Subgroups — C₁₃×#33426 #15719765

Order 3328.

520 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×#2875	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2166	C₂	1	4	yes	maximal, complemented
3	C₁₃×#2165	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2153	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2115	C₂	1	2	yes	maximal
6	C₂×C₁₃×#347	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#131	C₄	1	1	yes	characteristic
8	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
9	C₄×C₁₃×#41	C₄	1	1	yes	characteristic
10	C₂×C₁₃×M₃₂	C₂²	1	4	yes	—
11	C₁₃×#405	C₄	1	2	yes	—
12	C₁₃×#405	C₄	1	1	yes	characteristic
13	C₁₃×#405	C₂²	1	2	yes	—
14	C₁₃×#405	C₂²	1	1	yes	characteristic
15	C₄×C₁₃×M₁₆	C₄	1	1	yes	characteristic
16	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×#97	C₄	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₁₃×#364	C₂²	1	8	yes	—
20	C₁₃×#349	C₂²	1	8	yes	—
21	C₁₃×#347	C₂²	1	4	yes	—
22	C₁₃×#347	C₂²	1	4	yes	—
23	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
24	C₁₃×#131	—	2	4	no	—
25	C₁₃×#131	—	2	4	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
29	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
34	C₄×Q₈×C₁₃	—	2	1	no	—
35	C₄×D₄×C₁₃	—	2	2	no	—
36	C₄×D₄×C₁₃	—	2	1	no	—
37	C₁₃×#117	C₂×C₄	1	2	yes	—
38	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₁₃×M₃₂	—	2	4	no	—
42	C₁₃×M₃₂	C₂³	1	8	yes	—
43	C₂×C₁₃×C₁₆	—	2	4	no	—
44	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
45	C₁₃×#97	C₂×C₄	1	2	yes	—
46	C₁₃×#97	C₂×C₄	1	2	yes	—
47	C₁₃×#97	C₂×C₄	1	2	yes	—
48	C₁₃×#97	C₂×C₄	1	1	yes	characteristic
49	C₁₃×#97	C₂³	1	1	yes	characteristic
50	C₁₃×#97	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
52	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
53	C₄×C₈×C₁₃	C₂³	1	2	yes	—
54	#33426	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₁₃×#41	—	2	2	no	—
56	C₁₃×#41	—	2	2	no	—
57	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₁₃	M₁₆	1	2	yes	—
59	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
64	C₁₃×M₁₆	—	2	2	no	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	M₁₆	1	4	yes	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	4	no	—
71	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₁₃	#41	1	4	yes	—
77	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×C₄⋊C₄	—	2	2	no	—
79	C₁₃×C₄⋊C₄	—	2	1	no	—
80	C₁₃×C₂²⋊C₄	—	2	2	no	—
81	C₁₃×C₂²⋊C₄	—	2	1	no	—
82	C₄²×C₁₃	—	2	2	no	—
83	C₄²×C₁₃	—	2	1	no	—
84	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
86	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
87	C₁₃×C₁₆	—	2	8	no	—
88	#2875	C₂×C₁₃	1	1	yes	characteristic
89	#2166	C₂×C₁₃	1	4	yes	complemented
90	#2165	C₂×C₁₃	1	4	yes	complemented
91	#2153	C₂×C₁₃	1	2	yes	complemented
92	#2115	C₂×C₁₃	1	2	yes	—
93	C₂×#347	C₂×C₁₃	1	2	yes	complemented
94	C₂³×C₁₃	C₂×M₁₆	1	2	yes	—
95	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
96	Q₈×C₁₃	—	2	2	no	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	2	no	—
104	C₂×C₄×C₁₃	—	8	1	no	—
105	C₂×C₄×C₁₃	—	8	2	no	—
106	C₂×C₄×C₁₃	#117	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
111	C₈×C₁₃	—	2	2	no	—
112	C₈×C₁₃	—	2	1	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	#117	1	2	yes	—
115	C₈×C₁₃	C₂×M₁₆	1	4	yes	—
116	C₈×C₁₃	C₂×#41	1	2	yes	—
117	C₂×#131	C₄×C₁₃	1	1	yes	characteristic
118	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
119	C₄×#41	C₄×C₁₃	1	1	yes	characteristic
120	C₂×M₃₂	C₂²×C₁₃	1	4	yes	—
121	#405	C₄×C₁₃	1	2	yes	—
122	#405	C₄×C₁₃	1	1	yes	characteristic
123	#405	C₂²×C₁₃	1	2	yes	—
124	#405	C₂²×C₁₃	1	1	yes	characteristic
125	C₄×M₁₆	C₄×C₁₃	1	1	yes	characteristic
126	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
127	C₂×#97	C₄×C₁₃	1	2	yes	—
128	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
129	#364	C₂²×C₁₃	1	8	yes	—
130	#349	C₂²×C₁₃	1	8	yes	—
131	#347	C₂²×C₁₃	1	4	yes	—
132	#347	C₂²×C₁₃	1	4	yes	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	#432	1	2	yes	—
137	C₂²×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	4	1	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	—	4	1	no	—
142	C₄×C₁₃	#432	1	2	yes	—
143	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
144	C₄×C₁₃	C₂²×M₁₆	1	1	yes	characteristic
145	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
