Subgroups — C₂×#13116669 #16016084

Order 3328.

742 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×#88038	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₅₂	C₂	1	1	yes	maximal, characteristic
3	#13116669	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#11718099	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₁₃	C₂²	1	2	yes	—
8	C₂×C₄×D₅₂	C₄	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×#88060	C₄	1	2	yes	—
13	C₂²×#6090	C₂²	1	2	yes	—
14	C₂×C₄⋊#1055	C₄	1	1	yes	characteristic
15	C₂×#88038	—	2	8	no	—
16	C₂×#88038	C₂²	1	8	yes	complemented
17	C₈×D₅₂	C₄	1	4	yes	—
18	C₈×D₅₂	C₂²	1	4	yes	—
19	C₄×C₁₃×C₁₆	C₂²	1	4	yes	complemented
20	#11718099	C₂²	1	8	yes	complemented
21	C₄⋊#6090	C₂²	1	4	yes	complemented
22	C₂²×D₅₂	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₁₃	C₂×C₄	1	2	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#6103	C₈	1	2	yes	—
28	C₂×C₄⋊Dic₁₃	C₈	1	1	yes	characteristic
29	C₂²×#1055	C₂×C₄	1	2	yes	—
30	C₂×C₈×D₁₃	—	2	4	no	—
31	C₂×C₈×D₁₃	—	2	4	no	—
32	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
33	C₂×C₈×D₁₃	C₂³	1	4	yes	—
34	C₄×D₅₂	C₈	1	4	yes	—
35	C₄×D₅₂	C₂×C₄	1	4	yes	—
36	C₂×C₁₃×C₁₆	—	2	2	no	—
37	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
38	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
39	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
40	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
41	C₄×C₈×C₁₃	C₂³	1	2	yes	—
42	#88060	C₂×C₄	1	8	yes	—
43	C₂×#6090	—	2	4	no	—
44	C₂×#6090	C₂³	1	4	yes	—
45	C₄⋊#1055	C₂×C₄	1	4	yes	—
46	#88038	—	2	16	no	—
47	C₂×#2169	—	13	1	no	maximal, Sylow
48	C₂³×D₁₃	C₂×C₈	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
53	C₂×D₅₂	—	2	4	no	—
54	C₂×D₅₂	C₂×C₈	1	4	yes	—
55	C₂×C₄×D₁₃	—	2	4	no	—
56	C₂×C₄×D₁₃	—	2	4	no	—
57	C₂×C₄×D₁₃	C₂×C₈	1	4	yes	—
58	C₂×C₄×D₁₃	C₂²×C₄	1	4	yes	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
63	C₂×C₈×C₁₃	#41	1	2	yes	—
64	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
65	C₄²×C₁₃	C₂×C₈	1	2	yes	—
66	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
67	#6103	C₂×C₈	1	8	yes	—
68	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
69	C₂×#1055	—	2	4	no	—
70	C₂×#1055	C₂²×C₄	1	4	yes	—
71	C₈×D₁₃	—	2	8	no	—
72	C₈×D₁₃	—	2	4	no	—
73	C₈×D₁₃	—	2	4	no	—
74	C₁₃×C₁₆	—	2	2	no	—
75	C₁₃×C₁₆	C₂×D₄	1	4	yes	complemented
76	#6090	—	2	4	no	—
77	C₂²×M₃₂	—	13	2	no	—
78	C₂×C₈×D₄	—	13	1	no	—
79	#2169	—	13	8	no	—
80	C₂×C₄⋊C₁₆	—	13	1	no	—
81	C₂×C₂²⋊C₁₆	—	13	2	no	—
82	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic, complemented
83	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
84	C₂²×D₁₃	—	2	4	no	—
85	C₂²×D₁₃	—	2	4	no	—
86	C₂²×D₁₃	—	2	4	no	—
87	C₂²×D₁₃	C₂²×C₈	1	4	yes	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	M₃₂	1	4	yes	complemented
90	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
