Subgroups — C₂²×D₁₃×#127 #16769604

Order 3328.

24030 conjugacy classes of subgroups.

The subgroup lattice has 216 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₄×D₁₃	C₂	1	6	yes	maximal, complemented
2	C₂²×#88128	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#88194	C₂	1	6	yes	maximal, complemented
4	C₂×D₁₃×#127	C₂	1	48	yes	maximal, direct product factor
5	C₂²×C₄²×D₁₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₁₃×#127	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂⁵×D₁₃	C₂²	1	4	yes	—
8	C₂³×D₄×C₁₃	C₂²	1	6	yes	complemented
9	C₂³×C₂²⋊D₁₃	C₂²	1	12	yes	complemented
10	C₂²×D₄×D₁₃	—	2	96	no	—
11	C₂²×D₄×D₁₃	C₂²	1	144	yes	complemented
12	C₂³×D₅₂	C₂²	1	6	yes	complemented
13	C₂³×C₄×D₁₃	C₂²	1	3	yes	—
14	C₂×C₁₃×#127	C₂²	1	24	yes	complemented
15	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
16	C₂×#88194	C₂²	1	144	yes	complemented
17	C₂²×C₄×Dic₁₃	C₂²	1	3	yes	complemented
18	D₁₃×#127	C₂²	1	256	yes	direct product factor
19	C₂×#88128	C₂²	1	24	yes	complemented
20	C₂×C₄²×D₁₃	C₂²	1	24	yes	complemented
21	C₂⁵×C₁₃	C₂³	1	4	yes	—
22	C₂⁴×D₁₃	—	2	96	no	—
23	C₂⁴×D₁₃	C₂³	1	4	yes	—
24	C₂⁴×D₁₃	C₂³	1	48	yes	—
25	C₂⁴×D₁₃	C₂³	1	1	yes	characteristic
26	C₂²×D₄×C₁₃	—	2	48	no	—
27	C₂²×D₄×C₁₃	C₂³	1	72	yes	complemented
28	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
29	C₂²×C₂²⋊D₁₃	—	2	96	no	—
30	C₂²×C₂²⋊D₁₃	C₂³	1	144	yes	complemented
31	C₂³×Dic₁₃	C₂³	1	3	yes	—
32	C₂×D₄×D₁₃	—	2	1152	no	—
33	C₂×D₄×D₁₃	C₂³	1	384	yes	complemented
34	C₂²×D₅₂	—	2	48	no	—
35	C₂²×D₅₂	C₂³	1	72	yes	complemented
36	C₂²×C₄×D₁₃	D₄	1	48	yes	complemented
37	C₂²×C₄×D₁₃	C₂³	1	36	yes	—
38	C₁₃×#127	C₂³	1	64	yes	complemented
39	C₂×C₄²×C₁₃	C₂³	1	12	yes	complemented
40	#88194	C₂³	1	384	yes	complemented
41	C₂×C₄×Dic₁₃	C₂³	1	36	yes	complemented
42	#88128	C₂³	1	64	yes	complemented
43	C₄²×D₁₃	C₂³	1	64	yes	complemented
44	C₂³×#127	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	—	2	48	no	—
46	C₂⁴×C₁₃	C₂⁴	1	24	yes	—
47	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
48	C₂³×D₁₃	—	2	48	no	—
49	C₂³×D₁₃	—	2	576	no	—
50	C₂³×D₁₃	—	4	256	no	—
51	C₂³×D₁₃	C₂×D₄	1	24	yes	—
52	C₂³×D₁₃	C₂⁴	1	24	yes	—
53	C₂³×D₁₃	C₂⁴	1	64	yes	—
54	C₂³×D₁₃	C₂⁴	1	6	yes	—
55	C₂×D₄×C₁₃	—	2	288	no	—
56	C₂×D₄×C₁₃	C₂⁴	1	96	yes	complemented
57	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	complemented
