Subgroups — C₁₃×#3285 #13112343

Order 1664.

482 conjugacy classes of subgroups.

The subgroup lattice has 274 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#548	C₂	1	2	yes	maximal, complemented
4	C₁₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#467	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#466	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#464	C₂	1	2	yes	maximal, complemented
8	C₁₃×#452	C₂	1	2	yes	maximal, complemented
9	C₁₃×#448	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	2	yes	—
13	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
14	C₁₃×#135	—	2	4	no	—
15	C₂×C₁₃×Q₁₆	—	2	4	no	—
16	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
20	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
21	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#124	—	2	2	no	—
25	C₁₃×#124	C₂²	1	2	yes	—
26	C₁₃×C₂²⋊Q₈	—	2	2	no	—
27	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
28	C₁₃×#121	—	2	2	no	—
29	C₁₃×#121	C₂²	1	2	yes	—
30	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
31	C₄×D₄×C₁₃	—	2	2	no	—
32	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
33	C₁₃×#107	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×#103	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
39	C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₁₃×#41	—	2	4	no	—
42	C₁₃×#41	—	4	2	no	—
43	C₂×Q₈×C₁₃	—	2	2	no	—
44	C₂×Q₈×C₁₃	—	2	4	no	—
45	C₂×Q₈×C₁₃	—	4	2	no	—
46	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
51	C₂×D₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	—	2	2	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×Q₁₆	—	2	4	no	—
57	C₁₃×Q₁₆	—	2	2	no	—
58	C₁₃×Q₁₆	D₄	1	4	yes	complemented
59	C₁₃×SD₁₆	—	2	4	no	—
60	C₁₃×SD₁₆	—	4	2	no	—
61	C₁₃×D₈	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	—	2	2	no	—
69	C₁₃×C₄⋊C₄	—	2	2	no	—
70	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₁₃×C₂²⋊C₄	—	2	2	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₁₃×C₂²⋊C₄	—	2	2	no	—
75	C₁₃×C₂²⋊C₄	—	2	2	no	—
76	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
77	C₁₃×C₂²⋊C₄	D₄	1	1	yes	characteristic
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	C₂³	1	1	yes	characteristic
80	#3285	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₃	—	2	2	no	—
85	Q₈×C₁₃	—	2	2	no	—
86	Q₈×C₁₃	—	2	2	no	—
87	Q₈×C₁₃	—	4	4	no	—
88	Q₈×C₁₃	C₂×D₄	1	4	yes	—
89	D₄×C₁₃	—	2	2	no	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	4	4	no	—
93	D₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	—	2	2	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₈×C₁₃	—	2	1	no	—
109	C₈×C₁₃	—	2	1	no	—
110	C₈×C₁₃	—	2	1	no	—
111	C₈×C₁₃	C₂×D₄	1	2	yes	—
112	C₂×#135	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂²×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#548	C₂×C₁₃	1	2	yes	complemented
115	#495	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#467	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#466	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#464	C₂×C₁₃	1	2	yes	complemented
119	#452	C₂×C₁₃	1	2	yes	complemented
120	#448	C₂×C₁₃	1	2	yes	complemented
121	C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	#135	1	2	yes	complemented
132	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	2	4	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	2	1	no	—
138	C₄×C₁₃	—	4	2	no	—
139	C₄×C₁₃	—	4	2	no	—
140	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
142	C₂×#41	C₂²×C₁₃	1	2	yes	—
143	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
144	#135	—	2	4	no	—
145	C₂×Q₁₆	—	2	4	no	—
146	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
147	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
