Subgroups — C₂×C₁₃×#2232 #15719978

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#466	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×#107	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2232	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#366	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#357	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#121	—	2	1	no	—
10	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
11	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
12	C₁₃×#466	—	2	2	no	—
13	C₁₃×#466	C₂²	1	4	yes	—
14	C₂×C₁₃×#107	—	2	2	no	—
15	C₂×C₁₃×#107	—	2	1	no	—
16	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#107	C₂²	1	2	yes	—
18	C₂×C₁₃×#107	C₂²	1	4	yes	complemented
19	C₂×C₁₃×#102	—	2	1	no	—
20	C₁₃×#366	C₂²	1	8	yes	complemented
21	C₁₃×#357	C₂²	1	8	yes	complemented
22	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
23	C₂²×D₄×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	4	1	no	—
25	C₂³×C₄×C₁₃	—	4	1	no	—
26	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
27	C₂×C₁₃×D₈	—	2	1	no	—
28	C₂×C₁₃×D₈	—	2	4	no	—
29	C₂×C₁₃×D₈	C₂³	1	2	yes	—
30	C₂²×C₈×C₁₃	—	2	4	no	—
31	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₁₃	D₄	1	2	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#121	—	2	4	no	—
36	C₁₃×#121	—	4	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	4	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	4	1	no	—
43	C₂×C₁₃×C₁₆	—	2	4	no	—
44	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
45	C₁₃×#107	—	2	2	no	—
46	C₁₃×#107	—	2	4	no	—
47	C₁₃×#107	—	2	1	no	—
48	C₁₃×#107	C₂³	1	4	yes	—
49	C₁₃×#107	C₂³	1	2	yes	—
50	C₁₃×#102	—	2	4	no	—
51	C₂×#2232	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₁₃	—	4	1	no	—
53	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
54	C₂×D₄×C₁₃	—	2	2	no	—
55	C₂×D₄×C₁₃	—	4	4	no	—
56	C₂×D₄×C₁₃	—	4	2	no	—
57	C₂×D₄×C₁₃	—	4	1	no	—
58	C₂×D₄×C₁₃	—	4	2	no	—
59	C₂×D₄×C₁₃	—	8	4	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	4	4	no	—
63	C₂²×C₄×C₁₃	—	4	1	no	—
64	C₂²×C₄×C₁₃	—	4	2	no	—
65	C₂²×C₄×C₁₃	—	8	4	no	—
66	C₂²×C₄×C₁₃	D₈	1	2	yes	—
67	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
68	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
70	C₁₃×D₈	—	2	4	no	—
71	C₁₃×D₈	—	4	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
76	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₁₃	#41	1	4	yes	—
78	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₁₃×C₄⋊C₄	—	2	2	no	—
80	C₁₃×C₄⋊C₄	—	2	4	no	—
81	C₁₃×C₄⋊C₄	—	4	4	no	—
82	C₁₃×C₄⋊C₄	—	4	2	no	—
83	C₁₃×C₄⋊C₄	—	4	1	no	—
84	C₁₃×C₂²⋊C₄	—	4	4	no	—
85	C₁₃×C₁₆	—	2	4	no	—
86	C₂×#466	C₂×C₁₃	1	1	yes	characteristic
87	C₂²×#107	C₂×C₁₃	1	1	yes	characteristic, complemented
88	#2232	C₂×C₁₃	1	8	yes	direct product factor
89	C₂×#366	C₂×C₁₃	1	2	yes	complemented
90	C₂×#357	C₂×C₁₃	1	2	yes	complemented
91	C₂×C₂²⋊C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
92	C₂³×C₁₃	—	2	2	no	—
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	—	8	2	no	—
96	C₂³×C₁₃	—	8	2	no	—
97	C₂³×C₁₃	—	8	2	no	—
98	C₂³×C₁₃	D₁₆	1	4	yes	—
