Subgroups — C₂×#13114953 #15863062

Order 3328.

1246 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊#1055	C₂	1	2	yes	maximal
2	#13114953	C₂	1	8	yes	maximal, direct product factor
3	C₂×#11718116	C₂	1	4	yes	maximal
4	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₄²×C₁₃	C₄	1	2	yes	—
6	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
9	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×#1055	C₂²	1	4	yes	—
11	C₄³×C₁₃	C₄	1	4	yes	—
12	C₄³×C₁₃	C₂²	1	4	yes	—
13	C₂×C₄⋊#1055	—	2	8	no	—
14	C₂×C₄⋊#1055	C₂²	1	8	yes	—
15	#11718116	C₂²	1	16	yes	—
16	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
18	C₂³×C₄×C₁₃	C₂×C₄	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₄²×C₁₃	—	2	8	no	—
22	C₂×C₄²×C₁₃	—	2	4	no	—
23	C₂×C₄²×C₁₃	—	2	8	no	—
24	C₂×C₄²×C₁₃	C₈	1	8	yes	complemented
25	C₂×C₄²×C₁₃	C₂×C₄	1	8	yes	—
26	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
27	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
29	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
30	C₂×C₄²×C₁₃	D₄	1	4	yes	—
31	C₂×C₄²×C₁₃	Q₈	1	4	yes	—
32	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
33	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
34	C₂²×#1055	—	2	8	no	—
35	C₂²×#1055	—	2	16	no	—
36	C₂²×#1055	C₂³	1	8	yes	—
37	C₄⋊#1055	—	2	16	no	—
38	C₂×#1831	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	C₂²×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	—	2	4	no	—
41	C₂²×C₄×C₁₃	—	2	4	no	—
42	C₂²×C₄×C₁₃	—	2	4	no	—
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
46	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
47	C₂²×C₄×C₁₃	C₂×C₈	1	8	yes	—
48	C₂²×C₄×C₁₃	C₂×C₈	1	4	yes	—
49	C₂²×C₄×C₁₃	M₁₆	1	8	yes	—
50	C₂²×C₄×C₁₃	M₁₆	1	4	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
58	C₂²×C₄×C₁₃	#41	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
60	C₄²×C₁₃	—	2	8	no	—
61	C₄²×C₁₃	—	2	16	no	—
62	C₄²×C₁₃	—	2	8	no	—
63	C₄²×C₁₃	—	2	8	no	—
64	C₄²×C₁₃	C₄⋊C₄	1	8	yes	—
65	C₄²×C₁₃	C₂×C₈	1	16	yes	complemented
66	C₄²×C₁₃	C₂×D₄	1	4	yes	—
67	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
68	C₂×#1055	—	2	8	no	—
69	C₂×#1055	—	2	16	no	—
70	C₂×#1055	—	4	16	no	—
71	C₂²×C₄⋊C₈	—	13	2	no	—
72	C₂×C₄³	D₁₃	1	1	yes	characteristic
73	#1831	—	13	8	no	—
74	C₂×#336	—	13	4	no	—
75	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
76	C₂³×C₁₃	#117	1	4	yes	—
77	C₂³×C₁₃	C₂²×C₈	1	2	yes	—
78	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
79	C₂³×C₁₃	C₂×M₁₆	1	4	yes	—
80	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	—	2	4	no	—
82	C₂×C₄×C₁₃	—	2	4	no	—
83	C₂×C₄×C₁₃	—	2	4	no	—
84	C₂×C₄×C₁₃	—	2	4	no	—
85	C₂×C₄×C₁₃	—	2	4	no	—
86	C₂×C₄×C₁₃	—	2	8	no	—
87	C₂×C₄×C₁₃	—	2	16	no	—
88	C₂×C₄×C₁₃	—	2	16	no	—
89	C₂×C₄×C₁₃	C₄⋊C₈	1	16	yes	—
90	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
91	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
92	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
93	C₂×C₄×C₁₃	#117	1	4	yes	—
94	C₂×C₄×C₁₃	#124	1	8	yes	—
95	C₂×C₄×C₁₃	#125	1	8	yes	—
