Subgroups — C₂×#11929141 #72276186

Order 2496.

1796 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×C₁₃⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#11929141	C₂	1	16	yes	maximal, direct product factor
3	C₂×#10597707	C₂	1	2	yes	maximal, complemented
4	C₂×D₄×D₁₃⋊C₃	C₂	1	6	yes	maximal, complemented
5	C₂×#6127⋊C₃	C₂	1	3	yes	maximal, complemented
6	C₂²×C₄⋊(D₁₃⋊C₃)	C₂	1	3	yes	maximal, complemented
7	C₂×#88256	C₃	1	1	yes	maximal, characteristic, complemented, Hall
8	C₂³×D₁₃⋊C₃	C₂²	1	6	yes	complemented
9	#41×C₁₃⋊C₃	C₂²	1	8	yes	complemented
10	C₂×Q₈×C₁₃⋊C₃	C₂²	1	1	yes	characteristic, complemented
11	C₂×D₄×C₁₃⋊C₃	C₂²	1	3	yes	complemented
12	C₂²×C₄×C₁₃⋊C₃	C₂²	1	3	yes	complemented
13	C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	1	6	yes	complemented
14	#10597707	C₂²	1	16	yes	complemented
15	D₄×D₁₃⋊C₃	C₂²	1	48	yes	complemented
16	#6127⋊C₃	C₂²	1	24	yes	complemented
17	C₂×C₄⋊(D₁₃⋊C₃)	C₂²	1	3	yes	complemented
18	C₂×C₄⋊(D₁₃⋊C₃)	C₂²	1	6	yes	complemented
19	C₂×C₄⋊(D₁₃⋊C₃)	C₂²	1	24	yes	complemented
20	C₂×C₄×D₁₃⋊C₃	C₂²	1	6	yes	complemented
21	C₂×(C₁₃⋊C₃)⋊Q₈	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₁₃×#41	C₂×C₃	1	1	yes	characteristic, complemented
23	#88256	C₂×C₃	1	16	yes	complemented
24	C₂×#6131	C₂×C₃	1	2	yes	complemented
25	C₂×D₄×D₁₃	C₂×C₃	1	6	yes	complemented
26	C₂×#6127	C₂×C₃	1	3	yes	complemented
27	C₂²×D₅₂	C₂×C₃	1	3	yes	complemented
28	C₂³×C₁₃⋊C₃	C₂³	1	3	yes	—
29	C₂²×D₁₃⋊C₃	—	4	12	no	—
30	C₂²×D₁₃⋊C₃	C₂³	1	6	yes	—
31	C₂²×D₁₃⋊C₃	C₂³	1	24	yes	complemented
32	Q₈×C₁₃⋊C₃	C₂³	1	4	yes	complemented
33	D₄×C₁₃⋊C₃	C₂³	1	12	yes	complemented
34	C₂×C₄×C₁₃⋊C₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₁₃⋊C₃	C₂³	1	3	yes	—
36	C₂×C₄×C₁₃⋊C₃	C₂³	1	12	yes	complemented
37	(C₂²⋊D₁₃)⋊C₃	C₂³	1	24	yes	complemented
38	C₂×Dic₁₃⋊C₃	C₂³	1	2	yes	—
39	C₄⋊(D₁₃⋊C₃)	C₂³	1	12	yes	complemented
40	C₄⋊(D₁₃⋊C₃)	C₂³	1	24	yes	complemented
