Subgroups — C₁₃×#22733 #15709072

Order 3328.

638 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×C₁₃×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₁₃×#446	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1950	C₂	1	2	yes	maximal
4	C₁₃×#1950	C₂	1	4	yes	maximal
5	C₁₃×#1849	C₂	1	2	yes	maximal
6	C₁₃×#1848	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1778	C₂	1	2	yes	maximal
8	C₄×C₁₃×#108	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₁₃×#446	—	2	2	no	—
11	C₁₃×#446	—	2	4	no	—
12	C₁₃×#446	C₄	1	4	yes	—
13	C₁₃×#446	C₄	1	4	yes	—
14	C₈×Q₈×C₁₃	—	2	2	no	—
15	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#108	C₂²	1	2	yes	—
17	C₂×C₁₃×#108	C₂²	1	2	yes	—
18	C₂×C₁₃×#108	C₂²	1	4	yes	—
19	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
20	C₁₃×#423	C₂²	1	2	yes	—
21	C₁₃×#423	C₂²	1	4	yes	—
22	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
24	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×M₁₆	C₂²	1	4	yes	—
26	C₂×C₁₃×#97	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#341	C₂²	1	2	yes	—
29	C₁₃×#341	C₂²	1	4	yes	—
30	C₂²×Q₈×C₁₃	Q₈	1	1	yes	characteristic
31	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
32	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
33	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
38	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×Q₈×C₁₃	D₄	1	2	yes	—
40	C₄×Q₈×C₁₃	D₄	1	2	yes	—
41	C₂×C₁₃×C₄⋊C₄	Q₈	1	1	yes	characteristic
42	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
43	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₁₃×#108	—	2	2	no	—
46	C₁₃×#108	—	2	4	no	—
47	C₁₃×#108	—	4	2	no	—
48	C₁₃×#108	—	4	4	no	—
49	C₁₃×#108	Q₈	1	4	yes	complemented
50	C₁₃×C₄⋊C₈	—	2	2	no	—
51	C₁₃×C₄⋊C₈	—	2	4	no	—
52	C₁₃×C₄⋊C₈	—	2	4	no	—
53	C₁₃×C₄⋊C₈	—	2	2	no	—
54	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
55	C₁₃×C₄⋊C₈	C₂×C₄	1	2	yes	—
56	C₁₃×C₄⋊C₈	C₂×C₄	1	8	yes	—
57	C₁₃×#97	—	2	4	no	—
58	C₁₃×#97	—	2	2	no	—
59	C₁₃×#97	C₂×C₄	1	8	yes	—
60	C₄×C₈×C₁₃	—	2	2	no	—
61	C₄×C₈×C₁₃	—	2	4	no	—
62	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
63	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
64	#22733	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
67	C₂×Q₈×C₁₃	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×M₁₆	—	2	2	no	—
74	C₁₃×M₁₆	—	2	2	no	—
75	C₁₃×M₁₆	—	2	4	no	—
76	C₁₃×M₁₆	—	2	4	no	—
77	C₁₃×M₁₆	—	2	2	no	—
78	C₁₃×M₁₆	C₂×Q₈	1	4	yes	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	2	no	—
85	C₂×C₈×C₁₃	—	2	4	no	—
86	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
88	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
92	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
93	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
94	C₁₃×C₄⋊C₄	C₄⋊C₄	1	2	yes	—
95	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₁₃	C₂×D₄	1	4	yes	—
100	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
101	Q₈×M₁₆	C₂×C₁₃	1	2	yes	—
102	C₂×#446	C₂×C₁₃	1	1	yes	characteristic
103	#1950	C₂×C₁₃	1	4	yes	—
104	#1950	C₂×C₁₃	1	2	yes	—
105	#1849	C₂×C₁₃	1	2	yes	—
106	#1848	C₂×C₁₃	1	1	yes	characteristic
107	#1778	C₂×C₁₃	1	2	yes	—
108	C₄×#108	C₂×C₁₃	1	1	yes	characteristic
109	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
110	Q₈×C₁₃	—	2	4	no	—
111	Q₈×C₁₃	—	4	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	4	1	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
119	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
121	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
126	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
127	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	C₈×C₁₃	—	2	2	no	—
129	C₈×C₁₃	—	2	4	no	—
130	C₈×C₁₃	—	2	4	no	—
131	C₈×C₁₃	—	4	2	no	—
132	C₈×C₁₃	—	4	4	no	—
133	C₈×C₁₃	C₄×Q₈	1	4	yes	—
134	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
135	#446	—	2	2	no	—
136	#446	—	2	4	no	—
137	#446	C₄×C₁₃	1	4	yes	—
138	#446	C₄×C₁₃	1	4	yes	—
139	C₈×Q₈	—	2	2	no	—
140	C₂×#108	C₂²×C₁₃	1	2	yes	—
141	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×#108	C₂²×C₁₃	1	2	yes	—
143	C₂×#108	C₂²×C₁₃	1	4	yes	—
144	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
145	#423	C₂²×C₁₃	1	2	yes	—
146	#423	C₂²×C₁₃	1	4	yes	—
147	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
149	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
150	C₄×M₁₆	C₂²×C₁₃	1	4	yes	—
151	C₂×#97	C₂²×C₁₃	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
153	#341	C₂²×C₁₃	1	2	yes	—
154	#341	C₂²×C₁₃	1	4	yes	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	2	1	no	—
157	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₂²×C₁₃	#557	1	1	yes	characteristic
159	C₂²×C₁₃	Q₈²	1	1	yes	characteristic
160	C₄×C₁₃	—	2	1	no	—
161	C₄×C₁₃	—	2	4	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	1	no	—
