₂×C₄×C₅	—	2	4	no	—
108	C₂×C₄×C₅	—	2	8	no	—
109	C₂×C₄×C₅	C₂×C₁₆	1	4	yes	complemented
110	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
112	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
113	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
115	C₂×C₄×C₅	#131	1	2	yes	—
116	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
117	#168	—	2	4	no	—
118	#168	—	2	2	no	—
119	#168	C₄×D₄	1	2	yes	—
120	#168	C₂²×D₄	1	1	yes	characteristic
121	#168	C₂×#41	1	1	yes	characteristic
122	C₂²×D₄	#170	1	1	yes	characteristic
123	C₂³×C₄	C₂×AGL(1,𝔽₅)	1	2	yes	—
124	C₂²×C₈	—	5	1	no	—
125	C₂²×C₈	—	5	4	no	—
126	C₂²×C₈	—	5	4	no	—
127	C₂²×C₈	—	5	2	no	—
128	C₂²×C₈	—	5	1	no	—
129	C₂²×C₈	—	10	4	no	—
130	C₂²×C₈	—	10	4	no	—
131	C₄×D₄	#170	1	4	yes	—
132	C₄×D₄	C₂×AGL(1,𝔽₅)	1	4	yes	—
133	C₂×C₄⋊C₄	#170	1	1	yes	characteristic
134	C₂×C₂²⋊C₄	#170	1	2	yes	—
135	C₂×C₄²	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
136	C₂×C₁₆	—	5	2	no	—
137	C₂×C₁₆	—	5	4	no	—
138	C₂×C₁₆	—	5	2	no	—
139	C₂×C₁₆	—	5	4	no	—
140	C₂×C₁₆	—	10	16	no	—
141	C₂×C₁₆	—	10	2	no	—
142	C₂×C₁₆	—	10	4	no	—
143	C₄⋊C₈	—	5	4	no	—
144	C₂²⋊C₈	—	5	8	no	—
145	C₄×C₈	—	5	2	no	—
146	C₄×C₈	—	5	2	no	—
147	C₂²×C₅	—	2	4	no	—
148	C₂²×C₅	—	2	4	no	—
149	C₂²×C₅	—	2	16	no	—
150	C₂²×C₅	C₈×D₄	1	2	yes	—
151	C₂²×C₅	C₂²×C₁₆	1	2	yes	—
152	C₂²×C₅	C₂²×C₁₆	1	8	yes	complemented
153	C₂²×C₅	#504	1	2	yes	—
154	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₅	—	2	2	no	—
156	C₄×C₅	—	2	4	no	—
157	C₄×C₅	C₈×D₄	1	2	yes	—
158	C₄×C₅	C₂²×C₁₆	1	4	yes	complemented
159	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₅	C₂×#131	1	1	yes	characteristic
161	C₂⁴	C₂×#170	1	2	yes	—
162	C₂×D₄	#674	1	8	yes	direct product factor
163	C₂×D₄	C₂×#170	1	4	yes	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₂×#170	1	1	yes	characteristic
167	C₂²×C₄	C₂×#170	1	2	yes	—
168	C₂²×C₄	C₂×#170	1	4	yes	—
169	C₂²×C₄	C₂×#170	1	1	yes	characteristic
170	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	4	yes	—
171	C₂²×C₄	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
172	C₂×C₈	—	5	2	no	—
173	C₂×C₈	—	5	2	no	—
174	C₂×C₈	—	5	1	no	—
175	C₂×C₈	—	5	1	no	—
176	C₂×C₈	—	5	2	no	—
177	C₂×C₈	—	5	4	no	—
178	C₂×C₈	—	10	4	no	—
179	C₂×C₈	—	10	4	no	—
180	C₂×C₈	—	10	8	no	—
181	C₂×C₈	—	10	2	no	—
182	C₂×C₈	—	10	4	no	—
183	C₄⋊C₄	C₂×#170	1	4	yes	—
184	C₂²⋊C₄	C₂×#170	1	8	yes	—
185	C₄²	C₂×#170	1	2	yes	—
186	C₄²	C₂²×AGL(1,𝔽₅)	1	2	yes	—
187	C₁₆	—	5	4	no	—
188	C₁₆	—	10	4	no	—
189	C₁₆	—	10	2	no	—
190	C₂×C₅	—	2	8	no	—
191	C₂×C₅	D₄×C₁₆	1	4	yes	complemented
192	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
193	C₂×C₅	C₂³×C₁₆	1	1	yes	characteristic
194	C₂×C₅	C₂×#504	1	1	yes	characteristic