146	#131	—	2	4	no	—
147	#131	—	2	4	no	—
148	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
149	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
150	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
151	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
152	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
153	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
154	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
155	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
156	C₄×Q₈	—	2	1	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	1	no	—
159	#117	C₂×C₄×C₁₃	1	2	yes	—
160	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
161	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
162	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
163	M₃₂	—	2	4	no	—
164	M₃₂	C₂³×C₁₃	1	8	yes	—
165	C₂×C₁₆	—	2	4	no	—
166	#97	C₂×C₄×C₁₃	1	1	yes	characteristic
167	#97	C₂×C₄×C₁₃	1	2	yes	—
168	#97	C₂×C₄×C₁₃	1	2	yes	—
169	#97	C₂×C₄×C₁₃	1	2	yes	—
170	#97	C₂×C₄×C₁₃	1	1	yes	characteristic
171	#97	C₂³×C₁₃	1	1	yes	characteristic
172	#97	C₂³×C₁₃	1	1	yes	characteristic
173	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
174	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
175	C₄×C₈	C₂³×C₁₃	1	2	yes	—
176	C₂×C₁₃	—	2	1	no	—
177	C₂×C₁₃	—	2	2	no	—
178	C₂×C₁₃	—	4	2	no	—
179	C₂×C₁₃	C₂×#432	1	1	yes	characteristic
180	#41	—	2	2	no	—
181	#41	—	2	2	no	—
182	C₂×Q₈	C₁₃×M₁₆	1	1	yes	characteristic
183	C₂×D₄	C₁₃×M₁₆	1	2	yes	—
184	C₂×D₄	C₁₃×M₁₆	1	1	yes	characteristic
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
188	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
189	M₁₆	—	2	2	no	—
190	M₁₆	—	2	4	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	C₁₃×M₁₆	1	4	yes	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	2	4	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
197	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
198	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
199	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
200	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
201	C₂×C₈	C₁₃×#41	1	4	yes	—
202	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	2	2	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
210	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
211	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	C₁₆	—	2	8	no	—
213	C₁₃	#33426	1	1	yes	characteristic, direct product factor, Sylow
214	C₂³	C₂×C₁₃×M₁₆	1	2	yes	—
215	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
216	Q₈	—	2	2	no	—
217	D₄	—	2	4	no	—
218	D₄	—	2	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	C₁₃×#117	1	2	yes	—
227	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
228	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
229	C₂×C₄	C₂×C₁₃×M₁₆	1	2	yes	—
230	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
231	C₈	—	2	2	no	—
232	C₈	—	2	1	no	—
233	C₈	—	2	1	no	—
234	C₈	C₁₃×#117	1	2	yes	—
235	C₈	C₂×C₁₃×M₁₆	1	4	yes	—
236	C₈	C₂×C₁₃×#41	1	2	yes	—
237	C₂²	—	2	2	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₁₃×#432	1	2	yes	—
241	C₂²	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
242	C₄	—	2	2	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	2	no	—
245	C₄	—	4	1	no	—
246	C₄	C₁₃×#432	1	2	yes	—
247	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
248	C₄	C₂²×C₁₃×M₁₆	1	1	yes	characteristic
249	C₂	—	2	1	no	—
250	C₂	—	2	2	no	—
251	C₂	—	4	2	no	—
252	C₂	C₂×C₁₃×#432	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	C₂×#117
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×#432
prime radical	C₂×C₁₃×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂×C₁₃×#97 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2166	C₂	semidirect
3. C₁₃×#2165	C₂	semidirect
4. C₁₃×#2153	C₂	semidirect
6. C₂×C₁₃×#347	C₂	semidirect
54. #33426	C₁₃	direct (N × H)
89. #2166	C₂×C₁₃	semidirect
90. #2165	C₂×C₁₃	semidirect
91. #2153	C₂×C₁₃	semidirect
93. C₂×#347	C₂×C₁₃	semidirect
213. C₁₃	#33426	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×#2875	2
C₁₃×#2166	2
C₁₃×#2165	2
C₁₃×#2153	2
C₁₃×#2115	2
C₂×C₁₃×#347	2
#33426	13

Subgroups — C13×#33426 #15719765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33426 #15719765