Subgroups — C₁₃×#9724 #15696063

Order 3328.

444 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#1525	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1484	C₂	1	2	yes	maximal, complemented
5	C₈×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#532	C₂²	1	2	yes	—
7	C₁₃×#500	C₄	1	1	yes	characteristic
8	C₁₃×#493	C₄	1	1	yes	characteristic
9	C₁₃×#486	C₄	1	2	yes	—
10	C₁₃×#466	—	2	1	no	—
11	C₁₃×#466	—	2	1	no	—
12	C₄×C₁₃×D₈	—	2	1	no	—
13	C₄×C₁₃×D₈	—	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×#329	—	2	4	no	—
21	C₁₃×D₄⋊C₈	—	2	4	no	—
22	C₁₃×#322	—	2	1	no	—
23	C₈²×C₁₃	—	2	1	no	—
24	C₂×C₁₃×#41	—	2	2	no	—
25	C₂×C₁₃×D₈	—	2	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×M₁₆	—	2	1	no	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₁₃×C₄⋊Q₈	C₂×C₄	1	2	yes	—
32	C₁₃×#127	C₂×C₄	1	2	yes	—
33	C₁₃×#124	—	2	2	no	—
34	C₁₃×#121	—	2	2	no	—
35	C₁₃×#121	—	2	2	no	—
36	C₄×D₄×C₁₃	—	2	2	no	—
37	C₄×D₄×C₁₃	—	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×#102	—	2	2	no	—
44	C₁₃×#102	—	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	#9724	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₃×#41	—	8	4	no	—
57	C₂×Q₈×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	2	no	—
59	C₂×D₄×C₁₃	—	2	2	no	—
60	C₂×D₄×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	4	2	no	—
62	C₂²×C₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₁₃×D₈	—	4	2	no	—
67	C₁₃×D₈	—	4	2	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	2	4	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	4	1	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₂²⋊C₄	—	4	2	no	—
83	C₁₃×C₂²⋊C₄	—	4	2	no	—
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
88	#3169	C₂×C₁₃	1	1	yes	characteristic
89	C₄×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
90	#1525	C₂×C₁₃	1	2	yes	complemented
91	#1484	C₂×C₁₃	1	2	yes	complemented
92	C₈×M₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
93	C₂³×C₁₃	—	4	2	no	—
94	C₂³×C₁₃	#124	1	1	yes	characteristic
95	Q₈×C₁₃	—	8	2	no	—
96	D₄×C₁₃	—	4	2	no	—
97	D₄×C₁₃	—	4	2	no	—
98	D₄×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	8	2	no	—
100	D₄×C₁₃	—	8	4	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	8	4	no	—
107	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
112	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
113	C₈×C₁₃	—	2	4	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	—	2	1	no	—
116	C₈×C₁₃	—	2	1	no	—
117	C₈×C₁₃	—	4	4	no	—
118	C₈×C₁₃	—	4	2	no	—
119	C₈×C₁₃	#104	1	2	yes	—
120	#532	C₂²×C₁₃	1	2	yes	—
121	#500	C₄×C₁₃	1	1	yes	characteristic
122	#493	C₄×C₁₃	1	1	yes	characteristic
123	#486	C₄×C₁₃	1	2	yes	—
124	#466	—	2	1	no	—
125	#466	—	2	1	no	—
126	C₄×D₈	—	2	1	no	—
127	C₄×D₈	—	2	1	no	—
128	#431	C₂²×C₁₃	1	1	yes	characteristic
129	C₂×#108	—	2	1	no	—
130	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
131	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
133	#341	—	2	1	no	—
134	#329	—	2	4	no	—
135	D₄⋊C₈	—	2	4	no	—
136	#322	—	2	1	no	—
137	C₈²	—	2	1	no	—
138	C₂²×C₁₃	—	2	1	no	—
139	C₂²×C₁₃	—	2	1	no	—
140	C₂²×C₁₃	—	2	1	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	#390	1	1	yes	characteristic
145	C₄×C₁₃	—	2	1	no	—
146	C₄×C₁₃	—	2	1	no	—
147	C₄×C₁₃	—	2	1	no	—
148	C₄×C₁₃	—	8	2	no	—
149	C₄×C₁₃	C₂×#104	1	1	yes	characteristic
150	C₄×C₁₃	#421	1	1	yes	characteristic
151	C₄×C₁₃	#443	1	2	yes	—
152	C₄×C₁₃	#472	1	2	yes	—
153	C₂×#41	—	2	2	no	—
154	C₂×D₈	—	2	1	no	—
155	C₂×D₈	—	2	1	no	—
156	C₂×M₁₆	—	2	1	no	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
