Subgroups — #15771192

Order 3328.

634 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88058	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718146	C₂	1	2	yes	maximal
3	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
4	#13114557	C₂	1	2	yes	maximal
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×C₄⋊Dic₁₃	C₂²	1	2	yes	—
10	C₂²×#6099	C₂²	1	2	yes	—
11	C₂×#88058	—	2	2	no	—
12	C₂×#88058	—	2	4	no	—
13	C₁₃×#336	—	2	2	no	—
14	#11718146	—	2	4	no	—
15	#11718144	C₄	1	4	yes	complemented
16	#11718121	—	2	4	no	—
17	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
18	C₂³×Dic₁₃	—	2	2	no	—
19	C₂×C₁₃×M₁₆	—	2	1	no	—
20	C₂×C₁₃×M₁₆	—	2	2	no	—
21	C₂×C₁₃×M₁₆	—	2	2	no	—
22	C₂×C₁₃×M₁₆	Q₈	1	2	yes	—
23	C₂²×C₈×C₁₃	—	2	2	no	—
24	C₂²×C₈×C₁₃	—	2	2	no	—
25	C₂²×C₈×C₁₃	—	2	1	no	—
26	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₁₃	D₄	1	2	yes	—
28	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
29	C₁₃×#117	—	2	2	no	—
30	C₁₃×#117	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
32	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
33	C₂×C₄²×C₁₃	—	2	1	no	—
34	C₂×C₄⋊Dic₁₃	—	2	2	no	—
35	C₂×C₄⋊Dic₁₃	—	2	2	no	—
36	C₂×C₄⋊Dic₁₃	—	2	2	no	—
37	C₂×C₄⋊Dic₁₃	—	2	2	no	—
38	C₂×C₄⋊Dic₁₃	—	2	2	no	—
39	C₂×C₄⋊Dic₁₃	D₄	1	4	yes	—
40	C₂×#6099	—	2	2	no	—
41	C₂×#6099	—	2	2	no	—
42	C₂×#6099	—	2	2	no	—
43	C₂×#6099	—	2	2	no	—
44	C₂×#6099	—	2	2	no	—
45	C₂×#6099	C₂×C₄	1	4	yes	—
46	C₂²×#1055	—	2	2	no	—
47	#88058	—	2	8	no	—
48	#11270	—	13	1	no	maximal, Sylow
49	C₂⁴×C₁₃	#41	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	C₂×Q₈	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	C₂²×Dic₁₃	—	2	4	no	—
62	C₂²×Dic₁₃	—	4	2	no	—
63	C₂²×Dic₁₃	—	4	2	no	—
64	C₂²×Dic₁₃	—	8	4	no	—
65	C₁₃×M₁₆	—	2	4	no	—
66	C₁₃×M₁₆	—	4	2	no	—
67	C₂×C₈×C₁₃	—	2	4	no	—
68	C₂×C₈×C₁₃	—	2	2	no	—
69	C₂×C₈×C₁₃	—	2	4	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	4	1	no	—
73	C₂×C₈×C₁₃	—	4	1	no	—
74	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
75	C₁₃×C₄⋊C₄	—	2	4	no	—
76	C₁₃×C₂²⋊C₄	—	4	2	no	—
77	C₄²×C₁₃	—	4	1	no	—
78	C₄²×C₁₃	—	4	1	no	—
79	C₄⋊Dic₁₃	—	2	4	no	—
80	C₄⋊Dic₁₃	—	2	4	no	—
81	C₄⋊Dic₁₃	—	4	2	no	—
82	C₄⋊Dic₁₃	—	4	2	no	—
83	#6099	—	2	4	no	—
84	#6099	—	2	4	no	—
85	#6099	—	4	2	no	—
86	#6099	—	4	2	no	—
87	C₂×#1055	—	2	4	no	—
88	C₂×#1055	—	4	2	no	—
89	C₂×#1055	—	4	2	no	—
90	C₂²×#107	—	13	1	no	—
91	C₂×#343	—	13	2	no	—
92	C₂×#341	—	13	1	no	—
93	#1736	—	13	2	no	—
94	#1729	D₁₃	1	1	yes	characteristic
95	C₂³×C₁₃	—	2	2	no	—
96	C₂³×C₁₃	—	2	2	no	—
97	C₂³×C₁₃	C₄×Q₈	1	1	yes	characteristic
98	C₂³×C₁₃	#121	1	1	yes	characteristic
99	C₂³×C₁₃	#123	1	2	yes	—
100	C₂³×C₁₃	#125	1	1	yes	characteristic
101	C₂³×C₁₃	#126	1	2	yes	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	2	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	#117	1	2	yes	—
107	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#121	1	2	yes	—
