Subgroups — C₂×#13119865 #16034673

Order 3328.

3182 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88142	C₂	1	4	yes	maximal
2	#13119865	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718640	C₂	1	4	yes	maximal
4	C₂×#11718634	C₂	1	4	yes	maximal
5	C₂×C₄×C₄⋊Dic₁₃	C₂	1	2	yes	maximal
6	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
7	C₂³×Dic₂₆	C₂²	1	4	yes	—
8	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
12	C₂²×C₄⋊Dic₁₃	C₂²	1	2	yes	—
13	C₂²×#6101	C₂²	1	8	yes	—
14	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	—
15	C₂×#88142	—	2	32	no	—
16	C₂×#88142	C₂²	1	32	yes	—
17	C₁₃×#384	C₂²	1	8	yes	—
18	C₂×#88071	C₂²	1	4	yes	—
19	C₂×#88071	C₂²	1	4	yes	—
20	#11718640	C₂²	1	32	yes	—
21	#11718634	C₂²	1	32	yes	—
22	C₄×C₄⋊Dic₁₃	C₂²	1	16	yes	—
23	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
24	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
25	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
26	C₂³×Dic₁₃	C₂³	1	4	yes	—
27	C₂³×Dic₁₃	C₂³	1	4	yes	—
28	C₂²×Dic₂₆	—	2	32	no	—
29	C₂²×Dic₂₆	—	4	8	no	—
30	C₂²×Dic₂₆	C₂³	1	16	yes	—
31	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
32	C₂×C₁₃×C₄⋊C₄	—	4	4	no	—
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
34	C₂×C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
35	C₂×C₄²×C₁₃	Q₈	1	8	yes	complemented
36	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
37	C₂×C₄⋊Dic₁₃	D₄	1	16	yes	—
38	C₂×C₄⋊Dic₁₃	Q₈	1	16	yes	complemented
39	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	8	yes	—
41	C₂×#6101	—	2	32	no	—
42	C₂×#6101	C₂³	1	32	yes	—
43	C₂×C₄×Dic₁₃	—	2	16	no	—
44	C₂×C₄×Dic₁₃	C₂³	1	16	yes	—
45	#88142	—	2	128	no	—
46	#88071	C₂³	1	16	yes	—
47	#88071	C₂³	1	16	yes	—
48	C₂×#2551	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	16	no	—
53	C₂²×C₄×C₁₃	—	2	8	no	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
55	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
57	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
58	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
61	C₂²×Dic₁₃	—	2	8	no	—
62	C₂²×Dic₁₃	—	2	8	no	—
63	C₂²×Dic₁₃	—	2	8	no	—
64	C₂²×Dic₁₃	—	2	8	no	—
65	C₂²×Dic₁₃	C₂×D₄	1	16	yes	—
66	C₂²×Dic₁₃	C₂×Q₈	1	16	yes	—
67	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
68	C₂²×Dic₁₃	C₂⁴	1	8	yes	—
69	C₂×Dic₂₆	—	2	64	no	—
70	C₂×Dic₂₆	—	4	16	no	—
71	C₂×Dic₂₆	—	4	64	no	—
72	C₁₃×C₄⋊C₄	—	2	32	no	—
73	C₁₃×C₄⋊C₄	—	4	8	no	—
74	C₄²×C₁₃	C₂×Q₈	1	16	yes	complemented
75	C₄⋊Dic₁₃	C₂×D₄	1	32	yes	—
76	C₄⋊Dic₁₃	C₂×Q₈	1	32	yes	complemented
77	#6101	—	2	64	no	—
78	C₄×Dic₁₃	—	2	32	no	—
79	C₂²×C₄⋊Q₈	—	13	4	no	—
80	#2551	—	13	16	no	—
81	C₂×#395	—	13	4	no	—
82	C₂×#389	—	13	4	no	—
83	C₂×#384	D₁₃	1	1	yes	characteristic
84	C₂×C₄×C₄⋊C₄	—	13	2	no	—
85	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
86	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
87	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
88	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
89	C₂³×C₁₃	2₋¹⁺⁴	1	2	yes	—
