Subgroups — C₁₃×#13200 #15699539

Order 3328.

438 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	C₁₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2117	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2112	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2110	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₁₃×#525	C₄	1	1	yes	characteristic
9	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
10	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
11	C₁₃×#435	—	2	2	no	—
12	C₈×D₄×C₁₃	—	2	2	no	—
13	C₁₃×#424	C₄	1	1	yes	characteristic
14	C₁₃×#408	C₄	1	1	yes	characteristic
15	C₁₃×#407	C₄	1	1	yes	characteristic
16	C₁₃×#405	C₂²	1	1	yes	characteristic
17	C₁₃×#364	—	2	2	no	—
18	C₁₃×#350	—	2	4	no	—
19	C₁₃×C₂²⋊C₁₆	—	2	4	no	—
20	C₁₃×#347	—	2	2	no	—
21	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₁₃×#131	—	2	2	no	—
24	C₁₃×#131	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	1	no	—
26	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×M₁₆	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	1	no	—
30	C₂²×C₈×C₁₃	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×#123	—	2	1	no	—
36	C₁₃×C₂²⋊Q₈	—	2	1	no	—
37	C₁₃×#121	—	2	1	no	—
38	C₁₃×#120	—	2	1	no	—
39	C₄×D₄×C₁₃	—	2	2	no	—
40	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
41	C₁₃×M₃₂	—	2	2	no	—
42	C₁₃×M₃₂	—	2	2	no	—
43	C₁₃×M₃₂	—	2	2	no	—
44	C₁₃×M₃₂	—	2	2	no	—
45	C₂×C₁₃×C₁₆	—	2	2	no	—
46	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
47	C₁₃×C₄⋊C₈	—	2	2	no	—
48	C₁₃×C₂²⋊C₈	—	2	2	no	—
49	C₁₃×C₂²⋊C₈	—	2	2	no	—
50	C₁₃×#97	D₄	1	2	yes	—
51	C₄×C₈×C₁₃	D₄	1	2	yes	—
52	#13200	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂⁴×C₁₃	M₁₆	1	1	yes	characteristic
54	C₁₃×#41	—	2	2	no	—
55	C₁₃×#41	—	2	2	no	—
56	C₂×Q₈×C₁₃	M₁₆	1	1	yes	characteristic
57	C₂×D₄×C₁₃	—	2	2	no	—
58	C₂×D₄×C₁₃	—	2	1	no	—
59	C₂×D₄×C₁₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	—	2	1	no	—
61	C₂²×C₄×C₁₃	—	2	1	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	1	no	—
64	C₂²×C₄×C₁₃	—	2	2	no	—
65	C₂²×C₄×C₁₃	M₁₆	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
67	C₁₃×M₁₆	—	2	2	no	—
68	C₁₃×M₁₆	—	2	2	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₂×C₈×C₁₃	—	2	1	no	—
71	C₂×C₈×C₁₃	—	2	1	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	4	2	no	—
74	C₂×C₈×C₁₃	—	4	1	no	—
75	C₂×C₈×C₁₃	—	4	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₂×C₈×C₁₃	—	4	1	no	—
78	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	#41	1	2	yes	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	1	no	—
85	C₁₃×C₂²⋊C₄	—	2	2	no	—
86	C₁₃×C₂²⋊C₄	—	2	2	no	—
87	C₁₃×C₂²⋊C₄	—	2	1	no	—
88	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
89	C₁₃×C₁₆	—	2	4	no	—
90	C₁₃×C₁₆	—	4	2	no	—
91	#2929	C₂×C₁₃	1	1	yes	characteristic
92	#2153	C₂×C₁₃	1	1	yes	characteristic, complemented
93	#2117	C₂×C₁₃	1	2	yes	complemented
94	#2112	C₂×C₁₃	1	2	yes	complemented
