Subgroups — C₂×#13511 #499419

Order 512.

586 conjugacy classes of subgroups.

The subgroup lattice has 287 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#2155	C₂	1	2	yes	maximal
2	C₂×#1946	C₂	1	1	yes	maximal, characteristic
3	C₂×#1849	C₂	1	1	yes	maximal, characteristic
4	#13511	C₂	1	8	yes	maximal, direct product factor
5	C₂×#1385	C₂	1	2	yes	maximal
6	C₂×#1375	C₂	1	1	yes	maximal, characteristic
7	C₂²×M₃₂	C₂²	1	2	yes	—
8	C₂²×#108	—	2	1	no	—
9	C₂²×#108	C₂²	1	1	yes	characteristic
10	C₂²×#108	C₂²	1	2	yes	—
11	C₂×#427	C₂²	1	1	yes	characteristic
12	C₂×#424	—	2	1	no	—
13	C₂×#423	—	2	1	no	—
14	C₂²×#97	C₂²	1	1	yes	characteristic
15	#2155	—	2	4	no	—
16	#2155	C₂²	1	8	yes	—
17	C₂×#365	—	2	4	no	—
18	#1946	—	2	2	no	—
19	#1946	C₂²	1	4	yes	—
20	#1849	—	2	2	no	—
21	#1849	C₂²	1	4	yes	—
22	C₂×#339	—	2	1	no	—
23	#1385	C₂²	1	8	yes	—
24	#1375	C₂²	1	4	yes	—
25	C₂²×M₁₆	—	2	2	no	—
26	C₂²×M₁₆	—	2	1	no	—
27	C₂³×C₈	D₄	1	1	yes	characteristic
28	C₂³×C₈	C₂³	1	1	yes	characteristic
29	C₂×#117	—	2	1	no	—
30	C₂²×C₄²	D₄	1	1	yes	characteristic
31	C₂×M₃₂	—	2	2	no	—
32	C₂×M₃₂	—	2	2	no	—
33	C₂×M₃₂	—	2	2	no	—
34	C₂×M₃₂	—	2	2	no	—
35	C₂×M₃₂	Q₈	1	4	yes	—
36	C₂×M₃₂	C₂³	1	4	yes	—
37	C₂²×C₁₆	Q₈	1	4	yes	—
38	C₂×#108	—	2	2	no	—
39	C₂×#108	—	2	2	no	—
40	C₂×#108	—	2	2	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	4	no	—
46	C₂×#108	—	2	1	no	—
47	C₂×#108	—	4	2	no	—
48	C₂×#108	—	4	2	no	—
49	C₂×#108	C₂³	1	4	yes	—
50	C₂×#108	C₂³	1	2	yes	—
51	#427	—	2	1	no	—
52	#427	—	2	1	no	—
53	#427	D₄	1	2	yes	—
54	#427	C₂³	1	2	yes	—
55	C₂×#107	—	2	1	no	—
56	C₂×#106	D₄	1	2	yes	—
57	#424	—	2	4	no	—
58	#424	—	4	2	no	—
59	#423	—	2	4	no	—
60	#423	—	4	2	no	—
61	C₂×C₄⋊C₈	—	4	2	no	—
62	C₂×C₄⋊C₈	—	4	2	no	—
63	C₂×#97	—	2	2	no	—
64	C₂×#97	—	2	2	no	—
65	C₂×#97	D₄	1	2	yes	—
66	C₂×#97	C₂³	1	2	yes	—
67	#365	—	2	16	no	—
68	#339	—	2	4	no	—
69	C₂³×C₄	—	2	1	no	—
70	C₂³×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×M₁₆	—	2	2	no	—
72	C₂×M₁₆	—	2	4	no	—
73	C₂×M₁₆	—	4	2	no	—
74	C₂×M₁₆	—	4	1	no	—
75	C₂×M₁₆	—	4	2	no	—
76	C₂×M₁₆	—	4	2	no	—
77	C₂×M₁₆	—	4	1	no	—
78	C₂×M₁₆	—	4	1	no	—
79	C₂×M₁₆	—	4	1	no	—
80	C₂×M₁₆	—	4	1	no	—
81	C₂×M₁₆	—	4	2	no	—
82	C₂×M₁₆	—	4	2	no	—
83	C₂²×C₈	—	2	1	no	—
84	C₂²×C₈	—	2	1	no	—
85	C₂²×C₈	—	2	1	no	—
86	C₂²×C₈	—	2	1	no	—
87	C₂²×C₈	—	4	2	no	—
88	C₂²×C₈	—	4	1	no	—
89	C₂²×C₈	SD₁₆	1	2	yes	—
90	C₂²×C₈	SD₁₆	1	2	yes	—
91	C₂²×C₈	C₂×D₄	1	2	yes	—
92	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₈	C₂×Q₈	1	2	yes	—
96	C₂²×C₈	C₂×Q₈	1	2	yes	—
97	C₂²×C₈	#41	1	1	yes	characteristic
98	C₂²×C₈	#41	1	1	yes	characteristic
99	C₂²×C₈	#41	1	2	yes	—
100	C₂²×C₈	#41	1	2	yes	—
101	C₂²×C₈	C₂⁴	1	1	yes	characteristic
102	#117	—	2	2	no	—
103	#117	—	2	2	no	—
104	#117	—	4	1	no	—
105	#117	—	4	1	no	—
106	C₂×C₄⋊C₄	—	2	2	no	—
107	C₂×C₂²⋊C₄	—	4	1	no	—
108	C₂×C₄²	—	2	1	no	—
109	C₂×C₄²	—	2	1	no	—
110	C₂×C₄²	—	2	1	no	—
111	C₂×C₄²	—	4	1	no	—
