Subgroups — C₂×#13117454 #16021533

Order 3328.

3502 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88180	C₂	1	2	yes	maximal
2	#13117454	C₂	1	16	yes	maximal, direct product factor
3	C₂×#11718646	C₂	1	8	yes	maximal, complemented
4	C₂×#11718873	C₂	1	4	yes	maximal
5	C₂³×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
6	C₂⁴×C₄×C₁₃	C₂²	1	2	yes	—
7	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₁₃×C₄⋊C₄	C₄	1	16	yes	complemented
9	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	8	yes	—
11	C₂²×C₂²⋊Dic₁₃	C₂²	1	4	yes	—
12	C₂×#88180	—	2	16	no	—
13	C₂×#88180	C₂²	1	16	yes	—
14	C₂²×C₄⋊Dic₁₃	C₂²	1	4	yes	—
15	C₂²×C₄⋊Dic₁₃	C₂²	1	8	yes	—
16	C₂²×C₄×Dic₁₃	C₂²	1	4	yes	—
17	#11718873	C₂²	1	32	yes	—
18	C₂×#88071	C₂²	1	8	yes	—
19	#11718646	C₂²	1	64	yes	complemented
20	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
21	C₂³×C₄×C₁₃	—	4	8	no	—
22	C₂³×C₄×C₁₃	C₂×C₄	1	16	yes	—
23	C₂³×C₄×C₁₃	C₂×C₄	1	8	yes	—
24	C₂³×C₄×C₁₃	D₄	1	8	yes	—
25	C₂³×C₄×C₁₃	Q₈	1	8	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	8	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
30	C₂³×Dic₁₃	C₂³	1	8	yes	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	32	yes	—
32	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	64	yes	complemented
33	C₂×C₁₃×C₄⋊C₄	C₂³	1	16	yes	—
34	C₂×C₂²⋊Dic₁₃	—	2	16	no	—
35	C₂×C₂²⋊Dic₁₃	C₂³	1	16	yes	—
36	#88180	—	2	64	no	—
37	C₂×C₄⋊Dic₁₃	—	2	16	no	—
38	C₂×C₄⋊Dic₁₃	—	4	16	no	—
39	C₂×C₄⋊Dic₁₃	C₂³	1	16	yes	—
40	C₂×C₄⋊Dic₁₃	C₂³	1	32	yes	—
41	C₂×C₄×Dic₁₃	—	2	16	no	—
42	C₂×C₄×Dic₁₃	C₂³	1	16	yes	—
43	#88071	C₂³	1	32	yes	—
44	C₂×#2316	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	—	2	4	no	—
46	C₂⁴×C₁₃	—	2	4	no	—
47	C₂⁴×C₁₃	C₂²×C₄	1	4	yes	—
48	C₂⁴×C₁₃	C₂×D₄	1	4	yes	—
49	C₂⁴×C₁₃	C₂×Q₈	1	4	yes	—
50	C₂⁴×C₁₃	C₂⁴	1	2	yes	—
51	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	16	no	—
53	C₂²×C₄×C₁₃	—	4	32	no	—
54	C₂²×C₄×C₁₃	—	4	16	no	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	64	yes	complemented
56	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	32	yes	—
58	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	16	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	16	yes	—
62	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
63	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
