Subgroups — #15771351

Order 3328.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718786	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718125	C₂	1	2	yes	maximal
3	C₂×#11718144	C₂	1	2	yes	maximal
4	C₂×#11718152	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	#11718786	—	2	4	no	—
12	C₂×#88059	—	2	2	no	—
13	C₁₃×#336	—	2	2	no	—
14	#11718152	C₄	1	4	yes	complemented
15	#11718144	—	2	4	no	—
16	#11718125	—	2	4	no	—
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂×C₁₃×M₁₆	—	2	2	no	—
19	C₂×C₁₃×M₁₆	—	2	2	no	—
20	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
22	C₂²×C₈×C₁₃	—	2	1	no	—
23	C₂²×C₈×C₁₃	—	2	1	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	2	no	—
26	C₂²×C₈×C₁₃	D₄	1	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₁₃×#117	—	2	2	no	—
29	C₁₃×#117	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
31	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
32	C₂×C₄²×C₁₃	—	2	1	no	—
33	C₂×#6099	—	2	2	no	—
34	C₂×#6099	—	2	2	no	—
35	C₂×#6099	—	2	2	no	—
36	C₂×#6099	—	2	2	no	—
37	C₂×#6099	—	2	2	no	—
38	C₂×#6099	—	2	2	no	—
39	C₂×#6099	—	2	2	no	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	C₂×C₄	1	4	yes	—
44	C₂×#6099	D₄	1	4	yes	—
45	C₂²×#1055	—	2	2	no	—
46	C₂²×#1055	—	2	2	no	—
47	#88059	—	2	8	no	—
48	#11295	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	#41	1	2	yes	—
59	C₂²×C₄×C₁₃	#41	1	2	yes	—
60	C₂²×C₄×C₁₃	#41	1	2	yes	—
61	C₁₃×M₁₆	—	2	4	no	—
62	C₁₃×M₁₆	—	2	4	no	—
63	C₂×C₈×C₁₃	—	2	2	no	—
64	C₂×C₈×C₁₃	—	2	2	no	—
65	C₂×C₈×C₁₃	—	2	2	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	—	4	1	no	—
68	C₂×C₈×C₁₃	—	4	1	no	—
69	C₂×C₈×C₁₃	—	4	2	no	—
70	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	C₁₃×C₂²⋊C₄	—	2	4	no	—
73	C₄²×C₁₃	—	4	1	no	—
74	C₄²×C₁₃	—	4	1	no	—
75	#6099	—	2	4	no	—
76	#6099	—	2	4	no	—
77	#6099	—	2	4	no	—
78	#6099	—	2	4	no	—
79	#6099	—	4	2	no	—
80	#6099	—	4	2	no	—
81	#6099	—	4	2	no	—
82	#6099	—	4	2	no	—
83	C₂×#1055	—	2	4	no	—
84	C₂×#1055	—	2	4	no	—
85	C₂×#1055	—	4	2	no	—
86	C₂×#1055	—	4	2	no	—
87	C₂×#1055	—	4	2	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×#430	—	13	1	no	—
90	C₂×#344	—	13	2	no	—
91	C₂×#341	—	13	2	no	—
92	C₂×#338	—	13	1	no	—
93	#1729	D₁₃	1	1	yes	characteristic
94	C₂³×C₁₃	—	4	2	no	—
95	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
96	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
97	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
98	C₂³×C₁₃	#125	1	2	yes	—
99	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₁₃	—	4	4	no	—
101	C₂×C₄×C₁₃	—	4	2	no	—
102	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	#117	1	2	yes	—
104	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
107	C₂×C₄×C₁₃	#121	1	2	yes	—
108	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
110	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#123	1	2	yes	—
112	C₂×C₄×C₁₃	#123	1	2	yes	—
113	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#124	1	2	yes	—
116	C₂×C₄×C₁₃	#125	1	2	yes	—
117	C₂×C₄×C₁₃	#125	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#126	1	2	yes	—
119	C₂×C₄×C₁₃	#126	1	2	yes	—
120	C₈×C₁₃	—	2	4	no	—
121	C₈×C₁₃	—	4	2	no	—
122	#1055	—	4	4	no	—
123	#1055	—	4	4	no	—
124	C₂²×M₁₆	—	13	2	no	—
125	C₂²×M₁₆	—	13	2	no	—
126	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
127	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
128	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
129	#430	—	26	4	no	—
130	C₂×#108	—	26	2	no	—
131	#344	—	26	4	no	—
132	#341	—	26	4	no	—
133	#338	—	13	4	no	—
134	#336	—	2	2	no	—
135	C₂²×C₁₃	—	2	2	no	—
136	C₂²×C₁₃	—	2	4	no	—
137	C₂²×C₁₃	—	4	1	no	—
138	C₂²×C₁₃	—	4	2	no	—
139	C₂²×C₁₃	—	4	1	no	—
140	C₂²×C₁₃	#387	1	2	yes	—
141	C₂²×C₁₃	#389	1	1	yes	characteristic
142	C₂²×C₁₃	#393	1	1	yes	characteristic
143	C₂²×C₁₃	#398	1	2	yes	—
144	C₂²×C₁₃	#400	1	1	yes	characteristic
145	C₄×C₁₃	—	8	2	no	—
146	C₄×C₁₃	#382	1	1	yes	characteristic
147	C₄×C₁₃	#385	1	1	yes	characteristic
148	C₄×C₁₃	#388	1	2	yes	—
149	C₄×C₁₃	#394	1	1	yes	characteristic
