Subgroups — C₂×#13116922 #16017337

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₂₀₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#11718776	C₂	1	2	yes	maximal, complemented
3	#13116922	C₂	1	8	yes	maximal, direct product factor
4	C₂×#11718096	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718101	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×#88184	—	2	2	no	—
9	C₂²×D₁₀₄	C₂²	1	2	yes	—
10	C₂²×C₁₃×C₁₆	—	2	4	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	complemented
14	#11718776	—	2	4	no	—
15	#11718776	C₂²	1	8	yes	complemented
16	C₂×#88061	—	2	2	no	—
17	C₂×#88058	C₂²	1	2	yes	—
18	C₂×D₂₀₈	—	2	4	no	—
19	C₂×D₂₀₈	C₂²	1	4	yes	complemented
20	#11718101	C₂²	1	8	yes	complemented
21	#11718096	C₂²	1	4	yes	complemented
22	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
23	C₂²×C₂²⋊D₁₃	—	4	2	no	—
24	C₂²×D₅₂	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	—	2	2	no	—
27	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	D₄	1	2	yes	—
29	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
31	#88184	—	2	8	no	—
32	#88184	—	4	4	no	—
33	C₂×#6103	—	4	2	no	—
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₂×D₁₀₄	—	2	4	no	—
36	C₂×D₁₀₄	—	2	4	no	—
37	C₂×D₁₀₄	—	2	2	no	—
38	C₂×D₁₀₄	C₂³	1	4	yes	—
39	C₂×C₁₃×C₁₆	—	2	2	no	—
40	C₂×C₁₃×C₁₆	—	2	8	no	—
41	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
42	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	#88061	—	2	8	no	—
45	#88058	—	2	2	no	—
46	#88058	C₂³	1	4	yes	—
47	D₂₀₈	—	2	8	no	—
48	C₂×#2215	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
50	C₂³×D₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	D₈	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₁₃	—	4	8	no	—
58	C₂×C₂²⋊D₁₃	—	8	8	no	—
59	C₂²×Dic₁₃	—	4	2	no	—
60	C₂×D₅₂	—	2	4	no	—
61	C₂×D₅₂	—	4	4	no	—
62	C₂×D₅₂	—	4	4	no	—
63	C₂×D₅₂	—	4	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	#41	1	2	yes	—
72	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
73	#6103	—	4	8	no	—
74	C₄⋊Dic₁₃	—	2	4	no	—
75	C₄⋊Dic₁₃	—	4	2	no	—
76	D₁₀₄	—	2	4	no	—
77	D₁₀₄	—	2	4	no	—
78	D₁₀₄	—	4	4	no	—
79	C₁₃×C₁₆	—	2	2	no	—
80	C₁₃×C₁₆	C₂×D₄	1	4	yes	complemented
81	C₂²×D₁₆	—	13	1	no	—
82	C₂³×C₁₆	D₁₃	1	1	yes	characteristic, complemented
83	C₂×#466	—	13	2	no	—
84	#2215	—	13	8	no	—
85	C₂×#366	—	13	1	no	—
86	C₂×#357	—	13	2	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	D₁₆	1	4	yes	complemented
90	C₂³×C₁₃	C₂×D₈	1	2	yes	—
91	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
92	C₂²×D₁₃	—	4	4	no	—
93	C₂²×D₁₃	—	8	4	no	—
94	C₂²×D₁₃	—	8	4	no	—
95	C₂²×D₁₃	—	8	4	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	#121	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#135	1	2	yes	—
104	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₂²⋊D₁₃	—	8	16	no	—
106	C₂×Dic₁₃	—	4	4	no	—
107	C₂×Dic₁₃	—	8	4	no	—
108	D₅₂	—	4	4	no	—
109	D₅₂	—	4	4	no	—
110	D₅₂	—	8	4	no	—
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	#121	1	2	yes	—
113	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₂²×D₈	—	13	2	no	—
116	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
117	C₂×#121	—	26	2	no	—
118	C₂×D₁₆	—	13	4	no	—
119	C₂×D₁₆	—	26	4	no	—
120	C₂²×C₁₆	—	2	4	no	—
121	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
123	C₂²×C₁₆	C₂×D₁₃	1	4	yes	complemented
124	#466	—	13	8	no	—
125	#466	—	26	4	no	—
126	C₂×#107	—	13	2	no	—
127	C₂×#102	—	26	2	no	—
128	#366	—	13	4	no	—
129	#357	—	13	8	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	8	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	#466	1	2	yes	—
134	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
135	C₂²×C₁₃	C₂×D₁₆	1	4	yes	complemented
136	C₂²×C₁₃	#508	1	2	yes	—
137	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
138	C₂×D₁₃	—	8	4	no	—
139	C₂×D₁₃	—	8	4	no	—
140	C₂×D₁₃	—	8	4	no	—
141	C₂×D₁₃	—	16	8	no	—
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	#466	1	2	yes	—
144	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
145	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
146	Dic₁₃	—	8	4	no	—
147	C₂²×D₄	—	26	2	no	—
148	C₂²×D₄	—	52	2	no	—
149	C₂³×C₄	D₅₂	1	1	yes	characteristic
150	C₂×D₈	—	13	4	no	—
151	C₂×D₈	—	26	2	no	—
152	C₂×D₈	—	26	4	no	—
153	C₂×D₈	—	26	4	no	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	D₅₂	1	1	yes	characteristic
