Subgroups — C₁₃×#12982 #15699321

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2931	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2117	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2115	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1376	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
7	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
8	C₁₃×#532	C₄	1	1	yes	characteristic
9	C₁₃×#504	—	2	4	no	—
10	C₂×C₁₃×M₃₂	—	2	2	no	—
11	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#436	—	2	2	no	—
14	C₈×D₄×C₁₃	—	2	2	no	—
15	C₁₃×#423	C₄	1	1	yes	characteristic
16	C₁₃×#408	C₄	1	1	yes	characteristic
17	C₁₃×#408	C₄	1	1	yes	characteristic
18	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₁₃×#350	—	2	4	no	—
20	C₁₃×#349	—	2	2	no	—
21	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
23	C₁₃×#131	—	2	2	no	—
24	C₁₃×#131	—	2	2	no	—
25	C₁₃×#131	—	2	2	no	—
26	C₁₃×#131	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	1	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	D₄	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	C₈	1	1	yes	characteristic
36	C₁₃×#127	C₈	1	1	yes	characteristic
37	C₁₃×#124	C₈	1	1	yes	characteristic
38	C₁₃×#124	C₈	1	1	yes	characteristic
39	C₁₃×#121	—	2	2	no	—
40	C₄×D₄×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₁₃×M₃₂	—	2	4	no	—
43	C₁₃×M₃₂	—	2	2	no	—
44	C₁₃×M₃₂	—	2	2	no	—
45	C₁₃×M₃₂	—	2	2	no	—
46	C₁₃×M₃₂	D₄	1	4	yes	complemented
47	C₂×C₁₃×C₁₆	—	2	4	no	—
48	C₂×C₁₃×C₁₆	—	2	2	no	—
49	C₂×C₁₃×C₁₆	—	2	2	no	—
50	C₁₃×C₄⋊C₈	—	2	2	no	—
51	C₁₃×C₂²⋊C₈	—	2	2	no	—
52	C₁₃×C₂²⋊C₈	—	2	2	no	—
53	C₄×C₈×C₁₃	—	2	1	no	—
54	C₄×C₈×C₁₃	—	2	1	no	—
55	#12982	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₃×#41	—	2	2	no	—
57	C₁₃×#41	—	2	2	no	—
58	C₁₃×#41	—	2	2	no	—
59	C₁₃×#41	—	2	2	no	—
60	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
61	C₂×Q₈×C₁₃	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	2	1	no	—
64	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
65	C₂×D₄×C₁₃	C₂×C₈	1	1	yes	characteristic
66	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
67	C₂²×C₄×C₁₃	—	2	1	no	—
68	C₂²×C₄×C₁₃	—	2	1	no	—
69	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
71	C₁₃×M₁₆	—	2	2	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	2	2	no	—
74	C₁₃×M₁₆	—	4	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	4	1	no	—
81	C₂×C₈×C₁₃	—	4	1	no	—
82	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₂²⋊C₄	—	2	2	no	—
89	C₁₃×C₂²⋊C₄	—	2	2	no	—
90	C₄²×C₁₃	—	2	1	no	—
91	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
92	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
93	C₁₃×C₁₆	—	2	4	no	—
94	C₁₃×C₁₆	—	4	2	no	—
95	#3407	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#2931	C₂×C₁₃	1	1	yes	characteristic
97	#2117	C₂×C₁₃	1	2	yes	complemented
98	#2115	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#1376	C₂×C₁₃	1	2	yes	complemented
100	C₂³×C₁₃	—	2	1	no	—
101	C₂³×C₁₃	C₂²⋊C₈	1	2	yes	—
102	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
103	Q₈×C₁₃	—	2	2	no	—
104	Q₈×C₁₃	—	2	2	no	—
105	D₄×C₁₃	—	2	4	no	—
106	D₄×C₁₃	—	2	2	no	—
107	D₄×C₁₃	—	2	2	no	—
108	D₄×C₁₃	—	4	4	no	—
109	D₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	—	4	1	no	—
118	C₂×C₄×C₁₃	C₂²⋊C₈	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#131	1	2	yes	—
123	C₈×C₁₃	—	2	2	no	—
124	C₈×C₁₃	—	4	1	no	—
125	C₈×C₁₃	—	4	1	no	—
126	C₈×C₁₃	#120	1	1	yes	characteristic
127	C₈×C₁₃	#121	1	2	yes	—
128	C₈×C₁₃	#123	1	1	yes	characteristic
129	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
131	#532	C₄×C₁₃	1	1	yes	characteristic
132	#504	—	2	4	no	—
133	C₂×M₃₂	—	2	2	no	—
134	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
135	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
136	#436	—	2	2	no	—
137	C₈×D₄	—	2	2	no	—
138	#423	C₄×C₁₃	1	1	yes	characteristic
139	#408	C₄×C₁₃	1	1	yes	characteristic
140	#408	C₄×C₁₃	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	#350	—	2	4	no	—
143	#349	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	—	4	1	no	—
149	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂²×C₁₃	#435	1	2	yes	—
151	C₄×C₁₃	—	2	2	no	—
152	C₄×C₁₃	—	2	2	no	—
153	C₄×C₁₃	—	4	1	no	—
154	C₄×C₁₃	—	4	1	no	—
