Subgroups — C₁₃×#33607 #15719946

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 260 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	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3058	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2218	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2215	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2140	C₂	1	2	yes	maximal, complemented
8	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
9	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
10	C₁₃×#535	—	2	2	no	—
11	C₂×C₁₃×D₁₆	—	2	4	no	—
12	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
14	C₁₃×#504	—	2	4	no	—
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	2	yes	complemented
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	complemented
19	C₁₃×#469	—	2	4	no	—
20	C₁₃×#466	C₂²	1	8	yes	complemented
21	C₁₃×#427	C₂²	1	2	yes	—
22	C₁₃×#418	—	2	2	no	—
23	C₁₃×#367	C₂²	1	2	yes	complemented
24	C₁₃×#366	C₂²	1	2	yes	complemented
25	C₁₃×#357	C₂²	1	8	yes	complemented
26	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂×C₁₃×D₈	—	2	4	no	—
29	C₂×C₁₃×D₈	—	2	2	no	—
30	C₂×C₁₃×D₈	—	2	4	no	—
31	C₂×C₁₃×D₈	C₂³	1	4	yes	—
32	C₁₃×#131	—	2	2	no	—
33	C₁₃×#131	—	2	2	no	—
34	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
35	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×#127	—	4	2	no	—
39	C₁₃×#124	—	4	2	no	—
40	C₁₃×#121	—	2	8	no	—
41	C₁₃×#120	—	4	4	no	—
42	C₁₃×#117	—	2	2	no	—
43	C₁₃×D₁₆	—	2	4	no	—
44	C₁₃×D₁₆	—	2	4	no	—
45	C₁₃×M₃₂	—	2	2	no	—
46	C₁₃×M₃₂	—	2	4	no	—
47	C₂×C₁₃×C₁₆	—	2	4	no	—
48	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×#107	C₂³	1	4	yes	—
53	C₁₃×#106	—	2	2	no	—
54	C₁₃×#102	—	2	8	no	—
55	#33607	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	—	4	2	no	—
57	C₁₃×#41	—	2	2	no	—
58	C₁₃×#41	—	2	2	no	—
59	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
60	C₂×D₄×C₁₃	—	2	4	no	—
61	C₂×D₄×C₁₃	—	4	8	no	—
62	C₂×D₄×C₁₃	—	4	4	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂×D₄×C₁₃	—	4	4	no	—
65	C₂×D₄×C₁₃	D₈	1	2	yes	—
66	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
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	—	2	2	no	—
73	C₁₃×M₁₆	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃	#41	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	8	no	—
84	C₄²×C₁₃	—	4	2	no	—
85	C₁₃×C₁₆	—	2	2	no	—
86	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
87	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
88	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₂×#504	C₂×C₁₃	1	1	yes	characteristic, complemented
90	#3058	C₂×C₁₃	1	2	yes	complemented
91	#2218	C₂×C₁₃	1	4	yes	complemented
92	#2215	C₂×C₁₃	1	4	yes	complemented
93	#2158	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#2140	C₂×C₁₃	1	2	yes	complemented
95	C₂³×C₁₃	—	4	4	no	—
96	C₂³×C₁₃	—	8	4	no	—
97	C₂³×C₁₃	—	16	4	no	—
98	C₂³×C₁₃	C₂×D₈	1	2	yes	—
99	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	2	no	—
101	D₄×C₁₃	—	2	4	no	—
102	D₄×C₁₃	—	2	2	no	—
103	D₄×C₁₃	—	4	4	no	—
104	D₄×C₁₃	—	4	4	no	—
105	D₄×C₁₃	—	8	4	no	—
106	D₄×C₁₃	—	16	8	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	#121	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#135	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	#121	1	2	yes	—
117	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
118	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
120	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
121	#535	—	2	2	no	—
122	C₂×D₁₆	—	2	4	no	—
123	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
124	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
125	#504	—	2	4	no	—
126	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×M₃₂	C₂²×C₁₃	1	2	yes	complemented
128	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
129	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	complemented
130	#469	—	2	4	no	—
131	#466	C₂²×C₁₃	1	8	yes	complemented
132	#427	C₂²×C₁₃	1	2	yes	—
133	#418	—	2	2	no	—
134	#367	C₂²×C₁₃	1	2	yes	complemented
135	#366	C₂²×C₁₃	1	2	yes	complemented
136	#357	C₂²×C₁₃	1	8	yes	complemented
137	C₂²×C₁₃	—	2	2	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	8	8	no	—
141	C₂²×C₁₃	—	8	4	no	—
142	C₂²×C₁₃	—	16	8	no	—
143	C₂²×C₁₃	#466	1	2	yes	—
144	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	8	4	no	—
147	C₄×C₁₃	#466	1	2	yes	—
148	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
149	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
150	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
