Subgroups — C₂³×C₁₃×#137 #15749421

Order 3328.

11166 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	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×#137	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₁₃×Q₁₆	C₂	1	2	yes	maximal, complemented
5	C₂³×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
6	C₂³×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×#41	—	2	14	no	—
8	C₂²×C₁₃×#41	C₂²	1	28	yes	—
9	C₂³×Q₈×C₁₃	—	2	14	no	—
10	C₂³×Q₈×C₁₃	—	2	1	no	—
11	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
12	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₁₃	C₂²	1	28	yes	complemented
14	C₂³×D₄×C₁₃	—	2	1	no	—
15	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
16	C₂⁴×C₄×C₁₃	—	2	1	no	—
17	C₂⁴×C₄×C₁₃	—	2	1	no	—
18	C₂⁴×C₄×C₁₃	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×#137	C₂²	1	448	yes	direct product factor
20	C₂²×C₁₃×Q₁₆	C₂²	1	56	yes	complemented
21	C₂²×C₁₃×SD₁₆	C₂²	1	56	yes	complemented
22	C₂²×C₁₃×M₁₆	C₂²	1	28	yes	complemented
23	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
24	C₂⁵×C₁₃	—	2	1	no	—
25	C₂⁵×C₁₃	D₄	1	1	yes	characteristic
26	C₂×C₁₃×#41	—	2	168	no	—
27	C₂×C₁₃×#41	C₂³	1	112	yes	—
28	C₂²×Q₈×C₁₃	—	2	14	no	—
29	C₂²×Q₈×C₁₃	—	2	28	no	—
30	C₂²×Q₈×C₁₃	—	2	168	no	—
31	C₂²×Q₈×C₁₃	—	2	7	no	—
32	C₂²×Q₈×C₁₃	C₂³	1	28	yes	—
33	C₂²×Q₈×C₁₃	C₂³	1	14	yes	—
34	C₂²×Q₈×C₁₃	C₂³	1	112	yes	complemented
35	C₂²×D₄×C₁₃	—	2	28	no	—
36	C₂²×D₄×C₁₃	—	2	7	no	—
37	C₂²×D₄×C₁₃	C₂³	1	14	yes	—
38	C₂³×C₄×C₁₃	—	2	28	no	—
39	C₂³×C₄×C₁₃	—	2	1	no	—
40	C₂³×C₄×C₁₃	—	2	2	no	—
41	C₂³×C₄×C₁₃	—	2	28	no	—
42	C₂³×C₄×C₁₃	D₄	1	14	yes	—
43	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
44	C₂³×C₄×C₁₃	C₂³	1	14	yes	—
45	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#137	C₂³	1	512	yes	direct product factor
47	C₂×C₁₃×Q₁₆	C₂³	1	224	yes	complemented
48	C₂×C₁₃×SD₁₆	C₂³	1	224	yes	complemented
49	C₂×C₁₃×M₁₆	C₂³	1	112	yes	complemented
50	C₂²×C₈×C₁₃	C₂³	1	28	yes	—
51	C₂³×#137	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	2	8	no	—
53	C₂⁴×C₁₃	—	2	14	no	—
54	C₂⁴×C₁₃	—	4	8	no	—
55	C₂⁴×C₁₃	C₂×D₄	1	14	yes	—
56	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₁₃×#41	—	2	224	no	—
58	C₁₃×#41	C₂⁴	1	64	yes	—
59	C₂×Q₈×C₁₃	—	2	84	no	—
60	C₂×Q₈×C₁₃	—	2	112	no	—
61	C₂×Q₈×C₁₃	—	2	224	no	—
62	C₂×Q₈×C₁₃	—	2	42	no	—
63	C₂×Q₈×C₁₃	C₂⁴	1	56	yes	—
64	C₂×Q₈×C₁₃	C₂⁴	1	28	yes	—
65	C₂×Q₈×C₁₃	C₂⁴	1	64	yes	complemented
66	C₂×D₄×C₁₃	—	2	112	no	—
67	C₂×D₄×C₁₃	—	2	42	no	—