111	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₁₃	#123	1	2	yes	—
116	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
117	C₂×C₄×C₁₃	#124	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#124	1	2	yes	—
119	C₂×C₄×C₁₃	#125	1	2	yes	—
120	C₂×C₄×C₁₃	#125	1	2	yes	—
121	C₂×C₄×C₁₃	#126	1	2	yes	—
122	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
123	C₂×Dic₁₃	—	4	4	no	—
124	C₂×Dic₁₃	—	8	4	no	—
125	C₂×Dic₁₃	—	8	4	no	—
126	C₂×Dic₁₃	—	8	4	no	—
127	C₈×C₁₃	—	2	4	no	—
128	C₈×C₁₃	—	4	2	no	—
129	#1055	—	4	4	no	—
130	C₂²×M₁₆	—	13	2	no	—
131	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
132	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
134	C₂²×C₄⋊C₄	—	13	2	no	—
135	C₂×#107	—	26	4	no	—
136	C₂×#107	—	26	2	no	—
137	#343	—	26	4	no	—
138	#341	—	13	4	no	—
139	#340	—	26	4	no	—
140	#336	—	2	2	no	—
141	C₂²×C₁₃	—	2	4	no	—
142	C₂²×C₁₃	—	2	2	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	2	no	—
146	C₂²×C₁₃	#387	1	2	yes	—
147	C₂²×C₁₃	#389	1	1	yes	characteristic
148	C₂²×C₁₃	#394	1	1	yes	characteristic
149	C₂²×C₁₃	#397	1	1	yes	characteristic
150	C₂²×C₁₃	#400	1	2	yes	—
151	C₂²×C₁₃	#472	1	2	yes	—
152	C₂²×C₁₃	#473	1	2	yes	—
153	C₄×C₁₃	—	8	2	no	—
154	C₄×C₁₃	#382	1	1	yes	characteristic
155	C₄×C₁₃	#385	1	1	yes	characteristic
156	C₄×C₁₃	#388	1	2	yes	—
157	C₄×C₁₃	#393	1	1	yes	characteristic
158	C₄×C₁₃	#398	1	2	yes	—
159	C₄×C₁₃	#400	1	1	yes	characteristic
160	Dic₁₃	—	8	4	no	—
161	C₂³×C₄	—	26	2	no	—
162	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
163	C₂×M₁₆	—	2	1	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	13	4	no	—
167	C₂×M₁₆	—	26	2	no	—
168	C₂×M₁₆	—	26	2	no	—
169	C₂×M₁₆	—	26	2	no	—
170	C₂×M₁₆	—	26	2	no	—
171	C₂×M₁₆	—	26	2	no	—
172	C₂×M₁₆	Dic₂₆	1	2	yes	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	26	2	no	—
177	C₂²×C₈	Dic₂₆	1	2	yes	—
178	C₂²×C₈	C₄×D₁₃	1	2	yes	—
179	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
180	#117	—	2	2	no	—
181	#117	—	2	2	no	—
182	C₂×C₄⋊C₄	—	13	4	no	—
183	C₂×C₄⋊C₄	—	26	2	no	—
184	C₂×C₄⋊C₄	—	26	2	no	—
185	C₂×C₄⋊C₄	—	26	2	no	—
186	C₂×C₄⋊C₄	—	26	2	no	—
187	C₂×C₄⋊C₄	—	26	2	no	—
188	C₂×C₄⋊C₄	D₅₂	1	2	yes	—
189	C₂×C₂²⋊C₄	—	2	1	no	—
190	C₂×C₄²	—	2	1	no	—
191	#107	—	26	8	no	—
192	C₂×C₁₃	—	2	2	no	—
193	C₂×C₁₃	—	2	2	no	—
194	C₂×C₁₃	#1442	1	1	yes	characteristic
195	C₂×C₁₃	#1952	1	2	yes	—
196	C₂×C₁₃	#2045	1	1	yes	characteristic
197	C₂×C₁₃	#2047	1	1	yes	characteristic
198	C₂×C₁₃	#2078	1	2	yes	—
199	C₂⁴	#6127	1	1	yes	characteristic
200	C₂²×C₄	—	2	2	no	—
201	C₂²×C₄	—	26	4	no	—
202	C₂²×C₄	—	52	2	no	—
203	C₂²×C₄	—	52	2	no	—
204	C₂²×C₄	—	104	4	no	—
205	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
206	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	C₂×D₅₂	1	1	yes	characteristic
209	C₂²×C₄	#6127	1	1	yes	characteristic
210	C₂²×C₄	D₄×D₁₃	1	2	yes	—
211	C₂²×C₄	#6129	1	2	yes	—
212	C₂²×C₄	#6129	1	2	yes	—
213	C₂²×C₄	#6131	1	2	yes	—
