Subgroups — C₂×#13130772 #16798268

Order 3328.

1358 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	C₂²×#88102	C₂	1	4	yes	maximal
2	C₂×#11719059	C₂	1	2	yes	maximal
3	#13130772	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄⋊#261	—	2	4	no	—
6	C₂²×C₄⋊#261	C₂²	1	4	yes	—
7	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
8	C₂×C₄²×D₁₃	C₂²	1	1	yes	characteristic
9	C₂×#88102	—	2	16	no	—
10	C₂×#88102	C₂²	1	16	yes	—
11	#11719059	—	2	4	no	—
12	#11719059	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic, complemented
14	C₂×C₈×Dic₁₃	C₄	1	2	yes	complemented
15	C₂×C₄×#1055	C₄	1	1	yes	characteristic, complemented
16	C₄×C₈×D₁₃	C₄	1	4	yes	complemented
17	C₄×C₈×D₁₃	C₂²	1	4	yes	—
18	C₂³×#261	—	2	4	no	—
19	C₂²×C₄×D₁₃	D₄	1	2	yes	—
20	C₂²×C₄×D₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄⋊#261	—	2	16	no	—
22	C₂×C₄⋊#261	—	2	4	no	—
23	C₂×C₄⋊#261	—	2	16	no	—
24	C₂×C₄⋊#261	—	4	16	no	—
25	C₂×C₄⋊#261	C₂³	1	8	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	Q₈	1	2	yes	—
30	C₂²×#1055	C₂×C₄	1	2	yes	—
31	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₁₃	D₄	1	8	yes	—
33	C₂×C₈×D₁₃	Q₈	1	8	yes	—
34	C₂×C₈×D₁₃	C₂³	1	4	yes	—
35	C₄²×D₁₃	C₂×C₄	1	2	yes	—
36	C₄²×D₁₃	D₄	1	2	yes	—
37	C₄²×D₁₃	Q₈	1	2	yes	—
38	C₄²×D₁₃	C₂³	1	2	yes	—
39	#88102	—	2	32	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
41	C₈×Dic₁₃	C₂×C₄	1	8	yes	complemented
42	C₄×#1055	C₂×C₄	1	4	yes	complemented
43	C₂×#1845	—	13	1	no	maximal, Sylow
44	C₂³×D₁₃	C₂×D₄	1	1	yes	characteristic
45	C₂²×#261	—	2	8	no	—
46	C₂²×#261	—	4	8	no	—
47	C₂²×#261	—	4	16	no	—
48	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	C₄⋊C₄	1	2	yes	—
51	C₂²×Dic₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₄×D₁₃	C₄⋊C₄	1	4	yes	—
53	C₂×C₄×D₁₃	D₈	1	8	yes	—
54	C₂×C₄×D₁₃	Q₁₆	1	8	yes	—
55	C₂×C₄×D₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×D₁₃	C₂×D₄	1	4	yes	—
57	C₂×C₄×D₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×D₁₃	C₂×D₄	1	4	yes	—
59	C₂×C₄×D₁₃	C₂×Q₈	1	4	yes	—
60	C₂×C₄×D₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₄×D₁₃	C₂⁴	1	1	yes	characteristic
62	C₄⋊#261	—	2	16	no	—
63	C₄⋊#261	—	4	8	no	—
64	C₄⋊#261	—	4	32	no	—
65	C₂×C₈×C₁₃	C₄⋊C₄	1	8	yes	complemented
66	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
67	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
69	C₄×Dic₁₃	C₄⋊C₄	1	4	yes	—
