Subgroups — C₂×#13123255 #16726318

Order 3328.

3342 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#6099	C₂	1	2	yes	maximal
2	#13123255	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718658	C₂	1	8	yes	maximal, complemented
4	C₂×C₄×#6099	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×#117	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#117	C₄	1	4	yes	—
8	C₂×C₁₃×#117	C₄	1	8	yes	—
9	C₂×C₁₃×#117	C₂²	1	4	yes	—
10	C₂×C₁₃×#117	C₂²	1	8	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₄	1	2	yes	—
12	C₂²×C₁₃×C₂²⋊C₄	C₄	1	2	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
14	C₂²×#6099	C₄	1	32	yes	complemented
15	C₂²×#6099	C₂²	1	4	yes	—
16	C₂²×#6099	C₂²	1	4	yes	—
17	C₂²×#6099	C₂²	1	16	yes	—
18	C₂³×#1055	C₂²	1	4	yes	—
19	#11718658	C₂²	1	64	yes	complemented
20	C₄×#6099	C₂²	1	32	yes	complemented
21	C₂×#88043	C₂²	1	8	yes	complemented
22	C₂×C₄×#1055	C₂²	1	8	yes	complemented
23	C₂⁵×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	—
25	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
26	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₃	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
32	C₁₃×#117	C₂×C₄	1	16	yes	—
33	C₁₃×#117	C₂×C₄	1	32	yes	—
34	C₁₃×#117	C₂³	1	16	yes	—
35	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
36	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
37	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	16	yes	—
39	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	8	yes	—
43	C₂×#6099	C₂×C₄	1	32	yes	—
44	C₂×#6099	C₂×C₄	1	32	yes	complemented
45	C₂×#6099	C₂×C₄	1	128	yes	complemented
46	C₂×#6099	C₂³	1	16	yes	—
47	C₂×#6099	C₂³	1	16	yes	—
48	C₂²×#1055	C₂×C₄	1	32	yes	complemented
49	C₂²×#1055	C₂³	1	8	yes	—
50	C₂²×#1055	C₂³	1	16	yes	—
51	#88043	C₂³	1	32	yes	complemented
52	C₄×#1055	C₂³	1	32	yes	complemented
53	C₂×#2873	—	13	1	no	maximal, Sylow
54	C₂⁴×C₁₃	—	4	4	no	—
55	C₂⁴×C₁₃	C₄²	1	8	yes	—
56	C₂⁴×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂⁴×C₁₃	C₂²×C₄	1	4	yes	—
59	C₂²×C₄×C₁₃	—	4	8	no	—
60	C₂²×C₄×C₁₃	C₄²	1	8	yes	—
61	C₂²×C₄×C₁₃	C₄²	1	16	yes	—
62	C₂²×C₄×C₁₃	C₄²	1	32	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
64	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	32	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
68	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
69	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	C₂²×C₄	1	32	yes	—
79	C₁₃×C₂²⋊C₄	C₂²×C₄	1	32	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	8	yes	—
81	C₄²×C₁₃	C₂²×C₄	1	16	yes	—
82	C₄²×C₁₃	C₂⁴	1	8	yes	—
83	#6099	C₂²×C₄	1	64	yes	—
84	#6099	C₂²×C₄	1	64	yes	complemented
85	C₂×#1055	C₂²×C₄	1	32	yes	—
86	C₂×#1055	C₂²×C₄	1	64	yes	complemented
87	C₂×#1055	C₂⁴	1	16	yes	—
88	C₂³×M₁₆	—	13	2	no	—
89	C₂²×#117	D₁₃	1	1	yes	characteristic
90	#2873	—	13	16	no	—
91	C₂×#405	—	13	8	no	—
92	C₂×C₄×M₁₆	—	13	4	no	—
93	C₂³×C₁₃	—	2	4	no	—
94	C₂³×C₁₃	—	2	8	no	—
95	C₂³×C₁₃	—	4	16	no	—
96	C₂³×C₁₃	—	4	8	no	—
97	C₂³×C₁₃	C₂×C₄²	1	8	yes	—
98	C₂³×C₁₃	C₂×C₄²	1	4	yes	—
99	C₂³×C₁₃	C₂×C₄²	1	16	yes	—
100	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
101	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
102	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	—	2	16	no	—
104	C₂×C₄×C₁₃	—	4	16	no	—
105	C₂×C₄×C₁₃	C₂×C₄²	1	16	yes	—
106	C₂×C₄×C₁₃	C₂×C₄²	1	8	yes	—
107	C₂×C₄×C₁₃	C₂×C₄²	1	16	yes	—
108	C₂×C₄×C₁₃	C₂×C₄²	1	32	yes	—
