Subgroups — C₁₃×#29226 #15715565

Order 3328.

714 conjugacy classes of subgroups.

The subgroup lattice has 248 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#125	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2099	C₂	1	4	yes	maximal
4	C₁₃×#1992	C₂	1	2	yes	maximal
5	C₁₃×#1987	C₂	1	4	yes	maximal
6	C₁₃×#1836	C₂	1	2	yes	maximal
7	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊Q₈	—	2	2	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₁₃×#389	C₂²	1	2	yes	—
14	C₁₃×#387	—	2	2	no	—
15	C₁₃×#384	—	2	2	no	—
16	C₁₃×#384	C₂²	1	2	yes	—
17	C₁₃×#384	C₂²	1	2	yes	—
18	C₁₃×#382	C₂²	1	1	yes	characteristic
19	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
20	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄³×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#340	C₂²	1	8	yes	—
23	C₁₃×#340	C₂²	1	8	yes	—
24	C₁₃×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
27	C₁₃×C₄⋊Q₈	—	2	8	no	—
28	C₁₃×C₄⋊Q₈	—	4	8	no	—
29	C₁₃×#125	—	2	2	no	—
30	C₁₃×#125	Q₈	1	4	yes	complemented
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃	—	2	2	no	—
40	C₂×C₄²×C₁₃	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	—	2	4	no	—
46	C₁₃×#97	—	2	4	no	—
47	C₁₃×#95	—	2	2	no	—
48	C₁₃×#95	—	2	2	no	—
49	C₁₃×#95	—	2	2	no	—
50	#29226	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂×Q₈×C₁₃	—	4	8	no	—
52	C₂×Q₈×C₁₃	—	4	4	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₈×C₁₃	—	2	4	no	—
62	C₂×C₈×C₁₃	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	4	no	—
64	C₁₃×C₄⋊C₄	—	2	8	no	—
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	2	8	no	—
67	C₁₃×C₄⋊C₄	—	2	4	no	—
68	C₁₃×C₄⋊C₄	—	4	8	no	—
69	C₁₃×C₄⋊C₄	—	4	4	no	—
70	C₁₃×C₄⋊C₄	—	4	4	no	—
71	C₁₃×C₄⋊C₄	—	4	8	no	—
72	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
73	C₄²×C₁₃	—	2	4	no	—
74	C₄²×C₁₃	—	2	4	no	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	4	4	no	—
77	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
78	C₄²×C₁₃	#41	1	2	yes	—
79	C₄²×C₁₃	#41	1	4	yes	—
80	#2868	C₂×C₁₃	1	1	yes	characteristic
81	C₄×#125	C₂×C₁₃	1	1	yes	characteristic
82	#2099	C₂×C₁₃	1	4	yes	—
83	#1992	C₂×C₁₃	1	2	yes	—
84	#1987	C₂×C₁₃	1	4	yes	—
85	#1836	C₂×C₁₃	1	2	yes	—
86	#1767	C₂×C₁₃	1	1	yes	characteristic
87	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
88	Q₈×C₁₃	—	8	8	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	4	2	no	—
96	C₂×C₄×C₁₃	—	4	2	no	—
97	C₂×C₄×C₁₃	—	4	4	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	8	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
103	C₂×C₄×C₁₃	#124	1	2	yes	—
104	C₂×C₄×C₁₃	#124	1	2	yes	—
105	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
106	C₂×C₄×C₁₃	#137	1	4	yes	—
107	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
110	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
111	C₈×C₁₃	—	4	4	no	—
112	C₂×C₄⋊Q₈	—	2	2	no	—
113	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
114	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
115	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
116	C₂×#97	C₂²×C₁₃	1	2	yes	—
117	#389	C₂²×C₁₃	1	2	yes	—
118	#387	—	2	2	no	—
119	#384	—	2	2	no	—
120	#384	C₂²×C₁₃	1	2	yes	—
121	#384	C₂²×C₁₃	1	2	yes	—
122	#382	C₂²×C₁₃	1	1	yes	characteristic
123	C₄×C₄⋊C₄	—	2	2	no	—
124	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
125	C₄³	C₂²×C₁₃	1	1	yes	characteristic
126	#340	C₂²×C₁₃	1	8	yes	—
127	#340	C₂²×C₁₃	1	8	yes	—
128	#336	C₂²×C₁₃	1	2	yes	—
129	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
130	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
131	C₂²×C₁₃	#536	1	1	yes	characteristic
132	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
