Subgroups — C₂×#11719540 #13129587

Order 1664.

1512 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#88256	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719540	C₂	1	16	yes	maximal, direct product factor
4	C₂×#88204	C₂	1	3	yes	maximal, complemented
5	C₂×#88203	C₂	1	3	yes	maximal, complemented
6	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#88186	C₂	1	3	yes	maximal, complemented
8	C₂²×#6104	C₂	1	3	yes	maximal, complemented
9	C₁₃×H(5,𝔽₂)	—	2	4	no	—
10	C₁₃×H(5,𝔽₂)	C₂²	1	8	yes	complemented
11	C₂×C₁₃×#41	—	2	1	no	—
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#41	C₂²	1	3	yes	complemented
14	C₂²×D₄×C₁₃	—	2	3	no	—
15	C₂²×D₄×C₁₃	C₂²	1	3	yes	complemented
16	#88256	—	2	4	no	—
17	#88256	C₂²	1	8	yes	complemented
18	C₂×#6131	—	2	1	no	—
19	C₂×D₄×D₁₃	—	2	3	no	—
20	C₂×#6127	C₂²	1	3	yes	complemented
21	C₂²×D₅₂	C₂²	1	3	yes	complemented
22	#88204	C₂²	1	24	yes	complemented
23	#88203	C₂²	1	24	yes	complemented
24	#88202	C₂²	1	8	yes	complemented
25	C₂×#6107	C₂²	1	1	yes	characteristic, complemented
26	C₂×#6106	C₂²	1	3	yes	complemented
27	C₂×#6105	C₂²	1	3	yes	complemented
28	#88186	C₂²	1	24	yes	complemented
29	C₂×#6104	C₂²	1	3	yes	complemented
30	C₂×#6104	C₂²	1	6	yes	complemented
31	C₂×#6104	C₂²	1	24	yes	complemented
32	C₂×#6099	C₂²	1	3	yes	complemented
33	C₂²×#1055	C₂²	1	3	yes	complemented
34	C₂⁴×C₁₃	—	2	3	no	—
35	C₂³×D₁₃	—	2	3	no	—
36	C₁₃×#41	—	2	8	no	—
37	C₁₃×#41	—	2	6	no	—
38	C₁₃×#41	D₄	1	4	yes	—
39	C₁₃×#41	C₂³	1	4	yes	—
40	C₁₃×#41	C₂³	1	12	yes	complemented
41	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
42	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×D₄×C₁₃	—	2	3	no	—
44	C₂×D₄×C₁₃	—	2	24	no	—
45	C₂×D₄×C₁₃	—	2	3	no	—
46	C₂×D₄×C₁₃	—	2	6	no	—
47	C₂×D₄×C₁₃	D₄	1	3	yes	—
48	C₂×D₄×C₁₃	C₂³	1	3	yes	—
49	C₂×D₄×C₁₃	C₂³	1	12	yes	complemented
50	C₂²×C₄×C₁₃	—	2	3	no	—
51	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
52	C₂×C₂²⋊D₁₃	—	2	3	no	—
53	#6131	—	2	8	no	—
54	D₄×D₁₃	—	2	24	no	—
55	#6127	—	2	6	no	—
56	#6127	C₂³	1	12	yes	complemented
57	C₂×D₅₂	—	2	3	no	—
58	C₂×D₅₂	—	2	6	no	—
59	C₂×D₅₂	C₂³	1	3	yes	—
60	C₂×D₅₂	C₂³	1	12	yes	complemented
61	C₂×C₄×D₁₃	—	2	3	no	—
62	C₂×Dic₂₆	C₂³	1	1	yes	characteristic
63	#6107	C₂³	1	4	yes	complemented
64	#6106	C₂³	1	12	yes	complemented
