Subgroups — C₁₃×#2947 #13112005

Order 1664.

470 conjugacy classes of subgroups.

The subgroup lattice has 262 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	1	yes	maximal, characteristic
2	C₁₃×#520	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#519	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#441	C₂	1	2	yes	maximal, complemented
5	C₁₃×#440	C₂	1	2	yes	maximal, complemented
6	C₄×C₁₃×Q₁₆	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
8	C₁₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#416	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#103	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#405	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₁₃×#137	C₄	1	8	yes	complemented
15	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
17	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
19	C₄×Q₈×C₁₃	—	2	1	no	—
20	C₄×Q₈×C₁₃	—	2	1	no	—
21	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
24	C₄×D₄×C₁₃	—	2	2	no	—
25	C₄×D₄×C₁₃	C₂²	1	2	yes	—
26	C₁₃×#117	—	2	1	no	—
27	C₁₃×#117	—	2	2	no	—
28	C₁₃×#117	C₂²	1	1	yes	characteristic
29	C₁₃×#117	C₂²	1	1	yes	characteristic
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
32	C₁₃×#107	C₂²	1	2	yes	—
33	C₁₃×#106	C₂²	1	2	yes	—
34	C₁₃×#103	C₂²	1	2	yes	—
35	C₁₃×#103	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	1	yes	characteristic
37	C₁₃×#103	C₂²	1	1	yes	characteristic
38	C₁₃×#102	C₂²	1	1	yes	characteristic
39	C₁₃×#102	C₂²	1	1	yes	characteristic
40	C₁₃×#97	C₂²	1	2	yes	—
41	C₄×C₈×C₁₃	C₂²	1	2	yes	—
42	C₁₃×#41	—	2	2	no	—
43	C₁₃×#41	C₂×C₄	1	4	yes	—
44	C₂×Q₈×C₁₃	—	2	1	no	—
45	C₂×Q₈×C₁₃	—	2	2	no	—
46	C₂×Q₈×C₁₃	C₂×C₄	1	4	yes	complemented
47	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	—	2	1	no	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×Q₁₆	C₂×C₄	1	8	yes	complemented
57	C₁₃×SD₁₆	C₂×C₄	1	8	yes	complemented
58	C₁₃×M₁₆	C₂×C₄	1	4	yes	complemented
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	C₂×C₈×C₁₃	C₂³	1	2	yes	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	2	no	—
66	C₁₃×C₄⋊C₄	—	2	2	no	—
67	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
70	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
75	C₄²×C₁₃	—	2	2	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	1	no	—
79	C₄²×C₁₃	C₂³	1	2	yes	—
80	#2947	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₁₃	—	2	1	no	—
82	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₁₃	—	2	4	no	—
84	Q₈×C₁₃	—	2	2	no	—
85	Q₈×C₁₃	—	2	1	no	—
86	Q₈×C₁₃	C₂²×C₄	1	4	yes	—
87	Q₈×C₁₃	C₂²×C₄	1	2	yes	—
88	D₄×C₁₃	—	2	4	no	—
89	D₄×C₁₃	—	2	1	no	—
90	D₄×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₁₃	—	2	4	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	4	no	—
95	C₂×C₄×C₁₃	—	2	1	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	8	2	no	—
99	C₂×C₄×C₁₃	—	8	1	no	—
100	C₂×C₄×C₁₃	C₂²×C₄	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	#41	1	2	yes	—
105	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₈×C₁₃	—	2	1	no	—
107	C₈×C₁₃	—	2	1	no	—
108	C₈×C₁₃	C₂²×C₄	1	4	yes	—
109	C₂×#137	C₂×C₁₃	1	1	yes	characteristic
110	#520	C₂×C₁₃	1	1	yes	characteristic
111	#519	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#441	C₂×C₁₃	1	2	yes	complemented
113	#440	C₂×C₁₃	1	2	yes	complemented
114	C₄×Q₁₆	C₂×C₁₃	1	2	yes	complemented
115	C₄×SD₁₆	C₂×C₁₃	1	2	yes	complemented
116	#428	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#416	C₂×C₁₃	1	1	yes	characteristic
118	C₂×#103	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#405	C₂×C₁₃	1	1	yes	characteristic, complemented
120	C₂²×C₁₃	—	2	1	no	—
121	C₂²×C₁₃	—	2	1	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	—	4	1	no	—
124	C₂²×C₁₃	—	4	1	no	—
125	C₂²×C₁₃	C₄×D₄	1	2	yes	—
126	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₄×C₁₃	—	2	2	no	—
128	C₄×C₁₃	—	2	4	no	—
129	C₄×C₁₃	—	2	2	no	—
130	C₄×C₁₃	—	2	2	no	—
131	C₄×C₁₃	—	4	1	no	—
132	C₄×C₁₃	—	4	2	no	—
133	C₄×C₁₃	—	4	1	no	—
134	C₄×C₁₃	C₄×D₄	1	2	yes	—
135	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
138	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
139	#137	C₄×C₁₃	1	8	yes	complemented
140	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
141	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
142	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
143	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₄×Q₈	—	2	1	no	—
145	C₄×Q₈	—	2	1	no	—
146	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