195	C₂³	—	2	4	no	—
196	C₂³	—	2	4	no	—
197	C₂³	C₂×#674	1	8	yes	complemented
198	C₂³	C₂²×#170	1	4	yes	—
199	C₂³	C₂²×#170	1	1	yes	characteristic
200	D₄	C₂×#674	1	16	yes	direct product factor
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	4	no	—
205	C₂×C₄	—	2	8	no	—
206	C₂×C₄	C₂×#674	1	4	yes	complemented
207	C₂×C₄	#4131	1	2	yes	—
208	C₂×C₄	D₄×AGL(1,𝔽₅)	1	2	yes	—
209	C₂×C₄	C₂²×#170	1	2	yes	—
210	C₂×C₄	C₂²×#170	1	4	yes	—
211	C₂×C₄	C₂²×#170	1	2	yes	—
212	C₂×C₄	C₂²×#170	1	1	yes	characteristic
213	C₂×C₄	C₂³×AGL(1,𝔽₅)	1	1	yes	characteristic
214	C₈	—	5	1	no	—
215	C₈	—	5	1	no	—
216	C₈	—	5	2	no	—
217	C₈	—	10	2	no	—
218	C₈	—	10	4	no	—
219	C₅	C₂×D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
220	C₂²	—	2	16	no	—
221	C₂²	—	2	4	no	—
222	C₂²	—	2	4	no	—
223	C₂²	C₂²×#674	1	2	yes	—
224	C₂²	C₂²×#674	1	8	yes	complemented
225	C₂²	D₄×#170	1	2	yes	—
226	C₂²	#65791	1	2	yes	—
227	C₂²	C₂³×#170	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	2	no	—
230	C₄	C₂²×#674	1	4	yes	complemented
231	C₄	D₄×#170	1	2	yes	—
232	C₄	C₂×#4131	1	1	yes	characteristic
233	C₄	C₂×D₄×AGL(1,𝔽₅)	1	1	yes	characteristic
234	C₂	—	2	8	no	—
235	C₂	D₄×#674	1	4	yes	direct product factor
236	C₂	C₂³×#674	1	1	yes	characteristic
237	C₂	C₂×D₄×#170	1	1	yes	characteristic
238	C₂	C₂×#65791	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×AGL(1,𝔽₅)
commutator	C₂×C₅	C₂³×C₁₆
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₅)
Fitting	C₂×C₄×C₅×D₄	C₄
socle	C₂³×C₅	C₂²×C₈
prime radical	C₂²×C₅×D₄	C₈
cosocle	C₂×#168	C₂⁴
hypercenter	C₂×C₄×D₄	AGL(1,𝔽₅)
hypocenter	C₅	C₂×D₄×C₁₆
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₂³×#674	C₂	semidirect
3. C₂×C₂²⋊#674	C₂	semidirect
4. D₄×#674	C₂	direct (N × H)
5. C₂×C₄⋊#674	C₂	semidirect
6. C₂×C₄×#674	C₂	semidirect
13. C₂²×#674	C₂²	semidirect
19. C₂²⋊#674	C₂²	semidirect
20. C₄⋊#674	C₂²	semidirect
21. C₄×#674	C₂²	semidirect
40. C₂×#674	D₄	direct (N × H)
55. C₂×C₅×D₄	C₁₆	semidirect
82. #674	C₂×D₄	direct (N × H)
100. C₂³×C₅	C₂×C₁₆	semidirect
103. C₅×D₄	C₂×C₁₆	semidirect
109. C₂×C₄×C₅	C₂×C₁₆	semidirect
152. C₂²×C₅	C₂²×C₁₆	semidirect
158. C₄×C₅	C₂²×C₁₆	semidirect
162. C₂×D₄	#674	direct (N × H)
191. C₂×C₅	D₄×C₁₆	semidirect
197. C₂³	C₂×#674	semidirect
200. D₄	C₂×#674	direct (N × H)
206. C₂×C₄	C₂×#674	semidirect
219. C₅	C₂×D₄×C₁₆	semidirect
224. C₂²	C₂²×#674	semidirect
230. C₄	C₂²×#674	semidirect
235. C₂	D₄×#674	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×#674	2
C₂×D₄×#168	2
C₂×C₂²⋊#674	2
D₄×#674	2
C₂×C₄⋊#674	2
C₂×C₄×#674	2
C₂×D₄×C₁₆	5

Subgroups — C2×D4×#674 #13043499

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×D₄×#674 #13043499