Subgroups — C₂³×#88257 #16776808

Order 3328.

59806 conjugacy classes of subgroups.

The subgroup lattice has 192 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88257	C₂	1	112	yes	maximal, direct product factor
2	C₂³×#6127	C₂	1	3	yes	maximal, complemented
3	C₂³×#6129	C₂	1	6	yes	maximal, complemented
4	C₂⁴×Dic₂₆	C₂	1	3	yes	maximal, complemented
5	C₂³×Q₈×D₁₃	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×#41	C₂²	1	56	yes	complemented
8	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
10	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
11	C₂³×C₂²⋊D₁₃	C₂²	1	6	yes	—
12	C₂⁴×Dic₁₃	C₂²	1	6	yes	—
13	C₂×#88257	C₂²	1	1792	yes	direct product factor
14	C₂²×Q₈×D₁₃	C₂²	1	112	yes	complemented
15	C₂²×#6129	C₂²	1	336	yes	complemented
16	C₂²×#6127	C₂²	1	168	yes	complemented
17	C₂³×D₅₂	C₂²	1	1	yes	characteristic
18	C₂³×C₄×D₁₃	C₂²	1	6	yes	—
19	C₂³×Dic₂₆	C₂²	1	6	yes	—
20	C₂³×Dic₂₆	C₂²	1	3	yes	—
21	C₂³×Dic₂₆	C₂²	1	168	yes	complemented
22	C₂⁵×C₁₃	C₂³	1	3	yes	—
23	C₂⁴×D₁₃	C₂³	1	2	yes	—
24	C₂×C₁₃×#41	C₂³	1	448	yes	complemented
25	C₂²×Q₈×C₁₃	C₂³	1	28	yes	—
26	C₂²×D₄×C₁₃	C₂³	1	84	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
28	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
29	C₂³×C₄×C₁₃	C₂³	1	84	yes	—
30	C₂²×C₂²⋊D₁₃	C₂³	1	168	yes	—
31	C₂³×Dic₁₃	C₂³	1	6	yes	—
32	C₂³×Dic₁₃	C₂³	1	168	yes	—
33	#88257	C₂³	1	4096	yes	direct product factor
34	C₂×Q₈×D₁₃	C₂³	1	896	yes	complemented
35	C₂×#6129	C₂³	1	2688	yes	complemented
36	C₂×#6127	C₂³	1	1344	yes	complemented
37	C₂²×D₅₂	C₂³	1	28	yes	—
38	C₂²×C₄×D₁₃	C₂³	1	168	yes	—
39	C₂²×Dic₂₆	C₂³	1	84	yes	—
40	C₂²×Dic₂₆	C₂³	1	168	yes	—
41	C₂²×Dic₂₆	C₂³	1	1344	yes	complemented
42	C₂³×2₋¹⁺⁴	—	13	1	no	maximal, Sylow
43	C₂⁴×C₁₃	—	2	24	no	—
44	C₂⁴×C₁₃	C₂⁴	1	42	yes	—
45	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
46	C₂³×D₁₃	—	2	16	no	—
47	C₂³×D₁₃	C₂⁴	1	28	yes	—
48	C₁₃×#41	C₂⁴	1	512	yes	complemented
49	C₂×Q₈×C₁₃	C₂⁴	1	112	yes	—
50	C₂×D₄×C₁₃	C₂⁴	1	336	yes	—
51	C₂²×C₄×C₁₃	C₂⁴	1	14	yes	—
52	C₂²×C₄×C₁₃	C₂⁴	1	42	yes	—
53	C₂²×C₄×C₁₃	C₂⁴	1	336	yes	—
54	C₂×C₂²⋊D₁₃	C₂⁴	1	672	yes	—
55	C₂²×Dic₁₃	C₂⁴	1	84	yes	—
56	C₂²×Dic₁₃	C₂⁴	1	672	yes	—
57	Q₈×D₁₃	C₂⁴	1	1024	yes	complemented
58	#6129	C₂⁴	1	3072	yes	complemented
59	#6127	C₂⁴	1	1536	yes	complemented
60	C₂×D₅₂	C₂⁴	1	112	yes	—
61	C₂×C₄×D₁₃	C₂⁴	1	672	yes	—
62	C₂×Dic₂₆	C₂⁴	1	336	yes	—
63	C₂×Dic₂₆	C₂⁴	1	672	yes	—
64	C₂×Dic₂₆	C₂⁴	1	1536	yes	complemented
65	C₂²×2₋¹⁺⁴	—	13	112	no	—
66	C₂³×#41	—	13	6	no	—
67	C₂³×#41	—	13	3	no	—
68	C₂³×#41	D₁₃	1	1	yes	characteristic, complemented
69	C₂⁴×Q₈	—	13	2	no	—
70	C₂⁴×Q₈	—	13	3	no	—
71	C₂³×C₁₃	—	2	168	no	—
72	C₂³×C₁₃	2₋¹⁺⁴	1	8	yes	complemented
73	C₂³×C₁₃	C₂⁵	1	84	yes	—
74	C₂³×C₁₃	C₂⁵	1	7	yes	—
75	C₂²×D₁₃	—	2	112	no	—
76	C₂²×D₁₃	C₂⁵	1	56	yes	—
77	Q₈×C₁₃	C₂⁵	1	64	yes	—
78	D₄×C₁₃	C₂⁵	1	192	yes	—
79	C₂×C₄×C₁₃	C₂⁵	1	28	yes	—
80	C₂×C₄×C₁₃	C₂⁵	1	84	yes	—
81	C₂×C₄×C₁₃	C₂⁵	1	192	yes	—
82	C₂²⋊D₁₃	C₂⁵	1	384	yes	—
83	C₂×Dic₁₃	C₂⁵	1	168	yes	—
84	C₂×Dic₁₃	C₂⁵	1	384	yes	—
85	D₅₂	C₂⁵	1	64	yes	—
86	C₄×D₁₃	C₂⁵	1	384	yes	—
87	Dic₂₆	C₂⁵	1	192	yes	—
88	Dic₂₆	C₂⁵	1	384	yes	—
89	C₂×2₋¹⁺⁴	—	13	1792	no	—
