Subgroups — C₂×#13520 #499446

Order 512.

556 conjugacy classes of subgroups.

The subgroup lattice has 239 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2161	C₂	1	1	yes	maximal, characteristic
2	C₂×#2151	C₂	1	1	yes	maximal, characteristic
3	C₂×#1953	C₂	1	2	yes	maximal
4	#13520	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1385	C₂	1	1	yes	maximal, characteristic
6	C₂×#1381	C₂	1	2	yes	maximal
7	C₂²×M₃₂	C₂²	1	1	yes	characteristic
8	C₂³×C₁₆	C₂²	1	1	yes	characteristic
9	C₂×#430	—	2	2	no	—
10	C₂²×#108	C₂²	1	2	yes	—
11	C₂²×#108	C₂²	1	2	yes	—
12	C₂×#405	C₂²	1	1	yes	characteristic
13	#2161	—	2	2	no	—
14	#2161	C₂²	1	4	yes	—
15	C₂×#368	—	2	2	no	—
16	#2151	—	2	2	no	—
17	#2151	C₂²	1	4	yes	—
18	C₂×C₄⋊C₁₆	—	2	2	no	—
19	#1953	—	2	4	no	—
20	#1953	C₂²	1	8	yes	—
21	C₂×#344	—	2	2	no	—
22	#1385	C₂²	1	4	yes	—
23	#1381	C₂²	1	8	yes	—
24	C₂²×M₁₆	—	2	2	no	—
25	C₂²×M₁₆	—	2	2	no	—
26	C₂²×M₁₆	D₄	1	1	yes	characteristic
27	C₂³×C₈	C₂³	1	1	yes	characteristic
28	C₂×#117	D₄	1	1	yes	characteristic
29	C₂×M₃₂	—	2	2	no	—
30	C₂×M₃₂	—	2	2	no	—
31	C₂×M₃₂	D₄	1	2	yes	—
32	C₂×M₃₂	C₂³	1	2	yes	—
33	C₂²×C₁₆	—	2	1	no	—
34	C₂²×C₁₆	—	2	1	no	—
35	C₂²×C₁₆	—	2	2	no	—
36	C₂²×C₁₆	—	2	2	no	—
37	C₂²×C₁₆	D₄	1	2	yes	—
38	C₂²×C₁₆	C₂³	1	2	yes	—
39	#430	—	2	8	no	—
40	#430	—	4	4	no	—
41	C₂×#108	—	2	2	no	—
42	C₂×#108	—	2	2	no	—
43	C₂×#108	—	2	2	no	—
44	C₂×#108	—	2	2	no	—
45	C₂×#108	—	2	2	no	—
46	C₂×#108	—	2	2	no	—
47	C₂×#108	—	2	2	no	—
48	C₂×#108	—	2	2	no	—
49	C₂×#108	—	4	4	no	—
50	C₂×#108	Q₈	1	4	yes	—
51	C₂×#108	Q₈	1	4	yes	—
52	C₂×#108	C₂³	1	4	yes	—
53	C₂×#108	C₂³	1	4	yes	—
54	#405	—	2	1	no	—
55	#405	—	2	1	no	—
56	#405	D₄	1	2	yes	—
57	#405	C₂³	1	2	yes	—
58	C₂×#97	—	2	1	no	—
59	C₂×C₄×C₈	—	2	1	no	—
60	#368	—	2	8	no	—
61	C₄⋊C₁₆	—	2	8	no	—
62	#344	—	2	8	no	—
63	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×M₁₆	—	2	1	no	—
65	C₂×M₁₆	—	2	4	no	—
66	C₂×M₁₆	—	2	4	no	—
67	C₂×M₁₆	—	2	2	no	—
68	C₂×M₁₆	—	2	2	no	—
69	C₂×M₁₆	—	2	4	no	—
70	C₂×M₁₆	—	2	4	no	—
71	C₂×M₁₆	—	4	2	no	—
72	C₂×M₁₆	—	4	2	no	—
73	C₂×M₁₆	—	4	2	no	—
74	C₂×M₁₆	—	4	2	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂×M₁₆	—	4	2	no	—
77	C₂×M₁₆	—	4	2	no	—
78	C₂×M₁₆	—	4	2	no	—
79	C₂×M₁₆	C₂×D₄	1	2	yes	—
80	C₂²×C₈	—	2	1	no	—
81	C₂²×C₈	—	4	2	no	—
82	C₂²×C₈	—	4	2	no	—
83	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₈	C₂×Q₈	1	2	yes	—