150	C₄×C₁₃	#397	1	1	yes	characteristic
151	C₄×C₁₃	#400	1	2	yes	—
152	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
153	C₂×M₁₆	—	2	2	no	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂×M₁₆	—	13	4	no	—
156	C₂×M₁₆	—	13	4	no	—
157	C₂×M₁₆	—	26	2	no	—
158	C₂×M₁₆	—	26	2	no	—
159	C₂×M₁₆	—	26	2	no	—
160	C₂×M₁₆	—	26	2	no	—
161	C₂×M₁₆	—	26	2	no	—
162	C₂×M₁₆	—	26	2	no	—
163	C₂×M₁₆	—	26	2	no	—
164	C₂×M₁₆	—	26	2	no	—
165	C₂×M₁₆	—	26	2	no	—
166	C₂×M₁₆	—	26	2	no	—
167	C₂×M₁₆	Dic₂₆	1	2	yes	—
168	C₂×M₁₆	C₄×D₁₃	1	2	yes	—
169	C₂²×C₈	—	2	1	no	—
170	C₂²×C₈	—	2	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	26	2	no	—
174	C₂²×C₈	—	26	2	no	—
175	C₂²×C₈	D₅₂	1	2	yes	—
176	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₂²⋊C₄	Dic₂₆	1	2	yes	—
181	C₂×C₄²	—	2	1	no	—
182	#108	—	26	8	no	—
183	C₂×C₁₃	—	2	1	no	—
184	C₂×C₁₃	—	2	2	no	—
185	C₂×C₁₃	—	2	1	no	—
186	C₂×C₁₃	—	2	2	no	—
187	C₂×C₁₃	#1442	1	1	yes	characteristic
188	C₂×C₁₃	#2103	1	2	yes	—
189	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
192	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
194	C₂²×C₄	#6127	1	1	yes	characteristic
195	C₂²×C₄	#6127	1	1	yes	characteristic
196	C₂²×C₄	D₄×D₁₃	1	2	yes	—
197	C₂²×C₄	#6129	1	2	yes	—
198	C₂²×C₄	#6129	1	2	yes	—
199	C₂²×C₄	#6131	1	2	yes	—
200	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
201	M₁₆	—	2	4	no	—
202	M₁₆	—	2	4	no	—
203	M₁₆	—	26	4	no	—
204	M₁₆	—	26	4	no	—
205	M₁₆	—	26	4	no	—
206	M₁₆	—	26	4	no	—
207	M₁₆	—	52	2	no	—
208	M₁₆	—	52	2	no	—
209	M₁₆	—	52	2	no	—
210	M₁₆	—	52	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	4	1	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	26	4	no	—
219	C₂×C₈	—	26	4	no	—
220	C₂×C₈	—	52	2	no	—
221	C₂×C₈	—	52	2	no	—
222	C₂×C₈	—	52	2	no	—
223	C₂×C₈	—	52	2	no	—
224	C₂×C₈	#6101	1	4	yes	—
225	C₄⋊C₄	—	4	2	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₄²	—	4	1	no	—
228	C₄²	—	4	1	no	—
229	C₁₃	#11295	1	1	yes	characteristic, complemented, Sylow
230	C₂³	—	4	2	no	—
231	C₂³	C₄×Dic₂₆	1	1	yes	characteristic
232	C₂³	#88123	1	1	yes	characteristic
233	C₂³	#88132	1	2	yes	—
234	C₂³	#88144	1	2	yes	—
235	C₂³	#88178	1	1	yes	characteristic
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	#88124	1	1	yes	characteristic
239	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
240	C₂×C₄	#88132	1	2	yes	—
241	C₂×C₄	#88133	1	2	yes	—
242	C₂×C₄	#88135	1	2	yes	—
243	C₂×C₄	#88139	1	2	yes	—
244	C₂×C₄	#88140	1	2	yes	—
245	C₂×C₄	#88144	1	2	yes	—
246	C₂×C₄	#88146	1	2	yes	—
247	C₂×C₄	#88148	1	2	yes	—
248	C₂×C₄	#88149	1	2	yes	—
249	C₂×C₄	#88152	1	2	yes	—
250	C₂×C₄	C₂×#6101	1	1	yes	characteristic
251	C₂×C₄	#88184	1	1	yes	characteristic
252	C₂×C₄	#88189	1	1	yes	characteristic
253	C₂×C₄	#88190	1	1	yes	characteristic
254	C₂×C₄	#88191	1	1	yes	characteristic
255	C₂×C₄	#88201	1	1	yes	characteristic
256	C₈	—	2	4	no	—
257	C₈	—	4	2	no	—
258	C₈	—	52	4	no	—
259	C₈	—	52	4	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	1	no	—
264	C₂²	—	4	2	no	—
265	C₂²	#11718316	1	2	yes	—
266	C₂²	#11718322	1	2	yes	—
267	C₂²	#11718589	1	1	yes	characteristic
268	C₂²	#11718608	1	1	yes	characteristic
269	C₂²	#11718647	1	1	yes	characteristic
270	C₄	—	8	2	no	—
271	C₄	#11718321	1	2	yes	—
272	C₄	#11718331	1	2	yes	—
273	C₄	#11718607	1	1	yes	characteristic
274	C₄	#11718614	1	1	yes	characteristic
275	C₄	#11718638	1	1	yes	characteristic
276	C₄	#11718652	1	1	yes	characteristic
277	C₂	—	2	1	no	—
278	C₂	—	2	2	no	—
279	C₂	—	2	2	no	—
280	C₂	—	2	1	no	—
281	C₂	#13113092	1	1	yes	characteristic
282	C₂	#13116416	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#88201
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#1729	C₂
socle	C₂²×C₁₃	#400
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#1729	D₁₃
hypocenter	C₁₃	#11295
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
14. #11718152	C₄	semidirect
229. C₁₃	#11295	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718786	2
C₂×#11718125	2
C₂×#11718144	2
C₂×#11718152	2
C₁₃×#1729	2
#11295	13