Subgroups — C₂×#13116645 #16016038

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
2	#13116645	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718103	C₂	1	2	yes	maximal, complemented
4	C₂×#11718145	C₂	1	1	yes	maximal, characteristic
5	C₂×#11718104	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂²×#6099	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	M₁₆×Dic₁₃	C₄	1	4	yes	—
12	M₁₆×Dic₁₃	C₂²	1	4	yes	—
13	C₂×#88055	C₄	1	1	yes	characteristic
14	C₂×#88055	C₂²	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂×#88052	C₂²	1	2	yes	—
18	C₁₃×#349	C₂²	1	4	yes	—
19	#11718145	C₂²	1	4	yes	—
20	#11718104	C₂²	1	8	yes	complemented
21	#11718103	C₂²	1	8	yes	complemented
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂³×Dic₁₃	—	2	1	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
27	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	—	2	1	no	—
31	C₂×C₄×Dic₁₃	—	2	2	no	—
32	C₂×C₄×Dic₁₃	C₂×C₄	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×#6099	—	2	2	no	—
36	C₂×#6099	—	2	2	no	—
37	C₂×#6099	C₂×C₄	1	4	yes	—
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
40	C₁₃×M₃₂	—	2	2	no	—
41	C₁₃×M₃₂	C₂³	1	4	yes	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	#88055	C₂×C₄	1	4	yes	—
44	#88055	C₂×C₄	1	2	yes	—
45	#88055	C₂³	1	2	yes	—
46	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
47	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
48	C₈×Dic₁₃	C₂³	1	2	yes	—
49	#88052	—	2	2	no	—
50	#88052	C₂³	1	4	yes	—
51	C₂×#6090	—	2	2	no	—
52	C₂×#2165	—	13	1	no	maximal, Sylow
53	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	—	2	1	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	4	no	—
64	C₂²×Dic₁₃	—	2	1	no	—
65	C₂²×Dic₁₃	—	4	4	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	1	no	—
69	C₂×C₈×C₁₃	—	2	1	no	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	#41	1	2	yes	—
73	C₂×C₈×C₁₃	#41	1	2	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₄×Dic₁₃	—	2	1	no	—
76	C₄×Dic₁₃	—	2	4	no	—
77	C₄×Dic₁₃	—	2	1	no	—
78	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
79	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
80	#6099	—	2	8	no	—
81	C₂×#1055	—	2	1	no	—
82	C₂×#1055	—	2	1	no	—
83	C₂×#1055	M₁₆	1	2	yes	—
84	C₂×#1055	M₁₆	1	2	yes	—
85	C₂×#1055	C₂²×C₄	1	2	yes	—
86	C₂×#1055	C₂²×C₄	1	2	yes	—
87	C₁₃×C₁₆	—	2	4	no	—
88	#6090	—	2	4	no	—
89	C₂×C₄×M₁₆	—	13	1	no	—
90	#2165	—	13	8	no	—
91	C₂×#364	—	13	2	no	—
92	C₂×#349	—	13	1	no	—
93	C₂×#349	D₁₃	1	1	yes	characteristic
94	C₂×#347	—	13	2	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	4	2	no	—
98	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×Dic₁₃	—	2	4	no	—
114	C₂×Dic₁₃	—	2	2	no	—
115	C₂×Dic₁₃	—	2	2	no	—
116	C₂×Dic₁₃	—	4	8	no	—
117	C₂×Dic₁₃	—	4	2	no	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	#117	1	2	yes	—
120	C₈×C₁₃	C₂×#41	1	2	yes	—
121	#1055	—	2	2	no	—
122	#1055	C₂×M₁₆	1	4	yes	—
123	C₂²×M₁₆	—	13	1	no	—
124	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
125	C₂²×C₄²	—	13	1	no	—
126	C₂×M₃₂	—	13	2	no	—
127	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
128	C₄×M₁₆	—	13	4	no	—
129	C₄×M₁₆	—	13	4	no	—
130	C₂×#97	—	13	1	no	—
131	C₂×#97	—	13	1	no	—
132	C₂×C₄×C₈	—	13	1	no	—
133	C₂×C₄×C₈	—	13	1	no	—
134	#364	—	13	8	no	—
135	#349	—	13	4	no	—
136	#349	C₂×D₁₃	1	4	yes	—
137	#347	—	13	8	no	—
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	2	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	#432	1	1	yes	characteristic
144	C₂²×C₁₃	#432	1	1	yes	characteristic
145	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	#432	1	1	yes	characteristic
148	C₄×C₁₃	#432	1	1	yes	characteristic
149	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
150	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
151	Dic₁₃	—	2	4	no	—
152	Dic₁₃	—	4	2	no	—
153	C₂³×C₄	—	26	1	no	—
