Subgroups — C₁₃×#22618 #15708957

Order 3328.

810 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2345	C₂	1	2	yes	maximal
3	C₁₃×#1943	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1918	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1824	C₂	1	2	yes	maximal
6	C₁₃×#1807	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1776	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1752	C₂	1	1	yes	maximal, characteristic
9	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	2	yes	—
14	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×#106	C₂²	1	2	yes	—
16	C₂×C₁₃×C₂²⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#97	C₂²	1	2	yes	—
18	C₁₃×#389	C₂²	1	4	yes	—
19	C₁₃×#387	—	2	4	no	—
20	C₁₃×#380	—	2	2	no	—
21	C₁₃×#380	C₂²	1	2	yes	—
22	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
23	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
24	C₄×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₁₃×#340	C₂²	1	4	yes	—
26	C₁₃×#340	C₂²	1	2	yes	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂³×C₄×C₁₃	—	2	2	no	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	—	2	4	no	—
32	C₂²×C₈×C₁₃	C₂×C₄	1	4	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	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
37	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
39	C₂×C₁₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
45	C₂×C₄²×C₁₃	—	2	4	no	—
46	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
47	C₁₃×#107	C₂×C₄	1	8	yes	—
48	C₁₃×#107	C₂×C₄	1	4	yes	—
49	C₁₃×#107	C₂×C₄	1	4	yes	—
50	C₁₃×#106	—	2	4	no	—
51	C₁₃×C₂²⋊C₈	—	2	2	no	—
52	C₁₃×#97	—	2	4	no	—
53	C₁₃×#95	—	2	4	no	—
54	C₁₃×#95	—	2	4	no	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	D₄	1	2	yes	—
57	#22618	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	—	2	2	no	—
60	C₂²×C₄×C₁₃	—	2	2	no	—
61	C₂²×C₄×C₁₃	—	2	2	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	8	no	—
64	C₂²×C₄×C₁₃	—	2	4	no	—
65	C₂²×C₄×C₁₃	—	8	4	no	—
66	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
68	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₁₃	#41	1	2	yes	—
70	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	8	no	—
73	C₂×C₈×C₁₃	—	2	2	no	—
74	C₂×C₈×C₁₃	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
78	C₂×C₈×C₁₃	#41	1	4	yes	—
79	C₁₃×C₄⋊C₄	—	2	8	no	—
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	2	4	no	—
84	C₁₃×C₄⋊C₄	—	8	4	no	—
85	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
86	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
87	C₁₃×C₂²⋊C₄	—	2	4	no	—
88	C₁₃×C₂²⋊C₄	—	4	4	no	—
89	C₄²×C₁₃	—	2	2	no	—
90	C₄²×C₁₃	—	2	2	no	—
91	C₄²×C₁₃	—	8	4	no	—
92	C₂²×#107	C₂×C₁₃	1	1	yes	characteristic
93	#2345	C₂×C₁₃	1	2	yes	—
94	#1943	C₂×C₁₃	1	2	yes	complemented
95	#1918	C₂×C₁₃	1	4	yes	complemented
96	#1824	C₂×C₁₃	1	2	yes	—
97	#1807	C₂×C₁₃	1	1	yes	characteristic
98	#1776	C₂×C₁₃	1	2	yes	complemented
99	#1752	C₂×C₁₃	1	1	yes	characteristic
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	C₄×D₄	1	2	yes	—
103	C₂³×C₁₃	#137	1	4	yes	—
104	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
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	—	2	4	no	—
110	C₂×C₄×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	2	4	no	—
113	C₂×C₄×C₁₃	—	2	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	—	4	4	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	8	8	no	—
120	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
121	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
124	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
125	C₈×C₁₃	—	2	2	no	—
126	C₈×C₁₃	—	4	2	no	—
127	C₈×C₁₃	#117	1	4	yes	—
128	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
129	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
130	C₂×#107	C₄×C₁₃	1	8	yes	complemented
131	C₂×#107	C₂²×C₁₃	1	2	yes	—
132	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
134	C₂×#106	C₂²×C₁₃	1	2	yes	—
135	C₂×C₂²⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
136	C₂×#97	C₂²×C₁₃	1	2	yes	—
137	#389	C₂²×C₁₃	1	4	yes	—
138	#387	—	2	4	no	—
139	#380	—	2	2	no	—
140	#380	C₂²×C₁₃	1	2	yes	—
141	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
142	C₄×C₂²⋊C₄	—	2	2	no	—
143	C₄×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
144	#340	C₂²×C₁₃	1	2	yes	—
145	#340	C₂²×C₁₃	1	4	yes	—
146	#340	C₂²×C₁₃	1	4	yes	—
147	#336	C₂²×C₁₃	1	2	yes	—
