Subgroups — #13123888

Order 1664.

422 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11719533	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#408	C₂	1	1	yes	maximal, characteristic, complemented
3	#11718895	C₂	1	1	yes	maximal, characteristic, complemented
4	#11718788	C₂	1	2	yes	maximal, complemented
5	C₈×C₂²⋊D₁₃	C₂	1	2	yes	maximal, complemented
6	#11718389	C₂	1	2	yes	maximal, complemented
7	#11718386	C₂	1	2	yes	maximal, complemented
8	D₁₃×C₂²⋊C₈	C₂	1	2	yes	maximal, complemented
9	#11718381	C₂	1	2	yes	maximal, complemented
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₂²×C₄×D₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic, complemented
13	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
14	#88200	C₄	1	2	yes	—
15	#88192	C₄	1	2	yes	—
16	#88191	C₄	1	1	yes	characteristic
17	#88191	C₄	1	1	yes	characteristic
18	#88189	C₄	1	1	yes	characteristic
19	#88189	C₄	1	1	yes	characteristic
20	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
21	C₄×C₂²⋊D₁₃	C₂²	1	2	yes	—
22	#88180	C₂²	1	1	yes	characteristic
23	C₂×#6099	C₂²	1	1	yes	characteristic, complemented
24	C₂²×#1055	C₂²	1	1	yes	characteristic, complemented
25	C₂×C₈×D₁₃	C₂²	1	2	yes	—
26	C₂×C₈×D₁₃	C₂²	1	2	yes	—
27	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
28	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	complemented
29	C₂²⋊#1055	C₂²	1	2	yes	—
30	C₂²⋊#1055	C₂²	1	2	yes	complemented
31	#88060	C₂²	1	2	yes	—
32	#88060	C₂²	1	2	yes	—
33	#88055	C₂²	1	2	yes	complemented
34	#88054	C₂²	1	2	yes	complemented
35	#88054	C₂²	1	2	yes	complemented
36	C₈×Dic₁₃	C₂²	1	2	yes	complemented
37	C₂³×D₁₃	C₂×C₄	1	1	yes	characteristic
38	C₁₃×#41	—	4	1	no	—
39	C₁₃×#41	—	4	1	no	—
40	C₂×Q₈×C₁₃	C₂×C₄	1	1	yes	characteristic
41	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
43	C₂×D₄×C₁₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₁₃	C₂×C₄	1	2	yes	—
49	C₂²×Dic₁₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₁₃	—	4	1	no	—
51	C₂×C₄×D₁₃	—	4	1	no	—
52	C₂×C₄×D₁₃	C₂×C₄	1	2	yes	—
53	C₂×C₄×D₁₃	C₂³	1	2	yes	—
54	C₁₃×M₁₆	—	4	1	no	—
55	C₂×C₈×C₁₃	—	4	1	no	—
56	C₂×C₈×C₁₃	C₂³	1	2	yes	—
57	C₂×C₈×C₁₃	C₂³	1	2	yes	—
58	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
59	C₂²⋊Dic₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
61	C₂²⋊Dic₁₃	C₂×C₄	1	2	yes	—
62	#6103	C₂×C₄	1	2	yes	—
63	#6103	C₂×C₄	1	2	yes	—
64	C₄⋊Dic₁₃	C₂×C₄	1	2	yes	—
65	#6101	C₂×C₄	1	2	yes	—
66	#6101	C₂×C₄	1	2	yes	—
67	C₄×Dic₁₃	C₂³	1	2	yes	—
68	#6099	—	4	1	no	—
69	C₂×#1055	—	4	1	no	—
70	C₂×#1055	C₂³	1	2	yes	—
71	C₂×#1055	C₂³	1	2	yes	—
72	C₈×D₁₃	—	4	2	no	—
73	C₈×D₁₃	—	4	2	no	—
74	#2977	—	13	1	no	maximal, Sylow
75	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
76	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
77	C₂³×C₁₃	C₂²×C₄	1	1	yes	characteristic
78	C₂²×D₁₃	—	2	4	no	—
79	C₂²×D₁₃	—	4	1	no	—
80	C₂²×D₁₃	C₂²×C₄	1	2	yes	—
81	Q₈×C₁₃	—	4	1	no	—
82	D₄×C₁₃	—	4	1	no	—
83	D₄×C₁₃	—	4	1	no	—
84	D₄×C₁₃	—	4	1	no	—
85	C₂×C₄×C₁₃	—	2	1	no	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊D₁₃	—	2	4	no	—
96	C₂²⋊D₁₃	—	2	4	no	—
97	C₂×Dic₁₃	—	4	1	no	—
98	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
99	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
100	C₂×Dic₁₃	C₂²×C₄	1	2	yes	—
101	C₄×D₁₃	—	2	2	no	—
102	C₄×D₁₃	—	2	2	no	—
103	C₄×D₁₃	—	4	2	no	—
104	C₈×C₁₃	—	2	2	no	—
105	C₈×C₁₃	—	2	2	no	—
106	#1055	—	2	2	no	—
107	#1055	—	2	2	no	—
108	#525	—	13	1	no	—
109	#435	—	13	2	no	—
110	#435	—	13	2	no	—
111	C₈×D₄	—	13	2	no	—
112	C₈×D₄	—	13	2	no	—
113	#433	—	13	2	no	—
114	#408	—	13	1	no	—
115	#408	D₁₃	1	1	yes	characteristic, complemented
116	C₂×C₂²⋊C₈	—	13	2	no	—
117	C₂²×C₁₃	—	2	1	no	—
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	2	2	no	—
121	C₂²×C₁₃	—	2	2	no	—
122	C₂²×C₁₃	—	2	2	no	—
123	C₂²×C₁₃	C₂³×C₄	1	1	yes	characteristic
124	C₂×D₁₃	—	2	2	no	—
125	C₂×D₁₃	—	2	2	no	—
126	C₂×D₁₃	—	4	4	no	—
