Subgroups — C₂×C₁₃×#2209 #15719916

Order 3328.

702 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×Q₃₂	C₂	1	2	yes	maximal
2	C₂×C₁₃×#494	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2209	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×C₁₃×#358	C₂	1	2	yes	maximal
7	C₂²×C₁₃×Q₁₆	—	2	1	no	—
8	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
11	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
12	C₂×C₁₃×Q₃₂	—	2	8	no	—
13	C₂×C₁₃×Q₃₂	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	—
17	C₄×C₁₃×Q₁₆	—	2	2	no	—
18	C₄×C₁₃×Q₁₆	C₂²	1	4	yes	—
19	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×#103	—	2	1	no	—
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×C₄⋊C₁₆	C₂²	1	4	yes	—
23	C₁₃×#358	C₂²	1	8	yes	—
24	C₂²×Q₈×C₁₃	—	2	1	no	—
25	C₂²×Q₈×C₁₃	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	—	2	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	2	no	—
31	C₂×C₁₃×Q₁₆	—	2	4	no	—
32	C₂×C₁₃×Q₁₆	—	4	4	no	—
33	C₂×C₁₃×Q₁₆	D₄	1	4	yes	—
34	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
35	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₁₃×C₄⋊Q₈	—	2	4	no	—
39	C₁₃×C₄⋊Q₈	—	4	2	no	—
40	C₄×Q₈×C₁₃	—	2	4	no	—
41	C₄×Q₈×C₁₃	—	4	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
44	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
45	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
46	C₂×C₄²×C₁₃	—	4	1	no	—
47	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
48	C₁₃×Q₃₂	—	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×#103	—	2	4	no	—
54	C₄×C₈×C₁₃	D₄	1	2	yes	—
55	C₄×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×#2209	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂×Q₈×C₁₃	—	2	4	no	—
58	C₂×Q₈×C₁₃	—	2	2	no	—
59	C₂×Q₈×C₁₃	—	2	4	no	—
60	C₂×Q₈×C₁₃	—	4	1	no	—
61	C₂×Q₈×C₁₃	—	4	2	no	—
62	C₂×Q₈×C₁₃	—	4	2	no	—
63	C₂×Q₈×C₁₃	—	4	4	no	—
64	C₂×Q₈×C₁₃	—	4	4	no	—
65	C₂²×C₄×C₁₃	—	4	1	no	—
66	C₂²×C₄×C₁₃	—	4	1	no	—
67	C₂²×C₄×C₁₃	—	4	2	no	—
68	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₁₃×Q₁₆	—	2	4	no	—
72	C₁₃×Q₁₆	—	2	4	no	—
73	C₁₃×Q₁₆	—	2	2	no	—
74	C₁₃×Q₁₆	—	2	4	no	—
75	C₁₃×Q₁₆	—	4	8	no	—
76	C₁₃×Q₁₆	—	4	4	no	—
77	C₁₃×Q₁₆	C₂×D₄	1	4	yes	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₈×C₁₃	#41	1	2	yes	—
83	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×C₄⋊C₄	—	2	2	no	—
85	C₁₃×C₄⋊C₄	—	4	1	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₄⋊C₄	—	4	4	no	—
89	C₄²×C₁₃	—	4	4	no	—
90	C₄²×C₁₃	D₈	1	2	yes	—
91	C₄²×C₁₃	C₂×D₄	1	2	yes	—
92	C₁₃×C₁₆	—	2	4	no	—
93	C₂²×Q₃₂	C₂×C₁₃	1	2	yes	—
94	C₂×#494	C₂×C₁₃	1	1	yes	characteristic
95	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
96	#2209	C₂×C₁₃	1	8	yes	direct product factor
97	C₂×C₄⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic
98	C₂×#358	C₂×C₁₃	1	2	yes	—
99	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	4	no	—
101	Q₈×C₁₃	—	4	2	no	—
102	Q₈×C₁₃	—	4	4	no	—
103	Q₈×C₁₃	—	4	4	no	—
104	Q₈×C₁₃	—	4	4	no	—
105	Q₈×C₁₃	—	8	4	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	2	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	8	4	no	—
112	C₂×C₄×C₁₃	—	8	2	no	—
113	C₂×C₄×C₁₃	Q₃₂	1	4	yes	complemented
114	C₂×C₄×C₁₃	#121	1	2	yes	—
115	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
118	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
119	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
120	C₈×C₁₃	—	2	2	no	—
121	C₈×C₁₃	#121	1	2	yes	—
122	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂²×Q₁₆	—	2	1	no	—
125	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
126	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
127	C₂×C₄⋊Q₈	—	2	1	no	—
128	C₂×C₄×Q₈	—	2	1	no	—
129	C₂×Q₃₂	—	2	8	no	—
130	C₂×Q₃₂	C₂²×C₁₃	1	8	yes	—
131	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
132	#494	—	2	2	no	—
133	#494	C₂²×C₁₃	1	4	yes	—
134	C₄×Q₁₆	—	2	2	no	—
135	C₄×Q₁₆	C₂²×C₁₃	1	4	yes	—
136	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×#103	—	2	1	no	—
138	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
139	C₄⋊C₁₆	C₂²×C₁₃	1	4	yes	—
140	#358	C₂²×C₁₃	1	8	yes	—
141	C₂²×C₁₃	#459	1	2	yes	—
142	C₂²×C₁₃	C₂×Q₃₂	1	2	yes	—
143	C₂²×C₁₃	#510	1	2	yes	—
144	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	4	4	no	—
147	C₄×C₁₃	—	8	2	no	—
148	C₄×C₁₃	—	8	4	no	—
149	C₄×C₁₃	#459	1	2	yes	—