99	C₂³×C₁₃	C₂×D₈	1	2	yes	—
100	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
101	D₄×C₁₃	—	4	2	no	—
102	D₄×C₁₃	—	4	2	no	—
103	D₄×C₁₃	—	8	8	no	—
104	D₄×C₁₃	—	8	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	4	4	no	—
111	C₂×C₄×C₁₃	—	8	8	no	—
112	C₂×C₄×C₁₃	—	8	4	no	—
113	C₂×C₄×C₁₃	—	8	2	no	—
114	C₂×C₄×C₁₃	#123	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	#137	1	2	yes	—
118	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₈×C₁₃	—	2	2	no	—
120	C₈×C₁₃	#123	1	2	yes	—
121	C₈×C₁₃	C₂×#41	1	2	yes	—
122	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
123	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
124	C₂×#121	—	2	1	no	—
125	C₂²×C₄⋊C₄	—	2	1	no	—
126	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
127	#466	—	2	2	no	—
128	#466	C₂²×C₁₃	1	4	yes	—
129	C₂×#107	—	2	2	no	—
130	C₂×#107	—	2	1	no	—
131	C₂×#107	C₂²×C₁₃	1	2	yes	—
132	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×#107	C₂²×C₁₃	1	4	yes	complemented
134	C₂×#102	—	2	1	no	—
135	#366	C₂²×C₁₃	1	8	yes	complemented
136	#357	C₂²×C₁₃	1	8	yes	complemented
137	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	complemented
138	C₂²×C₁₃	—	2	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	2	8	no	—
141	C₂²×C₁₃	—	8	2	no	—
142	C₂²×C₁₃	—	8	2	no	—
143	C₂²×C₁₃	—	8	2	no	—
144	C₂²×C₁₃	—	16	4	no	—
145	C₂²×C₁₃	#480	1	2	yes	—
146	C₂²×C₁₃	C₂×D₁₆	1	4	yes	—
147	C₂²×C₁₃	C₂×D₁₆	1	2	yes	—
148	C₂²×C₁₃	#510	1	2	yes	—
149	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
150	C₄×C₁₃	—	2	2	no	—
151	C₄×C₁₃	—	8	4	no	—
152	C₄×C₁₃	—	8	2	no	—
153	C₄×C₁₃	#480	1	2	yes	—
154	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
155	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
156	C₂²×D₄	—	2	1	no	—
157	C₂²×D₄	—	4	1	no	—
158	C₂³×C₄	—	4	1	no	—
159	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
160	C₂×D₈	—	2	1	no	—
161	C₂×D₈	—	2	4	no	—
162	C₂×D₈	C₂³×C₁₃	1	2	yes	—
163	C₂²×C₈	—	2	4	no	—
164	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
165	C₂²×C₈	D₄×C₁₃	1	2	yes	—
166	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
167	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
168	#121	—	2	4	no	—
169	#121	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	4	no	—
172	C₂×C₄⋊C₄	—	2	1	no	—
173	C₂×C₄⋊C₄	—	4	1	no	—
174	C₂×C₄⋊C₄	—	4	2	no	—
175	C₂×C₂²⋊C₄	—	4	1	no	—
176	C₂×C₁₆	—	2	4	no	—
177	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
178	#107	—	2	2	no	—
179	#107	—	2	4	no	—
180	#107	—	2	1	no	—
181	#107	C₂³×C₁₃	1	4	yes	—
182	#107	C₂³×C₁₃	1	2	yes	—
183	#102	—	2	4	no	—
184	C₂×C₁₃	—	2	4	no	—
185	C₂×C₁₃	—	16	2	no	—
186	C₂×C₁₃	#2232	1	4	yes	direct product factor
187	C₂×C₁₃	C₂×#480	1	1	yes	characteristic
188	C₂×C₁₃	C₂²×D₁₆	1	1	yes	characteristic
189	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
190	C₂⁴	—	4	1	no	—
191	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	4	4	no	—
194	C₂×D₄	—	4	2	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	8	4	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	4	1	no	—
201	C₂²×C₄	—	4	4	no	—
202	C₂²×C₄	—	4	2	no	—
203	C₂²×C₄	—	8	4	no	—
