Subgroups — #15768829

Order 3328.

766 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88062	C₂	1	2	yes	maximal
2	C₂²×#88055	C₂	1	1	yes	maximal, characteristic
3	C₂×#11718144	C₂	1	1	yes	maximal, characteristic
4	#13114529	C₂	1	2	yes	maximal
5	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₁₃	C₂²	1	1	yes	characteristic
9	C₂²×#6099	C₂²	1	2	yes	—
10	C₂³×#1055	C₂²	1	1	yes	characteristic
11	C₂×#88062	—	2	4	no	—
12	C₂×#88062	—	2	4	no	—
13	C₂×#88062	C₄	1	8	yes	complemented
14	C₂×#88055	—	2	4	no	—
15	C₂×#88055	C₄	1	4	yes	—
16	C₁₃×#341	—	2	2	no	—
17	#11718144	—	2	2	no	—
18	#11718142	—	2	4	no	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂³×Dic₁₃	Q₈	1	2	yes	—
21	C₂×C₁₃×M₁₆	—	2	2	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	C₂×C₄	1	4	yes	—
27	C₂²×C₈×C₁₃	—	2	2	no	—
28	C₂²×C₈×C₁₃	—	2	2	no	—
29	C₂²×C₈×C₁₃	—	2	2	no	—
30	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₂²×C₈×C₁₃	D₄	1	2	yes	—
32	C₂²×C₈×C₁₃	Q₈	1	2	yes	—
33	C₂×C₄×Dic₁₃	—	2	2	no	—
34	C₂×C₄×Dic₁₃	—	2	2	no	—
35	C₂×C₄×Dic₁₃	C₂×C₄	1	2	yes	—
36	C₂×C₄×Dic₁₃	D₄	1	2	yes	—
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	C₂×C₄	1	4	yes	—
43	C₂²×#1055	—	2	2	no	—
44	C₂²×#1055	C₂×C₄	1	2	yes	—
45	C₂²×#1055	C₂×C₄	1	4	yes	—
46	C₂²×#1055	D₄	1	4	yes	—
47	C₂²×#1055	D₄	1	2	yes	—
48	C₂²×#1055	Q₈	1	2	yes	—
49	#88062	—	2	16	no	—
50	#88055	—	2	8	no	—
51	#10865	—	13	1	no	maximal, Sylow
52	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	2	yes	—
62	C₂²×C₄×C₁₃	#41	1	2	yes	—
63	C₂²×Dic₁₃	—	2	4	no	—
64	C₂²×Dic₁₃	—	4	4	no	—
65	C₂²×Dic₁₃	C₄⋊C₄	1	4	yes	—
66	C₁₃×M₁₆	—	2	4	no	—
67	C₁₃×M₁₆	—	2	4	no	—
68	C₁₃×M₁₆	—	4	2	no	—
69	C₁₃×M₁₆	—	4	2	no	—
70	C₂×C₈×C₁₃	—	2	2	no	—
71	C₂×C₈×C₁₃	—	2	2	no	—
72	C₂×C₈×C₁₃	—	2	4	no	—
73	C₂×C₈×C₁₃	—	2	4	no	—
74	C₂×C₈×C₁₃	—	2	4	no	—
75	C₂×C₈×C₁₃	—	4	2	no	—
76	C₂×C₈×C₁₃	—	4	2	no	—
77	C₂×C₈×C₁₃	C₄⋊C₄	1	4	yes	—
78	C₄×Dic₁₃	—	2	2	no	—
79	C₄×Dic₁₃	—	2	2	no	—
80	C₄×Dic₁₃	—	2	4	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	—	4	2	no	—
87	C₂×#1055	—	4	2	no	—
88	C₂×#1055	C₄²	1	4	yes	—
89	C₂×#1055	C₂²⋊C₄	1	4	yes	—
90	C₂×#1055	C₂²⋊C₄	1	8	yes	—
91	C₂×#1055	C₄⋊C₄	1	4	yes	—
92	C₂×#1055	C₄⋊C₄	1	8	yes	—
93	C₂²×#108	—	13	2	no	—
94	C₂²×#97	—	13	1	no	—
95	C₂×#341	—	13	1	no	—
96	C₂×#341	D₁₃	1	1	yes	characteristic
97	#1728	—	13	2	no	—
98	C₂³×C₁₃	—	2	4	no	—
99	C₂³×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₁₃	C₄×Q₈	1	2	yes	—
101	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
102	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
103	C₂³×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
109	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
110	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
115	C₂×C₄×C₁₃	#121	1	2	yes	—
116	C₂×C₄×C₁₃	#121	1	2	yes	—