150	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
151	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
152	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
153	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
154	#124	—	2	2	no	—
155	#124	C₂²×C₁₃	1	2	yes	—
156	C₂²⋊Q₈	—	2	2	no	—
157	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
158	#121	—	2	2	no	—
159	#121	C₂²×C₁₃	1	2	yes	—
160	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
161	C₄×D₄	—	2	2	no	—
162	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
163	#107	C₂²×C₁₃	1	1	yes	characteristic
164	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
165	#103	C₂²×C₁₃	1	2	yes	—
166	#103	C₂²×C₁₃	1	2	yes	—
167	#102	C₂²×C₁₃	1	2	yes	—
168	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
169	C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×C₁₃	—	2	2	no	—
172	C₂×C₁₃	—	4	1	no	—
173	C₂×C₁₃	—	8	2	no	—
174	C₂×C₁₃	D₄²	1	1	yes	characteristic
175	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
176	C₂×C₁₃	#578	1	1	yes	characteristic
177	#41	—	2	4	no	—
178	#41	—	4	2	no	—
179	C₂×Q₈	—	2	2	no	—
180	C₂×Q₈	—	2	4	no	—
181	C₂×Q₈	—	4	2	no	—
182	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
183	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
187	C₂×D₄	C₂³×C₁₃	1	2	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
192	Q₁₆	—	2	4	no	—
193	Q₁₆	—	2	2	no	—
194	Q₁₆	D₄×C₁₃	1	4	yes	complemented
195	SD₁₆	—	2	4	no	—
196	SD₁₆	—	4	2	no	—
197	D₈	—	2	2	no	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
207	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
208	C₂²⋊C₄	—	2	2	no	—
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
213	C₂²⋊C₄	D₄×C₁₃	1	1	yes	characteristic
214	C₄²	—	2	2	no	—
215	C₄²	C₂³×C₁₃	1	1	yes	characteristic
216	C₁₃	#3285	1	1	yes	characteristic, direct product factor, Sylow
217	C₂³	—	2	2	no	—
218	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
220	Q₈	—	2	2	no	—
221	Q₈	—	2	2	no	—
222	Q₈	—	2	2	no	—
223	Q₈	—	4	4	no	—
224	Q₈	C₂×D₄×C₁₃	1	4	yes	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	4	no	—
239	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	2	1	no	—
247	C₈	C₂×D₄×C₁₃	1	2	yes	—
248	C₂²	—	2	1	no	—
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	C₁₃×#135	1	2	yes	complemented
257	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	2	4	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
267	C₂	—	2	2	no	—
268	C₂	—	4	1	no	—
269	C₂	—	8	2	no	—
270	C₂	D₄²×C₁₃	1	1	yes	characteristic
271	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
272	C₂	C₁₃×#578	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#135	C₂	semidirect
2. C₂²×C₁₃×Q₁₆	C₂	semidirect
3. C₁₃×#548	C₂	semidirect
4. C₁₃×#495	C₂	semidirect
5. C₁₃×#467	C₂	semidirect
6. C₁₃×#466	C₂	semidirect
7. C₁₃×#464	C₂	semidirect
8. C₁₃×#452	C₂	semidirect
9. C₁₃×#448	C₂	semidirect
10. C₄×C₁₃×Q₁₆	C₂	semidirect
11. C₈×D₄×C₁₃	C₂	semidirect
58. C₁₃×Q₁₆	D₄	semidirect
80. #3285	C₁₃	direct (N × H)
112. C₂×#135	C₂×C₁₃	semidirect
113. C₂²×Q₁₆	C₂×C₁₃	semidirect
114. #548	C₂×C₁₃	semidirect
115. #495	C₂×C₁₃	semidirect
116. #467	C₂×C₁₃	semidirect
117. #466	C₂×C₁₃	semidirect
118. #464	C₂×C₁₃	semidirect
119. #452	C₂×C₁₃	semidirect
120. #448	C₂×C₁₃	semidirect
121. C₄×Q₁₆	C₂×C₁₃	semidirect
122. C₈×D₄	C₂×C₁₃	semidirect
131. C₂²×C₁₃	#135	semidirect
194. Q₁₆	D₄×C₁₃	semidirect
216. C₁₃	#3285	direct (N × H)
256. C₂²	C₁₃×#135	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#135	2
C₂²×C₁₃×Q₁₆	2
C₁₃×#548	2
C₁₃×#495	2
C₁₃×#467	2
C₁₃×#466	2
C₁₃×#464	2
C₁₃×#452	2
C₁₃×#448	2
C₄×C₁₃×Q₁₆	2
C₈×D₄×C₁₃	2
#3285	13

Subgroups — C13×#3285 #13112343

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3285 #13112343