41	C₄×D₁₃⋊C₃	C₂³	1	24	yes	complemented
42	(C₁₃⋊C₃)⋊Q₈	C₂³	1	4	yes	complemented
43	C₂³×D₁₃	C₂²×C₃	1	6	yes	complemented
44	C₁₃×#41	C₂²×C₃	1	8	yes	complemented
45	C₂×Q₈×C₁₃	C₂²×C₃	1	1	yes	characteristic, complemented
46	C₂×D₄×C₁₃	C₂²×C₃	1	3	yes	complemented
47	C₂²×C₄×C₁₃	C₂²×C₃	1	3	yes	complemented
48	C₂×C₂²⋊D₁₃	C₂²×C₃	1	6	yes	complemented
49	#6131	C₂²×C₃	1	16	yes	complemented
50	D₄×D₁₃	C₂²×C₃	1	48	yes	complemented
51	#6127	C₂²×C₃	1	24	yes	complemented
52	C₂×D₅₂	C₂²×C₃	1	3	yes	complemented
53	C₂×D₅₂	C₂²×C₃	1	6	yes	complemented
54	C₂×D₅₂	C₂²×C₃	1	24	yes	complemented
55	C₂×C₄×D₁₃	C₂²×C₃	1	6	yes	complemented
56	C₂×Dic₂₆	C₂²×C₃	1	1	yes	characteristic, complemented
57	C₂×C₃×H(5,𝔽₂)	—	13	1	no	maximal, Hall
58	C₂²×C₁₃⋊C₃	—	2	6	no	—
59	C₂²×C₁₃⋊C₃	C₂⁴	1	6	yes	—
60	C₂²×C₁₃⋊C₃	C₂⁴	1	1	yes	characteristic
61	C₂×D₁₃⋊C₃	—	2	12	no	—
62	C₂×D₁₃⋊C₃	—	4	24	no	—
63	C₂×D₁₃⋊C₃	C₂⁴	1	12	yes	—
64	C₄×C₁₃⋊C₃	C₂⁴	1	2	yes	—
65	C₄×C₁₃⋊C₃	C₂⁴	1	6	yes	—
66	Dic₁₃⋊C₃	C₂⁴	1	4	yes	—
67	C₂³×C₁₃	C₂³×C₃	1	3	yes	—
68	C₂²×D₁₃	—	4	12	no	—
69	C₂²×D₁₃	C₂³×C₃	1	6	yes	—
70	C₂²×D₁₃	C₂³×C₃	1	24	yes	complemented
71	Q₈×C₁₃	C₂³×C₃	1	4	yes	complemented
72	D₄×C₁₃	C₂³×C₃	1	12	yes	complemented
73	C₂×C₄×C₁₃	C₂³×C₃	1	1	yes	characteristic
74	C₂×C₄×C₁₃	C₂³×C₃	1	3	yes	—
75	C₂×C₄×C₁₃	C₂³×C₃	1	12	yes	complemented
76	C₂²⋊D₁₃	C₂³×C₃	1	24	yes	complemented
77	C₂×Dic₁₃	C₂³×C₃	1	2	yes	—
78	D₅₂	C₂³×C₃	1	12	yes	complemented
79	D₅₂	C₂³×C₃	1	24	yes	complemented
80	C₄×D₁₃	C₂³×C₃	1	24	yes	complemented
81	Dic₂₆	C₂³×C₃	1	4	yes	complemented
82	C₃×H(5,𝔽₂)	—	13	16	no	—
83	C₂×C₃×#41	—	13	3	no	—
84	C₂×C₃×#41	—	13	2	no	—
85	C₂×C₃×#41	—	13	1	no	—
86	C₂²×C₃×D₄	—	13	3	no	—
87	C₂²×C₃×D₄	—	13	6	no	—
88	C₂×C₁₃⋊C₃	—	2	6	no	—
89	C₂×C₁₃⋊C₃	H(5,𝔽₂)	1	2	yes	complemented
90	C₂×C₁₃⋊C₃	C₂⁵	1	1	yes	characteristic
91	D₁₃⋊C₃	—	2	12	no	—
92	C₂×H(5,𝔽₂)	—	13	1	no	Sylow
93	C₂²×C₁₃	—	2	6	no	—
94	C₂²×C₁₃	C₂⁴×C₃	1	6	yes	—
