Subgroups — C₁₃×#3278 #13112336

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×#137	C₂	1	2	yes	maximal, complemented
2	C₁₃×#550	C₂	1	1	yes	maximal, characteristic
3	D₄×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#468	C₂	1	2	yes	maximal, complemented
6	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#460	C₂	1	1	yes	maximal, characteristic, 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	C₁₃×#436	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×#137	—	2	8	no	—
15	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
18	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
19	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
20	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
21	C₁₃×C₄⋊Q₈	—	2	1	no	—
22	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
23	C₁₃×#127	—	2	1	no	—
24	C₁₃×#127	C₂²	1	1	yes	characteristic
25	C₁₃×C₂²⋊Q₈	—	2	2	no	—
26	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
27	C₁₃×#121	—	2	2	no	—
28	C₁₃×#121	C₂²	1	2	yes	—
29	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₃	—	2	1	no	—
32	C₄×D₄×C₁₃	—	2	1	no	—
33	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
34	C₁₃×#107	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	1	yes	characteristic
38	C₁₃×#103	C₂²	1	2	yes	—
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	—	4	2	no	—
44	C₂×Q₈×C₁₃	—	2	2	no	—
45	C₂×Q₈×C₁₃	—	2	4	no	—
46	C₂×Q₈×C₁₃	—	4	2	no	—
47	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₁₃	—	2	2	no	—
51	C₂×D₄×C₁₃	—	2	2	no	—
52	C₂×D₄×C₁₃	—	2	1	no	—
53	C₂×D₄×C₁₃	—	2	1	no	—
54	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
55	C₂×D₄×C₁₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₃	—	2	2	no	—
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
59	C₁₃×Q₁₆	—	2	4	no	—
60	C₁₃×Q₁₆	D₄	1	4	yes	complemented
61	C₁₃×SD₁₆	—	2	4	no	—
62	C₁₃×SD₁₆	—	2	4	no	—
63	C₁₃×SD₁₆	—	4	2	no	—
64	C₁₃×M₁₆	—	2	4	no	—
65	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂³	1	2	yes	—
67	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	—	2	1	no	—
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	—	2	1	no	—
71	C₁₃×C₄⋊C₄	—	2	1	no	—
72	C₁₃×C₄⋊C₄	—	2	2	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	D₄	1	2	yes	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	—	2	1	no	—
81	C₄²×C₁₃	C₂³	1	1	yes	characteristic
82	#3278	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₁₃	—	2	2	no	—
84	C₂³×C₁₃	C₂×D₄	1	2	yes	—
85	Q₈×C₁₃	—	2	1	no	—
86	Q₈×C₁₃	—	2	1	no	—
87	Q₈×C₁₃	—	2	2	no	—
88	Q₈×C₁₃	—	2	2	no	—
89	Q₈×C₁₃	—	4	4	no	—
90	Q₈×C₁₃	C₂×D₄	1	2	yes	—
91	Q₈×C₁₃	C₂×D₄	1	2	yes	—
92	D₄×C₁₃	—	2	2	no	—
93	D₄×C₁₃	—	2	2	no	—
94	D₄×C₁₃	—	2	2	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	C₂×C₄×C₁₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	—	2	1	no	—
114	C₈×C₁₃	C₂×D₄	1	2	yes	—
115	C₂×#137	C₂×C₁₃	1	2	yes	complemented
116	#550	C₂×C₁₃	1	1	yes	characteristic
117	D₄×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#468	C₂×C₁₃	1	2	yes	complemented
120	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#460	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#452	C₂×C₁₃	1	2	yes	complemented
123	#448	C₂×C₁₃	1	2	yes	complemented
124	C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	2	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	4	2	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	—	4	2	no	—
132	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₁₃	—	2	2	no	—
134	C₄×C₁₃	—	2	2	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	4	2	no	—
138	C₄×C₁₃	—	4	1	no	—
139	C₄×C₁₃	—	4	1	no	—
140	C₄×C₁₃	#137	1	2	yes	complemented
141	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
143	C₂×#41	C₂²×C₁₃	1	2	yes	—
144	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
145	#137	—	2	8	no	—
146	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
147	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
149	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
150	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
151	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
152	C₄⋊Q₈	—	2	1	no	—
153	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