68	C₂×D₄×C₁₃	C₂⁴	1	28	yes	—
69	C₂²×C₄×C₁₃	—	2	14	no	—
70	C₂²×C₄×C₁₃	—	2	28	no	—
71	C₂²×C₄×C₁₃	—	2	112	no	—
72	C₂²×C₄×C₁₃	—	2	112	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	14	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	7	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	84	yes	—
76	C₂²×C₄×C₁₃	C₂⁴	1	7	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	28	yes	—
78	C₁₃×Q₁₆	C₂⁴	1	128	yes	complemented
79	C₁₃×SD₁₆	C₂⁴	1	128	yes	complemented
80	C₁₃×M₁₆	C₂⁴	1	64	yes	complemented
81	C₂×C₈×C₁₃	C₂⁴	1	56	yes	—
82	C₂³×#41	C₂×C₁₃	1	1	yes	characteristic
83	C₂⁴×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
84	C₂²×#137	C₂×C₁₃	1	56	yes	direct product factor
85	C₂³×Q₁₆	C₂×C₁₃	1	2	yes	complemented
86	C₂³×SD₁₆	C₂×C₁₃	1	2	yes	complemented
87	C₂³×M₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂³×C₁₃	—	2	56	no	—
89	C₂³×C₁₃	—	2	28	no	—
90	C₂³×C₁₃	—	4	56	no	—
91	C₂³×C₁₃	#137	1	8	yes	direct product factor
92	C₂³×C₁₃	C₂²×D₄	1	28	yes	—
93	C₂³×C₁₃	C₂²×D₄	1	7	yes	—
94	Q₈×C₁₃	—	2	56	no	—
95	Q₈×C₁₃	—	2	64	no	—
96	Q₈×C₁₃	—	2	28	no	—
97	Q₈×C₁₃	C₂⁵	1	16	yes	—
98	Q₈×C₁₃	C₂⁵	1	8	yes	—
99	D₄×C₁₃	—	2	64	no	—
100	D₄×C₁₃	—	2	28	no	—
101	D₄×C₁₃	C₂⁵	1	8	yes	—
102	C₂×C₄×C₁₃	—	2	28	no	—
103	C₂×C₄×C₁₃	—	2	56	no	—
104	C₂×C₄×C₁₃	—	2	64	no	—
105	C₂×C₄×C₁₃	—	2	64	no	—
106	C₂×C₄×C₁₃	C₂²×D₄	1	28	yes	—
107	C₂×C₄×C₁₃	C₂²×D₄	1	21	yes	—
108	C₂×C₄×C₁₃	C₂²×D₄	1	56	yes	—
109	C₂×C₄×C₁₃	C₂⁵	1	7	yes	—
110	C₂×C₄×C₁₃	C₂⁵	1	8	yes	—
111	C₈×C₁₃	C₂⁵	1	16	yes	—
112	C₂²×#41	—	2	14	no	—
113	C₂²×#41	C₂²×C₁₃	1	28	yes	—
114	C₂³×Q₈	—	2	1	no	—
115	C₂³×Q₈	—	2	14	no	—
116	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
117	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
118	C₂³×Q₈	C₂²×C₁₃	1	28	yes	complemented
119	C₂³×D₄	—	2	1	no	—
120	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
121	C₂⁴×C₄	—	2	1	no	—
122	C₂⁴×C₄	—	2	1	no	—
123	C₂⁴×C₄	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×#137	C₂²×C₁₃	1	448	yes	direct product factor
125	C₂²×Q₁₆	C₂²×C₁₃	1	56	yes	complemented
126	C₂²×SD₁₆	C₂²×C₁₃	1	56	yes	complemented
127	C₂²×M₁₆	C₂²×C₁₃	1	28	yes	complemented
128	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
129	C₂²×C₁₃	—	2	56	no	—
130	C₂²×C₁₃	—	2	8	no	—
131	C₂²×C₁₃	—	4	56	no	—
132	C₂²×C₁₃	C₂×#137	1	28	yes	direct product factor
133	C₂²×C₁₃	C₂³×D₄	1	8	yes	—
134	C₂²×C₁₃	C₂³×D₄	1	7	yes	—
135	C₄×C₁₃	—	2	8	no	—
136	C₄×C₁₃	—	2	16	no	—
137	C₄×C₁₃	C₂³×D₄	1	8	yes	—
138	C₄×C₁₃	C₂³×D₄	1	7	yes	—