95	C₂²×C₁₃	C₂⁴×C₃	1	1	yes	characteristic
96	C₂×D₁₃	—	2	12	no	—
97	C₂×D₁₃	—	4	24	no	—
98	C₂×D₁₃	C₂⁴×C₃	1	12	yes	—
99	C₄×C₁₃	C₂⁴×C₃	1	2	yes	—
100	C₄×C₁₃	C₂⁴×C₃	1	6	yes	—
101	Dic₁₃	C₂⁴×C₃	1	4	yes	—
102	C₂⁴×C₃	—	13	6	no	—
103	C₃×#41	—	13	24	no	—
104	C₃×#41	—	13	16	no	—
105	C₃×#41	—	13	8	no	—
106	C₂×C₃×Q₈	—	13	1	no	—
107	C₂×C₃×Q₈	—	13	1	no	—
108	C₂×C₃×D₄	—	13	3	no	—
109	C₂×C₃×D₄	—	13	6	no	—
110	C₂×C₃×D₄	—	13	24	no	—
111	C₂×C₃×D₄	—	13	48	no	—
112	C₂×C₃×D₄	—	13	6	no	—
113	C₂×C₃×D₄	—	13	3	no	—
114	C₂²×C₃×C₄	—	13	6	no	—
115	C₂²×C₃×C₄	—	13	3	no	—
116	C₁₃⋊C₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Hall
117	H(5,𝔽₂)	—	13	16	no	—
118	C₂×#41	—	13	2	no	—
119	C₂×#41	—	13	3	no	—
120	C₂×#41	D₁₃⋊C₃	1	1	yes	characteristic, complemented
121	C₂²×D₄	—	13	6	no	—
122	C₂²×D₄	—	13	3	no	—
123	C₂×C₁₃	—	2	6	no	—
124	C₂×C₁₃	C₃×H(5,𝔽₂)	1	2	yes	complemented
125	C₂×C₁₃	C₂⁵×C₃	1	1	yes	characteristic
126	D₁₃	—	2	12	no	—
127	C₂³×C₃	—	13	6	no	—
128	C₂³×C₃	—	13	24	no	—
129	C₂³×C₃	—	13	3	no	—
130	C₂³×C₃	—	52	12	no	—
131	C₃×Q₈	—	13	4	no	—
132	C₃×Q₈	—	13	4	no	—
133	C₃×D₄	—	13	12	no	—
134	C₃×D₄	—	13	24	no	—
135	C₃×D₄	—	13	24	no	—
136	C₃×D₄	—	13	12	no	—
137	C₂×C₃×C₄	—	13	24	no	—
138	C₂×C₃×C₄	—	13	2	no	—
139	C₂×C₃×C₄	—	13	3	no	—
140	C₂×C₃×C₄	—	13	1	no	—
141	C₂×C₃×C₄	—	13	12	no	—
142	C₂⁴	—	13	6	no	—
143	#41	—	13	16	no	—
144	#41	—	13	24	no	—
145	#41	C₂×D₁₃⋊C₃	1	8	yes	complemented
146	C₂×Q₈	—	13	1	no	—
147	C₂×Q₈	C₂×D₁₃⋊C₃	1	1	yes	characteristic, complemented
148	C₂×D₄	—	13	6	no	—
149	C₂×D₄	—	13	48	no	—
150	C₂×D₄	—	13	6	no	—
151	C₂×D₄	—	13	3	no	—
152	C₂×D₄	—	13	24	no	—
153	C₂×D₄	C₂×D₁₃⋊C₃	1	3	yes	complemented
154	C₂²×C₄	—	13	6	no	—
155	C₂²×C₄	C₂×D₁₃⋊C₃	1	3	yes	complemented
156	C₁₃	C₂×C₃×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
157	C₂²×C₃	—	13	12	no	—
158	C₂²×C₃	—	13	6	no	—
159	C₂²×C₃	—	13	1	no	—
160	C₂²×C₃	—	26	6	no	—
161	C₂²×C₃	—	26	12	no	—
