Subgroups — C₁₃×#33460 #15719799

Order 3328.

674 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	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2951	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2177	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2176	C₂	1	2	yes	maximal, complemented
5	C₄×C₁₃×SD₃₂	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#367	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2139	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₁₃×#520	—	2	2	no	—
11	C₁₃×#508	C₄	1	8	yes	complemented
12	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
14	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
17	C₁₃×#440	—	2	4	no	—
18	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
19	C₄×C₁₃×D₈	C₂²	1	4	yes	—
20	C₂×C₁₃×#107	C₂²	1	2	yes	—
21	C₁₃×#417	—	2	2	no	—
22	C₁₃×#405	C₂²	1	1	yes	characteristic
23	C₁₃×#367	C₂²	1	2	yes	—
24	C₁₃×#367	C₂²	1	2	yes	—
25	C₁₃×#358	C₂²	1	4	yes	—
26	C₁₃×#357	C₂²	1	4	yes	—
27	C₁₃×#346	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	2	2	no	—
30	C₁₃×#135	—	2	4	no	—
31	C₁₃×#135	C₂×C₄	1	8	yes	complemented
32	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
33	C₂×C₁₃×SD₁₆	—	2	2	no	—
34	C₂×C₁₃×D₈	C₂³	1	2	yes	—
35	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	—	2	4	no	—
38	C₄×D₄×C₁₃	—	2	4	no	—
39	C₄×D₄×C₁₃	—	4	4	no	—
40	C₁₃×#117	—	4	2	no	—
41	C₁₃×#117	D₄	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
44	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
45	C₁₃×SD₃₂	C₂×C₄	1	8	yes	complemented
46	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
47	C₁₃×M₃₂	—	2	2	no	—
48	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
52	C₁₃×#107	—	2	2	no	—
53	C₁₃×#107	C₂³	1	4	yes	—
54	C₁₃×#103	—	2	4	no	—
55	C₁₃×#102	—	2	4	no	—
56	C₁₃×#97	D₄	1	2	yes	—
57	C₄×C₈×C₁₃	C₂³	1	2	yes	—
58	#33460	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₁₃×#41	—	2	8	no	—
60	C₁₃×#41	—	4	4	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	4	2	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	4	4	no	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₁₃×Q₁₆	—	2	2	no	—
69	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
70	C₁₃×SD₁₆	—	2	8	no	—
71	C₁₃×D₈	—	2	2	no	—
72	C₁₃×D₈	C₂²×C₄	1	4	yes	—
73	C₁₃×M₁₆	—	2	2	no	—
74	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	2	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	4	no	—
80	C₁₃×C₄⋊C₄	—	4	2	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
83	C₁₃×C₂²⋊C₄	—	4	4	no	—
84	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
85	C₄²×C₁₃	—	4	4	no	—
86	C₄²×C₁₃	C₂×D₄	1	2	yes	—
87	C₁₃×C₁₆	—	2	2	no	—
88	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	C₂×#508	C₂×C₁₃	1	1	yes	characteristic
91	#2951	C₂×C₁₃	1	2	yes	complemented
92	#2177	C₂×C₁₃	1	2	yes	complemented
93	#2176	C₂×C₁₃	1	2	yes	complemented
94	C₄×SD₃₂	C₂×C₁₃	1	4	yes	complemented
95	C₂×#367	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#2139	C₂×C₁₃	1	2	yes	complemented
97	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂³×C₁₃	—	4	2	no	—
99	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	2	4	no	—
103	D₄×C₁₃	—	4	8	no	—
104	D₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	2	no	—
106	C₂×C₄×C₁₃	—	4	8	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	16	4	no	—
109	C₂×C₄×C₁₃	—	16	2	no	—
110	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
112	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	#135	1	2	yes	—
114	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	C₈×C₁₃	—	2	2	no	—
116	C₈×C₁₃	C₄×D₄	1	2	yes	—
117	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
118	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
119	C₂×#135	C₂²×C₁₃	1	2	yes	—
120	#520	—	2	2	no	—
121	#508	C₄×C₁₃	1	8	yes	complemented
122	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
123	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
124	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
125	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
126	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
127	#440	—	2	4	no	—
128	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
129	C₄×D₈	C₂²×C₁₃	1	4	yes	—
130	C₂×#107	C₂²×C₁₃	1	2	yes	—
131	#417	—	2	2	no	—
132	#405	C₂²×C₁₃	1	1	yes	characteristic
133	#367	C₂²×C₁₃	1	2	yes	—
134	#367	C₂²×C₁₃	1	2	yes	—
135	#358	C₂²×C₁₃	1	4	yes	—
136	#357	C₂²×C₁₃	1	4	yes	—
137	#346	C₂²×C₁₃	1	2	yes	—
138	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	4	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	C₄×D₈	1	2	yes	—
144	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	—	4	4	no	—