164	C₄×C₁₃	—	2	4	no	—
165	C₄×C₁₃	#430	1	2	yes	—
166	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
167	C₄×C₁₃	#520	1	1	yes	characteristic
168	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
169	C₄×C₁₃	#550	1	1	yes	characteristic
170	C₂²×Q₈	Q₈×C₁₃	1	1	yes	characteristic
171	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
172	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
173	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
174	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
176	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₄×Q₈	C₂×C₄×C₁₃	1	2	yes	—
178	C₄×Q₈	C₂×C₄×C₁₃	1	2	yes	—
179	C₄×Q₈	D₄×C₁₃	1	2	yes	—
180	C₄×Q₈	D₄×C₁₃	1	2	yes	—
181	C₂×C₄⋊C₄	Q₈×C₁₃	1	1	yes	characteristic
182	C₂×C₄⋊C₄	Q₈×C₁₃	1	2	yes	—
183	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
185	#108	—	2	4	no	—
186	#108	—	2	2	no	—
187	#108	—	4	2	no	—
188	#108	—	4	4	no	—
189	#108	Q₈×C₁₃	1	4	yes	complemented
190	C₄⋊C₈	—	2	2	no	—
191	C₄⋊C₈	—	2	4	no	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	2	no	—
194	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
195	C₄⋊C₈	C₂×C₄×C₁₃	1	2	yes	—
196	C₄⋊C₈	C₂×C₄×C₁₃	1	8	yes	—
197	#97	—	2	4	no	—
198	#97	—	2	2	no	—
199	#97	C₂×C₄×C₁₃	1	8	yes	—
200	C₄×C₈	—	2	2	no	—
201	C₄×C₈	—	2	4	no	—
202	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
203	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
204	C₂×C₁₃	—	2	1	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	Q₈×C₄⋊C₄	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#430	1	1	yes	characteristic
208	C₂×C₁₃	#2916	1	1	yes	characteristic
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	C₁₃×C₄⋊C₄	1	2	yes	—
211	C₂×Q₈	C₁₃×C₄⋊C₄	1	2	yes	—
212	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
215	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
216	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
217	M₁₆	—	2	2	no	—
218	M₁₆	—	2	4	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	2	4	no	—
221	M₁₆	—	2	2	no	—
222	M₁₆	C₂×Q₈×C₁₃	1	4	yes	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	1	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
231	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
232	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
233	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
234	C₂×C₈	C₁₃×#41	1	2	yes	—
235	C₄⋊C₄	C₁₃×C₄⋊C₄	1	4	yes	—
236	C₄⋊C₄	C₁₃×C₄⋊C₄	1	4	yes	—
237	C₄⋊C₄	C₁₃×C₄⋊C₄	1	2	yes	—
238	C₄⋊C₄	C₁₃×C₄⋊C₄	1	2	yes	—
239	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
240	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
241	C₄²	C₂²×C₄×C₁₃	1	4	yes	—
242	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₄²	C₂×D₄×C₁₃	1	4	yes	—
244	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₁₃	#22733	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	C₂²×Q₈×C₁₃	1	1	yes	characteristic
247	Q₈	—	2	4	no	—
248	Q₈	—	4	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	1	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
254	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
255	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
256	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
257	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
258	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
259	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
260	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
263	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
264	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
265	C₈	—	2	2	no	—
266	C₈	—	2	4	no	—
267	C₈	—	2	4	no	—
268	C₈	—	4	2	no	—
269	C₈	—	4	4	no	—
270	C₈	C₄×Q₈×C₁₃	1	4	yes	—
271	C₂²	—	2	1	no	—
272	C₂²	—	2	1	no	—
273	C₂²	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
274	C₂²	C₁₃×#557	1	1	yes	characteristic
275	C₂²	Q₈²×C₁₃	1	1	yes	characteristic
276	C₄	—	2	1	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	4	no	—
281	C₄	C₁₃×#430	1	2	yes	—
282	C₄	C₂×C₄×Q₈×C₁₃	1	1	yes	characteristic
283	C₄	C₁₃×#520	1	1	yes	characteristic
284	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
285	C₄	C₁₃×#550	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	Q₈×C₁₃×C₄⋊C₄	1	1	yes	characteristic
289	C₂	C₂×C₁₃×#430	1	1	yes	characteristic
290	C₂	C₁₃×#2916	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂³×C₁₃	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
49. C₁₃×#108	Q₈	semidirect
64. #22733	C₁₃	direct (N × H)
189. #108	Q₈×C₁₃	semidirect
245. C₁₃	#22733	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×C₁₃×M₁₆	2
C₂×C₁₃×#446	2
C₁₃×#1950	2
C₁₃×#1849	2
C₁₃×#1848	2
C₁₃×#1778	2
C₄×C₁₃×#108	2
#22733	13

Subgroups — C13×#22733 #15709072

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22733 #15709072