Subgroups — C₁₃×#12788 #15699127

Order 3328.

538 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#364	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2112	C₂	1	2	yes	maximal
4	C₁₃×#1376	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
9	C₂×C₄×C₈×C₁₃	—	2	4	no	—
10	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#364	—	2	4	no	—
15	C₁₃×C₂²⋊C₁₆	—	2	4	no	—
16	C₂³×C₄×C₁₃	—	2	1	no	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	C₈	1	4	yes	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₁₃	—	2	1	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₁₃×M₃₂	—	2	4	no	—
35	C₁₃×M₃₂	—	2	4	no	—
36	C₂×C₁₃×C₁₆	—	2	4	no	—
37	C₄×C₈×C₁₃	—	2	2	no	—
38	C₄×C₈×C₁₃	—	2	2	no	—
39	C₄×C₈×C₁₃	—	2	2	no	—
40	C₄×C₈×C₁₃	D₄	1	4	yes	—
41	#12788	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	—	2	4	no	—
44	C₂²×C₄×C₁₃	—	2	8	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	1	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	—	2	1	no	—
51	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
52	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
53	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	8	no	—
60	C₂×C₈×C₁₃	—	2	4	no	—
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₁₃	#41	1	2	yes	—
71	C₄²×C₁₃	—	2	4	no	—
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	—	2	1	no	—
74	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
75	C₄²×C₁₃	M₁₆	1	2	yes	—
76	C₁₃×C₁₆	—	4	4	no	—
77	C₂²×C₄×C₈	C₂×C₁₃	1	1	yes	characteristic
78	C₂×#364	C₂×C₁₃	1	2	yes	complemented
79	#2112	C₂×C₁₃	1	2	yes	—
80	#1376	C₂×C₁₃	1	2	yes	complemented
81	C₂³×C₁₃	—	2	4	no	—
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	C₄⋊C₈	1	2	yes	—
85	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	16	no	—
87	C₂×C₄×C₁₃	—	2	4	no	—
88	C₂×C₄×C₁₃	—	2	4	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	C₄⋊C₈	1	2	yes	—
98	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
104	C₈×C₁₃	—	2	2	no	—
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	C₂²⋊C₈	1	4	yes	—
107	C₈×C₁₃	#121	1	2	yes	—
108	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
109	C₂³×C₈	C₄×C₁₃	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
111	C₂²×C₄²	C₄×C₁₃	1	1	yes	characteristic
112	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
113	C₂×C₄×C₈	—	2	4	no	—
114	C₂×C₄×C₈	C₄×C₁₃	1	1	yes	characteristic
115	C₂×C₄×C₈	C₄×C₁₃	1	1	yes	characteristic
116	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
118	#364	—	2	4	no	—
119	C₂²⋊C₁₆	—	2	4	no	—
120	C₂²×C₁₃	—	2	4	no	—
121	C₂²×C₁₃	—	2	4	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	2	2	no	—
124	C₂²×C₁₃	—	2	2	no	—
125	C₂²×C₁₃	#364	1	4	yes	complemented
126	C₂²×C₁₃	#407	1	1	yes	characteristic
127	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
128	C₂²×C₁₃	#432	1	1	yes	characteristic
129	C₄×C₁₃	—	2	4	no	—
130	C₄×C₁₃	—	2	4	no	—
131	C₄×C₁₃	—	2	2	no	—
132	C₄×C₁₃	#380	1	1	yes	characteristic
133	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₁₃	#423	1	1	yes	characteristic
135	C₄×C₁₃	#431	1	1	yes	characteristic
136	C₂³×C₄	—	2	1	no	—
137	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	—	2	2	no	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₂²×C₈	C₈×C₁₃	1	4	yes	—
143	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
144	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
145	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
146	C₂²×C₈	D₄×C₁₃	1	2	yes	—
147	C₂²×C₈	Q₈×C₁₃	1	2	yes	—
