Subgroups — C₄×C₁₃×#444 #15706850

Order 3328.

1022 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#444	C₂	1	2	yes	maximal
2	C₄×C₁₃×#135	C₂	1	1	yes	maximal, characteristic
3	C₄×C₁₃×#131	C₂	1	2	yes	maximal, complemented
4	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#1773	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1765	C₂	1	4	yes	maximal, complemented
7	C₄×C₁₃×#104	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	4	yes	—
11	C₄×C₁₃×#41	C₂²	1	2	yes	—
12	C₁₃×#444	C₄	1	16	yes	direct product factor
13	C₄×C₁₃×Q₁₆	C₄	1	2	yes	complemented
14	C₄×C₁₃×SD₁₆	C₄	1	4	yes	complemented
15	C₄×C₁₃×D₈	C₄	1	2	yes	complemented
16	C₂×C₁₃×#108	C₂²	1	2	yes	—
17	C₁₃×#427	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#104	C₂²	1	4	yes	—
19	C₁₃×#416	C₂²	1	2	yes	—
20	C₁₃×#405	—	2	4	no	—
21	C₁₃×#405	C₂²	1	4	yes	—
22	C₄×C₁₃×M₁₆	—	2	2	no	—
23	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
24	C₂×C₁₃×#97	C₂²	1	2	yes	—
25	C₂×C₁₃×#97	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃	—	2	2	no	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₄³×C₁₃	C₂²	1	1	yes	characteristic
30	C₁₃×#341	C₂²	1	2	yes	—
31	C₁₃×#339	C₂²	1	4	yes	—
32	C₂×C₁₃×#41	C₂³	1	2	yes	—
33	C₁₃×#135	C₂×C₄	1	8	yes	—
34	C₂×C₁₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
35	C₂×C₁₃×SD₁₆	C₂×C₄	1	2	yes	—
36	C₂×C₁₃×D₈	C₂×C₄	1	1	yes	characteristic
37	C₁₃×#131	—	2	8	no	—
38	C₁₃×#131	C₂×C₄	1	16	yes	complemented
39	C₂×C₁₃×M₁₆	—	2	4	no	—
40	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
41	C₂²×C₈×C₁₃	—	2	4	no	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂×C₄	1	4	yes	—
46	C₄×D₄×C₁₃	—	2	4	no	—
47	C₄×D₄×C₁₃	C₂×C₄	1	4	yes	—
48	C₁₃×#117	—	2	2	no	—
49	C₁₃×#117	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	—	2	2	no	—
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#108	C₂×C₄	1	8	yes	complemented
55	C₁₃×#107	C₂×C₄	1	2	yes	—
56	C₁₃×#106	C₂×C₄	1	2	yes	—
57	C₁₃×#104	C₂×C₄	1	16	yes	complemented
58	C₁₃×#103	C₂×C₄	1	4	yes	—
59	C₁₃×#102	C₂×C₄	1	4	yes	—
60	C₁₃×#97	—	2	4	no	—
61	C₁₃×#97	—	2	4	no	—
62	C₁₃×#97	—	2	2	no	—
63	C₁₃×#97	C₂×C₄	1	8	yes	complemented
64	C₄×C₈×C₁₃	—	2	4	no	—
65	C₄×C₈×C₁₃	—	2	2	no	—
66	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
67	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
68	C₄×C₈×C₁₃	D₄	1	2	yes	—
69	C₄×C₈×C₁₃	D₄	1	2	yes	—
70	C₄×#444	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₃×#41	—	2	4	no	—
72	C₁₃×#41	C₂²×C₄	1	8	yes	—
73	C₂×Q₈×C₁₃	C₂²×C₄	1	2	yes	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	#41	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
81	C₁₃×Q₁₆	C₄²	1	4	yes	complemented
82	C₁₃×SD₁₆	C₄²	1	8	yes	complemented
83	C₁₃×D₈	C₄²	1	4	yes	complemented
84	C₁₃×M₁₆	—	2	4	no	—
85	C₁₃×M₁₆	—	2	16	no	—
86	C₁₃×M₁₆	C₂²×C₄	1	8	yes	—
87	C₂×C₈×C₁₃	—	2	4	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	16	no	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
96	C₂×C₈×C₁₃	#41	1	4	yes	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
99	C₁₃×C₂²⋊C₄	—	2	2	no	—
100	C₁₃×C₂²⋊C₄	—	2	4	no	—
101	C₄²×C₁₃	—	2	4	no	—
102	C₄²×C₁₃	—	2	2	no	—
103	C₄²×C₁₃	—	2	2	no	—
104	C₄²×C₁₃	—	4	4	no	—
105	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
106	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×#444	C₂×C₁₃	1	2	yes	—
110	C₄×#135	C₂×C₁₃	1	1	yes	characteristic
111	C₄×#131	C₂×C₁₃	1	2	yes	complemented
112	C₄×#108	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#1773	C₂×C₁₃	1	2	yes	complemented
114	#1765	C₂×C₁₃	1	4	yes	complemented