65	#6105	C₂³	1	12	yes	complemented
66	#6104	C₂³	1	12	yes	complemented
67	#6104	C₂³	1	24	yes	complemented
68	#6099	C₂³	1	12	yes	complemented
69	C₂×#1055	C₂³	1	1	yes	characteristic
70	C₂×#1055	C₂³	1	3	yes	—
71	C₂×#1055	C₂³	1	12	yes	complemented
72	C₂×#576	—	13	1	no	maximal, Sylow
73	C₂³×C₁₃	—	2	3	no	—
74	C₂³×C₁₃	—	2	12	no	—
75	C₂³×C₁₃	—	8	6	no	—
76	C₂³×C₁₃	C₂×D₄	1	3	yes	—
77	C₂²×D₁₃	—	2	3	no	—
78	C₂²×D₁₃	—	2	12	no	—
79	C₂²×D₁₃	—	8	6	no	—
80	Q₈×C₁₃	—	2	1	no	—
81	Q₈×C₁₃	C₂×D₄	1	4	yes	—
82	Q₈×C₁₃	C₂⁴	1	2	yes	—
83	D₄×C₁₃	—	2	12	no	—
84	D₄×C₁₃	—	2	3	no	—
85	D₄×C₁₃	—	2	12	no	—
86	D₄×C₁₃	C₂×D₄	1	12	yes	—
87	D₄×C₁₃	C₂⁴	1	6	yes	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	12	no	—
90	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
91	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
92	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
94	C₂²⋊D₁₃	—	2	12	no	—
95	C₂×Dic₁₃	—	2	1	no	—
96	D₅₂	—	2	3	no	—
97	D₅₂	—	2	12	no	—
98	D₅₂	C₂⁴	1	6	yes	—
99	C₄×D₁₃	—	2	12	no	—
100	Dic₂₆	—	2	1	no	—
101	Dic₂₆	C₂⁴	1	2	yes	—
102	#1055	C₂⁴	1	2	yes	—
103	#1055	C₂⁴	1	6	yes	—
104	C₂×H(5,𝔽₂)	—	13	1	no	—
105	C₂×H(5,𝔽₂)	D₁₃	1	1	yes	characteristic, complemented
106	#576	—	13	16	no	—
107	C₂×AGL(1,ℤ/8)	—	13	3	no	—
108	C₂×AGL(1,ℤ/8)	—	13	3	no	—
109	C₂×#135	—	13	3	no	—
110	C₂²×D₈	—	13	3	no	—
111	C₂×#131	—	13	1	no	—
112	C₂²×C₁₃	—	2	6	no	—
113	C₂²×C₁₃	—	2	6	no	—
114	C₂²×C₁₃	—	4	6	no	—
115	C₂²×C₁₃	—	8	12	no	—
116	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
117	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₂×D₁₃	—	2	6	no	—
119	C₂×D₁₃	—	4	6	no	—
120	C₂×D₁₃	—	8	12	no	—
121	C₄×C₁₃	—	2	2	no	—
122	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₁₃	C₂²×D₄	1	6	yes	—
124	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
125	Dic₁₃	—	2	2	no	—
126	H(5,𝔽₂)	—	2	4	no	—
127	H(5,𝔽₂)	—	13	8	no	—
128	H(5,𝔽₂)	—	26	4	no	—
129	H(5,𝔽₂)	C₂×D₁₃	1	8	yes	complemented
130	C₂×#41	—	2	1	no	—
131	C₂×#41	—	13	3	no	—
132	C₂×#41	—	26	1	no	—
133	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
134	C₂×#41	C₂×D₁₃	1	3	yes	complemented
135	C₂²×D₄	—	2	3	no	—
136	C₂²×D₄	—	13	3	no	—
137	C₂²×D₄	—	26	3	no	—
138	C₂²×D₄	C₂×D₁₃	1	3	yes	complemented
139	AGL(1,ℤ/8)	—	13	24	no	—
140	AGL(1,ℤ/8)	—	13	24	no	—
141	#135	—	13	24	no	—
142	C₂×Q₁₆	—	13	1	no	—