154	#127	—	2	1	no	—
155	#127	C₂²×C₁₃	1	1	yes	characteristic
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	1	yes	characteristic
161	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₄×D₄	—	2	1	no	—
163	C₄×D₄	—	2	1	no	—
164	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
165	#107	C₂²×C₁₃	1	1	yes	characteristic
166	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
167	#103	C₂²×C₁₃	1	1	yes	characteristic
168	#103	C₂²×C₁₃	1	1	yes	characteristic
169	#103	C₂²×C₁₃	1	2	yes	—
170	#102	C₂²×C₁₃	1	2	yes	—
171	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
172	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
173	C₂×C₁₃	—	4	2	no	—
174	C₂×C₁₃	—	8	2	no	—
175	C₂×C₁₃	D₄²	1	1	yes	characteristic
176	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
177	C₂×C₁₃	#577	1	1	yes	characteristic
178	#41	—	2	4	no	—
179	#41	—	4	2	no	—
180	C₂×Q₈	—	2	2	no	—
181	C₂×Q₈	—	2	4	no	—
182	C₂×Q₈	—	4	2	no	—
183	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
184	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
185	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	—	2	1	no	—
189	C₂×D₄	—	2	1	no	—
190	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
191	C₂×D₄	C₂³×C₁₃	1	2	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
195	Q₁₆	—	2	4	no	—
196	Q₁₆	D₄×C₁₃	1	4	yes	complemented
197	SD₁₆	—	2	4	no	—
198	SD₁₆	—	2	4	no	—
199	SD₁₆	—	4	2	no	—
200	M₁₆	—	2	4	no	—
201	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₈	C₂³×C₁₃	1	2	yes	—
203	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	2	1	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	—	2	1	no	—
208	C₄⋊C₄	—	2	2	no	—
209	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
210	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
211	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	2	no	—
214	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
215	C₄²	—	2	1	no	—
216	C₄²	—	2	1	no	—
217	C₄²	C₂³×C₁₃	1	1	yes	characteristic
218	C₁₃	#3278	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³	—	2	2	no	—
220	C₂³	C₂×D₄×C₁₃	1	2	yes	—
221	Q₈	—	2	1	no	—
222	Q₈	—	2	1	no	—
223	Q₈	—	2	2	no	—
224	Q₈	—	2	2	no	—
225	Q₈	—	4	4	no	—
226	Q₈	C₂×D₄×C₁₃	1	2	yes	—
227	Q₈	C₂×D₄×C₁₃	1	2	yes	—
228	D₄	—	2	2	no	—
229	D₄	—	2	2	no	—
230	D₄	—	2	2	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	2	no	—
233	D₄	—	8	1	no	—
234	D₄	—	8	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
246	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
247	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
248	C₈	—	2	2	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×D₄×C₁₃	1	2	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	2	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	C₁₃×#137	1	2	yes	complemented
266	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
267	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂	—	4	2	no	—
269	C₂	—	8	2	no	—
270	C₂	D₄²×C₁₃	1	1	yes	characteristic
271	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
272	C₂	C₁₃×#577	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#550	C₂
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×#137	C₂	semidirect
3. D₄×Q₈×C₁₃	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#468	C₂	semidirect
6. C₁₃×#462	C₂	semidirect
7. C₁₃×#460	C₂	semidirect
8. C₁₃×#452	C₂	semidirect
9. C₁₃×#448	C₂	semidirect
10. C₄×C₁₃×Q₁₆	C₂	semidirect
11. C₁₃×#436	C₂	semidirect
60. C₁₃×Q₁₆	D₄	semidirect
82. #3278	C₁₃	direct (N × H)
115. C₂×#137	C₂×C₁₃	semidirect
117. D₄×Q₈	C₂×C₁₃	semidirect
118. #492	C₂×C₁₃	semidirect
119. #468	C₂×C₁₃	semidirect
120. #462	C₂×C₁₃	semidirect
121. #460	C₂×C₁₃	semidirect
122. #452	C₂×C₁₃	semidirect
123. #448	C₂×C₁₃	semidirect
124. C₄×Q₁₆	C₂×C₁₃	semidirect
125. #436	C₂×C₁₃	semidirect
140. C₄×C₁₃	#137	semidirect
196. Q₁₆	D₄×C₁₃	semidirect
218. C₁₃	#3278	direct (N × H)
265. C₄	C₁₃×#137	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#137	2
C₁₃×#550	2
D₄×Q₈×C₁₃	2
C₁₃×#492	2
C₁₃×#468	2
C₁₃×#462	2
C₁₃×#460	2
C₁₃×#452	2
C₁₃×#448	2
C₄×C₁₃×Q₁₆	2
C₁₃×#436	2
#3278	13

Subgroups — C13×#3278 #13112336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3278 #13112336