Subgroups — C₁₃×#13274 #15699613

Order 3328.

338 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1382	C₂	1	4	yes	maximal
5	C₁₃×#533	—	2	1	no	—
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₁₃×#500	C₄	1	1	yes	characteristic
8	C₁₃×#500	C₄	1	1	yes	characteristic
9	C₁₃×#499	—	2	1	no	—
10	C₁₃×#499	C₄	1	2	yes	—
11	C₁₃×#498	—	2	1	no	—
12	C₂×C₁₃×#108	C₂²	1	2	yes	—
13	C₁₃×#427	—	2	1	no	—
14	C₁₃×#427	C₂²	1	2	yes	—
15	C₄×C₁₃×M₁₆	—	2	1	no	—
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊C₁₆	—	2	1	no	—
18	C₁₃×C₄⋊C₁₆	—	2	1	no	—
19	C₁₃×#341	—	2	1	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₁₃×C₄⋊Q₈	—	2	1	no	—
24	C₁₃×C₄⋊Q₈	—	2	1	no	—
25	C₁₃×#125	—	2	2	no	—
26	C₁₃×C₂²⋊Q₈	—	4	1	no	—
27	C₁₃×C₂²⋊Q₈	—	4	1	no	—
28	C₄×Q₈×C₁₃	—	4	1	no	—
29	C₄×Q₈×C₁₃	—	4	1	no	—
30	C₁₃×#117	—	2	2	no	—
31	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
32	C₂×C₁₃×C₁₆	—	2	2	no	—
33	C₂×C₁₃×C₁₆	—	2	4	no	—
34	C₁₃×#108	—	2	2	no	—
35	C₁₃×#108	Q₈	1	4	yes	complemented
36	C₁₃×#107	—	2	1	no	—
37	C₁₃×#107	—	2	1	no	—
38	C₁₃×#107	C₂×C₄	1	2	yes	—
39	C₁₃×#107	C₂×C₄	1	2	yes	—
40	C₁₃×#106	—	2	2	no	—
41	C₁₃×#106	—	2	2	no	—
42	C₁₃×#97	—	4	1	no	—
43	C₄×C₈×C₁₃	—	4	1	no	—
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	#13274	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
49	C₂×Q₈×C₁₃	—	4	1	no	—
50	C₂×Q₈×C₁₃	—	4	1	no	—
51	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₁₃×M₁₆	—	2	4	no	—
55	C₁₃×M₁₆	—	4	2	no	—
56	C₂×C₈×C₁₃	—	2	2	no	—
57	C₂×C₈×C₁₃	—	4	2	no	—
58	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
63	C₂×C₈×C₁₃	#41	1	2	yes	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	4	2	no	—
67	C₁₃×C₄⋊C₄	—	4	2	no	—
68	C₁₃×C₄⋊C₄	—	4	1	no	—
69	C₁₃×C₄⋊C₄	—	4	1	no	—
70	C₁₃×C₂²⋊C₄	—	4	2	no	—
71	C₄²×C₁₃	—	4	2	no	—
72	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₄²×C₁₃	D₈	1	1	yes	characteristic
75	C₄²×C₁₃	D₈	1	1	yes	characteristic
76	C₁₃×C₁₆	—	4	2	no	—
77	#3160	C₂×C₁₃	1	1	yes	characteristic
78	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
79	C₄×#108	C₂×C₁₃	1	1	yes	characteristic
80	#1382	C₂×C₁₃	1	4	yes	—
81	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
82	Q₈×C₁₃	—	8	2	no	—
83	Q₈×C₁₃	—	8	2	no	—
84	C₂×C₄×C₁₃	—	2	2	no	—
85	C₂×C₄×C₁₃	—	4	1	no	—
86	C₂×C₄×C₁₃	—	4	2	no	—
87	C₂×C₄×C₁₃	—	4	2	no	—
88	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
89	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
90	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
91	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
95	C₂×C₄×C₁₃	#137	1	2	yes	—
96	C₈×C₁₃	—	2	1	no	—
97	C₈×C₁₃	—	2	1	no	—
98	C₈×C₁₃	—	4	4	no	—
99	C₈×C₁₃	C₄×Q₈	1	2	yes	—
100	C₈×C₁₃	#124	1	1	yes	characteristic
101	C₈×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
102	#533	—	2	1	no	—
103	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
104	#500	C₄×C₁₃	1	1	yes	characteristic
105	#500	C₄×C₁₃	1	1	yes	characteristic
106	#499	—	2	1	no	—
107	#499	C₄×C₁₃	1	2	yes	—
108	#498	—	2	1	no	—
109	C₂×#108	C₂²×C₁₃	1	2	yes	—
110	#427	—	2	1	no	—
111	#427	C₂²×C₁₃	1	2	yes	—
112	C₄×M₁₆	—	2	1	no	—
113	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
114	C₄⋊C₁₆	—	2	1	no	—
115	C₄⋊C₁₆	—	2	1	no	—
116	#341	—	2	1	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
120	C₂²×C₁₃	#475	1	2	yes	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	2	1	no	—
123	C₄×C₁₃	—	2	1	no	—
124	C₄×C₁₃	—	8	2	no	—
125	C₄×C₁₃	—	8	2	no	—
126	C₄×C₁₃	#359	1	2	yes	—
127	C₄×C₁₃	#391	1	1	yes	characteristic
128	C₄×C₁₃	#419	1	1	yes	characteristic
129	C₄×C₁₃	#474	1	2	yes	—
130	C₂×M₁₆	—	2	2	no	—
131	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
132	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