58	C₂²×C₄×C₁₃	C₂⁴	1	18	yes	—
59	C₂×C₂²⋊D₁₃	—	2	576	no	—
60	C₂×C₂²⋊D₁₃	C₂⁴	1	192	yes	complemented
61	C₂²×Dic₁₃	C₂×D₄	1	24	yes	complemented
62	C₂²×Dic₁₃	C₂⁴	1	18	yes	—
63	D₄×D₁₃	—	2	1536	no	—
64	C₂×D₅₂	—	2	288	no	—
65	C₂×D₅₂	C₂⁴	1	96	yes	complemented
66	C₂×C₄×D₁₃	C₂×D₄	1	288	yes	complemented
67	C₂×C₄×D₁₃	C₂⁴	1	48	yes	—
68	C₄²×C₁₃	C₂⁴	1	16	yes	complemented
69	C₄×Dic₁₃	C₂⁴	1	48	yes	complemented
70	C₂⁴×D₄	—	13	6	no	—
71	C₂²×#127	—	13	1	no	—
72	C₂²×#127	—	13	6	no	—
73	C₂²×#127	—	13	48	no	—
74	C₂²×#127	D₁₃	1	1	yes	characteristic, direct product factor
75	C₂³×C₄²	—	13	1	no	—
76	C₂³×C₁₃	—	2	144	no	—
77	C₂³×C₁₃	—	4	64	no	—
78	C₂³×C₁₃	C₂²×D₄	1	12	yes	—
79	C₂³×C₁₃	C₂⁵	1	16	yes	—
80	C₂³×C₁₃	C₂⁵	1	3	yes	—
81	C₂²×D₁₃	—	2	144	no	—
82	C₂²×D₁₃	—	2	384	no	—
83	C₂²×D₁₃	—	4	64	no	—
84	C₂²×D₁₃	—	4	768	no	—
85	C₂²×D₁₃	#127	1	64	yes	direct product factor
86	C₂²×D₁₃	C₂²×D₄	1	72	yes	—
87	C₂²×D₁₃	C₂⁵	1	16	yes	—
88	C₂²×D₁₃	C₂⁵	1	4	yes	—
89	D₄×C₁₃	—	2	192	no	—
90	C₂×C₄×C₁₃	C₂²×D₄	1	72	yes	complemented
91	C₂×C₄×C₁₃	C₂⁵	1	12	yes	—
92	C₂²⋊D₁₃	—	2	384	no	—
93	C₂×Dic₁₃	C₂²×D₄	1	72	yes	complemented
94	C₂×Dic₁₃	C₂⁵	1	12	yes	—
95	D₅₂	—	2	192	no	—
96	C₄×D₁₃	C₂²×D₄	1	192	yes	complemented
97	C₂⁶	—	13	4	no	—
98	C₂³×D₄	—	13	12	no	—
99	C₂³×D₄	—	13	144	no	—
100	C₂³×D₄	—	13	6	no	—
101	C₂³×D₄	—	26	96	no	—
102	C₂³×D₄	C₂×D₁₃	1	6	yes	complemented
103	C₂⁴×C₄	—	13	3	no	—
104	C₂×#127	—	13	144	no	—
105	C₂×#127	—	13	256	no	—
106	C₂×#127	—	13	24	no	—
107	C₂×#127	C₂×D₁₃	1	24	yes	direct product factor
108	C₂²×C₄²	—	13	3	no	—
109	C₂²×C₄²	—	13	24	no	—
110	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic, complemented
111	C₂²×C₁₃	—	2	48	no	—
112	C₂²×C₁₃	—	4	96	no	—
113	C₂²×C₁₃	C₂×#127	1	16	yes	complemented
114	C₂²×C₁₃	C₂³×D₄	1	18	yes	—
115	C₂²×C₁₃	C₂⁶	1	1	yes	characteristic
116	C₂×D₁₃	—	2	48	no	—
117	C₂×D₁₃	—	4	96	no	—
118	C₂×D₁₃	—	4	256	no	—
119	C₂×D₁₃	C₂×#127	1	96	yes	direct product factor
120	C₂×D₁₃	C₂³×D₄	1	24	yes	—
121	C₄×C₁₃	C₂³×D₄	1	24	yes	complemented
122	Dic₁₃	C₂³×D₄	1	24	yes	complemented
123	C₂⁵	—	13	1	no	—
124	C₂⁵	—	13	48	no	—
125	C₂⁵	—	13	4	no	—
126	C₂⁵	—	26	96	no	—
127	C₂⁵	C₂²×D₁₃	1	4	yes	—
128	C₂²×D₄	—	2	48	no	—
129	C₂²×D₄	—	13	144	no	—
130	C₂²×D₄	—	13	384	no	—
131	C₂²×D₄	—	13	72	no	—
132	C₂²×D₄	—	26	1152	no	—
133	C₂²×D₄	—	26	96	no	—
