Subgroups — C₄×C₁₃×#441 #15706841

Order 3328.

1086 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×#441	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄²×Q₈×C₁₃	C₂	1	2	yes	maximal
4	C₁₃×#1776	C₂	1	2	yes	maximal
5	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1763	C₂	1	4	yes	maximal
7	C₄×C₁₃×#103	C₂	1	2	yes	maximal
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
12	C₁₃×#441	C₄	1	16	yes	direct product factor
13	C₄×C₁₃×Q₁₆	C₄	1	8	yes	complemented
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×#103	C₂²	1	4	yes	—
17	C₂×C₁₃×#103	C₂²	1	2	yes	—
18	C₂×C₁₃×#97	C₂²	1	2	yes	—
19	C₂×C₁₃×#97	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
22	C₄×C₁₃×C₄⋊C₄	—	2	4	no	—
23	C₄×C₁₃×C₄⋊C₄	—	2	2	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
26	C₄³×C₁₃	—	2	2	no	—
27	C₄³×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#340	C₂²	1	4	yes	—
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
31	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
32	C₂×C₁₃×Q₁₆	C₂×C₄	1	8	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₃	—	2	4	no	—
37	C₄×Q₈×C₁₃	—	2	8	no	—
38	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
39	C₄×Q₈×C₁₃	C₂×C₄	1	16	yes	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
43	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
44	C₂×C₄²×C₁₃	—	2	4	no	—
45	C₂×C₄²×C₁₃	—	2	2	no	—
46	C₂×C₄²×C₁₃	—	2	4	no	—
47	C₂×C₄²×C₁₃	—	2	2	no	—
48	C₂×C₄²×C₁₃	D₄	1	2	yes	—
49	C₂×C₄²×C₁₃	D₄	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×#107	C₂×C₄	1	4	yes	—
53	C₁₃×#106	C₂×C₄	1	8	yes	—
54	C₁₃×#103	C₂×C₄	1	8	yes	—
55	C₁₃×#103	C₂×C₄	1	16	yes	—
56	C₁₃×#97	—	2	4	no	—
57	C₁₃×#97	C₂×C₄	1	8	yes	—
58	C₄×C₈×C₁₃	—	2	2	no	—
59	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	4	no	—
62	C₄×#441	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	2	4	no	—
65	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
66	C₂×Q₈×C₁₃	C₂²×C₄	1	8	yes	—
67	C₂²×C₄×C₁₃	—	2	2	no	—
68	C₂²×C₄×C₁₃	—	2	4	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
74	C₁₃×Q₁₆	C₄²	1	16	yes	complemented
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
80	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
82	C₁₃×C₄⋊C₄	—	2	16	no	—
83	C₁₃×C₄⋊C₄	—	2	8	no	—
84	C₁₃×C₄⋊C₄	—	2	4	no	—
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	C₂²×C₄	1	8	yes	—
87	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
88	C₄²×C₁₃	—	2	16	no	—
89	C₄²×C₁₃	—	2	8	no	—
90	C₄²×C₁₃	—	2	4	no	—
91	C₄²×C₁₃	—	4	2	no	—
92	C₄²×C₁₃	—	4	8	no	—
93	C₄²×C₁₃	—	4	4	no	—
94	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
96	C₄²×C₁₃	#41	1	2	yes	—
97	C₄²×C₁₃	#41	1	4	yes	—
98	C₂×#441	C₂×C₁₃	1	2	yes	—
99	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
100	C₄²×Q₈	C₂×C₁₃	1	2	yes	—
101	#1776	C₂×C₁₃	1	2	yes	—
102	C₄×#106	C₂×C₁₃	1	1	yes	characteristic
103	#1763	C₂×C₁₃	1	4	yes	—
104	C₄×#103	C₂×C₁₃	1	2	yes	—
105	C₄×#97	C₂×C₁₃	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	4	no	—
108	Q₈×C₁₃	—	2	8	no	—
109	Q₈×C₁₃	C₂×C₄²	1	8	yes	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	8	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	8	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	8	yes	—
121	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	#137	1	4	yes	—
124	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
128	C₈×C₁₃	—	2	4	no	—
129	C₈×C₁₃	—	2	2	no	—
130	C₈×C₁₃	C₂×C₄²	1	4	yes	—
131	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
133	C₂×C₄×Q₈	C₂²×C₁₃	1	4	yes	—
134	#441	C₄×C₁₃	1	16	yes	direct product factor
135	C₄×Q₁₆	C₄×C₁₃	1	8	yes	complemented
136	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
137	C₂×#106	C₂²×C₁₃	1	2	yes	—
138	C₂×#103	C₂²×C₁₃	1	2	yes	—
139	C₂×#103	C₂²×C₁₃	1	4	yes	—
140	C₂×#97	C₂²×C₁₃	1	2	yes	—
141	C₂×#97	C₂²×C₁₃	1	2	yes	—
142	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
144	C₄×C₄⋊C₄	—	2	2	no	—
145	C₄×C₄⋊C₄	—	2	4	no	—
146	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
147	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
148	C₄³	—	2	2	no	—
149	C₄³	C₂²×C₁₃	1	1	yes	characteristic
150	#340	C₂²×C₁₃	1	4	yes	—
151	#340	C₂²×C₁₃	1	2	yes	—
152	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₁₃	C₂×C₄×D₄	1	2	yes	—
154	C₂²×C₁₃	C₂×#137	1	2	yes	—
155	C₂²×C₁₃	#575	1	2	yes	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	8	no	—
158	C₄×C₁₃	—	2	8	no	—
159	C₄×C₁₃	—	4	4	no	—
160	C₄×C₁₃	—	4	8	no	—
161	C₄×C₁₃	#441	1	4	yes	direct product factor
162	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
163	C₄×C₁₃	C₄×#41	1	2	yes	—
164	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
