Subgroups — C₂×#13131206 #16807092

Order 3328.

570 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131206	C₂	1	8	yes	maximal, direct product factor
2	C₂×#11718261	C₂	1	2	yes	maximal, complemented
3	C₂×#11718284	C₂	1	2	yes	maximal
4	C₂×#11718262	C₂	1	2	yes	maximal, complemented
5	C₂×M₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂²×#6099	C₂²	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	M₁₆×Dic₁₃	C₄	1	4	yes	—
11	M₁₆×Dic₁₃	C₂²	1	4	yes	—
12	C₂×#88055	C₄	1	1	yes	characteristic
13	C₂×#88055	C₂²	1	1	yes	characteristic
14	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
15	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
16	#11718284	C₂²	1	8	yes	—
17	#11718262	C₂²	1	8	yes	complemented
18	#11718261	C₂²	1	8	yes	complemented
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×Dic₁₃	—	2	1	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₁₃	—	2	1	no	—
27	C₂×C₄×Dic₁₃	—	2	2	no	—
28	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×Dic₁₃	C₂×C₄	1	4	yes	—
31	C₂×#6099	—	2	2	no	—
32	C₂×#6099	—	2	2	no	—
33	C₂×#6099	C₂×C₄	1	2	yes	—
34	C₂×#6099	C₂³	1	2	yes	—
35	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
36	C₂²×#1055	C₂³	1	1	yes	characteristic
37	#88106	—	2	4	no	—
38	#88106	C₂³	1	8	yes	—
39	C₂×#6092	—	2	4	no	—
40	#88055	C₂×C₄	1	4	yes	—
41	#88055	C₂×C₄	1	2	yes	—
42	#88055	C₂³	1	2	yes	—
43	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
44	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
45	C₈×Dic₁₃	C₂³	1	2	yes	—
46	C₂×#2165	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×Dic₁₃	—	2	1	no	—
57	C₂²×Dic₁₃	—	2	4	no	—
58	C₂²×Dic₁₃	—	2	1	no	—
59	C₂²×Dic₁₃	—	4	4	no	—
60	C₁₃×M₁₆	—	2	8	no	—
61	C₂×C₈×C₁₃	—	2	1	no	—
62	C₂×C₈×C₁₃	—	2	1	no	—
63	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
64	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
67	C₄×Dic₁₃	—	2	1	no	—
68	C₄×Dic₁₃	—	2	4	no	—
69	C₄×Dic₁₃	—	2	1	no	—
70	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
71	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
72	#6099	—	2	4	no	—
73	#6099	—	2	4	no	—
74	C₂×#1055	—	2	1	no	—
75	C₂×#1055	—	2	1	no	—
76	C₂×#1055	C₂²×C₄	1	2	yes	—
77	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
78	C₂×#1055	#41	1	2	yes	—
79	C₂×#1055	#41	1	2	yes	—
80	C₂×#1055	C₂⁴	1	1	yes	characteristic
81	#6092	—	2	8	no	—
82	C₂×C₄×M₁₆	—	13	1	no	—
83	#2165	—	13	8	no	—
84	C₂×#364	—	13	2	no	—
85	C₂×#349	—	13	2	no	—
86	C₂×#347	—	13	2	no	—
87	C₂³×C₁₃	—	2	1	no	—
88	C₂³×C₁₃	—	2	1	no	—
89	C₂³×C₁₃	—	4	2	no	—
90	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
91	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×Dic₁₃	—	2	4	no	—
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	—	4	8	no	—
109	C₂×Dic₁₃	—	4	2	no	—
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
112	#1055	—	2	2	no	—
113	#1055	#117	1	2	yes	—
114	#1055	C₂×#41	1	2	yes	—
115	C₂²×M₁₆	—	13	1	no	—
116	C₂²×M₁₆	#261	1	1	yes	characteristic
117	C₂²×C₄²	—	13	1	no	—
118	C₂×M₃₂	—	13	4	no	—
119	C₄×M₁₆	—	13	4	no	—
120	C₄×M₁₆	—	13	4	no	—
121	C₂×#97	—	13	1	no	—
122	C₂×#97	—	13	1	no	—
123	C₂×C₄×C₈	—	13	1	no	—
124	C₂×C₄×C₈	—	13	1	no	—
125	#364	—	13	8	no	—
126	#349	—	13	8	no	—
127	#347	—	13	8	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	—	4	2	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	#432	1	1	yes	characteristic
134	C₂²×C₁₃	#432	1	1	yes	characteristic
135	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	#432	1	1	yes	characteristic
138	C₄×C₁₃	#432	1	1	yes	characteristic
139	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
140	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
141	Dic₁₃	—	2	4	no	—
142	Dic₁₃	—	4	2	no	—
143	C₂³×C₄	—	26	1	no	—
144	C₂³×C₄	C₂×#261	1	1	yes	characteristic
145	C₂×M₁₆	—	2	2	no	—
146	C₂×M₁₆	—	2	2	no	—
