Subgroups — C₂×#13117201 #16021143

Order 3328.

3654 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×Q₈×Dic₁₃	C₂	1	2	yes	maximal
2	#13117201	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₄⋊C₄×Dic₁₃	C₂	1	6	yes	maximal
4	C₂×C₄×C₄⋊Dic₁₃	C₂	1	3	yes	maximal
5	C₂×#11718593	C₂	1	3	yes	maximal
6	C₂²×C₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄×Q₈×C₁₃	C₄	1	16	yes	complemented
9	C₂×C₄×Q₈×C₁₃	C₂²	1	8	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	3	yes	—
11	C₂²×C₄²×C₁₃	C₂²	1	3	yes	—
12	C₂×Q₈×Dic₁₃	C₄	1	32	yes	complemented
13	C₂×Q₈×Dic₁₃	C₂²	1	16	yes	—
14	C₂²×C₄⋊Dic₁₃	C₂²	1	6	yes	—
15	C₂²×C₄×Dic₁₃	C₂²	1	6	yes	—
16	C₂²×C₄×Dic₁₃	C₂²	1	6	yes	—
17	C₂×#88071	C₂²	1	1	yes	characteristic
18	C₂×#88071	C₂²	1	3	yes	—
19	C₂×#88071	C₂²	1	6	yes	—
20	C₄⋊C₄×Dic₁₃	C₂²	1	48	yes	—
21	C₄×C₄⋊Dic₁₃	C₂²	1	24	yes	—
22	#11718593	C₂²	1	24	yes	—
23	C₂²×Q₈×C₁₃	C₂×C₄	1	16	yes	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
29	C₂³×Dic₁₃	C₂³	1	6	yes	—
30	C₂³×Dic₁₃	C₂³	1	2	yes	—
31	C₄×Q₈×C₁₃	C₂×C₄	1	64	yes	complemented
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	24	yes	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
34	C₂×C₄²×C₁₃	C₂×C₄	1	24	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	12	yes	—
36	Q₈×Dic₁₃	C₂×C₄	1	128	yes	complemented
37	C₂×C₄⋊Dic₁₃	C₂×C₄	1	48	yes	—
38	C₂×C₄⋊Dic₁₃	C₂³	1	24	yes	—
39	C₂×C₄×Dic₁₃	—	4	12	no	—
40	C₂×C₄×Dic₁₃	C₂×C₄	1	48	yes	—
41	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
42	C₂×C₄×Dic₁₃	C₂³	1	24	yes	—
43	#88071	C₂³	1	4	yes	—
44	#88071	C₂³	1	12	yes	—
45	#88071	C₂³	1	24	yes	—
46	C₂×#2276	—	13	1	no	maximal, Sylow
47	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₂×Q₈×C₁₃	C₄²	1	64	yes	complemented
49	C₂×Q₈×C₁₃	C₂²×C₄	1	32	yes	—
50	C₂×Q₈×C₁₃	C₂²×C₄	1	16	yes	—
51	C₂²×C₄×C₁₃	—	2	6	no	—
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	2	6	no	—
54	C₂²×C₄×C₁₃	C₂²×C₄	1	24	yes	—
55	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	12	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
61	C₂²×Dic₁₃	—	2	12	no	—
62	C₂²×Dic₁₃	—	2	4	no	—
63	C₂²×Dic₁₃	—	2	4	no	—
64	C₂²×Dic₁₃	—	2	12	no	—
65	C₂²×Dic₁₃	C₂²×C₄	1	8	yes	—
66	C₂²×Dic₁₃	C₂²×C₄	1	24	yes	—
67	C₂²×Dic₁₃	C₂⁴	1	4	yes	—
68	C₂²×Dic₁₃	C₂⁴	1	12	yes	—
69	C₁₃×C₄⋊C₄	C₂²×C₄	1	48	yes	—
70	C₄²×C₁₃	C₂²×C₄	1	48	yes	—
71	C₄⋊Dic₁₃	C₂²×C₄	1	96	yes	—
72	C₄×Dic₁₃	—	4	24	no	—
73	C₄×Dic₁₃	C₂²×C₄	1	96	yes	—
74	C₂²×C₄×Q₈	—	13	2	no	—
75	C₂²×C₄×Q₈	D₁₃	1	1	yes	characteristic
76	#2276	—	13	16	no	—
77	C₂×C₄×C₄⋊C₄	—	13	3	no	—
78	C₂×C₄×C₄⋊C₄	—	13	6	no	—
79	C₂×#376	—	13	3	no	—
80	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
81	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
82	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
83	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
84	C₂³×C₁₃	2₋¹⁺⁴	1	2	yes	—
85	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
86	Q₈×C₁₃	C₂×C₄²	1	64	yes	complemented
87	C₂×C₄×C₁₃	—	2	8	no	—
88	C₂×C₄×C₁₃	—	2	24	no	—
89	C₂×C₄×C₁₃	—	2	6	no	—
90	C₂×C₄×C₁₃	—	2	4	no	—
91	C₂×C₄×C₁₃	—	2	6	no	—
92	C₂×C₄×C₁₃	C₂×C₄²	1	48	yes	—
93	C₂×C₄×C₁₃	C₂³×C₄	1	4	yes	—
94	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	24	yes	—
96	C₂×C₄×C₁₃	C₂³×C₄	1	12	yes	—
97	C₂×Dic₁₃	—	2	16	no	—
98	C₂×Dic₁₃	—	2	48	no	—
99	C₂×Dic₁₃	—	2	12	no	—
100	C₂×Dic₁₃	—	2	4	no	—
101	C₂×Dic₁₃	—	2	4	no	—
102	C₂×Dic₁₃	—	2	12	no	—
103	C₂×Dic₁₃	C₂³×C₄	1	8	yes	—
104	C₂×Dic₁₃	C₂³×C₄	1	24	yes	—
105	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
106	C₂×C₄×Q₈	—	13	32	no	—
107	C₂×C₄×Q₈	—	13	16	no	—
108	C₂×C₄×Q₈	Dic₁₃	1	16	yes	complemented
109	C₂×C₄×Q₈	C₂×D₁₃	1	8	yes	—
110	C₂²×C₄⋊C₄	—	13	6	no	—
111	C₂²×C₄⋊C₄	C₂×D₁₃	1	3	yes	—
112	C₂²×C₄²	—	13	6	no	—
113	C₂²×C₄²	—	13	6	no	—
114	C₂²×C₄²	C₂×D₁₃	1	3	yes	—
115	C₄×C₄⋊C₄	—	13	48	no	—
116	C₄×C₄⋊C₄	—	13	24	no	—
117	#376	—	13	24	no	—
118	C₂×#95	—	13	1	no	—
119	C₂×#95	—	13	3	no	—
120	C₂×#95	—	13	6	no	—
121	C₂²×C₁₃	C₂²×C₄²	1	4	yes	—
122	C₂²×C₁₃	#519	1	4	yes	—
123	C₂²×C₁₃	#519	1	8	yes	—
124	C₂²×C₁₃	#520	1	4	yes	—
