Subgroups — C₂×#10598667 #13025484

Order 1280.

574 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#3996	C₂	1	2	yes	maximal
2	C₂²×D₅×C₁₆	C₂	1	1	yes	maximal, characteristic
3	#10598667	C₂	1	4	yes	maximal, direct product factor
4	C₂²×#700	—	2	2	no	—
5	C₂²×C₈×D₅	C₂²	1	1	yes	characteristic
6	C₂×#3996	—	2	4	no	—
7	C₂×#3996	—	2	2	no	—
8	C₂×#3996	C₄	1	8	yes	—
9	C₂×#3996	C₂²	1	4	yes	—
10	C₂²×C₅×C₁₆	C₄	1	1	yes	characteristic
11	C₂²×#672	C₄	1	1	yes	characteristic
12	C₂×D₅×C₁₆	—	2	4	no	—
13	C₂×D₅×C₁₆	C₄	1	2	yes	—
14	C₂×D₅×C₁₆	C₂²	1	2	yes	—
15	C₂×#1381	—	5	1	no	maximal, Sylow
16	C₂²×C₄×D₅	D₄	1	1	yes	characteristic
17	C₂²×#170	—	4	2	no	—
18	C₂×#700	—	2	8	no	—
19	C₂×#700	—	2	4	no	—
20	C₂×#700	—	4	2	no	—
21	C₂×#700	—	4	4	no	—
22	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
23	C₂²×#168	Q₈	1	1	yes	characteristic
24	C₂×C₈×D₅	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×D₅	C₂×C₄	1	4	yes	—
26	C₂×C₈×D₅	D₄	1	1	yes	characteristic
27	C₂×C₈×D₅	D₄	1	4	yes	—
28	C₂×C₈×D₅	Q₈	1	1	yes	characteristic
29	C₂×C₈×D₅	C₂³	1	1	yes	characteristic
30	#3996	—	2	8	no	—
31	#3996	—	2	4	no	—
32	#3996	C₂×C₄	1	8	yes	—
33	C₂×C₅×C₁₆	—	2	2	no	—
34	C₂×C₅×C₁₆	C₂×C₄	1	2	yes	—
35	C₂×#672	—	2	2	no	—
36	C₂×#672	C₂×C₄	1	2	yes	—
37	D₅×C₁₆	—	2	4	no	—
38	D₅×C₁₆	—	2	4	no	—
39	C₂³×C₁₆	—	5	1	no	—
40	C₂²×#108	—	5	2	no	—
41	#1381	—	5	4	no	—
42	C₂³×D₅	Q₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₅	D₈	1	1	yes	characteristic
45	C₂×C₄×D₅	C₂²⋊C₄	1	4	yes	—
46	C₂×C₄×D₅	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×D₅	D₈	1	1	yes	characteristic
48	C₂×C₄×D₅	SD₁₆	1	4	yes	—
49	C₂×C₄×D₅	Q₁₆	1	1	yes	characteristic
50	C₂×C₄×D₅	C₂×D₄	1	1	yes	characteristic
51	C₂×#170	—	4	8	no	—
52	C₂×#170	—	4	4	no	—
53	#700	—	2	8	no	—
54	#700	—	4	8	no	—
55	#700	—	4	4	no	—
56	C₂×C₅×C₈	C₄²	1	2	yes	—
57	C₂×C₅×C₈	C₂²⋊C₄	1	2	yes	—
58	C₂×C₅×C₈	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
60	C₂×#168	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#168	C₄⋊C₄	1	4	yes	—
62	C₂×#168	C₂×Q₈	1	1	yes	characteristic
63	C₈×D₅	C₄²	1	2	yes	—
64	C₈×D₅	C₂²⋊C₄	1	4	yes	—
65	C₈×D₅	C₂²⋊C₄	1	2	yes	—
66	C₈×D₅	C₄⋊C₄	1	4	yes	—
67	C₈×D₅	C₂²×C₄	1	2	yes	—
68	C₈×D₅	C₂×D₄	1	2	yes	—
69	C₅×C₁₆	—	2	2	no	—
70	#672	—	2	2	no	—
71	C₂²×M₁₆	—	10	2	no	—
72	C₂³×C₈	—	5	1	no	—
73	C₂²×C₁₆	—	5	1	no	—
74	C₂²×C₁₆	—	5	2	no	—
75	C₂²×C₁₆	—	5	2	no	—
76	C₂²×C₁₆	—	10	4	no	—
77	C₂²×C₁₆	AGL(1,𝔽₅)	1	1	yes	characteristic
78	C₂×#108	—	5	4	no	—
