Subgroups — C₂×#13116672 #16016146

Order 3328.

662 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	C₂×#11718568	C₂	1	2	yes	maximal, complemented
2	C₂×#11718361	C₂	1	1	yes	maximal, characteristic
3	#13116672	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718107	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×#6090	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#364	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88158	C₄	1	1	yes	characteristic
8	C₂×#88155	C₂²	1	2	yes	—
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×#88059	C₄	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	1	yes	characteristic
13	C₂²×#6090	C₂²	1	1	yes	characteristic
14	C₂×#88045	C₄	1	2	yes	—
15	#11718568	—	2	4	no	—
16	#11718568	C₂²	1	8	yes	complemented
17	C₂×#88038	—	2	2	no	—
18	C₂×C₁₆×D₁₃	—	2	2	no	—
19	#11718361	C₄	1	4	yes	—
20	#11718361	C₂²	1	4	yes	—
21	C₁₃×#364	C₂²	1	4	yes	complemented
22	#11718107	C₂²	1	8	yes	complemented
23	C₄×#6090	C₂²	1	4	yes	complemented
24	C₂×#6127	C₂×C₄	1	2	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂×#6099	C₂×C₄	1	2	yes	—
29	C₂×Dic₅₂	C₈	1	1	yes	characteristic
30	#88158	C₈	1	4	yes	—
31	#88158	C₂×C₄	1	4	yes	—
32	C₂×D₁₀₄	C₈	1	1	yes	characteristic
33	C₂×#6095	C₈	1	2	yes	—
34	#88155	—	2	2	no	—
35	#88155	—	2	2	no	—
36	#88155	C₂×C₄	1	4	yes	—
37	#88155	C₂³	1	4	yes	—
38	C₂×#6094	—	2	2	no	—
39	C₂×C₈×D₁₃	—	2	2	no	—
40	C₁₃×M₃₂	—	2	2	no	—
41	C₁₃×M₃₂	C₂³	1	4	yes	—
42	C₂×C₁₃×C₁₆	—	2	2	no	—
43	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
44	C₄×C₈×C₁₃	C₂³	1	2	yes	—
45	#88059	C₂×C₄	1	4	yes	—
46	C₂×#6090	—	2	1	no	—
47	C₂×#6090	—	2	1	no	—
48	C₂×#6090	D₄	1	1	yes	characteristic
49	C₂×#6090	D₄	1	1	yes	characteristic
50	C₂×#6090	D₄	1	2	yes	—
51	C₂×#6090	C₂³	1	2	yes	—
52	C₂×#6090	C₂³	1	2	yes	—
53	#88045	C₂×C₄	1	8	yes	—
54	#88038	—	2	8	no	—
55	C₁₆×D₁₃	—	2	8	no	—
56	C₂×#2170	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	—	2	1	no	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₂²⋊D₁₃	—	2	2	no	—
60	#6127	—	2	2	no	—
61	#6127	—	2	2	no	—
62	#6127	C₂×C₈	1	4	yes	—
63	#6127	C₂²×C₄	1	4	yes	—
64	C₂×D₅₂	C₂×C₈	1	2	yes	—
65	C₂×C₄×D₁₃	—	2	2	no	—
66	C₂×Dic₂₆	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
70	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
75	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
76	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₄²×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
79	#6099	—	2	2	no	—
80	#6099	C₂²×C₄	1	4	yes	—
81	C₂×#1055	—	2	2	no	—
82	Dic₅₂	C₂×C₈	1	4	yes	—
83	D₁₀₄	C₂×C₈	1	4	yes	—
84	#6095	C₂×C₈	1	8	yes	—
85	#6094	—	2	4	no	—
86	#6094	—	2	4	no	—
87	C₈×D₁₃	—	2	4	no	—
88	C₈×D₁₃	—	2	4	no	—
89	C₁₃×C₁₆	—	2	4	no	—
90	#6090	—	2	2	no	—
91	#6090	C₂×D₄	1	2	yes	—
92	#6090	C₂×D₄	1	2	yes	—
93	C₂×#504	—	13	2	no	—
94	C₂×#443	—	13	1	no	—
