Subgroups — C₂×#11719541 #13129596

Order 1664.

1460 conjugacy classes of subgroups.

The subgroup lattice has 242 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×#88257	C₂	1	1	yes	maximal, characteristic, complemented
3	#11719541	C₂	1	16	yes	maximal, direct product factor
4	C₂×#88205	C₂	1	3	yes	maximal, complemented
5	C₂×#88204	C₂	1	3	yes	maximal, complemented
6	C₂×#88202	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#6105	C₂	1	3	yes	maximal, complemented
8	C₂×#88186	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	3	yes	—
13	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₃	—	2	3	no	—
15	C₂²×D₄×C₁₃	C₂²	1	3	yes	—
16	#88257	—	2	4	no	—
17	#88257	C₂²	1	8	yes	complemented
18	C₂×Q₈×D₁₃	—	2	1	no	—
19	C₂×#6129	—	2	3	no	—
20	C₂×#6127	C₂²	1	3	yes	complemented
21	C₂²×Dic₂₆	C₂²	1	3	yes	complemented
22	#88205	C₂²	1	24	yes	complemented
23	#88204	C₂²	1	24	yes	complemented
24	#88202	C₂²	1	8	yes	complemented
25	C₂×#6107	C₂²	1	3	yes	complemented
26	C₂×#6106	C₂²	1	1	yes	characteristic, complemented
27	C₂×#6105	C₂²	1	6	yes	complemented
28	C₂×#6105	C₂²	1	3	yes	complemented
29	C₂×#6105	C₂²	1	24	yes	complemented
30	#88186	C₂²	1	24	yes	complemented
31	C₂×#6104	C₂²	1	3	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	C₁₃×#41	—	2	8	no	—
36	C₁₃×#41	—	2	6	no	—
37	C₁₃×#41	D₄	1	4	yes	—
38	C₁₃×#41	C₂³	1	12	yes	—
39	C₁₃×#41	C₂³	1	4	yes	—
40	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×D₄×C₁₃	—	2	3	no	—
43	C₂×D₄×C₁₃	—	2	24	no	—
44	C₂×D₄×C₁₃	—	2	3	no	—
45	C₂×D₄×C₁₃	—	2	6	no	—
46	C₂×D₄×C₁₃	D₄	1	3	yes	—
47	C₂×D₄×C₁₃	C₂³	1	3	yes	—
48	C₂×D₄×C₁₃	C₂³	1	12	yes	—
49	C₂²×C₄×C₁₃	—	2	3	no	—
50	C₂²×C₄×C₁₃	C₂³	1	3	yes	—
51	C₂×C₂²⋊D₁₃	—	2	3	no	—
52	C₂²×Dic₁₃	—	2	3	no	—
53	Q₈×D₁₃	—	2	8	no	—
54	#6129	—	2	24	no	—
55	#6127	—	2	6	no	—
56	#6127	C₂³	1	12	yes	complemented
57	C₂×D₅₂	C₂³	1	1	yes	characteristic
58	C₂×C₄×D₁₃	—	2	3	no	—
59	C₂×Dic₂₆	—	2	3	no	—
60	C₂×Dic₂₆	—	2	6	no	—
61	C₂×Dic₂₆	C₂³	1	3	yes	—
62	C₂×Dic₂₆	C₂³	1	12	yes	complemented
63	#6107	C₂³	1	12	yes	complemented
64	#6106	C₂³	1	4	yes	complemented
65	#6105	C₂³	1	24	yes	complemented
66	#6105	C₂³	1	12	yes	complemented
67	#6104	C₂³	1	12	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×#577	—	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	1	no	—
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	C₂×D₄	1	4	yes	—
80	Q₈×C₁₃	C₂⁴	1	2	yes	—
81	D₄×C₁₃	—	2	12	no	—
82	D₄×C₁₃	—	2	3	no	—
83	D₄×C₁₃	—	2	12	no	—
84	D₄×C₁₃	C₂×D₄	1	12	yes	—
85	D₄×C₁₃	C₂⁴	1	6	yes	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	12	no	—
88	C₂×C₄×C₁₃	C₂×D₄	1	3	yes	—
89	C₂×C₄×C₁₃	C₂×D₄	1	6	yes	—