162	C₂²×C₃	—	52	24	no	—
163	C₃×C₄	—	13	4	no	—
164	C₃×C₄	—	13	6	no	—
165	C₃×C₄	—	13	2	no	—
166	C₂³	—	13	6	no	—
167	C₂³	—	13	24	no	—
168	C₂³	—	52	12	no	—
169	C₂³	C₂²×D₁₃⋊C₃	1	3	yes	—
170	Q₈	—	13	4	no	—
171	Q₈	C₂²×D₁₃⋊C₃	1	4	yes	complemented
172	D₄	—	13	24	no	—
173	D₄	—	13	24	no	—
174	D₄	—	13	12	no	—
175	D₄	C₂²×D₁₃⋊C₃	1	12	yes	complemented
176	C₂×C₄	—	13	2	no	—
177	C₂×C₄	—	13	24	no	—
178	C₂×C₄	C₂²×D₁₃⋊C₃	1	3	yes	—
179	C₂×C₄	C₂²×D₁₃⋊C₃	1	1	yes	characteristic
180	C₂×C₄	C₂²×D₁₃⋊C₃	1	12	yes	complemented
181	C₂×C₃	—	13	2	no	—
182	C₂×C₃	—	13	1	no	—
183	C₂×C₃	—	26	6	no	—
184	C₂×C₃	—	26	12	no	—
185	C₂²	—	2	6	no	—
186	C₂²	—	13	12	no	—
187	C₂²	—	26	12	no	—
188	C₂²	—	52	24	no	—
189	C₂²	C₂³×D₁₃⋊C₃	1	6	yes	—
190	C₂²	C₂³×D₁₃⋊C₃	1	1	yes	characteristic
191	C₄	—	13	4	no	—
192	C₄	C₂³×D₁₃⋊C₃	1	6	yes	—
193	C₄	C₂³×D₁₃⋊C₃	1	2	yes	—
194	C₃	—	13	1	no	Sylow
195	C₂	—	2	6	no	—
196	C₂	—	26	12	no	—
197	C₂	#11929141	1	2	yes	direct product factor
198	C₂	C₂⁴×D₁₃⋊C₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂³×D₁₃⋊C₃
commutator, cosocle	C₂×C₁₃	C₂⁵×C₃
Frattini	C₂	C₂⁴×D₁₃⋊C₃
Fitting	C₂×C₁₃×#41	C₂×C₃
socle	C₂²×C₁₃	C₂⁴×C₃
hypercenter	C₂×#41	D₁₃⋊C₃
hypocenter	C₁₃	C₂×C₃×H(5,𝔽₂)
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#41×C₁₃⋊C₃	C₂	semidirect
2. #11929141	C₂	direct (N × H)
3. C₂×#10597707	C₂	semidirect
4. C₂×D₄×D₁₃⋊C₃	C₂	semidirect
5. C₂×#6127⋊C₃	C₂	semidirect
6. C₂²×C₄⋊(D₁₃⋊C₃)	C₂	semidirect
7. C₂×#88256	C₃	semidirect
8. C₂³×D₁₃⋊C₃	C₂²	semidirect
9. #41×C₁₃⋊C₃	C₂²	semidirect
10. C₂×Q₈×C₁₃⋊C₃	C₂²	semidirect
11. C₂×D₄×C₁₃⋊C₃	C₂²	semidirect
12. C₂²×C₄×C₁₃⋊C₃	C₂²	semidirect
13. C₂×(C₂²⋊D₁₃)⋊C₃	C₂²	semidirect
14. #10597707	C₂²	semidirect
15. D₄×D₁₃⋊C₃	C₂²	semidirect
16. #6127⋊C₃	C₂²	semidirect
17. C₂×C₄⋊(D₁₃⋊C₃)	C₂²	semidirect
18. C₂×C₄⋊(D₁₃⋊C₃)	C₂²	semidirect
19. C₂×C₄⋊(D₁₃⋊C₃)	C₂²	semidirect
20. C₂×C₄×D₁₃⋊C₃	C₂²	semidirect