159	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
160	C₄⋊Q₈	C₂×C₄×C₁₃	1	2	yes	—
161	#127	C₂×C₄×C₁₃	1	2	yes	—
162	#124	—	2	2	no	—
163	#121	—	2	2	no	—
164	#121	—	2	2	no	—
165	C₄×D₄	—	2	2	no	—
166	C₄×D₄	—	2	2	no	—
167	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
168	#108	—	2	4	no	—
169	#107	—	2	1	no	—
170	#107	—	2	1	no	—
171	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
172	#102	—	2	2	no	—
173	#102	—	2	2	no	—
174	C₂²⋊C₈	—	2	1	no	—
175	#97	—	2	2	no	—
176	#97	D₄×C₁₃	1	2	yes	—
177	C₄×C₈	—	2	1	no	—
178	C₄×C₈	—	2	1	no	—
179	C₄×C₈	—	2	2	no	—
180	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
182	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
183	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
184	C₂×C₁₃	—	4	1	no	—
185	C₂×C₁₃	—	16	2	no	—
186	C₂×C₁₃	#1974	1	1	yes	characteristic
187	C₂×C₁₃	#1978	1	1	yes	characteristic
188	C₂×C₁₃	#1980	1	1	yes	characteristic
189	#41	—	8	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	2	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
198	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
199	D₈	—	4	2	no	—
200	D₈	—	4	2	no	—
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	4	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	2	2	no	—
206	C₂×C₈	—	2	2	no	—
207	C₂×C₈	—	4	1	no	—
208	C₂×C₈	—	4	2	no	—
209	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
210	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
211	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
212	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	2	2	no	—
215	C₂²⋊C₄	—	4	2	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
218	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
221	C₁₃	#9724	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	—	4	2	no	—
223	C₂³	C₁₃×#124	1	1	yes	characteristic
224	Q₈	—	8	2	no	—
225	D₄	—	4	2	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	4	no	—
228	D₄	—	8	2	no	—
229	D₄	—	8	4	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	8	4	no	—
236	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
239	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
240	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
241	C₂×C₄	C₁₃×#127	1	1	yes	characteristic
242	C₈	—	2	4	no	—
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	4	no	—
247	C₈	—	4	2	no	—
248	C₈	C₁₃×#104	1	2	yes	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	8	2	no	—
253	C₂²	—	8	2	no	—
254	C₂²	—	8	2	no	—
255	C₂²	C₁₃×#390	1	1	yes	characteristic
256	C₄	—	2	1	no	—
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	8	2	no	—
260	C₄	C₂×C₁₃×#104	1	1	yes	characteristic
261	C₄	C₁₃×#421	1	1	yes	characteristic
262	C₄	C₁₃×#443	1	2	yes	—
263	C₄	C₁₃×#472	1	2	yes	—
264	C₂	—	4	1	no	—
265	C₂	—	16	2	no	—
266	C₂	C₁₃×#1974	1	1	yes	characteristic
267	C₂	C₁₃×#1978	1	1	yes	characteristic
268	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	C₁₃×#3169	C₂
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
2. C₄×C₁₃×#108	C₂	semidirect
3. C₁₃×#1525	C₂	semidirect
4. C₁₃×#1484	C₂	semidirect
5. C₈×C₁₃×M₁₆	C₂	semidirect
55. #9724	C₁₃	direct (N × H)
89. C₄×#108	C₂×C₁₃	semidirect
90. #1525	C₂×C₁₃	semidirect
91. #1484	C₂×C₁₃	semidirect
92. C₈×M₁₆	C₂×C₁₃	semidirect
221. C₁₃	#9724	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×#3169	2
C₄×C₁₃×#108	2
C₁₃×#1525	2
C₁₃×#1484	2
C₈×C₁₃×M₁₆	2
#9724	13

Subgroups — C13×#9724 #15696063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#9724 #15696063