96	C₂×C₄×C₁₃	C₂²×C₈	1	8	yes	—
97	C₂×C₄×C₁₃	C₂²×C₈	1	4	yes	—
98	C₂×C₄×C₁₃	C₂×M₁₆	1	8	yes	—
99	C₂×C₄×C₁₃	C₂×M₁₆	1	4	yes	—
100	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
103	#1055	—	4	8	no	—
104	C₂³×C₈	—	13	4	no	—
105	C₂²×C₄²	Dic₁₃	1	2	yes	—
106	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
107	C₂²×C₄²	Dic₁₃	1	1	yes	characteristic
108	C₂²×C₄²	C₂×D₁₃	1	2	yes	—
109	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
110	C₂×C₄⋊C₈	—	13	8	no	—
111	C₂×C₄⋊C₈	—	26	8	no	—
112	C₄³	Dic₁₃	1	4	yes	—
113	C₄³	C₂×D₁₃	1	4	yes	—
114	#336	—	13	16	no	—
115	C₂²×C₁₃	#382	1	4	yes	—
116	C₂²×C₁₃	C₂×C₄⋊C₈	1	4	yes	—
117	C₂²×C₁₃	#423	1	4	yes	—
118	C₂²×C₁₃	#431	1	8	yes	—
119	C₂²×C₁₃	#432	1	8	yes	—
120	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
121	C₂²×C₁₃	C₂×#117	1	2	yes	—
122	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
123	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
124	C₂²×C₁₃	C₂²×M₁₆	1	2	yes	—
125	C₄×C₁₃	—	2	4	no	—
126	C₄×C₁₃	—	2	8	no	—
127	C₄×C₁₃	—	2	8	no	—
128	C₄×C₁₃	#382	1	4	yes	—
129	C₄×C₁₃	C₂×C₄⋊C₈	1	8	yes	—
130	C₄×C₁₃	C₂×#124	1	2	yes	—
131	C₄×C₁₃	C₂×#125	1	2	yes	—
132	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
133	C₂³×C₄	C₂×Dic₁₃	1	2	yes	—
134	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
135	C₂³×C₄	C₂×Dic₁₃	1	1	yes	characteristic
136	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
137	C₂²×C₈	—	13	8	no	—
138	C₂²×C₈	—	26	8	no	—
139	C₂²×C₈	—	26	16	no	—
140	C₂×C₄²	—	2	8	no	—
141	C₂×C₄²	—	2	4	no	—
142	C₂×C₄²	—	2	8	no	—
143	C₂×C₄²	#1055	1	8	yes	complemented
144	C₂×C₄²	Dic₂₆	1	4	yes	—
145	C₂×C₄²	D₅₂	1	4	yes	—
146	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
147	C₂×C₄²	C₂×Dic₁₃	1	8	yes	—
148	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
149	C₂×C₄²	C₂×Dic₁₃	1	4	yes	—
150	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
151	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
152	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
153	C₄⋊C₈	—	26	16	no	—
154	C₂×C₁₃	#1831	1	8	yes	complemented
155	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
156	C₂×C₁₃	C₂²×C₄⋊C₈	1	1	yes	characteristic
157	C₂×C₁₃	C₂×#423	1	1	yes	characteristic
158	C₂×C₁₃	C₂×#431	1	2	yes	—
159	C₂×C₁₃	C₂×#432	1	2	yes	—
160	C₂⁴	C₂²×Dic₁₃	1	1	yes	characteristic
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	4	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	4	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₂×#1055	1	8	yes	—
167	C₂²×C₄	C₂×#1055	1	4	yes	—
168	C₂²×C₄	#6099	1	8	yes	—
169	C₂²×C₄	#6099	1	4	yes	—
170	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
171	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
172	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
173	C₂²×C₄	C₂×D₅₂	1	2	yes	—
174	C₂²×C₄	#6127	1	8	yes	—
175	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
176	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
177	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