134	C₂²×D₄	—	26	48	no	—
135	C₂²×D₄	C₂²×D₁₃	1	72	yes	complemented
136	C₂³×C₄	—	13	3	no	—
137	C₂³×C₄	—	13	48	no	—
138	C₂³×C₄	—	13	36	no	—
139	C₂³×C₄	C₂²×D₁₃	1	3	yes	—
140	#127	—	13	384	no	—
141	#127	—	13	64	no	—
142	#127	C₂²×D₁₃	1	64	yes	direct product factor
143	C₂×C₄²	—	13	36	no	—
144	C₂×C₄²	—	13	64	no	—
145	C₂×C₄²	C₂²×D₁₃	1	12	yes	complemented
146	C₂×C₁₃	—	4	16	no	—
147	C₂×C₁₃	C₂²×#127	1	12	yes	complemented
148	C₂×C₁₃	C₂⁴×D₄	1	3	yes	—
149	D₁₃	—	4	16	no	—
150	D₁₃	C₂²×#127	1	16	yes	direct product factor
151	C₂⁴	—	2	48	no	—
152	C₂⁴	—	13	24	no	—
153	C₂⁴	—	13	6	no	—
154	C₂⁴	—	13	64	no	—
155	C₂⁴	—	13	24	no	—
156	C₂⁴	—	26	576	no	—
157	C₂⁴	—	26	48	no	—
158	C₂⁴	—	52	256	no	—
159	C₂⁴	C₂³×D₁₃	1	24	yes	—
160	C₂⁴	C₂³×D₁₃	1	1	yes	characteristic
161	C₂×D₄	—	2	288	no	—
162	C₂×D₄	—	13	192	no	—
163	C₂×D₄	—	13	96	no	—
164	C₂×D₄	—	26	1536	no	—
165	C₂×D₄	—	26	576	no	—
166	C₂×D₄	—	26	288	no	—
167	C₂×D₄	C₂³×D₁₃	1	96	yes	complemented
168	C₂²×C₄	—	13	24	no	—
169	C₂²×C₄	—	13	18	no	—
170	C₂²×C₄	—	13	288	no	—
171	C₂²×C₄	—	13	48	no	—
172	C₂²×C₄	D₄×D₁₃	1	24	yes	complemented
173	C₂²×C₄	C₂³×D₁₃	1	18	yes	—
174	C₄²	—	13	48	no	—
175	C₄²	C₂³×D₁₃	1	16	yes	complemented
176	C₁₃	C₂³×#127	1	1	yes	characteristic, complemented, Sylow
177	C₂³	—	2	144	no	—
178	C₂³	—	4	64	no	—
179	C₂³	—	13	72	no	—
180	C₂³	—	13	4	no	—
181	C₂³	—	13	64	no	—
182	C₂³	—	13	16	no	—
183	C₂³	—	26	384	no	—
184	C₂³	—	26	144	no	—
185	C₂³	—	52	768	no	—
186	C₂³	—	52	64	no	—
187	C₂³	C₂×D₄×D₁₃	1	12	yes	—
188	C₂³	C₂⁴×D₁₃	1	16	yes	—
189	C₂³	C₂⁴×D₁₃	1	3	yes	—
190	D₄	—	2	192	no	—
191	D₄	—	26	384	no	—
192	D₄	—	26	192	no	—
193	C₂×C₄	—	13	72	no	—
194	C₂×C₄	—	13	12	no	—
195	C₂×C₄	—	13	192	no	—
196	C₂×C₄	C₂×D₄×D₁₃	1	72	yes	complemented
197	C₂×C₄	C₂⁴×D₁₃	1	12	yes	—
198	C₂²	—	2	48	no	—
199	C₂²	—	4	96	no	—
200	C₂²	—	13	24	no	—
201	C₂²	—	13	96	no	—
202	C₂²	—	26	48	no	—
203	C₂²	—	52	256	no	—
204	C₂²	—	52	96	no	—
205	C₂²	D₁₃×#127	1	16	yes	direct product factor
206	C₂²	C₂²×D₄×D₁₃	1	18	yes	—
207	C₂²	C₂⁵×D₁₃	1	1	yes	characteristic
208	C₄	—	13	24	no	—
209	C₄	C₂²×D₄×D₁₃	1	24	yes	complemented
210	C₂	—	4	16	no	—
211	C₂	—	13	16	no	—
212	C₂	—	52	16	no	—
213	C₂	C₂×D₁₃×#127	1	12	yes	direct product factor
214	C₂	C₂³×D₄×D₁₃	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂³×D₁₃
commutator, cosocle	C₂²×C₁₃	C₂⁶
Frattini	C₂²	C₂⁵×D₁₃
