Subgroups — C₂×#11718788 #13123513

Order 1664.

564 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×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₂²⋊D₁₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊#1055	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718788	C₂	1	8	yes	maximal, direct product factor
6	C₂×#88060	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88055	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#88054	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₂²⋊D₁₃	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	—	2	4	no	—
13	C₂×C₁₃×M₁₆	C₂²	1	4	yes	complemented
14	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₂×C₂²⋊Dic₁₃	C₄	1	1	yes	characteristic
17	C₄×C₂²⋊D₁₃	C₄	1	4	yes	—
18	C₄×C₂²⋊D₁₃	C₂²	1	4	yes	—
19	C₂×#6103	C₄	1	1	yes	characteristic
20	C₂×#6101	C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
22	C₂²×#1055	C₂²	1	1	yes	characteristic
23	C₂²×#1055	C₂²	1	1	yes	characteristic
24	C₂×C₈×D₁₃	—	2	4	no	—
25	C₂×C₈×D₁₃	C₂²	1	4	yes	complemented
26	C₂²⋊#1055	C₂²	1	4	yes	complemented
27	#88060	C₂²	1	4	yes	complemented
28	#88055	C₂²	1	4	yes	complemented
29	#88054	C₂²	1	4	yes	complemented
30	C₂⁴×C₁₃	C₂×C₄	1	1	yes	characteristic
31	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	—	2	2	no	—
34	C₂²×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
38	C₂×C₂²⋊D₁₃	—	2	4	no	—
39	C₂×C₂²⋊D₁₃	C₂×C₄	1	4	yes	—
40	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄×D₁₃	—	2	2	no	—
44	C₂×C₄×D₁₃	—	2	2	no	—
45	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
46	C₂×C₄×D₁₃	C₂³	1	2	yes	—
47	C₁₃×M₁₆	—	2	8	no	—
48	C₂×C₈×C₁₃	—	2	2	no	—
49	C₂×C₈×C₁₃	D₄	1	4	yes	complemented
50	C₂×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂×C₈×C₁₃	C₂³	1	2	yes	—
52	C₂²⋊Dic₁₃	C₂×C₄	1	4	yes	—
53	#6103	C₂×C₄	1	4	yes	—
54	#6101	C₂×C₄	1	4	yes	—
55	C₄×Dic₁₃	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₃	C₂³	1	2	yes	—
57	C₂×#1055	—	2	2	no	—
58	C₂×#1055	—	2	2	no	—
59	C₂×#1055	C₂³	1	2	yes	—
60	C₂×#1055	C₂³	1	2	yes	—
61	C₈×D₁₃	—	2	8	no	—
62	C₂×#435	—	13	1	no	maximal, Sylow
63	C₂³×C₁₃	—	2	2	no	—
64	C₂³×C₁₃	—	2	2	no	—
65	C₂³×C₁₃	—	2	2	no	—
66	C₂³×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×D₁₃	—	2	2	no	—
69	C₂²×D₁₃	—	2	2	no	—
70	C₂²×D₁₃	M₁₆	1	4	yes	complemented
71	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₄×C₁₃	—	2	2	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₁₃	#41	1	2	yes	—
80	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₁₃	—	2	8	no	—
82	C₂×Dic₁₃	—	2	2	no	—
83	C₂×Dic₁₃	—	2	2	no	—
84	C₂×Dic₁₃	—	2	2	no	—
85	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
87	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