90	C₂²×#41	—	13	336	no	—
91	C₂²×#41	—	13	168	no	—
92	C₂²×#41	C₂×D₁₃	1	56	yes	complemented
93	C₂³×Q₈	—	13	112	no	—
94	C₂³×Q₈	—	13	6	no	—
95	C₂³×Q₈	—	13	3	no	—
96	C₂³×Q₈	—	13	168	no	—
97	C₂³×Q₈	C₂×D₁₃	1	1	yes	characteristic
98	C₂³×D₄	—	13	6	no	—
99	C₂³×D₄	—	13	1	no	—
100	C₂³×D₄	C₂×D₁₃	1	3	yes	—
101	C₂⁴×C₄	—	13	6	no	—
102	C₂⁴×C₄	—	13	6	no	—
103	C₂⁴×C₄	C₂×D₁₃	1	3	yes	—
104	C₂²×C₁₃	—	2	168	no	—
105	C₂²×C₁₃	C₂×2₋¹⁺⁴	1	28	yes	complemented
106	C₂²×C₁₃	C₂⁶	1	24	yes	—
107	C₂²×C₁₃	C₂⁶	1	7	yes	—
108	C₂×D₁₃	—	2	112	no	—
109	C₂×D₁₃	C₂⁶	1	16	yes	—
110	C₄×C₁₃	C₂⁶	1	8	yes	—
111	C₄×C₁₃	C₂⁶	1	24	yes	—
112	Dic₁₃	C₂⁶	1	48	yes	—
113	C₂⁵	—	13	2	no	—
114	C₂⁵	C₂²×D₁₃	1	3	yes	—
115	2₋¹⁺⁴	—	13	4096	no	—
116	C₂×#41	—	13	2688	no	—
117	C₂×#41	—	13	1344	no	—
118	C₂×#41	C₂²×D₁₃	1	448	yes	complemented
119	C₂²×Q₈	—	13	896	no	—
120	C₂²×Q₈	—	13	168	no	—
121	C₂²×Q₈	—	13	84	no	—
122	C₂²×Q₈	—	13	1344	no	—
123	C₂²×Q₈	C₂²×D₁₃	1	28	yes	—
124	C₂²×D₄	—	13	168	no	—
125	C₂²×D₄	—	13	28	no	—
126	C₂²×D₄	C₂²×D₁₃	1	84	yes	—
127	C₂³×C₄	—	13	168	no	—
128	C₂³×C₄	—	13	6	no	—
129	C₂³×C₄	—	13	168	no	—
130	C₂³×C₄	C₂²×D₁₃	1	84	yes	—
131	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
132	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
133	C₂×C₁₃	—	2	24	no	—
134	C₂×C₁₃	C₂²×2₋¹⁺⁴	1	14	yes	complemented
135	C₂×C₁₃	C₂⁷	1	1	yes	characteristic
136	D₁₃	—	2	16	no	—
137	C₂⁴	—	2	24	no	—
138	C₂⁴	—	13	28	no	—
139	C₂⁴	—	26	16	no	—
140	C₂⁴	C₂³×D₁₃	1	42	yes	—
141	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
142	#41	—	13	3072	no	—
143	#41	—	13	1536	no	—
144	#41	C₂³×D₁₃	1	512	yes	complemented
145	C₂×Q₈	—	13	1024	no	—
146	C₂×Q₈	—	13	672	no	—
147	C₂×Q₈	—	13	336	no	—
148	C₂×Q₈	—	13	1536	no	—
149	C₂×Q₈	C₂³×D₁₃	1	112	yes	—
150	C₂×D₄	—	13	672	no	—
151	C₂×D₄	—	13	112	no	—
152	C₂×D₄	C₂³×D₁₃	1	336	yes	—
153	C₂²×C₄	—	13	84	no	—
154	C₂²×C₄	—	13	672	no	—
155	C₂²×C₄	—	13	672	no	—
156	C₂²×C₄	C₂³×D₁₃	1	42	yes	—
157	C₂²×C₄	C₂³×D₁₃	1	14	yes	—
158	C₂²×C₄	C₂³×D₁₃	1	336	yes	—
159	C₁₃	C₂³×2₋¹⁺⁴	1	1	yes	characteristic, complemented, Sylow
160	C₂³	—	2	168	no	—
161	C₂³	—	13	56	no	—
162	C₂³	—	26	112	no	—
163	C₂³	#88257	1	8	yes	direct product factor
164	C₂³	C₂⁴×D₁₃	1	84	yes	—
165	C₂³	C₂⁴×D₁₃	1	7	yes	—
166	Q₈	—	13	384	no	—
167	Q₈	—	13	192	no	—
168	Q₈	C₂⁴×D₁₃	1	64	yes	—
169	D₄	—	13	384	no	—
170	D₄	—	13	64	no	—
171	D₄	C₂⁴×D₁₃	1	192	yes	—
172	C₂×C₄	—	13	168	no	—
173	C₂×C₄	—	13	384	no	—
174	C₂×C₄	—	13	384	no	—
175	C₂×C₄	C₂⁴×D₁₃	1	84	yes	—
176	C₂×C₄	C₂⁴×D₁₃	1	28	yes	—
177	C₂×C₄	C₂⁴×D₁₃	1	192	yes	—
178	C₂²	—	2	168	no	—
179	C₂²	—	13	16	no	—
180	C₂²	—	26	112	no	—
181	C₂²	C₂×#88257	1	28	yes	direct product factor
182	C₂²	C₂⁵×D₁₃	1	24	yes	—
183	C₂²	C₂⁵×D₁₃	1	7	yes	—
184	C₄	—	13	48	no	—
185	C₄	C₂⁵×D₁₃	1	24	yes	—
186	C₄	C₂⁵×D₁₃	1	8	yes	—
187	C₂	—	2	24	no	—
188	C₂	—	26	16	no	—
189	C₂	C₂²×#88257	1	14	yes	direct product factor
190	C₂	C₂⁶×D₁₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂×C₁₃	C₂⁷
Frattini	C₂	C₂⁶×D₁₃
Fitting	C₂³×C₁₃×#41	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂³×#41	D₁₃
hypocenter	C₁₃	C₂³×2₋¹⁺⁴
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41