117	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
118	C₂×C₄×C₁₃	#124	1	2	yes	—
119	C₂×C₄×C₁₃	#125	1	2	yes	—
120	C₂×C₄×C₁₃	#127	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₄⋊Q₈	1	1	yes	characteristic
122	C₂×Dic₁₃	—	2	4	no	—
123	C₂×Dic₁₃	—	2	8	no	—
124	C₂×Dic₁₃	—	4	8	no	—
125	C₈×C₁₃	—	2	4	no	—
126	C₈×C₁₃	—	4	4	no	—
127	#1055	—	4	4	no	—
128	#1055	#95	1	8	yes	—
129	C₂²×M₁₆	—	13	2	no	—
130	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
131	C₂³×C₈	—	13	1	no	—
132	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
133	C₂²×C₄²	—	13	1	no	—
134	C₂×#108	—	13	8	no	—
135	C₂×#108	—	26	4	no	—
136	C₂×#108	—	26	4	no	—
137	C₂×#97	—	13	4	no	—
138	C₂×#97	—	26	4	no	—
139	#341	—	2	2	no	—
140	#341	—	26	2	no	—
141	#336	—	26	4	no	—
142	C₂²×C₁₃	—	2	4	no	—
143	C₂²×C₁₃	—	2	2	no	—
144	C₂²×C₁₃	—	2	2	no	—
145	C₂²×C₁₃	—	2	4	no	—
146	C₂²×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
147	C₂²×C₁₃	#380	1	1	yes	characteristic
148	C₂²×C₁₃	#384	1	1	yes	characteristic
149	C₂²×C₁₃	#389	1	2	yes	—
150	C₂²×C₁₃	#391	1	2	yes	—
151	C₂²×C₁₃	#430	1	4	yes	—
152	C₄×C₁₃	C₂×#95	1	1	yes	characteristic
153	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₁₃	#380	1	1	yes	characteristic
155	C₄×C₁₃	#382	1	1	yes	characteristic
156	C₄×C₁₃	#389	1	2	yes	—
157	C₄×C₁₃	#390	1	2	yes	—
158	Dic₁₃	—	4	4	no	—
159	C₂³×C₄	—	13	2	no	—
160	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	2	no	—
163	C₂×M₁₆	—	2	2	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	C₄×D₁₃	1	4	yes	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	2	no	—
176	C₂²×C₈	—	13	2	no	—
177	C₂²×C₈	—	13	4	no	—
178	C₂²×C₈	—	13	4	no	—
179	C₂²×C₈	—	13	2	no	—
180	C₂²×C₈	—	13	2	no	—
181	C₂²×C₈	—	26	2	no	—
182	C₂²×C₈	Dic₂₆	1	2	yes	—
183	C₂²×C₈	C₄×D₁₃	1	2	yes	—
184	C₂²×C₈	C₂²⋊D₁₃	1	2	yes	—
185	C₂×C₄²	—	13	2	no	—
186	C₂×C₄²	—	13	2	no	—
187	C₂×C₄²	—	26	2	no	—
188	C₂×C₄²	—	26	2	no	—
189	#108	—	26	16	no	—
190	#97	—	26	8	no	—
191	C₂×C₁₃	—	2	2	no	—
192	C₂×C₁₃	—	2	2	no	—
193	C₂×C₁₃	C₄⋊#95	1	1	yes	characteristic
194	C₂×C₁₃	#1778	1	2	yes	—
195	C₂×C₁₃	#1811	1	2	yes	—
196	C₂×C₁₃	#1849	1	2	yes	—
197	C₂⁴	Q₈×D₁₃	1	1	yes	characteristic
198	C₂²×C₄	—	13	4	no	—
199	C₂²×C₄	—	26	4	no	—
200	C₂²×C₄	—	52	4	no	—
201	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₄×D₁₃	1	2	yes	—
203	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
204	C₂²×C₄	#6127	1	2	yes	—
205	C₂²×C₄	D₄×D₁₃	1	2	yes	—
206	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
207	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
208	C₂²×C₄	#6129	1	2	yes	—
209	C₂²×C₄	Q₈×D₁₃	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
211	M₁₆	—	2	4	no	—
212	M₁₆	—	2	4	no	—
213	M₁₆	—	4	2	no	—
214	M₁₆	—	4	2	no	—
215	M₁₆	—	26	4	no	—
216	M₁₆	—	26	4	no	—