143	C₂×SD₁₆	—	13	3	no	—
144	C₂×SD₁₆	—	13	3	no	—
145	C₂×D₈	—	13	3	no	—
146	C₂×D₈	—	13	6	no	—
147	C₂×D₈	—	13	24	no	—
148	#131	—	13	8	no	—
149	C₂×M₁₆	—	13	3	no	—
150	C₂²×C₈	—	13	3	no	—
151	C₂×C₁₃	—	2	6	no	—
152	C₂×C₁₃	—	4	6	no	—
153	C₂×C₁₃	#576	1	2	yes	complemented
154	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
155	D₁₃	—	4	6	no	—
156	C₂⁴	—	2	3	no	—
157	C₂⁴	—	26	3	no	—
158	#41	—	2	8	no	—
159	#41	—	2	6	no	—
160	#41	—	13	12	no	—
161	#41	—	26	8	no	—
162	#41	—	26	6	no	—
163	#41	C₂²⋊D₁₃	1	4	yes	—
164	#41	C₂²×D₁₃	1	4	yes	—
165	#41	C₂²×D₁₃	1	12	yes	complemented
166	C₂×Q₈	—	13	1	no	—
167	C₂×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
168	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
169	C₂×D₄	—	2	3	no	—
170	C₂×D₄	—	2	24	no	—
171	C₂×D₄	—	2	3	no	—
172	C₂×D₄	—	2	6	no	—
173	C₂×D₄	—	13	3	no	—
174	C₂×D₄	—	13	12	no	—
175	C₂×D₄	—	26	3	no	—
176	C₂×D₄	—	26	24	no	—
177	C₂×D₄	—	26	3	no	—
178	C₂×D₄	—	26	6	no	—
179	C₂×D₄	C₂²⋊D₁₃	1	3	yes	—
180	C₂×D₄	C₂²×D₁₃	1	3	yes	—
181	C₂×D₄	C₂²×D₁₃	1	12	yes	complemented
182	C₂²×C₄	—	2	3	no	—
183	C₂²×C₄	—	26	3	no	—
184	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
185	Q₁₆	—	13	4	no	—
186	SD₁₆	—	13	12	no	—
187	SD₁₆	—	13	12	no	—
188	D₈	—	13	12	no	—
189	D₈	—	13	24	no	—
190	M₁₆	—	13	12	no	—
191	C₂×C₈	—	13	1	no	—
192	C₂×C₈	—	13	3	no	—
193	C₂×C₈	—	13	12	no	—
194	C₁₃	C₂×#576	1	1	yes	characteristic, complemented, Sylow
195	C₂³	—	2	3	no	—
196	C₂³	—	2	12	no	—
197	C₂³	—	8	6	no	—
198	C₂³	—	26	3	no	—
199	C₂³	—	26	12	no	—
200	C₂³	—	104	6	no	—
201	C₂³	C₂×C₂²⋊D₁₃	1	3	yes	—
202	Q₈	—	2	1	no	—
203	Q₈	—	13	2	no	—
204	Q₈	—	26	1	no	—
205	Q₈	C₂×C₂²⋊D₁₃	1	4	yes	—
206	Q₈	C₂³×D₁₃	1	2	yes	—
207	D₄	—	2	12	no	—
208	D₄	—	2	3	no	—
209	D₄	—	2	12	no	—
210	D₄	—	13	6	no	—
211	D₄	—	26	12	no	—
212	D₄	—	26	12	no	—
213	D₄	—	26	3	no	—
214	D₄	C₂×C₂²⋊D₁₃	1	12	yes	—
215	D₄	C₂³×D₁₃	1	6	yes	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	12	no	—
218	C₂×C₄	—	26	1	no	—
219	C₂×C₄	—	26	12	no	—
220	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
221	C₂×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
222	C₂×C₄	C₂³×D₁₃	1	6	yes	—
223	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
224	C₈	—	13	2	no	—
225	C₈	—	13	6	no	—
226	C₂²	—	2	6	no	—
227	C₂²	—	2	6	no	—
228	C₂²	—	4	6	no	—
229	C₂²	—	8	12	no	—