148	C₂²×C₁₃	—	2	8	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	2	no	—
151	C₂²×C₁₃	#441	1	4	yes	—
152	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
153	C₂²×C₁₃	#547	1	2	yes	—
154	C₂²×C₁₃	C₂×#137	1	2	yes	—
155	C₂²×C₁₃	#576	1	2	yes	—
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	4	4	no	—
158	C₄×C₁₃	—	4	4	no	—
159	C₄×C₁₃	—	4	4	no	—
160	C₄×C₁₃	—	8	4	no	—
161	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
162	C₄×C₁₃	#519	1	1	yes	characteristic
163	C₄×C₁₃	#552	1	2	yes	—
164	C₂³×C₄	—	2	2	no	—
165	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
166	C₂²×C₈	—	2	4	no	—
167	C₂²×C₈	C₂×C₄×C₁₃	1	4	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	C₂×C₄⋊C₄	—	2	2	no	—
172	C₂×C₄⋊C₄	—	2	4	no	—
173	C₂×C₄⋊C₄	—	2	4	no	—
174	C₂×C₄⋊C₄	C₂×C₄×C₁₃	1	8	yes	—
175	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
176	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
177	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
178	C₂×C₂²⋊C₄	—	2	2	no	—
179	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
180	C₂×C₄²	—	2	4	no	—
181	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
182	#107	C₂×C₄×C₁₃	1	8	yes	—
183	#107	C₂×C₄×C₁₃	1	4	yes	—
184	#107	C₂×C₄×C₁₃	1	4	yes	—
185	#106	—	2	4	no	—
186	C₂²⋊C₈	—	2	2	no	—
187	#97	—	2	4	no	—
188	#95	—	2	4	no	—
189	#95	—	2	4	no	—
190	#95	—	2	2	no	—
191	#95	D₄×C₁₃	1	2	yes	—
192	C₂×C₁₃	—	2	4	no	—
193	C₂×C₁₃	#2344	1	1	yes	characteristic
194	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
195	C₂×C₁₃	#2948	1	1	yes	characteristic
196	C₂×C₁₃	#3346	1	4	yes	—
197	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	2	2	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	2	no	—
202	C₂²×C₄	—	2	8	no	—
203	C₂²×C₄	—	2	4	no	—
204	C₂²×C₄	—	8	4	no	—
205	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
206	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
207	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₁₃×#41	1	2	yes	—
209	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	2	8	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
216	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
217	C₂×C₈	C₁₃×#41	1	4	yes	—
218	C₄⋊C₄	—	2	8	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	8	4	no	—
224	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
225	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	4	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	8	4	no	—
231	C₁₃	#22618	1	1	yes	characteristic, direct product factor, Sylow
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	C₄×D₄×C₁₃	1	2	yes	—
235	C₂³	C₁₃×#137	1	4	yes	—
236	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	4	no	—
242	C₂×C₄	—	2	4	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	4	no	—
246	C₂×C₄	—	4	4	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	4	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	—	8	8	no	—
252	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
253	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
255	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
256	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
257	C₈	—	2	2	no	—
258	C₈	—	4	2	no	—
259	C₈	C₁₃×#117	1	4	yes	—
260	C₂²	—	2	8	no	—
261	C₂²	—	2	2	no	—
262	C₂²	—	2	2	no	—
263	C₂²	C₁₃×#441	1	4	yes	—
264	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
265	C₂²	C₁₃×#547	1	2	yes	—
266	C₂²	C₂×C₁₃×#137	1	2	yes	—
267	C₂²	C₁₃×#576	1	2	yes	—
268	C₄	—	2	2	no	—
269	C₄	—	4	4	no	—
270	C₄	—	4	4	no	—
271	C₄	—	4	4	no	—
272	C₄	—	8	4	no	—
273	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
274	C₄	C₁₃×#519	1	1	yes	characteristic
275	C₄	C₁₃×#552	1	2	yes	—
276	C₂	—	2	4	no	—
277	C₂	C₁₃×#2344	1	1	yes	characteristic
278	C₂	C₂×C₁₃×#441	1	1	yes	characteristic
279	C₂	C₁₃×#2948	1	1	yes	characteristic
280	C₂	C₁₃×#3346	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1943	C₂	semidirect
4. C₁₃×#1918	C₂	semidirect
7. C₁₃×#1776	C₂	semidirect
11. C₂×C₁₃×#107	C₄	semidirect
57. #22618	C₁₃	direct (N × H)
94. #1943	C₂×C₁₃	semidirect
95. #1918	C₂×C₁₃	semidirect
98. #1776	C₂×C₁₃	semidirect
130. C₂×#107	C₄×C₁₃	semidirect
231. C₁₃	#22618	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×#107	2
C₁₃×#2345	2
C₁₃×#1943	2
C₁₃×#1918	2
C₁₃×#1824	2
C₁₃×#1807	2
C₁₃×#1776	2
C₁₃×#1752	2
#22618	13

Subgroups — C13×#22618 #15708957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22618 #15708957