178	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
179	C₂²×C₄	C₂²×Dic₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
181	C₂×C₈	—	26	8	no	—
182	C₂×C₈	—	26	16	no	—
183	C₂×C₈	—	52	16	no	—
184	C₄²	—	2	16	no	—
185	C₄²	—	2	8	no	—
186	C₄²	—	2	8	no	—
187	C₄²	—	2	8	no	—
188	C₄²	C₂×#1055	1	16	yes	complemented
189	C₄²	C₄⋊Dic₁₃	1	8	yes	—
190	C₄²	C₂×Dic₂₆	1	4	yes	—
191	C₄²	C₂×D₅₂	1	4	yes	—
192	C₁₃	C₂×#1831	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂²×#1055	1	2	yes	—
194	C₂³	C₂×#6099	1	2	yes	—
195	C₂³	C₂×#6099	1	4	yes	—
196	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
197	C₂³	#88180	1	4	yes	—
198	C₂³	C₂³×Dic₁₃	1	1	yes	characteristic
199	C₂×C₄	—	2	4	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	4	no	—
204	C₂×C₄	—	2	8	no	—
205	C₂×C₄	—	2	16	no	—
206	C₂×C₄	—	2	16	no	—
207	C₂×C₄	C₄⋊#1055	1	16	yes	—
208	C₂×C₄	#88124	1	8	yes	—
209	C₂×C₄	#88129	1	8	yes	—
210	C₂×C₄	C₂²×#1055	1	8	yes	—
211	C₂×C₄	C₂²×#1055	1	4	yes	—
212	C₂×C₄	C₂×#6099	1	8	yes	—
213	C₂×C₄	C₂×#6099	1	4	yes	—
214	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
215	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
216	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
217	C₂×C₄	#88180	1	4	yes	—
218	C₂×C₄	C₂²×Dic₂₆	1	1	yes	characteristic
219	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
220	C₂×C₄	C₂×#6127	1	4	yes	—
221	C₈	—	52	8	no	—
222	C₂²	C₂×C₄⋊#1055	1	4	yes	—
223	C₂²	#11718585	1	4	yes	—
224	C₂²	#11718586	1	8	yes	—
225	C₂²	#11718587	1	8	yes	—
226	C₂²	#11718597	1	4	yes	—
227	C₂²	C₂³×#1055	1	1	yes	characteristic
228	C₂²	C₂²×#6099	1	1	yes	characteristic
229	C₂²	C₂²×#6099	1	2	yes	—
230	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
231	C₂²	C₂×#88180	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	2	8	no	—
234	C₄	—	2	8	no	—
235	C₄	C₂×C₄⋊#1055	1	8	yes	—
236	C₄	#11718597	1	4	yes	—
237	C₄	C₂×#88124	1	2	yes	—
238	C₄	C₂×#88129	1	2	yes	—
239	C₂	#13114953	1	8	yes	direct product factor
240	C₂	C₂×#11718597	1	1	yes	characteristic
241	C₂	C₂²×C₄⋊#1055	1	1	yes	characteristic
242	C₂	C₂×#11718585	1	1	yes	characteristic
243	C₂	C₂×#11718586	1	2	yes	—
244	C₂	C₂×#11718587	1	2	yes	—

	subgroup	G/H
center	C₂³×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄³×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂²×C₄
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄³	D₁₃
hypocenter	C₁₃	C₂×#1831
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³×C₄ ◁ C₂×C₄³

Complements

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

N	H ≅ G/N	type
2. #13114953	C₂	direct (N × H)
24. C₂×C₄²×C₁₃	C₈	semidirect
65. C₄²×C₁₃	C₂×C₈	semidirect
143. C₂×C₄²	#1055	semidirect
154. C₂×C₁₃	#1831	semidirect
188. C₄²	C₂×#1055	semidirect
192. C₁₃	C₂×#1831	semidirect
239. C₂	#13114953	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×C₄⋊#1055	2
#13114953	2
C₂×#11718116	2
C₂×C₄³×C₁₃	2
C₂×#1831	13

Subgroups — C2×#13114953 #15863062

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13114953 #15863062