Subgroups — C₁₃×#3284 #13112342

Order 1664.

494 conjugacy classes of subgroups.

The subgroup lattice has 282 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	2	yes	maximal, complemented
2	C₁₃×#550	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#547	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#465	C₂	1	2	yes	maximal, complemented
6	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#452	C₂	1	2	yes	maximal, complemented
9	C₁₃×#449	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×SD₁₆	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	C₂×C₁₃×#41	C₂²	1	2	yes	—
14	C₁₃×#135	—	2	8	no	—
15	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×D₈	C₂²	1	2	yes	—
20	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
21	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₁₃×#127	—	2	1	no	—
23	C₁₃×#127	C₂²	1	1	yes	characteristic
24	C₁₃×#123	—	2	2	no	—
25	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
26	C₁₃×#121	—	2	1	no	—
27	C₁₃×#121	—	2	2	no	—
28	C₁₃×#121	C₂²	1	2	yes	—
29	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₄×D₄×C₁₃	—	2	1	no	—
31	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
32	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
33	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
34	C₁₃×#106	C₂²	1	1	yes	characteristic
35	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₁₃×#103	C₂²	1	1	yes	characteristic
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	1	yes	characteristic
39	C₁₃×#102	C₂²	1	2	yes	—
40	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
41	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
42	C₁₃×#41	—	2	4	no	—
43	C₁₃×#41	—	2	4	no	—
44	C₁₃×#41	—	4	2	no	—
45	C₁₃×#41	—	4	2	no	—
46	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
47	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	—	2	2	no	—
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	1	no	—
52	C₂×D₄×C₁₃	—	2	1	no	—
53	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
54	C₂×D₄×C₁₃	C₂³	1	2	yes	—
55	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	—	2	1	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	2	2	no	—
59	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
60	C₁₃×Q₁₆	—	2	4	no	—
61	C₁₃×SD₁₆	—	2	4	no	—
62	C₁₃×SD₁₆	—	4	2	no	—
63	C₁₃×SD₁₆	D₄	1	4	yes	complemented
64	C₁₃×D₈	—	2	4	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₁₃	C₂³	1	2	yes	—
68	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	—	2	2	no	—
71	C₁₃×C₄⋊C₄	—	2	1	no	—
72	C₁₃×C₄⋊C₄	—	2	1	no	—
73	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
74	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
76	C₁₃×C₂²⋊C₄	—	2	2	no	—
77	C₁₃×C₂²⋊C₄	—	2	2	no	—
78	C₁₃×C₂²⋊C₄	—	2	1	no	—
79	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	C₂³	1	1	yes	characteristic
82	#3284	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	C₂×D₄	1	2	yes	—
86	Q₈×C₁₃	—	2	1	no	—
87	Q₈×C₁₃	—	2	2	no	—
88	Q₈×C₁₃	—	2	2	no	—
89	Q₈×C₁₃	C₂×D₄	1	2	yes	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	2	1	no	—
93	D₄×C₁₃	—	2	2	no	—
94	D₄×C₁₃	—	4	4	no	—
95	D₄×C₁₃	—	4	4	no	—
96	D₄×C₁₃	—	4	2	no	—
97	D₄×C₁₃	—	8	1	no	—
98	D₄×C₁₃	—	8	2	no	—
99	D₄×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₄×C₁₃	—	2	1	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	4	4	no	—
107	C₂×C₄×C₁₃	—	4	4	no	—
108	C₂×C₄×C₁₃	—	4	2	no	—
109	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
113	C₈×C₁₃	—	2	2	no	—
114	C₈×C₁₃	—	2	1	no	—
115	C₈×C₁₃	C₂×D₄	1	2	yes	—
116	C₂×#135	C₂×C₁₃	1	2	yes	complemented
117	#550	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#547	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#465	C₂×C₁₃	1	2	yes	complemented
121	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#459	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#452	C₂×C₁₃	1	2	yes	complemented
124	#449	C₂×C₁₃	1	2	yes	complemented
125	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
126	C₈×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	2	no	—
130	C₂²×C₁₃	—	2	2	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	—	4	1	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₄×C₁₃	—	2	2	no	—
138	C₄×C₁₃	—	2	2	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	4	2	no	—
141	C₄×C₁₃	—	4	1	no	—
142	C₄×C₁₃	—	4	1	no	—
143	C₄×C₁₃	#135	1	2	yes	complemented
144	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
145	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂×#41	C₂²×C₁₃	1	2	yes	—
147	C₂×#41	C₂²×C₁₃	1	2	yes	—
148	#135	—	2	8	no	—
