Subgroups — C₁₃×#32684 #15719023

Order 3328.

648 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×#2567	C₂	1	2	yes	maximal
2	C₁₃×#2087	C₂	1	2	yes	maximal
3	C₁₃×#2057	C₂	1	4	yes	maximal
4	C₁₃×#1984	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1942	C₂	1	2	yes	maximal
6	C₁₃×#1921	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×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₄×C₈×C₁₃	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×#107	Q₈	1	8	yes	complemented
46	C₁₃×C₄⋊C₈	—	4	1	no	—
47	C₄×C₈×C₁₃	—	4	1	no	—
48	C₁₃×#95	—	2	2	no	—
49	C₁₃×#95	—	2	4	no	—
50	C₁₃×#95	—	2	4	no	—
51	#32684	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂×Q₈×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	1	no	—
54	C₂×Q₈×C₁₃	—	4	4	no	—
55	C₂×Q₈×C₁₃	—	8	4	no	—
56	C₂×Q₈×C₁₃	Q₁₆	1	4	yes	—
57	C₂²×C₄×C₁₃	—	2	4	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	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	2	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
69	C₁₃×C₄⋊C₄	—	2	4	no	—
70	C₁₃×C₄⋊C₄	—	2	4	no	—
71	C₁₃×C₄⋊C₄	—	2	4	no	—
72	C₁₃×C₄⋊C₄	—	4	2	no	—
73	C₁₃×C₄⋊C₄	—	4	1	no	—
74	C₁₃×C₄⋊C₄	—	4	4	no	—
75	C₁₃×C₄⋊C₄	—	8	4	no	—
76	C₁₃×C₄⋊C₄	—	8	4	no	—
77	C₁₃×C₄⋊C₄	C₂×Q₈	1	8	yes	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	C₄²×C₁₃	—	2	2	no	—
82	C₄²×C₁₃	—	4	4	no	—
83	#2567	C₂×C₁₃	1	2	yes	—
84	#2087	C₂×C₁₃	1	2	yes	—
85	#2057	C₂×C₁₃	1	4	yes	—
86	#1984	C₂×C₁₃	1	1	yes	characteristic
87	#1942	C₂×C₁₃	1	2	yes	—
88	#1921	C₂×C₁₃	1	4	yes	—
89	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
90	Q₈×C₁₃	—	4	4	no	—
91	Q₈×C₁₃	—	8	8	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	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	8	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	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	—	4	4	no	—
106	C₂×C₄×C₁₃	C₂×Q₁₆	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×Q₁₆	1	4	yes	—
108	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
109	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
110	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
111	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
112	C₈×C₁₃	—	2	4	no	—
113	C₈×C₁₃	—	4	2	no	—
114	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
115	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
116	C₂×#107	C₂²×C₁₃	1	2	yes	—
117	C₂×#107	C₂²×C₁₃	1	2	yes	—
118	C₂×#107	C₂²×C₁₃	1	2	yes	—
119	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
120	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
121	#395	—	2	2	no	—
122	#395	—	2	2	no	—
123	#391	—	2	2	no	—
124	#389	C₂²×C₁₃	1	4	yes	—
125	#389	C₂²×C₁₃	1	4	yes	—
126	#384	—	2	2	no	—
127	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
128	#340	C₂²×C₁₃	1	4	yes	—
129	#340	C₂²×C₁₃	1	4	yes	—
130	#340	C₂²×C₁₃	1	2	yes	—
131	#336	C₂²×C₁₃	1	4	yes	—
132	C₂²×C₁₃	#534	1	1	yes	characteristic
133	C₂²×C₁₃	D₄×Q₈	1	2	yes	—
134	C₂²×C₁₃	C₂²×Q₁₆	1	1	yes	characteristic
135	C₂²×C₁₃	#575	1	1	yes	characteristic
136	C₂²×C₁₃	#576	1	2	yes	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	4	8	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	—	4	2	no	—
142	C₄×C₁₃	—	8	4	no	—
143	C₄×C₁₃	#544	1	2	yes	—
144	C₄×C₁₃	Q₈²	1	1	yes	characteristic
145	C₄×C₁₃	#563	1	1	yes	characteristic
146	C₂²×Q₈	—	2	2	no	—