95	#2110	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂³×C₁₃	—	2	1	no	—
97	C₂³×C₁₃	—	2	2	no	—
98	C₂³×C₁₃	—	2	1	no	—
99	C₂³×C₁₃	—	4	2	no	—
100	C₂³×C₁₃	C₂×M₁₆	1	1	yes	characteristic
101	Q₈×C₁₃	—	2	2	no	—
102	D₄×C₁₃	—	2	2	no	—
103	D₄×C₁₃	—	4	4	no	—
104	D₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	1	no	—
107	C₂×C₄×C₁₃	—	2	1	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	4	1	no	—
110	C₂×C₄×C₁₃	—	4	1	no	—
111	C₂×C₄×C₁₃	—	4	1	no	—
112	C₂×C₄×C₁₃	—	4	1	no	—
113	C₂×C₄×C₁₃	—	4	2	no	—
114	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#131	1	2	yes	—
118	C₈×C₁₃	—	2	2	no	—
119	C₈×C₁₃	—	4	1	no	—
120	C₈×C₁₃	—	4	1	no	—
121	C₈×C₁₃	#121	1	2	yes	—
122	C₈×C₁₃	#124	1	1	yes	characteristic
123	C₈×C₁₃	#127	1	1	yes	characteristic
124	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
125	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
126	#525	C₄×C₁₃	1	1	yes	characteristic
127	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
128	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
129	#435	—	2	2	no	—
130	C₈×D₄	—	2	2	no	—
131	#424	C₄×C₁₃	1	1	yes	characteristic
132	#408	C₄×C₁₃	1	1	yes	characteristic
133	#407	C₄×C₁₃	1	1	yes	characteristic
134	#405	C₂²×C₁₃	1	1	yes	characteristic
135	#364	—	2	2	no	—
136	#350	—	2	4	no	—
137	C₂²⋊C₁₆	—	2	4	no	—
138	#347	—	2	2	no	—
139	C₂²×C₁₃	—	2	2	no	—
140	C₂²×C₁₃	—	2	4	no	—
141	C₂²×C₁₃	—	4	1	no	—
142	C₂²×C₁₃	—	4	1	no	—
143	C₂²×C₁₃	—	4	2	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	4	1	no	—
146	C₂²×C₁₃	—	4	1	no	—
147	C₂²×C₁₃	#407	1	1	yes	characteristic
148	C₂²×C₁₃	#435	1	2	yes	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	—	4	1	no	—
151	C₄×C₁₃	—	4	1	no	—
152	C₄×C₁₃	#390	1	1	yes	characteristic
153	C₄×C₁₃	#408	1	1	yes	characteristic
154	C₄×C₁₃	#435	1	2	yes	—
155	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
156	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
157	#131	—	2	2	no	—
158	#131	—	2	2	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
161	C₂×M₁₆	C₂×C₄×C₁₃	1	1	yes	characteristic
162	C₂²×C₈	—	2	2	no	—
163	C₂²×C₈	—	2	1	no	—
164	C₂²×C₈	—	2	1	no	—
165	C₂²×C₈	—	2	1	no	—
166	C₂²×C₈	—	2	2	no	—
167	C₂²×C₈	C₂×C₄×C₁₃	1	1	yes	characteristic
168	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
169	#123	—	2	1	no	—
170	C₂²⋊Q₈	—	2	1	no	—
171	#121	—	2	1	no	—
172	#120	—	2	1	no	—
173	C₄×D₄	—	2	2	no	—
174	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
175	M₃₂	—	2	2	no	—
176	M₃₂	—	2	2	no	—
177	M₃₂	—	2	2	no	—
178	M₃₂	—	2	2	no	—
179	C₂×C₁₆	—	2	2	no	—
180	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
181	C₄⋊C₈	—	2	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	#97	D₄×C₁₃	1	2	yes	—
185	C₄×C₈	D₄×C₁₃	1	2	yes	—
186	C₂×C₁₃	—	2	1	no	—
187	C₂×C₁₃	—	2	2	no	—
188	C₂×C₁₃	—	4	2	no	—
189	C₂×C₁₃	—	8	1	no	—
190	C₂×C₁₃	#1955	1	1	yes	characteristic
191	C₂⁴	C₁₃×M₁₆	1	1	yes	characteristic
192	#41	—	2	2	no	—
193	#41	—	2	2	no	—
194	C₂×Q₈	C₁₃×M₁₆	1	1	yes	characteristic
