Subgroups — C₂×#13116632 #16015981

Order 3328.

726 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	C₂²×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116632	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718093	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊#6090	C₂	1	1	yes	maximal, characteristic
5	C₂×C₁₆×Dic₁₃	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂³×#1055	C₄	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₄	1	4	yes	—
13	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	4	yes	—
17	C₂²×#6090	C₂²	1	2	yes	—
18	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
19	C₂²⋊#6090	C₂²	1	4	yes	—
20	#11718093	C₂²	1	8	yes	complemented
21	C₁₆×Dic₁₃	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₁₃	C₈	1	1	yes	characteristic
24	C₂³×Dic₁₃	C₈	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	—	2	4	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
31	C₂×C₄×Dic₁₃	C₈	1	4	yes	—
32	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
33	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	—	2	4	no	—
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	4	yes	—
39	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
40	C₂×C₁₃×C₁₆	—	2	4	no	—
41	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
42	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
43	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
44	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
45	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
46	C₈×Dic₁₃	C₂³	1	2	yes	—
47	C₈×Dic₁₃	C₂³	1	2	yes	—
48	C₂×#6090	—	2	4	no	—
49	C₂×#6090	C₂³	1	4	yes	—
50	C₂×#2162	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂×C₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₁₃	—	2	4	no	—
59	C₂²×Dic₁₃	C₁₆	1	8	yes	complemented
60	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
62	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
64	C₂²×Dic₁₃	C₂×C₈	1	4	yes	—
65	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	#41	1	4	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
74	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
75	C₄×Dic₁₃	C₂×C₈	1	4	yes	—
76	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
77	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
78	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₂×#1055	—	2	2	no	—
80	C₂×#1055	—	2	8	no	—
81	C₂×#1055	M₁₆	1	4	yes	—
82	C₂×#1055	C₂²×C₄	1	2	yes	—
83	C₂×#1055	C₂²×C₄	1	2	yes	—
84	C₁₃×C₁₆	—	2	4	no	—
85	#6090	—	2	4	no	—
86	C₂²×C₄×C₈	—	13	1	no	—
87	#2162	—	13	8	no	—
88	C₂×C₄⋊C₁₆	—	13	2	no	—
89	C₂×C₂²⋊C₁₆	—	13	1	no	—
90	C₂×C₂²⋊C₁₆	D₁₃	1	1	yes	characteristic
91	C₂×C₄×C₁₆	—	13	2	no	—
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	C₂×C₁₆	1	4	yes	—
95	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
96	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	#117	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	C₂×M₁₆	1	2	yes	—
105	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	8	no	—
108	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
109	C₂×Dic₁₃	C₂×C₁₆	1	4	yes	—
110	C₂×Dic₁₃	C₂×C₁₆	1	16	yes	complemented
111	C₂×Dic₁₃	M₃₂	1	4	yes	—
112	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
113	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
114	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
115	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	#117	1	2	yes	—
118	C₈×C₁₃	C₂×#41	1	2	yes	—
119	#1055	—	2	2	no	—
120	#1055	C₂×M₁₆	1	4	yes	—
121	C₂³×C₈	—	13	1	no	—
122	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
123	C₂²×C₄²	—	13	1	no	—
124	C₂²×C₁₆	—	13	2	no	—
125	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
126	C₂×C₄×C₈	—	13	1	no	—
127	C₂×C₄×C₈	—	13	4	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂×C₄×C₈	—	13	4	no	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₄⋊C₁₆	—	13	8	no	—
133	C₂²⋊C₁₆	—	13	4	no	—
134	C₂²⋊C₁₆	C₂×D₁₃	1	4	yes	—
135	C₄×C₁₆	—	13	8	no	—
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	8	no	—
139	C₂²×C₁₃	#431	1	2	yes	—
140	C₂²×C₁₃	C₂²×C₁₆	1	4	yes	—
141	C₂²×C₁₃	C₂²×C₁₆	1	2	yes	—
142	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