139	C₄×C₁₃	C₂⁶	1	1	yes	characteristic
140	C₂⁵	—	2	1	no	—
141	C₂⁵	D₄×C₁₃	1	1	yes	characteristic
142	C₂×#41	—	2	168	no	—
143	C₂×#41	C₂³×C₁₃	1	112	yes	—
144	C₂²×Q₈	—	2	14	no	—
145	C₂²×Q₈	—	2	28	no	—
146	C₂²×Q₈	—	2	168	no	—
147	C₂²×Q₈	—	2	7	no	—
148	C₂²×Q₈	C₂³×C₁₃	1	28	yes	—
149	C₂²×Q₈	C₂³×C₁₃	1	14	yes	—
150	C₂²×Q₈	C₂³×C₁₃	1	112	yes	complemented
151	C₂²×D₄	—	2	28	no	—
152	C₂²×D₄	—	2	7	no	—
153	C₂²×D₄	C₂³×C₁₃	1	14	yes	—
154	C₂³×C₄	—	2	1	no	—
155	C₂³×C₄	—	2	2	no	—
156	C₂³×C₄	—	2	28	no	—
157	C₂³×C₄	—	2	28	no	—
158	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
159	C₂³×C₄	D₄×C₁₃	1	14	yes	—
160	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
161	C₂³×C₄	C₂³×C₁₃	1	14	yes	—
162	#137	C₂³×C₁₃	1	512	yes	direct product factor
163	C₂×Q₁₆	C₂³×C₁₃	1	224	yes	complemented
164	C₂×SD₁₆	C₂³×C₁₃	1	224	yes	complemented
165	C₂×M₁₆	C₂³×C₁₃	1	112	yes	complemented
166	C₂²×C₈	C₂³×C₁₃	1	28	yes	—
167	C₂×C₁₃	—	2	8	no	—
168	C₂×C₁₃	—	4	8	no	—
169	C₂×C₁₃	C₂²×#137	1	14	yes	direct product factor
170	C₂×C₁₃	C₂⁴×D₄	1	1	yes	characteristic
171	C₂⁴	—	2	8	no	—
172	C₂⁴	—	2	14	no	—
173	C₂⁴	—	4	8	no	—
174	C₂⁴	C₂×D₄×C₁₃	1	14	yes	—
175	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
176	#41	—	2	224	no	—
177	#41	C₂⁴×C₁₃	1	64	yes	—
178	C₂×Q₈	—	2	84	no	—
179	C₂×Q₈	—	2	112	no	—
180	C₂×Q₈	—	2	224	no	—
181	C₂×Q₈	—	2	42	no	—
182	C₂×Q₈	C₂⁴×C₁₃	1	56	yes	—
183	C₂×Q₈	C₂⁴×C₁₃	1	28	yes	—
184	C₂×Q₈	C₂⁴×C₁₃	1	64	yes	complemented
185	C₂×D₄	—	2	112	no	—
186	C₂×D₄	—	2	42	no	—
187	C₂×D₄	C₂⁴×C₁₃	1	28	yes	—
188	C₂²×C₄	—	2	14	no	—
189	C₂²×C₄	—	2	28	no	—
190	C₂²×C₄	—	2	112	no	—
191	C₂²×C₄	—	2	112	no	—
192	C₂²×C₄	C₂×D₄×C₁₃	1	14	yes	—
193	C₂²×C₄	C₂×D₄×C₁₃	1	7	yes	—
194	C₂²×C₄	C₂×D₄×C₁₃	1	84	yes	—
195	C₂²×C₄	C₂⁴×C₁₃	1	7	yes	—
196	C₂²×C₄	C₂⁴×C₁₃	1	28	yes	—
197	Q₁₆	C₂⁴×C₁₃	1	128	yes	complemented
198	SD₁₆	C₂⁴×C₁₃	1	128	yes	complemented
199	M₁₆	C₂⁴×C₁₃	1	64	yes	complemented
200	C₂×C₈	C₂⁴×C₁₃	1	56	yes	—
201	C₁₃	C₂³×#137	1	1	yes	characteristic, direct product factor, Sylow
202	C₂³	—	2	56	no	—
203	C₂³	—	2	28	no	—
204	C₂³	—	4	56	no	—
205	C₂³	C₁₃×#137	1	8	yes	direct product factor
206	C₂³	C₂²×D₄×C₁₃	1	28	yes	—
207	C₂³	C₂²×D₄×C₁₃	1	7	yes	—
208	Q₈	—	2	56	no	—
209	Q₈	—	2	64	no	—
210	Q₈	—	2	28	no	—
211	Q₈	C₂⁵×C₁₃	1	16	yes	—
212	Q₈	C₂⁵×C₁₃	1	8	yes	—
213	D₄	—	2	64	no	—
214	D₄	—	2	28	no	—
215	D₄	C₂⁵×C₁₃	1	8	yes	—
216	C₂×C₄	—	2	28	no	—
217	C₂×C₄	—	2	56	no	—
218	C₂×C₄	—	2	64	no	—
219	C₂×C₄	—	2	64	no	—
220	C₂×C₄	C₂²×D₄×C₁₃	1	28	yes	—
