Subgroups — C₁₃×#32695 #15719034

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2567	C₂	1	2	yes	maximal
2	C₁₃×#2086	C₂	1	2	yes	maximal
3	C₁₃×#2057	C₂	1	4	yes	maximal
4	C₁₃×#1985	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1944	C₂	1	2	yes	maximal
6	C₁₃×#1920	C₂	1	4	yes	maximal
7	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
8	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#107	C₂²	1	2	yes	—
10	C₂×C₁₃×#107	C₂²	1	2	yes	—
11	C₂×C₁₃×#106	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₁₃×#395	—	2	2	no	—
15	C₁₃×#395	—	2	2	no	—
16	C₁₃×#391	—	2	2	no	—
17	C₁₃×#389	C₂²	1	4	yes	—
18	C₁₃×#389	C₂²	1	4	yes	—
19	C₁₃×#384	—	2	2	no	—
20	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
21	C₁₃×#340	C₂²	1	4	yes	—
22	C₁₃×#340	C₂²	1	4	yes	—
23	C₁₃×#340	C₂²	1	2	yes	—
24	C₁₃×#336	C₂²	1	4	yes	—
25	C₂²×Q₈×C₁₃	—	2	2	no	—
26	C₂²×Q₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
28	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
29	C₁₃×C₄⋊Q₈	—	4	2	no	—
30	C₁₃×C₄⋊Q₈	—	4	2	no	—
31	C₄×Q₈×C₁₃	—	4	1	no	—
32	C₄×Q₈×C₁₃	—	4	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
34	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
35	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
36	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
37	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
38	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
39	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
40	C₂×C₄²×C₁₃	—	2	2	no	—
41	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
42	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#107	—	4	2	no	—
44	C₁₃×#107	—	4	2	no	—
45	C₁₃×#106	Q₈	1	8	yes	complemented
46	C₁₃×C₄⋊C₈	—	4	1	no	—
47	C₁₃×#97	—	4	1	no	—
48	C₁₃×#95	—	2	2	no	—
49	C₁₃×#95	—	2	4	no	—
50	C₁₃×#95	—	2	4	no	—
51	#32695	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×Q₈×C₁₃	—	2	4	no	—
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	4	no	—
55	C₂×Q₈×C₁₃	—	8	4	no	—
56	C₂²×C₄×C₁₃	—	2	4	no	—
57	C₂²×C₄×C₁₃	—	2	4	no	—
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
62	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	2	4	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₄⋊C₄	—	4	1	no	—
73	C₁₃×C₄⋊C₄	—	4	4	no	—
74	C₁₃×C₄⋊C₄	—	8	4	no	—
75	C₁₃×C₄⋊C₄	—	8	4	no	—
76	C₁₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
77	C₄²×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	4	4	no	—
82	#2567	C₂×C₁₃	1	2	yes	—
83	#2086	C₂×C₁₃	1	2	yes	—
84	#2057	C₂×C₁₃	1	4	yes	—
85	#1985	C₂×C₁₃	1	1	yes	characteristic
86	#1944	C₂×C₁₃	1	2	yes	—
87	#1920	C₂×C₁₃	1	4	yes	—
88	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
89	Q₈×C₁₃	—	4	4	no	—
90	Q₈×C₁₃	—	8	8	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	2	1	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	8	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	4	no	—
100	C₂×C₄×C₁₃	—	4	4	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	#137	1	4	yes	—
106	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
107	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
108	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
109	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
110	C₈×C₁₃	—	2	4	no	—
111	C₈×C₁₃	—	4	2	no	—
112	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
113	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
114	C₂×#107	C₂²×C₁₃	1	2	yes	—
115	C₂×#107	C₂²×C₁₃	1	2	yes	—
116	C₂×#106	C₂²×C₁₃	1	2	yes	—
117	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
118	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
119	#395	—	2	2	no	—
120	#395	—	2	2	no	—
121	#391	—	2	2	no	—
122	#389	C₂²×C₁₃	1	4	yes	—
123	#389	C₂²×C₁₃	1	4	yes	—
124	#384	—	2	2	no	—
125	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
126	#340	C₂²×C₁₃	1	4	yes	—