91	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
92	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
94	C₂×C₄×C₁₃	#131	1	2	yes	—
95	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
96	C₂×Dic₁₃	—	2	4	no	—
97	C₂×Dic₁₃	C₂²×C₈	1	4	yes	—
98	D₅₂	—	2	8	no	—
99	C₄×D₁₃	—	2	4	no	—
100	C₄×D₁₃	—	2	4	no	—
101	C₄×D₁₃	—	2	8	no	—
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	C₄×D₄	1	2	yes	—
104	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
106	#1055	—	2	4	no	—
107	C₂³×C₈	—	13	2	no	—
108	C₂×C₄×D₄	—	13	1	no	—
109	C₂×M₃₂	—	13	8	no	—
110	C₂×M₃₂	—	26	8	no	—
111	C₂²×C₁₆	—	13	2	no	—
112	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
113	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
114	C₈×D₄	—	13	4	no	—
115	C₈×D₄	—	13	4	no	—
116	C₂×C₄⋊C₈	—	13	1	no	—
117	C₂×C₂²⋊C₈	—	13	2	no	—
118	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
119	C₄⋊C₁₆	—	13	4	no	—
120	C₂²⋊C₁₆	—	13	8	no	—
121	C₄×C₁₆	C₂×D₁₃	1	4	yes	complemented
122	C₂²×C₁₃	C₈×D₄	1	2	yes	—
123	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
124	C₂²×C₁₃	#504	1	2	yes	—
125	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
126	C₂×D₁₃	—	2	4	no	—
127	C₂×D₁₃	—	2	4	no	—
128	C₂×D₁₃	—	2	4	no	—
129	C₂×D₁₃	—	4	8	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	C₈×D₄	1	2	yes	—
132	C₄×C₁₃	C₂×M₃₂	1	4	yes	complemented
133	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
134	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
135	Dic₁₃	—	2	4	no	—
136	C₂²×D₄	—	13	1	no	—
137	C₂³×C₄	—	13	2	no	—
138	C₂²×C₈	—	13	2	no	—
139	C₂²×C₈	—	13	4	no	—
140	C₂²×C₈	—	13	4	no	—
141	C₂²×C₈	—	26	4	no	—
142	C₂²×C₈	—	26	4	no	—
143	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
144	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
145	C₄×D₄	—	13	4	no	—
146	C₄×D₄	—	13	4	no	—
147	C₂×C₄⋊C₄	—	13	1	no	—
148	C₂×C₂²⋊C₄	—	13	2	no	—
149	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
150	M₃₂	—	26	16	no	—
151	C₂×C₁₆	—	2	2	no	—
152	C₂×C₁₆	—	13	4	no	—
153	C₂×C₁₆	—	26	4	no	—
154	C₂×C₁₆	D₅₂	1	4	yes	complemented
155	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
156	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
157	C₄⋊C₈	—	13	4	no	—
158	C₂²⋊C₈	—	13	8	no	—
159	C₄×C₈	C₄×D₁₃	1	2	yes	—
160	C₄×C₈	C₂²×D₁₃	1	2	yes	—
161	C₂×C₁₃	#2169	1	4	yes	complemented
162	C₂×C₁₃	C₂×C₈×D₄	1	1	yes	characteristic
163	C₂×C₁₃	C₂²×M₃₂	1	1	yes	characteristic
164	C₂×C₁₃	C₂×#504	1	1	yes	characteristic
165	D₁₃	—	4	4	no	—
166	C₂⁴	—	13	2	no	—
167	C₂×D₄	—	13	4	no	—
168	C₂×D₄	—	26	4	no	—
169	C₂²×C₄	—	13	2	no	—
170	C₂²×C₄	—	13	4	no	—
171	C₂²×C₄	—	13	4	no	—
172	C₂²×C₄	—	26	4	no	—
173	C₂²×C₄	—	26	4	no	—
174	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
175	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
177	C₂×C₈	—	2	2	no	—
178	C₂×C₈	—	13	4	no	—
179	C₂×C₈	—	26	4	no	—
