Subgroups — #15781185

Order 3328.

490 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718157	C₂	1	2	yes	maximal, complemented
2	C₂×#11718155	C₂	1	2	yes	maximal
3	C₄×#88052	C₂	1	1	yes	maximal, characteristic, complemented
4	#13115008	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×SD₁₆	—	2	1	no	—
8	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#124	—	2	1	no	—
10	C₁₃×#495	—	2	1	no	—
11	C₁₃×#495	—	2	1	no	—
12	C₁₃×#495	C₄	1	2	yes	—
13	C₁₃×#495	C₄	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
15	C₂×#88059	—	2	1	no	—
16	C₂×#88059	C₂²	1	2	yes	—
17	C₂×#88052	C₂²	1	2	yes	—
18	#11718585	—	2	1	no	—
19	#11718157	—	2	4	no	—
20	#11718155	—	2	4	no	—
21	#11718053	—	2	1	no	—
22	C₄×#6090	—	2	1	no	—
23	C₂²×Q₈×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	2	1	no	—
25	C₂×C₁₃×Q₁₆	—	2	2	no	—
26	C₂×C₁₃×Q₁₆	—	2	1	no	—
27	C₂×C₁₃×Q₁₆	—	2	2	no	—
28	C₂×C₁₃×Q₁₆	C₂×C₄	1	2	yes	—
29	C₂×C₁₃×SD₁₆	—	2	2	no	—
30	C₂×C₁₃×SD₁₆	—	2	2	no	—
31	C₂×C₁₃×SD₁₆	—	4	2	no	—
32	C₂×C₁₃×SD₁₆	—	4	2	no	—
33	C₂×C₁₃×D₈	—	2	2	no	—
34	C₂×C₁₃×D₈	—	2	1	no	—
35	C₂×C₁₃×D₈	—	2	2	no	—
36	C₂×C₁₃×D₈	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#124	—	2	2	no	—
40	C₁₃×#124	—	2	2	no	—
41	C₁₃×#124	—	4	2	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
43	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
44	C₂×#6099	—	2	2	no	—
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
49	#88059	—	2	2	no	—
50	#88059	—	2	2	no	—
51	#88059	—	4	2	no	—
52	#88052	—	2	2	no	—
53	#88052	D₄	1	4	yes	—
54	C₂×#6090	—	2	2	no	—
55	C₄⋊#1055	—	4	2	no	—
56	#13214	—	13	1	no	maximal, Sylow
57	C₂⁴×C₁₃	—	4	1	no	—
58	C₂×Q₈×C₁₃	—	2	2	no	—
59	C₂×Q₈×C₁₃	—	4	1	no	—
60	C₂×Q₈×C₁₃	—	4	1	no	—
61	C₂×Q₈×C₁₃	—	4	1	no	—
62	C₂×Q₈×C₁₃	—	4	1	no	—
63	C₂×Q₈×C₁₃	—	4	1	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	4	1	no	—
66	C₂×D₄×C₁₃	—	4	1	no	—
67	C₂×D₄×C₁₃	—	4	1	no	—
68	C₂×D₄×C₁₃	—	4	1	no	—
69	C₂×D₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	4	1	no	—
71	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₁₃×Q₁₆	—	2	4	no	—
74	C₁₃×Q₁₆	—	4	2	no	—
75	C₁₃×SD₁₆	—	4	4	no	—
76	C₁₃×SD₁₆	—	4	4	no	—
77	C₁₃×D₈	—	2	4	no	—
78	C₁₃×D₈	—	4	2	no	—
79	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₈×C₁₃	D₈	1	2	yes	—
82	C₂×C₈×C₁₃	SD₁₆	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₁₃×C₂²⋊C₄	—	4	4	no	—
88	C₁₃×C₂²⋊C₄	—	4	4	no	—
89	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
90	C₄²×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₄²×C₁₃	D₈	1	2	yes	—
92	#6099	—	4	2	no	—
93	#6099	—	4	2	no	—
94	C₂×#1055	—	4	2	no	—
95	#6090	—	2	4	no	—
96	C₂×#495	D₁₃	1	1	yes	characteristic
97	C₂×#362	—	13	2	no	—
98	C₂×#361	—	13	2	no	—
99	C₄×M₃₂	—	13	1	no	—
100	#1848	—	13	1	no	—
101	C₂³×C₁₃	—	4	2	no	—
102	C₂³×C₁₃	—	8	4	no	—
103	C₂³×C₁₃	—	8	1	no	—
104	C₂³×C₁₃	—	8	1	no	—
105	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