214	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
215	M₁₆	—	2	4	no	—
216	M₁₆	—	4	2	no	—
217	M₁₆	—	26	4	no	—
218	M₁₆	—	26	4	no	—
219	M₁₆	—	52	2	no	—
220	M₁₆	—	52	2	no	—
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	4	1	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	26	4	no	—
229	C₂×C₈	—	52	2	no	—
230	C₂×C₈	—	52	2	no	—
231	C₂×C₈	#6101	1	4	yes	—
232	C₄⋊C₄	—	2	4	no	—
233	C₄⋊C₄	—	26	4	no	—
234	C₄⋊C₄	—	26	4	no	—
235	C₄⋊C₄	—	52	2	no	—
236	C₄⋊C₄	—	52	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	1	no	—
240	C₁₃	#11270	1	1	yes	characteristic, complemented, Sylow
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	C₄×Dic₂₆	1	1	yes	characteristic
244	C₂³	#88124	1	1	yes	characteristic
245	C₂³	#88133	1	2	yes	—
246	C₂³	#88148	1	2	yes	—
247	C₂³	#88184	1	1	yes	characteristic
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	52	4	no	—
252	C₂×C₄	—	104	4	no	—
253	C₂×C₄	—	104	4	no	—
254	C₂×C₄	—	104	4	no	—
255	C₂×C₄	#88123	1	1	yes	characteristic
256	C₂×C₄	C₄×D₅₂	1	1	yes	characteristic
257	C₂×C₄	#88132	1	2	yes	—
258	C₂×C₄	#88132	1	2	yes	—
259	C₂×C₄	#88135	1	2	yes	—
260	C₂×C₄	#88139	1	2	yes	—
261	C₂×C₄	#88140	1	2	yes	—
262	C₂×C₄	#88144	1	2	yes	—
263	C₂×C₄	#88144	1	2	yes	—
264	C₂×C₄	#88146	1	2	yes	—
265	C₂×C₄	#88149	1	2	yes	—
266	C₂×C₄	#88152	1	2	yes	—
267	C₂×C₄	C₂×#6101	1	1	yes	characteristic
268	C₂×C₄	#88178	1	1	yes	characteristic
269	C₂×C₄	#88189	1	1	yes	characteristic
270	C₂×C₄	#88190	1	1	yes	characteristic
271	C₂×C₄	#88191	1	1	yes	characteristic
272	C₂×C₄	#88201	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	2	no	—
275	C₈	—	52	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	#11718316	1	2	yes	—
282	C₂²	#11718321	1	2	yes	—
283	C₂²	#11718589	1	1	yes	characteristic
284	C₂²	#11718614	1	1	yes	characteristic
285	C₂²	#11718652	1	1	yes	characteristic
286	C₂²	#11718803	1	2	yes	—
287	C₂²	#11718804	1	2	yes	—
288	C₄	—	8	2	no	—
289	C₄	—	104	4	no	—
290	C₄	#11718322	1	2	yes	—
291	C₄	#11718331	1	2	yes	—
292	C₄	#11718607	1	1	yes	characteristic
293	C₄	#11718608	1	1	yes	characteristic
294	C₄	#11718638	1	1	yes	characteristic
295	C₄	#11718647	1	1	yes	characteristic
296	C₂	—	2	2	no	—
297	C₂	—	2	2	no	—
298	C₂	#13113092	1	1	yes	characteristic
299	C₂	#13115621	1	2	yes	—
300	C₂	#13116092	1	1	yes	characteristic
301	C₂	#13116104	1	1	yes	characteristic
302	C₂	#13116276	1	2	yes	—

	subgroup	G/H
center	C₂³	#88184
commutator	C₂²×C₄×C₁₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₁₃×#1729	C₂
socle	C₂³×C₁₃	#121
prime radical	C₂⁴×C₁₃	#41
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	#1729	D₁₃
hypocenter	C₁₃	#11270
LCS	G ▷ C₂²×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
15. #11718144	C₄	semidirect
240. C₁₃	#11270	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88058	2
C₂×#11718146	2
C₂×#11718144	2
#13114557	2
C₁₃×#1729	2
#11270	13