Subgroups — C₂×#13116722 #16016325

Order 3328.

654 conjugacy classes of subgroups.

The subgroup lattice has 266 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	#13116722	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×#11718094	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	#11718094	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×#2179	—	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	2	yes	—
86	C₂×#1055	D₈	1	2	yes	—
87	C₂×#1055	C₂×D₄	1	2	yes	—
88	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	#6090	—	2	2	no	—
91	C₂×#508	D₁₃	1	1	yes	characteristic
92	C₂×#443	—	13	2	no	—
93	#2179	—	13	8	no	—
94	C₂×#368	—	13	1	no	—
95	C₂×#359	—	13	2	no	—
96	C₂×#346	—	13	1	no	—
97	C₂³×C₁₃	—	4	2	no	—
98	C₂³×C₁₃	#135	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	4	no	—
100	Q₈×C₁₃	—	4	2	no	—
101	D₄×C₁₃	—	2	4	no	—
102	D₄×C₁₃	—	4	4	no	—
103	D₄×C₁₃	—	4	4	no	—
104	D₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₁₃	—	2	2	no	—
115	C₂×Dic₁₃	—	4	2	no	—
116	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
119	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
120	#1055	—	4	4	no	—
121	#1055	C₂×D₈	1	4	yes	—
122	C₂×#135	C₂×D₁₃	1	2	yes	—
123	C₂×#131	—	26	2	no	—
124	#508	Dic₁₃	1	4	yes	complemented
125	#508	C₂×D₁₃	1	4	yes	—
126	C₂×Q₃₂	Dic₁₃	1	1	yes	characteristic, complemented
127	C₂×SD₃₂	Dic₁₃	1	2	yes	complemented
128	C₂×D₁₆	Dic₁₃	1	1	yes	characteristic, complemented
129	C₂²×C₁₆	—	13	1	no	—
130	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
131	#443	—	13	8	no	—
132	#443	—	26	4	no	—
133	C₂×#108	—	13	2	no	—
134	C₂×#104	—	26	2	no	—
135	C₂×C₄×C₈	—	13	1	no	—
136	#368	—	13	4	no	—
137	#359	—	13	8	no	—
138	#346	—	13	4	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	4	no	—
141	C₂²×C₁₃	—	8	4	no	—
142	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
143	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
145	C₄×C₁₃	—	4	4	no	—
146	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
147	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
148	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
149	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	Dic₁₃	—	4	2	no	—
151	C₂×#41	—	2	2	no	—
152	#135	—	2	2	no	—
153	#135	—	2	2	no	—
154	#135	C₂×Dic₁₃	1	4	yes	—
155	#135	C₂²×D₁₃	1	4	yes	—
156	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
157	C₂×SD₁₆	—	2	2	no	—
158	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
159	#131	—	26	8	no	—
160	#131	—	52	4	no	—
161	C₂×M₁₆	—	26	2	no	—
162	C₂×M₁₆	—	52	2	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	—	52	2	no	—
165	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×C₄²	—	13	1	no	—
167	Q₃₂	C₂×Dic₁₃	1	4	yes	complemented
168	SD₃₂	C₂×Dic₁₃	1	8	yes	complemented
169	D₁₆	C₂×Dic₁₃	1	4	yes	complemented
170	C₂×C₁₆	—	13	2	no	—
171	C₂×C₁₆	—	26	1	no	—
172	C₂×C₁₆	—	26	1	no	—
173	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
174	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂×Dic₁₃	1	2	yes	—
176	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
177	#108	—	13	4	no	—
178	#108	—	26	2	no	—
179	#104	—	26	8	no	—
180	C₄×C₈	—	13	2	no	—
181	C₄×C₈	—	13	2	no	—
182	C₂×C₁₃	—	2	1	no	—
183	C₂×C₁₃	—	2	1	no	—
184	C₂×C₁₃	—	8	4	no	—
185	C₂×C₁₃	#2179	1	2	yes	complemented
186	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
187	#41	—	2	4	no	—
188	#41	—	2	4	no	—
189	#41	—	4	2	no	—
190	#41	—	4	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂²×C₄	—	4	2	no	—
195	C₂²×C₄	—	26	1	no	—
196	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
197	Q₁₆	—	2	2	no	—
198	Q₁₆	C₂²×Dic₁₃	1	4	yes	—
199	SD₁₆	—	2	4	no	—
200	SD₁₆	—	2	4	no	—
201	D₈	—	2	2	no	—
202	D₈	C₂²×Dic₁₃	1	4	yes	—
203	M₁₆	—	26	4	no	—
204	M₁₆	—	52	8	no	—
205	M₁₆	—	52	2	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	52	8	no	—
210	C₂×C₈	—	52	2	no	—
211	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
212	C₂×C₈	#6129	1	1	yes	characteristic
213	C₂×C₈	#6129	1	1	yes	characteristic
214	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
216	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
217	C₄²	—	13	2	no	—
218	C₄²	—	26	1	no	—
219	C₁₆	—	26	2	no	—
220	C₁₆	C₂²×Dic₁₃	1	2	yes	—
221	C₁₆	C₂²×Dic₁₃	1	2	yes	—
222	C₁₃	C₂×#2179	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	4	2	no	—
224	C₂³	#88166	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	4	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	26	2	no	—
235	C₂×C₄	—	52	2	no	—
236	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
237	C₂×C₄	D₈×D₁₃	1	1	yes	characteristic
238	C₂×C₄	#88166	1	1	yes	characteristic
239	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
242	C₈	—	13	4	no	—
243	C₈	—	52	4	no	—
244	C₈	D₄×Dic₁₃	1	1	yes	characteristic
245	C₈	D₄×Dic₁₃	1	1	yes	characteristic
246	C₈	C₂×#6129	1	1	yes	characteristic
247	C₈	C₂³×Dic₁₃	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
252	C₂²	D₈×Dic₁₃	1	1	yes	characteristic
253	C₂²	C₂×#88166	1	1	yes	characteristic
254	C₄	—	4	4	no	—
255	C₄	—	52	2	no	—
256	C₄	D₈×Dic₁₃	1	1	yes	characteristic
257	C₄	D₈×Dic₁₃	1	1	yes	characteristic
258	C₄	C₂×D₈×D₁₃	1	1	yes	characteristic
259	C₄	C₂×D₄×Dic₁₃	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	8	4	no	—
263	C₂	#13116722	1	2	yes	direct product factor
264	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×#2179
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. #13116722	C₂	direct (N × H)
3. C₂×#11718098	C₂	semidirect
4. C₂×#11718159	C₂	semidirect
5. C₂×#11718094	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. #11718094	C₂²	semidirect
40. C₁₃×Q₃₂	C₂×C₄	semidirect
41. C₁₃×SD₃₂	C₂×C₄	semidirect
42. C₁₃×D₁₆	C₂×C₄	semidirect
124. #508	Dic₁₃	semidirect
126. C₂×Q₃₂	Dic₁₃	semidirect
127. C₂×SD₃₂	Dic₁₃	semidirect
128. C₂×D₁₆	Dic₁₃	semidirect
167. Q₃₂	C₂×Dic₁₃	semidirect
168. SD₃₂	C₂×Dic₁₃	semidirect
169. D₁₆	C₂×Dic₁₃	semidirect
185. C₂×C₁₃	#2179	semidirect
222. C₁₃	C₂×#2179	semidirect
263. C₂	#13116722	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
#13116722	2
C₂×#11718098	2
C₂×#11718159	2
C₂×#11718094	2
C₂×C₁₃×#508	2
C₂×#2179	13

Subgroups — C2×#13116722 #16016325

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116722 #16016325