115	C₄×#104	C₂×C₁₃	1	2	yes	complemented
116	C₄×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂³×C₁₃	—	2	2	no	—
118	C₂³×C₁₃	C₂×#41	1	1	yes	characteristic
119	Q₈×C₁₃	—	2	2	no	—
120	Q₈×C₁₃	C₂×C₄²	1	4	yes	—
121	D₄×C₁₃	—	2	8	no	—
122	D₄×C₁₃	—	2	2	no	—
123	D₄×C₁₃	C₂×C₄²	1	4	yes	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	8	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	8	no	—
131	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
134	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
137	C₈×C₁₃	—	2	8	no	—
138	C₈×C₁₃	—	2	2	no	—
139	C₈×C₁₃	C₂×C₄²	1	2	yes	—
140	C₈×C₁₃	C₂×C₄²	1	2	yes	—
141	C₈×C₁₃	C₄×D₄	1	8	yes	—
142	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#131	C₂²×C₁₃	1	4	yes	—
144	C₄×#41	C₂²×C₁₃	1	2	yes	—
145	#444	C₄×C₁₃	1	16	yes	direct product factor
146	C₄×Q₁₆	C₄×C₁₃	1	2	yes	complemented
147	C₄×SD₁₆	C₄×C₁₃	1	4	yes	complemented
148	C₄×D₈	C₄×C₁₃	1	2	yes	complemented
149	C₂×#108	C₂²×C₁₃	1	2	yes	—
150	#427	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#104	C₂²×C₁₃	1	4	yes	—
152	#416	C₂²×C₁₃	1	2	yes	—
153	#405	—	2	4	no	—
154	#405	C₂²×C₁₃	1	4	yes	—
155	C₄×M₁₆	—	2	2	no	—
156	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
157	C₂×#97	C₂²×C₁₃	1	2	yes	—
158	C₂×#97	C₂²×C₁₃	1	2	yes	—
159	C₂×C₄×C₈	—	2	2	no	—
160	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₄³	C₂²×C₁₃	1	1	yes	characteristic
163	#341	C₂²×C₁₃	1	2	yes	—
164	#339	C₂²×C₁₃	1	4	yes	—
165	C₂²×C₁₃	—	2	2	no	—
166	C₂²×C₁₃	—	2	4	no	—
167	C₂²×C₁₃	—	4	4	no	—
168	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂²×C₁₃	C₄×#41	1	2	yes	—
170	C₄×C₁₃	—	2	2	no	—
171	C₄×C₁₃	—	2	4	no	—
172	C₄×C₁₃	—	4	4	no	—
173	C₄×C₁₃	—	4	4	no	—
174	C₄×C₁₃	#444	1	4	yes	direct product factor
175	C₄×C₁₃	C₂²×C₄²	1	1	yes	characteristic
176	C₄×C₁₃	C₂×C₄×D₄	1	2	yes	—
177	C₄×C₁₃	C₄×#41	1	1	yes	characteristic
178	C₂×#41	C₂³×C₁₃	1	2	yes	—
179	#135	C₂×C₄×C₁₃	1	8	yes	—
180	C₂×Q₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
181	C₂×SD₁₆	C₂×C₄×C₁₃	1	2	yes	—
182	C₂×D₈	C₂×C₄×C₁₃	1	1	yes	characteristic
183	#131	—	2	8	no	—
184	#131	C₂×C₄×C₁₃	1	16	yes	complemented
185	C₂×M₁₆	—	2	4	no	—
186	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
187	C₂²×C₈	—	2	4	no	—
188	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
190	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
191	C₄×Q₈	C₂×C₄×C₁₃	1	4	yes	—
192	C₄×D₄	—	2	4	no	—
193	C₄×D₄	C₂×C₄×C₁₃	1	4	yes	—
194	#117	—	2	2	no	—
195	#117	C₂³×C₁₃	1	2	yes	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	2	no	—
198	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
199	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
200	#108	C₂×C₄×C₁₃	1	8	yes	complemented
201	#107	C₂×C₄×C₁₃	1	2	yes	—
202	#106	C₂×C₄×C₁₃	1	2	yes	—
203	#104	C₂×C₄×C₁₃	1	16	yes	complemented
204	#103	C₂×C₄×C₁₃	1	4	yes	—
205	#102	C₂×C₄×C₁₃	1	4	yes	—
206	#97	—	2	4	no	—
207	#97	—	2	4	no	—
208	#97	—	2	2	no	—
209	#97	C₂×C₄×C₁₃	1	8	yes	complemented
210	C₄×C₈	—	2	4	no	—
211	C₄×C₈	—	2	2	no	—
212	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
213	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
214	C₄×C₈	D₄×C₁₃	1	2	yes	—
215	C₄×C₈	D₄×C₁₃	1	2	yes	—
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	—	4	4	no	—
218	C₂×C₁₃	C₄²×D₄	1	1	yes	characteristic
219	C₂×C₁₃	C₂×#444	1	2	yes	—
220	#41	—	2	4	no	—
221	#41	C₂²×C₄×C₁₃	1	8	yes	—
222	C₂×Q₈	C₂²×C₄×C₁₃	1	2	yes	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	C₂²×C₄×C₁₃	1	2	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	4	no	—