155	C₄×C₁₃	#386	1	1	yes	characteristic
156	C₄×C₁₃	#408	1	1	yes	characteristic
157	C₄×C₁₃	C₈×D₄	1	2	yes	—
158	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
159	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
160	#131	—	2	2	no	—
161	#131	—	2	2	no	—
162	#131	—	2	2	no	—
163	#131	—	2	2	no	—
164	C₂×M₁₆	—	2	1	no	—
165	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
166	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
167	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
168	C₂²×C₈	—	2	1	no	—
169	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
170	C₂²×C₈	D₄×C₁₃	1	2	yes	—
171	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
172	C₄⋊Q₈	C₈×C₁₃	1	1	yes	characteristic
173	#127	C₈×C₁₃	1	1	yes	characteristic
174	#124	C₈×C₁₃	1	1	yes	characteristic
175	#124	C₈×C₁₃	1	1	yes	characteristic
176	#121	—	2	2	no	—
177	C₄×D₄	—	2	2	no	—
178	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
179	M₃₂	—	2	4	no	—
180	M₃₂	—	2	2	no	—
181	M₃₂	—	2	2	no	—
182	M₃₂	—	2	2	no	—
183	M₃₂	D₄×C₁₃	1	4	yes	complemented
184	C₂×C₁₆	—	2	4	no	—
185	C₂×C₁₆	—	2	2	no	—
186	C₂×C₁₆	—	2	2	no	—
187	C₄⋊C₈	—	2	2	no	—
188	C₂²⋊C₈	—	2	2	no	—
189	C₂²⋊C₈	—	2	2	no	—
190	C₄×C₈	—	2	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₂×C₁₃	—	2	1	no	—
193	C₂×C₁₃	—	2	2	no	—
194	C₂×C₁₃	—	4	2	no	—
195	C₂×C₁₃	—	8	1	no	—
196	C₂×C₁₃	#1869	1	1	yes	characteristic
197	#41	—	2	2	no	—
198	#41	—	2	2	no	—
199	#41	—	2	2	no	—
200	#41	—	2	2	no	—
201	C₂×Q₈	C₂×C₈×C₁₃	1	1	yes	characteristic
202	C₂×Q₈	C₂×C₈×C₁₃	1	1	yes	characteristic
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	2	1	no	—
205	C₂×D₄	C₂×C₈×C₁₃	1	1	yes	characteristic
206	C₂×D₄	C₂×C₈×C₁₃	1	1	yes	characteristic
207	C₂×D₄	C₁₃×M₁₆	1	2	yes	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
211	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	M₁₆	—	2	2	no	—
213	M₁₆	—	2	4	no	—
214	M₁₆	—	2	2	no	—
215	M₁₆	—	4	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
224	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
225	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₁₃×#41	1	2	yes	—
228	C₄⋊C₄	—	2	2	no	—
229	C₂²⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₄²	—	2	1	no	—
232	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
233	C₄²	C₂×C₈×C₁₃	1	1	yes	characteristic
234	C₁₆	—	2	4	no	—
235	C₁₆	—	4	2	no	—
236	C₁₃	#12982	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	—	2	1	no	—
238	C₂³	C₁₃×C₂²⋊C₈	1	2	yes	—
239	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
240	Q₈	—	2	2	no	—
241	Q₈	—	2	2	no	—
242	D₄	—	2	4	no	—
243	D₄	—	2	2	no	—
244	D₄	—	2	2	no	—
245	D₄	—	4	4	no	—
246	D₄	—	4	2	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	1	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	4	1	no	—
255	C₂×C₄	C₁₃×C₂²⋊C₈	1	2	yes	—
256	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
258	C₂×C₄	C₂²×C₈×C₁₃	1	1	yes	characteristic
259	C₂×C₄	C₁₃×#131	1	2	yes	—
260	C₈	—	2	2	no	—
261	C₈	—	4	1	no	—
262	C₈	—	4	1	no	—
263	C₈	C₁₃×#120	1	1	yes	characteristic
264	C₈	C₁₃×#121	1	2	yes	—
265	C₈	C₁₃×#123	1	1	yes	characteristic
266	C₂²	—	2	2	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂×C₁₃×C₂²⋊C₈	1	1	yes	characteristic
272	C₂²	C₁₃×#435	1	2	yes	—
273	C₄	—	2	2	no	—
274	C₄	—	2	2	no	—
275	C₄	—	4	1	no	—
276	C₄	—	4	1	no	—
277	C₄	C₁₃×#386	1	1	yes	characteristic
278	C₄	C₁₃×#408	1	1	yes	characteristic
279	C₄	C₈×D₄×C₁₃	1	2	yes	—
280	C₂	—	2	1	no	—
281	C₂	—	2	2	no	—
282	C₂	—	4	2	no	—
283	C₂	—	8	1	no	—
284	C₂	C₁₃×#1869	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#120
commutator	C₂×C₄	C₂²×C₈×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1869
prime radical	C₁₃×#532	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3407	C₂	semidirect
3. C₁₃×#2117	C₂	semidirect
4. C₁₃×#2115	C₂	semidirect
5. C₁₃×#1376	C₂	semidirect
46. C₁₃×M₃₂	D₄	semidirect
55. #12982	C₁₃	direct (N × H)
95. #3407	C₂×C₁₃	semidirect
97. #2117	C₂×C₁₃	semidirect
98. #2115	C₂×C₁₃	semidirect
99. #1376	C₂×C₁₃	semidirect
183. M₃₂	D₄×C₁₃	semidirect
236. C₁₃	#12982	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×#3407	2
C₁₃×#2931	2
C₁₃×#2117	2
C₁₃×#2115	2
C₁₃×#1376	2
#12982	13

Subgroups — C13×#12982 #15699321

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12982 #15699321