Subgroups — C₂×#13116635 #16015984

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116635	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#6090	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊#6090	C₂	1	2	yes	maximal
4	C₂×C₄×#6090	C₂	1	2	yes	maximal, complemented
5	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
9	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
10	C₂×C₄×C₈×C₁₃	C₄	1	4	yes	—
11	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×#6090	C₂²	1	4	yes	—
16	C₂²⋊#6090	C₂²	1	8	yes	—
17	C₄⋊#6090	C₂²	1	8	yes	complemented
18	C₄×#6090	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₈	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	—	2	4	no	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₈	1	4	yes	—
33	C₂×C₄²×C₁₃	C₈	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
36	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
39	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	C₂³	1	2	yes	—
42	C₄×C₈×C₁₃	C₂³	1	2	yes	—
43	C₂×#6090	—	2	8	no	—
44	C₂×#6090	C₂³	1	8	yes	—
45	C₂×#2162	—	13	1	no	maximal, Sylow
46	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	—	2	4	no	—
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	C₁₆	1	8	yes	complemented
51	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
55	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
58	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₂×C₈×C₁₃	—	2	8	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	M₁₆	1	4	yes	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	#41	1	4	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₄²×C₁₃	C₂×C₈	1	4	yes	—
74	C₄²×C₁₃	C₂×C₈	1	4	yes	—
75	C₄²×C₁₃	C₂×C₈	1	2	yes	—
76	C₄²×C₁₃	C₂×C₈	1	2	yes	—
77	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
78	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
79	#6090	—	2	8	no	—
80	C₂²×C₄×C₈	D₁₃	1	1	yes	characteristic
81	#2162	—	13	8	no	—
82	C₂×C₄⋊C₁₆	—	13	2	no	—
83	C₂×C₂²⋊C₁₆	—	13	2	no	—
84	C₂×C₄×C₁₆	—	13	2	no	—
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
88	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
89	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	8	no	—
96	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
97	C₂×C₄×C₁₃	C₂×C₁₆	1	4	yes	—
98	C₂×C₄×C₁₃	C₂×C₁₆	1	16	yes	complemented
99	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
100	C₂×C₄×C₁₃	#117	1	2	yes	—
101	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
105	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₈×C₁₃	—	2	2	no	—
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	#117	1	2	yes	—
112	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
113	C₈×C₁₃	C₂×#41	1	2	yes	—
114	C₂³×C₈	Dic₁₃	1	1	yes	characteristic
115	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
116	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
117	C₂²×C₁₆	—	13	4	no	—
118	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
119	C₂×C₄×C₈	Dic₁₃	1	1	yes	characteristic
120	C₂×C₄×C₈	Dic₁₃	1	4	yes	—
121	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
122	C₂×C₄×C₈	C₂×D₁₃	1	4	yes	—
123	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
124	C₄⋊C₁₆	—	13	8	no	—
125	C₂²⋊C₁₆	—	13	8	no	—
126	C₄×C₁₆	—	13	8	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	8	no	—
130	C₂²×C₁₃	#431	1	2	yes	—
131	C₂²×C₁₃	C₂²×C₁₆	1	4	yes	—
132	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
133	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
134	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	#431	1	2	yes	—
138	C₄×C₁₃	C₂²×C₁₆	1	4	yes	—
139	C₄×C₁₃	C₂²×C₁₆	1	4	yes	—
140	C₄×C₁₃	C₂×M₃₂	1	4	yes	—
141	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
142	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
143	C₂³×C₄	#1055	1	1	yes	characteristic
144	C₂³×C₄	#1055	1	1	yes	characteristic
145	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
146	C₂²×C₈	—	2	4	no	—
147	C₂²×C₈	—	2	4	no	—
148	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
149	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
150	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
151	C₂²×C₈	C₂×Dic₁₃	1	2	yes	—
152	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
154	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
155	C₂×C₄²	#1055	1	1	yes	characteristic
156	C₂×C₄²	#1055	1	4	yes	—
157	C₂×C₄²	#1055	1	1	yes	characteristic