64	C₂²×C₄×C₁₃	C₂⁴	1	8	yes	—
65	C₂²×C₄×C₁₃	C₂⁴	1	4	yes	—
66	C₂²×Dic₁₃	—	2	16	no	—
67	C₂²×Dic₁₃	—	2	16	no	—
68	C₂²×Dic₁₃	—	2	16	no	—
69	C₂²×Dic₁₃	C₂⁴	1	16	yes	—
70	C₁₃×C₄⋊C₄	C₂²×C₄	1	64	yes	—
71	C₂²⋊Dic₁₃	—	2	32	no	—
72	C₄⋊Dic₁₃	—	2	32	no	—
73	C₄⋊Dic₁₃	—	4	32	no	—
74	C₄×Dic₁₃	—	2	32	no	—
75	C₂²×#117	—	13	2	no	—
76	C₂³×C₄⋊C₄	D₁₃	1	1	yes	characteristic
77	#2316	—	13	16	no	—
78	C₂×#389	—	13	8	no	—
79	C₂×#380	—	13	4	no	—
80	C₂³×C₁₃	—	2	8	no	—
81	C₂³×C₁₃	—	2	8	no	—
82	C₂³×C₁₃	—	2	16	no	—
83	C₂³×C₁₃	—	2	16	no	—
84	C₂³×C₁₃	—	2	4	no	—
85	C₂³×C₁₃	—	2	4	no	—
86	C₂³×C₁₃	C₂×C₄⋊C₄	1	16	yes	—
87	C₂³×C₁₃	C₂³×C₄	1	4	yes	—
88	C₂³×C₁₃	C₂³×C₄	1	2	yes	—
89	C₂³×C₁₃	C₂²×D₄	1	4	yes	—
90	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
91	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
92	C₂³×C₁₃	C₂²×Q₈	1	2	yes	—
93	C₂³×C₁₃	H(5,𝔽₂)	1	4	yes	—
94	C₂³×C₁₃	2₋¹⁺⁴	1	4	yes	—
95	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
96	C₂×C₄×C₁₃	—	2	32	no	—
97	C₂×C₄×C₁₃	—	2	16	no	—
98	C₂×C₄×C₁₃	—	4	32	no	—
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	32	yes	—
100	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	64	yes	complemented
101	C₂×C₄×C₁₃	C₂³×C₄	1	16	yes	—
102	C₂×C₄×C₁₃	C₂³×C₄	1	8	yes	—
103	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
104	C₂×C₄×C₁₃	C₂²×Q₈	1	8	yes	—
105	C₂×Dic₁₃	—	2	16	no	—
106	C₂×Dic₁₃	—	2	16	no	—
107	C₂×Dic₁₃	—	2	16	no	—
108	C₂×Dic₁₃	—	4	32	no	—
109	C₂⁴×C₄	C₂×D₁₃	1	2	yes	—
110	C₂⁴×C₄	C₂×D₁₃	1	1	yes	characteristic
111	C₂×#117	—	13	16	no	—
112	C₂×#117	—	26	16	no	—
113	C₂²×C₄⋊C₄	—	13	4	no	—
114	C₂²×C₄⋊C₄	—	13	8	no	—
115	C₂²×C₄⋊C₄	Dic₁₃	1	16	yes	complemented
116	C₂²×C₄⋊C₄	C₂×D₁₃	1	4	yes	—
117	C₂²×C₄⋊C₄	C₂×D₁₃	1	8	yes	—
118	C₂²×C₂²⋊C₄	—	13	4	no	—
119	C₂²×C₄²	—	13	4	no	—
120	#389	—	13	64	no	—
121	#380	—	13	32	no	—
122	C₂×#95	—	13	8	no	—
123	C₂²×C₁₃	—	2	32	no	—
124	C₂²×C₁₃	—	2	4	no	—
125	C₂²×C₁₃	—	2	4	no	—
126	C₂²×C₁₃	—	2	8	no	—
127	C₂²×C₁₃	—	2	8	no	—
128	C₂²×C₁₃	C₂²×C₄⋊C₄	1	8	yes	—
129	C₂²×C₁₃	C₂²×C₄⋊C₄	1	4	yes	—
130	C₂²×C₁₃	#518	1	4	yes	—
131	C₂²×C₁₃	#519	1	4	yes	—
132	C₂²×C₁₃	#537	1	8	yes	—
133	C₂²×C₁₃	#544	1	8	yes	—
134	C₂²×C₁₃	C₂⁴×C₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
136	C₂²×C₁₃	C₂³×Q₈	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
