Subgroups — C₁₃×#14714 #15701053

Order 3328.

2078 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#391	C₂	1	4	yes	maximal
3	C₂×C₁₃×#384	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄³×C₁₃	C₂	1	1	yes	maximal, characteristic
5	C₁₃×C₄⋊#95	C₂	1	8	yes	maximal
6	C₂³×Q₈×C₁₃	C₂²	1	2	yes	—
7	C₂×C₁₃×C₄⋊Q₈	—	2	4	no	—
8	C₂×C₁₃×C₄⋊Q₈	—	2	4	no	—
9	C₂×C₁₃×C₄⋊Q₈	C₄	1	8	yes	complemented
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂²×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
13	C₂²×C₄²×C₁₃	C₂²	1	2	yes	—
14	C₂²×C₄²×C₁₃	C₂²	1	4	yes	—
15	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#391	—	2	16	no	—
17	C₁₃×#384	—	2	4	no	—
18	C₂×C₁₃×#95	C₂²	1	16	yes	—
19	C₄³×C₁₃	—	2	4	no	—
20	C₂²×Q₈×C₁₃	—	2	8	no	—
21	C₂²×Q₈×C₁₃	—	2	4	no	—
22	C₂²×Q₈×C₁₃	—	2	4	no	—
23	C₂²×Q₈×C₁₃	—	2	8	no	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	8	yes	—
25	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
26	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
27	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
28	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
29	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
30	C₁₃×C₄⋊Q₈	—	2	32	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
38	C₂×C₄²×C₁₃	—	2	4	no	—
39	C₂×C₄²×C₁₃	—	2	8	no	—
40	C₂×C₄²×C₁₃	—	2	4	no	—
41	C₂×C₄²×C₁₃	—	2	8	no	—
42	C₂×C₄²×C₁₃	C₂×C₄	1	4	yes	—
43	C₂×C₄²×C₁₃	D₄	1	8	yes	—
44	C₂×C₄²×C₁₃	D₄	1	4	yes	—
45	C₂×C₄²×C₁₃	D₄	1	4	yes	—
46	C₂×C₄²×C₁₃	Q₈	1	16	yes	complemented
47	C₁₃×#95	—	2	32	no	—
48	#14714	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₂×Q₈×C₁₃	—	2	16	no	—
51	C₂×Q₈×C₁₃	—	2	32	no	—
52	C₂×Q₈×C₁₃	—	4	16	no	—
53	C₂×Q₈×C₁₃	—	4	8	no	—
54	C₂×Q₈×C₁₃	—	4	8	no	—
55	C₂²×C₄×C₁₃	—	2	4	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	2	8	no	—
58	C₂²×C₄×C₁₃	—	2	8	no	—
59	C₂²×C₄×C₁₃	—	2	4	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	2	8	no	—
62	C₂²×C₄×C₁₃	—	2	8	no	—
63	C₂²×C₄×C₁₃	—	2	4	no	—
64	C₂²×C₄×C₁₃	—	2	16	no	—
65	C₂²×C₄×C₁₃	C₂²×C₄	1	8	yes	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
74	C₂²×C₄×C₁₃	C₂×Q₈	1	16	yes	—
75	C₂²×C₄×C₁₃	C₂×Q₈	1	8	yes	—
76	C₂²×C₄×C₁₃	#41	1	4	yes	—
77	C₂²×C₄×C₁₃	#41	1	8	yes	—
78	C₁₃×C₄⋊C₄	—	2	32	no	—
79	C₁₃×C₄⋊C₄	—	4	16	no	—
80	C₁₃×C₄⋊C₄	—	4	8	no	—
81	C₄²×C₁₃	—	2	16	no	—
82	C₄²×C₁₃	—	2	32	no	—
83	C₄²×C₁₃	C₂²⋊C₄	1	16	yes	—
84	C₂²×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
85	C₂×#391	C₂×C₁₃	1	4	yes	—
86	C₂×#384	C₂×C₁₃	1	1	yes	characteristic
87	C₂×C₄³	C₂×C₁₃	1	1	yes	characteristic
88	C₄⋊#95	C₂×C₁₃	1	8	yes	—
89	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
90	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
91	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
92	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
93	C₂³×C₁₃	C₂²×Q₈	1	4	yes	—
94	C₂³×C₁₃	C₂×#41	1	2	yes	—
95	C₂³×C₁₃	C₂×#41	1	4	yes	—
96	Q₈×C₁₃	—	4	32	no	—
97	C₂×C₄×C₁₃	—	2	8	no	—
98	C₂×C₄×C₁₃	—	2	16	no	—
99	C₂×C₄×C₁₃	—	2	8	no	—
100	C₂×C₄×C₁₃	—	2	16	no	—
101	C₂×C₄×C₁₃	—	2	8	no	—
102	C₂×C₄×C₁₃	—	2	16	no	—
103	C₂×C₄×C₁₃	—	4	8	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	—	4	8	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	16	no	—
