Subgroups — C₁₃×#29144 #15715483

Order 3328.

722 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×#2859	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2026	C₂	1	4	yes	maximal
4	C₁₃×#1956	C₂	1	2	yes	maximal
5	C₁₃×#1834	C₂	1	2	yes	maximal
6	C₁₃×#1805	C₂	1	4	yes	maximal
7	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#125	C₂²	1	2	yes	—
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
12	C₂×C₁₃×#103	C₂²	1	8	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	C₁₃×#391	C₂²	1	2	yes	—
15	C₁₃×#389	—	2	2	no	—
16	C₁₃×#389	—	2	2	no	—
17	C₁₃×#389	—	2	2	no	—
18	C₁₃×#384	C₂²	1	1	yes	characteristic
19	C₁₃×#384	C₂²	1	2	yes	—
20	C₁₃×#382	C₂²	1	2	yes	—
21	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
22	C₄³×C₁₃	C₂²	1	1	yes	characteristic
23	C₁₃×#340	C₂²	1	8	yes	—
24	C₁₃×#336	C₂²	1	2	yes	—
25	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
27	C₁₃×C₄⋊Q₈	—	2	2	no	—
28	C₁₃×C₄⋊Q₈	Q₈	1	4	yes	—
29	C₁₃×#125	—	4	4	no	—
30	C₄×Q₈×C₁₃	—	2	8	no	—
31	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
32	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
39	C₂×C₄²×C₁₃	—	2	2	no	—
40	C₂×C₄²×C₁₃	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃	D₄	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×C₄⋊C₈	—	2	4	no	—
46	C₁₃×#103	—	4	8	no	—
47	C₄×C₈×C₁₃	—	2	4	no	—
48	C₁₃×#95	—	2	4	no	—
49	C₁₃×#95	—	2	2	no	—
50	C₁₃×#95	—	2	2	no	—
51	#29144	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×Q₈×C₁₃	—	2	2	no	—
53	C₂×Q₈×C₁₃	—	2	4	no	—
54	C₂×Q₈×C₁₃	—	2	4	no	—
55	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	4	no	—
59	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₁₃	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₁₃×C₄⋊C₄	—	2	8	no	—
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	2	8	no	—
70	C₁₃×C₄⋊C₄	—	4	8	no	—
71	C₁₃×C₄⋊C₄	—	4	4	no	—
72	C₁₃×C₄⋊C₄	—	4	4	no	—
73	C₁₃×C₄⋊C₄	—	4	4	no	—
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₄²×C₁₃	—	2	4	no	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	4	4	no	—
79	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
80	C₄²×C₁₃	#41	1	2	yes	—
81	C₄²×C₁₃	#41	1	4	yes	—
82	#2859	C₂×C₁₃	1	1	yes	characteristic
83	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
84	#2026	C₂×C₁₃	1	4	yes	—
85	#1956	C₂×C₁₃	1	2	yes	—
86	#1834	C₂×C₁₃	1	2	yes	—
87	#1805	C₂×C₁₃	1	4	yes	—
88	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
89	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₁₃	—	2	8	no	—
91	Q₈×C₁₃	—	4	4	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
106	C₂×C₄×C₁₃	#124	1	2	yes	—
107	C₂×C₄×C₁₃	#124	1	2	yes	—
108	C₂×C₄×C₁₃	#135	1	4	yes	—
109	C₂×C₄×C₁₃	#137	1	4	yes	—
110	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
112	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
113	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
114	C₈×C₁₃	—	4	4	no	—
115	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
116	C₂×#125	C₂²×C₁₃	1	2	yes	—
117	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
118	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
119	C₂×#103	C₂²×C₁₃	1	8	yes	—
120	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
121	#391	C₂²×C₁₃	1	2	yes	—
122	#389	—	2	2	no	—
123	#389	—	2	2	no	—
124	#389	—	2	2	no	—
125	#384	C₂²×C₁₃	1	1	yes	characteristic
126	#384	C₂²×C₁₃	1	2	yes	—
127	#382	C₂²×C₁₃	1	2	yes	—
128	C₄×C₄⋊C₄	—	2	2	no	—
129	C₄³	C₂²×C₁₃	1	1	yes	characteristic
130	#340	C₂²×C₁₃	1	8	yes	—
131	#336	C₂²×C₁₃	1	2	yes	—
132	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
133	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
134	C₂²×C₁₃	#535	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#135	1	2	yes	—
