Subgroups — C₁₃×#33765 #15720104

Order 3328.

616 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	C₁₃×#3058	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2234	C₂	1	4	yes	maximal, complemented
3	C₁₃×#2218	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2215	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2184	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₈	C₂²	1	2	yes	complemented
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	C₁₃×#535	—	2	2	no	—
10	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
11	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₁₃×#469	—	2	2	no	—
15	C₁₃×#469	—	2	2	no	—
16	C₁₃×#466	C₂²	1	4	yes	—
17	C₁₃×#466	C₂²	1	4	yes	complemented
18	C₁₃×#427	C₂²	1	2	yes	complemented
19	C₁₃×#418	—	2	2	no	—
20	C₁₃×#367	C₂²	1	2	yes	complemented
21	C₁₃×#366	C₂²	1	2	yes	complemented
22	C₁₃×#357	C₂²	1	4	yes	complemented
23	C₁₃×#357	C₂²	1	4	yes	complemented
24	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
25	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
26	C₂²×D₄×C₁₃	—	2	2	no	—
27	C₂×C₁₃×D₈	—	2	2	no	—
28	C₂×C₁₃×D₈	—	4	2	no	—
29	C₂×C₁₃×D₈	—	4	2	no	—
30	C₂×C₁₃×D₈	C₂³	1	4	yes	—
31	C₁₃×#131	—	4	1	no	—
32	C₁₃×#131	—	4	1	no	—
33	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×#127	—	4	2	no	—
38	C₁₃×#124	—	4	2	no	—
39	C₁₃×#121	—	2	4	no	—
40	C₁₃×#121	—	2	4	no	—
41	C₁₃×#120	—	4	2	no	—
42	C₁₃×#120	—	4	2	no	—
43	C₁₃×#117	—	2	2	no	—
44	C₁₃×D₁₆	—	4	2	no	—
45	C₁₃×D₁₆	—	4	2	no	—
46	C₁₃×M₃₂	—	4	1	no	—
47	C₂×C₁₃×C₁₆	—	4	1	no	—
48	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₁₃×#107	C₂³	1	4	yes	—
51	C₁₃×#106	—	2	2	no	—
52	C₁₃×#102	—	2	4	no	—
53	C₁₃×#102	—	2	4	no	—
54	#33765	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂⁴×C₁₃	—	4	2	no	—
56	C₁₃×#41	—	4	1	no	—
57	C₁₃×#41	—	4	1	no	—
58	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
59	C₂×D₄×C₁₃	—	2	4	no	—
60	C₂×D₄×C₁₃	—	4	4	no	—
61	C₂×D₄×C₁₃	—	4	4	no	—
62	C₂×D₄×C₁₃	—	4	4	no	—
63	C₂×D₄×C₁₃	—	4	4	no	—
64	C₂×D₄×C₁₃	—	4	2	no	—
65	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
66	C₂×D₄×C₁₃	D₈	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₁₃×D₈	—	2	4	no	—
70	C₁₃×D₈	—	2	4	no	—
71	C₁₃×D₈	—	4	4	no	—
72	C₁₃×M₁₆	—	4	2	no	—
73	C₁₃×M₁₆	—	4	1	no	—
74	C₂×C₈×C₁₃	—	2	1	no	—
75	C₂×C₈×C₁₃	—	2	1	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₁₃×C₄⋊C₄	—	2	4	no	—
82	C₁₃×C₂²⋊C₄	—	4	2	no	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₄²×C₁₃	—	4	2	no	—
86	C₁₃×C₁₆	—	2	2	no	—
87	C₁₃×C₁₆	—	2	2	no	—
88	#3058	C₂×C₁₃	1	2	yes	complemented
89	#2234	C₂×C₁₃	1	4	yes	complemented
90	#2218	C₂×C₁₃	1	2	yes	complemented
91	#2215	C₂×C₁₃	1	2	yes	complemented
92	#2184	C₂×C₁₃	1	4	yes	complemented
93	#2113	C₂×C₁₃	1	1	yes	characteristic, complemented
94	C₂³×C₁₃	—	4	4	no	—
95	C₂³×C₁₃	—	8	4	no	—
96	C₂³×C₁₃	—	16	4	no	—
97	C₂³×C₁₃	C₂×D₈	1	2	yes	—
98	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
99	Q₈×C₁₃	—	4	1	no	—
100	D₄×C₁₃	—	4	2	no	—
101	D₄×C₁₃	—	4	1	no	—
102	D₄×C₁₃	—	4	4	no	—
103	D₄×C₁₃	—	4	4	no	—
104	D₄×C₁₃	—	8	4	no	—
105	D₄×C₁₃	—	16	4	no	—
106	D₄×C₁₃	—	16	4	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
118	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
119	C₂²×D₈	C₂²×C₁₃	1	2	yes	complemented
120	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
121	#535	—	2	2	no	—
122	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
123	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
124	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic, complemented
125	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
126	#469	—	2	2	no	—
127	#469	—	2	2	no	—
128	#466	C₂²×C₁₃	1	4	yes	—
129	#466	C₂²×C₁₃	1	4	yes	complemented
130	#427	C₂²×C₁₃	1	2	yes	complemented
131	#418	—	2	2	no	—
132	#367	C₂²×C₁₃	1	2	yes	complemented
133	#366	C₂²×C₁₃	1	2	yes	complemented
134	#357	C₂²×C₁₃	1	4	yes	complemented
135	#357	C₂²×C₁₃	1	4	yes	complemented
136	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	8	4	no	—
142	C₂²×C₁₃	—	8	4	no	—
143	C₂²×C₁₃	—	8	4	no	—
144	C₂²×C₁₃	—	16	8	no	—
145	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
146	C₄×C₁₃	—	2	2	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	8	4	no	—
149	C₄×C₁₃	#534	1	1	yes	characteristic