125	C₂²×C₁₃	#520	1	8	yes	—
126	C₂²×C₁₃	C₂⁴×C₄	1	2	yes	—
127	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
128	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
129	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
130	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
131	C₄×C₁₃	—	2	4	no	—
132	C₄×C₁₃	—	2	12	no	—
133	C₄×C₁₃	C₂²×C₄²	1	24	yes	—
134	Dic₁₃	—	2	8	no	—
135	Dic₁₃	—	2	24	no	—
136	C₂²×Q₈	C₄×D₁₃	1	16	yes	—
137	C₂²×Q₈	C₂×Dic₁₃	1	8	yes	—
138	C₂²×Q₈	C₂²×D₁₃	1	4	yes	—
139	C₂³×C₄	—	13	6	no	—
140	C₂³×C₄	—	13	2	no	—
141	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
142	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
143	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
144	C₄×Q₈	—	13	128	no	—
145	C₄×Q₈	C₂×Dic₁₃	1	64	yes	complemented
146	C₂×C₄⋊C₄	—	13	48	no	—
147	C₂×C₄⋊C₄	—	13	24	no	—
148	C₂×C₄⋊C₄	C₂×Dic₁₃	1	24	yes	—
149	C₂×C₄⋊C₄	C₂²×D₁₃	1	12	yes	—
150	C₂×C₄²	—	13	48	no	—
151	C₂×C₄²	—	13	24	no	—
152	C₂×C₄²	—	13	24	no	—
153	C₂×C₄²	—	52	12	no	—
154	C₂×C₄²	C₂×Dic₁₃	1	24	yes	—
155	C₂×C₄²	C₂²×D₁₃	1	12	yes	—
156	#95	—	13	4	no	—
157	#95	—	13	24	no	—
158	#95	—	13	12	no	—
159	C₂×C₁₃	#2276	1	8	yes	complemented
160	C₂×C₁₃	C₂³×C₄²	1	1	yes	characteristic
161	C₂×C₁₃	C₂×#519	1	2	yes	—
162	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
163	C₂×C₁₃	C₂×#520	1	2	yes	—
164	C₂×C₁₃	C₂×#520	1	1	yes	characteristic
165	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
166	C₂×Q₈	C₄×Dic₁₃	1	64	yes	complemented
167	C₂×Q₈	C₂×C₄×D₁₃	1	32	yes	—
168	C₂×Q₈	C₂²×Dic₁₃	1	16	yes	—
169	C₂²×C₄	—	2	4	no	—
170	C₂²×C₄	—	2	6	no	—
171	C₂²×C₄	—	2	6	no	—
172	C₂²×C₄	—	13	8	no	—
173	C₂²×C₄	—	13	24	no	—
174	C₂²×C₄	—	13	4	no	—
175	C₂²×C₄	—	13	12	no	—
176	C₂²×C₄	—	26	4	no	—
177	C₂²×C₄	—	26	12	no	—
178	C₂²×C₄	—	26	12	no	—
179	C₂²×C₄	—	26	4	no	—
180	C₂²×C₄	C₂×C₄×D₁₃	1	24	yes	—
181	C₂²×C₄	C₂²×Dic₁₃	1	4	yes	—
182	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
183	C₂²×C₄	C₂²×Dic₁₃	1	12	yes	—
184	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
185	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
186	C₂²×C₄	C₂³×D₁₃	1	6	yes	—
187	C₄⋊C₄	—	13	96	no	—
188	C₄⋊C₄	C₂²×Dic₁₃	1	48	yes	—
189	C₄²	—	13	96	no	—
190	C₄²	—	52	24	no	—
191	C₄²	C₂²×Dic₁₃	1	48	yes	—
192	C₁₃	C₂×#2276	1	1	yes	characteristic, complemented, Sylow
193	C₂³	C₂²×C₄×D₁₃	1	4	yes	—
194	C₂³	#88254	1	4	yes	—
195	C₂³	#88256	1	2	yes	—
196	C₂³	#88257	1	2	yes	—
197	C₂³	C₂³×Dic₁₃	1	2	yes	—
198	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
199	Q₈	C₂×C₄×Dic₁₃	1	64	yes	complemented
200	C₂×C₄	—	2	8	no	—
201	C₂×C₄	—	2	24	no	—
202	C₂×C₄	—	2	4	no	—
203	C₂×C₄	—	2	6	no	—
204	C₂×C₄	—	2	6	no	—
205	C₂×C₄	—	13	8	no	—
206	C₂×C₄	—	13	24	no	—
207	C₂×C₄	—	26	16	no	—
208	C₂×C₄	—	26	48	no	—
209	C₂×C₄	—	26	4	no	—
210	C₂×C₄	—	26	12	no	—
211	C₂×C₄	—	26	12	no	—
212	C₂×C₄	—	26	4	no	—
213	C₂×C₄	C₂×C₄×Dic₁₃	1	48	yes	—
214	C₂×C₄	C₂²×C₄×D₁₃	1	24	yes	—
215	C₂×C₄	C₂³×Dic₁₃	1	4	yes	—
216	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
217	C₂×C₄	C₂³×Dic₁₃	1	12	yes	—
218	C₂²	#11719277	1	8	yes	—
219	C₂²	#11719279	1	8	yes	—
220	C₂²	C₂²×C₄×Dic₁₃	1	4	yes	—
221	C₂²	#11719517	1	4	yes	—
222	C₂²	#11719532	1	4	yes	—
223	C₂²	C₂³×C₄×D₁₃	1	2	yes	—
224	C₂²	C₂×#88254	1	2	yes	—
225	C₂²	C₂×#88256	1	1	yes	characteristic
226	C₂²	C₂×#88257	1	1	yes	characteristic
227	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
228	C₄	—	2	4	no	—
229	C₄	—	2	12	no	—
230	C₄	—	26	8	no	—
231	C₄	—	26	24	no	—
232	C₄	C₂²×C₄×Dic₁₃	1	24	yes	—
233	C₂	#13117201	1	8	yes	direct product factor
234	C₂	C₂³×C₄×Dic₁₃	1	1	yes	characteristic
235	C₂	C₂×#11719277	1	2	yes	—
236	C₂	C₂×#11719517	1	1	yes	characteristic
237	C₂	C₂×#11719279	1	2	yes	—
238	C₂	C₂×#11719532	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator	C₂×C₁₃	C₂³×C₄²
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂²×C₄×Q₈×C₁₃	C₂
socle, prime radical	C₂⁴×C₁₃	C₂⁴
cosocle	C₂³×C₁₃	C₂⁵
hypercenter	C₂²×C₄×Q₈	D₁₃
hypocenter	C₁₃	C₂×#2276
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×Q₈

