Subgroups — C₂×#13131211 #16807098

Order 3328.

538 conjugacy classes of subgroups.

The subgroup lattice has 256 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13131211	C₂	1	8	yes	maximal, direct product factor
2	C₂×#11718261	C₂	1	2	yes	maximal, complemented
3	C₂×#11718263	C₂	1	2	yes	maximal
4	C₂×#11718232	C₂	1	2	yes	maximal, complemented
5	C₂×#11718350	C₂	1	1	yes	maximal, characteristic
6	C₂²×#6094	C₂²	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
8	C₂×#88126	C₄	1	1	yes	characteristic
9	C₂×#88106	C₂²	1	4	yes	—
10	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×#88055	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂×#88043	C₂²	1	1	yes	characteristic
14	#11718350	C₄	1	4	yes	—
15	#11718350	C₂²	1	4	yes	—
16	#11718263	C₂²	1	8	yes	—
17	#11718261	C₂²	1	8	yes	complemented
18	#11718232	C₂²	1	8	yes	complemented
19	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂×#6103	—	2	1	no	—
24	C₂×#6101	—	2	1	no	—
25	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂³	1	1	yes	characteristic
28	C₂×#6094	—	2	2	no	—
29	C₂×#6094	—	2	2	no	—
30	C₂×#6094	C₂×C₄	1	2	yes	—
31	C₂×#6094	C₂³	1	2	yes	—
32	C₂×C₈×D₁₃	—	2	2	no	—
33	C₂×C₈×D₁₃	—	2	2	no	—
34	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
35	#88126	—	2	2	no	—
36	#88126	C₂×C₄	1	4	yes	—
37	#88106	—	2	4	no	—
38	#88106	C₂³	1	8	yes	—
39	C₂×#6092	—	2	4	no	—
40	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
41	#88055	C₂×C₄	1	4	yes	—
42	C₈×Dic₁₃	C₂×C₄	1	2	yes	—
43	C₈×Dic₁₃	C₂³	1	2	yes	—
44	#88043	C₂×C₄	1	2	yes	—
45	#88043	C₂³	1	2	yes	—
46	C₂×#2166	—	13	1	no	maximal, Sylow
47	C₂³×D₁₃	M₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
50	C₂²×Dic₁₃	—	2	1	no	—
51	C₂²×Dic₁₃	M₁₆	1	1	yes	characteristic
52	C₂×C₄×D₁₃	—	2	1	no	—
53	C₂×C₄×D₁₃	—	2	1	no	—
54	C₂×C₄×D₁₃	—	2	1	no	—
55	C₂×C₄×D₁₃	—	2	1	no	—
56	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
57	C₂×C₄×D₁₃	C₂²×C₄	1	2	yes	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
61	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
62	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
64	C₄²×C₁₃	—	2	1	no	—
65	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
66	#6103	—	2	4	no	—
67	#6101	—	2	4	no	—
68	C₄×Dic₁₃	—	2	1	no	—
69	C₄×Dic₁₃	C₂²×C₄	1	2	yes	—
70	C₂×#1055	—	2	1	no	—
71	C₂×#1055	—	2	1	no	—
72	C₂×#1055	C₂²×C₄	1	2	yes	—
73	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
74	C₂×#1055	#41	1	2	yes	—
75	C₂×#1055	#41	1	2	yes	—
76	C₂×#1055	C₂⁴	1	1	yes	characteristic
77	#6094	—	2	4	no	—
78	#6094	—	2	4	no	—
79	C₈×D₁₃	—	2	8	no	—
80	#6092	—	2	8	no	—
81	C₂×#405	—	13	1	no	—
82	#2166	—	13	8	no	—
83	C₂×#364	—	13	2	no	—
84	C₂×#349	—	13	2	no	—
85	C₂×#347	—	13	2	no	—