90	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
91	C₂×C₄×C₁₃	—	2	8	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	16	no	—
95	C₂×C₄×C₁₃	—	2	8	no	—
96	C₂×C₄×C₁₃	—	4	16	no	—
97	C₂×C₄×C₁₃	C₄⋊Q₈	1	32	yes	complemented
98	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
99	C₂×C₄×C₁₃	C₂²×Q₈	1	4	yes	—
100	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
101	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
102	C₂×Dic₁₃	—	2	32	no	—
103	C₂×Dic₁₃	—	2	32	no	—
104	C₂×Dic₁₃	—	2	8	no	—
105	C₂×Dic₁₃	—	2	8	no	—
106	C₂×Dic₁₃	—	2	8	no	—
107	C₂×Dic₁₃	—	2	8	no	—
108	C₂×Dic₁₃	C₂²×D₄	1	16	yes	—
109	C₂×Dic₁₃	C₂²×Q₈	1	16	yes	—
110	Dic₂₆	—	4	64	no	—
111	C₂³×Q₈	—	13	4	no	—
112	C₂×C₄⋊Q₈	—	13	32	no	—
113	C₂×C₄⋊Q₈	—	26	32	no	—
114	C₂²×C₄⋊C₄	—	13	2	no	—
115	C₂²×C₄⋊C₄	—	13	2	no	—
116	C₂²×C₄⋊C₄	—	13	8	no	—
117	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
118	C₂²×C₄⋊C₄	C₂×D₁₃	1	2	yes	—
119	C₂²×C₄²	—	13	4	no	—
120	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
121	#395	—	13	32	no	—
122	#389	—	13	32	no	—
123	#384	C₂×D₁₃	1	8	yes	—
124	C₄×C₄⋊C₄	—	13	16	no	—
125	C₂×#95	—	13	4	no	—
126	C₂×#95	—	13	4	no	—
127	C₂²×C₁₃	C₂×C₄⋊Q₈	1	8	yes	—
128	C₂²×C₁₃	#544	1	4	yes	—
129	C₂²×C₁₃	D₄²	1	4	yes	—
130	C₂²×C₁₃	D₄×Q₈	1	8	yes	—
131	C₂²×C₁₃	Q₈²	1	4	yes	—
132	C₂²×C₁₃	C₂³×D₄	1	2	yes	—
133	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
134	C₂²×C₁₃	C₂³×Q₈	1	2	yes	—
135	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	4	8	no	—
139	C₄×C₁₃	C₂×C₄⋊Q₈	1	16	yes	complemented
140	Dic₁₃	—	2	16	no	—
141	Dic₁₃	—	2	16	no	—
142	C₂²×Q₈	—	13	16	no	—
143	C₂²×Q₈	—	26	32	no	—
144	C₂²×Q₈	—	52	8	no	—
145	C₂³×C₄	—	13	4	no	—
146	C₂³×C₄	—	13	4	no	—
147	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
148	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
149	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
150	C₄⋊Q₈	—	26	128	no	—
151	C₂×C₄⋊C₄	—	2	16	no	—
152	C₂×C₄⋊C₄	—	4	4	no	—
153	C₂×C₄⋊C₄	—	13	16	no	—
154	C₂×C₄⋊C₄	—	13	16	no	—
155	C₂×C₄⋊C₄	—	13	8	no	—
156	C₂×C₄⋊C₄	—	13	8	no	—
157	C₂×C₄⋊C₄	—	13	32	no	—
158	C₂×C₄⋊C₄	—	26	32	no	—
159	C₂×C₄⋊C₄	C₂²×D₁₃	1	16	yes	—
160	C₂×C₄⋊C₄	C₂²×D₁₃	1	8	yes	—
161	C₂×C₄²	—	13	16	no	—
162	C₂×C₄²	—	26	16	no	—
163	C₂×C₄²	Dic₂₆	1	8	yes	complemented
164	C₂×C₄²	C₂²×D₁₃	1	4	yes	—
165	#95	—	13	16	no	—
166	#95	—	13	16	no	—
167	C₂×C₁₃	#2551	1	8	yes	complemented
168	C₂×C₁₃	C₂²×C₄⋊Q₈	1	2	yes	—
169	C₂×C₁₃	C₂×#544	1	1	yes	characteristic
170	C₂×C₁₃	C₂×D₄²	1	1	yes	characteristic
171	C₂×C₁₃	C₂×D₄×Q₈	1	2	yes	—
172	C₂×C₁₃	C₂×Q₈²	1	1	yes	characteristic
173	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
174	C₂×Q₈	—	26	64	no	—
175	C₂×Q₈	—	52	16	no	—
176	C₂×Q₈	—	52	64	no	—
177	C₂²×C₄	—	2	2	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	16	no	—
180	C₂²×C₄	—	2	8	no	—
181	C₂²×C₄	—	13	16	no	—
182	C₂²×C₄	—	13	16	no	—
183	C₂²×C₄	—	13	8	no	—
184	C₂²×C₄	—	13	8	no	—
185	C₂²×C₄	—	26	8	no	—
186	C₂²×C₄	—	26	8	no	—
187	C₂²×C₄	—	26	8	no	—
188	C₂²×C₄	—	26	8	no	—
189	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
