Subgroups — #15777757

Order 3328.

678 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	C₂²×#88038	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×D₁₃	C₂	1	1	yes	maximal, characteristic
3	#13112914	C₂	1	1	yes	maximal, characteristic, complemented
4	#13112913	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1374	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₁₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₁₃	C₄	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
13	C₂²×#6090	C₂²	1	2	yes	—
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₂×#88038	—	2	4	no	—
16	C₂×#88038	—	2	4	no	—
17	C₂×#88038	C₄	1	8	yes	complemented
18	C₄×C₈×D₁₃	—	2	4	no	—
19	C₂²×C₄×D₁₃	C₈	1	2	yes	—
20	C₂²×C₄×D₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
23	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
25	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×D₁₃	—	2	2	no	—
29	C₂×C₈×D₁₃	—	2	2	no	—
30	C₂×C₈×D₁₃	C₈	1	4	yes	—
31	C₂×C₈×D₁₃	C₂×C₄	1	4	yes	—
32	C₂×C₈×D₁₃	D₄	1	2	yes	—
33	C₂×C₈×D₁₃	D₄	1	4	yes	—
34	C₂×C₈×D₁₃	Q₈	1	2	yes	—
35	C₄²×D₁₃	—	2	2	no	—
36	C₄²×D₁₃	—	2	2	no	—
37	C₂×C₁₃×C₁₆	—	2	2	no	—
38	C₂×C₁₃×C₁₆	—	2	2	no	—
39	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	complemented
40	C₄×C₈×C₁₃	—	2	2	no	—
41	C₈×Dic₁₃	—	2	2	no	—
42	C₈×Dic₁₃	—	2	2	no	—
43	C₂×#6090	—	2	2	no	—
44	C₂×#6090	—	2	2	no	—
45	C₂×#6090	C₂×C₄	1	4	yes	complemented
46	C₄×#1055	—	2	2	no	—
47	#88038	—	2	16	no	—
48	#12544	—	13	1	no	maximal, Sylow
49	C₂³×D₁₃	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
53	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
54	C₂×C₄×D₁₃	—	2	4	no	—
55	C₂×C₄×D₁₃	—	2	4	no	—
56	C₂×C₄×D₁₃	—	2	4	no	—
57	C₂×C₄×D₁₃	C₄²	1	2	yes	—
58	C₂×C₄×D₁₃	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×D₁₃	C₄⋊C₄	1	2	yes	—
60	C₂×C₄×D₁₃	C₂×C₈	1	2	yes	—
61	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
62	C₂×C₄×D₁₃	M₁₆	1	2	yes	—
63	C₂×C₈×C₁₃	—	2	1	no	—
64	C₂×C₈×C₁₃	—	2	1	no	—
65	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
67	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	—	2	1	no	—
72	C₄×Dic₁₃	—	2	1	no	—
73	C₄×Dic₁₃	—	2	1	no	—
74	C₄×Dic₁₃	—	2	4	no	—
75	C₂×#1055	—	2	1	no	—
76	C₂×#1055	—	2	1	no	—
77	C₂×#1055	C₄²	1	2	yes	—
78	C₂×#1055	C₂²⋊C₄	1	2	yes	—
79	C₂×#1055	C₄⋊C₄	1	2	yes	—
80	C₂×#1055	C₂×C₈	1	2	yes	—
81	C₈×D₁₃	—	2	8	no	—
82	C₈×D₁₃	C₂²⋊C₄	1	8	yes	—
83	C₈×D₁₃	C₄⋊C₄	1	8	yes	—
84	C₁₃×C₁₆	—	2	4	no	—
85	#6090	—	2	4	no	—
86	C₂²×M₃₂	—	13	2	no	—
87	C₂²×C₄×C₈	—	13	1	no	—
88	#1374	—	13	2	no	—
89	#1374	—	13	1	no	—
90	#1374	D₁₃	1	1	yes	characteristic, complemented
91	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
92	C₂²×D₁₃	—	2	2	no	—
93	C₂²×D₁₃	—	2	2	no	—
94	C₂²×D₁₃	C₄×C₈	1	2	yes	—
95	C₂²×D₁₃	C₂²⋊C₈	1	2	yes	—
96	C₂²×D₁₃	C₄⋊C₈	1	2	yes	—
97	C₂×C₄×C₁₃	—	2	2	no	—
98	C₂×C₄×C₁₃	—	2	2	no	—
99	C₂×C₄×C₁₃	M₃₂	1	4	yes	complemented
100	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×Dic₁₃	—	2	2	no	—
107	C₂×Dic₁₃	—	2	2	no	—