138	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	2	yes	—
139	C₄×C₁₃	—	2	16	no	—
140	C₄×C₁₃	C₂²×C₄⋊C₄	1	16	yes	—
141	Dic₁₃	—	4	16	no	—
142	C₂⁵	C₂²×D₁₃	1	1	yes	characteristic
143	C₂³×C₄	—	4	8	no	—
144	C₂³×C₄	—	13	8	no	—
145	C₂³×C₄	Dic₂₆	1	8	yes	—
146	C₂³×C₄	D₅₂	1	8	yes	—
147	C₂³×C₄	C₂×Dic₁₃	1	16	yes	—
148	C₂³×C₄	C₂×Dic₁₃	1	8	yes	—
149	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
150	C₂³×C₄	C₂²×D₁₃	1	8	yes	—
151	C₂³×C₄	C₂²×D₁₃	1	4	yes	—
152	C₂³×C₄	C₂²×D₁₃	1	2	yes	—
153	#117	—	26	64	no	—
154	C₂×C₄⋊C₄	—	13	16	no	—
155	C₂×C₄⋊C₄	—	13	32	no	—
156	C₂×C₄⋊C₄	—	26	16	no	—
157	C₂×C₄⋊C₄	—	52	16	no	—
158	C₂×C₄⋊C₄	C₂×Dic₁₃	1	32	yes	—
159	C₂×C₄⋊C₄	C₂×Dic₁₃	1	64	yes	complemented
160	C₂×C₄⋊C₄	C₂²×D₁₃	1	16	yes	—
161	C₂×C₂²⋊C₄	—	13	16	no	—
162	C₂×C₂²⋊C₄	—	26	16	no	—
163	C₂×C₄²	—	13	16	no	—
164	C₂×C₄²	—	26	16	no	—
165	#95	—	13	32	no	—
166	C₂×C₁₃	—	2	8	no	—
167	C₂×C₁₃	#2316	1	8	yes	complemented
168	C₂×C₁₃	C₂³×C₄⋊C₄	1	1	yes	characteristic
169	C₂×C₁₃	C₂×#518	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#519	1	1	yes	characteristic
171	C₂×C₁₃	C₂×#537	1	2	yes	—
172	C₂×C₁₃	C₂×#544	1	2	yes	—
173	C₂⁴	—	2	4	no	—
174	C₂⁴	—	2	4	no	—
175	C₂⁴	C₂×Dic₂₆	1	4	yes	—
176	C₂⁴	C₂×D₅₂	1	4	yes	—
177	C₂⁴	C₂²×Dic₁₃	1	4	yes	—
178	C₂⁴	C₂³×D₁₃	1	2	yes	—
179	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
180	C₂²×C₄	—	2	16	no	—
181	C₂²×C₄	—	4	32	no	—
182	C₂²×C₄	—	4	16	no	—
183	C₂²×C₄	—	13	16	no	—
184	C₂²×C₄	—	26	16	no	—
185	C₂²×C₄	—	26	16	no	—
186	C₂²×C₄	—	26	16	no	—
187	C₂²×C₄	C₄⋊Dic₁₃	1	64	yes	complemented
188	C₂²×C₄	C₂×Dic₂₆	1	8	yes	—
189	C₂²×C₄	C₂×Dic₂₆	1	16	yes	—
190	C₂²×C₄	C₂×D₅₂	1	8	yes	—
191	C₂²×C₄	C₂×D₅₂	1	16	yes	—
192	C₂²×C₄	C₂²×Dic₁₃	1	16	yes	—
193	C₂²×C₄	C₂²×Dic₁₃	1	32	yes	—
194	C₂²×C₄	C₂²×Dic₁₃	1	8	yes	—
195	C₂²×C₄	C₂²×Dic₁₃	1	16	yes	—
196	C₂²×C₄	C₂³×D₁₃	1	8	yes	—
197	C₂²×C₄	C₂³×D₁₃	1	4	yes	—
198	C₄⋊C₄	—	26	32	no	—
199	C₄⋊C₄	—	52	32	no	—
200	C₄⋊C₄	C₂²×Dic₁₃	1	64	yes	—
201	C₂²⋊C₄	—	26	32	no	—
202	C₄²	—	26	32	no	—
203	C₁₃	C₂×#2316	1	1	yes	characteristic, complemented, Sylow
204	C₂³	—	2	16	no	—
205	C₂³	—	2	16	no	—
206	C₂³	—	2	8	no	—
207	C₂³	—	2	8	no	—
208	C₂³	—	2	4	no	—
209	C₂³	—	2	4	no	—
210	C₂³	C₂×C₄⋊Dic₁₃	1	16	yes	—
211	C₂³	C₂²×Dic₂₆	1	4	yes	—
212	C₂³	C₂²×Dic₂₆	1	2	yes	—
213	C₂³	C₂²×D₅₂	1	4	yes	—
