Subgroups — C₂×C₁₃×#2226 #15719960

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2226	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#367	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#357	C₂	1	2	yes	maximal
7	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₄×D₄×C₁₃	—	2	1	no	—
10	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
11	C₁₃×#500	—	2	2	no	—
12	C₁₃×#500	C₂²	1	4	yes	complemented
13	C₄×C₁₃×D₈	—	2	2	no	—
14	C₄×C₁₃×D₈	C₂²	1	4	yes	—
15	C₂×C₁₃×#107	—	2	1	no	—
16	C₂×C₁₃×#107	C₂²	1	2	yes	—
17	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#102	—	2	1	no	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#367	C₂²	1	8	yes	complemented
21	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
22	C₁₃×#357	C₂²	1	8	yes	—
23	C₂²×Q₈×C₁₃	—	4	1	no	—
24	C₂²×D₄×C₁₃	—	2	1	no	—
25	C₂³×C₄×C₁₃	—	4	1	no	—
26	C₂×C₁₃×D₈	—	2	1	no	—
27	C₂×C₁₃×D₈	—	2	2	no	—
28	C₂×C₁₃×D₈	Q₈	1	4	yes	complemented
29	C₂×C₁₃×D₈	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
32	C₁₃×C₄⋊Q₈	—	2	4	no	—
33	C₁₃×C₄⋊Q₈	—	4	2	no	—
34	C₄×D₄×C₁₃	—	2	4	no	—
35	C₄×D₄×C₁₃	—	4	2	no	—
36	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
38	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
39	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
40	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
41	C₂×C₁₃×C₁₆	—	2	4	no	—
42	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
43	C₁₃×#107	—	2	4	no	—
44	C₁₃×#107	—	2	1	no	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	C₂³	1	4	yes	—
47	C₁₃×#107	C₂³	1	2	yes	—
48	C₁₃×#102	—	2	4	no	—
49	C₄×C₈×C₁₃	D₄	1	2	yes	—
50	C₄×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂×#2226	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	4	1	no	—
53	C₂×Q₈×C₁₃	—	4	4	no	—
54	C₂×Q₈×C₁₃	—	8	4	no	—
55	C₂×D₄×C₁₃	—	2	4	no	—
56	C₂×D₄×C₁₃	—	2	2	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	—	4	1	no	—
60	C₂²×C₄×C₁₃	—	4	2	no	—
61	C₂²×C₄×C₁₃	—	4	4	no	—
62	C₂²×C₄×C₁₃	—	8	4	no	—
63	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
66	C₁₃×D₈	—	2	4	no	—
67	C₁₃×D₈	—	2	2	no	—
68	C₁₃×D₈	C₂×Q₈	1	4	yes	complemented
69	C₂×C₈×C₁₃	—	2	2	no	—
70	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
71	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₁₃	#41	1	2	yes	—
74	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
75	C₁₃×C₄⋊C₄	—	2	2	no	—
76	C₁₃×C₄⋊C₄	—	2	4	no	—
77	C₁₃×C₄⋊C₄	—	4	1	no	—
78	C₁₃×C₄⋊C₄	—	4	2	no	—
79	C₁₃×C₄⋊C₄	—	4	4	no	—
80	C₁₃×C₂²⋊C₄	—	4	4	no	—
81	C₄²×C₁₃	D₈	1	2	yes	—
82	C₄²×C₁₃	C₂×D₄	1	2	yes	—
83	C₁₃×C₁₆	—	2	4	no	—
84	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
85	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
86	#2226	C₂×C₁₃	1	8	yes	direct product factor
87	C₂×#367	C₂×C₁₃	1	2	yes	complemented
88	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
89	C₂×#357	C₂×C₁₃	1	2	yes	—
90	C₂³×C₁₃	—	4	2	no	—
91	C₂³×C₁₃	—	4	4	no	—
92	C₂³×C₁₃	—	8	2	no	—
93	C₂³×C₁₃	—	8	2	no	—
94	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
95	Q₈×C₁₃	—	8	8	no	—
96	D₄×C₁₃	—	2	4	no	—
97	D₄×C₁₃	—	4	2	no	—
98	D₄×C₁₃	—	4	4	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	4	2	no	—
101	C₂×C₄×C₁₃	—	4	4	no	—
102	C₂×C₄×C₁₃	—	8	2	no	—
103	C₂×C₄×C₁₃	—	8	4	no	—
104	C₂×C₄×C₁₃	—	8	8	no	—
105	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
106	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
110	C₂×C₄×C₁₃	#137	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	C₈×C₁₃	—	2	2	no	—
113	C₈×C₁₃	C₂²⋊Q₈	1	2	yes	—
114	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
115	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
116	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×C₄⋊Q₈	—	2	1	no	—
118	C₂×C₄×D₄	—	2	1	no	—
119	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
120	#500	—	2	2	no	—
121	#500	C₂²×C₁₃	1	4	yes	complemented
122	C₄×D₈	—	2	2	no	—
123	C₄×D₈	C₂²×C₁₃	1	4	yes	—
124	C₂×#107	—	2	1	no	—
125	C₂×#107	C₂²×C₁₃	1	2	yes	—
126	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×#102	—	2	1	no	—
128	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
129	#367	C₂²×C₁₃	1	8	yes	complemented
130	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
131	#357	C₂²×C₁₃	1	8	yes	—
132	C₂²×C₁₃	—	4	4	no	—
133	C₂²×C₁₃	—	8	2	no	—
134	C₂²×C₁₃	—	8	2	no	—
135	C₂²×C₁₃	—	8	8	no	—
136	C₂²×C₁₃	#474	1	2	yes	—
137	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
138	C₂²×C₁₃	#509	1	2	yes	—
139	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
140	C₄×C₁₃	—	2	2	no	—
141	C₄×C₁₃	—	8	2	no	—
142	C₄×C₁₃	—	8	4	no	—
143	C₄×C₁₃	#474	1	2	yes	—
144	C₄×C₁₃	C₂×SD₃₂	1	4	yes	—
145	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