127	#340	C₂²×C₁₃	1	4	yes	—
128	#340	C₂²×C₁₃	1	2	yes	—
129	#336	C₂²×C₁₃	1	4	yes	—
130	C₂²×C₁₃	#534	1	1	yes	characteristic
131	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
132	C₂²×C₁₃	C₂×#137	1	2	yes	—
133	C₂²×C₁₃	#576	1	2	yes	—
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	—	4	8	no	—
136	C₄×C₁₃	—	4	1	no	—
137	C₄×C₁₃	—	4	1	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	—	8	4	no	—
140	C₄×C₁₃	#544	1	2	yes	—
141	C₄×C₁₃	Q₈²	1	1	yes	characteristic
142	C₄×C₁₃	#563	1	1	yes	characteristic
143	C₂²×Q₈	—	2	2	no	—
144	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
145	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
146	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
147	C₄⋊Q₈	—	4	2	no	—
148	C₄⋊Q₈	—	4	2	no	—
149	C₄×Q₈	—	4	1	no	—
150	C₄×Q₈	—	4	1	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	4	no	—
153	C₂×C₄⋊C₄	—	2	4	no	—
154	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
155	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
157	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
158	C₂×C₄²	—	2	2	no	—
159	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
160	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
161	#107	—	4	2	no	—
162	#107	—	4	2	no	—
163	#106	Q₈×C₁₃	1	8	yes	complemented
164	C₄⋊C₈	—	4	1	no	—
165	#97	—	4	1	no	—
166	#95	—	2	2	no	—
167	#95	—	2	4	no	—
168	#95	—	2	4	no	—
169	C₂×C₁₃	#2734	1	1	yes	characteristic
170	C₂×C₁₃	#3195	1	1	yes	characteristic
171	C₂×C₁₃	#3199	1	1	yes	characteristic
172	C₂×C₁₃	#3385	1	4	yes	—
173	C₂×Q₈	—	2	4	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	4	no	—
176	C₂×Q₈	—	8	4	no	—
177	C₂²×C₄	—	2	4	no	—
178	C₂²×C₄	—	2	4	no	—
179	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
181	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
183	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
189	C₄⋊C₄	—	2	4	no	—
190	C₄⋊C₄	—	2	4	no	—
191	C₄⋊C₄	—	2	4	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	1	no	—
194	C₄⋊C₄	—	4	4	no	—
195	C₄⋊C₄	—	8	4	no	—
196	C₄⋊C₄	—	8	4	no	—
197	C₄⋊C₄	C₂×Q₈×C₁₃	1	8	yes	—
198	C₄²	—	2	1	no	—
199	C₄²	—	2	1	no	—
200	C₄²	—	2	2	no	—
201	C₄²	—	2	2	no	—
202	C₄²	—	4	4	no	—
203	C₁₃	#32695	1	1	yes	characteristic, direct product factor, Sylow
204	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
205	Q₈	—	4	4	no	—
206	Q₈	—	8	8	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	1	no	—
212	C₂×C₄	—	2	8	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	4	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	—	4	4	no	—
219	C₂×C₄	—	4	4	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	C₁₃×#137	1	4	yes	—
222	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
223	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
224	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
225	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
226	C₈	—	2	4	no	—
227	C₈	—	4	2	no	—
228	C₂²	C₁₃×#534	1	1	yes	characteristic
229	C₂²	D₄×Q₈×C₁₃	1	2	yes	—
230	C₂²	C₂×C₁₃×#137	1	2	yes	—
231	C₂²	C₁₃×#576	1	2	yes	—
232	C₄	—	2	4	no	—
233	C₄	—	4	8	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	—	8	4	no	—
238	C₄	C₁₃×#544	1	2	yes	—
239	C₄	Q₈²×C₁₃	1	1	yes	characteristic
240	C₄	C₁₃×#563	1	1	yes	characteristic
241	C₂	C₁₃×#2734	1	1	yes	characteristic
242	C₂	C₁₃×#3195	1	1	yes	characteristic
243	C₂	C₁₃×#3199	1	1	yes	characteristic
244	C₂	C₁₃×#3385	1	4	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 ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
45. C₁₃×#106	Q₈	semidirect
51. #32695	C₁₃	direct (N × H)
163. #106	Q₈×C₁₃	semidirect
203. C₁₃	#32695	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×#2567	2
C₁₃×#2086	2
C₁₃×#2057	2
C₁₃×#1985	2
C₁₃×#1944	2
C₁₃×#1920	2
#32695	13

Subgroups — C13×#32695 #15719034

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32695 #15719034