Subgroups — C₂×#13116434 #16015555

Order 3328.

854 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×#88038	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
4	#13116434	C₂	1	8	yes	maximal, direct product factor
5	C₂×#11718094	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#6094	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88126	C₄	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88055	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
17	C₂²×#6090	C₂²	1	1	yes	characteristic
18	C₂²×#6090	C₂²	1	1	yes	characteristic
19	C₂×#88043	C₄	1	1	yes	characteristic
20	C₂×#88038	C₄	1	8	yes	complemented
21	C₂×#88038	C₂²	1	4	yes	—
22	C₂×C₁₆×D₁₃	C₄	1	8	yes	complemented
23	C₂×C₁₆×D₁₃	C₂²	1	4	yes	—
24	#11718350	C₄	1	4	yes	—
25	#11718350	C₂²	1	4	yes	—
26	C₁₃×#346	C₂²	1	4	yes	complemented
27	#11718094	C₂²	1	4	yes	complemented
28	C₁₆×Dic₁₃	C₂²	1	4	yes	complemented
29	C₄×#6090	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×#6103	C₈	1	2	yes	—
35	C₂×#6101	C₈	1	2	yes	—
36	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
38	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
39	C₂×#6094	C₈	1	8	yes	—
40	C₂×#6094	C₂×C₄	1	4	yes	—
41	C₂×C₈×D₁₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
44	C₂×C₈×D₁₃	C₂³	1	2	yes	—
45	#88126	C₈	1	4	yes	—
46	#88126	C₂×C₄	1	4	yes	—
47	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
48	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
51	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂³	1	2	yes	—
53	#88055	C₂×C₄	1	4	yes	—
54	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₃	C₂³	1	2	yes	—
56	C₂×#6090	C₂×C₄	1	4	yes	—
57	C₂×#6090	C₂×C₄	1	4	yes	—
58	C₂×#6090	C₂³	1	2	yes	—
59	C₂×#6090	C₂³	1	2	yes	—
60	#88043	C₂×C₄	1	4	yes	—
61	#88038	C₂×C₄	1	16	yes	complemented
62	C₁₆×D₁₃	C₂×C₄	1	16	yes	complemented
63	C₂×#2108	—	13	1	no	maximal, Sylow
64	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
68	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₁₃	C₄²	1	4	yes	—
70	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
72	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₁₃	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₄²×C₁₃	C₂×C₈	1	2	yes	—
80	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
81	#6103	C₂×C₈	1	8	yes	—
82	#6101	C₂×C₈	1	8	yes	—
83	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
84	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
85	C₂×#1055	C₄²	1	4	yes	—
86	C₂×#1055	C₂×C₈	1	4	yes	—
87	C₂×#1055	C₂²×C₄	1	2	yes	—
88	C₂×#1055	C₂²×C₄	1	2	yes	—
89	#6094	C₂×C₈	1	16	yes	—
90	C₈×D₁₃	C₄²	1	8	yes	—
91	C₈×D₁₃	C₂²×C₄	1	4	yes	—
92	C₈×D₁₃	C₂²×C₄	1	4	yes	—
93	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
94	C₁₃×C₁₆	C₂²×C₄	1	4	yes	—
95	#6090	C₂²×C₄	1	4	yes	—
96	#6090	C₂²×C₄	1	4	yes	—
97	C₂²×M₃₂	—	13	1	no	—
98	C₂³×C₁₆	—	13	1	no	—
99	C₂×#405	—	13	1	no	—
100	#2108	—	13	8	no	—
101	C₂×#346	—	13	1	no	—
102	C₂×#346	D₁₃	1	1	yes	characteristic, complemented
103	C₂×C₄×C₁₆	—	13	1	no	—