221	C₂×C₄	C₂²×D₄×C₁₃	1	21	yes	—
222	C₂×C₄	C₂²×D₄×C₁₃	1	56	yes	—
223	C₂×C₄	C₂⁵×C₁₃	1	7	yes	—
224	C₂×C₄	C₂⁵×C₁₃	1	8	yes	—
225	C₈	C₂⁵×C₁₃	1	16	yes	—
226	C₂²	—	2	56	no	—
227	C₂²	—	2	8	no	—
228	C₂²	—	4	56	no	—
229	C₂²	C₂×C₁₃×#137	1	28	yes	direct product factor
230	C₂²	C₂³×D₄×C₁₃	1	8	yes	—
231	C₂²	C₂³×D₄×C₁₃	1	7	yes	—
232	C₄	—	2	8	no	—
233	C₄	—	2	16	no	—
234	C₄	C₂³×D₄×C₁₃	1	8	yes	—
235	C₄	C₂³×D₄×C₁₃	1	7	yes	—
236	C₄	C₂⁶×C₁₃	1	1	yes	characteristic
237	C₂	—	2	8	no	—
238	C₂	—	4	8	no	—
239	C₂	C₂²×C₁₃×#137	1	14	yes	direct product factor
240	C₂	C₂⁴×D₄×C₁₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁶×C₁₃
prime radical	C₂³×C₁₃×#41	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂⁴×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂⁴×Q₈×C₁₃	C₂	semidirect
3. C₂²×C₁₃×#137	C₂	direct (N × H)
4. C₂³×C₁₃×Q₁₆	C₂	semidirect
5. C₂³×C₁₃×SD₁₆	C₂	semidirect
6. C₂³×C₁₃×M₁₆	C₂	semidirect
13. C₂³×Q₈×C₁₃	C₂²	semidirect
19. C₂×C₁₃×#137	C₂²	direct (N × H)
20. C₂²×C₁₃×Q₁₆	C₂²	semidirect
21. C₂²×C₁₃×SD₁₆	C₂²	semidirect
22. C₂²×C₁₃×M₁₆	C₂²	semidirect
34. C₂²×Q₈×C₁₃	C₂³	semidirect
46. C₁₃×#137	C₂³	direct (N × H)
47. C₂×C₁₃×Q₁₆	C₂³	semidirect
48. C₂×C₁₃×SD₁₆	C₂³	semidirect
49. C₂×C₁₃×M₁₆	C₂³	semidirect
51. C₂³×#137	C₁₃	direct (N × H)
65. C₂×Q₈×C₁₃	C₂⁴	semidirect
78. C₁₃×Q₁₆	C₂⁴	semidirect
79. C₁₃×SD₁₆	C₂⁴	semidirect
80. C₁₃×M₁₆	C₂⁴	semidirect
83. C₂⁴×Q₈	C₂×C₁₃	semidirect
84. C₂²×#137	C₂×C₁₃	direct (N × H)
85. C₂³×Q₁₆	C₂×C₁₃	semidirect
86. C₂³×SD₁₆	C₂×C₁₃	semidirect
87. C₂³×M₁₆	C₂×C₁₃	semidirect
91. C₂³×C₁₃	#137	direct (N × H)
118. C₂³×Q₈	C₂²×C₁₃	semidirect
124. C₂×#137	C₂²×C₁₃	direct (N × H)
125. C₂²×Q₁₆	C₂²×C₁₃	semidirect
126. C₂²×SD₁₆	C₂²×C₁₃	semidirect
127. C₂²×M₁₆	C₂²×C₁₃	semidirect
132. C₂²×C₁₃	C₂×#137	direct (N × H)
150. C₂²×Q₈	C₂³×C₁₃	semidirect
162. #137	C₂³×C₁₃	direct (N × H)
163. C₂×Q₁₆	C₂³×C₁₃	semidirect
164. C₂×SD₁₆	C₂³×C₁₃	semidirect
165. C₂×M₁₆	C₂³×C₁₃	semidirect
169. C₂×C₁₃	C₂²×#137	direct (N × H)
184. C₂×Q₈	C₂⁴×C₁₃	semidirect
197. Q₁₆	C₂⁴×C₁₃	semidirect
198. SD₁₆	C₂⁴×C₁₃	semidirect
199. M₁₆	C₂⁴×C₁₃	semidirect
201. C₁₃	C₂³×#137	direct (N × H)
205. C₂³	C₁₃×#137	direct (N × H)
229. C₂²	C₂×C₁₃×#137	direct (N × H)
239. C₂	C₂²×C₁₃×#137	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×C₁₃×#41	2
C₂⁴×Q₈×C₁₃	2
C₂²×C₁₃×#137	2
C₂³×C₁₃×Q₁₆	2
C₂³×C₁₃×SD₁₆	2
C₂³×C₁₃×M₁₆	2
C₂³×#137	13

Subgroups — C23×C13×#137 #15749421

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₁₃×#137 #15749421