Subgroups — C₁₃×#29126 #15715465

Order 3328.

686 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2864	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2099	C₂	1	2	yes	maximal
4	C₁₃×#2043	C₂	1	2	yes	maximal
5	C₁₃×#1933	C₂	1	4	yes	maximal
6	C₁₃×#1805	C₂	1	2	yes	maximal
7	C₁₃×#1805	C₂	1	2	yes	maximal
8	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#103	C₂²	1	4	yes	—
13	C₂×C₁₃×#103	C₂²	1	4	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#391	C₂²	1	2	yes	—
16	C₁₃×#389	—	2	2	no	—
17	C₁₃×#389	C₂²	1	2	yes	—
18	C₁₃×#389	C₂²	1	2	yes	—
19	C₁₃×#387	—	2	2	no	—
20	C₁₃×#387	—	2	2	no	—
21	C₁₃×#384	C₂²	1	1	yes	characteristic
22	C₁₃×#382	C₂²	1	1	yes	characteristic
23	C₁₃×#382	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₄³×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#340	C₂²	1	4	yes	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
31	C₁₃×C₄⋊Q₈	—	2	4	no	—
32	C₄×Q₈×C₁₃	—	2	4	no	—
33	C₄×Q₈×C₁₃	—	2	4	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₃	D₄	1	2	yes	—
44	C₂×C₄²×C₁₃	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×C₄⋊C₈	—	2	4	no	—
49	C₁₃×#103	—	2	8	no	—
50	C₁₃×#103	—	4	4	no	—
51	C₄×C₈×C₁₃	—	4	2	no	—
52	C₁₃×#95	—	2	4	no	—
53	C₁₃×#95	—	2	1	no	—
54	C₁₃×#95	—	2	1	no	—
55	C₁₃×#95	—	2	1	no	—
56	C₁₃×#95	—	2	1	no	—
57	C₁₃×#95	—	2	2	no	—
58	C₁₃×#95	—	2	2	no	—
59	#29126	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	#41	1	4	yes	—
66	C₂²×C₄×C₁₃	—	2	2	no	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₂×C₈×C₁₃	—	2	4	no	—
76	C₂×C₈×C₁₃	—	2	4	no	—
77	C₂×C₈×C₁₃	—	2	4	no	—
78	C₁₃×C₄⋊C₄	—	2	8	no	—
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	4	4	no	—
85	C₁₃×C₄⋊C₄	—	4	4	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₄²×C₁₃	—	2	4	no	—
89	C₄²×C₁₃	—	2	4	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	2	1	no	—
92	C₄²×C₁₃	—	2	1	no	—
93	C₄²×C₁₃	—	4	4	no	—
94	C₄²×C₁₃	#41	1	4	yes	—
95	#2864	C₂×C₁₃	1	1	yes	characteristic
96	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
97	#2099	C₂×C₁₃	1	2	yes	—
98	#2043	C₂×C₁₃	1	2	yes	—
99	#1933	C₂×C₁₃	1	4	yes	—
100	#1805	C₂×C₁₃	1	2	yes	—
101	#1805	C₂×C₁₃	1	2	yes	—
102	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
104	Q₈×C₁₃	—	2	4	no	—
105	Q₈×C₁₃	—	2	4	no	—
106	Q₈×C₁₃	—	4	2	no	—
107	Q₈×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	4	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	#124	1	2	yes	—
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#135	1	4	yes	—
128	C₂×C₄×C₁₃	#137	1	4	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
131	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
133	C₈×C₁₃	—	4	4	no	—
134	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
135	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
136	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
137	C₂×#103	C₂²×C₁₃	1	4	yes	—
138	C₂×#103	C₂²×C₁₃	1	4	yes	—
139	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
140	#391	C₂²×C₁₃	1	2	yes	—
141	#389	—	2	2	no	—
142	#389	C₂²×C₁₃	1	2	yes	—
143	#389	C₂²×C₁₃	1	2	yes	—
144	#387	—	2	2	no	—
145	#387	—	2	2	no	—
146	#384	C₂²×C₁₃	1	1	yes	characteristic
147	#382	C₂²×C₁₃	1	1	yes	characteristic
148	#382	C₂²×C₁₃	1	1	yes	characteristic
149	C₄×C₄⋊C₄	—	2	2	no	—
150	C₄³	C₂²×C₁₃	1	1	yes	characteristic
151	#340	C₂²×C₁₃	1	4	yes	—
152	#340	C₂²×C₁₃	1	4	yes	—
153	#336	C₂²×C₁₃	1	2	yes	—
154	C₂²×C₁₃	#525	1	1	yes	characteristic
155	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
156	C₂²×C₁₃	#537	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#135	1	2	yes	—