180	C₂×C₈	—	26	8	no	—
181	C₂×C₈	—	26	4	no	—
182	C₂×C₈	—	26	4	no	—
183	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
184	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
185	C₂×C₈	C₂×D₅₂	1	2	yes	—
186	C₂×C₈	#6127	1	2	yes	—
187	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
188	C₄⋊C₄	—	13	4	no	—
189	C₂²⋊C₄	—	13	8	no	—
190	C₄²	C₈×D₁₃	1	2	yes	—
191	C₄²	C₂×C₄×D₁₃	1	2	yes	—
192	C₁₆	—	2	2	no	—
193	C₁₆	—	26	4	no	—
194	C₁₆	C₂×D₅₂	1	4	yes	complemented
195	C₁₃	C₂×#2169	1	1	yes	characteristic, complemented, Sylow
196	C₂³	—	13	4	no	—
197	C₂³	—	26	4	no	—
198	C₂³	—	26	4	no	—
199	C₂³	—	26	4	no	—
200	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
201	D₄	—	26	8	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	13	4	no	—
204	C₂×C₄	—	26	4	no	—
205	C₂×C₄	—	26	4	no	—
206	C₂×C₄	—	26	4	no	—
207	C₂×C₄	—	26	8	no	—
208	C₂×C₄	#88038	1	4	yes	complemented
209	C₂×C₄	C₄×D₅₂	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
212	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
213	C₂×C₄	#88155	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	26	4	no	—
217	C₈	C₄×D₅₂	1	2	yes	—
218	C₈	C₂²×D₅₂	1	1	yes	characteristic
219	C₈	C₂×#6127	1	1	yes	characteristic
220	C₂²	—	26	4	no	—
221	C₂²	—	26	4	no	—
222	C₂²	—	26	4	no	—
223	C₂²	—	52	8	no	—
224	C₂²	C₈×D₅₂	1	2	yes	—
225	C₂²	C₂×#88038	1	2	yes	—
226	C₂²	#11718562	1	2	yes	—
227	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	26	4	no	—
230	C₄	C₈×D₅₂	1	2	yes	—
231	C₄	C₂×#88038	1	4	yes	complemented
232	C₄	C₂×C₄×D₅₂	1	1	yes	characteristic
233	C₄	C₂×#88155	1	1	yes	characteristic
234	C₂	—	52	4	no	—
235	C₂	#13116669	1	4	yes	direct product factor
236	C₂	C₂×C₈×D₅₂	1	1	yes	characteristic
237	C₂	C₂²×#88038	1	1	yes	characteristic
238	C₂	C₂×#11718562	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂²×D₅₂	C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	C₂×#2169
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#88038	C₂	semidirect
3. #13116669	C₂	direct (N × H)
4. C₂×C₄⋊#6090	C₂	semidirect
5. C₂×#11718099	C₂	semidirect
6. C₂×C₄×C₁₃×C₁₆	C₂	semidirect
16. C₂×#88038	C₂²	semidirect
19. C₄×C₁₃×C₁₆	C₂²	semidirect
20. #11718099	C₂²	semidirect
21. C₄⋊#6090	C₂²	semidirect
37. C₂×C₁₃×C₁₆	D₄	semidirect
75. C₁₃×C₁₆	C₂×D₄	semidirect
82. C₂×C₄×C₁₆	D₁₃	semidirect
89. C₂×C₄×C₁₃	M₃₂	semidirect
121. C₄×C₁₆	C₂×D₁₃	semidirect
132. C₄×C₁₃	C₂×M₃₂	semidirect
154. C₂×C₁₆	D₅₂	semidirect
161. C₂×C₁₃	#2169	semidirect
194. C₁₆	C₂×D₅₂	semidirect
195. C₁₃	C₂×#2169	semidirect
208. C₂×C₄	#88038	semidirect
231. C₄	C₂×#88038	semidirect
235. C₂	#13116669	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×#88038	2
C₂×C₈×D₅₂	2
#13116669	2
C₂×C₄⋊#6090	2
C₂×#11718099	2
C₂×C₄×C₁₃×C₁₆	2
C₂×#2169	13

Subgroups — C2×#13116669 #16016084

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13116669 #16016084