70	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
71	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
72	C₄×Dic₁₃	C₂×Q₈	1	4	yes	—
73	C₂×#1055	C₄⋊C₄	1	8	yes	complemented
74	C₂×#1055	C₂²×C₄	1	4	yes	—
75	C₈×D₁₃	C₄⋊C₄	1	16	yes	complemented
76	C₈×D₁₃	C₂×D₄	1	8	yes	—
77	C₈×D₁₃	C₂×Q₈	1	8	yes	—
78	C₂²×#107	—	13	4	no	—
79	C₂²×C₄×C₈	—	13	1	no	—
80	C₂×#384	—	13	2	no	—
81	#1845	—	13	8	no	—
82	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂²×D₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂²×D₁₃	#127	1	2	yes	—
85	C₂²×D₁₃	C₄⋊Q₈	1	2	yes	—
86	C₂²×D₁₃	C₂×D₈	1	4	yes	—
87	C₂²×D₁₃	C₂×Q₁₆	1	4	yes	—
88	C₂²×D₁₃	C₂²×D₄	1	1	yes	characteristic
89	C₂×#261	—	4	8	no	—
90	C₂×#261	—	4	16	no	—
91	C₂×#261	—	8	16	no	—
92	C₂×C₄×C₁₃	#107	1	8	yes	complemented
93	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
94	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
96	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
97	C₂×Dic₁₃	#107	1	8	yes	complemented
98	C₂×Dic₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
99	C₂×Dic₁₃	C₂×C₄⋊C₄	1	4	yes	—
100	C₂×Dic₁₃	C₄⋊Q₈	1	4	yes	—
101	C₂×Dic₁₃	C₂²×Q₈	1	1	yes	characteristic
102	C₄×D₁₃	#107	1	16	yes	complemented
103	C₄×D₁₃	C₂×C₄⋊C₄	1	4	yes	—
104	C₄×D₁₃	#127	1	2	yes	—
105	C₄×D₁₃	C₄⋊Q₈	1	4	yes	—
106	C₄×D₁₃	C₄⋊Q₈	1	2	yes	—
107	C₄×D₁₃	C₂×D₈	1	8	yes	—
108	C₄×D₁₃	C₂×Q₁₆	1	8	yes	—
109	C₄×D₁₃	C₂²×D₄	1	2	yes	—
110	C₄×D₁₃	C₂²×Q₈	1	2	yes	—
111	C₈×C₁₃	C₂×C₄⋊C₄	1	8	yes	complemented
112	#1055	C₂×C₄⋊C₄	1	8	yes	complemented
113	C₂³×C₈	—	13	2	no	—
114	C₂²×C₄⋊C₄	—	13	4	no	—
115	C₂²×C₄⋊C₄	—	26	4	no	—
116	C₂²×C₄²	—	13	1	no	—
117	C₂×#107	—	13	16	no	—
118	C₂×#107	—	26	16	no	—
119	C₂×C₄×C₈	—	13	2	no	—
120	C₂×C₄×C₈	—	13	1	no	—
121	C₂×C₄×C₈	—	13	4	no	—
122	C₂×C₄×C₈	—	13	4	no	—
123	C₂×C₄×C₈	#261	1	1	yes	characteristic, complemented
124	#384	—	13	8	no	—
125	#384	—	26	4	no	—
126	C₂²×C₁₃	#384	1	2	yes	—
127	C₂²×C₁₃	C₂×#107	1	4	yes	—
128	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
129	C₂×D₁₃	#384	1	2	yes	—
130	C₂×D₁₃	C₂×#107	1	4	yes	—
131	C₂×D₁₃	#493	1	4	yes	—
132	C₂×D₁₃	#494	1	4	yes	—
133	C₂×D₁₃	#500	1	8	yes	—
134	C₂×D₁₃	C₂×#127	1	1	yes	characteristic
135	C₂×D₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
136	C₂×D₁₃	C₂²×D₈	1	2	yes	—
137	C₂×D₁₃	C₂²×Q₁₆	1	2	yes	—
138	#261	—	8	8	no	—
139	C₄×C₁₃	#384	1	2	yes	—
140	C₄×C₁₃	C₂×#107	1	8	yes	complemented