90	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₂×C₄×C₁₃	C₂⁴	1	6	yes	—
92	C₂²⋊D₁₃	—	2	12	no	—
93	C₂×Dic₁₃	—	2	3	no	—
94	C₂×Dic₁₃	—	2	12	no	—
95	D₅₂	—	2	1	no	—
96	D₅₂	C₂⁴	1	2	yes	—
97	C₄×D₁₃	—	2	12	no	—
98	Dic₂₆	—	2	3	no	—
99	Dic₂₆	—	2	12	no	—
100	Dic₂₆	C₂⁴	1	6	yes	—
101	#1055	C₂⁴	1	2	yes	—
102	#1055	C₂⁴	1	6	yes	—
103	C₂×2₋¹⁺⁴	—	13	1	no	—
104	C₂×H(5,𝔽₂)	D₁₃	1	1	yes	characteristic, complemented
105	#577	—	13	16	no	—
106	C₂×#137	—	13	3	no	—
107	C₂×AGL(1,ℤ/8)	—	13	3	no	—
108	C₂×#135	—	13	3	no	—
109	C₂²×SD₁₆	—	13	3	no	—
110	C₂×#131	—	13	1	no	—
111	C₂²×C₁₃	—	2	6	no	—
112	C₂²×C₁₃	—	2	6	no	—
113	C₂²×C₁₃	—	4	6	no	—
114	C₂²×C₁₃	—	8	12	no	—
115	C₂²×C₁₃	C₂²×D₄	1	6	yes	—
116	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
117	C₂×D₁₃	—	2	2	no	—
118	C₂×D₁₃	—	4	2	no	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₁₃	C₂²×D₄	1	6	yes	—
122	C₄×C₁₃	C₂⁵	1	1	yes	characteristic
123	Dic₁₃	—	2	6	no	—
124	2₋¹⁺⁴	—	13	8	no	—
125	2₋¹⁺⁴	—	26	4	no	—
126	H(5,𝔽₂)	—	2	4	no	—
127	H(5,𝔽₂)	C₂×D₁₃	1	8	yes	complemented
128	C₂×#41	—	2	1	no	—
129	C₂×#41	—	13	3	no	—
130	C₂×#41	—	26	3	no	—
131	C₂×#41	C₂×D₁₃	1	3	yes	—
132	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
133	C₂²×Q₈	—	13	3	no	—
134	C₂²×Q₈	—	26	1	no	—
135	C₂²×D₄	—	2	3	no	—
136	C₂²×D₄	C₂×D₁₃	1	3	yes	—
137	#137	—	13	24	no	—
138	AGL(1,ℤ/8)	—	13	24	no	—
139	#135	—	13	24	no	—
140	C₂×Q₁₆	—	13	3	no	—
141	C₂×SD₁₆	—	13	1	no	—
142	C₂×SD₁₆	—	13	3	no	—
143	C₂×SD₁₆	—	13	6	no	—
144	C₂×SD₁₆	—	13	24	no	—
145	C₂×D₈	—	13	3	no	—
146	#131	—	13	8	no	—
147	C₂×M₁₆	—	13	3	no	—
148	C₂²×C₈	—	13	3	no	—
149	C₂×C₁₃	—	2	6	no	—
150	C₂×C₁₃	—	4	6	no	—
151	C₂×C₁₃	#577	1	2	yes	complemented
152	C₂×C₁₃	C₂³×D₄	1	1	yes	characteristic
153	D₁₃	—	4	2	no	—
154	C₂⁴	—	2	3	no	—
155	#41	—	2	8	no	—
156	#41	—	2	6	no	—
157	#41	—	13	12	no	—
158	#41	—	26	24	no	—
159	#41	—	26	6	no	—
160	#41	C₂²⋊D₁₃	1	4	yes	—
161	#41	C₂²×D₁₃	1	12	yes	—
162	#41	C₂²×D₁₃	1	4	yes	—
163	C₂×Q₈	—	13	3	no	—
164	C₂×Q₈	—	13	12	no	—
165	C₂×Q₈	—	26	8	no	—
166	C₂×Q₈	—	26	3	no	—
167	C₂×Q₈	—	26	6	no	—
168	C₂×Q₈	C₂²⋊D₁₃	1	1	yes	characteristic
169	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×D₄	—	2	3	no	—
171	C₂×D₄	—	2	24	no	—
172	C₂×D₄	—	2	3	no	—
173	C₂×D₄	—	2	6	no	—
174	C₂×D₄	—	13	1	no	—
175	C₂×D₄	—	26	3	no	—
176	C₂×D₄	C₂²⋊D₁₃	1	3	yes	—
177	C₂×D₄	C₂²×D₁₃	1	3	yes	—
178	C₂×D₄	C₂²×D₁₃	1	12	yes	—
179	C₂²×C₄	—	2	3	no	—
180	C₂²×C₄	—	26	3	no	—
181	C₂²×C₄	—	26	3	no	—
182	C₂²×C₄	C₂²×D₁₃	1	3	yes	—
183	Q₁₆	—	13	12	no	—
184	SD₁₆	—	13	4	no	—
185	SD₁₆	—	13	12	no	—
186	SD₁₆	—	13	24	no	—
187	D₈	—	13	12	no	—
188	M₁₆	—	13	12	no	—
189	C₂×C₈	—	13	1	no	—
190	C₂×C₈	—	13	3	no	—
191	C₂×C₈	—	13	12	no	—
