Subgroups — C₁₃×#23837 #15710176

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₁₃×C₄⋊Q₈	C₂	1	2	yes	maximal
2	C₁₃×#2084	C₂	1	4	yes	maximal
3	C₁₃×#2058	C₂	1	4	yes	maximal
4	C₄×C₁₃×#106	C₂	1	4	yes	maximal
5	C₁₃×#1744	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂×C₄×Q₈×C₁₃	—	2	2	no	—
8	C₂×C₁₃×#106	C₂²	1	8	yes	—
9	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
10	C₁₃×#391	—	2	2	no	—
11	C₁₃×#389	—	2	2	no	—
12	C₁₃×#389	C₂²	1	4	yes	—
13	C₁₃×#384	C₂²	1	2	yes	—
14	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
15	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
16	C₄³×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#340	C₂²	1	4	yes	—
18	C₁₃×#340	C₂²	1	4	yes	—
19	C₁₃×#336	C₂²	1	4	yes	—
20	C₂²×Q₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
22	C₁₃×C₄⋊Q₈	—	2	8	no	—
23	C₄×Q₈×C₁₃	—	4	8	no	—
24	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
25	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
26	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
27	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
28	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
30	C₂×C₄²×C₁₃	—	2	4	no	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	D₄	1	2	yes	—
34	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
35	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#106	—	2	16	no	—
37	C₄×C₈×C₁₃	Q₈	1	8	yes	complemented
38	C₁₃×#95	—	2	4	no	—
39	C₁₃×#95	—	2	4	no	—
40	#23837	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂×Q₈×C₁₃	—	2	8	no	—
42	C₂×Q₈×C₁₃	—	4	4	no	—
43	C₂×Q₈×C₁₃	—	4	4	no	—
44	C₂²×C₄×C₁₃	—	2	4	no	—
45	C₂²×C₄×C₁₃	—	2	4	no	—
46	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
47	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
49	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
51	C₂×C₈×C₁₃	—	2	8	no	—
52	C₂×C₈×C₁₃	C₂×Q₈	1	8	yes	—
53	C₁₃×C₄⋊C₄	—	2	8	no	—
54	C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₁₃×C₄⋊C₄	—	2	8	no	—
56	C₁₃×C₄⋊C₄	—	4	4	no	—
57	C₁₃×C₄⋊C₄	—	4	8	no	—
58	C₁₃×C₄⋊C₄	—	4	4	no	—
59	C₁₃×C₄⋊C₄	#41	1	8	yes	—
60	C₄²×C₁₃	—	2	4	no	—
61	C₄²×C₁₃	—	4	1	no	—
62	C₄²×C₁₃	—	4	1	no	—
63	C₄²×C₁₃	—	4	8	no	—
64	C₄²×C₁₃	SD₁₆	1	8	yes	—
65	C₄²×C₁₃	C₂×Q₈	1	4	yes	—
66	C₄×C₄⋊Q₈	C₂×C₁₃	1	2	yes	—
67	#2084	C₂×C₁₃	1	4	yes	—
68	#2058	C₂×C₁₃	1	4	yes	—
69	C₄×#106	C₂×C₁₃	1	4	yes	—
70	#1744	C₂×C₁₃	1	1	yes	characteristic
71	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
72	Q₈×C₁₃	—	4	16	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	1	no	—
77	C₂×C₄×C₁₃	—	2	1	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	8	no	—
80	C₂×C₄×C₁₃	—	4	4	no	—
81	C₂×C₄×C₁₃	—	4	4	no	—
82	C₂×C₄×C₁₃	—	4	4	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	—	4	4	no	—
85	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
86	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
87	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
88	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
89	C₂×C₄×C₁₃	C₂×SD₁₆	1	4	yes	—
90	C₂×C₄×C₁₃	C₂²×Q₈	1	2	yes	—
91	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
92	C₈×C₁₃	—	2	8	no	—
93	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
94	C₂×C₄×Q₈	—	2	2	no	—
95	C₂×#106	C₂²×C₁₃	1	8	yes	—
96	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
97	#391	—	2	2	no	—
98	#389	—	2	2	no	—
99	#389	C₂²×C₁₃	1	4	yes	—
100	#384	C₂²×C₁₃	1	2	yes	—
101	C₄×C₄⋊C₄	—	2	2	no	—
102	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
103	C₄³	C₂²×C₁₃	1	1	yes	characteristic
104	#340	C₂²×C₁₃	1	4	yes	—
105	#340	C₂²×C₁₃	1	4	yes	—
106	#336	C₂²×C₁₃	1	4	yes	—
