Subgroups — #15771187

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718150	C₂	1	2	yes	maximal
3	C₂×#11718150	C₂	1	2	yes	maximal
4	C₂×#11718116	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂³×#1055	C₂²	1	2	yes	—
11	#11718585	—	2	4	no	—
12	C₂×C₄⋊#1055	—	2	2	no	—
13	C₁₃×#336	—	2	2	no	—
14	#11718150	—	2	4	no	—
15	#11718150	—	2	4	no	—
16	#11718116	C₄	1	2	yes	—
17	#11718116	C₄	1	2	yes	—
18	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	1	no	—
21	C₂×C₁₃×M₁₆	—	2	1	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
27	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
28	C₂²×C₈×C₁₃	—	2	1	no	—
29	C₂²×C₈×C₁₃	—	2	1	no	—
30	C₂×C₄²×C₁₃	—	2	2	no	—
31	C₂×C₄²×C₁₃	—	2	2	no	—
32	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₁₃	D₄	1	2	yes	—
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	D₄	1	4	yes	—
41	C₂²×#1055	—	2	2	no	—
42	C₂²×#1055	—	2	2	no	—
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	—	2	2	no	—
45	C₂²×#1055	—	2	2	no	—
46	C₂²×#1055	—	2	2	no	—
47	C₂²×#1055	—	2	2	no	—
48	C₂²×#1055	C₂×C₄	1	4	yes	—
49	C₄⋊#1055	—	2	8	no	—
50	#11269	—	13	1	no	maximal, Sylow
51	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₁₃	—	2	4	no	—
53	C₂²×C₄×C₁₃	—	2	4	no	—
54	C₂²×C₄×C₁₃	—	4	2	no	—
55	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₄⋊C₄	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	#41	1	2	yes	—
65	C₂²×C₄×C₁₃	#41	1	2	yes	—
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	2	4	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₁₃×M₁₆	—	4	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	2	no	—
73	C₂×C₈×C₁₃	—	4	1	no	—
74	C₂×C₈×C₁₃	—	4	1	no	—
75	C₂×C₈×C₁₃	—	4	1	no	—
76	C₂×C₈×C₁₃	—	4	1	no	—
77	C₄²×C₁₃	—	2	2	no	—
78	C₄²×C₁₃	—	2	2	no	—
79	C₄²×C₁₃	—	2	2	no	—
80	C₄²×C₁₃	—	2	2	no	—
81	#6099	—	2	4	no	—
82	#6099	—	2	4	no	—
83	#6099	—	4	2	no	—
84	#6099	—	4	2	no	—
85	C₂×#1055	—	2	4	no	—
86	C₂×#1055	—	2	4	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	2	4	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂×#1055	—	4	2	no	—
91	C₂×#1055	—	4	2	no	—
92	C₂×#1055	—	4	2	no	—
93	C₂×#1055	—	4	2	no	—
94	C₂×#1055	—	4	2	no	—
95	C₂×#423	—	13	1	no	—
96	C₂×#341	—	13	2	no	—
97	C₂×#341	—	13	2	no	—
98	#1728	D₁₃	1	1	yes	characteristic
99	C₂×#336	—	13	1	no	—
100	C₂³×C₁₃	—	2	2	no	—
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
104	C₂³×C₁₃	#125	1	2	yes	—
105	C₂³×C₁₃	#125	1	1	yes	characteristic
106	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	4	8	no	—
113	C₂×C₄×C₁₃	#108	1	4	yes	—
114	C₂×C₄×C₁₃	#117	1	2	yes	—
115	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₄×Q₈	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#121	1	2	yes	—
120	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
122	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₁₃	#123	1	2	yes	—