147	C₄×C₁₃	—	8	4	no	—
148	C₄×C₁₃	C₄×D₈	1	2	yes	—
149	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
151	C₂×#41	—	2	2	no	—
152	#135	—	2	4	no	—
153	#135	C₂×C₄×C₁₃	1	8	yes	complemented
154	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
155	C₂×SD₁₆	—	2	2	no	—
156	C₂×D₈	C₂³×C₁₃	1	2	yes	—
157	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
159	C₄×Q₈	—	2	4	no	—
160	C₄×D₄	—	2	4	no	—
161	C₄×D₄	—	4	4	no	—
162	#117	—	4	2	no	—
163	#117	D₄×C₁₃	1	1	yes	characteristic
164	C₂×C₄⋊C₄	—	2	2	no	—
165	C₂×C₄⋊C₄	—	4	2	no	—
166	Q₃₂	C₂×C₄×C₁₃	1	4	yes	complemented
167	SD₃₂	C₂×C₄×C₁₃	1	8	yes	complemented
168	D₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
169	M₃₂	—	2	2	no	—
170	C₂×C₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
171	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
172	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
173	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
174	#107	—	2	2	no	—
175	#107	C₂³×C₁₃	1	4	yes	—
176	#103	—	2	4	no	—
177	#102	—	2	4	no	—
178	#97	D₄×C₁₃	1	2	yes	—
179	C₄×C₈	C₂³×C₁₃	1	2	yes	—
180	C₂×C₁₃	—	2	2	no	—
181	C₂×C₁₃	—	8	4	no	—
182	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
183	C₂×C₁₃	#3416	1	2	yes	—
184	#41	—	2	8	no	—
185	#41	—	4	4	no	—
186	C₂×Q₈	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂²×C₄	—	4	2	no	—
190	C₂²×C₄	—	4	2	no	—
191	C₂²×C₄	—	4	4	no	—
192	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
193	Q₁₆	—	2	2	no	—
194	Q₁₆	C₂²×C₄×C₁₃	1	4	yes	—
195	SD₁₆	—	2	8	no	—
196	D₈	—	2	2	no	—
197	D₈	C₂²×C₄×C₁₃	1	4	yes	—
198	M₁₆	—	2	2	no	—
199	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
200	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
201	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
202	C₂×C₈	C₁₃×#41	1	2	yes	—
203	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	2	no	—
206	C₄⋊C₄	—	4	4	no	—
207	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
210	C₄²	—	4	4	no	—
211	C₄²	C₂×D₄×C₁₃	1	2	yes	—
212	C₁₆	—	2	2	no	—
213	C₁₆	C₂²×C₄×C₁₃	1	2	yes	—
214	C₁₆	C₂²×C₄×C₁₃	1	2	yes	—
215	C₁₃	#33460	1	1	yes	characteristic, direct product factor, Sylow
216	C₂³	—	4	2	no	—
217	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	4	2	no	—
220	D₄	—	2	4	no	—
221	D₄	—	4	8	no	—
222	D₄	—	4	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	8	no	—
225	C₂×C₄	—	4	4	no	—
226	C₂×C₄	—	16	4	no	—
227	C₂×C₄	—	16	2	no	—
228	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
229	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
230	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
231	C₂×C₄	C₁₃×#135	1	2	yes	—
232	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
233	C₈	—	2	2	no	—
234	C₈	C₄×D₄×C₁₃	1	2	yes	—
235	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
236	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
237	C₂²	—	2	2	no	—
238	C₂²	—	4	4	no	—
239	C₂²	—	8	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	C₄×C₁₃×D₈	1	2	yes	—
242	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
243	C₄	—	2	4	no	—
244	C₄	—	4	4	no	—
245	C₄	—	8	4	no	—
246	C₄	C₄×C₁₃×D₈	1	2	yes	—
247	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
248	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
249	C₂	—	2	2	no	—
250	C₂	—	8	4	no	—
251	C₂	C₂×C₄×C₁₃×D₈	1	1	yes	characteristic
252	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator	C₈	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#508	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2951	C₂	semidirect
3. C₁₃×#2177	C₂	semidirect
4. C₁₃×#2176	C₂	semidirect
5. C₄×C₁₃×SD₃₂	C₂	semidirect
6. C₂×C₁₃×#367	C₂	semidirect
7. C₁₃×#2139	C₂	semidirect
8. C₁₃×#2108	C₂	semidirect
11. C₁₃×#508	C₄	semidirect
31. C₁₃×#135	C₂×C₄	semidirect
44. C₁₃×Q₃₂	C₂×C₄	semidirect
45. C₁₃×SD₃₂	C₂×C₄	semidirect
46. C₁₃×D₁₆	C₂×C₄	semidirect
48. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
58. #33460	C₁₃	direct (N × H)
91. #2951	C₂×C₁₃	semidirect
92. #2177	C₂×C₁₃	semidirect
93. #2176	C₂×C₁₃	semidirect
94. C₄×SD₃₂	C₂×C₁₃	semidirect
95. C₂×#367	C₂×C₁₃	semidirect
96. #2139	C₂×C₁₃	semidirect
97. #2108	C₂×C₁₃	semidirect
121. #508	C₄×C₁₃	semidirect
153. #135	C₂×C₄×C₁₃	semidirect
166. Q₃₂	C₂×C₄×C₁₃	semidirect
167. SD₃₂	C₂×C₄×C₁₃	semidirect
168. D₁₆	C₂×C₄×C₁₃	semidirect
170. C₂×C₁₆	C₂×C₄×C₁₃	semidirect
215. C₁₃	#33460	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#508	2
C₁₃×#2951	2
C₁₃×#2177	2
C₁₃×#2176	2
C₄×C₁₃×SD₃₂	2
C₂×C₁₃×#367	2
C₁₃×#2139	2
C₁₃×#2108	2
#33460	13

Subgroups — C13×#33460 #15719799

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33460 #15719799