204	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
205	C₂²×C₄	C₁₃×D₈	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	D₈	—	2	4	no	—
209	D₈	—	4	2	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	4	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
214	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₈	C₁₃×#41	1	4	yes	—
216	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	4	no	—
219	C₄⋊C₄	—	4	4	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	1	no	—
222	C₂²⋊C₄	—	4	4	no	—
223	C₁₆	—	2	4	no	—
224	C₁₃	C₂×#2232	1	1	yes	characteristic, direct product factor, Sylow
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	—	8	2	no	—
229	C₂³	—	8	2	no	—
230	C₂³	—	8	2	no	—
231	C₂³	C₁₃×D₁₆	1	4	yes	—
232	C₂³	C₂×C₁₃×D₈	1	2	yes	—
233	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
234	D₄	—	4	2	no	—
235	D₄	—	4	2	no	—
236	D₄	—	8	8	no	—
237	D₄	—	8	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	4	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	8	8	no	—
245	C₂×C₄	—	8	4	no	—
246	C₂×C₄	—	8	2	no	—
247	C₂×C₄	C₁₃×#123	1	2	yes	—
248	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
249	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
250	C₂×C₄	C₁₃×#137	1	2	yes	—
251	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	C₁₃×#123	1	2	yes	—
254	C₈	C₂×C₁₃×#41	1	2	yes	—
255	C₂²	—	2	2	no	—
256	C₂²	—	2	2	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	8	2	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	16	4	no	—
262	C₂²	C₁₃×#480	1	2	yes	—
263	C₂²	C₂×C₁₃×D₁₆	1	4	yes	—
264	C₂²	C₂×C₁₃×D₁₆	1	2	yes	—
265	C₂²	C₁₃×#510	1	2	yes	—
266	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
267	C₄	—	2	2	no	—
268	C₄	—	8	4	no	—
269	C₄	—	8	2	no	—
270	C₄	C₁₃×#480	1	2	yes	—
271	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
272	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
273	C₂	—	2	4	no	—
274	C₂	—	16	2	no	—
275	C₂	C₁₃×#2232	1	4	yes	direct product factor
276	C₂	C₂×C₁₃×#480	1	1	yes	characteristic
277	C₂	C₂²×C₁₃×D₁₆	1	1	yes	characteristic
278	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	C₂×C₁₃×#466	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₁₃×#107	C₂	semidirect
3. C₁₃×#2232	C₂	direct (N × H)
4. C₂×C₁₃×#366	C₂	semidirect
5. C₂×C₁₃×#357	C₂	semidirect
6. C₂×C₁₃×C₂²⋊C₁₆	C₂	semidirect
18. C₂×C₁₃×#107	C₂²	semidirect
20. C₁₃×#366	C₂²	semidirect
21. C₁₃×#357	C₂²	semidirect
22. C₁₃×C₂²⋊C₁₆	C₂²	semidirect
51. C₂×#2232	C₁₃	direct (N × H)
87. C₂²×#107	C₂×C₁₃	semidirect
88. #2232	C₂×C₁₃	direct (N × H)
89. C₂×#366	C₂×C₁₃	semidirect
90. C₂×#357	C₂×C₁₃	semidirect
91. C₂×C₂²⋊C₁₆	C₂×C₁₃	semidirect
133. C₂×#107	C₂²×C₁₃	semidirect
135. #366	C₂²×C₁₃	semidirect
136. #357	C₂²×C₁₃	semidirect
137. C₂²⋊C₁₆	C₂²×C₁₃	semidirect
186. C₂×C₁₃	#2232	direct (N × H)
224. C₁₃	C₂×#2232	direct (N × H)
275. C₂	C₁₃×#2232	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×#466	2
C₂²×C₁₃×#107	2
C₁₃×#2232	2
C₂×C₁₃×#366	2
C₂×C₁₃×#357	2
C₂×C₁₃×C₂²⋊C₁₆	2
C₂×#2232	13

Subgroups — C2×C13×#2232 #15719978

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2232 #15719978