Subgroups — C₁₃×#9701 #15696040

Order 3328.

486 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×#2862	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#1842	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₃×#333	C₂	1	2	yes	maximal
5	C₂×C₁₃×#332	C₂	1	2	yes	maximal
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊C₈	—	2	2	no	—
9	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
11	C₂×C₁₃×#97	—	2	1	no	—
12	C₁₃×#391	—	2	2	no	—
13	C₁₃×#391	C₄	1	4	yes	—
14	C₁₃×#382	—	2	1	no	—
15	C₁₃×#382	—	2	1	no	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#336	—	2	1	no	—
18	C₁₃×#333	—	2	4	no	—
19	C₁₃×#332	—	2	4	no	—
20	C₂²×Q₈×C₁₃	—	2	1	no	—
21	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₁₃×C₄⋊Q₈	—	2	2	no	—
25	C₁₃×C₄⋊Q₈	—	2	2	no	—
26	C₁₃×C₄⋊Q₈	—	2	2	no	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
35	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
37	C₁₃×C₄⋊C₈	—	2	2	no	—
38	C₁₃×C₄⋊C₈	—	2	2	no	—
39	C₁₃×C₄⋊C₈	—	2	2	no	—
40	C₁₃×C₄⋊C₈	—	4	4	no	—
41	C₁₃×C₄⋊C₈	D₄	1	4	yes	—
42	C₁₃×#97	—	4	2	no	—
43	C₁₃×#95	—	2	4	no	—
44	C₁₃×#95	—	2	4	no	—
45	#9701	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×Q₈×C₁₃	—	2	4	no	—
47	C₂×Q₈×C₁₃	—	4	4	no	—
48	C₂×Q₈×C₁₃	—	4	1	no	—
49	C₂×Q₈×C₁₃	—	4	1	no	—
50	C₂×Q₈×C₁₃	—	4	2	no	—
51	C₂×Q₈×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	—	2	4	no	—
60	C₂×C₈×C₁₃	—	4	2	no	—
61	C₂×C₈×C₁₃	—	4	2	no	—
62	C₂×C₈×C₁₃	—	4	2	no	—
63	C₂×C₈×C₁₃	—	4	4	no	—
64	C₁₃×C₄⋊C₄	—	2	4	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₁₃×C₄⋊C₄	—	4	1	no	—
68	C₁₃×C₄⋊C₄	—	4	1	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	C₄²×C₁₃	—	2	2	no	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	4	2	no	—
73	C₄²×C₁₃	SD₁₆	1	2	yes	—
74	C₄²×C₁₃	Q₁₆	1	2	yes	—
75	C₄²×C₁₃	C₂×D₄	1	2	yes	—
76	C₄²×C₁₃	#41	1	2	yes	—
77	#2862	C₂×C₁₃	1	1	yes	characteristic
78	#1842	C₂×C₁₃	1	1	yes	characteristic
79	#1767	C₂×C₁₃	1	1	yes	characteristic
80	C₂×#333	C₂×C₁₃	1	2	yes	—
81	C₂×#332	C₂×C₁₃	1	2	yes	—
82	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₁₃	—	8	4	no	—
84	Q₈×C₁₃	—	8	2	no	—
85	Q₈×C₁₃	—	8	1	no	—
86	Q₈×C₁₃	—	8	1	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	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	—	4	2	no	—
94	C₂×C₄×C₁₃	—	4	2	no	—
95	C₂×C₄×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	—	4	4	no	—
97	C₂×C₄×C₁₃	#103	1	4	yes	—
98	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
99	C₂×C₄×C₁₃	#121	1	2	yes	—
100	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
101	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×Q₁₆	1	1	yes	characteristic
104	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₁₃	#137	1	2	yes	—
106	C₂×C₄×C₁₃	#137	1	2	yes	—
107	C₈×C₁₃	—	4	4	no	—
108	C₈×C₁₃	—	8	2	no	—
109	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
110	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
111	C₂×C₄⋊C₈	—	2	2	no	—
112	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
113	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
114	C₂×#97	—	2	1	no	—
115	#391	—	2	2	no	—
116	#391	C₄×C₁₃	1	4	yes	—
117	#382	—	2	1	no	—
118	#382	—	2	1	no	—
119	C₄³	C₂²×C₁₃	1	1	yes	characteristic
120	#336	—	2	1	no	—
121	#333	—	2	4	no	—
122	#332	—	2	4	no	—
123	C₂²×C₁₃	#390	1	1	yes	characteristic
124	C₂²×C₁₃	#413	1	2	yes	—
125	C₂²×C₁₃	C₂×#103	1	1	yes	characteristic
126	C₂²×C₁₃	#417	1	2	yes	—
127	C₂²×C₁₃	#419	1	1	yes	characteristic
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	4	2	no	—
