Subgroups — C₁₃×#9725 #15696064

Order 3328.

400 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1526	C₂	1	2	yes	maximal
4	C₁₃×#1485	C₂	1	2	yes	maximal
5	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#533	C₂²	1	2	yes	—
7	C₁₃×#500	C₄	1	1	yes	characteristic
8	C₁₃×#494	C₄	1	1	yes	characteristic
9	C₁₃×#487	C₄	1	2	yes	—
10	C₁₃×#467	—	2	1	no	—
11	C₁₃×#467	—	2	1	no	—
12	C₄×C₁₃×Q₁₆	—	2	1	no	—
13	C₄×C₁₃×Q₁₆	—	2	1	no	—
14	C₁₃×#431	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#108	—	2	1	no	—
16	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
17	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#341	—	2	1	no	—
20	C₁₃×#330	—	2	4	no	—
21	C₁₃×Q₈⋊C₈	—	2	4	no	—
22	C₁₃×#322	—	2	1	no	—
23	C₈²×C₁₃	—	2	1	no	—
24	C₂×C₁₃×Q₁₆	—	2	1	no	—
25	C₂×C₁₃×Q₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
31	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₁₃×#125	—	2	2	no	—
33	C₁₃×C₂²⋊Q₈	—	2	2	no	—
34	C₁₃×C₂²⋊Q₈	—	2	2	no	—
35	C₄×Q₈×C₁₃	—	2	2	no	—
36	C₄×Q₈×C₁₃	—	2	2	no	—
37	C₁₃×#117	—	2	2	no	—
38	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#108	—	2	4	no	—
40	C₁₃×#107	—	2	1	no	—
41	C₁₃×#107	—	2	1	no	—
42	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
43	C₁₃×#103	—	2	2	no	—
44	C₁₃×#103	—	2	2	no	—
45	C₁₃×C₂²⋊C₈	—	2	1	no	—
46	C₁₃×#97	—	2	2	no	—
47	C₁₃×#97	D₄	1	2	yes	—
48	C₄×C₈×C₁₃	—	2	1	no	—
49	C₄×C₈×C₁₃	—	2	1	no	—
50	C₄×C₈×C₁₃	—	2	2	no	—
51	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
52	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
53	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
54	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
55	#9725	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂×Q₈×C₁₃	—	2	2	no	—
57	C₂×Q₈×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
61	C₁₃×Q₁₆	—	4	2	no	—
62	C₁₃×Q₁₆	—	4	2	no	—
63	C₁₃×M₁₆	—	2	4	no	—
64	C₁₃×M₁₆	—	2	4	no	—
65	C₁₃×M₁₆	—	4	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	2	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	4	1	no	—
70	C₂×C₈×C₁₃	—	4	2	no	—
71	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	2	no	—
77	C₁₃×C₄⋊C₄	—	2	2	no	—
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	4	2	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₂²⋊C₄	—	4	2	no	—
82	C₄²×C₁₃	—	4	2	no	—
83	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	#3170	C₂×C₁₃	1	1	yes	characteristic
88	C₄×#108	C₂×C₁₃	1	1	yes	characteristic
89	#1526	C₂×C₁₃	1	2	yes	—
90	#1485	C₂×C₁₃	1	2	yes	—
91	C₈×M₁₆	C₂×C₁₃	1	1	yes	characteristic
92	C₂³×C₁₃	#124	1	1	yes	characteristic
93	Q₈×C₁₃	—	4	2	no	—
94	Q₈×C₁₃	—	4	2	no	—
95	Q₈×C₁₃	—	4	4	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
108	C₈×C₁₃	—	2	4	no	—
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	—	2	1	no	—
112	C₈×C₁₃	—	4	4	no	—
113	C₈×C₁₃	—	4	2	no	—
114	C₈×C₁₃	#104	1	2	yes	—
115	#533	C₂²×C₁₃	1	2	yes	—
116	#500	C₄×C₁₃	1	1	yes	characteristic
117	#494	C₄×C₁₃	1	1	yes	characteristic
118	#487	C₄×C₁₃	1	2	yes	—
119	#467	—	2	1	no	—
120	#467	—	2	1	no	—
121	C₄×Q₁₆	—	2	1	no	—
122	C₄×Q₁₆	—	2	1	no	—
123	#431	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×#108	—	2	1	no	—
125	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
126	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
128	#341	—	2	1	no	—
129	#330	—	2	4	no	—
130	Q₈⋊C₈	—	2	4	no	—
131	#322	—	2	1	no	—
132	C₈²	—	2	1	no	—
133	C₂²×C₁₃	—	2	1	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	#390	1	1	yes	characteristic
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	8	2	no	—
141	C₄×C₁₃	—	8	2	no	—
142	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
143	C₄×C₁₃	#421	1	1	yes	characteristic
144	C₄×C₁₃	#443	1	2	yes	—