127	C₂×D₁₃	#131	1	4	yes	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	2	1	no	—
131	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
132	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
133	Dic₁₃	—	2	2	no	—
134	Dic₁₃	—	2	2	no	—
135	Dic₁₃	—	2	2	no	—
136	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
137	C₂³×C₄	—	13	1	no	—
138	C₂×M₁₆	—	13	1	no	—
139	C₂×M₁₆	C₂×D₁₃	1	1	yes	characteristic, complemented
140	C₂²×C₈	—	13	1	no	—
141	C₂²×C₈	—	13	2	no	—
142	C₂²×C₈	—	13	2	no	—
143	C₂²×C₈	C₂×D₁₃	1	1	yes	characteristic, complemented
144	#123	—	13	1	no	—
145	#123	—	13	1	no	—
146	C₂²⋊Q₈	—	13	2	no	—
147	#121	—	13	2	no	—
148	#120	—	13	1	no	—
149	#120	—	13	1	no	—
150	C₄×D₄	—	13	2	no	—
151	C₄×D₄	—	13	2	no	—
152	#117	—	13	1	no	—
153	C₄⋊C₈	—	13	2	no	—
154	C₄⋊C₈	—	13	2	no	—
155	C₂²⋊C₈	—	13	2	no	—
156	C₂²⋊C₈	—	13	2	no	—
157	C₂²⋊C₈	—	13	2	no	—
158	C₂²⋊C₈	—	13	2	no	—
159	C₂²⋊C₈	C₂×D₁₃	1	2	yes	—
160	C₂²⋊C₈	C₂×D₁₃	1	2	yes	complemented
161	#97	—	13	2	no	—
162	C₄×C₈	—	13	2	no	—
163	C₂×C₁₃	—	4	1	no	—
164	C₂×C₁₃	—	4	1	no	—
165	C₂×C₁₃	—	4	1	no	—
166	C₂×C₁₃	#518	1	1	yes	characteristic
167	C₂×C₁₃	C₂×#131	1	2	yes	—
168	D₁₃	—	2	4	no	—
169	C₂⁴	—	13	1	no	—
170	#41	—	4	1	no	—
171	#41	—	4	1	no	—
172	C₂×Q₈	C₄×D₁₃	1	1	yes	characteristic
173	C₂×D₄	—	13	2	no	—
174	C₂×D₄	—	13	2	no	—
175	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
176	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
177	C₂×D₄	C₄×D₁₃	1	1	yes	characteristic
178	C₂²×C₄	—	13	1	no	—
179	C₂²×C₄	—	13	2	no	—
180	C₂²×C₄	—	13	2	no	—
181	C₂²×C₄	—	52	1	no	—
182	C₂²×C₄	—	52	1	no	—
183	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
184	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
185	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
186	M₁₆	—	4	1	no	—
187	M₁₆	—	52	1	no	—
188	C₂×C₈	—	4	1	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	52	2	no	—
192	C₂×C₈	—	52	2	no	—
193	C₂×C₈	—	52	1	no	—
194	C₂×C₈	C₂²×D₁₃	1	2	yes	—
195	C₂×C₈	C₂²×D₁₃	1	2	yes	—
196	C₄⋊C₄	—	13	2	no	—
197	C₄⋊C₄	—	13	2	no	—
198	C₄⋊C₄	—	13	2	no	—
199	C₂²⋊C₄	—	13	1	no	—
200	C₂²⋊C₄	—	13	1	no	—
201	C₂²⋊C₄	—	13	2	no	—
202	C₂²⋊C₄	—	13	2	no	—
203	C₂²⋊C₄	—	13	2	no	—
204	C₂²⋊C₄	—	13	2	no	—
205	C₄²	—	13	2	no	—
206	C₁₃	#2977	1	1	yes	characteristic, complemented, Sylow
207	C₂³	—	13	2	no	—
208	C₂³	—	26	4	no	—
209	C₂³	—	52	1	no	—
210	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
211	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
212	C₂³	C₂×C₄×D₁₃	1	1	yes	characteristic
213	Q₈	—	4	1	no	—
214	D₄	—	4	1	no	—
215	D₄	—	4	1	no	—
216	D₄	—	4	1	no	—
217	D₄	—	26	4	no	—
218	D₄	—	26	4	no	—
219	C₂×C₄	—	2	1	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	1	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	13	2	no	—
226	C₂×C₄	—	13	2	no	—
227	C₂×C₄	—	13	2	no	—
228	C₂×C₄	—	26	2	no	—
229	C₂×C₄	—	26	2	no	—
230	C₂×C₄	—	52	2	no	—
231	C₂×C₄	—	52	1	no	—
232	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
234	C₂×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
235	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
236	C₈	—	2	2	no	—
237	C₈	—	2	2	no	—
238	C₈	—	26	2	no	—
239	C₈	—	26	2	no	—
240	C₂²	—	2	1	no	—
241	C₂²	—	2	1	no	—
242	C₂²	—	2	1	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	13	4	no	—
247	C₂²	—	26	2	no	—
248	C₂²	—	26	2	no	—
249	C₂²	—	52	4	no	—
250	C₂²	C₂²×C₄×D₁₃	1	1	yes	characteristic
251	C₄	—	2	1	no	—
252	C₄	—	2	1	no	—
253	C₄	—	2	1	no	—
254	C₄	—	26	2	no	—
255	C₄	—	26	2	no	—
256	C₄	—	26	2	no	—
257	C₄	#88251	1	1	yes	characteristic
258	C₄	#88251	1	1	yes	characteristic
259	C₂	—	4	1	no	—
260	C₂	—	4	1	no	—
261	C₂	—	4	1	no	—
262	C₂	—	26	4	no	—
263	C₂	#11719191	1	1	yes	characteristic
264	C₂	D₁₃×#131	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₄
Fitting	C₁₃×#408	C₂
prime radical	#11719533	C₂
cosocle	C₂×C₄×C₁₃	C₂⁴
hypercenter	#408	D₁₃
hypocenter	C₁₃	#2977
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #408

