Subgroups — C₁₃×#33610 #15719949

Order 3328.

674 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3061	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2220	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2216	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2215	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2139	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
11	C₁₃×#537	—	2	2	no	—
12	C₁₃×#508	—	2	4	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#504	—	2	4	no	—
17	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
19	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
20	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
21	C₁₃×#468	—	2	4	no	—
22	C₁₃×#467	C₂²	1	4	yes	—
23	C₁₃×#466	C₂²	1	4	yes	—
24	C₂×C₁₃×#107	C₂²	1	2	yes	—
25	C₁₃×#417	—	2	2	no	—
26	C₁₃×#366	C₂²	1	2	yes	—
27	C₁₃×#366	C₂²	1	2	yes	—
28	C₁₃×#358	C₂²	1	4	yes	—
29	C₁₃×#357	C₂²	1	4	yes	—
30	C₂×C₁₃×#41	—	2	2	no	—
31	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
32	C₁₃×#135	—	2	4	no	—
33	C₁₃×#135	—	2	4	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×D₈	C₂³	1	2	yes	—
37	C₁₃×#131	—	2	2	no	—
38	C₁₃×#131	—	2	2	no	—
39	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
40	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×C₂²⋊Q₈	—	2	4	no	—
44	C₁₃×C₂²⋊Q₈	—	4	2	no	—
45	C₁₃×#121	—	2	4	no	—
46	C₁₃×#121	—	4	2	no	—
47	C₁₃×#121	—	4	4	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₁₃×Q₃₂	—	2	2	no	—
50	C₁₃×SD₃₂	—	2	4	no	—
51	C₁₃×D₁₆	—	2	2	no	—
52	C₁₃×M₃₂	—	2	2	no	—
53	C₁₃×M₃₂	—	2	4	no	—
54	C₂×C₁₃×C₁₆	—	2	4	no	—
55	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×#107	—	2	2	no	—
60	C₁₃×#107	C₂³	1	4	yes	—
61	C₁₃×#103	—	2	4	no	—
62	C₁₃×#102	—	2	4	no	—
63	#33610	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₁₃×#41	—	2	2	no	—
65	C₁₃×#41	—	2	2	no	—
66	C₁₃×#41	—	4	4	no	—
67	C₁₃×#41	—	4	4	no	—
68	C₂×Q₈×C₁₃	—	2	2	no	—
69	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
70	C₂×D₄×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	4	2	no	—
72	C₂×D₄×C₁₃	—	4	4	no	—
73	C₂×D₄×C₁₃	D₈	1	2	yes	—
74	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	—	4	2	no	—
76	C₂²×C₄×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×Q₁₆	—	2	2	no	—
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×SD₁₆	—	4	4	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₁₃×M₁₆	—	2	2	no	—
85	C₁₃×M₁₆	—	2	4	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₁₃	#41	1	2	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
93	C₁₃×C₄⋊C₄	—	2	4	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	4	no	—
96	C₁₃×C₂²⋊C₄	—	4	4	no	—
97	C₁₃×C₂²⋊C₄	—	4	4	no	—
98	C₁₃×C₁₆	—	2	2	no	—
99	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
100	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
101	C₂×#508	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂×#504	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#3061	C₂×C₁₃	1	2	yes	complemented
104	#2220	C₂×C₁₃	1	4	yes	complemented
105	#2216	C₂×C₁₃	1	2	yes	complemented
106	#2215	C₂×C₁₃	1	2	yes	complemented
107	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#2139	C₂×C₁₃	1	2	yes	complemented
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	C₂×D₈	1	2	yes	—
111	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₁₃	—	2	2	no	—
113	Q₈×C₁₃	—	4	2	no	—
114	Q₈×C₁₃	—	4	2	no	—
115	D₄×C₁₃	—	2	4	no	—
116	D₄×C₁₃	—	2	2	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	2	no	—
119	D₄×C₁₃	—	8	4	no	—
120	D₄×C₁₃	—	16	4	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	8	4	no	—
126	C₂×C₄×C₁₃	—	16	2	no	—
127	C₂×C₄×C₁₃	#121	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#135	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	#121	1	2	yes	—
134	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₂×#135	C₂²×C₁₃	1	2	yes	—
137	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
138	#537	—	2	2	no	—
139	#508	—	2	4	no	—
140	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
141	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
142	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
143	#504	—	2	4	no	—
144	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
146	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
147	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
148	#468	—	2	4	no	—
149	#467	C₂²×C₁₃	1	4	yes	—
150	#466	C₂²×C₁₃	1	4	yes	—
151	C₂×#107	C₂²×C₁₃	1	2	yes	—
152	#417	—	2	2	no	—
153	#366	C₂²×C₁₃	1	2	yes	—
154	#366	C₂²×C₁₃	1	2	yes	—
155	#358	C₂²×C₁₃	1	4	yes	—
156	#357	C₂²×C₁₃	1	4	yes	—
157	C₂²×C₁₃	—	2	1	no	—
158	C₂²×C₁₃	—	2	1	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	2	2	no	—
161	C₂²×C₁₃	—	2	2	no	—