145	C₄×C₁₃	#473	1	2	yes	—
146	C₂×Q₁₆	—	2	1	no	—
147	C₂×Q₁₆	—	2	1	no	—
148	C₂×M₁₆	—	2	1	no	—
149	C₂×M₁₆	—	2	2	no	—
150	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
151	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
152	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
153	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
154	#125	—	2	2	no	—
155	C₂²⋊Q₈	—	2	2	no	—
156	C₂²⋊Q₈	—	2	2	no	—
157	C₄×Q₈	—	2	2	no	—
158	C₄×Q₈	—	2	2	no	—
159	#117	—	2	2	no	—
160	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
161	#108	—	2	4	no	—
162	#107	—	2	1	no	—
163	#107	—	2	1	no	—
164	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
165	#103	—	2	2	no	—
166	#103	—	2	2	no	—
167	C₂²⋊C₈	—	2	1	no	—
168	#97	—	2	2	no	—
169	#97	D₄×C₁₃	1	2	yes	—
170	C₄×C₈	—	2	1	no	—
171	C₄×C₈	—	2	1	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
175	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
176	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
177	C₂×C₁₃	—	4	1	no	—
178	C₂×C₁₃	#1974	1	1	yes	characteristic
179	C₂×C₁₃	#1979	1	1	yes	characteristic
180	C₂×C₁₃	#1980	1	1	yes	characteristic
181	C₂×Q₈	—	2	2	no	—
182	C₂×Q₈	—	2	2	no	—
183	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
185	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
186	Q₁₆	—	4	2	no	—
187	Q₁₆	—	4	2	no	—
188	M₁₆	—	2	4	no	—
189	M₁₆	—	2	4	no	—
190	M₁₆	—	4	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	4	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
197	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
198	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
199	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
200	C₄⋊C₄	—	2	2	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₄⋊C₄	—	2	2	no	—
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	4	2	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₂²⋊C₄	—	4	2	no	—
207	C₄²	—	4	2	no	—
208	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
209	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
210	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
212	C₁₃	#9725	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³	C₁₃×#124	1	1	yes	characteristic
214	Q₈	—	4	2	no	—
215	Q₈	—	4	2	no	—
216	Q₈	—	4	4	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	2	1	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
224	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
226	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
227	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
228	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
229	C₈	—	2	4	no	—
230	C₈	—	2	1	no	—
231	C₈	—	2	1	no	—
232	C₈	—	2	1	no	—
233	C₈	—	4	4	no	—
234	C₈	—	4	2	no	—
235	C₈	C₁₃×#104	1	2	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	C₁₃×#390	1	1	yes	characteristic
240	C₄	—	2	1	no	—
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	2	no	—
245	C₄	C₂×C₁₃×#104	1	1	yes	characteristic
246	C₄	C₁₃×#421	1	1	yes	characteristic
247	C₄	C₁₃×#443	1	2	yes	—
248	C₄	C₁₃×#473	1	2	yes	—
249	C₂	—	4	1	no	—
250	C₂	C₁₃×#1974	1	1	yes	characteristic
251	C₂	C₁₃×#1979	1	1	yes	characteristic
252	C₂	C₁₃×#1980	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	#121
commutator	C₄²	C₂²×C₄×C₁₃
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃
socle	C₂²×C₁₃	#390
prime radical	C₂³×C₁₃	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
55. #9725	C₁₃	direct (N × H)
212. C₁₃	#9725	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×#3170	2
C₄×C₁₃×#108	2
C₁₃×#1526	2
C₁₃×#1485	2
C₈×C₁₃×M₁₆	2
#9725	13

Subgroups — C13×#9725 #15696064

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9725 #15696064