146	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
147	C₂²×Q₈	—	4	1	no	—
148	C₂²×D₄	—	2	1	no	—
149	C₂³×C₄	—	4	1	no	—
150	C₂×D₈	—	2	1	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	Q₈×C₁₃	1	4	yes	complemented
153	C₂×D₈	C₂³×C₁₃	1	2	yes	—
154	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
155	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
156	C₄⋊Q₈	—	2	4	no	—
157	C₄⋊Q₈	—	4	2	no	—
158	C₄×D₄	—	2	4	no	—
159	C₄×D₄	—	4	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	2	no	—
162	C₂×C₄⋊C₄	—	4	1	no	—
163	C₂×C₂²⋊C₄	—	4	1	no	—
164	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
165	C₂×C₁₆	—	2	4	no	—
166	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
167	#107	—	2	4	no	—
168	#107	—	2	1	no	—
169	#107	—	2	2	no	—
170	#107	C₂³×C₁₃	1	4	yes	—
171	#107	C₂³×C₁₃	1	2	yes	—
172	#102	—	2	4	no	—
173	C₄×C₈	D₄×C₁₃	1	2	yes	—
174	C₄×C₈	C₂³×C₁₃	1	2	yes	—
175	C₂×C₁₃	—	8	4	no	—
176	C₂×C₁₃	#2226	1	4	yes	direct product factor
177	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
178	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
179	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
180	C₂⁴	—	4	1	no	—
181	C₂×Q₈	—	4	4	no	—
182	C₂×Q₈	—	8	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	1	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂²×C₄	—	4	1	no	—
188	C₂²×C₄	—	4	2	no	—
189	C₂²×C₄	—	4	4	no	—
190	C₂²×C₄	—	8	4	no	—
191	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
192	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
194	D₈	—	2	4	no	—
195	D₈	—	2	2	no	—
196	D₈	C₂×Q₈×C₁₃	1	4	yes	complemented
197	C₂×C₈	—	2	2	no	—
198	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
201	C₂×C₈	C₁₃×#41	1	2	yes	—
202	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
203	C₄⋊C₄	—	2	2	no	—
204	C₄⋊C₄	—	2	4	no	—
205	C₄⋊C₄	—	4	1	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	4	no	—
208	C₂²⋊C₄	—	4	4	no	—
209	C₄²	C₁₃×D₈	1	2	yes	—
210	C₄²	C₂×D₄×C₁₃	1	2	yes	—
211	C₁₆	—	2	4	no	—
212	C₁₃	C₂×#2226	1	1	yes	characteristic, direct product factor, Sylow
213	C₂³	—	4	4	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	8	2	no	—
216	C₂³	—	8	2	no	—
217	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
218	Q₈	—	8	8	no	—
219	D₄	—	2	4	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	2	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	4	2	no	—
224	C₂×C₄	—	4	4	no	—
225	C₂×C₄	—	8	2	no	—
226	C₂×C₄	—	8	4	no	—
227	C₂×C₄	—	8	8	no	—
228	C₂×C₄	C₁₃×SD₃₂	1	4	yes	—
229	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
230	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
231	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
232	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
233	C₂×C₄	C₁₃×#137	1	2	yes	—
234	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
235	C₈	—	2	2	no	—
236	C₈	C₁₃×C₂²⋊Q₈	1	2	yes	—
237	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
238	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
239	C₂²	—	4	4	no	—
240	C₂²	—	8	2	no	—
241	C₂²	—	8	2	no	—
242	C₂²	—	8	8	no	—
243	C₂²	C₁₃×#474	1	2	yes	—
244	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
245	C₂²	C₁₃×#509	1	2	yes	—
246	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	8	2	no	—
249	C₄	—	8	4	no	—
250	C₄	C₁₃×#474	1	2	yes	—
251	C₄	C₂×C₁₃×SD₃₂	1	4	yes	—
252	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
253	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
254	C₂	—	8	4	no	—
255	C₂	C₁₃×#2226	1	4	yes	direct product factor
256	C₂	C₂×C₁₃×#474	1	1	yes	characteristic
257	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
258	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#500	C₂	semidirect
3. C₁₃×#2226	C₂	direct (N × H)
4. C₂×C₁₃×#367	C₂	semidirect
5. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
12. C₁₃×#500	C₂²	semidirect
20. C₁₃×#367	C₂²	semidirect
21. C₁₃×C₄⋊C₁₆	C₂²	semidirect
28. C₂×C₁₃×D₈	Q₈	semidirect
51. C₂×#2226	C₁₃	direct (N × H)
68. C₁₃×D₈	C₂×Q₈	semidirect
84. C₂×#500	C₂×C₁₃	semidirect
86. #2226	C₂×C₁₃	direct (N × H)
87. C₂×#367	C₂×C₁₃	semidirect
88. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
121. #500	C₂²×C₁₃	semidirect
129. #367	C₂²×C₁₃	semidirect
130. C₄⋊C₁₆	C₂²×C₁₃	semidirect
152. C₂×D₈	Q₈×C₁₃	semidirect
176. C₂×C₁₃	#2226	direct (N × H)
196. D₈	C₂×Q₈×C₁₃	semidirect
212. C₁₃	C₂×#2226	direct (N × H)
255. C₂	C₁₃×#2226	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×#500	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2226	2
C₂×C₁₃×#367	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#357	2
C₂×#2226	13

Subgroups — C2×C13×#2226 #15719960

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2226 #15719960