106	Q₈×C₁₃	—	4	2	no	—
107	Q₈×C₁₃	—	4	2	no	—
108	Q₈×C₁₃	—	8	1	no	—
109	Q₈×C₁₃	—	8	1	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	D₄×C₁₃	—	8	1	no	—
113	D₄×C₁₃	—	8	1	no	—
114	C₂×C₄×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	4	2	no	—
116	C₂×C₄×C₁₃	—	8	1	no	—
117	C₂×C₄×C₁₃	—	8	1	no	—
118	C₂×C₄×C₁₃	#102	1	2	yes	—
119	C₂×C₄×C₁₃	#102	1	2	yes	—
120	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
122	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
126	C₈×C₁₃	#102	1	2	yes	—
127	C₈×C₁₃	#102	1	2	yes	—
128	C₈×C₁₃	C₄×D₄	1	2	yes	—
129	C₈×C₁₃	#121	1	2	yes	—
130	#1055	—	8	2	no	—
131	C₂²×Q₁₆	C₂×D₁₃	1	1	yes	characteristic
132	C₂²×SD₁₆	—	2	1	no	—
133	C₂²×D₈	C₂×D₁₃	1	1	yes	characteristic
134	C₂×#124	—	2	1	no	—
135	C₂×M₃₂	—	13	2	no	—
136	#495	—	2	1	no	—
137	#495	—	2	1	no	—
138	#495	Dic₁₃	1	2	yes	—
139	#495	Dic₁₃	1	2	yes	—
140	C₂×#108	—	13	2	no	—
141	C₂×#108	—	26	1	no	—
142	#423	—	26	1	no	—
143	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
144	#362	—	26	4	no	—
145	#361	—	26	4	no	—
146	#346	—	26	1	no	—
147	C₄×C₁₆	—	26	1	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	8	4	no	—
150	C₂²×C₁₃	—	8	2	no	—
151	C₂²×C₁₃	—	16	2	no	—
152	C₂²×C₁₃	—	16	2	no	—
153	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
154	C₂²×C₁₃	#486	1	1	yes	characteristic
155	C₂²×C₁₃	#492	1	1	yes	characteristic
156	C₄×C₁₃	—	2	2	no	—
157	C₄×C₁₃	—	2	2	no	—
158	C₄×C₁₃	—	8	2	no	—
159	C₄×C₁₃	#390	1	1	yes	characteristic
160	C₄×C₁₃	C₂×#102	1	1	yes	characteristic
161	C₄×C₁₃	#486	1	1	yes	characteristic
162	C₄×C₁₃	#493	1	1	yes	characteristic
163	C₂²×Q₈	—	2	1	no	—
164	C₂²×D₄	—	2	1	no	—
165	C₂×Q₁₆	—	2	2	no	—
166	C₂×Q₁₆	—	2	2	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	C₂×Dic₁₃	1	2	yes	—
169	C₂×SD₁₆	—	2	2	no	—
170	C₂×SD₁₆	—	2	2	no	—
171	C₂×SD₁₆	—	4	2	no	—
172	C₂×SD₁₆	—	4	2	no	—
173	C₂×D₈	—	2	2	no	—
174	C₂×D₈	—	2	2	no	—
175	C₂×D₈	—	2	1	no	—
176	C₂×D₈	C₂×Dic₁₃	1	2	yes	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
179	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
180	#124	—	2	2	no	—
181	#124	—	2	2	no	—
182	#124	—	4	2	no	—
183	C₂×C₂²⋊C₄	—	4	2	no	—
184	C₂×C₄²	C₂²⋊D₁₃	1	1	yes	characteristic
185	M₃₂	—	13	4	no	—
186	M₃₂	—	26	2	no	—
187	C₂×C₁₆	—	26	2	no	—
188	#108	—	26	2	no	—
189	#108	—	26	2	no	—
190	#108	—	52	2	no	—
191	C₄⋊C₈	—	52	2	no	—
192	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
193	C₄×C₈	C₂×Dic₁₃	1	1	yes	characteristic
194	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
195	C₄×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
196	C₂×C₁₃	—	2	1	no	—
197	C₂×C₁₃	—	2	1	no	—
198	C₂×C₁₃	—	16	2	no	—
199	C₂×C₁₃	#1957	1	1	yes	characteristic
200	C₂×C₁₃	#2148	1	2	yes	—
201	C₂⁴	—	4	1	no	—
202	C₂×Q₈	—	2	2	no	—
203	C₂×Q₈	—	4	1	no	—
204	C₂×Q₈	—	4	1	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	1	no	—