192	C₁₃	C₂×#577	1	1	yes	characteristic, complemented, Sylow
193	C₂³	—	2	3	no	—
194	C₂³	—	2	12	no	—
195	C₂³	—	8	6	no	—
196	C₂³	—	26	1	no	—
197	C₂³	C₂×C₂²⋊D₁₃	1	3	yes	—
198	Q₈	—	2	1	no	—
199	Q₈	—	13	6	no	—
200	Q₈	—	26	12	no	—
201	Q₈	—	26	3	no	—
202	Q₈	C₂×C₂²⋊D₁₃	1	4	yes	—
203	Q₈	C₂³×D₁₃	1	2	yes	—
204	D₄	—	2	12	no	—
205	D₄	—	2	12	no	—
206	D₄	—	2	3	no	—
207	D₄	—	13	2	no	—
208	D₄	—	26	12	no	—
209	D₄	—	26	1	no	—
210	D₄	C₂×C₂²⋊D₁₃	1	12	yes	—
211	D₄	C₂³×D₁₃	1	6	yes	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	12	no	—
214	C₂×C₄	—	26	12	no	—
215	C₂×C₄	—	26	3	no	—
216	C₂×C₄	—	26	12	no	—
217	C₂×C₄	C₂×C₂²⋊D₁₃	1	6	yes	—
218	C₂×C₄	C₂×C₂²⋊D₁₃	1	3	yes	—
219	C₂×C₄	C₂³×D₁₃	1	6	yes	—
220	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
221	C₈	—	13	2	no	—
222	C₈	—	13	6	no	—
223	C₂²	—	2	6	no	—
224	C₂²	—	2	6	no	—
225	C₂²	—	4	6	no	—
226	C₂²	—	8	12	no	—
227	C₂²	—	26	2	no	—
228	C₂²	—	52	2	no	—
229	C₂²	C₂²×C₂²⋊D₁₃	1	6	yes	—
230	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	26	6	no	—
233	C₄	C₂²×C₂²⋊D₁₃	1	6	yes	—
234	C₄	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
235	C₄	C₂⁴×D₁₃	1	1	yes	characteristic
236	C₂	—	2	6	no	—
237	C₂	—	4	6	no	—
238	C₂	—	52	2	no	—
239	C₂	#11719541	1	2	yes	direct product factor
240	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×#577
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×#88257	C₂	semidirect
3. #11719541	C₂	direct (N × H)
4. C₂×#88205	C₂	semidirect
5. C₂×#88204	C₂	semidirect
6. C₂×#88202	C₂	semidirect
7. C₂²×#6105	C₂	semidirect
8. C₂×#88186	C₂	semidirect
10. C₁₃×H(5,𝔽₂)	C₂²	semidirect
17. #88257	C₂²	semidirect
20. C₂×#6127	C₂²	semidirect
21. C₂²×Dic₂₆	C₂²	semidirect
22. #88205	C₂²	semidirect
23. #88204	C₂²	semidirect
24. #88202	C₂²	semidirect
25. C₂×#6107	C₂²	semidirect
26. C₂×#6106	C₂²	semidirect
27. C₂×#6105	C₂²	semidirect
28. C₂×#6105	C₂²	semidirect
29. C₂×#6105	C₂²	semidirect
30. #88186	C₂²	semidirect
31. C₂×#6104	C₂²	semidirect
32. C₂×#6099	C₂²	semidirect
33. C₂²×#1055	C₂²	semidirect
56. #6127	C₂³	semidirect
62. C₂×Dic₂₆	C₂³	semidirect
63. #6107	C₂³	semidirect
64. #6106	C₂³	semidirect
65. #6105	C₂³	semidirect
66. #6105	C₂³	semidirect
67. #6104	C₂³	semidirect
68. #6099	C₂³	semidirect
71. C₂×#1055	C₂³	semidirect
104. C₂×H(5,𝔽₂)	D₁₃	semidirect
127. H(5,𝔽₂)	C₂×D₁₃	semidirect
151. C₂×C₁₃	#577	semidirect
192. C₁₃	C₂×#577	semidirect
239. C₂	#11719541	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×#88257	2
#11719541	2
C₂×#88205	2
C₂×#88204	2
C₂×#88202	2
C₂²×#6105	2
C₂×#88186	2
C₂×#577	13

Subgroups — C2×#11719541 #13129596

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11719541 #13129596