Subgroups — C₂×C₁₃×#2179 #15719801

Order 3328.

654 conjugacy classes of subgroups.

The subgroup lattice has 266 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	C₂×C₁₃×#443	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2179	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#368	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#359	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₃×#135	C₂²	1	2	yes	—
8	C₂×C₁₃×#131	—	2	2	no	—
9	C₁₃×#508	C₄	1	4	yes	complemented
10	C₁₃×#508	C₂²	1	4	yes	—
11	C₂×C₁₃×Q₃₂	C₄	1	1	yes	characteristic, complemented
12	C₂×C₁₃×SD₃₂	C₄	1	2	yes	complemented
13	C₂×C₁₃×D₁₆	C₄	1	1	yes	characteristic, complemented
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#443	—	2	4	no	—
17	C₁₃×#443	C₂²	1	8	yes	complemented
18	C₂×C₁₃×#108	C₂²	1	2	yes	—
19	C₂×C₁₃×#104	—	2	2	no	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#368	C₂²	1	4	yes	complemented
22	C₁₃×#359	C₂²	1	8	yes	complemented
23	C₁₃×#346	C₂²	1	4	yes	complemented
24	C₂×C₁₃×#41	—	2	2	no	—
25	C₁₃×#135	—	2	2	no	—
26	C₁₃×#135	—	2	2	no	—
27	C₁₃×#135	C₂×C₄	1	4	yes	—
28	C₁₃×#135	C₂³	1	4	yes	—
29	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×SD₁₆	—	2	2	no	—
31	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
32	C₁₃×#131	—	2	8	no	—
33	C₁₃×#131	—	4	4	no	—
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₁₃×M₁₆	—	4	2	no	—
36	C₂²×C₈×C₁₃	—	4	2	no	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
40	C₁₃×Q₃₂	C₂×C₄	1	4	yes	complemented
41	C₁₃×SD₃₂	C₂×C₄	1	8	yes	complemented
42	C₁₃×D₁₆	C₂×C₄	1	4	yes	complemented
43	C₂×C₁₃×C₁₆	—	2	1	no	—
44	C₂×C₁₃×C₁₆	—	2	1	no	—
45	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
46	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
48	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
49	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
50	C₁₃×#108	—	2	2	no	—
51	C₁₃×#108	C₂³	1	4	yes	—
52	C₁₃×#104	—	2	8	no	—
53	C₄×C₈×C₁₃	D₄	1	2	yes	—
54	C₄×C₈×C₁₃	C₂³	1	2	yes	—
55	C₂×#2179	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₁₃×#41	—	2	4	no	—
57	C₁₃×#41	—	2	4	no	—
58	C₁₃×#41	—	4	2	no	—
59	C₁₃×#41	—	4	2	no	—
60	C₂×Q₈×C₁₃	—	2	2	no	—
61	C₂×D₄×C₁₃	—	2	2	no	—
62	C₂×D₄×C₁₃	—	4	2	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₁₃×Q₁₆	—	2	2	no	—
67	C₁₃×Q₁₆	C₂²×C₄	1	4	yes	—
68	C₁₃×SD₁₆	—	2	4	no	—
69	C₁₃×SD₁₆	—	2	4	no	—
70	C₁₃×D₈	—	2	2	no	—
71	C₁₃×D₈	C₂²×C₄	1	4	yes	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	4	2	no	—
74	C₁₃×M₁₆	—	4	8	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	8	no	—
77	C₂×C₈×C₁₃	D₈	1	2	yes	—
78	C₂×C₈×C₁₃	D₈	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
83	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₄²×C₁₃	—	2	1	no	—
87	C₄²×C₁₃	C₂×D₄	1	2	yes	—
88	C₁₃×C₁₆	—	2	2	no	—
89	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
90	C₁₃×C₁₆	C₂²×C₄	1	2	yes	—
91	C₂×#508	C₂×C₁₃	1	1	yes	characteristic
92	C₂×#443	C₂×C₁₃	1	2	yes	complemented
93	#2179	C₂×C₁₃	1	8	yes	direct product factor
94	C₂×#368	C₂×C₁₃	1	1	yes	characteristic, complemented
95	C₂×#359	C₂×C₁₃	1	2	yes	complemented
96	C₂×#346	C₂×C₁₃	1	1	yes	characteristic, complemented
97	C₂³×C₁₃	—	4	2	no	—
98	C₂³×C₁₃	#135	1	1	yes	characteristic
99	Q₈×C₁₃	—	2	4	no	—
100	Q₈×C₁₃	—	4	2	no	—
101	D₄×C₁₃	—	2	4	no	—