207	C₂×Q₈	—	4	1	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	4	1	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂×D₄	—	4	1	no	—
213	C₂×D₄	—	4	1	no	—
214	C₂²×C₄	—	4	1	no	—
215	C₂²×C₄	#6105	1	2	yes	—
216	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
217	Q₁₆	—	2	4	no	—
218	Q₁₆	—	4	2	no	—
219	SD₁₆	—	4	4	no	—
220	SD₁₆	—	4	4	no	—
221	D₈	—	2	4	no	—
222	D₈	—	4	2	no	—
223	M₁₆	—	52	2	no	—
224	M₁₆	—	52	2	no	—
225	C₂×C₈	—	52	2	no	—
226	C₂×C₈	#6104	1	2	yes	—
227	C₂×C₈	#6105	1	2	yes	—
228	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
230	C₂×C₈	D₄×D₁₃	1	2	yes	—
231	C₂×C₈	#6129	1	2	yes	—
232	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
234	C₂²⋊C₄	—	4	4	no	—
235	C₂²⋊C₄	—	4	4	no	—
236	C₄²	#6104	1	2	yes	—
237	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
238	C₄²	C₂²⋊Dic₁₃	1	1	yes	characteristic
239	C₁₆	—	26	4	no	—
240	C₁₃	#13214	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	4	2	no	—
242	C₂³	—	8	4	no	—
243	C₂³	—	8	1	no	—
244	C₂³	—	8	1	no	—
245	C₂³	C₂×#6105	1	1	yes	characteristic
246	Q₈	—	4	2	no	—
247	Q₈	—	4	2	no	—
248	Q₈	—	8	1	no	—
249	Q₈	—	8	1	no	—
250	D₄	—	4	2	no	—
251	D₄	—	4	2	no	—
252	D₄	—	8	1	no	—
253	D₄	—	8	1	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	8	1	no	—
257	C₂×C₄	—	8	1	no	—
258	C₂×C₄	#88072	1	2	yes	—
259	C₂×C₄	#88072	1	2	yes	—
260	C₂×C₄	C₂×#6104	1	1	yes	characteristic
261	C₂×C₄	C₂×#6104	1	1	yes	characteristic
262	C₂×C₄	C₂×#6105	1	1	yes	characteristic
263	C₂×C₄	#88190	1	1	yes	characteristic
264	C₂×C₄	#88194	1	1	yes	characteristic
265	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
266	C₈	—	104	2	no	—
267	C₈	#88072	1	2	yes	—
268	C₈	#88072	1	2	yes	—
269	C₈	D₄×Dic₁₃	1	2	yes	—
270	C₈	#88192	1	2	yes	—
271	C₂²	—	2	2	no	—
272	C₂²	—	8	4	no	—
273	C₂²	—	8	2	no	—
274	C₂²	—	16	2	no	—
275	C₂²	—	16	2	no	—
276	C₂²	#11718731	1	1	yes	characteristic
277	C₂²	#11718737	1	1	yes	characteristic
278	C₂²	C₂×#88072	1	1	yes	characteristic
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	8	2	no	—
282	C₄	#11718731	1	1	yes	characteristic
283	C₄	#11718734	1	1	yes	characteristic
284	C₄	C₂×#88072	1	1	yes	characteristic
285	C₄	#11718882	1	1	yes	characteristic
286	C₂	—	2	1	no	—
287	C₂	—	2	1	no	—
288	C₂	—	16	2	no	—
289	C₂	#13115643	1	1	yes	characteristic
290	C₂	#13116585	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂×C₁₃×#495	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂²×C₁₃×D₈	C₂²
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#495	D₁₃
hypocenter	C₁₃	#13214
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#495

Complements

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

N	H ≅ G/N	type
1. C₂×#11718157	C₂	semidirect
3. C₄×#88052	C₂	semidirect
4. #13115008	C₂	semidirect
240. C₁₃	#13214	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718157	2
C₂×#11718155	2
C₄×#88052	2
#13115008	2
C₂×C₁₃×#495	2
#13214	13