Subgroups — C₁₃×#33922 #15720261

Order 3328.

488 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3339	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2234	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2232	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2218	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2159	C₂	1	2	yes	maximal, complemented
10	C₁₃×#553	—	2	2	no	—
11	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#499	C₂²	1	2	yes	—
14	C₁₃×#482	—	2	2	no	—
15	C₁₃×#481	—	2	2	no	—
16	C₁₃×#478	—	2	2	no	—
17	C₁₃×#466	C₂²	1	2	yes	—
18	C₁₃×#466	C₂²	1	2	yes	—
19	C₄×C₁₃×D₈	C₂²	1	2	yes	—
20	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
21	C₁₃×#427	C₂²	1	2	yes	—
22	C₂×C₁₃×#107	C₂²	1	2	yes	—
23	C₁₃×#367	C₂²	1	2	yes	—
24	C₁₃×#367	C₂²	1	2	yes	—
25	C₁₃×#366	C₂²	1	1	yes	characteristic
26	C₁₃×#366	C₂²	1	1	yes	characteristic
27	C₁₃×#366	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	2	yes	—
29	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
30	C₁₃×#357	C₂²	1	2	yes	—
31	C₁₃×#357	C₂²	1	2	yes	—
32	C₁₃×#357	C₂²	1	2	yes	—
33	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
35	C₂×C₁₃×D₈	C₂³	1	2	yes	—
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₁₃×#126	—	4	2	no	—
38	C₁₃×#125	—	2	2	no	—
39	C₁₃×#123	—	4	2	no	—
40	C₁₃×#121	—	2	2	no	—
41	C₁₃×#121	—	2	2	no	—
42	C₁₃×#121	—	4	2	no	—
43	C₄×D₄×C₁₃	—	2	2	no	—
44	C₄×D₄×C₁₃	—	4	2	no	—
45	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
46	C₁₃×#117	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₁₃×D₁₆	—	2	2	no	—
49	C₁₃×M₃₂	—	2	4	no	—
50	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
51	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
52	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
53	C₁₃×#107	—	2	2	no	—
54	C₁₃×#107	C₂³	1	2	yes	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#107	C₂³	1	2	yes	—
57	C₁₃×#106	—	2	2	no	—
58	C₁₃×#106	—	2	2	no	—
59	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
60	C₁₃×#102	—	2	2	no	—
61	C₁₃×#102	—	2	2	no	—
62	C₁₃×#102	—	2	2	no	—
63	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	#33922	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	4	2	no	—
68	C₂×D₄×C₁₃	—	4	2	no	—
69	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	4	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₁₃×D₈	—	2	2	no	—
74	C₁₃×D₈	—	2	2	no	—
75	C₂×C₈×C₁₃	—	2	2	no	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
80	C₁₃×C₄⋊C₄	—	2	2	no	—
81	C₁₃×C₄⋊C₄	—	2	2	no	—
82	C₁₃×C₄⋊C₄	—	2	2	no	—
83	C₁₃×C₄⋊C₄	—	4	2	no	—
84	C₁₃×C₄⋊C₄	—	4	2	no	—
85	C₁₃×C₄⋊C₄	—	4	2	no	—
86	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
87	C₁₃×C₂²⋊C₄	—	4	2	no	—
88	C₁₃×C₂²⋊C₄	—	4	2	no	—
89	C₁₃×C₂²⋊C₄	—	4	2	no	—
90	C₁₃×C₂²⋊C₄	—	4	2	no	—
91	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
92	C₄²×C₁₃	—	4	2	no	—
93	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₁₃×C₁₆	—	2	2	no	—
95	C₁₃×C₁₆	—	2	1	no	—
96	C₁₃×C₁₆	#41	1	2	yes	—
97	#3339	C₂×C₁₃	1	2	yes	complemented
98	#2253	C₂×C₁₃	1	1	yes	characteristic, complemented
99	#2234	C₂×C₁₃	1	2	yes	complemented
100	#2232	C₂×C₁₃	1	2	yes	complemented
101	#2228	C₂×C₁₃	1	2	yes	complemented
102	#2218	C₂×C₁₃	1	2	yes	complemented
103	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#2169	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#2159	C₂×C₁₃	1	2	yes	complemented
106	C₂³×C₁₃	—	4	2	no	—
107	C₂³×C₁₃	C₂×D₈	1	2	yes	—
108	D₄×C₁₃	—	4	1	no	—
109	D₄×C₁₃	—	4	2	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	8	4	no	—
112	D₄×C₁₃	—	16	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	4	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	16	2	no	—
119	C₂×C₄×C₁₃	—	16	2	no	—
120	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
122	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	C₈×C₁₃	—	2	2	no	—
125	C₈×C₁₃	—	2	1	no	—
126	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
127	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
128	#553	—	2	2	no	—
129	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
131	#499	C₂²×C₁₃	1	2	yes	—
132	#482	—	2	2	no	—
133	#481	—	2	2	no	—
134	#478	—	2	2	no	—
135	#466	C₂²×C₁₃	1	2	yes	—
136	#466	C₂²×C₁₃	1	2	yes	—
137	C₄×D₈	C₂²×C₁₃	1	2	yes	—
138	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
139	#427	C₂²×C₁₃	1	2	yes	—
140	C₂×#107	C₂²×C₁₃	1	2	yes	—
141	#367	C₂²×C₁₃	1	2	yes	—
142	#367	C₂²×C₁₃	1	2	yes	—
143	#366	C₂²×C₁₃	1	1	yes	characteristic