141	C₄×C₁₃	C₂²×C₄⋊C₄	1	2	yes	—
142	Dic₁₃	#384	1	2	yes	—
143	Dic₁₃	C₂×#107	1	8	yes	complemented
144	Dic₁₃	C₂×C₄⋊Q₈	1	2	yes	—
145	C₂³×C₄	—	13	2	no	—
146	C₂³×C₄	—	13	1	no	—
147	C₂³×C₄	—	26	4	no	—
148	C₂²×C₈	—	13	2	no	—
149	C₂²×C₈	—	13	8	no	—
150	C₂²×C₈	—	13	8	no	—
151	C₂²×C₈	—	13	4	no	—
152	C₂²×C₈	—	13	4	no	—
153	C₂²×C₈	C₂×#261	1	2	yes	—
154	C₂×C₄⋊C₄	—	13	8	no	—
155	C₂×C₄⋊C₄	—	26	16	no	—
156	C₂×C₄⋊C₄	—	26	4	no	—
157	C₂×C₄⋊C₄	—	26	16	no	—
158	C₂×C₄⋊C₄	—	52	16	no	—
159	C₂×C₄²	—	13	2	no	—
160	C₂×C₄²	—	13	1	no	—
161	C₂×C₄²	—	13	2	no	—
162	C₂×C₄²	—	13	2	no	—
163	C₂×C₄²	—	13	2	no	—
164	C₂×C₄²	—	13	2	no	—
165	C₂×C₄²	C₂×#261	1	1	yes	characteristic
166	#107	—	26	32	no	—
167	C₄×C₈	—	13	8	no	—
168	C₄×C₈	—	13	4	no	—
169	C₄×C₈	C₂×#261	1	4	yes	complemented
170	C₂×C₁₃	#1845	1	4	yes	complemented
171	C₂×C₁₃	C₂×#384	1	1	yes	characteristic
172	C₂×C₁₃	C₂²×#107	1	2	yes	—
173	D₁₃	#1845	1	4	yes	complemented
174	D₁₃	C₂×#493	1	1	yes	characteristic
175	D₁₃	C₂×#494	1	1	yes	characteristic
176	D₁₃	C₂×#500	1	2	yes	—
177	C₂⁴	—	13	1	no	—
178	C₂²×C₄	—	13	2	no	—
179	C₂²×C₄	—	13	1	no	—
180	C₂²×C₄	—	13	8	no	—
181	C₂²×C₄	—	13	8	no	—
182	C₂²×C₄	—	13	1	no	—
183	C₂²×C₄	—	13	4	no	—
184	C₂²×C₄	—	13	4	no	—
185	C₂²×C₄	—	13	1	no	—
186	C₂²×C₄	—	13	4	no	—
187	C₂²×C₄	—	13	4	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	26	8	no	—
191	C₂²×C₄	—	52	8	no	—
192	C₂²×C₄	—	52	16	no	—
193	C₂²×C₄	C₄⋊#261	1	2	yes	—
194	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
195	C₂×C₈	—	13	8	no	—
196	C₂×C₈	—	13	4	no	—
197	C₂×C₈	—	13	8	no	—
198	C₂×C₈	—	13	16	no	—
199	C₂×C₈	—	13	8	no	—
200	C₂×C₈	C₄⋊#261	1	8	yes	complemented
201	C₂×C₈	C₂²×#261	1	4	yes	—
202	C₄⋊C₄	—	26	16	no	—
203	C₄⋊C₄	—	52	8	no	—
204	C₄⋊C₄	—	52	32	no	—
205	C₄²	—	13	2	no	—
206	C₄²	—	13	4	no	—
207	C₄²	—	13	4	no	—
208	C₄²	—	13	2	no	—
209	C₄²	C₄⋊#261	1	2	yes	—
210	C₄²	C₂²×#261	1	2	yes	—
211	C₁₃	C₂×#1845	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	13	2	no	—
213	C₂³	—	13	2	no	—
214	C₂³	—	13	4	no	—
215	C₂³	—	13	4	no	—
216	C₂³	—	13	1	no	—
217	C₂³	—	13	1	no	—
218	C₂³	C₂×C₄⋊#261	1	1	yes	characteristic
219	C₂×C₄	—	13	4	no	—
220	C₂×C₄	—	13	8	no	—
221	C₂×C₄	—	13	1	no	—
222	C₂×C₄	—	13	4	no	—
223	C₂×C₄	—	13	1	no	—
224	C₂×C₄	—	13	2	no	—
225	C₂×C₄	—	13	4	no	—
226	C₂×C₄	—	13	2	no	—
227	C₂×C₄	—	13	8	no	—
228	C₂×C₄	—	13	16	no	—
229	C₂×C₄	—	13	8	no	—