217	M₁₆	—	52	2	no	—
218	M₁₆	—	52	2	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	4	no	—
221	C₂×C₈	—	2	2	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	4	2	no	—
225	C₂×C₈	—	4	2	no	—
226	C₂×C₈	—	13	4	no	—
227	C₂×C₈	—	13	4	no	—
228	C₂×C₈	—	13	8	no	—
229	C₂×C₈	—	13	8	no	—
230	C₂×C₈	—	13	4	no	—
231	C₂×C₈	—	26	4	no	—
232	C₂×C₈	—	52	2	no	—
233	C₂×C₈	—	52	2	no	—
234	C₂×C₈	#6101	1	4	yes	—
235	C₄²	—	26	2	no	—
236	C₄²	—	26	2	no	—
237	C₄²	—	26	4	no	—
238	C₁₃	#10865	1	1	yes	characteristic, complemented, Sylow
239	C₂³	—	2	4	no	—
240	C₂³	#88141	1	2	yes	—
241	C₂³	#88142	1	1	yes	characteristic
242	C₂³	C₄⋊C₄×D₁₃	1	1	yes	characteristic
243	C₂³	#88150	1	2	yes	—
244	C₂³	#88198	1	1	yes	characteristic
245	C₂×C₄	—	26	8	no	—
246	C₂×C₄	—	26	4	no	—
247	C₂×C₄	—	52	8	no	—
248	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
249	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
250	C₂×C₄	#88126	1	1	yes	characteristic
251	C₂×C₄	#88131	1	1	yes	characteristic
252	C₂×C₄	#88132	1	2	yes	—
253	C₂×C₄	C₂²⋊C₄×D₁₃	1	1	yes	characteristic
254	C₂×C₄	C₂²⋊C₄×D₁₃	1	2	yes	—
255	C₂×C₄	#88135	1	2	yes	—
256	C₂×C₄	#88138	1	2	yes	—
257	C₂×C₄	#88139	1	2	yes	—
258	C₂×C₄	#88142	1	1	yes	characteristic
259	C₂×C₄	#88143	1	2	yes	—
260	C₂×C₄	C₄⋊C₄×D₁₃	1	2	yes	—
261	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂×#6101	1	1	yes	characteristic
263	C₂×C₄	C₄×C₂²⋊D₁₃	1	2	yes	—
264	C₂×C₄	#88193	1	2	yes	—
265	C₂×C₄	#88194	1	1	yes	characteristic
266	C₈	—	2	4	no	—
267	C₈	—	4	4	no	—
268	C₈	—	13	8	no	—
269	C₈	—	52	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	4	no	—
274	C₂²	#11718307	1	2	yes	—
275	C₂²	#11718310	1	1	yes	characteristic
276	C₂²	#11718328	1	1	yes	characteristic
277	C₂²	#11718554	1	4	yes	—
278	C₂²	#11718633	1	2	yes	—
279	C₂²	#11718637	1	1	yes	characteristic
280	C₄	—	52	4	no	—
281	C₄	#11718308	1	2	yes	—
282	C₄	#11718311	1	1	yes	characteristic
283	C₄	D₁₃×#95	1	1	yes	characteristic
284	C₄	C₄×#6101	1	1	yes	characteristic
285	C₄	#11718603	1	1	yes	characteristic
286	C₄	#11718618	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	2	2	no	—
289	C₂	#13113064	1	1	yes	characteristic
290	C₂	#13114741	1	2	yes	—
291	C₂	#13114877	1	2	yes	—
292	C₂	#13115009	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₁₃
Fitting	C₂×C₁₃×#341	C₂
socle	C₂³×C₁₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂³×C₄×C₁₃	C₂³
hypercenter	C₂×#341	D₁₃
hypocenter	C₁₃	#10865
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#341

Complements

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

N	H ≅ G/N	type
13. C₂×#88062	C₄	semidirect
238. C₁₃	#10865	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88062	2
C₂²×#88055	2
C₂×#11718144	2
#13114529	2
C₂×C₁₃×#341	2
#10865	13