144	#366	C₂²×C₁₃	1	2	yes	—
145	#366	C₂²×C₁₃	1	1	yes	characteristic
146	#366	C₂²×C₁₃	1	1	yes	characteristic
147	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
148	#357	C₂²×C₁₃	1	2	yes	—
149	#357	C₂²×C₁₃	1	2	yes	—
150	#357	C₂²×C₁₃	1	2	yes	—
151	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
152	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	2	2	no	—
155	C₂²×C₁₃	—	2	2	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	8	2	no	—
159	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	—	8	2	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	#547	1	1	yes	characteristic
167	C₄×C₁₃	#578	1	1	yes	characteristic
168	C₂×D₈	C₂³×C₁₃	1	2	yes	—
169	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
170	#126	—	4	2	no	—
171	#125	—	2	2	no	—
172	#123	—	4	2	no	—
173	#121	—	2	2	no	—
174	#121	—	2	2	no	—
175	#121	—	4	2	no	—
176	C₄×D₄	—	2	2	no	—
177	C₄×D₄	—	4	2	no	—
178	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
179	#117	—	2	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	D₁₆	—	2	2	no	—
182	M₃₂	—	2	4	no	—
183	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
185	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
186	#107	—	2	2	no	—
187	#107	C₂³×C₁₃	1	2	yes	—
188	#107	C₂³×C₁₃	1	2	yes	—
189	#107	C₂³×C₁₃	1	2	yes	—
190	#106	—	2	2	no	—
191	#106	—	2	2	no	—
192	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
193	#102	—	2	2	no	—
194	#102	—	2	2	no	—
195	#102	—	2	2	no	—
196	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
197	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
198	C₂×C₁₃	—	4	2	no	—
199	C₂×C₁₃	—	16	2	no	—
200	C₂×C₁₃	#3334	1	1	yes	characteristic
201	C₂×C₁₃	#3414	1	1	yes	characteristic
202	C₂×C₁₃	#3415	1	1	yes	characteristic
203	C₂×D₄	—	2	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	4	2	no	—
209	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
210	D₈	—	2	2	no	—
211	D₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
216	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
217	C₄⋊C₄	—	2	2	no	—
218	C₄⋊C₄	—	2	2	no	—
219	C₄⋊C₄	—	2	2	no	—
220	C₄⋊C₄	—	4	2	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
224	C₂²⋊C₄	—	4	2	no	—
225	C₂²⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
229	C₄²	—	4	2	no	—
230	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₁₆	—	2	2	no	—
232	C₁₆	—	2	1	no	—
233	C₁₆	C₁₃×#41	1	2	yes	—
234	C₁₃	#33922	1	1	yes	characteristic, direct product factor, Sylow
235	C₂³	—	4	2	no	—
236	C₂³	C₂×C₁₃×D₈	1	2	yes	—
237	D₄	—	4	1	no	—
238	D₄	—	4	2	no	—
239	D₄	—	4	2	no	—
240	D₄	—	8	4	no	—
241	D₄	—	16	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	—	16	2	no	—
248	C₂×C₄	—	16	2	no	—
249	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
250	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
251	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
252	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
253	C₈	—	2	2	no	—
254	C₈	—	2	1	no	—
255	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
256	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
257	C₂²	—	2	2	no	—
258	C₂²	—	2	2	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
264	C₄	—	2	2	no	—
265	C₄	—	2	1	no	—
266	C₄	—	2	1	no	—
267	C₄	—	8	2	no	—
268	C₄	—	8	2	no	—
269	C₄	—	8	2	no	—
270	C₄	C₁₃×#547	1	1	yes	characteristic
271	C₄	C₁₃×#578	1	1	yes	characteristic
272	C₂	—	4	2	no	—
273	C₂	—	16	2	no	—
274	C₂	C₁₃×#3334	1	1	yes	characteristic
275	C₂	C₁₃×#3414	1	1	yes	characteristic
276	C₂	C₁₃×#3415	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 ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3339	C₂	semidirect
2. C₁₃×#2253	C₂	semidirect
3. C₁₃×#2234	C₂	semidirect
4. C₁₃×#2232	C₂	semidirect
5. C₁₃×#2228	C₂	semidirect
6. C₁₃×#2218	C₂	semidirect
7. C₄×C₁₃×D₁₆	C₂	semidirect
8. C₁₃×#2169	C₂	semidirect
9. C₁₃×#2159	C₂	semidirect
65. #33922	C₁₃	direct (N × H)
97. #3339	C₂×C₁₃	semidirect
98. #2253	C₂×C₁₃	semidirect
99. #2234	C₂×C₁₃	semidirect
100. #2232	C₂×C₁₃	semidirect
101. #2228	C₂×C₁₃	semidirect
102. #2218	C₂×C₁₃	semidirect
103. C₄×D₁₆	C₂×C₁₃	semidirect
104. #2169	C₂×C₁₃	semidirect
105. #2159	C₂×C₁₃	semidirect
234. C₁₃	#33922	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×#3339	2
C₁₃×#2253	2
C₁₃×#2234	2
C₁₃×#2232	2
C₁₃×#2228	2
C₁₃×#2218	2
C₄×C₁₃×D₁₆	2
C₁₃×#2169	2
C₁₃×#2159	2
#33922	13

Subgroups — C13×#33922 #15720261

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33922 #15720261