Subgroups — C₂×C₁₃×#2215 #15719936

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 264 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#466	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2215	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#121	—	2	2	no	—
10	C₂×C₁₃×D₁₆	—	2	4	no	—
11	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
12	C₂²×C₁₃×C₁₆	—	2	4	no	—
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	4	yes	complemented
16	C₁₃×#466	—	2	4	no	—
17	C₁₃×#466	C₂²	1	8	yes	complemented
18	C₂×C₁₃×#107	C₂²	1	2	yes	—
19	C₂×C₁₃×#102	—	2	2	no	—
20	C₁₃×#366	C₂²	1	4	yes	complemented
21	C₁₃×#357	C₂²	1	8	yes	complemented
22	C₂²×D₄×C₁₃	—	2	2	no	—
23	C₂²×D₄×C₁₃	—	4	2	no	—
24	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
25	C₂×C₁₃×D₈	—	2	4	no	—
26	C₂×C₁₃×D₈	—	2	2	no	—
27	C₂×C₁₃×D₈	—	2	4	no	—
28	C₂×C₁₃×D₈	C₂³	1	4	yes	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#121	—	2	8	no	—
36	C₁₃×#121	—	4	4	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
39	C₁₃×D₁₆	—	2	8	no	—
40	C₂×C₁₃×C₁₆	—	2	2	no	—
41	C₂×C₁₃×C₁₆	—	2	8	no	—
42	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
43	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×#102	—	2	8	no	—
48	C₂×#2215	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	—	4	2	no	—
50	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
51	C₂×D₄×C₁₃	—	2	4	no	—
52	C₂×D₄×C₁₃	—	4	8	no	—
53	C₂×D₄×C₁₃	—	4	4	no	—
54	C₂×D₄×C₁₃	—	4	2	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	8	8	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	—	4	2	no	—
60	C₂²×C₄×C₁₃	D₈	1	2	yes	—
61	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×D₈	—	2	4	no	—
65	C₁₃×D₈	—	2	4	no	—
66	C₁₃×D₈	—	4	4	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₁₃	#41	1	2	yes	—
75	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	4	2	no	—
78	C₁₃×C₂²⋊C₄	—	4	8	no	—
79	C₁₃×C₁₆	—	2	2	no	—
80	C₁₃×C₁₆	C₂×D₄	1	4	yes	complemented
81	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
82	C₂³×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
83	C₂×#466	C₂×C₁₃	1	2	yes	complemented
84	#2215	C₂×C₁₃	1	8	yes	direct product factor
85	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
86	C₂×#357	C₂×C₁₃	1	2	yes	complemented
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	—	4	4	no	—
90	C₂³×C₁₃	—	8	4	no	—
91	C₂³×C₁₃	—	8	4	no	—
92	C₂³×C₁₃	—	8	4	no	—
93	C₂³×C₁₃	D₁₆	1	4	yes	complemented
94	C₂³×C₁₃	C₂×D₈	1	2	yes	—
95	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
96	D₄×C₁₃	—	4	4	no	—
97	D₄×C₁₃	—	4	4	no	—
98	D₄×C₁₃	—	8	16	no	—
99	D₄×C₁₃	—	8	4	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	8	4	no	—
106	C₂×C₄×C₁₃	#121	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#135	1	2	yes	—
110	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	#121	1	2	yes	—
113	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
116	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×#121	—	2	2	no	—
118	C₂×D₁₆	—	2	4	no	—
119	C₂×D₁₆	C₂²×C₁₃	1	4	yes	complemented
120	C₂²×C₁₆	—	2	4	no	—
121	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
122	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
123	C₂²×C₁₆	C₂²×C₁₃	1	4	yes	complemented
124	#466	—	2	4	no	—
125	#466	C₂²×C₁₃	1	8	yes	complemented
126	C₂×#107	C₂²×C₁₃	1	2	yes	—
127	C₂×#102	—	2	2	no	—
128	#366	C₂²×C₁₃	1	4	yes	complemented
129	#357	C₂²×C₁₃	1	8	yes	complemented
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	2	8	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	8	4	no	—
134	C₂²×C₁₃	—	8	4	no	—
135	C₂²×C₁₃	—	8	4	no	—
136	C₂²×C₁₃	—	16	8	no	—
137	C₂²×C₁₃	#466	1	2	yes	—
138	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
139	C₂²×C₁₃	C₂×D₁₆	1	4	yes	complemented
140	C₂²×C₁₃	#508	1	2	yes	—
141	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
142	C₄×C₁₃	—	2	2	no	—
143	C₄×C₁₃	—	8	4	no	—
144	C₄×C₁₃	#466	1	2	yes	—
145	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
147	C₂²×D₄	—	2	2	no	—
148	C₂²×D₄	—	4	2	no	—
149	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
150	C₂×D₈	—	2	4	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	—	2	4	no	—
153	C₂×D₈	C₂³×C₁₃	1	4	yes	—
154	C₂²×C₈	—	2	2	no	—
155	C₂²×C₈	—	2	2	no	—
156	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
157	C₂²×C₈	D₄×C₁₃	1	2	yes	—
158	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
159	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
160	#121	—	2	8	no	—