150	C₄×C₁₃	#575	1	1	yes	characteristic
151	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
152	C₂²×D₄	—	2	2	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	4	2	no	—
155	C₂×D₈	—	4	2	no	—
156	C₂×D₈	C₂³×C₁₃	1	4	yes	—
157	#131	—	4	1	no	—
158	#131	—	4	1	no	—
159	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
160	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
161	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
162	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
163	#127	—	4	2	no	—
164	#124	—	4	2	no	—
165	#121	—	2	4	no	—
166	#121	—	2	4	no	—
167	#120	—	4	2	no	—
168	#120	—	4	2	no	—
169	#117	—	2	2	no	—
170	D₁₆	—	4	2	no	—
171	D₁₆	—	4	2	no	—
172	M₃₂	—	4	1	no	—
173	C₂×C₁₆	—	4	1	no	—
174	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
175	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
176	#107	C₂³×C₁₃	1	4	yes	—
177	#106	—	2	2	no	—
178	#102	—	2	4	no	—
179	#102	—	2	4	no	—
180	C₂×C₁₃	—	4	1	no	—
181	C₂×C₁₃	—	4	2	no	—
182	C₂×C₁₃	—	16	4	no	—
183	C₂×C₁₃	#3186	1	1	yes	characteristic
184	C₂×C₁₃	#3415	1	2	yes	—
185	C₂⁴	—	4	2	no	—
186	#41	—	4	1	no	—
187	#41	—	4	1	no	—
188	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
189	C₂×D₄	—	2	4	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	—	4	4	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
196	C₂×D₄	C₁₃×D₈	1	2	yes	—
197	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
198	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
199	D₈	—	2	4	no	—
200	D₈	—	2	4	no	—
201	D₈	—	4	4	no	—
202	M₁₆	—	4	2	no	—
203	M₁₆	—	4	1	no	—
204	C₂×C₈	—	2	1	no	—
205	C₂×C₈	—	2	1	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	2	2	no	—
208	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
210	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
211	C₄⋊C₄	—	2	4	no	—
212	C₂²⋊C₄	—	4	2	no	—
213	C₂²⋊C₄	—	4	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₄²	—	4	2	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	—	2	2	no	—
218	C₁₃	#33765	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	4	4	no	—
220	C₂³	—	8	4	no	—
221	C₂³	—	16	4	no	—
222	C₂³	C₂×C₁₃×D₈	1	2	yes	—
223	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
224	Q₈	—	4	1	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	1	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	8	4	no	—
230	D₄	—	16	4	no	—
231	D₄	—	16	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
238	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
243	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	1	no	—
248	C₂²	—	8	4	no	—
249	C₂²	—	8	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	16	8	no	—
252	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	8	4	no	—
256	C₄	C₁₃×#534	1	1	yes	characteristic
257	C₄	C₁₃×#575	1	1	yes	characteristic
258	C₂	—	4	2	no	—
259	C₂	—	4	1	no	—
260	C₂	—	16	4	no	—
261	C₂	C₁₃×#3186	1	1	yes	characteristic
262	C₂	C₁₃×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3058	C₂	semidirect
2. C₁₃×#2234	C₂	semidirect
3. C₁₃×#2218	C₂	semidirect
4. C₁₃×#2215	C₂	semidirect
5. C₁₃×#2184	C₂	semidirect
6. C₁₃×#2113	C₂	semidirect
7. C₂²×C₁₃×D₈	C₂²	semidirect
12. C₂×C₁₃×M₃₂	C₂²	semidirect
13. C₂²×C₁₃×C₁₆	C₂²	semidirect
17. C₁₃×#466	C₂²	semidirect
18. C₁₃×#427	C₂²	semidirect
20. C₁₃×#367	C₂²	semidirect
21. C₁₃×#366	C₂²	semidirect
22. C₁₃×#357	C₂²	semidirect
23. C₁₃×#357	C₂²	semidirect
24. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
54. #33765	C₁₃	direct (N × H)
88. #3058	C₂×C₁₃	semidirect
89. #2234	C₂×C₁₃	semidirect
90. #2218	C₂×C₁₃	semidirect
91. #2215	C₂×C₁₃	semidirect
92. #2184	C₂×C₁₃	semidirect
93. #2113	C₂×C₁₃	semidirect
119. C₂²×D₈	C₂²×C₁₃	semidirect
124. C₂×M₃₂	C₂²×C₁₃	semidirect
125. C₂²×C₁₆	C₂²×C₁₃	semidirect
129. #466	C₂²×C₁₃	semidirect
130. #427	C₂²×C₁₃	semidirect
132. #367	C₂²×C₁₃	semidirect
133. #366	C₂²×C₁₃	semidirect
134. #357	C₂²×C₁₃	semidirect
135. #357	C₂²×C₁₃	semidirect
136. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
218. C₁₃	#33765	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3058	2
C₁₃×#2234	2
C₁₃×#2218	2
C₁₃×#2215	2
C₁₃×#2184	2
C₁₃×#2113	2
#33765	13

Subgroups — C13×#33765 #15720104

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33765 #15720104