157	C₂²×C₈	D₅₂	1	2	yes	—
158	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
159	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
160	#121	—	26	8	no	—
161	#121	—	52	4	no	—
162	C₂×C₄⋊C₄	—	26	2	no	—
163	C₂×C₂²⋊C₄	—	52	2	no	—
164	D₁₆	—	26	8	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	8	no	—
167	C₂×C₁₆	C₂²⋊D₁₃	1	4	yes	complemented
168	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
169	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
170	#107	—	13	4	no	—
171	#107	—	26	2	no	—
172	#102	—	26	8	no	—
173	C₂×C₁₃	—	2	4	no	—
174	C₂×C₁₃	#2215	1	4	yes	complemented
175	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
176	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
177	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
178	D₁₃	—	16	4	no	—
179	C₂⁴	—	52	2	no	—
180	C₂⁴	D₁₀₄	1	1	yes	characteristic
181	C₂×D₄	—	26	4	no	—
182	C₂×D₄	—	52	8	no	—
183	C₂×D₄	—	52	2	no	—
184	C₂×D₄	—	52	4	no	—
185	C₂×D₄	—	52	4	no	—
186	C₂×D₄	—	104	8	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	52	2	no	—
190	C₂²×C₄	D₁₀₄	1	2	yes	—
191	C₂²×C₄	D₁₀₄	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₅₂	1	2	yes	—
193	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
194	D₈	—	26	4	no	—
195	D₈	—	26	4	no	—
196	D₈	—	52	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×D₅₂	1	2	yes	—
202	C₂×C₈	C₂×D₅₂	1	1	yes	characteristic
203	C₂×C₈	#6127	1	2	yes	—
204	C₂×C₈	C₂×C₂²⋊D₁₃	1	2	yes	—
205	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
206	C₄⋊C₄	—	26	4	no	—
207	C₄⋊C₄	—	52	2	no	—
208	C₂²⋊C₄	—	52	8	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	C₂×C₂²⋊D₁₃	1	4	yes	complemented
211	C₁₃	C₂×#2215	1	1	yes	characteristic, complemented, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	52	4	no	—
215	C₂³	—	104	4	no	—
216	C₂³	—	104	4	no	—
217	C₂³	—	104	4	no	—
218	C₂³	D₂₀₈	1	4	yes	complemented
219	C₂³	C₂×D₁₀₄	1	2	yes	—
220	C₂³	C₂×D₁₀₄	1	1	yes	characteristic
221	D₄	—	52	4	no	—
222	D₄	—	52	4	no	—
223	D₄	—	104	16	no	—
224	D₄	—	104	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	4	no	—
229	C₂×C₄	—	52	4	no	—
230	C₂×C₄	—	104	4	no	—
231	C₂×C₄	C₂×D₁₀₄	1	2	yes	—
232	C₂×C₄	C₂×D₁₀₄	1	1	yes	characteristic
233	C₂×C₄	#88158	1	2	yes	—
234	C₂×C₄	#88184	1	2	yes	—
235	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	#88184	1	2	yes	—
238	C₈	C₂×#6127	1	1	yes	characteristic
239	C₈	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	104	4	no	—
244	C₂²	—	104	4	no	—
245	C₂²	—	104	4	no	—
246	C₂²	—	208	8	no	—
247	C₂²	C₂×D₂₀₈	1	2	yes	—
248	C₂²	C₂×D₂₀₈	1	4	yes	complemented
249	C₂²	#11718565	1	2	yes	—
250	C₂²	#11718776	1	2	yes	—
251	C₂²	C₂²×D₁₀₄	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	104	4	no	—
254	C₄	#11718776	1	2	yes	—
255	C₄	C₂×#88158	1	1	yes	characteristic
256	C₄	C₂×#88184	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	208	4	no	—
259	C₂	#13116922	1	4	yes	direct product factor
260	C₂	C₂×#11718776	1	1	yes	characteristic
261	C₂	C₂²×D₂₀₈	1	1	yes	characteristic
262	C₂	C₂×#11718565	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₀₄
commutator, cosocle	C₂×C₈×C₁₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂³×C₁₃×C₁₆	C₂
socle	C₂³×C₁₃	C₂×D₈
hypercenter	C₂³×C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2215
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×D₂₀₈	C₂	semidirect
2. C₂×#11718776	C₂	semidirect
3. #13116922	C₂	direct (N × H)
4. C₂×#11718096	C₂	semidirect
5. C₂×#11718101	C₂	semidirect
6. C₂³×C₁₃×C₁₆	C₂	semidirect
13. C₂²×C₁₃×C₁₆	C₂²	semidirect
15. #11718776	C₂²	semidirect
19. C₂×D₂₀₈	C₂²	semidirect
20. #11718101	C₂²	semidirect
21. #11718096	C₂²	semidirect
41. C₂×C₁₃×C₁₆	D₄	semidirect
80. C₁₃×C₁₆	C₂×D₄	semidirect
82. C₂³×C₁₆	D₁₃	semidirect
89. C₂³×C₁₃	D₁₆	semidirect
123. C₂²×C₁₆	C₂×D₁₃	semidirect
135. C₂²×C₁₃	C₂×D₁₆	semidirect
167. C₂×C₁₆	C₂²⋊D₁₃	semidirect
174. C₂×C₁₃	#2215	semidirect
210. C₁₆	C₂×C₂²⋊D₁₃	semidirect
211. C₁₃	C₂×#2215	semidirect
218. C₂³	D₂₀₈	semidirect
248. C₂²	C₂×D₂₀₈	semidirect
259. C₂	#13116922	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×D₂₀₈	2
C₂×#11718776	2
#13116922	2
C₂×#11718096	2
C₂×#11718101	2
C₂³×C₁₃×C₁₆	2
C₂×#2215	13

Subgroups — C2×#13116922 #16017337

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116922 #16017337