88	C₄×D₁₃	—	2	4	no	—
89	C₄×D₁₃	—	2	2	no	—
90	C₄×D₁₃	—	2	2	no	—
91	C₈×C₁₃	—	2	2	no	—
92	C₈×C₁₃	C₂×D₄	1	4	yes	complemented
93	#1055	—	2	2	no	—
94	#1055	—	2	2	no	—
95	C₂²×M₁₆	D₁₃	1	1	yes	characteristic, complemented
96	C₂³×C₈	—	13	1	no	—
97	C₂×C₄×D₄	—	13	1	no	—
98	#435	—	13	8	no	—
99	C₂×C₄⋊C₈	—	13	1	no	—
100	C₂×C₂²⋊C₈	—	13	1	no	—
101	C₂×C₂²⋊C₈	—	13	1	no	—
102	C₂×#97	—	13	1	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	2	2	no	—
105	C₂²×C₁₃	—	2	2	no	—
106	C₂²×C₁₃	—	4	4	no	—
107	C₂²×C₁₃	C₄×D₄	1	2	yes	—
108	C₂²×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂²×C₁₃	#131	1	2	yes	—
110	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×D₁₃	—	2	2	no	—
112	C₂×D₁₃	—	2	2	no	—
113	C₂×D₁₃	—	2	8	no	—
114	C₂×D₁₃	C₂×M₁₆	1	4	yes	complemented
115	C₄×C₁₃	—	2	2	no	—
116	C₄×C₁₃	C₄×D₄	1	2	yes	—
117	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
119	Dic₁₃	—	2	2	no	—
120	Dic₁₃	—	2	2	no	—
121	Dic₁₃	—	2	2	no	—
122	C₂²×D₄	—	13	1	no	—
123	C₂³×C₄	—	13	1	no	—
124	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
125	C₂×M₁₆	—	2	4	no	—
126	C₂×M₁₆	C₂×D₁₃	1	4	yes	complemented
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	—	13	4	no	—
130	C₂²×C₈	—	26	4	no	—
131	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
132	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₄×D₄	—	13	4	no	—
134	C₄×D₄	—	13	4	no	—
135	C₂×C₄⋊C₄	—	13	1	no	—
136	C₂×C₂²⋊C₄	—	13	1	no	—
137	C₂×C₂²⋊C₄	—	13	1	no	—
138	C₂×C₄²	—	13	1	no	—
139	C₄⋊C₈	—	13	4	no	—
140	C₂²⋊C₈	—	13	4	no	—
141	C₂²⋊C₈	—	13	4	no	—
142	#97	—	13	4	no	—
143	C₂×C₁₃	—	4	2	no	—
144	C₂×C₁₃	#435	1	4	yes	complemented
145	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
147	C₂×C₁₃	C₂×#131	1	1	yes	characteristic
148	D₁₃	—	2	4	no	—
149	C₂⁴	—	13	1	no	—
150	C₂⁴	C₄×D₁₃	1	1	yes	characteristic
151	C₂×D₄	—	13	4	no	—
152	C₂×D₄	—	26	4	no	—
153	C₂²×C₄	—	2	2	no	—
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	13	2	no	—
159	C₂²×C₄	—	13	2	no	—
160	C₂²×C₄	—	26	2	no	—
161	C₂²×C₄	—	26	2	no	—
162	C₂²×C₄	C₄×D₁₃	1	1	yes	characteristic
163	C₂²×C₄	C₄×D₁₃	1	2	yes	—
164	C₂²×C₄	C₂²×D₁₃	1	2	yes	—
165	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
166	M₁₆	—	2	8	no	—
167	C₂×C₈	—	2	2	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	26	2	no	—
171	C₂×C₈	—	26	2	no	—
172	C₂×C₈	—	26	8	no	—
173	C₂×C₈	C₂²⋊D₁₃	1	4	yes	complemented
174	C₂×C₈	C₂²×D₁₃	1	2	yes	—
175	C₂×C₈	C₂²×D₁₃	1	2	yes	—
176	C₄⋊C₄	—	13	4	no	—
177	C₂²⋊C₄	—	13	4	no	—
178	C₂²⋊C₄	—	13	4	no	—
179	C₄²	—	13	2	no	—
180	C₄²	—	13	2	no	—
181	C₁₃	C₂×#435	1	1	yes	characteristic, complemented, Sylow
182	C₂³	—	2	2	no	—
183	C₂³	—	2	2	no	—
184	C₂³	—	2	2	no	—
185	C₂³	—	13	4	no	—