95	#2170	—	13	8	no	—
96	C₂×#364	D₁₃	1	1	yes	characteristic, complemented
97	C₂×#350	—	13	2	no	—
98	C₂×C₄×C₁₆	—	13	1	no	—
99	C₂³×C₁₃	#131	1	1	yes	characteristic
100	C₂²×D₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
109	C₂²⋊D₁₃	—	2	4	no	—
110	C₂²⋊D₁₃	—	2	4	no	—
111	C₂×Dic₁₃	—	2	2	no	—
112	D₅₂	—	2	2	no	—
113	D₅₂	C₂²×C₈	1	4	yes	—
114	C₄×D₁₃	—	2	4	no	—
115	C₄×D₁₃	—	2	4	no	—
116	Dic₂₆	—	2	2	no	—
117	Dic₂₆	C₂²×C₈	1	4	yes	—
118	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
120	C₈×C₁₃	C₂²×C₈	1	2	yes	—
121	C₈×C₁₃	C₂²×C₈	1	2	yes	—
122	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
124	#1055	—	2	4	no	—
125	C₂×#135	—	13	1	no	—
126	C₂×#131	—	13	2	no	—
127	#504	—	13	8	no	—
128	#504	—	26	4	no	—
129	C₂×M₃₂	—	26	2	no	—
130	C₂×M₃₂	C₂×D₁₃	1	2	yes	—
131	C₂²×C₁₆	—	13	1	no	—
132	C₂²×C₁₆	—	13	1	no	—
133	C₂²×C₁₆	—	26	2	no	—
134	#443	—	13	4	no	—
135	#443	—	13	4	no	—
136	C₂×#108	—	13	1	no	—
137	C₂×#104	—	13	2	no	—
138	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
139	#364	C₂×D₁₃	1	4	yes	complemented
140	#350	—	13	8	no	—
141	C₄×C₁₆	—	13	4	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₈×D₄	1	1	yes	characteristic
145	C₂²×C₁₃	C₂×#131	1	1	yes	characteristic
146	C₂×D₁₃	—	2	4	no	—
147	C₂×D₁₃	—	4	4	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
150	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
151	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
153	Dic₁₃	—	2	4	no	—
154	C₂×#41	—	13	2	no	—
155	#135	—	13	4	no	—
156	#135	—	13	4	no	—
157	C₂×Q₁₆	—	13	1	no	—
158	C₂×SD₁₆	—	13	2	no	—
159	C₂×D₈	—	13	1	no	—
160	#131	—	13	4	no	—
161	#131	—	13	4	no	—
162	#131	—	26	2	no	—
163	#131	—	26	2	no	—
164	C₂×M₁₆	—	13	2	no	—
165	C₂×M₁₆	—	26	2	no	—
166	C₂²×C₈	—	26	2	no	—
167	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
170	M₃₂	—	2	2	no	—
171	M₃₂	—	26	8	no	—
172	M₃₂	C₂²×D₁₃	1	4	yes	—
173	C₂×C₁₆	—	2	2	no	—
174	C₂×C₁₆	—	13	2	no	—
175	C₂×C₁₆	—	13	1	no	—
176	C₂×C₁₆	—	13	1	no	—
177	C₂×C₁₆	—	13	2	no	—
178	C₂×C₁₆	—	13	2	no	—
179	C₂×C₁₆	—	26	8	no	—
180	C₂×C₁₆	—	26	1	no	—
181	C₂×C₁₆	—	26	1	no	—
182	#108	—	13	4	no	—
183	#104	—	13	8	no	—
184	C₄×C₈	C₄×D₁₃	1	2	yes	—
185	C₄×C₈	C₂²×D₁₃	1	2	yes	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	1	no	—
188	C₂×C₁₃	#2170	1	2	yes	complemented
189	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
190	D₁₃	—	4	4	no	—
191	#41	—	13	4	no	—
192	#41	—	13	4	no	—
193	#41	—	26	2	no	—
194	#41	—	26	2	no	—
195	C₂×Q₈	—	13	2	no	—
196	C₂×D₄	—	13	2	no	—
197	C₂×D₄	—	26	2	no	—
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	26	2	no	—
200	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
201	Q₁₆	—	13	4	no	—
202	SD₁₆	—	13	8	no	—
203	D₈	—	13	4	no	—
204	M₁₆	—	13	4	no	—
205	M₁₆	—	26	2	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	—	26	4	no	—
208	C₂×C₈	—	26	2	no	—
209	C₂×C₈	—	26	4	no	—