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	2	1	no	—
197	C₂×D₄	C₁₃×M₁₆	1	1	yes	characteristic
198	C₂²×C₄	—	2	1	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₁₃×M₁₆	1	1	yes	characteristic
204	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
205	M₁₆	—	2	2	no	—
206	M₁₆	—	2	2	no	—
207	M₁₆	—	4	2	no	—
208	C₂×C₈	—	2	1	no	—
209	C₂×C₈	—	2	1	no	—
210	C₂×C₈	—	2	2	no	—
211	C₂×C₈	—	4	2	no	—
212	C₂×C₈	—	4	1	no	—
213	C₂×C₈	—	4	1	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
217	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
219	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
220	C₂×C₈	C₁₃×#41	1	2	yes	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	2	1	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	1	no	—
226	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
227	C₁₆	—	2	4	no	—
228	C₁₆	—	4	2	no	—
229	C₁₃	#13200	1	1	yes	characteristic, direct product factor, Sylow
230	C₂³	—	2	1	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	4	2	no	—
234	C₂³	C₂×C₁₃×M₁₆	1	1	yes	characteristic
235	Q₈	—	2	2	no	—
236	D₄	—	2	2	no	—
237	D₄	—	4	4	no	—
238	D₄	—	4	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	4	1	no	—
244	C₂×C₄	—	4	1	no	—
245	C₂×C₄	—	4	1	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	4	2	no	—
248	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
249	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
250	C₂×C₄	C₂×C₁₃×M₁₆	1	1	yes	characteristic
251	C₂×C₄	C₁₃×#131	1	2	yes	—
252	C₈	—	2	2	no	—
253	C₈	—	4	1	no	—
254	C₈	—	4	1	no	—
255	C₈	C₁₃×#121	1	2	yes	—
256	C₈	C₁₃×#124	1	1	yes	characteristic
257	C₈	C₁₃×#127	1	1	yes	characteristic
258	C₂²	—	2	2	no	—
259	C₂²	—	2	4	no	—
260	C₂²	—	4	1	no	—
261	C₂²	—	4	1	no	—
262	C₂²	—	4	2	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	C₁₃×#407	1	1	yes	characteristic
267	C₂²	C₁₃×#435	1	2	yes	—
268	C₄	—	4	2	no	—
269	C₄	—	4	1	no	—
270	C₄	—	4	1	no	—
271	C₄	C₁₃×#390	1	1	yes	characteristic
272	C₄	C₁₃×#408	1	1	yes	characteristic
273	C₄	C₁₃×#435	1	2	yes	—
274	C₂	—	2	1	no	—
275	C₂	—	2	2	no	—
276	C₂	—	4	2	no	—
277	C₂	—	8	1	no	—
278	C₂	C₁₃×#1955	1	1	yes	characteristic

	subgroup	G/H
center	C₈×C₁₃	#127
commutator	C₂²×C₄	C₂²×C₄×C₁₃
Frattini, cosocle	C₂²×C₈	C₂³×C₁₃
socle	C₂×C₁₃	#1955
prime radical	C₁₃×#525	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₃ ◁ C₂²×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2153	C₂	semidirect
3. C₁₃×#2117	C₂	semidirect
4. C₁₃×#2112	C₂	semidirect
5. C₁₃×#2110	C₂	semidirect
46. C₂×C₁₃×C₁₆	D₄	semidirect
52. #13200	C₁₃	direct (N × H)
92. #2153	C₂×C₁₃	semidirect
93. #2117	C₂×C₁₃	semidirect
94. #2112	C₂×C₁₃	semidirect
95. #2110	C₂×C₁₃	semidirect
180. C₂×C₁₆	D₄×C₁₃	semidirect
229. C₁₃	#13200	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#2929	2
C₁₃×#2153	2
C₁₃×#2117	2
C₁₃×#2112	2
C₁₃×#2110	2
#13200	13

Subgroups — C13×#13200 #15699539

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#13200 #15699539