109	C₂×C₄×C₁₃	C₂×C₄²	1	8	yes	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	4	yes	—
111	C₂×C₄×C₁₃	C₂³×C₄	1	16	yes	—
112	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
113	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
114	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
115	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
116	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
117	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
119	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
120	#1055	C₂³×C₄	1	32	yes	—
121	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
122	C₂²×M₁₆	—	13	32	no	—
123	C₂²×M₁₆	—	13	4	no	—
124	C₂²×M₁₆	—	13	4	no	—
125	C₂²×M₁₆	—	13	16	no	—
126	C₂³×C₈	—	13	4	no	—
127	C₂×#117	Dic₁₃	1	4	yes	—
128	C₂×#117	Dic₁₃	1	8	yes	—
129	C₂×#117	C₂×D₁₃	1	4	yes	—
130	C₂×#117	C₂×D₁₃	1	8	yes	—
131	C₂²×C₄⋊C₄	Dic₁₃	1	2	yes	—
132	C₂²×C₂²⋊C₄	Dic₁₃	1	2	yes	—
133	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
134	#405	—	13	64	no	—
135	C₄×M₁₆	—	13	32	no	—
136	C₂×#97	—	13	8	no	—
137	C₂×C₄×C₈	—	13	8	no	—
138	C₂²×C₁₃	—	2	8	no	—
139	C₂²×C₁₃	—	2	16	no	—
140	C₂²×C₁₃	—	2	4	no	—
141	C₂²×C₁₃	—	2	8	no	—
142	C₂²×C₁₃	—	4	16	no	—
143	C₂²×C₁₃	C₂²×C₄²	1	8	yes	—
144	C₂²×C₁₃	C₂²×C₄²	1	4	yes	—
145	C₂²×C₁₃	C₂²×C₄²	1	2	yes	—
146	C₂²×C₁₃	#568	1	4	yes	—
147	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
148	C₄×C₁₃	—	2	16	no	—
149	C₄×C₁₃	C₂²×C₄²	1	2	yes	—
150	C₄×C₁₃	C₂²×C₄²	1	8	yes	—
151	C₄×C₁₃	C₂²×C₄²	1	4	yes	—
152	C₄×C₁₃	C₂⁴×C₄	1	2	yes	—
153	C₂⁵	C₂×Dic₁₃	1	1	yes	characteristic
154	C₂³×C₄	C₄×D₁₃	1	16	yes	—
155	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
156	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
157	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
158	C₂³×C₄	C₂×Dic₁₃	1	4	yes	—
159	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
160	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
161	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
162	C₂×M₁₆	—	13	32	no	—
163	C₂×M₁₆	—	13	32	no	—
164	C₂×M₁₆	—	13	128	no	—
165	C₂×M₁₆	—	13	16	no	—
166	C₂×M₁₆	—	13	16	no	—
167	C₂²×C₈	—	13	32	no	—
168	C₂²×C₈	—	13	16	no	—
169	C₂²×C₈	—	13	8	no	—
170	#117	C₂×Dic₁₃	1	16	yes	—
171	#117	C₂×Dic₁₃	1	32	yes	—
172	#117	C₂²×D₁₃	1	16	yes	—
173	C₂×C₄⋊C₄	C₂×Dic₁₃	1	8	yes	—
174	C₂×C₄⋊C₄	C₂×Dic₁₃	1	16	yes	—
175	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	8	yes	—
176	C₂×C₂²⋊C₄	C₂×Dic₁₃	1	16	yes	—
177	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
178	C₂×C₄²	C₂×Dic₁₃	1	8	yes	—
179	C₂×C₄²	C₂²×D₁₃	1	8	yes	—
180	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
181	#97	—	13	32	no	—
182	C₄×C₈	—	13	32	no	—
183	C₂×C₁₃	—	2	4	no	—
184	C₂×C₁₃	—	2	8	no	—
185	C₂×C₁₃	#2873	1	4	yes	complemented
186	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
187	C₂×C₁₃	C₂×#568	1	2	yes	—
188	C₂⁴	—	4	4	no	—
189	C₂⁴	C₄×Dic₁₃	1	8	yes	—
190	C₂⁴	C₂²×Dic₁₃	1	2	yes	—
191	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
192	C₂⁴	C₂²×Dic₁₃	1	4	yes	—
193	C₂²×C₄	—	4	8	no	—
194	C₂²×C₄	C₄×Dic₁₃	1	8	yes	—
195	C₂²×C₄	C₄×Dic₁₃	1	16	yes	—
196	C₂²×C₄	C₄×Dic₁₃	1	32	yes	—
197	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
198	C₂²×C₄	C₂×C₄×D₁₃	1	32	yes	—
199	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
200	C₂²×C₄	C₂×C₄×D₁₃	1	8	yes	—
201	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
202	C₂²×C₄	C₂²×Dic₁₃	1	16	yes	—
203	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