86	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂²×D₁₃	—	2	1	no	—
88	C₂²×D₁₃	—	2	1	no	—
89	C₂²×D₁₃	—	4	2	no	—
90	C₂²×D₁₃	C₂×M₁₆	1	2	yes	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
94	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
95	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
99	C₂×Dic₁₃	—	2	1	no	—
100	C₂×Dic₁₃	—	2	1	no	—
101	C₂×Dic₁₃	—	2	2	no	—
102	C₂×Dic₁₃	—	4	2	no	—
103	C₂×Dic₁₃	C₂×M₁₆	1	2	yes	—
104	C₄×D₁₃	—	2	2	no	—
105	C₄×D₁₃	—	2	2	no	—
106	C₄×D₁₃	—	2	2	no	—
107	C₄×D₁₃	—	2	2	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
110	#1055	—	2	2	no	—
111	#1055	#117	1	2	yes	—
112	#1055	C₂×#41	1	2	yes	—
113	C₂²×M₁₆	—	13	1	no	—
114	C₂³×C₈	—	13	1	no	—
115	C₂×#117	—	13	1	no	—
116	C₂×M₃₂	—	13	4	no	—
117	#405	—	13	4	no	—
118	#405	—	13	4	no	—
119	C₂×#97	—	13	1	no	—
120	C₂×#97	—	13	1	no	—
121	C₂×C₄×C₈	—	13	1	no	—
122	C₂×C₄×C₈	#261	1	1	yes	characteristic
123	#364	—	13	8	no	—
124	#349	—	13	8	no	—
125	#347	—	13	8	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	#432	1	1	yes	characteristic
128	C₂²×C₁₃	#432	1	1	yes	characteristic
129	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
130	C₂×D₁₃	—	2	2	no	—
131	C₂×D₁₃	—	4	2	no	—
132	C₂×D₁₃	—	4	2	no	—
133	C₂×D₁₃	—	4	2	no	—
134	C₄×C₁₃	—	4	2	no	—
135	C₄×C₁₃	#432	1	1	yes	characteristic
136	C₄×C₁₃	#432	1	1	yes	characteristic
137	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
138	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
139	Dic₁₃	—	2	2	no	—
140	Dic₁₃	—	4	2	no	—
141	C₂³×C₄	—	13	1	no	—
142	C₂×M₁₆	—	13	2	no	—
143	C₂×M₁₆	—	13	2	no	—
144	C₂×M₁₆	—	26	2	no	—
145	C₂×M₁₆	—	26	2	no	—
146	C₂²×C₈	—	13	1	no	—
147	C₂²×C₈	—	13	1	no	—
148	C₂²×C₈	—	13	4	no	—
149	C₂²×C₈	—	26	2	no	—
150	C₂²×C₈	—	26	2	no	—
151	C₂²×C₈	C₂×#261	1	1	yes	characteristic
152	C₂²×C₈	C₂×#261	1	1	yes	characteristic
153	#117	—	13	4	no	—
154	#117	—	26	2	no	—
155	C₂×C₄⋊C₄	—	26	1	no	—
156	C₂×C₂²⋊C₄	—	26	1	no	—
157	C₂×C₄²	—	13	1	no	—
158	C₂×C₄²	C₂×#261	1	1	yes	characteristic
159	M₃₂	—	13	8	no	—
160	M₃₂	—	26	4	no	—
161	C₂×C₁₆	—	26	4	no	—
162	#97	—	13	4	no	—
163	#97	—	13	2	no	—
164	#97	—	13	2	no	—
165	C₄×C₈	—	13	2	no	—
166	C₄×C₈	—	13	2	no	—
167	C₄×C₈	C₂×#261	1	4	yes	—
168	C₂×C₁₃	—	2	1	no	—
169	C₂×C₁₃	—	2	1	no	—
170	C₂×C₁₃	#2166	1	2	yes	complemented
171	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
172	D₁₃	—	4	2	no	—
173	C₂⁴	—	13	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	—	13	1	no	—
176	C₂²×C₄	—	13	2	no	—
177	C₂²×C₄	—	13	2	no	—
178	C₂²×C₄	—	26	1	no	—
179	C₂²×C₄	—	26	1	no	—
180	C₂²×C₄	—	26	1	no	—
181	C₂²×C₄	—	26	1	no	—
182	C₂²×C₄	—	26	1	no	—
183	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
184	M₁₆	—	26	4	no	—