186	C₂³	—	13	2	no	—
187	C₂³	—	26	2	no	—
188	C₂³	—	26	2	no	—
189	C₂³	C₂×C₄×D₁₃	1	2	yes	—
190	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
191	D₄	—	26	8	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	4	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	13	2	no	—
199	C₂×C₄	—	26	2	no	—
200	C₂×C₄	—	26	2	no	—
201	C₂×C₄	—	26	2	no	—
202	C₂×C₄	—	26	4	no	—
203	C₂×C₄	—	26	2	no	—
204	C₂×C₄	—	26	2	no	—
205	C₂×C₄	C₂×C₄×D₁₃	1	2	yes	—
206	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
207	C₂×C₄	#6127	1	2	yes	—
208	C₂×C₄	C₂×C₂²⋊D₁₃	1	2	yes	—
209	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
210	C₈	—	2	2	no	—
211	C₈	—	26	2	no	—
212	C₈	—	26	2	no	—
213	C₈	C₂×C₂²⋊D₁₃	1	4	yes	complemented
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	4	no	—
218	C₂²	—	13	4	no	—
219	C₂²	—	26	2	no	—
220	C₂²	—	26	2	no	—
221	C₂²	—	26	8	no	—
222	C₂²	M₁₆×D₁₃	1	2	yes	—
223	C₂²	#88161	1	2	yes	—
224	C₂²	C₄×C₂²⋊D₁₃	1	2	yes	—
225	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
226	C₄	—	2	2	no	—
227	C₄	—	26	2	no	—
228	C₄	—	26	2	no	—
229	C₄	—	26	2	no	—
230	C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
231	C₄	C₂×#6127	1	1	yes	characteristic
232	C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
233	C₂	—	4	2	no	—
234	C₂	—	26	4	no	—
235	C₂	#11718788	1	4	yes	direct product factor
236	C₂	C₂×M₁₆×D₁₃	1	1	yes	characteristic
237	C₂	C₂×#88161	1	1	yes	characteristic
238	C₂	C₂×C₄×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₃
Fitting	C₂²×C₁₃×M₁₆	C₂
socle	C₂³×C₁₃	C₂²×C₄
prime radical	C₂²×C₂²⋊D₁₃	C₄
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	C₂²×M₁₆	D₁₃
hypocenter	C₁₃	C₂×#435
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×M₁₆	C₂	semidirect
3. C₂²×C₈×D₁₃	C₂	semidirect
4. C₂×C₂²⋊#1055	C₂	semidirect
5. #11718788	C₂	direct (N × H)
6. C₂×#88060	C₂	semidirect
7. C₂×#88055	C₂	semidirect
8. C₂×#88054	C₂	semidirect
13. C₂×C₁₃×M₁₆	C₂²	semidirect
25. C₂×C₈×D₁₃	C₂²	semidirect
26. C₂²⋊#1055	C₂²	semidirect
27. #88060	C₂²	semidirect
28. #88055	C₂²	semidirect
29. #88054	C₂²	semidirect
49. C₂×C₈×C₁₃	D₄	semidirect
70. C₂²×D₁₃	M₁₆	semidirect
92. C₈×C₁₃	C₂×D₄	semidirect
95. C₂²×M₁₆	D₁₃	semidirect
114. C₂×D₁₃	C₂×M₁₆	semidirect
126. C₂×M₁₆	C₂×D₁₃	semidirect
144. C₂×C₁₃	#435	semidirect
173. C₂×C₈	C₂²⋊D₁₃	semidirect
181. C₁₃	C₂×#435	semidirect
213. C₈	C₂×C₂²⋊D₁₃	semidirect
235. C₂	#11718788	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×C₁₃×M₁₆	2
C₂×C₄×C₂²⋊D₁₃	2
C₂²×C₈×D₁₃	2
C₂×C₂²⋊#1055	2
#11718788	2
C₂×#88060	2
C₂×#88055	2
C₂×#88054	2
C₂×#435	13

Subgroups — C2×#11718788 #13123513

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11718788 #13123513