102	D₄×C₁₃	—	4	4	no	—
103	D₄×C₁₃	—	4	4	no	—
104	D₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	4	4	no	—
110	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#135	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₁₃	—	4	4	no	—
117	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
118	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
119	C₈×C₁₃	C₂×D₈	1	4	yes	—
120	C₈×C₁₃	C₂³×C₄	1	1	yes	characteristic
121	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
122	C₂×#135	C₂²×C₁₃	1	2	yes	—
123	C₂×#131	—	2	2	no	—
124	#508	C₄×C₁₃	1	4	yes	complemented
125	#508	C₂²×C₁₃	1	4	yes	—
126	C₂×Q₃₂	C₄×C₁₃	1	1	yes	characteristic, complemented
127	C₂×SD₃₂	C₄×C₁₃	1	2	yes	complemented
128	C₂×D₁₆	C₄×C₁₃	1	1	yes	characteristic, complemented
129	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
131	#443	—	2	4	no	—
132	#443	C₂²×C₁₃	1	8	yes	complemented
133	C₂×#108	C₂²×C₁₃	1	2	yes	—
134	C₂×#104	—	2	2	no	—
135	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
136	#368	C₂²×C₁₃	1	4	yes	complemented
137	#359	C₂²×C₁₃	1	8	yes	complemented
138	#346	C₂²×C₁₃	1	4	yes	complemented
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	4	4	no	—
141	C₂²×C₁₃	—	8	4	no	—
142	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
143	C₂²×C₁₃	C₄×D₈	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
145	C₄×C₁₃	—	4	2	no	—
146	C₄×C₁₃	—	4	4	no	—
147	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
148	C₄×C₁₃	C₄×D₈	1	1	yes	characteristic
149	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
151	C₂×#41	—	2	2	no	—
152	#135	—	2	2	no	—
153	#135	—	2	2	no	—
154	#135	C₂×C₄×C₁₃	1	4	yes	—
155	#135	C₂³×C₁₃	1	4	yes	—
156	C₂×Q₁₆	C₂×C₄×C₁₃	1	2	yes	—
157	C₂×SD₁₆	—	2	2	no	—
158	C₂×D₈	C₂×C₄×C₁₃	1	2	yes	—
159	#131	—	2	8	no	—
160	#131	—	4	4	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	4	2	no	—
163	C₂²×C₈	—	4	2	no	—
164	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
166	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
167	Q₃₂	C₂×C₄×C₁₃	1	4	yes	complemented
168	SD₃₂	C₂×C₄×C₁₃	1	8	yes	complemented
169	D₁₆	C₂×C₄×C₁₃	1	4	yes	complemented
170	C₂×C₁₆	—	2	1	no	—
171	C₂×C₁₆	—	2	1	no	—
172	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
173	C₂×C₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
174	C₂×C₁₆	C₂×C₄×C₁₃	1	2	yes	—
175	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
176	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
177	#108	—	2	2	no	—
178	#108	C₂³×C₁₃	1	4	yes	—
179	#104	—	2	8	no	—
180	C₄×C₈	D₄×C₁₃	1	2	yes	—
181	C₄×C₈	C₂³×C₁₃	1	2	yes	—
182	C₂×C₁₃	—	2	1	no	—
183	C₂×C₁₃	—	2	1	no	—
184	C₂×C₁₃	—	8	4	no	—
185	C₂×C₁₃	#2179	1	2	yes	direct product factor
186	C₂×C₁₃	C₂×C₄×D₈	1	1	yes	characteristic
187	#41	—	2	4	no	—
188	#41	—	2	4	no	—
189	#41	—	4	2	no	—
190	#41	—	4	2	no	—
191	C₂×Q₈	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	2	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	—	4	2	no	—
196	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	Q₁₆	—	2	2	no	—
198	Q₁₆	C₂²×C₄×C₁₃	1	4	yes	—
199	SD₁₆	—	2	4	no	—
200	SD₁₆	—	2	4	no	—
201	D₈	—	2	2	no	—
202	D₈	C₂²×C₄×C₁₃	1	4	yes	—
203	M₁₆	—	2	4	no	—
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	8	no	—
206	C₂×C₈	—	4	2	no	—
207	C₂×C₈	—	4	8	no	—
208	C₂×C₈	C₁₃×D₈	1	2	yes	—
209	C₂×C₈	C₁₃×D₈	1	2	yes	—
