Subgroups — C₁₃×#12605 #15698944

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2112	C₂	1	2	yes	maximal
3	C₂×C₁₃×#347	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1375	C₂	1	2	yes	maximal, complemented
5	C₂³×C₈×C₁₃	C₄	1	1	yes	characteristic
6	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
7	C₂²×C₄²×C₁₃	C₄	1	1	yes	characteristic
8	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
9	C₂×C₁₃×#97	—	2	4	no	—
10	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
11	C₂×C₁₃×#97	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
14	C₁₃×C₂²⋊C₁₆	C₄	1	8	yes	complemented
15	C₁₃×#347	—	2	4	no	—
16	C₂³×C₄×C₁₃	—	2	1	no	—
17	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	—	2	2	no	—
19	C₂²×C₈×C₁₃	—	2	2	no	—
20	C₂²×C₈×C₁₃	—	2	2	no	—
21	C₂²×C₈×C₁₃	—	2	2	no	—
22	C₂²×C₈×C₁₃	—	2	2	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
25	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	—	2	1	no	—
29	C₂×C₄²×C₁₃	—	2	2	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₁₃×M₃₂	—	2	4	no	—
34	C₁₃×M₃₂	—	2	4	no	—
35	C₂×C₁₃×C₁₆	C₂×C₄	1	8	yes	complemented
36	C₁₃×#97	—	2	2	no	—
37	C₁₃×#97	—	2	2	no	—
38	C₁₃×#97	—	2	2	no	—
39	C₁₃×#97	D₄	1	4	yes	—
40	#12605	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
41	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	—	2	1	no	—
43	C₂²×C₄×C₁₃	—	2	1	no	—
44	C₂²×C₄×C₁₃	—	2	1	no	—
45	C₂²×C₄×C₁₃	—	2	1	no	—
46	C₂²×C₄×C₁₃	—	2	4	no	—
47	C₂²×C₄×C₁₃	—	2	1	no	—
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	4	4	no	—
50	C₂²×C₄×C₁₃	C₄²	1	2	yes	—
51	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	M₁₆	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	—	2	8	no	—
55	C₂×C₈×C₁₃	—	2	1	no	—
56	C₂×C₈×C₁₃	—	2	1	no	—
57	C₂×C₈×C₁₃	—	2	1	no	—
58	C₂×C₈×C₁₃	—	2	1	no	—
59	C₂×C₈×C₁₃	—	2	1	no	—
60	C₂×C₈×C₁₃	—	2	1	no	—
61	C₂×C₈×C₁₃	—	2	8	no	—
62	C₂×C₈×C₁₃	C₄²	1	2	yes	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
65	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
66	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
67	C₂×C₈×C₁₃	#41	1	2	yes	—
68	C₄²×C₁₃	—	2	4	no	—
69	C₄²×C₁₃	—	2	1	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
72	C₄²×C₁₃	M₁₆	1	2	yes	—
73	C₁₃×C₁₆	—	2	8	no	—
74	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic
75	#2112	C₂×C₁₃	1	2	yes	—
76	C₂×#347	C₂×C₁₃	1	2	yes	complemented
77	#1375	C₂×C₁₃	1	2	yes	complemented
78	C₂³×C₁₃	—	2	4	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	#97	1	2	yes	—
82	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	—	2	1	no	—
84	C₂×C₄×C₁₃	—	2	1	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	2	no	—
88	C₂×C₄×C₁₃	—	2	2	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	4	8	no	—
92	C₂×C₄×C₁₃	—	4	2	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
94	C₂×C₄×C₁₃	#97	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	C₈×C₁₃	—	2	2	no	—
102	C₈×C₁₃	—	2	2	no	—
103	C₈×C₁₃	—	2	2	no	—
104	C₈×C₁₃	C₄×D₄	1	4	yes	—
105	C₂³×C₈	C₄×C₁₃	1	1	yes	characteristic
106	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
107	C₂²×C₄²	C₄×C₁₃	1	1	yes	characteristic
108	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
109	C₂×#97	—	2	4	no	—
110	C₂×#97	C₄×C₁₃	1	1	yes	characteristic
111	C₂×#97	C₄×C₁₃	1	1	yes	characteristic
112	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
114	C₂²⋊C₁₆	C₄×C₁₃	1	8	yes	complemented
115	#347	—	2	4	no	—
116	C₂²×C₁₃	—	2	4	no	—
117	C₂²×C₁₃	—	2	4	no	—
118	C₂²×C₁₃	—	2	2	no	—
119	C₂²×C₁₃	—	2	2	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	#347	1	4	yes	complemented
122	C₂²×C₁₃	C₂×#97	1	1	yes	characteristic
123	C₂²×C₁₃	#407	1	1	yes	characteristic
124	C₂²×C₁₃	#432	1	1	yes	characteristic
125	C₄×C₁₃	—	2	2	no	—
126	C₄×C₁₃	—	4	2	no	—
127	C₄×C₁₃	—	4	2	no	—
128	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
130	C₄×C₁₃	#407	1	1	yes	characteristic
131	C₄×C₁₃	#432	1	1	yes	characteristic
132	C₂³×C₄	—	2	1	no	—
133	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
134	C₂²×C₈	—	2	2	no	—
135	C₂²×C₈	—	2	2	no	—