Complements

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

N	H ≅ G/N	type
1. C₂²×#88257	C₂	direct (N × H)
2. C₂³×#6127	C₂	semidirect
3. C₂³×#6129	C₂	semidirect
4. C₂⁴×Dic₂₆	C₂	semidirect
5. C₂³×Q₈×D₁₃	C₂	semidirect
6. C₂³×C₁₃×#41	C₂	semidirect
7. C₂²×C₁₃×#41	C₂²	semidirect
13. C₂×#88257	C₂²	direct (N × H)
14. C₂²×Q₈×D₁₃	C₂²	semidirect
15. C₂²×#6129	C₂²	semidirect
16. C₂²×#6127	C₂²	semidirect
21. C₂³×Dic₂₆	C₂²	semidirect
24. C₂×C₁₃×#41	C₂³	semidirect
33. #88257	C₂³	direct (N × H)
34. C₂×Q₈×D₁₃	C₂³	semidirect
35. C₂×#6129	C₂³	semidirect
36. C₂×#6127	C₂³	semidirect
41. C₂²×Dic₂₆	C₂³	semidirect
48. C₁₃×#41	C₂⁴	semidirect
57. Q₈×D₁₃	C₂⁴	semidirect
58. #6129	C₂⁴	semidirect
59. #6127	C₂⁴	semidirect
64. C₂×Dic₂₆	C₂⁴	semidirect
68. C₂³×#41	D₁₃	semidirect
72. C₂³×C₁₃	2₋¹⁺⁴	semidirect
92. C₂²×#41	C₂×D₁₃	semidirect
105. C₂²×C₁₃	C₂×2₋¹⁺⁴	semidirect
118. C₂×#41	C₂²×D₁₃	semidirect
134. C₂×C₁₃	C₂²×2₋¹⁺⁴	semidirect
144. #41	C₂³×D₁₃	semidirect
159. C₁₃	C₂³×2₋¹⁺⁴	semidirect
163. C₂³	#88257	direct (N × H)
181. C₂²	C₂×#88257	direct (N × H)
189. C₂	C₂²×#88257	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×#88257	2
C₂³×#6127	2
C₂³×#6129	2
C₂⁴×Dic₂₆	2
C₂³×Q₈×D₁₃	2
C₂³×C₁₃×#41	2
C₂³×2₋¹⁺⁴	13

Subgroups — C23×#88257 #16776808

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#88257 #16776808