Subgroups — #15779228

Order 3328.

442 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13116447	C₂	1	1	yes	maximal, characteristic
2	#13114874	C₂	1	1	yes	maximal, characteristic
3	#13112944	C₂	1	4	yes	maximal, complemented
4	C₂²×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
5	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
6	C₂²×C₁₃×C₄⋊C₄	—	2	1	no	—
7	C₂×C₁₃×#107	—	2	1	no	—
8	C₂×C₁₃×#107	—	2	1	no	—
9	C₂×C₁₃×#107	—	2	1	no	—
10	C₂×C₁₃×#107	C₄	1	2	yes	—
11	C₂×C₁₃×#107	C₄	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	#11718868	—	2	1	no	—
14	C₂×#88059	C₂²	1	2	yes	—
15	C₂×#88052	C₂²	1	2	yes	—
16	#11718144	—	2	1	no	—
17	C₂²⋊#6090	—	2	1	no	—
18	C₂²⋊#6090	—	2	1	no	—
19	C₂³×C₄×C₁₃	—	4	1	no	—
20	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
21	C₂²×C₈×C₁₃	—	2	4	no	—
22	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
26	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
27	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
29	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
30	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
31	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
32	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
33	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
34	C₂×#6099	—	2	2	no	—
35	C₁₃×#107	—	2	2	no	—
36	C₁₃×#107	—	2	2	no	—
37	C₁₃×#107	—	2	2	no	—
38	C₁₃×#107	C₂×C₄	1	4	yes	—
39	C₂²⋊#1055	—	4	1	no	—
40	C₂²⋊#1055	—	4	1	no	—
41	#88059	—	2	4	no	—
42	#88052	—	2	4	no	—
43	C₂×#6090	—	2	2	no	—
44	#12871	—	13	1	no	maximal, Sylow
45	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	—	2	2	no	—
47	C₂²×C₄×C₁₃	—	2	2	no	—
48	C₂²×C₄×C₁₃	—	4	2	no	—
49	C₂²×C₄×C₁₃	—	4	2	no	—
50	C₂²×C₄×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	8	4	no	—
52	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₁₃	—	2	2	no	—
59	C₂×C₈×C₁₃	—	2	2	no	—
60	C₂×C₈×C₁₃	—	2	2	no	—
61	C₂×C₈×C₁₃	—	2	2	no	—
62	C₂×C₈×C₁₃	—	2	2	no	—
63	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₁₃	#41	1	4	yes	—
68	C₁₃×C₄⋊C₄	—	2	4	no	—
69	C₁₃×C₄⋊C₄	—	4	2	no	—
70	C₁₃×C₄⋊C₄	—	4	2	no	—
71	C₁₃×C₄⋊C₄	—	4	2	no	—
72	#6099	—	4	2	no	—
73	#6099	—	4	2	no	—
74	C₂×#1055	—	4	2	no	—
75	#6090	—	4	2	no	—
76	C₂²×#107	D₁₃	1	1	yes	characteristic
77	#2112	—	13	1	no	—
78	#1810	—	13	1	no	—
79	#1387	—	13	4	no	—
80	C₂³×C₁₃	—	2	2	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	#102	1	1	yes	characteristic
84	C₂³×C₁₃	#102	1	1	yes	characteristic
85	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂×C₄×C₁₃	—	2	2	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	1	no	—
90	C₂×C₄×C₁₃	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	2	no	—
93	C₂×C₄×C₁₃	—	4	4	no	—
94	C₂×C₄×C₁₃	—	8	4	no	—
95	C₂×C₄×C₁₃	—	8	4	no	—
96	C₂×C₄×C₁₃	—	8	4	no	—
97	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
98	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#123	1	2	yes	—
101	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
102	C₂×C₄×C₁₃	#137	1	2	yes	—
103	C₈×C₁₃	—	2	1	no	—
104	C₈×C₁₃	—	2	1	no	—
105	C₈×C₁₃	#117	1	2	yes	—