109	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
110	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	8	yes	—
111	C₂×C₄×C₁₃	C₄×D₄	1	8	yes	—
112	C₂×C₄×C₁₃	C₄×Q₈	1	16	yes	—
113	C₂×C₄×C₁₃	#121	1	8	yes	—
114	C₂×C₄×C₁₃	C₂²⋊Q₈	1	16	yes	—
115	C₂×C₄×C₁₃	#124	1	4	yes	—
116	C₂×C₄×C₁₃	#124	1	8	yes	—
117	C₂×C₄×C₁₃	#127	1	4	yes	—
118	C₂×C₄×C₁₃	C₄⋊Q₈	1	8	yes	—
119	C₂×C₄×C₁₃	C₄⋊Q₈	1	16	yes	complemented
120	C₂³×Q₈	C₂²×C₁₃	1	2	yes	—
121	C₂×C₄⋊Q₈	—	2	4	no	—
122	C₂×C₄⋊Q₈	—	2	4	no	—
123	C₂×C₄⋊Q₈	C₄×C₁₃	1	8	yes	complemented
124	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
125	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
127	C₂²×C₄²	C₂²×C₁₃	1	2	yes	—
128	C₂²×C₄²	C₂²×C₁₃	1	4	yes	—
129	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
130	#391	—	2	16	no	—
131	#384	—	2	4	no	—
132	C₂×#95	C₂²×C₁₃	1	16	yes	—
133	C₄³	—	2	4	no	—
134	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×C₄×D₄	1	2	yes	—
136	C₂²×C₁₃	C₂×C₄×Q₈	1	4	yes	—
137	C₂²×C₁₃	C₂×#121	1	2	yes	—
138	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	4	yes	—
139	C₂²×C₁₃	C₂×#124	1	2	yes	—
140	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
141	C₂²×C₁₃	C₂×#127	1	1	yes	characteristic
142	C₂²×C₁₃	C₂×C₄⋊Q₈	1	4	yes	—
143	C₂²×C₁₃	C₂×C₄⋊Q₈	1	2	yes	—
144	C₂²×C₁₃	#532	1	4	yes	—
145	C₂²×C₁₃	#533	1	8	yes	—
146	C₄×C₁₃	—	2	16	no	—
147	C₄×C₁₃	—	4	8	no	—
148	C₄×C₁₃	—	8	4	no	—
149	C₄×C₁₃	#390	1	8	yes	—
150	C₄×C₁₃	#391	1	16	yes	complemented
151	C₂²×Q₈	—	2	4	no	—
152	C₂²×Q₈	—	2	4	no	—
153	C₂²×Q₈	—	2	8	no	—
154	C₂²×Q₈	—	2	8	no	—
155	C₂²×Q₈	C₂×C₄×C₁₃	1	8	yes	—
156	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
157	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
158	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
159	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
160	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
161	C₄⋊Q₈	—	2	32	no	—
162	C₂×C₄⋊C₄	—	2	8	no	—
163	C₂×C₄⋊C₄	—	2	8	no	—
164	C₂×C₄⋊C₄	—	2	8	no	—
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	8	no	—
167	C₂×C₄⋊C₄	—	2	16	no	—
168	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	16	yes	—
169	C₂×C₄²	—	2	4	no	—
170	C₂×C₄²	—	2	8	no	—
171	C₂×C₄²	—	2	4	no	—
172	C₂×C₄²	—	2	8	no	—
173	C₂×C₄²	C₂×C₄×C₁₃	1	4	yes	—
174	C₂×C₄²	D₄×C₁₃	1	8	yes	—
175	C₂×C₄²	D₄×C₁₃	1	4	yes	—
176	C₂×C₄²	D₄×C₁₃	1	4	yes	—
177	C₂×C₄²	Q₈×C₁₃	1	16	yes	complemented
178	#95	—	2	32	no	—
179	C₂×C₁₃	C₂×#390	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#391	1	2	yes	—
181	C₂×C₁₃	C₄×C₄⋊Q₈	1	4	yes	—
182	C₂×C₁₃	#2857	1	4	yes	—
183	C₂×C₁₃	#2858	1	1	yes	characteristic
184	C₂×C₁₃	#2862	1	2	yes	—
185	C₂×C₁₃	#2868	1	1	yes	characteristic
186	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
187	C₂×Q₈	—	2	16	no	—
188	C₂×Q₈	—	2	32	no	—
189	C₂×Q₈	—	4	16	no	—
190	C₂×Q₈	—	4	8	no	—
191	C₂×Q₈	—	4	8	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	8	no	—
195	C₂²×C₄	—	2	8	no	—
196	C₂²×C₄	—	2	4	no	—
197	C₂²×C₄	—	2	4	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	8	no	—
200	C₂²×C₄	—	2	4	no	—
201	C₂²×C₄	—	2	16	no	—