147	C₂×M₁₆	—	13	2	no	—
148	C₂×M₁₆	—	13	2	no	—
149	C₂×M₁₆	—	26	2	no	—
150	C₂×M₁₆	—	26	2	no	—
151	C₂×M₁₆	C₂×#261	1	4	yes	—
152	C₂²×C₈	—	13	1	no	—
153	C₂²×C₈	—	13	1	no	—
154	C₂²×C₈	C₂×#261	1	1	yes	characteristic
155	C₂²×C₈	C₂×#261	1	1	yes	characteristic
156	C₂×C₄²	—	13	1	no	—
157	C₂×C₄²	—	13	1	no	—
158	C₂×C₄²	—	13	4	no	—
159	C₂×C₄²	—	26	1	no	—
160	C₂×C₄²	—	26	2	no	—
161	M₃₂	—	13	8	no	—
162	M₃₂	—	26	4	no	—
163	C₂×C₁₆	—	26	4	no	—
164	#97	—	13	2	no	—
165	#97	—	13	2	no	—
166	#97	—	13	4	no	—
167	C₄×C₈	—	13	2	no	—
168	C₄×C₈	—	13	2	no	—
169	C₄×C₈	—	13	4	no	—
170	C₂×C₁₃	—	2	1	no	—
171	C₂×C₁₃	—	2	1	no	—
172	C₂×C₁₃	—	4	2	no	—
173	C₂×C₁₃	#2165	1	2	yes	complemented
174	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
175	C₂⁴	#6118	1	1	yes	characteristic
176	C₂²×C₄	—	2	1	no	—
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	—	2	1	no	—
179	C₂²×C₄	—	2	1	no	—
180	C₂²×C₄	—	26	1	no	—
181	C₂²×C₄	—	26	4	no	—
182	C₂²×C₄	—	26	1	no	—
183	C₂²×C₄	—	52	4	no	—
184	C₂²×C₄	#6118	1	1	yes	characteristic
185	C₂²×C₄	#6118	1	2	yes	—
186	C₂²×C₄	C₂²×#261	1	2	yes	—
187	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
188	M₁₆	—	2	8	no	—
189	M₁₆	—	26	4	no	—
190	M₁₆	—	26	4	no	—
191	C₂×C₈	—	2	1	no	—
192	C₂×C₈	—	2	1	no	—
193	C₂×C₈	—	13	2	no	—
194	C₂×C₈	—	13	2	no	—
195	C₂×C₈	—	13	1	no	—
196	C₂×C₈	—	13	1	no	—
197	C₂×C₈	—	13	2	no	—
198	C₂×C₈	—	26	1	no	—
199	C₂×C₈	—	26	1	no	—
200	C₂×C₈	#6122	1	2	yes	—
201	C₂×C₈	#6122	1	2	yes	—
202	C₂×C₈	C₂²×#261	1	2	yes	—
203	C₂×C₈	C₂²×#261	1	2	yes	—
204	C₄²	—	13	2	no	—
205	C₄²	—	13	2	no	—
206	C₄²	—	26	1	no	—
207	C₄²	—	26	4	no	—
208	C₄²	—	26	1	no	—
209	C₁₆	—	26	8	no	—
210	C₁₃	C₂×#2165	1	1	yes	characteristic, complemented, Sylow
211	C₂³	—	2	1	no	—
212	C₂³	—	2	1	no	—
213	C₂³	—	4	2	no	—
214	C₂³	C₂×#6118	1	1	yes	characteristic
215	C₂³	C₂×#6118	1	2	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	2	no	—
222	C₂×C₄	—	26	4	no	—
223	C₂×C₄	—	26	2	no	—
224	C₂×C₄	—	26	2	no	—
225	C₂×C₄	—	52	8	no	—
226	C₂×C₄	—	52	2	no	—
227	C₂×C₄	C₂×#6118	1	2	yes	—
228	C₂×C₄	C₂×#6118	1	1	yes	characteristic
229	C₂×C₄	#88237	1	1	yes	characteristic
230	C₂×C₄	#88237	1	1	yes	characteristic
231	C₂×C₄	C₂×#6122	1	1	yes	characteristic
232	C₂×C₄	C₂×#6122	1	1	yes	characteristic
233	C₂×C₄	C₂³×#261	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	—	13	2	no	—
236	C₈	—	13	2	no	—
237	C₈	—	26	2	no	—
238	C₈	C₂×#6122	1	4	yes	—
239	C₂²	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	4	2	no	—
244	C₂²	#11719130	1	1	yes	characteristic
245	C₂²	#11719130	1	1	yes	characteristic
246	C₂²	C₂²×#6118	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	26	4	no	—
249	C₄	—	52	2	no	—
250	C₄	#11719130	1	1	yes	characteristic
251	C₄	#11719130	1	1	yes	characteristic
252	C₄	C₂×#88237	1	1	yes	characteristic
253	C₄	C₂²×#6122	1	1	yes	characteristic
254	C₂	—	2	1	no	—
255	C₂	—	2	1	no	—
256	C₂	—	4	2	no	—
257	C₂	#13131206	1	2	yes	direct product factor
258	C₂	C₂×#11719130	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88237
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂²×C₁₃×M₁₆	C₄
socle	C₂²×C₁₃	#432
prime radical	C₂⁴×C₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂²×M₁₆	#261
hypocenter	C₁₃	C₂×#2165
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. #13131206	C₂	direct (N × H)
2. C₂×#11718261	C₂	semidirect
4. C₂×#11718262	C₂	semidirect
17. #11718262	C₂²	semidirect
18. #11718261	C₂²	semidirect
173. C₂×C₁₃	#2165	semidirect
210. C₁₃	C₂×#2165	semidirect
257. C₂	#13131206	direct (N × H)

Maximal subgroups

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

subgroup	index
#13131206	2
C₂×#11718261	2
C₂×#11718284	2
C₂×#11718262	2
C₂×M₁₆×Dic₁₃	2
C₂×#2165	13

Subgroups — C2×#13131206 #16807092

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131206 #16807092