79	C₂×#108	—	5	8	no	—
80	C₂×#108	—	10	2	no	—
81	C₂×#108	—	10	4	no	—
82	C₂³×C₅	#107	1	1	yes	characteristic
83	C₂²×D₅	—	2	2	no	—
84	C₂²×D₅	—	2	2	no	—
85	C₂²×D₅	#103	1	4	yes	—
86	C₂²×D₅	#107	1	1	yes	characteristic
87	C₂²×D₅	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅	#95	1	2	yes	—
89	C₂×C₄×C₅	#106	1	2	yes	—
90	C₂×C₄×C₅	#107	1	1	yes	characteristic
91	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
92	C₂×Dic₅	#102	1	4	yes	—
93	C₂×Dic₅	#107	1	1	yes	characteristic
94	C₂×Dic₅	C₂×D₈	1	1	yes	characteristic
95	C₄×D₅	#95	1	2	yes	—
96	C₄×D₅	#102	1	4	yes	—
97	C₄×D₅	#103	1	4	yes	—
98	C₄×D₅	#106	1	2	yes	—
99	C₄×D₅	C₂×C₂²⋊C₄	1	2	yes	—
100	C₄×D₅	C₂×SD₁₆	1	2	yes	—
101	#170	—	4	8	no	—
102	C₅×C₈	#95	1	2	yes	—
103	C₅×C₈	C₂×C₄²	1	1	yes	characteristic
104	C₅×C₈	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	#168	#95	1	2	yes	—
106	#168	C₂×C₄⋊C₄	1	2	yes	—
107	C₂³×C₄	—	5	1	no	—
108	C₂×M₁₆	—	10	4	no	—
109	C₂×M₁₆	—	10	8	no	—
110	C₂×M₁₆	—	20	4	no	—
111	C₂×M₁₆	—	20	2	no	—
112	C₂²×C₈	—	5	1	no	—
113	C₂²×C₈	—	5	4	no	—
114	C₂²×C₈	—	5	1	no	—
115	C₂²×C₈	—	5	1	no	—
116	C₂²×C₈	—	5	4	no	—
117	C₂²×C₈	—	5	1	no	—
118	C₂²×C₈	—	5	1	no	—
119	C₂²×C₈	—	20	2	no	—
120	C₂²×C₈	C₂×AGL(1,𝔽₅)	1	1	yes	characteristic
121	C₂×C₁₆	—	2	2	no	—
122	C₂×C₁₆	—	5	2	no	—
123	C₂×C₁₆	—	10	4	no	—
124	C₂×C₁₆	—	10	4	no	—
125	C₂×C₁₆	—	10	2	no	—
126	C₂×C₁₆	C₂×AGL(1,𝔽₅)	1	2	yes	—
127	#108	—	5	8	no	—
128	#108	—	10	4	no	—
129	#108	—	10	8	no	—
130	C₂²×C₅	#340	1	2	yes	—
131	C₂²×C₅	#368	1	2	yes	—
132	C₂²×C₅	#368	1	2	yes	—
133	C₂²×C₅	C₂×#107	1	1	yes	characteristic
134	C₂×D₅	—	2	8	no	—
135	C₂×D₅	—	2	1	no	—
136	C₂×D₅	—	2	1	no	—
137	C₂×D₅	—	2	1	no	—
138	C₂×D₅	—	2	1	no	—
139	C₂×D₅	#340	1	2	yes	—
140	C₂×D₅	C₂×#103	1	2	yes	—
141	C₄×C₅	#340	1	2	yes	—
142	C₄×C₅	C₂×#95	1	1	yes	characteristic
143	C₄×C₅	C₂×#106	1	1	yes	characteristic
144	Dic₅	#340	1	2	yes	—
145	Dic₅	C₂×#102	1	2	yes	—
146	C₂⁴	—	5	1	no	—
147	C₂²×C₄	—	5	1	no	—
148	C₂²×C₄	—	5	1	no	—
149	C₂²×C₄	—	5	1	no	—
150	C₂²×C₄	—	5	4	no	—
151	C₂²×C₄	—	5	1	no	—
152	C₂²×C₄	—	5	1	no	—
153	C₂²×C₄	—	5	4	no	—
154	C₂²×C₄	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
155	M₁₆	—	10	8	no	—
156	M₁₆	—	20	8	no	—
157	M₁₆	—	20	4	no	—
158	C₂×C₈	—	5	1	no	—
159	C₂×C₈	—	5	1	no	—
160	C₂×C₈	—	5	4	no	—
161	C₂×C₈	—	5	2	no	—
162	C₂×C₈	—	5	2	no	—
163	C₂×C₈	—	5	2	no	—
164	C₂×C₈	—	5	2	no	—
165	C₂×C₈	—	5	4	no	—
166	C₂×C₈	—	5	4	no	—
167	C₂×C₈	—	20	8	no	—
168	C₂×C₈	—	20	4	no	—