161	#121	—	4	4	no	—
162	C₂×C₄⋊C₄	—	2	2	no	—
163	C₂×C₂²⋊C₄	—	4	2	no	—
164	D₁₆	—	2	8	no	—
165	C₂×C₁₆	—	2	2	no	—
166	C₂×C₁₆	—	2	8	no	—
167	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
168	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
169	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
170	#107	—	2	2	no	—
171	#107	C₂³×C₁₃	1	4	yes	—
172	#102	—	2	8	no	—
173	C₂×C₁₃	—	2	4	no	—
174	C₂×C₁₃	—	16	4	no	—
175	C₂×C₁₃	#2215	1	4	yes	direct product factor
176	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
177	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
178	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
179	C₂⁴	—	4	2	no	—
180	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
181	C₂×D₄	—	2	4	no	—
182	C₂×D₄	—	4	8	no	—
183	C₂×D₄	—	4	4	no	—
184	C₂×D₄	—	4	2	no	—
185	C₂×D₄	—	4	4	no	—
186	C₂×D₄	—	8	8	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
191	C₂²×C₄	C₁₃×D₈	1	2	yes	—
192	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
193	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
194	D₈	—	2	4	no	—
195	D₈	—	2	4	no	—
196	D₈	—	4	4	no	—
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	—	2	4	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
202	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
204	C₂×C₈	C₁₃×#41	1	2	yes	—
205	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
206	C₄⋊C₄	—	2	4	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	4	8	no	—
209	C₁₆	—	2	2	no	—
210	C₁₆	C₂×D₄×C₁₃	1	4	yes	complemented
211	C₁₃	C₂×#2215	1	1	yes	characteristic, direct product factor, Sylow
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	4	no	—
215	C₂³	—	8	4	no	—
216	C₂³	—	8	4	no	—
217	C₂³	—	8	4	no	—
218	C₂³	C₁₃×D₁₆	1	4	yes	complemented
219	C₂³	C₂×C₁₃×D₈	1	2	yes	—
220	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
221	D₄	—	4	4	no	—
222	D₄	—	4	4	no	—
223	D₄	—	8	16	no	—
224	D₄	—	8	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	4	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	4	no	—
231	C₂×C₄	C₁₃×#121	1	2	yes	—
232	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
233	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
234	C₂×C₄	C₁₃×#135	1	2	yes	—
235	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	C₁₃×#121	1	2	yes	—
238	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
239	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	8	no	—
243	C₂²	—	8	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	4	no	—
246	C₂²	—	16	8	no	—
247	C₂²	C₁₃×#466	1	2	yes	—
248	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
249	C₂²	C₂×C₁₃×D₁₆	1	4	yes	complemented
250	C₂²	C₁₃×#508	1	2	yes	—
251	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	8	4	no	—
254	C₄	C₁₃×#466	1	2	yes	—
255	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
256	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
257	C₂	—	2	4	no	—
258	C₂	—	16	4	no	—
259	C₂	C₁₃×#2215	1	4	yes	direct product factor
260	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
261	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
262	C₂	C₂×C₁₃×#508	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×D₁₆	C₂	semidirect
2. C₂³×C₁₃×C₁₆	C₂	semidirect
3. C₂×C₁₃×#466	C₂	semidirect
4. C₁₃×#2215	C₂	direct (N × H)
5. C₂×C₁₃×#366	C₂	semidirect
6. C₂×C₁₃×#357	C₂	semidirect
11. C₂×C₁₃×D₁₆	C₂²	semidirect
15. C₂²×C₁₃×C₁₆	C₂²	semidirect
17. C₁₃×#466	C₂²	semidirect
20. C₁₃×#366	C₂²	semidirect
21. C₁₃×#357	C₂²	semidirect
42. C₂×C₁₃×C₁₆	D₄	semidirect
48. C₂×#2215	C₁₃	direct (N × H)
80. C₁₃×C₁₆	C₂×D₄	semidirect
81. C₂²×D₁₆	C₂×C₁₃	semidirect
82. C₂³×C₁₆	C₂×C₁₃	semidirect
83. C₂×#466	C₂×C₁₃	semidirect
84. #2215	C₂×C₁₃	direct (N × H)
85. C₂×#366	C₂×C₁₃	semidirect
86. C₂×#357	C₂×C₁₃	semidirect
93. C₂³×C₁₃	D₁₆	semidirect
119. C₂×D₁₆	C₂²×C₁₃	semidirect
123. C₂²×C₁₆	C₂²×C₁₃	semidirect
125. #466	C₂²×C₁₃	semidirect
128. #366	C₂²×C₁₃	semidirect
129. #357	C₂²×C₁₃	semidirect
139. C₂²×C₁₃	C₂×D₁₆	semidirect
167. C₂×C₁₆	D₄×C₁₃	semidirect
175. C₂×C₁₃	#2215	direct (N × H)
210. C₁₆	C₂×D₄×C₁₃	semidirect
211. C₁₃	C₂×#2215	direct (N × H)
218. C₂³	C₁₃×D₁₆	semidirect
249. C₂²	C₂×C₁₃×D₁₆	semidirect
259. C₂	C₁₃×#2215	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×C₁₃×D₁₆	2
C₂³×C₁₃×C₁₆	2
C₂×C₁₃×#466	2
C₁₃×#2215	2
C₂×C₁₃×#366	2
C₂×C₁₃×#357	2
C₂×#2215	13

Subgroups — C2×C13×#2215 #15719936

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2215 #15719936