Fitting	C₂²×C₁₃×#127	C₂
socle	C₂⁴×C₁₃	C₂⁴
hypercenter	C₂²×#127	D₁₃
hypocenter	C₁₃	C₂³×#127
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#127

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×D₁₃	C₂	semidirect
2. C₂²×#88128	C₂	semidirect
3. C₂²×#88194	C₂	semidirect
4. C₂×D₁₃×#127	C₂	direct (N × H)
5. C₂²×C₄²×D₁₃	C₂	semidirect
6. C₂²×C₁₃×#127	C₂	semidirect
8. C₂³×D₄×C₁₃	C₂²	semidirect
9. C₂³×C₂²⋊D₁₃	C₂²	semidirect
11. C₂²×D₄×D₁₃	C₂²	semidirect
12. C₂³×D₅₂	C₂²	semidirect
14. C₂×C₁₃×#127	C₂²	semidirect
15. C₂²×C₄²×C₁₃	C₂²	semidirect
16. C₂×#88194	C₂²	semidirect
17. C₂²×C₄×Dic₁₃	C₂²	semidirect
18. D₁₃×#127	C₂²	direct (N × H)
19. C₂×#88128	C₂²	semidirect
20. C₂×C₄²×D₁₃	C₂²	semidirect
27. C₂²×D₄×C₁₃	C₂³	semidirect
30. C₂²×C₂²⋊D₁₃	C₂³	semidirect
33. C₂×D₄×D₁₃	C₂³	semidirect
35. C₂²×D₅₂	C₂³	semidirect
36. C₂²×C₄×D₁₃	D₄	semidirect
38. C₁₃×#127	C₂³	semidirect
39. C₂×C₄²×C₁₃	C₂³	semidirect
40. #88194	C₂³	semidirect
41. C₂×C₄×Dic₁₃	C₂³	semidirect
42. #88128	C₂³	semidirect
43. C₄²×D₁₃	C₂³	semidirect
56. C₂×D₄×C₁₃	C₂⁴	semidirect
57. C₂²×C₄×C₁₃	C₂×D₄	semidirect
60. C₂×C₂²⋊D₁₃	C₂⁴	semidirect
61. C₂²×Dic₁₃	C₂×D₄	semidirect
65. C₂×D₅₂	C₂⁴	semidirect
66. C₂×C₄×D₁₃	C₂×D₄	semidirect
68. C₄²×C₁₃	C₂⁴	semidirect
69. C₄×Dic₁₃	C₂⁴	semidirect
74. C₂²×#127	D₁₃	direct (N × H)
85. C₂²×D₁₃	#127	direct (N × H)
90. C₂×C₄×C₁₃	C₂²×D₄	semidirect
93. C₂×Dic₁₃	C₂²×D₄	semidirect
96. C₄×D₁₃	C₂²×D₄	semidirect
102. C₂³×D₄	C₂×D₁₃	semidirect
107. C₂×#127	C₂×D₁₃	direct (N × H)
110. C₂²×C₄²	C₂×D₁₃	semidirect
113. C₂²×C₁₃	C₂×#127	semidirect
119. C₂×D₁₃	C₂×#127	direct (N × H)
121. C₄×C₁₃	C₂³×D₄	semidirect
122. Dic₁₃	C₂³×D₄	semidirect
135. C₂²×D₄	C₂²×D₁₃	semidirect
142. #127	C₂²×D₁₃	direct (N × H)
145. C₂×C₄²	C₂²×D₁₃	semidirect
147. C₂×C₁₃	C₂²×#127	semidirect
150. D₁₃	C₂²×#127	direct (N × H)
167. C₂×D₄	C₂³×D₁₃	semidirect
172. C₂²×C₄	D₄×D₁₃	semidirect
175. C₄²	C₂³×D₁₃	semidirect
176. C₁₃	C₂³×#127	semidirect
196. C₂×C₄	C₂×D₄×D₁₃	semidirect
205. C₂²	D₁₃×#127	direct (N × H)
209. C₄	C₂²×D₄×D₁₃	semidirect
213. C₂	C₂×D₁₃×#127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₄×D₁₃	2
C₂²×#88128	2
C₂²×#88194	2
C₂×D₁₃×#127	2
C₂²×C₄²×D₁₃	2
C₂²×C₁₃×#127	2
C₂³×#127	13

Subgroups — C22×D13×#127 #16769604

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₁₃×#127 #16769604