Complements

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

N	H ≅ G/N	type
2. #13117201	C₂	direct (N × H)
8. C₂×C₄×Q₈×C₁₃	C₄	semidirect
12. C₂×Q₈×Dic₁₃	C₄	semidirect
31. C₄×Q₈×C₁₃	C₂×C₄	semidirect
36. Q₈×Dic₁₃	C₂×C₄	semidirect
48. C₂×Q₈×C₁₃	C₄²	semidirect
86. Q₈×C₁₃	C₂×C₄²	semidirect
108. C₂×C₄×Q₈	Dic₁₃	semidirect
145. C₄×Q₈	C₂×Dic₁₃	semidirect
159. C₂×C₁₃	#2276	semidirect
166. C₂×Q₈	C₄×Dic₁₃	semidirect
192. C₁₃	C₂×#2276	semidirect
199. Q₈	C₂×C₄×Dic₁₃	semidirect
233. C₂	#13117201	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×Q₈×Dic₁₃	2
#13117201	2
C₂×C₄⋊C₄×Dic₁₃	2
C₂×C₄×C₄⋊Dic₁₃	2
C₂×#11718593	2
C₂²×C₄×Q₈×C₁₃	2
C₂×#2276	13

Subgroups — C2×#13117201 #16021143

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117201 #16021143