87	C₂²×C₈	C₂×Q₈	1	2	yes	—
88	C₂²×C₈	#41	1	1	yes	characteristic
89	C₂²×C₈	#41	1	1	yes	characteristic
90	C₂²×C₈	#41	1	2	yes	—
91	C₂²×C₈	#41	1	2	yes	—
92	C₂²×C₈	C₂⁴	1	1	yes	characteristic
93	#117	—	2	1	no	—
94	#117	—	2	1	no	—
95	#117	Q₁₆	1	2	yes	—
96	#117	C₂×D₄	1	2	yes	—
97	C₂×C₄⋊C₄	—	2	1	no	—
98	C₂×C₂²⋊C₄	Q₁₆	1	2	yes	—
99	C₂×C₄²	—	2	1	no	—
100	M₃₂	—	2	4	no	—
101	M₃₂	—	2	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	2	2	no	—
105	C₂×C₁₆	—	2	4	no	—
106	C₂×C₁₆	—	4	1	no	—
107	C₂×C₁₆	—	4	1	no	—
108	C₂×C₁₆	—	4	2	no	—
109	C₂×C₁₆	C₂×D₄	1	4	yes	—
110	#108	—	2	4	no	—
111	#108	—	2	4	no	—
112	#108	—	2	4	no	—
113	#108	—	2	4	no	—
114	#108	—	2	4	no	—
115	#108	—	4	16	no	—
116	#108	—	4	2	no	—
117	#108	—	4	2	no	—
118	#108	C₂×Q₈	1	8	yes	—
119	#97	—	2	2	no	—
120	#97	—	2	2	no	—
121	C₄×C₈	—	2	2	no	—
122	C₄×C₈	—	2	2	no	—
123	C₂⁴	C₂×Q₁₆	1	1	yes	characteristic
124	C₂²×C₄	—	2	2	no	—
125	C₂²×C₄	#121	1	1	yes	characteristic
126	C₂²×C₄	#121	1	1	yes	characteristic
127	C₂²×C₄	C₂²⋊Q₈	1	2	yes	—
128	C₂²×C₄	C₂²⋊Q₈	1	2	yes	—
129	C₂²×C₄	C₂×Q₁₆	1	1	yes	characteristic
130	C₂²×C₄	#135	1	1	yes	characteristic
131	C₂²×C₄	#135	1	1	yes	characteristic
132	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
133	C₂²×C₄	AGL(1,ℤ/8)	1	2	yes	—
134	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
135	M₁₆	—	2	8	no	—
136	M₁₆	—	2	4	no	—
137	M₁₆	—	4	4	no	—
138	M₁₆	—	4	2	no	—
139	M₁₆	—	4	4	no	—
140	M₁₆	—	4	4	no	—
141	M₁₆	—	4	4	no	—
142	M₁₆	—	4	4	no	—
143	M₁₆	—	8	2	no	—
144	M₁₆	—	8	2	no	—
145	C₂×C₈	—	2	2	no	—
146	C₂×C₈	—	4	4	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	—	4	1	no	—
149	C₂×C₈	—	4	4	no	—
150	C₂×C₈	—	4	4	no	—
151	C₂×C₈	—	4	4	no	—
152	C₂×C₈	—	8	2	no	—
153	C₂×C₈	—	8	2	no	—
154	C₂×C₈	#121	1	1	yes	characteristic
155	C₂×C₈	#121	1	1	yes	characteristic
156	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
157	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
158	C₂×C₈	#123	1	1	yes	characteristic
159	C₂×C₈	#123	1	1	yes	characteristic
160	C₂×C₈	#123	1	2	yes	—
161	C₂×C₈	#125	1	2	yes	—
162	C₂×C₈	#125	1	2	yes	—
163	C₂×C₈	#125	1	2	yes	—
164	C₂×C₈	#125	1	2	yes	—
165	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
166	C₂×C₈	C₄⋊Q₈	1	1	yes	characteristic
167	C₂×C₈	C₄⋊Q₈	1	2	yes	—
168	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
169	C₂×C₈	C₂²×Q₈	1	2	yes	—
