Subgroups — C₁₃×#29147 #15715486

Order 3328.

726 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2862	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2024	C₂	1	4	yes	maximal
4	C₁₃×#1960	C₂	1	2	yes	maximal
5	C₁₃×#1836	C₂	1	2	yes	maximal
6	C₁₃×#1805	C₂	1	4	yes	maximal
7	C₁₃×#1767	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×C₄⋊Q₈	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×C₁₃×#97	C₂²	1	2	yes	—
14	C₁₃×#391	—	2	2	no	—
15	C₁₃×#391	—	2	2	no	—
16	C₁₃×#391	C₂²	1	2	yes	—
17	C₁₃×#389	—	2	2	no	—
18	C₁₃×#384	C₂²	1	1	yes	characteristic
19	C₁₃×#382	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	—	2	2	no	—
26	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
27	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
28	C₁₃×C₄⋊Q₈	—	2	2	no	—
29	C₁₃×C₄⋊Q₈	—	4	4	no	—
30	C₁₃×C₄⋊Q₈	Q₈	1	4	yes	complemented
31	C₄×Q₈×C₁₃	—	2	8	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	C₂³	1	4	yes	—
37	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
38	C₂×C₄²×C₁₃	—	2	2	no	—
39	C₂×C₄²×C₁₃	D₄	1	2	yes	—
40	C₂×C₄²×C₁₃	D₄	1	2	yes	—
41	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊C₈	—	2	4	no	—
45	C₁₃×#103	—	4	8	no	—
46	C₁₃×#97	—	2	4	no	—
47	C₁₃×#95	—	2	4	no	—
48	C₁₃×#95	—	2	4	no	—
49	C₁₃×#95	—	2	2	no	—
50	C₁₃×#95	—	2	2	no	—
51	#29147	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	—	4	4	no	—
56	C₂×Q₈×C₁₃	—	4	4	no	—
57	C₂×Q₈×C₁₃	—	4	4	no	—
58	C₂×Q₈×C₁₃	C₂×Q₈	1	4	yes	—
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	4	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	4	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	2	8	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	2	8	no	—
73	C₁₃×C₄⋊C₄	—	4	4	no	—
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₁₃×C₄⋊C₄	—	4	4	no	—
76	C₄²×C₁₃	—	2	4	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	4	2	no	—
79	C₄²×C₁₃	—	4	4	no	—
80	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
81	C₄²×C₁₃	#41	1	2	yes	—
82	C₄²×C₁₃	#41	1	4	yes	—
83	#2862	C₂×C₁₃	1	1	yes	characteristic
84	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
85	#2024	C₂×C₁₃	1	4	yes	—
86	#1960	C₂×C₁₃	1	2	yes	—
87	#1836	C₂×C₁₃	1	2	yes	—
88	#1805	C₂×C₁₃	1	4	yes	—
89	#1767	C₂×C₁₃	1	1	yes	characteristic
90	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
91	Q₈×C₁₃	—	2	8	no	—
92	Q₈×C₁₃	—	4	4	no	—
93	Q₈×C₁₃	—	8	8	no	—
94	C₂×C₄×C₁₃	—	2	2	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	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	4	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	4	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	C₂²⋊Q₈	1	4	yes	—
109	C₂×C₄×C₁₃	#124	1	2	yes	—
110	C₂×C₄×C₁₃	#124	1	2	yes	—
111	C₂×C₄×C₁₃	#137	1	4	yes	—
112	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
114	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
116	C₈×C₁₃	—	4	4	no	—
117	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
118	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
119	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
120	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
121	C₂×#103	C₂²×C₁₃	1	8	yes	—
122	C₂×#97	C₂²×C₁₃	1	2	yes	—
123	#391	—	2	2	no	—
124	#391	—	2	2	no	—
125	#391	C₂²×C₁₃	1	2	yes	—
126	#389	—	2	2	no	—
127	#384	C₂²×C₁₃	1	1	yes	characteristic
128	#382	C₂²×C₁₃	1	2	yes	—
129	#382	C₂²×C₁₃	1	2	yes	—
130	C₄×C₄⋊C₄	—	2	2	no	—
131	C₄³	C₂²×C₁₃	1	1	yes	characteristic
132	#340	C₂²×C₁₃	1	8	yes	—
133	#336	C₂²×C₁₃	1	2	yes	—
134	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