214	C₂³	C₂²×D₅₂	1	2	yes	—
215	C₂³	#88251	1	4	yes	—
216	C₂³	#88254	1	4	yes	—
217	C₂³	C₂³×Dic₁₃	1	4	yes	—
218	C₂³	C₂³×Dic₁₃	1	2	yes	—
219	C₂³	C₂⁴×D₁₃	1	1	yes	characteristic
220	C₂×C₄	—	2	32	no	—
221	C₂×C₄	—	2	16	no	—
222	C₂×C₄	—	4	32	no	—
223	C₂×C₄	—	26	16	no	—
224	C₂×C₄	—	26	16	no	—
225	C₂×C₄	—	26	16	no	—
226	C₂×C₄	—	52	32	no	—
227	C₂×C₄	C₂×C₄⋊Dic₁₃	1	32	yes	—
228	C₂×C₄	C₂×C₄⋊Dic₁₃	1	64	yes	complemented
229	C₂×C₄	C₂²×Dic₂₆	1	8	yes	—
230	C₂×C₄	C₂²×D₅₂	1	8	yes	—
231	C₂×C₄	C₂³×Dic₁₃	1	16	yes	—
232	C₂×C₄	C₂³×Dic₁₃	1	8	yes	—
233	C₂²	—	2	32	no	—
234	C₂²	—	2	8	no	—
235	C₂²	—	2	4	no	—
236	C₂²	—	2	4	no	—
237	C₂²	—	2	8	no	—
238	C₂²	#11719205	1	8	yes	—
239	C₂²	#11719215	1	8	yes	—
240	C₂²	C₂²×C₄⋊Dic₁₃	1	8	yes	—
241	C₂²	C₂²×C₄⋊Dic₁₃	1	4	yes	—
242	C₂²	#11719503	1	4	yes	—
243	C₂²	#11719517	1	4	yes	—
244	C₂²	C₂³×Dic₂₆	1	1	yes	characteristic
245	C₂²	C₂³×D₅₂	1	1	yes	characteristic
246	C₂²	C₂×#88251	1	2	yes	—
247	C₂²	C₂×#88254	1	2	yes	—
248	C₂²	C₂⁴×Dic₁₃	1	1	yes	characteristic
249	C₄	—	2	16	no	—
250	C₄	—	52	16	no	—
251	C₄	C₂²×C₄⋊Dic₁₃	1	16	yes	—
252	C₂	—	2	8	no	—
253	C₂	#13117454	1	8	yes	direct product factor
254	C₂	C₂³×C₄⋊Dic₁₃	1	1	yes	characteristic
255	C₂	C₂×#11719503	1	1	yes	characteristic
256	C₂	C₂×#11719517	1	1	yes	characteristic
257	C₂	C₂×#11719215	1	2	yes	—
258	C₂	C₂×#11719205	1	2	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator	C₂²×C₁₃	C₂⁴×C₄
Frattini	C₂³	C₂⁴×D₁₃
Fitting	C₂³×C₁₃×C₄⋊C₄	C₂
socle	C₂⁴×C₁₃	C₂⁴
prime radical	C₂⁵×C₁₃	C₂³
cosocle	C₂³×C₁₃	C₂⁵
hypercenter	C₂³×C₄⋊C₄	D₁₃
hypocenter	C₁₃	C₂×#2316
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
2. #13117454	C₂	direct (N × H)
3. C₂×#11718646	C₂	semidirect
8. C₂²×C₁₃×C₄⋊C₄	C₄	semidirect
19. #11718646	C₂²	semidirect
32. C₂×C₁₃×C₄⋊C₄	C₂×C₄	semidirect
55. C₂²×C₄×C₁₃	C₄⋊C₄	semidirect
100. C₂×C₄×C₁₃	C₂×C₄⋊C₄	semidirect
115. C₂²×C₄⋊C₄	Dic₁₃	semidirect
159. C₂×C₄⋊C₄	C₂×Dic₁₃	semidirect
167. C₂×C₁₃	#2316	semidirect
187. C₂²×C₄	C₄⋊Dic₁₃	semidirect
203. C₁₃	C₂×#2316	semidirect
228. C₂×C₄	C₂×C₄⋊Dic₁₃	semidirect
253. C₂	#13117454	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×#88180	2
#13117454	2
C₂×#11718646	2
C₂×#11718873	2
C₂³×C₁₃×C₄⋊C₄	2
C₂×#2316	13

Subgroups — C2×#13117454 #16021533

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13117454 #16021533