112	C₂×C₄²	SD₁₆	1	2	yes	—
113	C₂×C₄²	SD₁₆	1	2	yes	—
114	C₂×C₄²	C₂×D₄	1	2	yes	—
115	M₃₂	—	2	4	no	—
116	M₃₂	—	2	4	no	—
117	M₃₂	—	2	4	no	—
118	M₃₂	—	2	4	no	—
119	C₂×C₁₆	—	2	4	no	—
120	C₂×C₁₆	—	2	4	no	—
121	C₂×C₁₆	C₂×Q₈	1	8	yes	—
122	#108	—	2	4	no	—
123	#108	—	2	4	no	—
124	#108	—	2	4	no	—
125	#108	—	4	8	no	—
126	#108	—	4	2	no	—
127	#108	—	4	2	no	—
128	#108	—	4	2	no	—
129	#107	—	2	2	no	—
130	#107	—	2	2	no	—
131	#106	—	2	2	no	—
132	#106	C₂×D₄	1	4	yes	—
133	C₄⋊C₈	—	4	8	no	—
134	C₄⋊C₈	—	4	8	no	—
135	#97	—	2	4	no	—
136	#97	—	2	4	no	—
137	C₂⁴	#137	1	1	yes	characteristic
138	C₂²×C₄	—	2	2	no	—
139	C₂²×C₄	—	2	2	no	—
140	C₂²×C₄	—	2	2	no	—
141	C₂²×C₄	—	4	2	no	—
142	C₂²×C₄	—	8	2	no	—
143	C₂²×C₄	#121	1	1	yes	characteristic
144	C₂²×C₄	#121	1	1	yes	characteristic
145	C₂²×C₄	#123	1	1	yes	characteristic
146	C₂²×C₄	#123	1	1	yes	characteristic
147	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
148	C₂²×C₄	C₄⋊Q₈	1	1	yes	characteristic
149	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
150	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
151	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
152	C₂²×C₄	C₂×SD₁₆	1	1	yes	characteristic
153	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
154	C₂²×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
155	C₂²×C₄	#137	1	1	yes	characteristic
156	C₂²×C₄	C₂²×D₄	1	1	yes	characteristic
157	M₁₆	—	4	4	no	—
158	M₁₆	—	4	2	no	—
159	M₁₆	—	4	4	no	—
160	M₁₆	—	4	2	no	—
161	M₁₆	—	8	2	no	—
162	M₁₆	—	8	2	no	—
163	M₁₆	—	8	1	no	—
164	M₁₆	—	8	1	no	—
165	C₂×C₈	—	2	2	no	—
166	C₂×C₈	—	2	2	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	2	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	2	2	no	—
171	C₂×C₈	—	4	2	no	—
172	C₂×C₈	—	4	4	no	—
173	C₂×C₈	—	8	2	no	—
174	C₂×C₈	—	8	2	no	—
175	C₂×C₈	—	8	1	no	—
176	C₂×C₈	—	8	1	no	—
177	C₂×C₈	#121	1	1	yes	characteristic
178	C₂×C₈	#121	1	1	yes	characteristic
179	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
180	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
181	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
182	C₂×C₈	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₈	#123	1	2	yes	—
184	C₂×C₈	#125	1	2	yes	—
185	C₂×C₈	#125	1	2	yes	—
186	C₂×C₈	#125	1	2	yes	—
187	C₂×C₈	#125	1	2	yes	—
188	C₂×C₈	C₄⋊Q₈	1	2	yes	—
189	C₂×C₈	C₂×SD₁₆	1	4	yes	—
190	C₂×C₈	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₈	C₂²×Q₈	1	2	yes	—
192	C₂×C₈	C₂×#41	1	2	yes	—
193	C₂×C₈	C₂×#41	1	1	yes	characteristic
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	4	2	no	—
196	C₂²⋊C₄	—	4	2	no	—
197	C₂²⋊C₄	—	4	2	no	—
198	C₄²	—	2	2	no	—
199	C₄²	—	2	2	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	4	2	no	—
202	C₄²	—	4	2	no	—
203	C₄²	C₂×SD₁₆	1	4	yes	—
204	C₁₆	—	2	8	no	—
205	C₂³	—	4	2	no	—
206	C₂³	#461	1	1	yes	characteristic
207	C₂³	#461	1	1	yes	characteristic
208	C₂³	#483	1	1	yes	characteristic