170	C₂×C₈	C₂×#41	1	2	yes	—
171	C₂×C₈	C₂×#41	1	1	yes	characteristic
172	C₄⋊C₄	—	2	2	no	—
173	C₄⋊C₄	—	2	2	no	—
174	C₂²⋊C₄	—	2	2	no	—
175	C₂²⋊C₄	C₂×Q₁₆	1	4	yes	—
176	C₄²	—	2	2	no	—
177	C₄²	—	2	2	no	—
178	C₁₆	—	2	4	no	—
179	C₁₆	—	4	2	no	—
180	C₂³	—	4	2	no	—
181	C₂³	#467	1	1	yes	characteristic
182	C₂³	#467	1	1	yes	characteristic
183	C₂³	#477	1	2	yes	—
184	C₂³	#477	1	2	yes	—
185	C₂³	C₂²×Q₁₆	1	1	yes	characteristic
186	C₂×C₄	—	2	4	no	—
187	C₂×C₄	—	4	4	no	—
188	C₂×C₄	#398	1	2	yes	—
189	C₂×C₄	#398	1	1	yes	characteristic
190	C₂×C₄	#398	1	1	yes	characteristic
191	C₂×C₄	#467	1	1	yes	characteristic
192	C₂×C₄	#467	1	1	yes	characteristic
193	C₂×C₄	#469	1	1	yes	characteristic
194	C₂×C₄	#469	1	2	yes	—
195	C₂×C₄	#469	1	1	yes	characteristic
196	C₂×C₄	#477	1	2	yes	—
197	C₂×C₄	#477	1	2	yes	—
198	C₂×C₄	#478	1	2	yes	—
199	C₂×C₄	#478	1	2	yes	—
200	C₂×C₄	#478	1	2	yes	—
201	C₂×C₄	#478	1	2	yes	—
202	C₂×C₄	C₂×#121	1	1	yes	characteristic
203	C₂×C₄	C₂×C₂²⋊Q₈	1	2	yes	—
204	C₂×C₄	C₂×#135	1	1	yes	characteristic
205	C₂×C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
206	C₈	—	4	8	no	—
207	C₈	—	4	2	no	—
208	C₈	—	8	4	no	—
209	C₈	#398	1	1	yes	characteristic
210	C₈	#398	1	1	yes	characteristic
211	C₈	#398	1	2	yes	—
212	C₈	C₂×#123	1	1	yes	characteristic
213	C₈	C₂×#125	1	2	yes	—
214	C₈	C₂×C₄⋊Q₈	1	1	yes	characteristic
215	C₂²	—	2	2	no	—
216	C₂²	—	2	4	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	2	no	—
220	C₂²	#2087	1	1	yes	characteristic
221	C₂²	#2087	1	2	yes	—
222	C₂²	#2087	1	1	yes	characteristic
223	C₂²	C₂×#467	1	1	yes	characteristic
224	C₂²	C₂×#477	1	2	yes	—
225	C₄	—	4	4	no	—
226	C₄	#2087	1	1	yes	characteristic
227	C₄	#2087	1	2	yes	—
228	C₄	#2087	1	1	yes	characteristic
229	C₄	C₂×#398	1	1	yes	characteristic
230	C₄	C₂×#469	1	1	yes	characteristic
231	C₄	C₂×#478	1	2	yes	—
232	C₂	—	2	1	no	—
233	C₂	—	2	2	no	—
234	C₂	—	2	1	no	—
235	C₂	—	2	2	no	—
236	C₂	#13520	1	2	yes	direct product factor
237	C₂	C₂×#2087	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#2087
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
4. #13520	C₂	direct (N × H)
236. C₂	#13520	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×#2161	2
C₂×#2151	2
C₂×#1953	2
#13520	2
C₂×#1385	2
C₂×#1381	2

Subgroups — C2×#13520 #499446

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13520 #499446