136	C₂²×C₁₃	C₂×#137	1	2	yes	—
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	4	4	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	—	8	2	no	—
142	C₄×C₁₃	—	8	4	no	—
143	C₄×C₁₃	#479	1	8	yes	—
144	C₄×C₁₃	#554	1	2	yes	—
145	C₄×C₁₃	#555	1	2	yes	—
146	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
147	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
148	C₄⋊Q₈	—	2	2	no	—
149	C₄⋊Q₈	Q₈×C₁₃	1	4	yes	—
150	#125	—	4	4	no	—
151	C₄×Q₈	—	2	8	no	—
152	C₂×C₄⋊C₄	—	2	2	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	—	2	4	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄⋊C₄	—	2	2	no	—
157	C₂×C₄⋊C₄	—	2	2	no	—
158	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
159	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
160	C₂×C₄²	—	2	2	no	—
161	C₂×C₄²	D₄×C₁₃	1	2	yes	—
162	C₂×C₄²	D₄×C₁₃	1	2	yes	—
163	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
164	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	4	no	—
167	#103	—	4	8	no	—
168	C₄×C₈	—	2	4	no	—
169	#95	—	2	4	no	—
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂×C₁₃	#2563	1	1	yes	characteristic
173	C₂×C₁₃	C₂×#479	1	2	yes	—
174	C₂×C₁₃	#3131	1	1	yes	characteristic
175	C₂×C₁₃	#3139	1	1	yes	characteristic
176	C₂×C₁₃	#3210	1	2	yes	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	C₂×Q₈×C₁₃	1	4	yes	—
181	C₂²×C₄	—	2	2	no	—
182	C₂²×C₄	—	2	2	no	—
183	C₂²×C₄	—	2	4	no	—
184	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
185	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
187	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
188	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	2	4	no	—
192	C₄⋊C₄	—	2	8	no	—
193	C₄⋊C₄	—	2	4	no	—
194	C₄⋊C₄	—	2	8	no	—
195	C₄⋊C₄	—	4	8	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	4	4	no	—
198	C₄⋊C₄	—	4	4	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄²	—	2	4	no	—
201	C₄²	—	2	4	no	—
202	C₄²	—	2	2	no	—
203	C₄²	—	4	4	no	—
204	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
205	C₄²	C₁₃×#41	1	2	yes	—
206	C₄²	C₁₃×#41	1	4	yes	—
207	C₁₃	#29144	1	1	yes	characteristic, direct product factor, Sylow
208	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
209	Q₈	—	2	8	no	—
210	Q₈	—	4	4	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	4	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
225	C₂×C₄	C₁₃×#124	1	2	yes	—
226	C₂×C₄	C₁₃×#124	1	2	yes	—
227	C₂×C₄	C₁₃×#135	1	4	yes	—
228	C₂×C₄	C₁₃×#137	1	4	yes	—
229	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
230	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
231	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
232	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
233	C₈	—	4	4	no	—
234	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
236	C₂²	C₁₃×#535	1	1	yes	characteristic
237	C₂²	C₂×C₁₃×#135	1	2	yes	—
238	C₂²	C₂×C₁₃×#137	1	2	yes	—
239	C₄	—	2	2	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	4	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	2	no	—
244	C₄	—	8	4	no	—
245	C₄	C₁₃×#479	1	8	yes	—
246	C₄	C₁₃×#554	1	2	yes	—
247	C₄	C₁₃×#555	1	2	yes	—
248	C₂	C₁₃×#2563	1	1	yes	characteristic
249	C₂	C₂×C₁₃×#479	1	2	yes	—
250	C₂	C₁₃×#3131	1	1	yes	characteristic
251	C₂	C₁₃×#3139	1	1	yes	characteristic
252	C₂	C₁₃×#3210	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
51. #29144	C₁₃	direct (N × H)
207. C₁₃	#29144	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×#2859	2
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#2026	2
C₁₃×#1956	2
C₁₃×#1834	2
C₁₃×#1805	2
C₄×C₁₃×C₄⋊C₈	2
#29144	13

Subgroups — C13×#29144 #15715483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29144 #15715483