104	C₂×C₄×C₁₆	—	13	1	no	—
105	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂²×D₁₃	—	2	4	no	—
107	C₂²×D₁₃	C₄×C₈	1	4	yes	—
108	C₂²×D₁₃	C₂²×C₈	1	2	yes	—
109	C₂×C₄×C₁₃	—	2	4	no	—
110	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×Dic₁₃	—	2	4	no	—
116	C₂×Dic₁₃	C₄×C₈	1	4	yes	—
117	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
118	C₂×Dic₁₃	C₂²×C₈	1	2	yes	—
119	C₄×D₁₃	C₄×C₈	1	8	yes	—
120	C₄×D₁₃	C₂×C₄²	1	4	yes	—
121	C₄×D₁₃	C₂²×C₈	1	4	yes	—
122	C₈×C₁₃	C₂×C₄²	1	2	yes	—
123	C₈×C₁₃	C₂²×C₈	1	4	yes	—
124	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
126	#1055	C₂×C₄²	1	4	yes	—
127	#1055	C₂²×C₈	1	4	yes	—
128	C₂²×M₁₆	—	13	1	no	—
129	C₂³×C₈	—	13	1	no	—
130	C₂×#117	—	13	1	no	—
131	C₂×M₃₂	—	13	8	no	—
132	C₂×M₃₂	—	13	4	no	—
133	C₂²×C₁₆	—	13	1	no	—
134	C₂²×C₁₆	—	13	1	no	—
135	C₂²×C₁₆	—	13	8	no	—
136	C₂²×C₁₆	—	13	4	no	—
137	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
138	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
139	#405	—	13	4	no	—
140	#405	—	13	4	no	—
141	C₂×#97	—	13	1	no	—
142	C₂×#97	—	13	1	no	—
143	C₂×C₄×C₈	—	13	1	no	—
144	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
145	#346	—	13	4	no	—
146	#346	C₂×D₁₃	1	4	yes	complemented
147	C₄×C₁₆	—	13	4	no	—
148	C₄×C₁₆	—	13	4	no	—
149	C₂²×C₁₃	C₂×C₄×C₈	1	2	yes	—
150	C₂²×C₁₃	#504	1	4	yes	—
151	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
152	C₂×D₁₃	—	2	8	no	—
153	C₂×D₁₃	—	2	4	no	—
154	C₂×D₁₃	C₂×C₄×C₈	1	4	yes	—
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	C₂×C₄×C₈	1	2	yes	—
157	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
158	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
159	Dic₁₃	—	2	4	no	—
160	Dic₁₃	C₂×C₄×C₈	1	4	yes	—
161	C₂³×C₄	—	13	1	no	—
162	C₂×M₁₆	—	13	8	no	—
163	C₂×M₁₆	—	13	4	no	—
164	C₂²×C₈	—	13	1	no	—
165	C₂²×C₈	—	13	1	no	—
166	C₂²×C₈	—	13	2	no	—
167	C₂²×C₈	—	13	4	no	—
168	C₂²×C₈	—	13	4	no	—
169	C₂²×C₈	—	13	2	no	—
170	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
171	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
172	#117	—	13	4	no	—
173	#117	—	13	4	no	—
174	C₂×C₄⋊C₄	—	13	2	no	—
175	C₂×C₂²⋊C₄	—	13	2	no	—
176	C₂×C₄²	—	13	1	no	—
177	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
178	M₃₂	—	13	16	no	—
179	C₂×C₁₆	—	13	4	no	—
180	C₂×C₁₆	—	13	4	no	—
181	C₂×C₁₆	—	13	2	no	—
182	C₂×C₁₆	—	13	2	no	—
183	C₂×C₁₆	—	13	16	no	—
184	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
185	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
186	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
187	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
188	#97	—	13	4	no	—
189	#97	—	13	4	no	—
190	C₄×C₈	—	13	2	no	—
191	C₄×C₈	—	13	2	no	—
192	C₄×C₈	C₄×D₁₃	1	2	yes	—
193	C₄×C₈	C₂²×D₁₃	1	2	yes	—
194	C₂×C₁₃	#2108	1	4	yes	complemented
195	C₂×C₁₃	C₂²×C₄×C₈	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#504	1	2	yes	—
197	D₁₃	—	2	4	no	—
198	C₂⁴	—	13	1	no	—
199	C₂²×C₄	—	13	1	no	—
200	C₂²×C₄	—	13	4	no	—
201	C₂²×C₄	—	13	2	no	—
202	C₂²×C₄	—	13	4	no	—
203	C₂²×C₄	—	13	2	no	—