162	C₂²×C₁₃	—	8	2	no	—
163	C₂²×C₁₃	—	8	2	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	#466	1	2	yes	—
166	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	8	4	no	—
169	C₄×C₁₃	—	8	2	no	—
170	C₄×C₁₃	#466	1	2	yes	—
171	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
172	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
173	C₂×#41	—	2	2	no	—
174	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
175	#135	—	2	4	no	—
176	#135	—	2	4	no	—
177	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
178	C₂×SD₁₆	—	2	2	no	—
179	C₂×D₈	C₂³×C₁₃	1	2	yes	—
180	#131	—	2	2	no	—
181	#131	—	2	2	no	—
182	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
183	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
184	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
186	C₂²⋊Q₈	—	2	4	no	—
187	C₂²⋊Q₈	—	4	2	no	—
188	#121	—	2	4	no	—
189	#121	—	4	2	no	—
190	#121	—	4	4	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	Q₃₂	—	2	2	no	—
193	SD₃₂	—	2	4	no	—
194	D₁₆	—	2	2	no	—
195	M₃₂	—	2	2	no	—
196	M₃₂	—	2	4	no	—
197	C₂×C₁₆	—	2	4	no	—
198	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
199	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
200	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
201	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
202	#107	—	2	2	no	—
203	#107	C₂³×C₁₃	1	4	yes	—
204	#103	—	2	4	no	—
205	#102	—	2	4	no	—
206	C₂×C₁₃	—	2	2	no	—
207	C₂×C₁₃	—	4	2	no	—
208	C₂×C₁₃	—	16	2	no	—
209	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
210	C₂×C₁₃	#3415	1	1	yes	characteristic
211	C₂×C₁₃	#3417	1	1	yes	characteristic
212	#41	—	2	2	no	—
213	#41	—	2	2	no	—
214	#41	—	4	4	no	—
215	#41	—	4	4	no	—
216	C₂×Q₈	—	2	2	no	—
217	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
218	C₂×D₄	—	2	2	no	—
219	C₂×D₄	—	4	2	no	—
220	C₂×D₄	—	4	4	no	—
221	C₂×D₄	C₁₃×D₈	1	2	yes	—
222	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	—	4	2	no	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
227	Q₁₆	—	2	2	no	—
228	Q₁₆	—	2	2	no	—
229	SD₁₆	—	4	4	no	—
230	D₈	—	2	2	no	—
231	D₈	—	2	2	no	—
232	M₁₆	—	2	2	no	—
233	M₁₆	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
239	C₂×C₈	C₁₃×#41	1	2	yes	—
240	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	4	2	no	—
243	C₄⋊C₄	—	4	4	no	—
244	C₂²⋊C₄	—	4	4	no	—
245	C₂²⋊C₄	—	4	4	no	—
246	C₁₆	—	2	2	no	—
247	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
248	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
249	C₁₃	#33610	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	4	2	no	—
251	C₂³	C₂×C₁₃×D₈	1	2	yes	—
252	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
253	Q₈	—	2	2	no	—
254	Q₈	—	4	2	no	—
255	Q₈	—	4	2	no	—
256	D₄	—	2	4	no	—
257	D₄	—	2	2	no	—
258	D₄	—	4	2	no	—
259	D₄	—	4	2	no	—
260	D₄	—	8	4	no	—
261	D₄	—	16	4	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	8	4	no	—
267	C₂×C₄	—	16	2	no	—
268	C₂×C₄	C₁₃×#121	1	2	yes	—
269	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
270	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
271	C₂×C₄	C₁₃×#135	1	2	yes	—
272	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	C₈	—	2	2	no	—
274	C₈	C₁₃×#121	1	2	yes	—
275	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
276	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	8	2	no	—
283	C₂²	—	8	2	no	—
284	C₂²	—	8	2	no	—
285	C₂²	C₁₃×#466	1	2	yes	—
286	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
287	C₄	—	2	2	no	—
288	C₄	—	8	2	no	—
289	C₄	—	8	4	no	—
290	C₄	C₁₃×#466	1	2	yes	—
291	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
292	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
293	C₂	—	2	2	no	—
294	C₂	—	4	2	no	—
295	C₂	—	16	2	no	—
296	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
297	C₂	C₁₃×#3415	1	1	yes	characteristic
298	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#508	C₂	semidirect
2. C₂×C₁₃×#504	C₂	semidirect
3. C₁₃×#3061	C₂	semidirect
4. C₁₃×#2220	C₂	semidirect
5. C₁₃×#2216	C₂	semidirect
6. C₁₃×#2215	C₂	semidirect
7. C₂×C₁₃×#366	C₂	semidirect
8. C₁₃×#2139	C₂	semidirect
55. C₂×C₁₃×C₁₆	D₄	semidirect
63. #33610	C₁₃	direct (N × H)
101. C₂×#508	C₂×C₁₃	semidirect
102. C₂×#504	C₂×C₁₃	semidirect
103. #3061	C₂×C₁₃	semidirect
104. #2220	C₂×C₁₃	semidirect
105. #2216	C₂×C₁₃	semidirect
106. #2215	C₂×C₁₃	semidirect
107. C₂×#366	C₂×C₁₃	semidirect
108. #2139	C₂×C₁₃	semidirect
198. C₂×C₁₆	D₄×C₁₃	semidirect
249. C₁₃	#33610	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×#508	2
C₂×C₁₃×#504	2
C₁₃×#3061	2
C₁₃×#2220	2
C₁₃×#2216	2
C₁₃×#2215	2
C₂×C₁₃×#366	2
C₁₃×#2139	2
#33610	13

Subgroups — C13×#33610 #15719949

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33610 #15719949