165	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
166	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
167	C₂×Q₁₆	C₂×C₄×C₁₃	1	8	yes	—
168	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
170	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
171	C₄×Q₈	—	2	4	no	—
172	C₄×Q₈	—	2	8	no	—
173	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
174	C₄×Q₈	C₂×C₄×C₁₃	1	16	yes	—
175	C₂×C₄⋊C₄	—	2	2	no	—
176	C₂×C₄⋊C₄	—	2	4	no	—
177	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
178	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
179	C₂×C₄²	—	2	4	no	—
180	C₂×C₄²	—	2	2	no	—
181	C₂×C₄²	—	2	4	no	—
182	C₂×C₄²	—	2	2	no	—
183	C₂×C₄²	D₄×C₁₃	1	2	yes	—
184	C₂×C₄²	D₄×C₁₃	1	2	yes	—
185	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
186	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
187	#107	C₂×C₄×C₁₃	1	4	yes	—
188	#106	C₂×C₄×C₁₃	1	8	yes	—
189	#103	C₂×C₄×C₁₃	1	8	yes	—
190	#103	C₂×C₄×C₁₃	1	16	yes	—
191	#97	—	2	4	no	—
192	#97	C₂×C₄×C₁₃	1	8	yes	—
193	C₄×C₈	—	2	2	no	—
194	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
195	#95	—	2	2	no	—
196	#95	—	2	4	no	—
197	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
198	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
199	C₂×C₁₃	#2943	1	1	yes	characteristic
200	C₂×C₁₃	C₄×#137	1	4	yes	—
201	C₂×Q₈	—	2	2	no	—
202	C₂×Q₈	—	2	4	no	—
203	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
204	C₂×Q₈	C₂²×C₄×C₁₃	1	8	yes	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	4	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
209	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
210	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
211	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
212	Q₁₆	C₄²×C₁₃	1	16	yes	complemented
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
217	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
218	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
219	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
220	C₄⋊C₄	—	2	16	no	—
221	C₄⋊C₄	—	2	8	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	2	2	no	—
224	C₄⋊C₄	C₂²×C₄×C₁₃	1	8	yes	—
225	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
226	C₄²	—	2	16	no	—
227	C₄²	—	2	8	no	—
228	C₄²	—	2	4	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	8	no	—
231	C₄²	—	4	4	no	—
232	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
233	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
234	C₄²	C₁₃×#41	1	2	yes	—
235	C₄²	C₁₃×#41	1	4	yes	—
236	C₁₃	C₄×#441	1	1	yes	characteristic, direct product factor, Sylow
237	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
238	Q₈	—	2	4	no	—
239	Q₈	—	2	8	no	—
240	Q₈	C₂×C₄²×C₁₃	1	8	yes	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	4	no	—
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₄×D₄×C₁₃	1	8	yes	—
252	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
253	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
254	C₂×C₄	C₁₃×#137	1	4	yes	—
255	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
257	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
258	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
259	C₈	—	2	4	no	—
260	C₈	—	2	2	no	—
261	C₈	C₂×C₄²×C₁₃	1	4	yes	—
262	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
263	C₂²	C₂×C₄×D₄×C₁₃	1	2	yes	—
264	C₂²	C₂×C₁₃×#137	1	2	yes	—
265	C₂²	C₁₃×#575	1	2	yes	—
266	C₄	—	2	8	no	—
267	C₄	—	2	2	no	—
268	C₄	—	2	8	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	8	no	—
271	C₄	C₁₃×#441	1	4	yes	direct product factor
272	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
273	C₄	C₄×C₁₃×#41	1	2	yes	—
274	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
275	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
276	C₂	C₂×C₁₃×#441	1	1	yes	characteristic
277	C₂	C₁₃×#2943	1	1	yes	characteristic
278	C₂	C₄×C₁₃×#137	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator	C₄	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle, prime radical	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#441	C₄	direct (N × H)
13. C₄×C₁₃×Q₁₆	C₄	semidirect
62. C₄×#441	C₁₃	direct (N × H)
74. C₁₃×Q₁₆	C₄²	semidirect
134. #441	C₄×C₁₃	direct (N × H)
135. C₄×Q₁₆	C₄×C₁₃	semidirect
161. C₄×C₁₃	#441	direct (N × H)
212. Q₁₆	C₄²×C₁₃	semidirect
236. C₁₃	C₄×#441	direct (N × H)
271. C₄	C₁₃×#441	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×#441	2
C₂×C₄×C₁₃×Q₁₆	2
C₄²×Q₈×C₁₃	2
C₁₃×#1776	2
C₄×C₁₃×#106	2
C₁₃×#1763	2
C₄×C₁₃×#103	2
C₄×C₁₃×#97	2
C₄×#441	13

Subgroups — C4×C13×#441 #15706841

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₃×#441 #15706841