Subgroups — C₂×C₁₃×#2208 #15719913

Order 3328.

766 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	2	yes	maximal
2	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2208	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#358	C₂	1	2	yes	maximal, complemented
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₄×D₄×C₁₃	—	2	1	no	—
12	C₂×C₁₃×SD₃₂	—	2	8	no	—
13	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	—
14	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
15	C₁₃×#494	—	2	2	no	—
16	C₁₃×#494	C₂²	1	4	yes	complemented
17	C₄×C₁₃×D₈	—	2	2	no	—
18	C₄×C₁₃×D₈	C₂²	1	4	yes	—
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#102	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	complemented
23	C₁₃×#358	C₂²	1	8	yes	complemented
24	C₂²×Q₈×C₁₃	—	2	2	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂³×C₄×C₁₃	—	4	1	no	—
27	C₂×C₁₃×Q₁₆	—	2	4	no	—
28	C₂×C₁₃×Q₁₆	—	2	2	no	—
29	C₂×C₁₃×Q₁₆	—	2	4	no	—
30	C₂×C₁₃×Q₁₆	—	2	4	no	—
31	C₂×C₁₃×Q₁₆	—	4	4	no	—
32	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
33	C₂×C₁₃×D₈	—	2	1	no	—
34	C₂×C₁₃×D₈	—	2	2	no	—
35	C₂×C₁₃×D₈	D₄	1	4	yes	—
36	C₂×C₁₃×D₈	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	—	2	4	no	—
40	C₁₃×C₄⋊Q₈	—	4	2	no	—
41	C₄×D₄×C₁₃	—	2	4	no	—
42	C₄×D₄×C₁₃	—	4	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₁₃×SD₃₂	—	2	16	no	—
49	C₂×C₁₃×C₁₆	—	2	4	no	—
50	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
51	C₁₃×#107	—	2	1	no	—
52	C₁₃×#107	C₂³	1	2	yes	—
53	C₁₃×#102	—	2	4	no	—
54	C₄×C₈×C₁₃	D₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×#2208	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	4	1	no	—
58	C₂×Q₈×C₁₃	—	2	4	no	—
59	C₂×Q₈×C₁₃	—	4	2	no	—
60	C₂×Q₈×C₁₃	—	4	4	no	—
61	C₂×Q₈×C₁₃	—	4	4	no	—
62	C₂×D₄×C₁₃	—	2	4	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	4	1	no	—
65	C₂×D₄×C₁₃	—	4	2	no	—
66	C₂²×C₄×C₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	—	4	4	no	—
69	C₂²×C₄×C₁₃	—	8	4	no	—
70	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₁₃×Q₁₆	—	2	4	no	—
74	C₁₃×Q₁₆	—	2	4	no	—
75	C₁₃×Q₁₆	—	4	8	no	—
76	C₁₃×Q₁₆	—	4	4	no	—
77	C₁₃×D₈	—	2	4	no	—
78	C₁₃×D₈	—	2	2	no	—
79	C₁₃×D₈	C₂×D₄	1	4	yes	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
84	C₂×C₈×C₁₃	#41	1	2	yes	—
85	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	4	1	no	—
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₁₃×C₄⋊C₄	—	4	4	no	—
90	C₁₃×C₂²⋊C₄	—	4	4	no	—
91	C₄²×C₁₃	D₈	1	2	yes	—
92	C₄²×C₁₃	C₂×D₄	1	2	yes	—
93	C₁₃×C₁₆	—	2	4	no	—
94	C₂²×SD₃₂	C₂×C₁₃	1	2	yes	—
95	C₂×#494	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
97	#2208	C₂×C₁₃	1	8	yes	direct product factor
98	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂×#358	C₂×C₁₃	1	2	yes	complemented
100	C₂³×C₁₃	—	4	2	no	—
101	C₂³×C₁₃	—	4	4	no	—
102	C₂³×C₁₃	—	8	2	no	—
103	C₂³×C₁₃	—	8	2	no	—
104	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
105	Q₈×C₁₃	—	4	4	no	—
106	Q₈×C₁₃	—	4	4	no	—
107	Q₈×C₁₃	—	8	4	no	—
108	D₄×C₁₃	—	2	4	no	—
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	8	4	no	—
115	C₂×C₄×C₁₃	—	8	2	no	—
116	C₂×C₄×C₁₃	—	8	8	no	—
117	C₂×C₄×C₁₃	SD₃₂	1	4	yes	complemented
118	C₂×C₄×C₁₃	#121	1	2	yes	—
119	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	#121	1	2	yes	—
126	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₂²×Q₁₆	—	2	1	no	—
129	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
130	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×C₄⋊Q₈	—	2	1	no	—
132	C₂×C₄×D₄	—	2	1	no	—
133	C₂×SD₃₂	—	2	8	no	—
134	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	—
135	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
136	#494	—	2	2	no	—
137	#494	C₂²×C₁₃	1	4	yes	complemented
138	C₄×D₈	—	2	2	no	—
139	C₄×D₈	C₂²×C₁₃	1	4	yes	—
140	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×#102	—	2	1	no	—
142	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
144	#358	C₂²×C₁₃	1	8	yes	complemented
145	C₂²×C₁₃	—	4	4	no	—
146	C₂²×C₁₃	—	8	2	no	—
147	C₂²×C₁₃	—	8	2	no	—
148	C₂²×C₁₃	—	8	8	no	—
149	C₂²×C₁₃	#459	1	2	yes	—
150	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
151	C₂²×C₁₃	#510	1	2	yes	—
152	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	8	2	no	—
155	C₄×C₁₃	—	8	4	no	—
156	C₄×C₁₃	#459	1	2	yes	—
157	C₄×C₁₃	C₂×SD₃₂	1	4	yes	complemented
158	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
159	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
160	C₂²×Q₈	—	2	2	no	—
161	C₂²×D₄	—	2	1	no	—
162	C₂³×C₄	—	4	1	no	—
163	C₂×Q₁₆	—	2	4	no	—
164	C₂×Q₁₆	—	2	4	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	4	no	—
167	C₂×Q₁₆	—	4	4	no	—
168	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