21. C₂×(C₁₃⋊C₃)⋊Q₈	C₂²	semidirect
22. C₂×C₁₃×#41	C₂×C₃	semidirect
23. #88256	C₂×C₃	semidirect
24. C₂×#6131	C₂×C₃	semidirect
25. C₂×D₄×D₁₃	C₂×C₃	semidirect
26. C₂×#6127	C₂×C₃	semidirect
27. C₂²×D₅₂	C₂×C₃	semidirect
31. C₂²×D₁₃⋊C₃	C₂³	semidirect
32. Q₈×C₁₃⋊C₃	C₂³	semidirect
33. D₄×C₁₃⋊C₃	C₂³	semidirect
36. C₂×C₄×C₁₃⋊C₃	C₂³	semidirect
37. (C₂²⋊D₁₃)⋊C₃	C₂³	semidirect
39. C₄⋊(D₁₃⋊C₃)	C₂³	semidirect
40. C₄⋊(D₁₃⋊C₃)	C₂³	semidirect
41. C₄×D₁₃⋊C₃	C₂³	semidirect
42. (C₁₃⋊C₃)⋊Q₈	C₂³	semidirect
43. C₂³×D₁₃	C₂²×C₃	semidirect
44. C₁₃×#41	C₂²×C₃	semidirect
45. C₂×Q₈×C₁₃	C₂²×C₃	semidirect
46. C₂×D₄×C₁₃	C₂²×C₃	semidirect
47. C₂²×C₄×C₁₃	C₂²×C₃	semidirect
48. C₂×C₂²⋊D₁₃	C₂²×C₃	semidirect
49. #6131	C₂²×C₃	semidirect
50. D₄×D₁₃	C₂²×C₃	semidirect
51. #6127	C₂²×C₃	semidirect
52. C₂×D₅₂	C₂²×C₃	semidirect
53. C₂×D₅₂	C₂²×C₃	semidirect
54. C₂×D₅₂	C₂²×C₃	semidirect
55. C₂×C₄×D₁₃	C₂²×C₃	semidirect
56. C₂×Dic₂₆	C₂²×C₃	semidirect
70. C₂²×D₁₃	C₂³×C₃	semidirect
71. Q₈×C₁₃	C₂³×C₃	semidirect
72. D₄×C₁₃	C₂³×C₃	semidirect
75. C₂×C₄×C₁₃	C₂³×C₃	semidirect
76. C₂²⋊D₁₃	C₂³×C₃	semidirect
78. D₅₂	C₂³×C₃	semidirect
79. D₅₂	C₂³×C₃	semidirect
80. C₄×D₁₃	C₂³×C₃	semidirect
81. Dic₂₆	C₂³×C₃	semidirect
89. C₂×C₁₃⋊C₃	H(5,𝔽₂)	semidirect
116. C₁₃⋊C₃	C₂×H(5,𝔽₂)	semidirect
120. C₂×#41	D₁₃⋊C₃	semidirect
124. C₂×C₁₃	C₃×H(5,𝔽₂)	semidirect
145. #41	C₂×D₁₃⋊C₃	semidirect
147. C₂×Q₈	C₂×D₁₃⋊C₃	semidirect
153. C₂×D₄	C₂×D₁₃⋊C₃	semidirect
155. C₂²×C₄	C₂×D₁₃⋊C₃	semidirect
156. C₁₃	C₂×C₃×H(5,𝔽₂)	semidirect
171. Q₈	C₂²×D₁₃⋊C₃	semidirect
175. D₄	C₂²×D₁₃⋊C₃	semidirect
180. C₂×C₄	C₂²×D₁₃⋊C₃	semidirect
197. C₂	#11929141	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×#41×C₁₃⋊C₃	2
#11929141	2
C₂×#10597707	2
C₂×D₄×D₁₃⋊C₃	2
C₂×#6127⋊C₃	2
C₂²×C₄⋊(D₁₃⋊C₃)	2
C₂×#88256	3
C₂×C₃×H(5,𝔽₂)	13

Subgroups — C2×#11929141 #72276186

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#11929141 #72276186