133	C₄⋊Q₈	—	2	1	no	—
134	C₄⋊Q₈	—	2	1	no	—
135	#125	—	2	2	no	—
136	C₂²⋊Q₈	—	4	1	no	—
137	C₂²⋊Q₈	—	4	1	no	—
138	C₄×Q₈	—	4	1	no	—
139	C₄×Q₈	—	4	1	no	—
140	#117	—	2	2	no	—
141	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
142	C₂×C₁₆	—	2	2	no	—
143	C₂×C₁₆	—	2	4	no	—
144	#108	—	2	2	no	—
145	#108	Q₈×C₁₃	1	4	yes	complemented
146	#107	—	2	1	no	—
147	#107	—	2	1	no	—
148	#107	C₂×C₄×C₁₃	1	2	yes	—
149	#107	C₂×C₄×C₁₃	1	2	yes	—
150	#106	—	2	2	no	—
151	#106	—	2	2	no	—
152	#97	—	4	1	no	—
153	C₄×C₈	—	4	1	no	—
154	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
155	C₄×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
156	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
157	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
158	C₂×C₁₃	—	2	2	no	—
159	C₂×C₁₃	#1988	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#359	1	1	yes	characteristic
161	C₂×C₁₃	#2143	1	1	yes	characteristic
162	C₂×Q₈	—	4	1	no	—
163	C₂×Q₈	—	4	1	no	—
164	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
165	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
166	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
167	M₁₆	—	2	4	no	—
168	M₁₆	—	4	2	no	—
169	C₂×C₈	—	2	2	no	—
170	C₂×C₈	—	4	2	no	—
171	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
172	C₂×C₈	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
174	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
175	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
176	C₂×C₈	C₁₃×#41	1	2	yes	—
177	C₄⋊C₄	—	2	2	no	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	4	2	no	—
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	1	no	—
182	C₄⋊C₄	—	4	1	no	—
183	C₂²⋊C₄	—	4	2	no	—
184	C₄²	—	4	2	no	—
185	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
186	C₄²	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
187	C₄²	C₁₃×D₈	1	1	yes	characteristic
188	C₄²	C₁₃×D₈	1	1	yes	characteristic
189	C₁₆	—	4	2	no	—
190	C₁₃	#13274	1	1	yes	characteristic, direct product factor, Sylow
191	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
192	Q₈	—	8	2	no	—
193	Q₈	—	8	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	1	no	—
196	C₂×C₄	—	4	2	no	—
197	C₂×C₄	—	4	2	no	—
198	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
199	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
200	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
201	C₂×C₄	C₁₃×#102	1	1	yes	characteristic
202	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
203	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
204	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
205	C₂×C₄	C₁₃×#137	1	2	yes	—
206	C₈	—	2	1	no	—
207	C₈	—	2	1	no	—
208	C₈	—	4	4	no	—
209	C₈	C₄×Q₈×C₁₃	1	2	yes	—
210	C₈	C₁₃×#124	1	1	yes	characteristic
211	C₈	C₁₃×C₄⋊Q₈	1	1	yes	characteristic
212	C₂²	—	2	1	no	—
213	C₂²	—	2	1	no	—
214	C₂²	C₂×C₁₃×#102	1	1	yes	characteristic
215	C₂²	C₁₃×#475	1	2	yes	—
216	C₄	—	2	1	no	—
217	C₄	—	2	1	no	—
218	C₄	—	2	1	no	—
219	C₄	—	8	2	no	—
220	C₄	—	8	2	no	—
221	C₄	C₁₃×#359	1	2	yes	—
222	C₄	C₁₃×#391	1	1	yes	characteristic
223	C₄	C₁₃×#419	1	1	yes	characteristic
224	C₄	C₁₃×#474	1	2	yes	—
225	C₂	—	2	2	no	—
226	C₂	C₁₃×#1988	1	1	yes	characteristic
227	C₂	C₂×C₁₃×#359	1	1	yes	characteristic
228	C₂	C₁₃×#2143	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂³×C₁₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
35. C₁₃×#108	Q₈	semidirect
48. #13274	C₁₃	direct (N × H)
145. #108	Q₈×C₁₃	semidirect
190. C₁₃	#13274	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×#3160	2
C₂×C₁₃×C₄⋊C₁₆	2
C₄×C₁₃×#108	2
C₁₃×#1382	2
#13274	13

Subgroups — C13×#13274 #15699613

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13274 #15699613