169	C₂×C₈	C₄×AGL(1,𝔽₅)	1	2	yes	—
170	C₂×C₈	C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
171	C₂×C₈	C₂²⋊AGL(1,𝔽₅)	1	2	yes	—
172	C₂×C₈	C₂²×AGL(1,𝔽₅)	1	1	yes	characteristic
173	C₁₆	—	2	2	no	—
174	C₁₆	—	10	2	no	—
175	C₂×C₅	#1381	1	4	yes	complemented
176	C₂×C₅	C₂×#340	1	1	yes	characteristic
177	C₂×C₅	C₂×#368	1	1	yes	characteristic
178	C₂×C₅	C₂×#368	1	1	yes	characteristic
179	D₅	—	2	2	no	—
180	D₅	—	2	2	no	—
181	C₂³	—	5	4	no	—
182	C₂³	—	5	1	no	—
183	C₂³	—	5	1	no	—
184	C₂³	—	10	2	no	—
185	C₂³	—	10	2	no	—
186	C₂³	#3994	1	1	yes	characteristic
187	C₂×C₄	—	5	4	no	—
188	C₂×C₄	—	5	1	no	—
189	C₂×C₄	—	5	1	no	—
190	C₂×C₄	—	5	4	no	—
191	C₂×C₄	—	5	4	no	—
192	C₂×C₄	—	5	2	no	—
193	C₂×C₄	—	5	2	no	—
194	C₂×C₄	—	5	2	no	—
195	C₂×C₄	—	5	2	no	—
196	C₂×C₄	#3993	1	2	yes	—
197	C₂×C₄	#3994	1	1	yes	characteristic
198	C₂×C₄	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
199	C₂×C₄	C₂×C₄⋊AGL(1,𝔽₅)	1	1	yes	characteristic
200	C₈	—	5	2	no	—
201	C₈	—	5	2	no	—
202	C₈	—	20	8	no	—
203	C₈	(C₂×AGL(1,𝔽₅))⋊C₄	1	2	yes	—
204	C₈	C₂×C₄×AGL(1,𝔽₅)	1	1	yes	characteristic
205	C₈	C₂×C₂²⋊AGL(1,𝔽₅)	1	1	yes	characteristic
206	C₅	C₂×#1381	1	1	yes	characteristic, complemented, Sylow
207	C₂²	—	5	2	no	—
208	C₂²	—	5	2	no	—
209	C₂²	—	10	8	no	—
210	C₂²	—	10	1	no	—
211	C₂²	—	10	1	no	—
212	C₂²	—	10	1	no	—
213	C₂²	—	10	1	no	—
214	C₂²	#64851	1	2	yes	—
215	C₂²	#64852	1	2	yes	—
216	C₂²	#64893	1	2	yes	—
217	C₂²	C₂×#3994	1	1	yes	characteristic
218	C₄	—	5	2	no	—
219	C₄	—	5	2	no	—
220	C₄	#64893	1	2	yes	—
221	C₄	C₂×#3993	1	1	yes	characteristic
222	C₄	C₂×(C₂×AGL(1,𝔽₅))⋊C₄	1	1	yes	characteristic
223	C₂	—	10	2	no	—
224	C₂	—	10	2	no	—
225	C₂	#10598667	1	4	yes	direct product factor
226	C₂	C₂×#64851	1	1	yes	characteristic
227	C₂	C₂×#64852	1	1	yes	characteristic
228	C₂	C₂×#64893	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#3994
commutator	C₅×C₈	C₂×C₄²
Frattini	C₂×C₈	C₂²×AGL(1,𝔽₅)
Fitting	C₂²×C₅×C₁₆	C₄
socle	C₂³×C₅	#107
prime radical	C₂³×D₅	Q₁₆
cosocle	C₂×C₈×D₅	C₂³
hypercenter	C₂²×C₁₆	AGL(1,𝔽₅)
hypocenter	C₅	C₂×#1381
LCS	G ▷ C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #10598667	C₂	direct (N × H)
175. C₂×C₅	#1381	semidirect
206. C₅	C₂×#1381	semidirect
225. C₂	#10598667	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×#3996	2
C₂²×D₅×C₁₆	2
#10598667	2
C₂×#1381	5

Subgroups — C2×#10598667 #13025484

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#10598667 #13025484