149	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
150	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
151	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
152	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×D₈	C₂²×C₁₃	1	2	yes	—
154	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
155	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
156	#127	—	2	1	no	—
157	#127	C₂²×C₁₃	1	1	yes	characteristic
158	#123	—	2	2	no	—
159	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
160	#121	—	2	1	no	—
161	#121	—	2	2	no	—
162	#121	C₂²×C₁₃	1	2	yes	—
163	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₄×D₄	—	2	1	no	—
165	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
166	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×C₄⋊C₄	—	2	1	no	—
168	#106	C₂²×C₁₃	1	1	yes	characteristic
169	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
170	#103	C₂²×C₁₃	1	1	yes	characteristic
171	#103	C₂²×C₁₃	1	2	yes	—
172	#102	C₂²×C₁₃	1	1	yes	characteristic
173	#102	C₂²×C₁₃	1	2	yes	—
174	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
175	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
176	C₂×C₁₃	—	4	2	no	—
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	—	8	2	no	—
179	C₂×C₁₃	D₄²	1	1	yes	characteristic
180	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
181	C₂×C₁₃	#577	1	1	yes	characteristic
182	#41	—	2	4	no	—
183	#41	—	2	4	no	—
184	#41	—	4	2	no	—
185	#41	—	4	2	no	—
186	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
187	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
188	C₂×D₄	—	2	2	no	—
189	C₂×D₄	—	2	2	no	—
190	C₂×D₄	—	2	2	no	—
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
194	C₂×D₄	C₂³×C₁₃	1	2	yes	—
195	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
200	Q₁₆	—	2	4	no	—
201	SD₁₆	—	2	4	no	—
202	SD₁₆	—	4	2	no	—
203	SD₁₆	D₄×C₁₃	1	4	yes	complemented
204	D₈	—	2	4	no	—
205	C₂×C₈	—	2	4	no	—
206	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
207	C₂×C₈	C₂³×C₁₃	1	2	yes	—
208	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	2	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
214	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	2	no	—
217	C₂²⋊C₄	—	2	2	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
220	C₄²	—	2	1	no	—
221	C₄²	C₂³×C₁₃	1	1	yes	characteristic
222	C₁₃	#3284	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	2	1	no	—
224	C₂³	—	2	2	no	—
225	C₂³	C₂×D₄×C₁₃	1	2	yes	—
226	Q₈	—	2	1	no	—
227	Q₈	—	2	2	no	—
228	Q₈	—	2	2	no	—
229	Q₈	C₂×D₄×C₁₃	1	2	yes	—
230	D₄	—	2	2	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	1	no	—
233	D₄	—	2	2	no	—
234	D₄	—	4	4	no	—
235	D₄	—	4	4	no	—
236	D₄	—	4	2	no	—
237	D₄	—	8	1	no	—
238	D₄	—	8	2	no	—
239	D₄	C₂×D₄×C₁₃	1	2	yes	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
251	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	1	no	—
255	C₈	C₂×D₄×C₁₃	1	2	yes	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	2	no	—
265	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
266	C₄	—	2	2	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	1	no	—
269	C₄	—	4	2	no	—
270	C₄	—	4	1	no	—
271	C₄	—	4	1	no	—
272	C₄	C₁₃×#135	1	2	yes	complemented
273	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
274	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
275	C₂	—	4	2	no	—
276	C₂	—	4	2	no	—
277	C₂	—	8	2	no	—
278	C₂	D₄²×C₁₃	1	1	yes	characteristic
279	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
280	C₂	C₁₃×#577	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. C₁₃×#550	C₂	semidirect
3. C₁₃×#547	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#465	C₂	semidirect
6. C₁₃×#462	C₂	semidirect
7. C₁₃×#459	C₂	semidirect
8. C₁₃×#452	C₂	semidirect
9. C₁₃×#449	C₂	semidirect
10. C₄×C₁₃×SD₁₆	C₂	semidirect
11. C₈×D₄×C₁₃	C₂	semidirect
63. C₁₃×SD₁₆	D₄	semidirect
82. #3284	C₁₃	direct (N × H)
116. C₂×#135	C₂×C₁₃	semidirect
117. #550	C₂×C₁₃	semidirect
118. #547	C₂×C₁₃	semidirect
119. #492	C₂×C₁₃	semidirect
120. #465	C₂×C₁₃	semidirect
121. #462	C₂×C₁₃	semidirect
122. #459	C₂×C₁₃	semidirect
123. #452	C₂×C₁₃	semidirect
124. #449	C₂×C₁₃	semidirect
125. C₄×SD₁₆	C₂×C₁₃	semidirect
126. C₈×D₄	C₂×C₁₃	semidirect
143. C₄×C₁₃	#135	semidirect
203. SD₁₆	D₄×C₁₃	semidirect
222. C₁₃	#3284	direct (N × H)
272. C₄	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
C₁₃×#550	2
C₁₃×#547	2
C₁₃×#492	2
C₁₃×#465	2
C₁₃×#462	2
C₁₃×#459	2
C₁₃×#452	2
C₁₃×#449	2
C₄×C₁₃×SD₁₆	2
C₈×D₄×C₁₃	2
#3284	13

Subgroups — C13×#3284 #13112342

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3284 #13112342