158	C₂²×C₁₃	C₂×#137	1	2	yes	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	4	no	—
163	C₄×C₁₃	—	4	4	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	—	8	2	no	—
167	C₄×C₁₃	#552	1	2	yes	—
168	C₄×C₁₃	#553	1	2	yes	—
169	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
170	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
171	C₄⋊Q₈	—	2	4	no	—
172	C₄×Q₈	—	2	4	no	—
173	C₄×Q₈	—	2	4	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	2	no	—
177	C₂×C₄⋊C₄	—	2	2	no	—
178	C₂×C₄⋊C₄	—	2	2	no	—
179	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
180	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
181	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	D₄×C₁₃	1	2	yes	—
184	C₂×C₄²	D₄×C₁₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
187	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	#103	—	2	8	no	—
190	#103	—	4	4	no	—
191	C₄×C₈	—	4	2	no	—
192	#95	—	2	4	no	—
193	#95	—	2	1	no	—
194	#95	—	2	1	no	—
195	#95	—	2	1	no	—
196	#95	—	2	1	no	—
197	#95	—	2	2	no	—
198	#95	—	2	2	no	—
199	C₂×C₁₃	#2562	1	1	yes	characteristic
200	C₂×C₁₃	#3108	1	2	yes	—
201	C₂×C₁₃	#3131	1	1	yes	characteristic
202	C₂×C₁₃	#3139	1	1	yes	characteristic
203	C₂×C₁₃	#3248	1	2	yes	—
204	C₂×Q₈	—	2	2	no	—
205	C₂×Q₈	—	2	2	no	—
206	C₂×Q₈	—	2	2	no	—
207	C₂×Q₈	—	2	2	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	C₁₃×#41	1	4	yes	—
210	C₂²×C₄	—	2	2	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	2	2	no	—
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
217	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
218	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	4	no	—
222	C₄⋊C₄	—	2	8	no	—
223	C₄⋊C₄	—	2	4	no	—
224	C₄⋊C₄	—	2	2	no	—
225	C₄⋊C₄	—	2	2	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	4	no	—
231	C₄⋊C₄	—	4	4	no	—
232	C₄²	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	2	no	—
235	C₄²	—	2	1	no	—
236	C₄²	—	2	1	no	—
237	C₄²	—	4	4	no	—
238	C₄²	C₁₃×#41	1	4	yes	—
239	C₁₃	#29126	1	1	yes	characteristic, direct product factor, Sylow
240	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
241	Q₈	—	2	4	no	—
242	Q₈	—	2	4	no	—
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	C₁₃×#124	1	2	yes	—
263	C₂×C₄	C₁₃×#124	1	2	yes	—
264	C₂×C₄	C₁₃×#135	1	4	yes	—
265	C₂×C₄	C₁₃×#137	1	4	yes	—
266	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
267	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
268	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
269	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
270	C₈	—	4	4	no	—
271	C₂²	C₁₃×#525	1	1	yes	characteristic
272	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
273	C₂²	C₁₃×#537	1	1	yes	characteristic
274	C₂²	C₂×C₁₃×#135	1	2	yes	—
275	C₂²	C₂×C₁₃×#137	1	2	yes	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	4	no	—
280	C₄	—	4	4	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	C₁₃×#552	1	2	yes	—
285	C₄	C₁₃×#553	1	2	yes	—
286	C₂	C₁₃×#2562	1	1	yes	characteristic
287	C₂	C₁₃×#3108	1	2	yes	—
288	C₂	C₁₃×#3131	1	1	yes	characteristic
289	C₂	C₁₃×#3139	1	1	yes	characteristic
290	C₂	C₁₃×#3248	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
59. #29126	C₁₃	direct (N × H)
239. C₁₃	#29126	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×#2864	2
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#2099	2
C₁₃×#2043	2
C₁₃×#1933	2
C₁₃×#1805	2
C₄×C₁₃×C₄⋊C₈	2
#29126	13

Subgroups — C13×#29126 #15715465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29126 #15715465