210	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
211	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
212	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
213	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
214	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
215	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
216	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
217	C₄²	—	2	1	no	—
218	C₄²	C₂×D₄×C₁₃	1	2	yes	—
219	C₁₆	—	2	2	no	—
220	C₁₆	C₂²×C₄×C₁₃	1	2	yes	—
221	C₁₆	C₂²×C₄×C₁₃	1	2	yes	—
222	C₁₃	C₂×#2179	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	4	2	no	—
224	C₂³	C₁₃×#135	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	4	2	no	—
227	D₄	—	2	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	D₄	—	4	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₄	C₄×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
239	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
240	C₂×C₄	C₁₃×#135	1	1	yes	characteristic
241	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
242	C₈	—	4	4	no	—
243	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
244	C₈	C₄×D₄×C₁₃	1	1	yes	characteristic
245	C₈	C₂×C₁₃×D₈	1	4	yes	—
246	C₈	C₂³×C₄×C₁₃	1	1	yes	characteristic
247	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
248	C₂²	—	2	2	no	—
249	C₂²	—	4	4	no	—
250	C₂²	—	8	4	no	—
251	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
252	C₂²	C₄×C₁₃×D₈	1	1	yes	characteristic
253	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
254	C₄	—	4	2	no	—
255	C₄	—	4	4	no	—
256	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
257	C₄	C₄×C₁₃×D₈	1	1	yes	characteristic
258	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
259	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
260	C₂	—	2	1	no	—
261	C₂	—	2	1	no	—
262	C₂	—	8	4	no	—
263	C₂	C₁₃×#2179	1	2	yes	direct product factor
264	C₂	C₂×C₄×C₁₃×D₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#443	C₂	semidirect
3. C₁₃×#2179	C₂	direct (N × H)
4. C₂×C₁₃×#368	C₂	semidirect
5. C₂×C₁₃×#359	C₂	semidirect
6. C₂×C₁₃×#346	C₂	semidirect
9. C₁₃×#508	C₄	semidirect
11. C₂×C₁₃×Q₃₂	C₄	semidirect
12. C₂×C₁₃×SD₃₂	C₄	semidirect
13. C₂×C₁₃×D₁₆	C₄	semidirect
17. C₁₃×#443	C₂²	semidirect
21. C₁₃×#368	C₂²	semidirect
22. C₁₃×#359	C₂²	semidirect
23. C₁₃×#346	C₂²	semidirect
40. C₁₃×Q₃₂	C₂×C₄	semidirect
41. C₁₃×SD₃₂	C₂×C₄	semidirect
42. C₁₃×D₁₆	C₂×C₄	semidirect
55. C₂×#2179	C₁₃	direct (N × H)
92. C₂×#443	C₂×C₁₃	semidirect
93. #2179	C₂×C₁₃	direct (N × H)
94. C₂×#368	C₂×C₁₃	semidirect
95. C₂×#359	C₂×C₁₃	semidirect
96. C₂×#346	C₂×C₁₃	semidirect
124. #508	C₄×C₁₃	semidirect
126. C₂×Q₃₂	C₄×C₁₃	semidirect
127. C₂×SD₃₂	C₄×C₁₃	semidirect
128. C₂×D₁₆	C₄×C₁₃	semidirect
132. #443	C₂²×C₁₃	semidirect
136. #368	C₂²×C₁₃	semidirect
137. #359	C₂²×C₁₃	semidirect
138. #346	C₂²×C₁₃	semidirect
167. Q₃₂	C₂×C₄×C₁₃	semidirect
168. SD₃₂	C₂×C₄×C₁₃	semidirect
169. D₁₆	C₂×C₄×C₁₃	semidirect
185. C₂×C₁₃	#2179	direct (N × H)
222. C₁₃	C₂×#2179	direct (N × H)
263. C₂	C₁₃×#2179	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
C₂×C₁₃×#443	2
C₁₃×#2179	2
C₂×C₁₃×#368	2
C₂×C₁₃×#359	2
C₂×C₁₃×#346	2
C₂×#2179	13

Subgroups — C2×C13×#2179 #15719801

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2179 #15719801