169	C₂×D₈	—	2	1	no	—
170	C₂×D₈	—	2	2	no	—
171	C₂×D₈	D₄×C₁₃	1	4	yes	—
172	C₂×D₈	C₂³×C₁₃	1	2	yes	—
173	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
174	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
175	C₄⋊Q₈	—	2	4	no	—
176	C₄⋊Q₈	—	4	2	no	—
177	C₄×D₄	—	2	4	no	—
178	C₄×D₄	—	4	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	4	1	no	—
181	C₂×C₄⋊C₄	—	4	1	no	—
182	C₂×C₂²⋊C₄	—	4	1	no	—
183	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
184	SD₃₂	—	2	16	no	—
185	C₂×C₁₆	—	2	4	no	—
186	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
187	#107	—	2	1	no	—
188	#107	C₂³×C₁₃	1	2	yes	—
189	#102	—	2	4	no	—
190	C₄×C₈	D₄×C₁₃	1	2	yes	—
191	C₄×C₈	C₂³×C₁₃	1	2	yes	—
192	C₂×C₁₃	—	8	4	no	—
193	C₂×C₁₃	#2208	1	4	yes	direct product factor
194	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
195	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
196	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
197	C₂⁴	—	4	1	no	—
198	C₂×Q₈	—	2	4	no	—
199	C₂×Q₈	—	4	2	no	—
200	C₂×Q₈	—	4	4	no	—
201	C₂×Q₈	—	4	4	no	—
202	C₂×D₄	—	2	4	no	—
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	1	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂²×C₄	—	4	1	no	—
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	4	no	—
209	C₂²×C₄	—	8	4	no	—
210	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
211	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
212	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
213	Q₁₆	—	2	4	no	—
214	Q₁₆	—	2	4	no	—
215	Q₁₆	—	4	8	no	—
216	Q₁₆	—	4	4	no	—
217	D₈	—	2	4	no	—
218	D₈	—	2	2	no	—
219	D₈	C₂×D₄×C₁₃	1	4	yes	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
222	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
224	C₂×C₈	C₁₃×#41	1	2	yes	—
225	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	4	1	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₂²⋊C₄	—	4	4	no	—
231	C₄²	C₁₃×D₈	1	2	yes	—
232	C₄²	C₂×D₄×C₁₃	1	2	yes	—
233	C₁₆	—	2	4	no	—
234	C₁₃	C₂×#2208	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	4	4	no	—
236	C₂³	—	4	2	no	—
237	C₂³	—	8	2	no	—
238	C₂³	—	8	2	no	—
239	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
240	Q₈	—	4	4	no	—
241	Q₈	—	4	4	no	—
242	Q₈	—	8	4	no	—
243	D₄	—	2	4	no	—
244	D₄	—	4	4	no	—
245	D₄	—	4	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	8	4	no	—
250	C₂×C₄	—	8	2	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₁₃×SD₃₂	1	4	yes	complemented
253	C₂×C₄	C₁₃×#121	1	2	yes	—
254	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
255	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
256	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
257	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
258	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
259	C₈	—	2	2	no	—
260	C₈	C₁₃×#121	1	2	yes	—
261	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
262	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
263	C₂²	—	4	4	no	—
264	C₂²	—	8	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	8	no	—
267	C₂²	C₁₃×#459	1	2	yes	—
268	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
269	C₂²	C₁₃×#510	1	2	yes	—
270	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	8	2	no	—
274	C₄	C₁₃×#459	1	2	yes	—
275	C₄	C₂×C₁₃×SD₃₂	1	4	yes	complemented
276	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
277	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
278	C₂	—	8	4	no	—
279	C₂	C₁₃×#2208	1	4	yes	direct product factor
280	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
281	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
282	C₂	C₂×C₁₃×#510	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
2. C₂×C₁₃×#494	C₂	semidirect
4. C₁₃×#2208	C₂	direct (N × H)
5. C₂×C₁₃×C₄⋊C₁₆	C₂	semidirect
6. C₂×C₁₃×#358	C₂	semidirect
16. C₁₃×#494	C₂²	semidirect
22. C₁₃×C₄⋊C₁₆	C₂²	semidirect
23. C₁₃×#358	C₂²	semidirect
56. C₂×#2208	C₁₃	direct (N × H)
95. C₂×#494	C₂×C₁₃	semidirect
97. #2208	C₂×C₁₃	direct (N × H)
98. C₂×C₄⋊C₁₆	C₂×C₁₃	semidirect
99. C₂×#358	C₂×C₁₃	semidirect
117. C₂×C₄×C₁₃	SD₃₂	semidirect
137. #494	C₂²×C₁₃	semidirect
143. C₄⋊C₁₆	C₂²×C₁₃	semidirect
144. #358	C₂²×C₁₃	semidirect
157. C₄×C₁₃	C₂×SD₃₂	semidirect
193. C₂×C₁₃	#2208	direct (N × H)
234. C₁₃	C₂×#2208	direct (N × H)
252. C₂×C₄	C₁₃×SD₃₂	semidirect
275. C₄	C₂×C₁₃×SD₃₂	semidirect
279. C₂	C₁₃×#2208	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×SD₃₂	2
C₂×C₁₃×#494	2
C₂×C₄×C₁₃×D₈	2
C₁₃×#2208	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#358	2
C₂×#2208	13

Subgroups — C2×C13×#2208 #15719913

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2208 #15719913