107	C₂²×C₁₃	#525	1	2	yes	—
108	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
109	C₂²×C₁₃	C₂²×SD₁₆	1	2	yes	—
110	C₂²×C₁₃	#575	1	1	yes	characteristic
111	C₂²×C₁₃	#575	1	1	yes	characteristic
112	C₄×C₁₃	—	2	4	no	—
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	—	2	2	no	—
115	C₄×C₁₃	—	4	8	no	—
116	C₄×C₁₃	—	8	4	no	—
117	C₄×C₁₃	#498	1	4	yes	—
118	C₄×C₁₃	#533	1	4	yes	—
119	C₂²×Q₈	—	2	2	no	—
120	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
121	C₄⋊Q₈	—	2	8	no	—
122	C₄×Q₈	—	4	8	no	—
123	C₂×C₄⋊C₄	—	2	2	no	—
124	C₂×C₄⋊C₄	—	2	2	no	—
125	C₂×C₄⋊C₄	—	2	4	no	—
126	C₂×C₄⋊C₄	—	2	2	no	—
127	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
128	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
129	C₂×C₄²	—	2	4	no	—
130	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
131	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
132	C₂×C₄²	D₄×C₁₃	1	2	yes	—
133	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
134	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
135	#106	—	2	16	no	—
136	C₄×C₈	Q₈×C₁₃	1	8	yes	complemented
137	#95	—	2	4	no	—
138	#95	—	2	4	no	—
139	C₂×C₁₃	#2399	1	1	yes	characteristic
140	C₂×C₁₃	#3101	1	2	yes	—
141	C₂×C₁₃	#3103	1	2	yes	—
142	C₂×C₁₃	C₂×#498	1	1	yes	characteristic
143	C₂×C₁₃	#3162	1	1	yes	characteristic
144	C₂×Q₈	—	2	8	no	—
145	C₂×Q₈	—	4	4	no	—
146	C₂×Q₈	—	4	4	no	—
147	C₂²×C₄	—	2	4	no	—
148	C₂²×C₄	—	2	4	no	—
149	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
150	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
151	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
152	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
153	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
154	C₂×C₈	—	2	8	no	—
155	C₂×C₈	C₂×Q₈×C₁₃	1	8	yes	—
156	C₄⋊C₄	—	2	8	no	—
157	C₄⋊C₄	—	2	4	no	—
158	C₄⋊C₄	—	2	8	no	—
159	C₄⋊C₄	—	4	4	no	—
160	C₄⋊C₄	—	4	8	no	—
161	C₄⋊C₄	—	4	4	no	—
162	C₄⋊C₄	C₁₃×#41	1	8	yes	—
163	C₄²	—	2	4	no	—
164	C₄²	—	4	1	no	—
165	C₄²	—	4	1	no	—
166	C₄²	—	4	8	no	—
167	C₄²	C₁₃×SD₁₆	1	8	yes	—
168	C₄²	C₂×Q₈×C₁₃	1	4	yes	—
169	C₁₃	#23837	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
171	Q₈	—	4	16	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	2	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	8	no	—
179	C₂×C₄	—	4	4	no	—
180	C₂×C₄	—	4	4	no	—
181	C₂×C₄	—	4	4	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	4	4	no	—
184	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
185	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
186	C₂×C₄	C₂×C₁₃×SD₁₆	1	4	yes	—
187	C₂×C₄	C₂×C₁₃×SD₁₆	1	4	yes	—
188	C₂×C₄	C₂×C₁₃×SD₁₆	1	4	yes	—
189	C₂×C₄	C₂²×Q₈×C₁₃	1	2	yes	—
190	C₂×C₄	C₂×C₁₃×#41	1	4	yes	—
191	C₈	—	2	8	no	—
192	C₂²	C₁₃×#525	1	2	yes	—
193	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
194	C₂²	C₂²×C₁₃×SD₁₆	1	2	yes	—
195	C₂²	C₁₃×#575	1	1	yes	characteristic
196	C₂²	C₁₃×#575	1	1	yes	characteristic
197	C₄	—	2	4	no	—
198	C₄	—	2	4	no	—
199	C₄	—	2	2	no	—
200	C₄	—	4	8	no	—
201	C₄	—	8	4	no	—
202	C₄	C₁₃×#498	1	4	yes	—
203	C₄	C₁₃×#533	1	4	yes	—
204	C₂	C₁₃×#2399	1	1	yes	characteristic
205	C₂	C₁₃×#3101	1	2	yes	—
206	C₂	C₁₃×#3103	1	2	yes	—
207	C₂	C₂×C₁₃×#498	1	1	yes	characteristic
208	C₂	C₁₃×#3162	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
37. C₄×C₈×C₁₃	Q₈	semidirect
40. #23837	C₁₃	direct (N × H)
136. C₄×C₈	Q₈×C₁₃	semidirect
169. C₁₃	#23837	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#2084	2
C₁₃×#2058	2
C₄×C₁₃×#106	2
C₁₃×#1744	2
#23837	13

Subgroups — C13×#23837 #15710176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23837 #15710176