148	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	2	no	—
151	C₂×C₄²	—	2	2	no	—
152	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
153	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
154	M₃₂	—	2	4	no	—
155	M₃₂	—	2	4	no	—
156	C₂×C₁₆	—	2	4	no	—
157	C₄×C₈	—	2	2	no	—
158	C₄×C₈	—	2	2	no	—
159	C₄×C₈	—	2	2	no	—
160	C₄×C₈	D₄×C₁₃	1	4	yes	—
161	C₂×C₁₃	—	2	1	no	—
162	C₂×C₁₃	—	2	4	no	—
163	C₂×C₁₃	—	2	1	no	—
164	C₂×C₁₃	#1797	1	1	yes	characteristic
165	C₂×C₁₃	C₂×#364	1	2	yes	—
166	C₂⁴	C₁₃×M₁₆	1	1	yes	characteristic
167	C₂²×C₄	—	2	4	no	—
168	C₂²×C₄	—	2	8	no	—
169	C₂²×C₄	—	2	1	no	—
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	—	2	1	no	—
173	C₂²×C₄	—	2	1	no	—
174	C₂²×C₄	—	2	1	no	—
175	C₂²×C₄	C₁₃×C₄⋊C₄	1	2	yes	—
176	C₂²×C₄	C₂×C₈×C₁₃	1	2	yes	—
177	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
178	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	2	8	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
187	C₂×C₈	C₁₃×C₄⋊C₄	1	2	yes	—
188	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
189	C₂×C₈	C₂×C₈×C₁₃	1	2	yes	—
190	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
191	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
193	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
194	C₂×C₈	C₁₃×#41	1	2	yes	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	1	no	—
197	C₄²	—	2	1	no	—
198	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
199	C₄²	C₁₃×M₁₆	1	2	yes	—
200	C₁₆	—	4	4	no	—
201	C₁₃	#12788	1	1	yes	characteristic, direct product factor, Sylow
202	C₂³	—	2	1	no	—
203	C₂³	—	2	1	no	—
204	C₂³	—	2	4	no	—
205	C₂³	C₁₃×C₄⋊C₈	1	2	yes	—
206	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
207	C₂×C₄	—	2	16	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	2	4	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	C₁₃×C₄⋊C₈	1	2	yes	—
219	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
221	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
222	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
223	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
224	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	2	2	no	—
227	C₈	C₁₃×C₂²⋊C₈	1	4	yes	—
228	C₈	C₁₃×#121	1	2	yes	—
229	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
230	C₂²	—	2	4	no	—
231	C₂²	—	2	4	no	—
232	C₂²	—	2	2	no	—
233	C₂²	—	2	2	no	—
234	C₂²	—	2	2	no	—
235	C₂²	C₁₃×#364	1	4	yes	complemented
236	C₂²	C₁₃×#407	1	1	yes	characteristic
237	C₂²	C₂×C₁₃×C₄⋊C₈	1	1	yes	characteristic
238	C₂²	C₁₃×#432	1	1	yes	characteristic
239	C₄	—	2	4	no	—
240	C₄	—	2	4	no	—
241	C₄	—	2	2	no	—
242	C₄	C₁₃×#380	1	1	yes	characteristic
243	C₄	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
244	C₄	C₁₃×#423	1	1	yes	characteristic
245	C₄	C₁₃×#431	1	1	yes	characteristic
246	C₂	—	2	1	no	—
247	C₂	—	2	1	no	—
248	C₂	—	2	4	no	—
249	C₂	C₁₃×#1797	1	1	yes	characteristic
250	C₂	C₂×C₁₃×#364	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×C₄⋊C₈
prime radical	C₂⁴×C₁₃	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#364	C₂	semidirect
4. C₁₃×#1376	C₂	semidirect
41. #12788	C₁₃	direct (N × H)
78. C₂×#364	C₂×C₁₃	semidirect
80. #1376	C₂×C₁₃	semidirect
125. C₂²×C₁₃	#364	semidirect
201. C₁₃	#12788	direct (N × H)
235. C₂²	C₁₃×#364	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₈×C₁₃	2
C₂×C₁₃×#364	2
C₁₃×#2112	2
C₁₃×#1376	2
#12788	13

Subgroups — C13×#12788 #15699127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12788 #15699127