147	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
148	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
149	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
150	C₄⋊Q₈	—	4	2	no	—
151	C₄⋊Q₈	—	4	2	no	—
152	C₄×Q₈	—	4	1	no	—
153	C₄×Q₈	—	4	1	no	—
154	C₂×C₄⋊C₄	—	2	2	no	—
155	C₂×C₄⋊C₄	—	2	4	no	—
156	C₂×C₄⋊C₄	—	2	4	no	—
157	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
158	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
160	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
161	C₂×C₄²	—	2	2	no	—
162	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
163	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
164	#107	—	4	2	no	—
165	#107	—	4	2	no	—
166	#107	Q₈×C₁₃	1	8	yes	complemented
167	C₄⋊C₈	—	4	1	no	—
168	C₄×C₈	—	4	1	no	—
169	#95	—	2	2	no	—
170	#95	—	2	4	no	—
171	#95	—	2	4	no	—
172	C₂×C₁₃	#2734	1	1	yes	characteristic
173	C₂×C₁₃	#3189	1	1	yes	characteristic
174	C₂×C₁₃	#3199	1	1	yes	characteristic
175	C₂×C₁₃	Q₈×Q₁₆	1	2	yes	—
176	C₂×C₁₃	#3389	1	2	yes	—
177	C₂×Q₈	—	4	1	no	—
178	C₂×Q₈	—	4	1	no	—
179	C₂×Q₈	—	4	4	no	—
180	C₂×Q₈	—	8	4	no	—
181	C₂×Q₈	C₁₃×Q₁₆	1	4	yes	—
182	C₂²×C₄	—	2	4	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	2	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	2	4	no	—
193	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
194	C₄⋊C₄	—	2	4	no	—
195	C₄⋊C₄	—	2	4	no	—
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	1	no	—
199	C₄⋊C₄	—	4	4	no	—
200	C₄⋊C₄	—	8	4	no	—
201	C₄⋊C₄	—	8	4	no	—
202	C₄⋊C₄	C₂×Q₈×C₁₃	1	8	yes	—
203	C₄²	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	4	4	no	—
208	C₁₃	#32684	1	1	yes	characteristic, direct product factor, Sylow
209	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
210	Q₈	—	4	4	no	—
211	Q₈	—	8	8	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	2	no	—
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	8	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	4	no	—
221	C₂×C₄	—	4	4	no	—
222	C₂×C₄	—	4	4	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	C₂×C₁₃×Q₁₆	1	2	yes	—
227	C₂×C₄	C₂×C₁₃×Q₁₆	1	4	yes	—
228	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
229	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
230	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
231	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	—	4	2	no	—
234	C₂²	C₁₃×#534	1	1	yes	characteristic
235	C₂²	D₄×Q₈×C₁₃	1	2	yes	—
236	C₂²	C₂²×C₁₃×Q₁₆	1	1	yes	characteristic
237	C₂²	C₁₃×#575	1	1	yes	characteristic
238	C₂²	C₁₃×#576	1	2	yes	—
239	C₄	—	2	4	no	—
240	C₄	—	2	2	no	—
241	C₄	—	4	8	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	2	no	—
244	C₄	—	8	4	no	—
245	C₄	C₁₃×#544	1	2	yes	—
246	C₄	Q₈²×C₁₃	1	1	yes	characteristic
247	C₄	C₁₃×#563	1	1	yes	characteristic
248	C₂	C₁₃×#2734	1	1	yes	characteristic
249	C₂	C₁₃×#3189	1	1	yes	characteristic
250	C₂	C₁₃×#3199	1	1	yes	characteristic
251	C₂	Q₈×C₁₃×Q₁₆	1	2	yes	—
252	C₂	C₁₃×#3389	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 ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
45. C₁₃×#107	Q₈	semidirect
51. #32684	C₁₃	direct (N × H)
166. #107	Q₈×C₁₃	semidirect
208. C₁₃	#32684	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×#2087	2
C₁₃×#2057	2
C₁₃×#1984	2
C₁₃×#1942	2
C₁₃×#1921	2
#32684	13

Subgroups — C13×#32684 #15719023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32684 #15719023