190	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
191	C₂²×C₄	D₄×D₁₃	1	8	yes	—
192	C₂²×C₄	Q₈×D₁₃	1	8	yes	—
193	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
194	C₂²×C₄	C₂³×D₁₃	1	2	yes	—
195	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
196	C₄⋊C₄	—	2	32	no	—
197	C₄⋊C₄	—	4	8	no	—
198	C₄⋊C₄	—	13	32	no	—
199	C₄⋊C₄	—	13	32	no	—
200	C₄⋊C₄	—	26	64	no	—
201	C₄²	—	26	32	no	—
202	C₄²	C₂×Dic₂₆	1	16	yes	complemented
203	C₁₃	C₂×#2551	1	1	yes	characteristic, complemented, Sylow
204	C₂³	C₂²×Dic₂₆	1	2	yes	—
205	C₂³	C₂×D₄×D₁₃	1	4	yes	—
206	C₂³	#88251	1	2	yes	—
207	C₂³	C₂×Q₈×D₁₃	1	4	yes	—
208	C₂³	#88254	1	2	yes	—
209	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
210	Q₈	—	52	64	no	—
211	C₂×C₄	—	2	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	16	no	—
215	C₂×C₄	—	2	8	no	—
216	C₂×C₄	—	4	16	no	—
217	C₂×C₄	—	13	16	no	—
218	C₂×C₄	—	13	16	no	—
219	C₂×C₄	—	26	32	no	—
220	C₂×C₄	—	26	8	no	—
221	C₂×C₄	—	26	8	no	—
222	C₂×C₄	—	26	8	no	—
223	C₂×C₄	—	26	8	no	—
224	C₂×C₄	—	26	32	no	—
225	C₂×C₄	#88142	1	32	yes	complemented
226	C₂×C₄	C₂²×Dic₂₆	1	4	yes	—
227	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
228	C₂×C₄	C₂×D₄×D₁₃	1	8	yes	—
229	C₂×C₄	C₂×Q₈×D₁₃	1	8	yes	—
230	C₂²	C₂×#88142	1	8	yes	—
231	C₂²	#11719205	1	4	yes	—
232	C₂²	#11719422	1	4	yes	—
233	C₂²	#11719427	1	4	yes	—
234	C₂²	#11719432	1	8	yes	—
235	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
236	C₂²	C₂²×D₄×D₁₃	1	2	yes	—
237	C₂²	C₂×#88251	1	1	yes	characteristic
238	C₂²	C₂²×Q₈×D₁₃	1	2	yes	—
239	C₂²	C₂×#88254	1	1	yes	characteristic
240	C₄	—	2	4	no	—
241	C₄	—	4	8	no	—
242	C₄	—	26	16	no	—
243	C₄	—	26	16	no	—
244	C₄	C₂×#88142	1	16	yes	complemented
245	C₂	#13119865	1	8	yes	direct product factor
246	C₂	C₂²×#88142	1	2	yes	—
247	C₂	C₂×#11719205	1	1	yes	characteristic
248	C₂	C₂×#11719422	1	1	yes	characteristic
249	C₂	C₂×#11719432	1	2	yes	—
250	C₂	C₂×#11719427	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂³×C₁₃	C₂⁵
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂×C₁₃×#384	C₂
socle, prime radical	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂×#384	D₁₃
hypocenter	C₁₃	C₂×#2551
LCS	G ▷ C₂³×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂×#384

Complements

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

N	H ≅ G/N	type
2. #13119865	C₂	direct (N × H)
35. C₂×C₄²×C₁₃	Q₈	semidirect
38. C₂×C₄⋊Dic₁₃	Q₈	semidirect
74. C₄²×C₁₃	C₂×Q₈	semidirect
76. C₄⋊Dic₁₃	C₂×Q₈	semidirect
97. C₂×C₄×C₁₃	C₄⋊Q₈	semidirect
139. C₄×C₁₃	C₂×C₄⋊Q₈	semidirect
163. C₂×C₄²	Dic₂₆	semidirect
167. C₂×C₁₃	#2551	semidirect
202. C₄²	C₂×Dic₂₆	semidirect
203. C₁₃	C₂×#2551	semidirect
225. C₂×C₄	#88142	semidirect
244. C₄	C₂×#88142	semidirect
245. C₂	#13119865	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×#88142	2
#13119865	2
C₂×#11718640	2
C₂×#11718634	2
C₂×C₄×C₄⋊Dic₁₃	2
C₂×C₁₃×#384	2
C₂×#2551	13

Subgroups — C2×#13119865 #16034673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13119865 #16034673