154	C₂³×C₄	C₄×D₁₃	1	1	yes	characteristic
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	13	4	no	—
158	C₂×M₁₆	—	26	2	no	—
159	C₂×M₁₆	—	26	2	no	—
160	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
161	C₂×M₁₆	C₂²×D₁₃	1	2	yes	—
162	C₂²×C₈	—	13	1	no	—
163	C₂²×C₈	—	13	1	no	—
164	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	13	1	no	—
168	C₂×C₄²	—	13	4	no	—
169	C₂×C₄²	—	26	1	no	—
170	C₂×C₄²	—	26	2	no	—
171	M₃₂	—	2	2	no	—
172	M₃₂	—	13	4	no	—
173	M₃₂	—	26	2	no	—
174	M₃₂	C₂²×D₁₃	1	4	yes	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	26	2	no	—
177	#97	—	13	4	no	—
178	#97	—	13	2	no	—
179	#97	—	13	2	no	—
180	C₄×C₈	—	13	2	no	—
181	C₄×C₈	—	13	4	no	—
182	C₄×C₈	—	13	2	no	—
183	C₂×C₁₃	—	2	1	no	—
184	C₂×C₁₃	—	2	1	no	—
185	C₂×C₁₃	—	4	2	no	—
186	C₂×C₁₃	#2165	1	2	yes	complemented
187	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
188	C₂⁴	#6094	1	1	yes	characteristic
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	1	no	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	26	1	no	—
194	C₂²×C₄	—	26	4	no	—
195	C₂²×C₄	—	26	1	no	—
196	C₂²×C₄	—	52	4	no	—
197	C₂²×C₄	#6094	1	1	yes	characteristic
198	C₂²×C₄	#6094	1	2	yes	—
199	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	26	8	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	13	2	no	—
207	C₂×C₈	—	13	2	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	13	2	no	—
210	C₂×C₈	—	26	1	no	—
211	C₂×C₈	—	26	1	no	—
212	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
214	C₂×C₈	#6129	1	2	yes	—
215	C₂×C₈	#6129	1	2	yes	—
216	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
217	C₄²	—	13	2	no	—
218	C₄²	—	13	2	no	—
219	C₄²	—	26	1	no	—
220	C₄²	—	26	4	no	—
221	C₄²	—	26	1	no	—
222	C₁₆	—	2	4	no	—
223	C₁₆	—	26	4	no	—
224	C₁₃	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	2	1	no	—
226	C₂³	—	2	1	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×#6094	1	1	yes	characteristic
229	C₂³	C₂×#6094	1	2	yes	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	26	4	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	52	8	no	—
240	C₂×C₄	—	52	2	no	—
241	C₂×C₄	#88131	1	1	yes	characteristic
242	C₂×C₄	#88131	1	1	yes	characteristic
243	C₂×C₄	C₂×#6094	1	2	yes	—
244	C₂×C₄	C₂×#6094	1	1	yes	characteristic
245	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
246	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	13	4	no	—
250	C₈	—	26	2	no	—
251	C₈	#88131	1	2	yes	—
252	C₈	C₂×#6129	1	2	yes	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	#11718380	1	1	yes	characteristic
259	C₂²	#11718380	1	1	yes	characteristic
260	C₂²	C₂²×#6094	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	26	4	no	—
263	C₄	—	52	2	no	—
264	C₄	#11718380	1	1	yes	characteristic
265	C₄	#11718380	1	1	yes	characteristic
266	C₄	C₂×#88131	1	1	yes	characteristic
267	C₄	C₂×M₁₆×D₁₃	1	1	yes	characteristic
268	C₂	—	2	1	no	—
269	C₂	—	2	1	no	—
270	C₂	—	4	2	no	—
271	C₂	#13116645	1	2	yes	direct product factor
272	C₂	C₂×#11718380	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88131
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#349	C₂
socle	C₂²×C₁₃	#432
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#349	D₁₃
hypocenter	C₁₃	C₂×#2165
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

Complements

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

N	H ≅ G/N	type
2. #13116645	C₂	direct (N × H)
3. C₂×#11718103	C₂	semidirect
5. C₂×#11718104	C₂	semidirect
20. #11718104	C₂²	semidirect
21. #11718103	C₂²	semidirect
186. C₂×C₁₃	#2165	semidirect
224. C₁₃	C₂×#2165	semidirect
271. C₂	#13116645	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×M₁₆×Dic₁₃	2
#13116645	2
C₂×#11718103	2
C₂×#11718145	2
C₂×#11718104	2
C₂×C₁₃×#349	2
C₂×#2165	13

Subgroups — C2×#13116645 #16016038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116645 #16016038