Subgroups — C₂×#13116717 #16016319

Order 3328.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718857	C₂	1	2	yes	maximal, complemented
2	#13116717	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718098	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718159	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic
7	C₂×C₁₃×#135	C₂²	1	2	yes	—
8	C₂×#88202	—	2	2	no	—
9	C₁₃×#508	C₄	1	4	yes	complemented
10	C₁₃×#508	C₂²	1	4	yes	—
11	C₂×C₁₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
13	C₂×C₁₃×D₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂×#88077	—	2	2	no	—
16	#11718857	—	2	4	no	—
17	#11718857	C₂²	1	8	yes	complemented
18	C₂×#88059	C₂²	1	2	yes	—
19	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
20	C₂²×#6090	C₂²	1	1	yes	characteristic
21	#11718159	C₂²	1	8	yes	complemented
22	#11718098	C₂²	1	4	yes	complemented
23	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
24	C₂×C₁₃×#41	—	2	2	no	—
25	C₁₃×#135	—	2	2	no	—
26	C₁₃×#135	—	2	2	no	—
27	C₁₃×#135	C₂×C₄	1	4	yes	—
28	C₁₃×#135	C₂³	1	4	yes	—
29	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×SD₁₆	—	2	2	no	—
31	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	#88202	—	2	8	no	—
34	#88202	—	4	4	no	—
35	C₂×C₄×Dic₁₃	D₄	1	1	yes	characteristic
36	C₂×#6099	—	2	2	no	—
37	C₂×#6099	—	4	2	no	—
38	C₂²×#1055	—	4	2	no	—
39	C₂²×#1055	D₄	1	1	yes	characteristic
40	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
41	C₁₃×SD₃₂	C₂×C₄	1	8	yes	complemented
42	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
45	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
47	#88077	—	2	8	no	—
48	#88059	—	2	2	no	—
49	#88059	C₂³	1	4	yes	—
50	C₈×Dic₁₃	D₄	1	2	yes	—
51	C₈×Dic₁₃	C₂³	1	2	yes	—
52	C₂×#6090	—	2	1	no	—
53	C₂×#6090	—	2	1	no	—
54	C₂×#6090	C₂³	1	2	yes	—
55	C₂×#2178	—	13	1	no	maximal, Sylow
56	C₁₃×#41	—	2	4	no	—
57	C₁₃×#41	—	2	4	no	—
58	C₁₃×#41	—	4	2	no	—
59	C₁₃×#41	—	4	2	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×Dic₁₃	—	2	1	no	—
66	C₁₃×Q₁₆	—	2	2	no	—
67	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
68	C₁₃×SD₁₆	—	2	4	no	—
69	C₁₃×SD₁₆	—	2	4	no	—
70	C₁₃×D₈	—	2	2	no	—
71	C₁₃×D₈	C₂²×C₄	1	4	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₄×Dic₁₃	—	2	1	no	—
79	C₄×Dic₁₃	C₂×D₄	1	2	yes	—
80	#6099	—	2	4	no	—
81	#6099	—	4	2	no	—
82	#6099	—	4	8	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	—	4	8	no	—
85	C₂×#1055	D₈	1	1	yes	characteristic
86	C₂×#1055	D₈	1	1	yes	characteristic
87	C₂×#1055	D₈	1	2	yes	—
88	C₂×#1055	C₂×D₄	1	2	yes	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
91	#6090	—	2	2	no	—
92	C₂×#508	D₁₃	1	1	yes	characteristic
93	C₂×#443	—	13	2	no	—
94	#2178	—	13	8	no	—
95	C₂×#368	—	13	1	no	—
96	C₂×#359	—	13	2	no	—
97	C₂×C₄×C₁₆	—	13	1	no	—
98	C₂³×C₁₃	—	4	2	no	—
99	C₂³×C₁₃	#135	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	2	4	no	—
103	D₄×C₁₃	—	4	4	no	—
104	D₄×C₁₃	—	4	4	no	—
105	D₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	4	no	—
116	C₂×Dic₁₃	—	2	2	no	—
117	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
120	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
121	#1055	—	4	4	no	—
122	#1055	C₂×D₈	1	2	yes	—
123	#1055	C₂×D₈	1	2	yes	—
124	C₂×#135	C₂×D₁₃	1	2	yes	—
125	C₂×#131	—	26	2	no	—
126	#508	Dic₁₃	1	4	yes	complemented
127	#508	C₂×D₁₃	1	4	yes	—
128	C₂×Q₃₂	Dic₁₃	1	1	yes	characteristic, complemented
129	C₂×SD₃₂	Dic₁₃	1	2	yes	complemented
130	C₂×D₁₆	Dic₁₃	1	1	yes	characteristic, complemented
131	C₂²×C₁₆	—	13	1	no	—
132	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
133	#443	—	13	8	no	—
134	#443	—	26	4	no	—
135	C₂×#108	—	13	2	no	—
136	C₂×#104	—	26	2	no	—
137	C₂×C₄×C₈	—	13	1	no	—
138	#368	—	13	4	no	—
139	#359	—	13	8	no	—
140	C₄×C₁₆	—	13	4	no	—
141	C₂²×C₁₃	—	2	2	no	—
142	C₂²×C₁₃	—	4	4	no	—
143	C₂²×C₁₃	—	8	4	no	—
144	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
145	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
147	C₄×C₁₃	—	4	4	no	—
148	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
149	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
150	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
152	Dic₁₃	—	2	4	no	—
153	C₂×#41	—	2	2	no	—
154	#135	—	2	2	no	—
155	#135	—	2	2	no	—
156	#135	C₂×Dic₁₃	1	4	yes	—
157	#135	C₂²×D₁₃	1	4	yes	—
158	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
159	C₂×SD₁₆	—	2	2	no	—
160	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
161	#131	—	26	8	no	—
162	#131	—	52	4	no	—
