Subgroups — C₂×#13131225 #16807169

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×#11719096	C₂	1	2	yes	maximal, complemented
2	#13131225	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718261	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#11718270	C₂	1	2	yes	maximal, complemented
5	C₂×#11718257	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11718556	C₂	1	1	yes	maximal, characteristic
7	C₂×#88161	C₂²	1	2	yes	—
8	C₂×#88158	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	2	yes	—
10	C₂²×#6092	C₂²	1	1	yes	characteristic
11	C₂²×#6092	C₂²	1	1	yes	characteristic
12	#11719096	—	2	4	no	—
13	#11719096	C₂²	1	8	yes	complemented
14	C₂×#88098	—	2	2	no	—
15	C₂×D₁₃⋊C₁₆	—	2	2	no	—
16	C₂×C₁₃×#108	C₄	1	1	yes	characteristic
17	C₂×#88065	C₄	1	2	yes	—
18	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
19	#11718556	C₄	1	4	yes	—
20	#11718556	C₂²	1	4	yes	—
21	#11718270	C₂²	1	8	yes	complemented
22	#11718261	C₂²	1	4	yes	complemented
23	#11718257	C₂²	1	4	yes	complemented
24	C₂×#6127	C₂×C₄	1	2	yes	—
25	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×#1055	C₂³	1	1	yes	characteristic
29	#88161	—	2	2	no	—
30	#88161	—	2	2	no	—
31	#88161	C₂×C₄	1	4	yes	—
32	#88161	C₂³	1	4	yes	—
33	C₂×Dic₅₂	C₈	1	1	yes	characteristic
34	#88158	C₈	1	4	yes	—
35	#88158	C₂×C₄	1	4	yes	—
36	C₂×D₁₀₄	C₈	1	1	yes	characteristic
37	C₂×#6095	C₈	1	2	yes	—
38	C₂×#6094	—	2	2	no	—
39	C₂×C₈×D₁₃	—	2	2	no	—
40	#88106	—	2	2	no	—
41	#88106	C₂³	1	4	yes	—
42	C₂×#6092	—	2	2	no	—
43	C₂×#6092	—	2	1	no	—
44	C₂×#6092	—	2	1	no	—
45	C₂×#6092	D₄	1	1	yes	characteristic
46	C₂×#6092	D₄	1	1	yes	characteristic
47	C₂×#6092	D₄	1	2	yes	—
48	C₂×#6092	C₂³	1	2	yes	—
49	C₂×#6092	C₂³	1	2	yes	—
50	#88098	—	2	8	no	—
51	D₁₃⋊C₁₆	—	2	8	no	—
52	C₁₃×#108	C₂×C₄	1	4	yes	—
53	#88065	C₂×C₄	1	8	yes	—
54	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
55	C₈×Dic₁₃	C₂³	1	2	yes	—
56	C₂×#2170	—	13	1	no	maximal, Sylow
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₃	—	2	2	no	—
59	C₂²×Dic₁₃	—	2	1	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	C₁₃×M₁₆	—	2	2	no	—
68	C₁₃×M₁₆	C₂²×C₄	1	4	yes	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₈×C₁₃	C₂×C₈	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
73	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₄×Dic₁₃	—	2	1	no	—
75	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
76	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
77	C₂×#1055	C₂×D₄	1	1	yes	characteristic
78	C₂×#1055	C₂×D₄	1	1	yes	characteristic
79	C₂×#1055	#41	1	1	yes	characteristic
80	C₂×#1055	#41	1	1	yes	characteristic
81	C₂×#1055	C₂⁴	1	1	yes	characteristic
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	#6092	—	2	4	no	—
90	#6092	—	2	2	no	—
91	#6092	C₂×D₄	1	2	yes	—
92	#6092	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	—	13	1	no	—
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	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#131	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
107	C₂²⋊D₁₃	—	2	4	no	—
108	C₂²⋊D₁₃	—	2	4	no	—
109	C₂×Dic₁₃	—	2	4	no	—
110	C₂×Dic₁₃	—	2	2	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	—	2	4	no	—
119	C₈×C₁₃	C₂²×C₈	1	2	yes	—
120	C₈×C₁₃	C₂²×C₈	1	2	yes	—
121	#1055	C₄×D₄	1	1	yes	characteristic
122	#1055	C₄×D₄	1	1	yes	characteristic
123	#1055	C₂²×D₄	1	1	yes	characteristic
124	#1055	C₂×#41	1	1	yes	characteristic
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	—	13	2	no	—
130	C₂×M₃₂	—	26	2	no	—
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	#261	1	1	yes	characteristic
137	C₂×#104	—	13	2	no	—
138	C₂×C₄×C₈	—	13	1	no	—
139	#364	—	13	4	no	—
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	C₈×D₄	1	1	yes	characteristic