124	C₂×C₄×C₁₃	#123	1	2	yes	—
125	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
126	C₂×C₄×C₁₃	#124	1	2	yes	—
127	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₁₃	#125	1	2	yes	—
130	C₂×C₄×C₁₃	#126	1	2	yes	—
131	C₂×C₄×C₁₃	#126	1	2	yes	—
132	C₈×C₁₃	—	4	2	no	—
133	C₈×C₁₃	—	4	2	no	—
134	#1055	—	4	4	no	—
135	#1055	—	4	4	no	—
136	C₂²×M₁₆	—	13	2	no	—
137	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
138	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
139	C₂³×C₈	—	13	2	no	—
140	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
141	#423	—	26	4	no	—
142	C₂×C₄⋊C₈	—	26	2	no	—
143	#341	—	26	4	no	—
144	#341	—	26	4	no	—
145	#336	—	2	2	no	—
146	#336	—	13	2	no	—
147	#336	—	13	2	no	—
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	2	no	—
150	C₂²×C₁₃	—	2	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	#382	1	1	yes	characteristic
154	C₂²×C₁₃	#385	1	1	yes	characteristic
155	C₂²×C₁₃	#389	1	1	yes	characteristic
156	C₂²×C₁₃	#398	1	2	yes	—
157	C₂²×C₁₃	#400	1	2	yes	—
158	C₂²×C₁₃	C₂×#108	1	2	yes	—
159	C₂²×C₁₃	#430	1	2	yes	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	—	4	2	no	—
162	C₄×C₁₃	#387	1	2	yes	—
163	C₄×C₁₃	#388	1	2	yes	—
164	C₄×C₁₃	#393	1	1	yes	characteristic
165	C₄×C₁₃	#394	1	1	yes	characteristic
166	C₄×C₁₃	#397	1	1	yes	characteristic
167	C₄×C₁₃	#400	1	1	yes	characteristic
168	C₂³×C₄	Dic₂₆	1	2	yes	—
169	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	1	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	13	4	no	—
177	C₂×M₁₆	—	26	2	no	—
178	C₂×M₁₆	—	26	2	no	—
179	C₂×M₁₆	—	26	2	no	—
180	C₂×M₁₆	—	26	2	no	—
181	C₂×M₁₆	—	26	2	no	—
182	C₂×M₁₆	Dic₂₆	1	2	yes	—
183	C₂×M₁₆	D₅₂	1	2	yes	—
184	C₂²×C₈	—	2	1	no	—
185	C₂²×C₈	—	2	1	no	—
186	C₂²×C₈	—	13	4	no	—
187	C₂²×C₈	—	26	2	no	—
188	C₂²×C₈	—	26	2	no	—
189	C₂²×C₈	—	26	2	no	—
190	C₂²×C₈	—	26	2	no	—
191	C₂²×C₈	—	26	2	no	—
192	C₂²×C₈	—	26	2	no	—
193	C₂²×C₈	—	26	2	no	—
194	C₂×C₄²	—	2	2	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
198	C₂×C₄²	C₂²⋊D₁₃	1	2	yes	—
199	C₄⋊C₈	—	26	8	no	—
200	C₂×C₁₃	—	2	2	no	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	#1442	1	1	yes	characteristic
203	C₂×C₁₃	#1837	1	1	yes	characteristic
204	C₂×C₁₃	#1838	1	1	yes	characteristic
205	C₂×C₁₃	#1855	1	2	yes	—
206	C₂×C₁₃	#1950	1	2	yes	—
207	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	#6101	1	2	yes	—
212	C₂²×C₄	#6101	1	2	yes	—
213	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
214	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
216	C₂²×C₄	#6127	1	1	yes	characteristic
217	C₂²×C₄	#6127	1	1	yes	characteristic
218	C₂²×C₄	D₄×D₁₃	1	2	yes	—
219	C₂²×C₄	#6129	1	2	yes	—
220	C₂²×C₄	#6129	1	2	yes	—
221	C₂²×C₄	#6131	1	2	yes	—
222	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