228	C₂²×C₄	C₁₃×#41	1	2	yes	—
229	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
230	Q₁₆	C₄²×C₁₃	1	4	yes	complemented
231	SD₁₆	C₄²×C₁₃	1	8	yes	complemented
232	D₈	C₄²×C₁₃	1	4	yes	complemented
233	M₁₆	—	2	4	no	—
234	M₁₆	—	2	16	no	—
235	M₁₆	C₂²×C₄×C₁₃	1	8	yes	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	16	no	—
239	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂²×C₄×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
244	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
245	C₂×C₈	C₁₃×#41	1	4	yes	—
246	C₄⋊C₄	—	2	4	no	—
247	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
255	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
256	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₁₃	C₄×#444	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	—	2	2	no	—
260	C₂³	C₂×C₁₃×#41	1	1	yes	characteristic
261	Q₈	—	2	2	no	—
262	Q₈	C₂×C₄²×C₁₃	1	4	yes	—
263	D₄	—	2	8	no	—
264	D₄	—	2	2	no	—
265	D₄	C₂×C₄²×C₁₃	1	4	yes	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	4	8	no	—
273	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
275	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
276	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
277	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
278	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
279	C₈	—	2	8	no	—
280	C₈	—	2	2	no	—
281	C₈	C₂×C₄²×C₁₃	1	2	yes	—
282	C₈	C₂×C₄²×C₁₃	1	2	yes	—
283	C₈	C₄×D₄×C₁₃	1	8	yes	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	4	no	—
286	C₂²	—	4	4	no	—
287	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
288	C₂²	C₄×C₁₃×#41	1	2	yes	—
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	4	4	no	—
292	C₄	—	4	4	no	—
293	C₄	C₁₃×#444	1	4	yes	direct product factor
294	C₄	C₂²×C₄²×C₁₃	1	1	yes	characteristic
295	C₄	C₂×C₄×D₄×C₁₃	1	2	yes	—
296	C₄	C₄×C₁₃×#41	1	1	yes	characteristic
297	C₂	—	2	2	no	—
298	C₂	—	4	4	no	—
299	C₂	C₄²×D₄×C₁₃	1	1	yes	characteristic
300	C₂	C₂×C₁₃×#444	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃	C₂×D₄
commutator	C₄	C₂²×C₄²×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×C₁₃×#131	C₂	semidirect
4. C₄×C₁₃×#108	C₂	semidirect
5. C₁₃×#1773	C₂	semidirect
6. C₁₃×#1765	C₂	semidirect
7. C₄×C₁₃×#104	C₂	semidirect
8. C₄×C₁₃×#97	C₂	semidirect
12. C₁₃×#444	C₄	direct (N × H)
13. C₄×C₁₃×Q₁₆	C₄	semidirect
14. C₄×C₁₃×SD₁₆	C₄	semidirect
15. C₄×C₁₃×D₈	C₄	semidirect
38. C₁₃×#131	C₂×C₄	semidirect
54. C₁₃×#108	C₂×C₄	semidirect
57. C₁₃×#104	C₂×C₄	semidirect
63. C₁₃×#97	C₂×C₄	semidirect
70. C₄×#444	C₁₃	direct (N × H)
81. C₁₃×Q₁₆	C₄²	semidirect
82. C₁₃×SD₁₆	C₄²	semidirect
83. C₁₃×D₈	C₄²	semidirect
111. C₄×#131	C₂×C₁₃	semidirect
112. C₄×#108	C₂×C₁₃	semidirect
113. #1773	C₂×C₁₃	semidirect
114. #1765	C₂×C₁₃	semidirect
115. C₄×#104	C₂×C₁₃	semidirect
116. C₄×#97	C₂×C₁₃	semidirect
145. #444	C₄×C₁₃	direct (N × H)
146. C₄×Q₁₆	C₄×C₁₃	semidirect
147. C₄×SD₁₆	C₄×C₁₃	semidirect
148. C₄×D₈	C₄×C₁₃	semidirect
174. C₄×C₁₃	#444	direct (N × H)
184. #131	C₂×C₄×C₁₃	semidirect
200. #108	C₂×C₄×C₁₃	semidirect
203. #104	C₂×C₄×C₁₃	semidirect
209. #97	C₂×C₄×C₁₃	semidirect
230. Q₁₆	C₄²×C₁₃	semidirect
231. SD₁₆	C₄²×C₁₃	semidirect
232. D₈	C₄²×C₁₃	semidirect
258. C₁₃	C₄×#444	direct (N × H)
293. C₄	C₁₃×#444	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×#444	2
C₄×C₁₃×#135	2
C₄×C₁₃×#131	2
C₄×C₁₃×#108	2
C₁₃×#1773	2
C₁₃×#1765	2
C₄×C₁₃×#104	2
C₄×C₁₃×#97	2
C₄×#444	13

Subgroups — C4×C13×#444 #15706850

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×C₁₃×#444 #15706850