151	C₂²×D₄	—	2	2	no	—
152	C₂×D₈	—	2	4	no	—
153	C₂×D₈	—	2	2	no	—
154	C₂×D₈	—	2	4	no	—
155	C₂×D₈	C₂³×C₁₃	1	4	yes	—
156	#131	—	2	2	no	—
157	#131	—	2	2	no	—
158	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
159	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
160	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
161	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
162	#127	—	4	2	no	—
163	#124	—	4	2	no	—
164	#121	—	2	8	no	—
165	#120	—	4	4	no	—
166	#117	—	2	2	no	—
167	D₁₆	—	2	4	no	—
168	D₁₆	—	2	4	no	—
169	M₃₂	—	2	2	no	—
170	M₃₂	—	2	4	no	—
171	C₂×C₁₆	—	2	4	no	—
172	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
173	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
174	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
175	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
176	#107	C₂³×C₁₃	1	4	yes	—
177	#106	—	2	2	no	—
178	#102	—	2	8	no	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	—	4	2	no	—
181	C₂×C₁₃	—	16	4	no	—
182	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
183	C₂×C₁₃	#3415	1	2	yes	—
184	C₂⁴	—	4	2	no	—
185	#41	—	2	2	no	—
186	#41	—	2	2	no	—
187	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
188	C₂×D₄	—	2	4	no	—
189	C₂×D₄	—	4	8	no	—
190	C₂×D₄	—	4	4	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	—	4	4	no	—
193	C₂×D₄	C₁₃×D₈	1	2	yes	—
194	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
195	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
196	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	D₈	—	2	4	no	—
198	D₈	—	2	4	no	—
199	D₈	—	4	4	no	—
200	M₁₆	—	2	2	no	—
201	M₁₆	—	2	4	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
205	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
207	C₂×C₈	C₁₃×#41	1	2	yes	—
208	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	4	no	—
210	C₂²⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	4	8	no	—
212	C₄²	—	4	2	no	—
213	C₁₆	—	2	2	no	—
214	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
215	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
216	C₁₃	#33607	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	4	4	no	—
218	C₂³	—	8	4	no	—
219	C₂³	—	16	4	no	—
220	C₂³	C₂×C₁₃×D₈	1	2	yes	—
221	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
222	Q₈	—	2	2	no	—
223	D₄	—	2	4	no	—
224	D₄	—	2	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	4	no	—
227	D₄	—	8	4	no	—
228	D₄	—	16	8	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	C₁₃×#121	1	2	yes	—
233	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
234	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
235	C₂×C₄	C₁₃×#135	1	2	yes	—
236	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
237	C₈	—	2	2	no	—
238	C₈	C₁₃×#121	1	2	yes	—
239	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
240	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	4	no	—
244	C₂²	—	8	4	no	—
245	C₂²	—	8	8	no	—
246	C₂²	—	16	8	no	—
247	C₂²	C₁₃×#466	1	2	yes	—
248	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
249	C₄	—	2	2	no	—
250	C₄	—	8	4	no	—
251	C₄	C₁₃×#466	1	2	yes	—
252	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
253	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
254	C₂	—	2	2	no	—
255	C₂	—	4	2	no	—
256	C₂	—	16	4	no	—
257	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
258	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. C₂²×C₁₃×D₁₆	C₂	semidirect
2. C₂×C₁₃×#504	C₂	semidirect
3. C₁₃×#3058	C₂	semidirect
4. C₁₃×#2218	C₂	semidirect
5. C₁₃×#2215	C₂	semidirect
6. C₁₃×#2158	C₂	semidirect
7. C₁₃×#2140	C₂	semidirect
16. C₂×C₁₃×M₃₂	C₂²	semidirect
18. C₂²×C₁₃×C₁₆	C₂²	semidirect
20. C₁₃×#466	C₂²	semidirect
23. C₁₃×#367	C₂²	semidirect
24. C₁₃×#366	C₂²	semidirect
25. C₁₃×#357	C₂²	semidirect
48. C₂×C₁₃×C₁₆	D₄	semidirect
55. #33607	C₁₃	direct (N × H)
88. C₂²×D₁₆	C₂×C₁₃	semidirect
89. C₂×#504	C₂×C₁₃	semidirect
90. #3058	C₂×C₁₃	semidirect
91. #2218	C₂×C₁₃	semidirect
92. #2215	C₂×C₁₃	semidirect
93. #2158	C₂×C₁₃	semidirect
94. #2140	C₂×C₁₃	semidirect
127. C₂×M₃₂	C₂²×C₁₃	semidirect
129. C₂²×C₁₆	C₂²×C₁₃	semidirect
131. #466	C₂²×C₁₃	semidirect
134. #367	C₂²×C₁₃	semidirect
135. #366	C₂²×C₁₃	semidirect
136. #357	C₂²×C₁₃	semidirect
172. C₂×C₁₆	D₄×C₁₃	semidirect
216. C₁₃	#33607	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
C₂×C₁₃×#504	2
C₁₃×#3058	2
C₁₃×#2218	2
C₁₃×#2215	2
C₁₃×#2158	2
C₁₃×#2140	2
#33607	13

Subgroups — C13×#33607 #15719946

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33607 #15719946