202	C₂²×C₄	C₂²×C₄×C₁₃	1	8	yes	—
203	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
204	C₂²×C₄	C₂²×C₄×C₁₃	1	4	yes	—
205	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
208	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
209	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
210	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
211	C₂²×C₄	C₂×Q₈×C₁₃	1	16	yes	—
212	C₂²×C₄	C₂×Q₈×C₁₃	1	8	yes	—
213	C₂²×C₄	C₁₃×#41	1	4	yes	—
214	C₂²×C₄	C₁₃×#41	1	8	yes	—
215	C₄⋊C₄	—	2	32	no	—
216	C₄⋊C₄	—	4	16	no	—
217	C₄⋊C₄	—	4	8	no	—
218	C₄²	—	2	16	no	—
219	C₄²	—	2	32	no	—
220	C₄²	C₁₃×C₂²⋊C₄	1	16	yes	—
221	C₁₃	#14714	1	1	yes	characteristic, direct product factor, Sylow
222	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
223	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
224	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
225	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
226	C₂³	C₂²×Q₈×C₁₃	1	4	yes	—
227	C₂³	C₂×C₁₃×#41	1	2	yes	—
228	C₂³	C₂×C₁₃×#41	1	4	yes	—
229	Q₈	—	4	32	no	—
230	C₂×C₄	—	2	8	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	16	no	—
234	C₂×C₄	—	2	8	no	—
235	C₂×C₄	—	2	16	no	—
236	C₂×C₄	—	4	8	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	—	4	8	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	—	4	16	no	—
242	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
243	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	8	yes	—
244	C₂×C₄	C₄×D₄×C₁₃	1	8	yes	—
245	C₂×C₄	C₄×Q₈×C₁₃	1	16	yes	—
246	C₂×C₄	C₁₃×#121	1	8	yes	—
247	C₂×C₄	C₁₃×C₂²⋊Q₈	1	16	yes	—
248	C₂×C₄	C₁₃×#124	1	4	yes	—
249	C₂×C₄	C₁₃×#124	1	8	yes	—
250	C₂×C₄	C₁₃×#127	1	4	yes	—
251	C₂×C₄	C₁₃×C₄⋊Q₈	1	8	yes	—
252	C₂×C₄	C₁₃×C₄⋊Q₈	1	16	yes	complemented
253	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
254	C₂²	C₂×C₄×D₄×C₁₃	1	2	yes	—
255	C₂²	C₂×C₄×Q₈×C₁₃	1	4	yes	—
256	C₂²	C₂×C₁₃×#121	1	2	yes	—
257	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	4	yes	—
258	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
259	C₂²	C₂×C₁₃×#124	1	2	yes	—
260	C₂²	C₂×C₁₃×#127	1	1	yes	characteristic
261	C₂²	C₂×C₁₃×C₄⋊Q₈	1	2	yes	—
262	C₂²	C₂×C₁₃×C₄⋊Q₈	1	4	yes	—
263	C₂²	C₁₃×#532	1	4	yes	—
264	C₂²	C₁₃×#533	1	8	yes	—
265	C₄	—	2	16	no	—
266	C₄	—	4	8	no	—
267	C₄	—	8	4	no	—
268	C₄	C₁₃×#390	1	8	yes	—
269	C₄	C₁₃×#391	1	16	yes	complemented
270	C₂	C₂×C₁₃×#390	1	1	yes	characteristic
271	C₂	C₂×C₁₃×#391	1	2	yes	—
272	C₂	C₄×C₁₃×C₄⋊Q₈	1	4	yes	—
273	C₂	C₁₃×#2857	1	4	yes	—
274	C₂	C₁₃×#2858	1	1	yes	characteristic
275	C₂	C₁₃×#2862	1	2	yes	—
276	C₂	C₁₃×#2868	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
9. C₂×C₁₃×C₄⋊Q₈	C₄	semidirect
46. C₂×C₄²×C₁₃	Q₈	semidirect
48. #14714	C₁₃	direct (N × H)
119. C₂×C₄×C₁₃	C₄⋊Q₈	semidirect
123. C₂×C₄⋊Q₈	C₄×C₁₃	semidirect
150. C₄×C₁₃	#391	semidirect
177. C₂×C₄²	Q₈×C₁₃	semidirect
221. C₁₃	#14714	direct (N × H)
252. C₂×C₄	C₁₃×C₄⋊Q₈	semidirect
269. C₄	C₁₃×#391	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₃×C₄⋊Q₈	2
C₂×C₁₃×#391	2
C₂×C₁₃×#384	2
C₂×C₄³×C₁₃	2
C₁₃×C₄⋊#95	2
#14714	13

Subgroups — C13×#14714 #15701053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#14714 #15701053