149	C₄×C₁₃	C₈×D₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
151	C₄×C₁₃	C₂³×C₈	1	1	yes	characteristic
152	Dic₁₃	—	2	4	no	—
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	—	26	2	no	—
165	C₂×M₁₆	C₂×#261	1	2	yes	—
166	C₂²×C₈	—	13	1	no	—
167	C₂²×C₈	—	26	2	no	—
168	C₂²×C₈	C₂×#261	1	1	yes	characteristic
169	C₂×C₄²	—	13	1	no	—
170	M₃₂	—	13	4	no	—
171	M₃₂	—	26	8	no	—
172	M₃₂	—	26	2	no	—
173	C₂×C₁₆	—	13	1	no	—
174	C₂×C₁₆	—	13	1	no	—
175	C₂×C₁₆	—	13	2	no	—
176	C₂×C₁₆	—	13	2	no	—
177	C₂×C₁₆	—	13	2	no	—
178	C₂×C₁₆	—	26	8	no	—
179	C₂×C₁₆	—	26	1	no	—
180	C₂×C₁₆	—	26	1	no	—
181	C₂×C₁₆	—	26	2	no	—
182	#108	C₂×#261	1	4	yes	—
183	#104	—	13	8	no	—
184	C₄×C₈	—	13	2	no	—
185	C₄×C₈	—	13	2	no	—
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	—	26	1	no	—
199	C₂²×C₄	—	26	2	no	—
200	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
201	Q₁₆	—	13	4	no	—
202	SD₁₆	—	13	8	no	—
203	D₈	—	13	4	no	—
204	M₁₆	—	2	2	no	—
205	M₁₆	—	26	4	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	C₂²×#261	1	4	yes	—
208	C₂×C₈	—	2	2	no	—
209	C₂×C₈	—	13	1	no	—
210	C₂×C₈	—	13	1	no	—
211	C₂×C₈	—	13	1	no	—
212	C₂×C₈	—	13	1	no	—
213	C₂×C₈	—	13	1	no	—
214	C₂×C₈	—	13	1	no	—
215	C₂×C₈	—	26	4	no	—
216	C₂×C₈	—	26	4	no	—
217	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
218	C₂×C₈	D₁₃⋊C₈	1	1	yes	characteristic
219	C₂×C₈	D₁₃⋊C₈	1	2	yes	—
220	C₂×C₈	C₂²×#261	1	2	yes	—
221	C₄²	—	13	2	no	—
222	C₄²	—	26	1	no	—
223	C₁₆	—	13	2	no	—
224	C₁₆	—	13	2	no	—
225	C₁₆	—	26	2	no	—
226	C₁₆	—	26	4	no	—
227	C₁₃	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
228	C₂³	—	26	2	no	—
229	C₂³	#88240	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	—	26	4	no	—
237	C₂×C₄	—	26	2	no	—
238	C₂×C₄	—	26	2	no	—
239	C₂×C₄	—	26	4	no	—
240	C₂×C₄	—	26	4	no	—
241	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₁₃⋊C₈	1	1	yes	characteristic
243	C₂×C₄	D₄×#261	1	1	yes	characteristic
244	C₂×C₄	D₄×#261	1	1	yes	characteristic
245	C₂×C₄	#88240	1	1	yes	characteristic
246	C₂×C₄	C₂³×#261	1	1	yes	characteristic
247	C₈	—	2	4	no	—
248	C₈	—	13	1	no	—
249	C₈	—	13	1	no	—
250	C₈	—	13	1	no	—
251	C₈	—	13	1	no	—
252	C₈	C₂×D₁₃⋊C₈	1	2	yes	—
253	C₈	C₂×D₁₃⋊C₈	1	2	yes	—
254	C₂²	—	2	2	no	—
255	C₂²	—	26	4	no	—
256	C₂²	—	52	4	no	—
257	C₂²	#11719073	1	1	yes	characteristic
258	C₂²	#11719073	1	1	yes	characteristic
259	C₂²	C₂×#88240	1	1	yes	characteristic
260	C₄	—	26	4	no	—
261	C₄	—	26	4	no	—
262	C₄	#11719073	1	1	yes	characteristic
263	C₄	#11719073	1	1	yes	characteristic
264	C₄	C₂²×D₁₃⋊C₈	1	1	yes	characteristic
265	C₄	C₂×D₄×#261	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	52	4	no	—
269	C₂	#13131225	1	2	yes	direct product factor
270	C₂	C₂×#11719073	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×#261
commutator	C₄×C₁₃	C₂³×C₈
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₁₃×#108	C₄
socle	C₂²×C₁₃	C₈×D₄
prime radical	C₂×#88158	C₄
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×#108	#261
hypocenter	C₁₃	C₂×#2170
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

Complements

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

N	H ≅ G/N	type
1. C₂×#11719096	C₂	semidirect
2. #13131225	C₂	direct (N × H)
3. C₂×#11718261	C₂	semidirect
4. C₂×#11718270	C₂	semidirect
5. C₂×#11718257	C₂	semidirect
13. #11719096	C₂²	semidirect
21. #11718270	C₂²	semidirect
22. #11718261	C₂²	semidirect
23. #11718257	C₂²	semidirect
188. C₂×C₁₃	#2170	semidirect
227. C₁₃	C₂×#2170	semidirect
269. C₂	#13131225	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×#11719096	2
#13131225	2
C₂×#11718261	2
C₂×#11718270	2
C₂×#11718257	2
C₂×#11718556	2
C₂×#2170	13

Subgroups — C2×#13131225 #16807169

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131225 #16807169