136	C₂²×C₁₃	#535	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×#137	1	2	yes	—
138	C₂²×C₁₃	#575	1	2	yes	—
139	C₄×C₁₃	—	2	4	no	—
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	2	2	no	—
142	C₄×C₁₃	—	2	4	no	—
143	C₄×C₁₃	—	4	4	no	—
144	C₄×C₁₃	—	8	2	no	—
145	C₄×C₁₃	—	8	4	no	—
146	C₄×C₁₃	#554	1	2	yes	—
147	C₄×C₁₃	#555	1	2	yes	—
148	C₂²×Q₈	—	2	2	no	—
149	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
150	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
151	C₄⋊Q₈	—	2	2	no	—
152	C₄⋊Q₈	—	4	4	no	—
153	C₄⋊Q₈	Q₈×C₁₃	1	4	yes	complemented
154	C₄×Q₈	—	2	8	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	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	D₄×C₁₃	1	2	yes	—
163	C₂×C₄²	D₄×C₁₃	1	2	yes	—
164	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
165	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
166	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
167	C₄⋊C₈	—	2	4	no	—
168	#103	—	4	8	no	—
169	#97	—	2	4	no	—
170	#95	—	2	4	no	—
171	#95	—	2	4	no	—
172	#95	—	2	2	no	—
173	#95	—	2	2	no	—
174	C₂×C₁₃	#2563	1	1	yes	characteristic
175	C₂×C₁₃	#3078	1	2	yes	—
176	C₂×C₁₃	#3135	1	1	yes	characteristic
177	C₂×C₁₃	#3139	1	1	yes	characteristic
178	C₂×C₁₃	#3215	1	2	yes	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	4	4	no	—
184	C₂×Q₈	—	4	4	no	—
185	C₂×Q₈	C₂×Q₈×C₁₃	1	4	yes	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	4	no	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
190	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
193	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	2	4	no	—
197	C₄⋊C₄	—	2	8	no	—
198	C₄⋊C₄	—	2	4	no	—
199	C₄⋊C₄	—	2	8	no	—
200	C₄⋊C₄	—	4	4	no	—
201	C₄⋊C₄	—	4	4	no	—
202	C₄⋊C₄	—	4	4	no	—
203	C₄²	—	2	4	no	—
204	C₄²	—	2	2	no	—
205	C₄²	—	4	2	no	—
206	C₄²	—	4	4	no	—
207	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
208	C₄²	C₁₃×#41	1	2	yes	—
209	C₄²	C₁₃×#41	1	4	yes	—
210	C₁₃	#29147	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
212	Q₈	—	2	8	no	—
213	Q₈	—	4	4	no	—
214	Q₈	—	8	8	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	4	4	no	—
227	C₂×C₄	—	4	4	no	—
228	C₂×C₄	—	4	4	no	—
229	C₂×C₄	C₁₃×C₂²⋊Q₈	1	4	yes	—
230	C₂×C₄	C₁₃×#124	1	2	yes	—
231	C₂×C₄	C₁₃×#124	1	2	yes	—
232	C₂×C₄	C₁₃×#137	1	4	yes	—
233	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
235	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
236	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
237	C₈	—	4	4	no	—
238	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
239	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
240	C₂²	C₁₃×#535	1	1	yes	characteristic
241	C₂²	C₂×C₁₃×#137	1	2	yes	—
242	C₂²	C₁₃×#575	1	2	yes	—
243	C₄	—	2	4	no	—
244	C₄	—	2	2	no	—
245	C₄	—	2	2	no	—
246	C₄	—	2	4	no	—
247	C₄	—	4	4	no	—
248	C₄	—	8	2	no	—
249	C₄	—	8	4	no	—
250	C₄	C₁₃×#554	1	2	yes	—
251	C₄	C₁₃×#555	1	2	yes	—
252	C₂	C₁₃×#2563	1	1	yes	characteristic
253	C₂	C₁₃×#3078	1	2	yes	—
254	C₂	C₁₃×#3135	1	1	yes	characteristic
255	C₂	C₁₃×#3139	1	1	yes	characteristic
256	C₂	C₁₃×#3215	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
30. C₁₃×C₄⋊Q₈	Q₈	semidirect
51. #29147	C₁₃	direct (N × H)
153. C₄⋊Q₈	Q₈×C₁₃	semidirect
210. C₁₃	#29147	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×#2862	2
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#2024	2
C₁₃×#1960	2
C₁₃×#1836	2
C₁₃×#1805	2
C₁₃×#1767	2
#29147	13

Subgroups — C13×#29147 #15715486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29147 #15715486