147	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
148	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
149	C₄×D₄	—	2	2	no	—
150	C₄×D₄	C₂²×C₁₃	1	2	yes	—
151	#117	—	2	1	no	—
152	#117	—	2	2	no	—
153	#117	C₂²×C₁₃	1	1	yes	characteristic
154	#117	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×C₄⋊C₄	—	2	1	no	—
156	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
157	#107	C₂²×C₁₃	1	2	yes	—
158	#106	C₂²×C₁₃	1	2	yes	—
159	#103	C₂²×C₁₃	1	2	yes	—
160	#103	C₂²×C₁₃	1	2	yes	—
161	#103	C₂²×C₁₃	1	1	yes	characteristic
162	#103	C₂²×C₁₃	1	1	yes	characteristic
163	#102	C₂²×C₁₃	1	1	yes	characteristic
164	#102	C₂²×C₁₃	1	1	yes	characteristic
165	#97	C₂²×C₁₃	1	2	yes	—
166	C₄×C₈	C₂²×C₁₃	1	2	yes	—
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	4	2	no	—
169	C₂×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂×C₁₃	#577	1	1	yes	characteristic
171	C₂×C₁₃	#578	1	1	yes	characteristic
172	#41	—	2	2	no	—
173	#41	C₂×C₄×C₁₃	1	4	yes	—
174	C₂×Q₈	—	2	1	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	C₂×C₄×C₁₃	1	4	yes	complemented
177	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
178	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
179	C₂×D₄	—	2	1	no	—
180	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
186	Q₁₆	C₂×C₄×C₁₃	1	8	yes	complemented
187	SD₁₆	C₂×C₄×C₁₃	1	8	yes	complemented
188	M₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂³×C₁₃	1	2	yes	—
193	C₄⋊C₄	—	2	1	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
198	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
199	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	2	no	—
204	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
205	C₄²	—	2	2	no	—
206	C₄²	—	2	2	no	—
207	C₄²	—	2	2	no	—
208	C₄²	—	2	1	no	—
209	C₄²	C₂³×C₁₃	1	2	yes	—
210	C₁₃	#2947	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
213	Q₈	—	2	4	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	1	no	—
216	Q₈	C₂²×C₄×C₁₃	1	4	yes	—
217	Q₈	C₂²×C₄×C₁₃	1	2	yes	—
218	D₄	—	2	4	no	—
219	D₄	—	2	1	no	—
220	D₄	C₂²×C₄×C₁₃	1	2	yes	—
221	C₂×C₄	—	2	4	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	1	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	8	2	no	—
229	C₂×C₄	—	8	1	no	—
230	C₂×C₄	C₂²×C₄×C₁₃	1	2	yes	—
231	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
232	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
233	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₂×C₄	C₁₃×#41	1	2	yes	—
235	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
236	C₈	—	2	1	no	—
237	C₈	—	2	1	no	—
238	C₈	C₂²×C₄×C₁₃	1	4	yes	—
239	C₂²	—	2	1	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	4	1	no	—
243	C₂²	—	4	1	no	—
244	C₂²	C₄×D₄×C₁₃	1	2	yes	—
245	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
246	C₄	—	2	2	no	—
247	C₄	—	2	4	no	—
248	C₄	—	2	2	no	—
249	C₄	—	2	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	2	no	—
252	C₄	—	4	1	no	—
253	C₄	C₄×D₄×C₁₃	1	2	yes	—
254	C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
255	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
256	C₂	—	2	2	no	—
257	C₂	—	4	2	no	—
258	C₂	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
259	C₂	C₁₃×#577	1	1	yes	characteristic
260	C₂	C₁₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator	C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#41	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#519	C₂	semidirect
4. C₁₃×#441	C₂	semidirect
5. C₁₃×#440	C₂	semidirect
6. C₄×C₁₃×Q₁₆	C₂	semidirect
7. C₄×C₁₃×SD₁₆	C₂	semidirect
8. C₁₃×#428	C₂	semidirect
10. C₂×C₁₃×#103	C₂	semidirect
11. C₁₃×#405	C₂	semidirect
14. C₁₃×#137	C₄	semidirect
46. C₂×Q₈×C₁₃	C₂×C₄	semidirect
56. C₁₃×Q₁₆	C₂×C₄	semidirect
57. C₁₃×SD₁₆	C₂×C₄	semidirect
58. C₁₃×M₁₆	C₂×C₄	semidirect
80. #2947	C₁₃	direct (N × H)
111. #519	C₂×C₁₃	semidirect
112. #441	C₂×C₁₃	semidirect
113. #440	C₂×C₁₃	semidirect
114. C₄×Q₁₆	C₂×C₁₃	semidirect
115. C₄×SD₁₆	C₂×C₁₃	semidirect
116. #428	C₂×C₁₃	semidirect
118. C₂×#103	C₂×C₁₃	semidirect
119. #405	C₂×C₁₃	semidirect
139. #137	C₄×C₁₃	semidirect
176. C₂×Q₈	C₂×C₄×C₁₃	semidirect
186. Q₁₆	C₂×C₄×C₁₃	semidirect
187. SD₁₆	C₂×C₄×C₁₃	semidirect
188. M₁₆	C₂×C₄×C₁₃	semidirect
210. C₁₃	#2947	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×#137	2
C₁₃×#520	2
C₁₃×#519	2
C₁₃×#441	2
C₁₃×#440	2
C₄×C₁₃×Q₁₆	2
C₄×C₁₃×SD₁₆	2
C₁₃×#428	2
C₁₃×#416	2
C₂×C₁₃×#103	2
C₁₃×#405	2
#2947	13

Subgroups — C13×#2947 #13112005

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2947 #13112005