210	C₂×C₈	—	26	4	no	—
211	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
212	C₂×C₈	C₈×D₁₃	1	1	yes	characteristic
213	C₂×C₈	C₈×D₁₃	1	2	yes	—
214	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
216	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
217	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
218	C₂×C₈	#6131	1	1	yes	characteristic
219	C₂×C₈	#6131	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
221	C₄²	—	2	1	no	—
222	C₄²	C₂×C₄×D₁₃	1	2	yes	—
223	C₁₆	—	2	4	no	—
224	C₁₆	—	13	2	no	—
225	C₁₆	—	13	2	no	—
226	C₁₆	—	26	2	no	—
227	C₁₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	26	2	no	—
229	C₂³	#88161	1	1	yes	characteristic
230	Q₈	—	13	4	no	—
231	Q₈	—	26	2	no	—
232	D₄	—	13	4	no	—
233	D₄	—	26	4	no	—
234	D₄	—	26	4	no	—
235	D₄	—	26	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	26	4	no	—
240	C₂×C₄	—	26	4	no	—
241	C₂×C₄	#88147	1	1	yes	characteristic
242	C₂×C₄	#88147	1	1	yes	characteristic
243	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
245	C₂×C₄	#88161	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
247	C₈	—	26	4	no	—
248	C₈	#88147	1	1	yes	characteristic
249	C₈	#88147	1	1	yes	characteristic
250	C₈	C₂×C₈×D₁₃	1	2	yes	—
251	C₈	C₂×C₈×D₁₃	1	2	yes	—
252	C₈	C₂×D₄×D₁₃	1	1	yes	characteristic
253	C₈	C₂×#6131	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	26	4	no	—
256	C₂²	—	52	4	no	—
257	C₂²	#11718495	1	1	yes	characteristic
258	C₂²	#11718495	1	1	yes	characteristic
259	C₂²	C₂×#88161	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	26	4	no	—
262	C₄	#11718495	1	1	yes	characteristic
263	C₄	#11718495	1	1	yes	characteristic
264	C₄	C₂×#88147	1	1	yes	characteristic
265	C₄	C₂²×C₈×D₁₃	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	52	4	no	—
269	C₂	#13116672	1	2	yes	direct product factor
270	C₂	C₂×#11718495	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₁₃
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₁₃×#364	C₂
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×#88158	C₄
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×#364	D₁₃
hypocenter	C₁₃	C₂×#2170
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
1. C₂×#11718568	C₂	semidirect
3. #13116672	C₂	direct (N × H)
4. C₂×#11718107	C₂	semidirect
5. C₂×C₄×#6090	C₂	semidirect
6. C₂×C₁₃×#364	C₂	semidirect
16. #11718568	C₂²	semidirect
21. C₁₃×#364	C₂²	semidirect
22. #11718107	C₂²	semidirect
23. C₄×#6090	C₂²	semidirect
96. C₂×#364	D₁₃	semidirect
139. #364	C₂×D₁₃	semidirect
188. C₂×C₁₃	#2170	semidirect
227. C₁₃	C₂×#2170	semidirect
269. C₂	#13116672	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×#11718568	2
C₂×#11718361	2
#13116672	2
C₂×#11718107	2
C₂×C₄×#6090	2
C₂×C₁₃×#364	2
C₂×#2170	13

Subgroups — C2×#13116672 #16016146

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116672 #16016146