223	M₁₆	—	2	4	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	M₁₆	—	26	4	no	—
228	M₁₆	—	26	4	no	—
229	M₁₆	—	52	2	no	—
230	M₁₆	—	52	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	4	1	no	—
234	C₂×C₈	—	4	1	no	—
235	C₂×C₈	—	4	1	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	26	4	no	—
238	C₂×C₈	—	26	4	no	—
239	C₂×C₈	—	26	4	no	—
240	C₂×C₈	—	26	4	no	—
241	C₂×C₈	—	52	2	no	—
242	C₂×C₈	—	52	2	no	—
243	C₂×C₈	—	52	2	no	—
244	C₂×C₈	—	52	2	no	—
245	C₂×C₈	—	52	2	no	—
246	C₂×C₈	—	52	2	no	—
247	C₄²	—	2	2	no	—
248	C₄²	—	2	2	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₁₃	#11269	1	1	yes	characteristic, complemented, Sylow
252	C₂³	—	2	2	no	—
253	C₂³	—	2	2	no	—
254	C₂³	#88123	1	1	yes	characteristic
255	C₂³	#88124	1	1	yes	characteristic
256	C₂³	#88132	1	2	yes	—
257	C₂³	#88144	1	2	yes	—
258	C₂³	C₂×#6101	1	1	yes	characteristic
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	#88062	1	4	yes	—
266	C₂×C₄	C₄×Dic₂₆	1	1	yes	characteristic
267	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
268	C₂×C₄	#88132	1	2	yes	—
269	C₂×C₄	#88133	1	2	yes	—
270	C₂×C₄	#88135	1	2	yes	—
271	C₂×C₄	#88139	1	2	yes	—
272	C₂×C₄	#88140	1	2	yes	—
273	C₂×C₄	#88144	1	2	yes	—
274	C₂×C₄	#88146	1	2	yes	—
275	C₂×C₄	#88148	1	2	yes	—
276	C₂×C₄	#88149	1	2	yes	—
277	C₂×C₄	#88152	1	2	yes	—
278	C₂×C₄	#88178	1	1	yes	characteristic
279	C₂×C₄	#88184	1	1	yes	characteristic
280	C₂×C₄	#88189	1	1	yes	characteristic
281	C₂×C₄	#88190	1	1	yes	characteristic
282	C₂×C₄	#88191	1	1	yes	characteristic
283	C₂×C₄	#88201	1	1	yes	characteristic
284	C₈	—	4	2	no	—
285	C₈	—	4	2	no	—
286	C₈	—	52	4	no	—
287	C₈	—	52	4	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	2	4	no	—
291	C₂²	—	2	2	no	—
292	C₂²	—	2	2	no	—
293	C₂²	#11718321	1	2	yes	—
294	C₂²	#11718322	1	2	yes	—
295	C₂²	#11718589	1	1	yes	characteristic
296	C₂²	#11718607	1	1	yes	characteristic
297	C₂²	#11718638	1	1	yes	characteristic
298	C₂²	C₂×#88062	1	2	yes	—
299	C₂²	#11718785	1	2	yes	—
300	C₄	—	2	4	no	—
301	C₄	—	4	2	no	—
302	C₄	#11718316	1	2	yes	—
303	C₄	#11718331	1	2	yes	—
304	C₄	#11718608	1	1	yes	characteristic
305	C₄	#11718614	1	1	yes	characteristic
306	C₄	#11718647	1	1	yes	characteristic
307	C₄	#11718652	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	2	2	no	—
310	C₂	#13113092	1	1	yes	characteristic
311	C₂	#13114973	1	1	yes	characteristic
312	C₂	#13114976	1	1	yes	characteristic
313	C₂	#13115040	1	2	yes	—
314	C₂	#13115611	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₁₃
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#1728	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#1728	D₁₃
hypocenter	C₁₃	#11269
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
251. C₁₃	#11269	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
C₂×#11718150	2
C₂×#11718116	2
C₁₃×#1728	2
#11269	13