108	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
109	C₂×Dic₁₃	C₂²⋊C₈	1	2	yes	—
110	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
111	C₂×Dic₁₃	M₃₂	1	4	yes	complemented
112	C₄×D₁₃	—	2	16	no	—
113	C₄×D₁₃	#95	1	4	yes	—
114	C₄×D₁₃	#97	1	4	yes	—
115	C₄×D₁₃	C₂²⋊C₈	1	4	yes	—
116	C₄×D₁₃	C₄⋊C₈	1	4	yes	—
117	C₈×C₁₃	—	2	2	no	—
118	C₈×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
119	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
120	#1055	—	2	2	no	—
121	#1055	#95	1	4	yes	—
122	C₂³×C₈	—	13	1	no	—
123	C₂³×C₈	—	13	1	no	—
124	C₂²×C₄²	—	13	1	no	—
125	C₂×M₃₂	—	13	8	no	—
126	C₂×M₃₂	—	26	4	no	—
127	C₂×M₃₂	—	26	4	no	—
128	C₂²×C₁₆	—	13	2	no	—
129	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
130	C₂×C₄×C₈	—	13	1	no	—
131	C₂×C₄×C₈	—	13	1	no	—
132	C₂×C₄×C₈	—	13	1	no	—
133	C₂×C₄×C₈	—	26	4	no	—
134	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
135	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂²×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
137	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
138	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
139	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
140	C₂×D₁₃	—	2	2	no	—
141	C₂×D₁₃	—	2	2	no	—
142	C₂×D₁₃	—	2	4	no	—
143	C₂×D₁₃	—	2	4	no	—
144	C₂×D₁₃	#336	1	4	yes	—
145	C₄×C₁₃	—	2	4	no	—
146	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
147	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
148	C₄×C₁₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
149	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
150	Dic₁₃	—	2	4	no	—
151	Dic₁₃	#336	1	4	yes	—
152	C₂³×C₄	—	13	1	no	—
153	C₂³×C₄	—	13	2	no	—
154	C₂²×C₈	—	13	1	no	—
155	C₂²×C₈	—	13	4	no	—
156	C₂²×C₈	—	13	4	no	—
157	C₂²×C₈	—	13	2	no	—
158	C₂²×C₈	—	13	2	no	—
159	C₂²×C₈	—	13	1	no	—
160	C₂²×C₈	—	13	4	no	—
161	C₂²×C₈	—	26	2	no	—
162	C₂²×C₈	—	26	2	no	—
163	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
164	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
165	C₂×C₄²	—	13	2	no	—
166	C₂×C₄²	—	13	1	no	—
167	C₂×C₄²	—	26	2	no	—
168	C₂×C₄²	—	26	2	no	—
169	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
170	M₃₂	—	26	16	no	—
171	C₂×C₁₆	—	2	2	no	—
172	C₂×C₁₆	—	2	2	no	—
173	C₂×C₁₆	—	13	4	no	—
174	C₂×C₁₆	—	26	2	no	—
175	C₂×C₁₆	—	26	2	no	—
176	C₂×C₁₆	C₄×D₁₃	1	4	yes	complemented
177	C₄×C₈	—	2	2	no	—
178	C₄×C₈	—	26	2	no	—
179	C₄×C₈	—	26	2	no	—
180	C₄×C₈	—	26	2	no	—
181	C₂×C₁₃	C₂×#336	1	1	yes	characteristic
182	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
183	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
184	C₂×C₁₃	#2112	1	2	yes	—
185	C₂×C₁₃	#2150	1	2	yes	—
186	D₁₃	—	2	4	no	—
187	C₂⁴	—	13	1	no	—
188	C₂²×C₄	—	13	1	no	—
189	C₂²×C₄	—	13	2	no	—
190	C₂²×C₄	—	13	2	no	—
191	C₂²×C₄	—	13	2	no	—
192	C₂²×C₄	—	13	2	no	—
193	C₂²×C₄	—	13	2	no	—
194	C₂²×C₄	—	13	2	no	—
195	C₂²×C₄	—	13	2	no	—
196	C₂²×C₄	—	26	4	no	—
197	C₂²×C₄	—	26	4	no	—
198	C₂²×C₄	—	26	4	no	—
199	C₂²×C₄	C₈×D₁₃	1	2	yes	—
200	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	13	2	no	—