158	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
159	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
160	C₂×C₄²	C₂×Dic₁₃	1	1	yes	characteristic
161	C₂×C₁₆	—	13	8	no	—
162	C₂×C₁₆	—	26	8	no	—
163	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
164	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
165	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
166	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
167	C₄×C₈	C₂²×D₁₃	1	2	yes	—
168	C₄×C₈	C₂²×D₁₃	1	2	yes	—
169	C₂×C₁₃	—	2	4	no	—
170	C₂×C₁₃	#2162	1	4	yes	complemented
171	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
172	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
173	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
174	C₂⁴	C₂×#1055	1	1	yes	characteristic
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	#6090	1	8	yes	complemented
179	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
180	C₂²×C₄	C₂×#1055	1	2	yes	—
181	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
182	C₂²×C₄	C₂×#1055	1	4	yes	—
183	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
184	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
185	C₂²×C₄	C₂×#1055	1	4	yes	—
186	C₂²×C₄	C₂×#1055	1	1	yes	characteristic
187	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
188	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	8	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	—	2	4	no	—
194	C₂×C₈	#6099	1	4	yes	—
195	C₂×C₈	#6127	1	4	yes	—
196	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
197	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
198	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
199	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
200	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
201	C₄²	C₂×#1055	1	4	yes	—
202	C₄²	C₂×#1055	1	4	yes	—
203	C₄²	C₂×#1055	1	2	yes	—
204	C₄²	C₂×#1055	1	2	yes	—
205	C₄²	C₂²×Dic₁₃	1	2	yes	—
206	C₄²	C₂²×Dic₁₃	1	2	yes	—
207	C₁₆	—	26	8	no	—
208	C₁₃	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
209	C₂³	—	2	2	no	—
210	C₂³	—	2	2	no	—
211	C₂³	C₂×#6090	1	4	yes	—
212	C₂³	C₂²×#1055	1	2	yes	—
213	C₂³	C₂²×#1055	1	1	yes	characteristic
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	4	no	—
220	C₂×C₄	C₂×#6090	1	4	yes	—
221	C₂×C₄	C₂×#6090	1	4	yes	—
222	C₂×C₄	C₂×#6090	1	16	yes	complemented
223	C₂×C₄	#88052	1	4	yes	—
224	C₂×C₄	C₂²×#1055	1	2	yes	—
225	C₂×C₄	C₂²×#1055	1	2	yes	—
226	C₂×C₄	C₂²×#1055	1	2	yes	—
227	C₂×C₄	C₂²×#1055	1	2	yes	—
228	C₂×C₄	C₂²×#1055	1	2	yes	—
229	C₂×C₄	C₂²×#1055	1	1	yes	characteristic
230	C₂×C₄	C₂×#6099	1	2	yes	—
231	C₂×C₄	#88180	1	2	yes	—
232	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	—	2	2	no	—
235	C₈	C₂×#6099	1	4	yes	—
236	C₈	#88180	1	2	yes	—
237	C₈	C₂×#6127	1	2	yes	—
238	C₂²	—	2	2	no	—
239	C₂²	—	2	8	no	—
240	C₂²	—	2	2	no	—
241	C₂²	#11718586	1	2	yes	—
242	C₂²	C₂²×#6090	1	4	yes	—
243	C₂²	C₂²×#6090	1	2	yes	—
244	C₂²	C₂×#88052	1	2	yes	—
245	C₂²	C₂³×#1055	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	2	no	—
248	C₄	#11718586	1	2	yes	—
249	C₄	C₂²×#6090	1	4	yes	—
250	C₄	C₂²×#6090	1	4	yes	—
251	C₄	C₂×#88052	1	4	yes	—
252	C₄	C₂²×#6099	1	1	yes	characteristic
253	C₄	C₂×#88180	1	1	yes	characteristic
254	C₂	—	2	4	no	—
255	C₂	#13116635	1	4	yes	direct product factor
256	C₂	C₂×#11718586	1	1	yes	characteristic
257	C₂	C₂³×#6090	1	1	yes	characteristic
258	C₂	C₂²×#88052	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈	C₂×D₁₃
commutator	C₂×C₁₃	C₂³×C₁₆
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂²×C₄×C₈×C₁₃	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂⁴×C₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₁₃
hypocenter	C₁₃	C₂×#2162
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #13116635	C₂	direct (N × H)
2. C₂×C₄⋊#6090	C₂	semidirect
4. C₂×C₄×#6090	C₂	semidirect
17. C₄⋊#6090	C₂²	semidirect
18. C₄×#6090	C₂²	semidirect
50. C₂²×C₄×C₁₃	C₁₆	semidirect
98. C₂×C₄×C₁₃	C₂×C₁₆	semidirect
170. C₂×C₁₃	#2162	semidirect
178. C₂²×C₄	#6090	semidirect
208. C₁₃	C₂×#2162	semidirect
222. C₂×C₄	C₂×#6090	semidirect
255. C₂	#13116635	direct (N × H)

Maximal subgroups

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

subgroup	index
#13116635	2
C₂×C₄⋊#6090	2
C₂×C₂²⋊#6090	2
C₂×C₄×#6090	2
C₂²×C₄×C₈×C₁₃	2
C₂×#2162	13

Subgroups — C2×#13116635 #16015984

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116635 #16015984