230	C₂×C₄	—	13	2	no	—
231	C₂×C₄	—	13	4	no	—
232	C₂×C₄	—	13	2	no	—
233	C₂×C₄	—	52	8	no	—
234	C₂×C₄	—	52	16	no	—
235	C₂×C₄	—	104	16	no	—
236	C₂×C₄	#88102	1	8	yes	complemented
237	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
239	C₂×C₄	C₂×C₄⋊#261	1	4	yes	—
240	C₂×C₄	C₂³×#261	1	1	yes	characteristic
241	C₈	—	13	8	no	—
242	C₈	C₂×C₄⋊#261	1	8	yes	complemented
243	C₂²	—	13	4	no	—
244	C₂²	—	13	4	no	—
245	C₂²	—	13	8	no	—
246	C₂²	—	13	1	no	—
247	C₂²	—	13	1	no	—
248	C₂²	—	13	2	no	—
249	C₂²	—	13	2	no	—
250	C₂²	—	13	2	no	—
251	C₂²	—	13	4	no	—
252	C₂²	#11719059	1	2	yes	—
253	C₂²	C₂×#88102	1	4	yes	—
254	C₂²	C₂²×C₄⋊#261	1	1	yes	characteristic
255	C₄	—	13	2	no	—
256	C₄	—	13	2	no	—
257	C₄	—	13	8	no	—
258	C₄	—	104	8	no	—
259	C₄	#11719059	1	2	yes	—
260	C₄	C₂×#88102	1	8	yes	complemented
261	C₄	C₂²×C₄⋊#261	1	2	yes	—
262	C₂	—	13	1	no	—
263	C₂	—	13	4	no	—
264	C₂	—	13	1	no	—
265	C₂	—	13	2	no	—
266	C₂	#13130772	1	4	yes	direct product factor
267	C₂	C₂×#11719059	1	1	yes	characteristic
268	C₂	C₂²×#88102	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×C₄⋊#261
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂³×D₁₃	C₂×D₄
cosocle	C₂×C₄×D₁₃	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#1845
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. #13130772	C₂	direct (N × H)
13. C₂×C₄×C₈×C₁₃	C₄	semidirect
14. C₂×C₈×Dic₁₃	C₄	semidirect
15. C₂×C₄×#1055	C₄	semidirect
16. C₄×C₈×D₁₃	C₄	semidirect
40. C₄×C₈×C₁₃	C₂×C₄	semidirect
41. C₈×Dic₁₃	C₂×C₄	semidirect
42. C₄×#1055	C₂×C₄	semidirect
65. C₂×C₈×C₁₃	C₄⋊C₄	semidirect
73. C₂×#1055	C₄⋊C₄	semidirect
75. C₈×D₁₃	C₄⋊C₄	semidirect
92. C₂×C₄×C₁₃	#107	semidirect
97. C₂×Dic₁₃	#107	semidirect
102. C₄×D₁₃	#107	semidirect
111. C₈×C₁₃	C₂×C₄⋊C₄	semidirect
112. #1055	C₂×C₄⋊C₄	semidirect
123. C₂×C₄×C₈	#261	semidirect
140. C₄×C₁₃	C₂×#107	semidirect
143. Dic₁₃	C₂×#107	semidirect
169. C₄×C₈	C₂×#261	semidirect
170. C₂×C₁₃	#1845	semidirect
173. D₁₃	#1845	semidirect
200. C₂×C₈	C₄⋊#261	semidirect
211. C₁₃	C₂×#1845	semidirect
236. C₂×C₄	#88102	semidirect
242. C₈	C₂×C₄⋊#261	semidirect
260. C₄	C₂×#88102	semidirect
266. C₂	#13130772	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#88102	2
C₂×#11719059	2
#13130772	2
C₂×C₄×C₈×D₁₃	2
C₂×#1845	13

Subgroups — C2×#13130772 #16798268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13130772 #16798268