131	C₄×C₁₃	—	8	2	no	—
132	C₄×C₁₃	—	8	2	no	—
133	C₄×C₁₃	#461	1	2	yes	—
134	C₄×C₁₃	#462	1	2	yes	—
135	C₄×C₁₃	#489	1	2	yes	—
136	C₄×C₁₃	#496	1	1	yes	characteristic
137	C₄×C₁₃	#497	1	1	yes	characteristic
138	C₂²×Q₈	—	2	1	no	—
139	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
140	C₂²×C₈	—	2	2	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₂×C₄⋊C₄	—	2	1	no	—
147	C₂×C₄⋊C₄	—	2	1	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	2	yes	—
150	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
151	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
152	C₂×C₄²	D₄×C₁₃	1	2	yes	—
153	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	C₄⋊C₈	—	2	2	no	—
158	C₄⋊C₈	—	4	4	no	—
159	C₄⋊C₈	D₄×C₁₃	1	4	yes	—
160	#97	—	4	2	no	—
161	#95	—	2	4	no	—
162	#95	—	2	4	no	—
163	C₂×C₁₃	#1550	1	2	yes	—
164	C₂×C₁₃	#1559	1	2	yes	—
165	C₂×C₁₃	#1959	1	1	yes	characteristic
166	C₂×C₁₃	#1967	1	1	yes	characteristic
167	C₂×C₁₃	#1981	1	1	yes	characteristic
168	C₂×Q₈	—	2	4	no	—
169	C₂×Q₈	—	4	4	no	—
170	C₂×Q₈	—	4	1	no	—
171	C₂×Q₈	—	4	1	no	—
172	C₂×Q₈	—	4	2	no	—
173	C₂×Q₈	—	4	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
177	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
178	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
181	C₂×C₈	—	2	4	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₂×C₈	—	4	2	no	—
185	C₂×C₈	—	4	4	no	—
186	C₄⋊C₄	—	2	4	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	1	no	—
190	C₄⋊C₄	—	4	1	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	2	2	no	—
193	C₄²	—	2	2	no	—
194	C₄²	—	4	2	no	—
195	C₄²	C₁₃×SD₁₆	1	2	yes	—
196	C₄²	C₁₃×Q₁₆	1	2	yes	—
197	C₄²	C₂×D₄×C₁₃	1	2	yes	—
198	C₄²	C₁₃×#41	1	2	yes	—
199	C₁₃	#9701	1	1	yes	characteristic, direct product factor, Sylow
200	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
201	Q₈	—	8	4	no	—
202	Q₈	—	8	2	no	—
203	Q₈	—	8	1	no	—
204	Q₈	—	8	1	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	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	4	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	C₁₃×#103	1	4	yes	—
216	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
217	C₂×C₄	C₁₃×#121	1	2	yes	—
218	C₂×C₄	C₁₃×#124	1	1	yes	characteristic
219	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
220	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
221	C₂×C₄	C₂×C₁₃×Q₁₆	1	1	yes	characteristic
222	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
223	C₂×C₄	C₁₃×#137	1	2	yes	—
224	C₂×C₄	C₁₃×#137	1	2	yes	—
225	C₈	—	4	4	no	—
226	C₈	—	8	2	no	—
227	C₂²	C₁₃×#390	1	1	yes	characteristic
228	C₂²	C₁₃×#413	1	2	yes	—
229	C₂²	C₂×C₁₃×#103	1	1	yes	characteristic
230	C₂²	C₁₃×#417	1	2	yes	—
231	C₂²	C₁₃×#419	1	1	yes	characteristic
232	C₄	—	2	4	no	—
233	C₄	—	2	2	no	—
234	C₄	—	4	2	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	C₁₃×#461	1	2	yes	—
238	C₄	C₁₃×#462	1	2	yes	—
239	C₄	C₁₃×#489	1	2	yes	—
240	C₄	C₁₃×#496	1	1	yes	characteristic
241	C₄	C₁₃×#497	1	1	yes	characteristic
242	C₂	C₁₃×#1550	1	2	yes	—
243	C₂	C₁₃×#1559	1	2	yes	—
244	C₂	C₁₃×#1959	1	1	yes	characteristic
245	C₂	C₁₃×#1967	1	1	yes	characteristic
246	C₂	C₁₃×#1981	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
45. #9701	C₁₃	direct (N × H)
199. C₁₃	#9701	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×#2862	2
C₁₃×#1842	2
C₁₃×#1767	2
C₂×C₁₃×#333	2
C₂×C₁₃×#332	2
#9701	13

Subgroups — C13×#9701 #15696040

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9701 #15696040