143	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
144	C₄×C₁₃	—	2	2	no	—
145	C₄×C₁₃	#431	1	2	yes	—
146	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
147	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
148	Dic₁₃	—	2	2	no	—
149	Dic₁₃	C₂²×C₁₆	1	4	yes	—
150	Dic₁₃	C₂²×C₁₆	1	4	yes	—
151	Dic₁₃	C₂×M₃₂	1	4	yes	—
152	C₂³×C₄	—	13	1	no	—
153	C₂³×C₄	—	13	1	no	—
154	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
155	C₂²×C₈	—	2	4	no	—
156	C₂²×C₈	—	13	1	no	—
157	C₂²×C₈	—	13	4	no	—
158	C₂²×C₈	—	13	1	no	—
159	C₂²×C₈	—	26	4	no	—
160	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₄×D₁₃	1	2	yes	—
162	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
163	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
164	C₂×C₄²	—	13	1	no	—
165	C₂×C₄²	—	13	4	no	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	4	no	—
169	C₂×C₄²	—	13	1	no	—
170	C₂×C₁₆	—	2	4	no	—
171	C₂×C₁₆	—	13	4	no	—
172	C₂×C₁₆	—	26	4	no	—
173	C₂×C₁₆	C₂²×D₁₃	1	4	yes	—
174	C₄×C₈	—	13	4	no	—
175	C₄×C₈	—	13	2	no	—
176	C₄×C₈	—	13	4	no	—
177	C₄×C₈	—	13	2	no	—
178	C₄×C₈	—	13	2	no	—
179	C₄×C₈	—	13	2	no	—
180	C₂×C₁₃	—	2	4	no	—
181	C₂×C₁₃	#2162	1	4	yes	complemented
182	C₂×C₁₃	C₂×#431	1	1	yes	characteristic
183	C₂×C₁₃	C₂³×C₁₆	1	1	yes	characteristic
184	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
185	C₂⁴	C₈×D₁₃	1	1	yes	characteristic
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	1	no	—
190	C₂²×C₄	—	13	4	no	—
191	C₂²×C₄	—	13	1	no	—
192	C₂²×C₄	—	13	8	no	—
193	C₂²×C₄	—	13	4	no	—
194	C₂²×C₄	—	13	1	no	—
195	C₂²×C₄	—	26	4	no	—
196	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₈×D₁₃	1	2	yes	—
198	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	2	4	no	—
202	C₂×C₈	—	2	4	no	—
203	C₂×C₈	—	13	4	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	26	8	no	—
207	C₂×C₈	—	26	2	no	—
208	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
209	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
210	C₂×C₈	#6129	1	4	yes	—
211	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
212	C₄²	—	13	4	no	—
213	C₄²	—	13	2	no	—
214	C₄²	—	13	4	no	—
215	C₄²	—	13	2	no	—
216	C₄²	—	13	2	no	—
217	C₄²	—	13	2	no	—
218	C₁₆	—	2	4	no	—
219	C₁₆	—	26	4	no	—
220	C₁₃	C₂×#2162	1	1	yes	characteristic, complemented, Sylow
221	C₂³	—	2	2	no	—
222	C₂³	—	2	2	no	—
223	C₂³	C₁₆×D₁₃	1	4	yes	—
224	C₂³	C₂×C₈×D₁₃	1	2	yes	—
225	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	13	2	no	—
231	C₂×C₄	—	13	2	no	—
232	C₂×C₄	—	13	2	no	—
233	C₂×C₄	—	13	4	no	—
234	C₂×C₄	—	13	4	no	—
235	C₂×C₄	—	13	16	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	2	no	—
238	C₂×C₄	—	26	8	no	—
239	C₂×C₄	—	26	2	no	—
240	C₂×C₄	#88131	1	2	yes	—
241	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
242	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
243	C₂×C₄	M₁₆×D₁₃	1	2	yes	—
244	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
245	C₈	—	2	2	no	—
246	C₈	—	13	4	no	—
247	C₈	—	26	2	no	—
248	C₈	#88131	1	2	yes	—
249	C₈	C₂×#6129	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	8	no	—
252	C₂²	—	2	2	no	—
253	C₂²	#11718379	1	2	yes	—
254	C₂²	C₂×C₁₆×D₁₃	1	4	yes	—
255	C₂²	C₂×C₁₆×D₁₃	1	2	yes	—
256	C₂²	D₁₃×M₃₂	1	2	yes	—
257	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	13	4	no	—
260	C₄	—	13	4	no	—
261	C₄	—	13	4	no	—
262	C₄	—	26	2	no	—
263	C₄	#11718379	1	2	yes	—
264	C₄	C₂×#88131	1	1	yes	characteristic
265	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
266	C₂	—	2	4	no	—
267	C₂	#13116632	1	4	yes	direct product factor
268	C₂	C₂×#11718379	1	1	yes	characteristic
269	C₂	C₂²×C₁₆×D₁₃	1	1	yes	characteristic
270	C₂	C₂×D₁₃×M₃₂	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #13116632	C₂	direct (N × H)
3. C₂×#11718093	C₂	semidirect
5. C₂×C₁₆×Dic₁₃	C₂	semidirect
20. #11718093	C₂²	semidirect
21. C₁₆×Dic₁₃	C₂²	semidirect
59. C₂²×Dic₁₃	C₁₆	semidirect
110. C₂×Dic₁₃	C₂×C₁₆	semidirect
181. C₂×C₁₃	#2162	semidirect
220. C₁₃	C₂×#2162	semidirect
267. C₂	#13116632	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×C₈×Dic₁₃	2
#13116632	2
C₂×#11718093	2
C₂×C₂²⋊#6090	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2162	13

Subgroups — C2×#13116632 #16015981

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116632 #16015981