204	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
205	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
206	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
207	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
208	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
209	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
210	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
211	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
212	M₁₆	—	13	64	no	—
213	M₁₆	—	13	64	no	—
214	C₂×C₈	—	13	32	no	—
215	C₂×C₈	—	13	64	no	—
216	C₂×C₈	—	13	16	no	—
217	C₄⋊C₄	C₂²×Dic₁₃	1	32	yes	—
218	C₂²⋊C₄	C₂²×Dic₁₃	1	32	yes	—
219	C₄²	C₂²×Dic₁₃	1	16	yes	—
220	C₄²	C₂²×Dic₁₃	1	8	yes	—
221	C₄²	C₂³×D₁₃	1	8	yes	—
222	C₁₃	C₂×#2873	1	1	yes	characteristic, complemented, Sylow
223	C₂³	—	2	4	no	—
224	C₂³	—	2	8	no	—
225	C₂³	—	4	16	no	—
226	C₂³	—	4	8	no	—
227	C₂³	C₂×C₄×Dic₁₃	1	8	yes	—
228	C₂³	C₂×C₄×Dic₁₃	1	4	yes	—
229	C₂³	C₂×C₄×Dic₁₃	1	16	yes	—
230	C₂³	C₂³×Dic₁₃	1	4	yes	—
231	C₂³	C₂³×Dic₁₃	1	2	yes	—
232	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	4	16	no	—
235	C₂×C₄	C₂×C₄×Dic₁₃	1	16	yes	—
236	C₂×C₄	C₂×C₄×Dic₁₃	1	8	yes	—
237	C₂×C₄	C₂×C₄×Dic₁₃	1	16	yes	—
238	C₂×C₄	C₂×C₄×Dic₁₃	1	32	yes	—
239	C₂×C₄	C₂×C₄×Dic₁₃	1	8	yes	—
240	C₂×C₄	C₂×C₄×Dic₁₃	1	4	yes	—
241	C₂×C₄	C₂²×C₄×D₁₃	1	8	yes	—
242	C₂×C₄	C₂²×C₄×D₁₃	1	4	yes	—
243	C₂×C₄	C₂²×C₄×D₁₃	1	8	yes	—
244	C₂×C₄	C₂²×C₄×D₁₃	1	8	yes	—
245	C₂×C₄	C₂³×Dic₁₃	1	16	yes	—
246	C₂×C₄	C₂³×Dic₁₃	1	2	yes	—
247	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
249	C₂×C₄	C₂⁴×D₁₃	1	1	yes	characteristic
250	C₈	—	13	32	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	8	no	—
253	C₂²	—	2	8	no	—
254	C₂²	—	2	16	no	—
255	C₂²	—	4	16	no	—
256	C₂²	C₂²×C₄×Dic₁₃	1	8	yes	—
257	C₂²	C₂²×C₄×Dic₁₃	1	4	yes	—
258	C₂²	C₂²×C₄×Dic₁₃	1	2	yes	—
259	C₂²	#11719524	1	4	yes	—
260	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
261	C₄	—	2	16	no	—
262	C₄	C₂²×C₄×Dic₁₃	1	8	yes	—
263	C₄	C₂²×C₄×Dic₁₃	1	2	yes	—
264	C₄	C₂²×C₄×Dic₁₃	1	4	yes	—
265	C₄	C₂³×C₄×D₁₃	1	2	yes	—
266	C₂	—	2	4	no	—
267	C₂	—	2	8	no	—
268	C₂	#13123255	1	4	yes	direct product factor
269	C₂	C₂³×C₄×Dic₁₃	1	1	yes	characteristic
270	C₂	C₂×#11719524	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂³×D₁₃
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂×C₄	C₂⁴×D₁₃
Fitting	C₂²×C₁₃×#117	C₂
socle	C₂³×C₁₃	C₂³×C₄
prime radical	C₂⁵×C₁₃	C₂×C₄
cosocle	C₂×C₄×C₁₃	C₂⁵
hypercenter	C₂²×#117	D₁₃
hypocenter	C₁₃	C₂×#2873
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117

Complements

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

N	H ≅ G/N	type
2. #13123255	C₂	direct (N × H)
3. C₂×#11718658	C₂	semidirect
4. C₂×C₄×#6099	C₂	semidirect
14. C₂²×#6099	C₄	semidirect
19. #11718658	C₂²	semidirect
20. C₄×#6099	C₂²	semidirect
21. C₂×#88043	C₂²	semidirect
22. C₂×C₄×#1055	C₂²	semidirect
44. C₂×#6099	C₂×C₄	semidirect
45. C₂×#6099	C₂×C₄	semidirect
48. C₂²×#1055	C₂×C₄	semidirect
51. #88043	C₂³	semidirect
52. C₄×#1055	C₂³	semidirect
84. #6099	C₂²×C₄	semidirect
86. C₂×#1055	C₂²×C₄	semidirect
185. C₂×C₁₃	#2873	semidirect
222. C₁₃	C₂×#2873	semidirect
268. C₂	#13123255	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#6099	2
#13123255	2
C₂×#11718658	2
C₂×C₄×#6099	2
C₂²×C₁₃×#117	2
C₂×#2873	13

Subgroups — C2×#13123255 #16726318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13123255 #16726318