230	C₂²	—	26	6	no	—
231	C₂²	—	52	6	no	—
232	C₂²	—	104	12	no	—
233	C₂²	C₂²×C₂²⋊D₁₃	1	6	yes	—
234	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₄	—	2	2	no	—
236	C₄	—	26	2	no	—
237	C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
238	C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
239	C₄	C₂⁴×D₁₃	1	1	yes	characteristic
240	C₂	—	2	6	no	—
241	C₂	—	4	6	no	—
242	C₂	—	52	6	no	—
243	C₂	#11719540	1	2	yes	direct product factor
244	C₂	C₂³×C₂²⋊D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₄×C₁₃	C₂⁵
Frattini	C₄	C₂⁴×D₁₃
Fitting	C₂×C₁₃×H(5,𝔽₂)	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	C₂×H(5,𝔽₂)	D₁₃
hypocenter	C₁₃	C₂×#576
LCS	G ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×H(5,𝔽₂)	C₂	semidirect
2. C₂×#88256	C₂	semidirect
3. #11719540	C₂	direct (N × H)
4. C₂×#88204	C₂	semidirect
5. C₂×#88203	C₂	semidirect
6. C₂×#88202	C₂	semidirect
7. C₂×#88186	C₂	semidirect
8. C₂²×#6104	C₂	semidirect
10. C₁₃×H(5,𝔽₂)	C₂²	semidirect
13. C₂×C₁₃×#41	C₂²	semidirect
15. C₂²×D₄×C₁₃	C₂²	semidirect
17. #88256	C₂²	semidirect
20. C₂×#6127	C₂²	semidirect
21. C₂²×D₅₂	C₂²	semidirect
22. #88204	C₂²	semidirect
23. #88203	C₂²	semidirect
24. #88202	C₂²	semidirect
25. C₂×#6107	C₂²	semidirect
26. C₂×#6106	C₂²	semidirect
27. C₂×#6105	C₂²	semidirect
28. #88186	C₂²	semidirect
29. C₂×#6104	C₂²	semidirect
30. C₂×#6104	C₂²	semidirect
31. C₂×#6104	C₂²	semidirect
32. C₂×#6099	C₂²	semidirect
33. C₂²×#1055	C₂²	semidirect
40. C₁₃×#41	C₂³	semidirect
49. C₂×D₄×C₁₃	C₂³	semidirect
56. #6127	C₂³	semidirect
60. C₂×D₅₂	C₂³	semidirect
63. #6107	C₂³	semidirect
64. #6106	C₂³	semidirect
65. #6105	C₂³	semidirect
66. #6104	C₂³	semidirect
67. #6104	C₂³	semidirect
68. #6099	C₂³	semidirect
71. C₂×#1055	C₂³	semidirect
105. C₂×H(5,𝔽₂)	D₁₃	semidirect
129. H(5,𝔽₂)	C₂×D₁₃	semidirect
134. C₂×#41	C₂×D₁₃	semidirect
138. C₂²×D₄	C₂×D₁₃	semidirect
153. C₂×C₁₃	#576	semidirect
165. #41	C₂²×D₁₃	semidirect
181. C₂×D₄	C₂²×D₁₃	semidirect
194. C₁₃	C₂×#576	semidirect
243. C₂	#11719540	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×H(5,𝔽₂)	2
C₂×#88256	2
#11719540	2
C₂×#88204	2
C₂×#88203	2
C₂×#88202	2
C₂×#88186	2
C₂²×#6104	2
C₂×#576	13

Subgroups — C2×#11719540 #13129587

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719540 #13129587