133	C₂²×C₁₃	C₂×#137	1	2	yes	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	4	no	—
137	C₄×C₁₃	—	4	4	no	—
138	C₄×C₁₃	—	8	4	no	—
139	C₄×C₁₃	—	8	2	no	—
140	C₄×C₁₃	#491	1	4	yes	—
141	C₄×C₁₃	#556	1	2	yes	—
142	C₄×C₁₃	#557	1	2	yes	—
143	C₂²×Q₈	—	2	2	no	—
144	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
145	C₄⋊Q₈	—	2	8	no	—
146	C₄⋊Q₈	—	4	8	no	—
147	#125	—	2	2	no	—
148	#125	Q₈×C₁₃	1	4	yes	complemented
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	4	no	—
151	C₂×C₄⋊C₄	—	2	4	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
155	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
156	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
157	C₂×C₄²	—	2	2	no	—
158	C₂×C₄²	D₄×C₁₃	1	2	yes	—
159	C₂×C₄²	D₄×C₁₃	1	2	yes	—
160	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
162	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
163	C₄⋊C₈	—	2	4	no	—
164	#97	—	2	4	no	—
165	#95	—	2	2	no	—
166	#95	—	2	2	no	—
167	#95	—	2	2	no	—
168	C₂×C₁₃	#2568	1	1	yes	characteristic
169	C₂×C₁₃	#3079	1	2	yes	—
170	C₂×C₁₃	C₂×#491	1	1	yes	characteristic
171	C₂×C₁₃	#3138	1	1	yes	characteristic
172	C₂×C₁₃	#3238	1	2	yes	—
173	C₂×Q₈	—	4	8	no	—
174	C₂×Q₈	—	4	4	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	—	2	2	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
179	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	—	2	4	no	—
185	C₂×C₈	—	2	4	no	—
186	C₄⋊C₄	—	2	8	no	—
187	C₄⋊C₄	—	2	2	no	—
188	C₄⋊C₄	—	2	8	no	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	4	8	no	—
191	C₄⋊C₄	—	4	4	no	—
192	C₄⋊C₄	—	4	8	no	—
193	C₄⋊C₄	—	4	4	no	—
194	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
195	C₄²	—	2	4	no	—
196	C₄²	—	2	4	no	—
197	C₄²	—	2	2	no	—
198	C₄²	—	4	4	no	—
199	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
200	C₄²	C₁₃×#41	1	2	yes	—
201	C₄²	C₁₃×#41	1	4	yes	—
202	C₁₃	#29226	1	1	yes	characteristic, direct product factor, Sylow
203	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
204	Q₈	—	8	8	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	2	no	—
210	C₂×C₄	—	2	4	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	8	no	—
216	C₂×C₄	—	4	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
219	C₂×C₄	C₁₃×#124	1	2	yes	—
220	C₂×C₄	C₁₃×#124	1	2	yes	—
221	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
222	C₂×C₄	C₁₃×#137	1	4	yes	—
223	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
224	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
225	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
226	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
227	C₈	—	4	4	no	—
228	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
229	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
230	C₂²	C₁₃×#536	1	1	yes	characteristic
231	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
232	C₂²	C₂×C₁₃×#137	1	2	yes	—
233	C₄	—	2	4	no	—
234	C₄	—	2	2	no	—
235	C₄	—	2	4	no	—
236	C₄	—	4	4	no	—
237	C₄	—	8	4	no	—
238	C₄	—	8	2	no	—
239	C₄	C₁₃×#491	1	4	yes	—
240	C₄	C₁₃×#556	1	2	yes	—
241	C₄	C₁₃×#557	1	2	yes	—
242	C₂	C₁₃×#2568	1	1	yes	characteristic
243	C₂	C₁₃×#3079	1	2	yes	—
244	C₂	C₂×C₁₃×#491	1	1	yes	characteristic
245	C₂	C₁₃×#3138	1	1	yes	characteristic
246	C₂	C₁₃×#3238	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
30. C₁₃×#125	Q₈	semidirect
50. #29226	C₁₃	direct (N × H)
148. #125	Q₈×C₁₃	semidirect
202. C₁₃	#29226	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×#2868	2
C₄×C₁₃×#125	2
C₁₃×#2099	2
C₁₃×#1992	2
C₁₃×#1987	2
C₁₃×#1836	2
C₁₃×#1767	2
#29226	13

Subgroups — C13×#29226 #15715565

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29226 #15715565