163	C₂×M₁₆	—	26	2	no	—
164	C₂×M₁₆	—	52	2	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	52	2	no	—
167	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
168	C₂×C₄²	—	13	1	no	—
169	Q₃₂	C₂×Dic₁₃	1	4	yes	complemented
170	SD₃₂	C₂×Dic₁₃	1	8	yes	complemented
171	D₁₆	C₂×Dic₁₃	1	4	yes	complemented
172	C₂×C₁₆	—	13	2	no	—
173	C₂×C₁₆	—	26	1	no	—
174	C₂×C₁₆	—	26	1	no	—
175	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
176	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
177	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
178	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
179	#108	—	13	4	no	—
180	#108	—	26	2	no	—
181	#104	—	26	8	no	—
182	C₄×C₈	—	13	2	no	—
183	C₄×C₈	—	13	2	no	—
184	C₂×C₁₃	—	2	1	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	—	8	4	no	—
187	C₂×C₁₃	#2178	1	2	yes	complemented
188	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
189	#41	—	2	4	no	—
190	#41	—	2	4	no	—
191	#41	—	4	2	no	—
192	#41	—	4	2	no	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×D₄	—	2	2	no	—
195	C₂×D₄	—	4	2	no	—
196	C₂²×C₄	—	4	2	no	—
197	C₂²×C₄	—	26	1	no	—
198	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
199	Q₁₆	—	2	2	no	—
200	Q₁₆	C₂²×Dic₁₃	1	4	yes	—
201	SD₁₆	—	2	4	no	—
202	SD₁₆	—	2	4	no	—
203	D₈	—	2	2	no	—
204	D₈	C₂²×Dic₁₃	1	4	yes	—
205	M₁₆	—	26	4	no	—
206	M₁₆	—	52	8	no	—
207	M₁₆	—	52	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	13	1	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	52	8	no	—
213	C₂×C₈	—	52	2	no	—
214	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	#6129	1	1	yes	characteristic
216	C₂×C₈	#6129	1	1	yes	characteristic
217	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
219	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
220	C₄²	—	13	2	no	—
221	C₄²	—	26	1	no	—
222	C₁₆	—	26	2	no	—
223	C₁₆	C₂²×Dic₁₃	1	2	yes	—
224	C₁₆	C₂²×Dic₁₃	1	2	yes	—
225	C₁₃	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
226	C₂³	—	4	2	no	—
227	C₂³	#88166	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	26	4	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
240	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
241	C₂×C₄	#88166	1	1	yes	characteristic
242	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
243	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
245	C₈	—	13	2	no	—
246	C₈	—	13	2	no	—
247	C₈	—	52	4	no	—
248	C₈	D₄×Dic₁₃	1	1	yes	characteristic
249	C₈	D₄×Dic₁₃	1	1	yes	characteristic
250	C₈	C₂×#6129	1	1	yes	characteristic
251	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	8	4	no	—
255	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
256	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
257	C₂²	C₂×#88166	1	1	yes	characteristic
258	C₄	—	4	4	no	—
259	C₄	—	26	4	no	—
260	C₄	D₈×Dic₁₃	1	1	yes	characteristic
261	C₄	D₈×Dic₁₃	1	1	yes	characteristic
262	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
263	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	8	4	no	—
267	C₂	#13116717	1	2	yes	direct product factor
268	C₂	C₂×D₈×Dic₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₁₃
commutator	C₈×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting, prime radical	C₂×C₁₃×#508	C₂
socle	C₂²×C₁₃	C₄×D₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#508	D₁₃
hypocenter	C₁₃	C₂×#2178
LCS	G ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#11718857	C₂	semidirect
2. #13116717	C₂	direct (N × H)
3. C₂×#11718098	C₂	semidirect
4. C₂×#11718159	C₂	semidirect
5. C₂×C₁₆×Dic₁₃	C₂	semidirect
9. C₁₃×#508	C₄	semidirect
11. C₂×C₁₃×Q₃₂	C₄	semidirect
12. C₂×C₁₃×SD₃₂	C₄	semidirect
13. C₂×C₁₃×D₁₆	C₄	semidirect
17. #11718857	C₂²	semidirect
21. #11718159	C₂²	semidirect
22. #11718098	C₂²	semidirect
23. C₁₆×Dic₁₃	C₂²	semidirect
40. C₁₃×Q₃₂	C₂×C₄	semidirect
41. C₁₃×SD₃₂	C₂×C₄	semidirect
42. C₁₃×D₁₆	C₂×C₄	semidirect
126. #508	Dic₁₃	semidirect
128. C₂×Q₃₂	Dic₁₃	semidirect
129. C₂×SD₃₂	Dic₁₃	semidirect
130. C₂×D₁₆	Dic₁₃	semidirect
169. Q₃₂	C₂×Dic₁₃	semidirect
170. SD₃₂	C₂×Dic₁₃	semidirect
171. D₁₆	C₂×Dic₁₃	semidirect
187. C₂×C₁₃	#2178	semidirect
225. C₁₃	C₂×#2178	semidirect
267. C₂	#13116717	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×#11718857	2
#13116717	2
C₂×#11718098	2
C₂×#11718159	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×#508	2
C₂×#2178	13

Subgroups — C2×#13116717 #16016319

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116717 #16016319