Complements

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

N	H ≅ G/N	type
2. C₁₃×#408	C₂	semidirect
3. #11718895	C₂	semidirect
4. #11718788	C₂	semidirect
5. C₈×C₂²⋊D₁₃	C₂	semidirect
6. #11718389	C₂	semidirect
7. #11718386	C₂	semidirect
8. D₁₃×C₂²⋊C₈	C₂	semidirect
9. #11718381	C₂	semidirect
12. C₂×C₁₃×M₁₆	C₂²	semidirect
13. C₂²×C₈×C₁₃	C₂²	semidirect
23. C₂×#6099	C₂²	semidirect
24. C₂²×#1055	C₂²	semidirect
28. C₁₃×C₂²⋊C₈	C₂²	semidirect
30. C₂²⋊#1055	C₂²	semidirect
33. #88055	C₂²	semidirect
34. #88054	C₂²	semidirect
35. #88054	C₂²	semidirect
36. C₈×Dic₁₃	C₂²	semidirect
115. #408	D₁₃	semidirect
139. C₂×M₁₆	C₂×D₁₃	semidirect
143. C₂²×C₈	C₂×D₁₃	semidirect
160. C₂²⋊C₈	C₂×D₁₃	semidirect
206. C₁₃	#2977	semidirect

Maximal subgroups

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

subgroup	index
#11719533	2
C₁₃×#408	2
#11718895	2
#11718788	2
C₈×C₂²⋊D₁₃	2
#11718389	2
#11718386	2
D₁₃×C₂²⋊C₈	2
#11718381	2
#2977	13