150	C₄×C₁₃	C₂×Q₃₂	1	4	yes	complemented
151	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
152	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₂²×Q₈	—	2	1	no	—
154	C₂²×Q₈	—	2	2	no	—
155	C₂×Q₁₆	—	2	1	no	—
156	C₂×Q₁₆	—	2	4	no	—
157	C₂×Q₁₆	—	2	4	no	—
158	C₂×Q₁₆	—	2	2	no	—
159	C₂×Q₁₆	—	2	2	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	4	4	no	—
162	C₂×Q₁₆	D₄×C₁₃	1	4	yes	—
163	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
164	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
165	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
167	C₄⋊Q₈	—	2	4	no	—
168	C₄⋊Q₈	—	4	2	no	—
169	C₄×Q₈	—	2	4	no	—
170	C₄×Q₈	—	4	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	—	4	1	no	—
173	C₂×C₄⋊C₄	—	4	1	no	—
174	C₂×C₄⋊C₄	—	4	1	no	—
175	C₂×C₄²	—	4	1	no	—
176	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
177	Q₃₂	—	2	16	no	—
178	C₂×C₁₆	—	2	4	no	—
179	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
180	#107	—	2	1	no	—
181	#107	C₂³×C₁₃	1	2	yes	—
182	#103	—	2	4	no	—
183	C₄×C₈	D₄×C₁₃	1	2	yes	—
184	C₄×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂×C₁₃	#2209	1	4	yes	direct product factor
186	C₂×C₁₃	C₂×#459	1	1	yes	characteristic
187	C₂×C₁₃	C₂²×Q₃₂	1	1	yes	characteristic
188	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
189	C₂×Q₈	—	2	4	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	—	2	4	no	—
192	C₂×Q₈	—	4	1	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	2	no	—
195	C₂×Q₈	—	4	4	no	—
196	C₂×Q₈	—	4	4	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	1	no	—
199	C₂²×C₄	—	4	2	no	—
200	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
201	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
203	Q₁₆	—	2	4	no	—
204	Q₁₆	—	2	4	no	—
205	Q₁₆	—	2	2	no	—
206	Q₁₆	—	2	4	no	—
207	Q₁₆	—	4	8	no	—
208	Q₁₆	—	4	4	no	—
209	Q₁₆	C₂×D₄×C₁₃	1	4	yes	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
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	2	yes	—
215	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
216	C₄⋊C₄	—	2	2	no	—
217	C₄⋊C₄	—	4	1	no	—
218	C₄⋊C₄	—	4	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	4	no	—
221	C₄²	—	4	4	no	—
222	C₄²	C₁₃×D₈	1	2	yes	—
223	C₄²	C₂×D₄×C₁₃	1	2	yes	—
224	C₁₆	—	2	4	no	—
225	C₁₃	C₂×#2209	1	1	yes	characteristic, direct product factor, Sylow
226	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
227	Q₈	—	2	4	no	—
228	Q₈	—	4	4	no	—
229	Q₈	—	4	2	no	—
230	Q₈	—	4	4	no	—
231	Q₈	—	4	4	no	—
232	Q₈	—	8	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	4	4	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	—	8	2	no	—
240	C₂×C₄	C₁₃×Q₃₂	1	4	yes	complemented
241	C₂×C₄	C₁₃×#121	1	2	yes	—
242	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
243	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
244	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
245	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
246	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	C₁₃×#121	1	2	yes	—
249	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
250	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
251	C₂²	C₁₃×#459	1	2	yes	—
252	C₂²	C₂×C₁₃×Q₃₂	1	2	yes	—
253	C₂²	C₁₃×#510	1	2	yes	—
254	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	4	4	no	—
257	C₄	—	8	4	no	—
258	C₄	—	8	2	no	—
259	C₄	C₁₃×#459	1	2	yes	—
260	C₄	C₂×C₁₃×Q₃₂	1	4	yes	complemented
261	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
262	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
263	C₂	C₁₃×#2209	1	4	yes	direct product factor
264	C₂	C₂×C₁₃×#459	1	1	yes	characteristic
265	C₂	C₂²×C₁₃×Q₃₂	1	1	yes	characteristic
266	C₂	C₂×C₁₃×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
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
4. C₁₃×#2209	C₂	direct (N × H)
56. C₂×#2209	C₁₃	direct (N × H)
96. #2209	C₂×C₁₃	direct (N × H)
113. C₂×C₄×C₁₃	Q₃₂	semidirect
150. C₄×C₁₃	C₂×Q₃₂	semidirect
185. C₂×C₁₃	#2209	direct (N × H)
225. C₁₃	C₂×#2209	direct (N × H)
240. C₂×C₄	C₁₃×Q₃₂	semidirect
260. C₄	C₂×C₁₃×Q₃₂	semidirect
263. C₂	C₁₃×#2209	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×Q₃₂	2
C₂×C₁₃×#494	2
C₂×C₄×C₁₃×Q₁₆	2
C₁₃×#2209	2
C₂×C₁₃×C₄⋊C₁₆	2
C₂×C₁₃×#358	2
C₂×#2209	13

Subgroups — C2×C13×#2209 #15719916

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2209 #15719916