136	C₂²×C₈	—	2	2	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
140	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
141	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
142	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
143	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
144	C₂×C₄²	—	2	1	no	—
145	C₂×C₄²	—	2	2	no	—
146	C₂×C₄²	—	2	2	no	—
147	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
148	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
149	M₃₂	—	2	4	no	—
150	M₃₂	—	2	4	no	—
151	C₂×C₁₆	C₂×C₄×C₁₃	1	8	yes	complemented
152	#97	—	2	2	no	—
153	#97	—	2	2	no	—
154	#97	—	2	2	no	—
155	#97	D₄×C₁₃	1	4	yes	—
156	C₂×C₁₃	—	2	1	no	—
157	C₂×C₁₃	—	2	4	no	—
158	C₂×C₁₃	—	2	1	no	—
159	C₂×C₁₃	#1749	1	1	yes	characteristic
160	C₂×C₁₃	C₂×#347	1	2	yes	—
161	C₂⁴	C₁₃×M₁₆	1	1	yes	characteristic
162	C₂²×C₄	—	2	1	no	—
163	C₂²×C₄	—	2	1	no	—
164	C₂²×C₄	—	2	1	no	—
165	C₂²×C₄	—	2	1	no	—
166	C₂²×C₄	—	2	4	no	—
167	C₂²×C₄	—	2	1	no	—
168	C₂²×C₄	—	2	1	no	—
169	C₂²×C₄	—	4	4	no	—
170	C₂²×C₄	C₄²×C₁₃	1	2	yes	—
171	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
172	C₂²×C₄	C₁₃×M₁₆	1	2	yes	—
173	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
174	C₂×C₈	—	2	8	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	1	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	8	no	—
182	C₂×C₈	C₄²×C₁₃	1	2	yes	—
183	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
184	C₂×C₈	C₁₃×M₁₆	1	2	yes	—
185	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
186	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
187	C₂×C₈	C₁₃×#41	1	2	yes	—
188	C₄²	—	2	4	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	2	1	no	—
191	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
192	C₄²	C₁₃×M₁₆	1	2	yes	—
193	C₁₆	—	2	8	no	—
194	C₁₃	#12605	1	1	yes	characteristic, direct product factor, Sylow
195	C₂³	—	2	1	no	—
196	C₂³	—	2	4	no	—
197	C₂³	—	2	1	no	—
198	C₂³	C₁₃×#97	1	2	yes	—
199	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
200	C₂×C₄	—	2	1	no	—
201	C₂×C₄	—	2	1	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	4	8	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₁₃×#97	1	2	yes	—
212	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
213	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
214	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
215	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
216	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
217	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
218	C₈	—	2	2	no	—
219	C₈	—	2	2	no	—
220	C₈	—	2	2	no	—
221	C₈	C₄×D₄×C₁₃	1	4	yes	—
222	C₂²	—	2	4	no	—
223	C₂²	—	2	4	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	2	no	—
227	C₂²	C₁₃×#347	1	4	yes	complemented
228	C₂²	C₂×C₁₃×#97	1	1	yes	characteristic
229	C₂²	C₁₃×#407	1	1	yes	characteristic
230	C₂²	C₁₃×#432	1	1	yes	characteristic
231	C₄	—	2	2	no	—
232	C₄	—	4	2	no	—
233	C₄	—	4	2	no	—
234	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
235	C₄	C₄×C₁₃×M₁₆	1	1	yes	characteristic
236	C₄	C₁₃×#407	1	1	yes	characteristic
237	C₄	C₁₃×#432	1	1	yes	characteristic
238	C₂	—	2	1	no	—
239	C₂	—	2	4	no	—
240	C₂	—	2	1	no	—
241	C₂	C₁₃×#1749	1	1	yes	characteristic
242	C₂	C₂×C₁₃×#347	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×C₄²
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂²×C₁₃	C₂×#97
prime radical	C₂⁴×C₁₃	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂×C₁₃×#347	C₂	semidirect
4. C₁₃×#1375	C₂	semidirect
14. C₁₃×C₂²⋊C₁₆	C₄	semidirect
35. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
40. #12605	C₁₃	direct (N × H)
76. C₂×#347	C₂×C₁₃	semidirect
77. #1375	C₂×C₁₃	semidirect
114. C₂²⋊C₁₆	C₄×C₁₃	semidirect
121. C₂²×C₁₃	#347	semidirect
151. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
194. C₁₃	#12605	direct (N × H)
227. C₂²	C₁₃×#347	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#97	2
C₁₃×#2112	2
C₂×C₁₃×#347	2
C₁₃×#1375	2
#12605	13

Subgroups — C13×#12605 #15698944

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#12605 #15698944