204	C₂×C₈	—	13	2	no	—
205	C₂×C₈	—	13	2	no	—
206	C₂×C₈	—	13	8	no	—
207	C₂×C₈	—	13	8	no	—
208	C₂×C₈	—	13	2	no	—
209	C₂×C₈	—	26	8	no	—
210	C₂×C₈	—	26	1	no	—
211	C₂×C₈	—	26	1	no	—
212	C₂×C₈	C₈×D₁₃	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
214	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
215	C₂×C₈	D₄×D₁₃	1	2	yes	—
216	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
217	C₄²	—	2	1	no	—
218	C₄²	—	2	1	no	—
219	C₄²	—	26	4	no	—
220	C₄²	—	26	1	no	—
221	C₄²	—	26	1	no	—
222	C₁₆	—	2	4	no	—
223	C₁₆	—	26	4	no	—
224	C₁₃	#12544	1	1	yes	characteristic, complemented, Sylow
225	C₂³	—	13	2	no	—
226	C₂³	—	13	2	no	—
227	C₂³	—	13	2	no	—
228	C₂³	—	26	2	no	—
229	C₂³	—	26	2	no	—
230	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	13	2	no	—
234	C₂×C₄	—	13	2	no	—
235	C₂×C₄	—	13	2	no	—
236	C₂×C₄	—	13	4	no	—
237	C₂×C₄	—	13	4	no	—
238	C₂×C₄	—	13	4	no	—
239	C₂×C₄	—	13	4	no	—
240	C₂×C₄	—	13	4	no	—
241	C₂×C₄	—	26	16	no	—
242	C₂×C₄	—	26	2	no	—
243	C₂×C₄	—	26	2	no	—
244	C₂×C₄	#88038	1	4	yes	complemented
245	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
247	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
248	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
249	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
250	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	13	4	no	—
253	C₈	—	26	2	no	—
254	C₈	C₂²⋊C₄×D₁₃	1	2	yes	—
255	C₈	C₄⋊C₄×D₁₃	1	2	yes	—
256	C₂²	—	13	4	no	—
257	C₂²	—	26	2	no	—
258	C₂²	—	26	2	no	—
259	C₂²	—	26	4	no	—
260	C₂²	—	26	4	no	—
261	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
262	C₂²	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
263	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
264	C₂²	C₂×#88038	1	2	yes	—
265	C₂²	D₁₃×M₃₂	1	2	yes	—
266	C₄	—	2	4	no	—
267	C₄	—	13	4	no	—
268	C₄	—	26	4	no	—
269	C₄	D₁₃×#95	1	1	yes	characteristic
270	C₄	D₁₃×#97	1	1	yes	characteristic
271	C₄	D₁₃×C₂²⋊C₈	1	1	yes	characteristic
272	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
273	C₂	—	26	4	no	—
274	C₂	D₁₃×#336	1	1	yes	characteristic
275	C₂	C₄×#88038	1	1	yes	characteristic
276	C₂	#13116429	1	1	yes	characteristic
277	C₂	#13116445	1	2	yes	—
278	C₂	#13116590	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₁₃×#1374	C₂
socle	C₂³×C₁₃	C₂²×C₈
prime radical	C₂³×D₁₃	C₂×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	#1374	D₁₃
hypocenter	C₁₃	#12544
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1374

Complements

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

N	H ≅ G/N	type
3. #13112914	C₂	semidirect
4. #13112913	C₂	semidirect
5. C₁₃×#1374	C₂	semidirect
17. C₂×#88038	C₄	semidirect
39. C₂×C₁₃×C₁₆	C₂×C₄	semidirect
45. C₂×#6090	C₂×C₄	semidirect
90. #1374	D₁₃	semidirect
99. C₂×C₄×C₁₃	M₃₂	semidirect
111. C₂×Dic₁₃	M₃₂	semidirect
176. C₂×C₁₆	C₄×D₁₃	semidirect
224. C₁₃	#12544	semidirect
244. C₂×C₄	#88038	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88038	2
C₂×C₄×C₈×D₁₃	2
#13112914	2
#13112913	2
C₁₃×#1374	2
#12544	13