209	C₂³	#483	1	1	yes	characteristic
210	C₂³	#501	1	1	yes	characteristic
211	C₂³	#501	1	1	yes	characteristic
212	C₂³	C₂×#137	1	1	yes	characteristic
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	8	2	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	#398	1	1	yes	characteristic
220	C₂×C₄	#398	1	1	yes	characteristic
221	C₂×C₄	#398	1	1	yes	characteristic
222	C₂×C₄	#398	1	1	yes	characteristic
223	C₂×C₄	#460	1	1	yes	characteristic
224	C₂×C₄	#460	1	1	yes	characteristic
225	C₂×C₄	#460	1	1	yes	characteristic
226	C₂×C₄	#460	1	1	yes	characteristic
227	C₂×C₄	#461	1	1	yes	characteristic
228	C₂×C₄	#461	1	1	yes	characteristic
229	C₂×C₄	#481	1	1	yes	characteristic
230	C₂×C₄	#481	1	1	yes	characteristic
231	C₂×C₄	#481	1	1	yes	characteristic
232	C₂×C₄	#481	1	1	yes	characteristic
233	C₂×C₄	#483	1	1	yes	characteristic
234	C₂×C₄	#483	1	1	yes	characteristic
235	C₂×C₄	#498	1	1	yes	characteristic
236	C₂×C₄	#498	1	1	yes	characteristic
237	C₂×C₄	#498	1	1	yes	characteristic
238	C₂×C₄	#498	1	1	yes	characteristic
239	C₂×C₄	#501	1	1	yes	characteristic
240	C₂×C₄	#501	1	1	yes	characteristic
241	C₂×C₄	C₂×#121	1	1	yes	characteristic
242	C₂×C₄	C₂×#123	1	1	yes	characteristic
243	C₂×C₄	C₂×C₄⋊Q₈	1	1	yes	characteristic
244	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
245	C₂×C₄	C₂²×SD₁₆	1	1	yes	characteristic
246	C₂×C₄	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	—	8	2	no	—
249	C₈	—	8	4	no	—
250	C₈	#398	1	2	yes	—
251	C₈	#398	1	2	yes	—
252	C₈	C₂×C₂²⋊Q₈	1	2	yes	—
253	C₈	C₂×#125	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	4	1	no	—
260	C₂²	—	4	1	no	—
261	C₂²	#2084	1	1	yes	characteristic
262	C₂²	#2084	1	1	yes	characteristic
263	C₂²	#2084	1	1	yes	characteristic
264	C₂²	#2084	1	1	yes	characteristic
265	C₂²	C₂×#461	1	1	yes	characteristic
266	C₂²	C₂×#483	1	1	yes	characteristic
267	C₂²	C₂×#501	1	1	yes	characteristic
268	C₄	—	4	4	no	—
269	C₄	—	8	4	no	—
270	C₄	#2084	1	1	yes	characteristic
271	C₄	#2084	1	1	yes	characteristic
272	C₄	#2084	1	1	yes	characteristic
273	C₄	#2084	1	1	yes	characteristic
274	C₄	C₂×#398	1	1	yes	characteristic
275	C₄	C₂×#460	1	1	yes	characteristic
276	C₄	C₂×#481	1	1	yes	characteristic
277	C₄	C₂×#498	1	1	yes	characteristic
278	C₂	—	2	1	no	—
279	C₂	—	2	1	no	—
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	1	no	—
284	C₂	#13511	1	2	yes	direct product factor
285	C₂	C₂×#2084	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#498
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	#2084
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ 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. #13511	C₂	direct (N × H)
284. C₂	#13511	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×#2155	2
C₂×#1946	2
C₂×#1849	2
#13511	2
C₂×#1385	2
C₂×#1375	2

Subgroups — C2×#13511 #499419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13511 #499419