185	M₁₆	—	26	4	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	13	2	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	1	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	2	no	—
193	C₂×C₈	—	26	1	no	—
194	C₂×C₈	—	26	1	no	—
195	C₂×C₈	—	26	8	no	—
196	C₂×C₈	#6118	1	2	yes	—
197	C₂×C₈	#6118	1	2	yes	—
198	C₂×C₈	C₂²×#261	1	2	yes	—
199	C₂×C₈	C₂²×#261	1	2	yes	—
200	C₄⋊C₄	—	26	4	no	—
201	C₂²⋊C₄	—	26	4	no	—
202	C₄²	—	2	1	no	—
203	C₄²	—	13	2	no	—
204	C₄²	—	26	1	no	—
205	C₄²	C₂²×#261	1	2	yes	—
206	C₁₆	—	26	8	no	—
207	C₁₃	C₂×#2166	1	1	yes	characteristic, complemented, Sylow
208	C₂³	—	13	2	no	—
209	C₂³	—	26	1	no	—
210	C₂³	—	26	1	no	—
211	C₂³	—	52	2	no	—
212	C₂³	D₁₃⋊M₁₆	1	1	yes	characteristic
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	13	2	no	—
216	C₂×C₄	—	26	1	no	—
217	C₂×C₄	—	26	1	no	—
218	C₂×C₄	—	26	2	no	—
219	C₂×C₄	—	26	2	no	—
220	C₂×C₄	—	26	2	no	—
221	C₂×C₄	—	26	2	no	—
222	C₂×C₄	—	26	2	no	—
223	C₂×C₄	—	52	2	no	—
224	C₂×C₄	C₂×#6118	1	1	yes	characteristic
225	C₂×C₄	C₂×#6118	1	1	yes	characteristic
226	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
227	C₂×C₄	#88237	1	1	yes	characteristic
228	C₂×C₄	#88237	1	1	yes	characteristic
229	C₂×C₄	C₂³×#261	1	1	yes	characteristic
230	C₈	—	2	2	no	—
231	C₈	—	13	2	no	—
232	C₈	—	13	2	no	—
233	C₈	—	26	2	no	—
234	C₈	C₂×#6118	1	4	yes	—
235	C₂²	—	2	2	no	—
236	C₂²	—	26	2	no	—
237	C₂²	—	52	2	no	—
238	C₂²	—	52	2	no	—
239	C₂²	—	52	2	no	—
240	C₂²	#11719054	1	1	yes	characteristic
241	C₂²	#11719054	1	1	yes	characteristic
242	C₂²	C₂×D₁₃⋊M₁₆	1	1	yes	characteristic
243	C₄	—	4	2	no	—
244	C₄	—	26	2	no	—
245	C₄	—	52	2	no	—
246	C₄	#11719054	1	1	yes	characteristic
247	C₄	#11719054	1	1	yes	characteristic
248	C₄	C₂²×#6118	1	1	yes	characteristic
249	C₄	C₂×#88237	1	1	yes	characteristic
250	C₂	—	2	1	no	—
251	C₂	—	2	1	no	—
252	C₂	—	52	2	no	—
253	C₂	#13131211	1	2	yes	direct product factor
254	C₂	C₂×#11719054	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#88237
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×#261
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle	C₂²×C₁₃	#432
prime radical	C₂³×D₁₃	M₁₆
cosocle	C₂×#1055	C₂⁴
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	C₂×#2166
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
1. #13131211	C₂	direct (N × H)
2. C₂×#11718261	C₂	semidirect
4. C₂×#11718232	C₂	semidirect
17. #11718261	C₂²	semidirect
18. #11718232	C₂²	semidirect
170. C₂×C₁₃	#2166	semidirect
207. C₁₃	C₂×#2166	semidirect
253. C₂	#13131211	direct (N × H)

Maximal subgroups

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

subgroup	index
#13131211	2
C₂×#11718261	2
C₂×#11718263	2
C₂×#11718232	2
C₂×#11718350	2
C₂×#2166	13

Subgroups — C2×#13131211 #16807098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13131211 #16807098