204	C₂²×C₄	—	13	2	no	—
205	C₂²×C₄	C₈×D₁₃	1	2	yes	—
206	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	M₁₆	—	13	16	no	—
208	C₂×C₈	—	13	4	no	—
209	C₂×C₈	—	13	4	no	—
210	C₂×C₈	—	13	2	no	—
211	C₂×C₈	—	13	2	no	—
212	C₂×C₈	—	13	8	no	—
213	C₂×C₈	—	13	4	no	—
214	C₂×C₈	—	13	4	no	—
215	C₂×C₈	C₈×D₁₃	1	4	yes	—
216	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
217	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
220	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
221	C₄⋊C₄	—	13	8	no	—
222	C₂²⋊C₄	—	13	8	no	—
223	C₄²	—	13	2	no	—
224	C₄²	—	13	2	no	—
225	C₄²	C₈×D₁₃	1	2	yes	—
226	C₄²	C₂×C₄×D₁₃	1	2	yes	—
227	C₁₆	—	13	4	no	—
228	C₁₆	—	13	4	no	—
229	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
230	C₁₆	C₂×C₄×D₁₃	1	4	yes	—
231	C₁₃	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
232	C₂³	—	13	2	no	—
233	C₂³	—	13	4	no	—
234	C₂³	—	26	4	no	—
235	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	13	2	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	4	no	—
240	C₂×C₄	—	13	4	no	—
241	C₂×C₄	—	13	8	no	—
242	C₂×C₄	—	13	4	no	—
243	C₂×C₄	—	26	4	no	—
244	C₂×C₄	C₄²×D₁₃	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₁₃	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
247	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
249	C₈	—	13	4	no	—
250	C₈	—	13	4	no	—
251	C₈	C₄²×D₁₃	1	2	yes	—
252	C₈	C₂×C₈×D₁₃	1	4	yes	—
253	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₁₃	1	1	yes	characteristic
255	C₂²	—	13	4	no	—
256	C₂²	—	26	4	no	—
257	C₂²	—	26	8	no	—
258	C₂²	C₄×C₈×D₁₃	1	2	yes	—
259	C₂²	#11718568	1	4	yes	—
260	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	13	4	no	—
263	C₄	—	26	4	no	—
264	C₄	C₄×C₈×D₁₃	1	2	yes	—
265	C₄	C₂×C₄²×D₁₃	1	1	yes	characteristic
266	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
267	C₂	—	26	4	no	—
268	C₂	#13116434	1	4	yes	direct product factor
269	C₂	C₂×C₄×C₈×D₁₃	1	1	yes	characteristic
270	C₂	C₂×#11718568	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂×C₁₃	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#346	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	C₂×#2108
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
4. #13116434	C₂	direct (N × H)
5. C₂×#11718094	C₂	semidirect
6. C₂×C₄×#6090	C₂	semidirect
7. C₂×C₁₆×Dic₁₃	C₂	semidirect
8. C₂×C₁₃×#346	C₂	semidirect
20. C₂×#88038	C₄	semidirect
22. C₂×C₁₆×D₁₃	C₄	semidirect
26. C₁₃×#346	C₂²	semidirect
27. #11718094	C₂²	semidirect
28. C₁₆×Dic₁₃	C₂²	semidirect
29. C₄×#6090	C₂²	semidirect
61. #88038	C₂×C₄	semidirect
62. C₁₆×D₁₃	C₂×C₄	semidirect
102. C₂×#346	D₁₃	semidirect
146. #346	C₂×D₁₃	semidirect
194. C₂×C₁₃	#2108	semidirect
231. C₁₃	C₂×#2108	semidirect
268. C₂	#13116434	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×#88038	2
C₂²×C₁₆×D₁₃	2
C₂×#11718350	2
#13116434	2
C₂×#11718094	2
C₂×C₄×#6090	2
C₂×C₁₆×Dic₁₃	2
C₂×C₁₃×#346	2
C₂×#2108	13

Subgroups — C2×#13116434 #16015555

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116434 #16015555