106	C₈×C₁₃	#123	1	2	yes	—
107	#1055	—	8	2	no	—
108	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
109	C₂²×C₄⋊C₄	—	2	1	no	—
110	C₂×M₃₂	—	13	2	no	—
111	C₂×#108	—	13	2	no	—
112	C₂×#107	—	2	1	no	—
113	C₂×#107	—	2	1	no	—
114	C₂×#107	—	2	1	no	—
115	C₂×#107	Dic₁₃	1	2	yes	—
116	C₂×#107	Dic₁₃	1	2	yes	—
117	C₂×#107	C₂×D₁₃	1	2	yes	—
118	#407	—	26	1	no	—
119	C₂²⋊C₁₆	—	26	1	no	—
120	C₂²⋊C₁₆	—	26	1	no	—
121	#341	—	26	1	no	—
122	C₂²×C₁₃	—	2	8	no	—
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	2	1	no	—
127	C₂²×C₁₃	—	2	1	no	—
128	C₂²×C₁₃	—	2	1	no	—
129	C₂²×C₁₃	#361	1	2	yes	—
130	C₂²×C₁₃	#362	1	2	yes	—
131	C₂²×C₁₃	C₂×#102	1	1	yes	characteristic
132	C₂²×C₁₃	#483	1	2	yes	—
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	8	4	no	—
136	C₄×C₁₃	#381	1	1	yes	characteristic
137	C₄×C₁₃	#419	1	1	yes	characteristic
138	C₄×C₁₃	#480	1	2	yes	—
139	C₂³×C₄	—	4	1	no	—
140	C₂³×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
141	C₂×M₁₆	—	26	2	no	—
142	C₂²×C₈	—	2	4	no	—
143	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
144	C₂²×C₈	C₂×Dic₁₃	1	1	yes	characteristic
145	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
147	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
148	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	4	1	no	—
153	C₂×C₄⋊C₄	—	4	1	no	—
154	C₂×C₄⋊C₄	—	4	1	no	—
155	M₃₂	—	26	4	no	—
156	C₂×C₁₆	—	26	2	no	—
157	#108	—	26	4	no	—
158	#107	—	2	2	no	—
159	#107	—	2	2	no	—
160	#107	—	2	2	no	—
161	#107	C₂×Dic₁₃	1	4	yes	—
162	C₂²⋊C₈	—	52	1	no	—
163	C₂²⋊C₈	—	52	1	no	—
164	C₂×C₁₃	—	2	2	no	—
165	C₂×C₁₃	—	2	2	no	—
166	C₂×C₁₃	—	2	2	no	—
167	C₂×C₁₃	#1823	1	1	yes	characteristic
168	C₂×C₁₃	C₂×#361	1	1	yes	characteristic
169	C₂×C₁₃	C₂×#362	1	1	yes	characteristic
170	C₂⁴	#6105	1	1	yes	characteristic
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	4	2	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	—	4	2	no	—
176	C₂²×C₄	—	8	4	no	—
177	C₂²×C₄	#6104	1	1	yes	characteristic
178	C₂²×C₄	#6104	1	1	yes	characteristic
179	C₂²×C₄	#6105	1	1	yes	characteristic
180	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
181	C₂²×C₄	C₂²⋊Dic₁₃	1	1	yes	characteristic
182	C₂²×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
183	M₁₆	—	52	2	no	—
184	M₁₆	—	52	2	no	—
185	C₂×C₈	—	2	2	no	—
186	C₂×C₈	—	2	2	no	—
187	C₂×C₈	—	2	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	—	52	2	no	—
191	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
192	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
193	C₂×C₈	#6127	1	4	yes	—
194	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
195	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
196	C₄⋊C₄	—	2	4	no	—
197	C₄⋊C₄	—	4	2	no	—
198	C₄⋊C₄	—	4	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₁₆	—	52	2	no	—
201	C₁₃	#12871	1	1	yes	characteristic, complemented, Sylow
202	C₂³	—	2	2	no	—
203	C₂³	—	2	2	no	—
204	C₂³	—	2	2	no	—
205	C₂³	#88072	1	1	yes	characteristic
206	C₂³	#88072	1	1	yes	characteristic
207	C₂³	C₂×#6105	1	1	yes	characteristic
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	4	4	no	—
216	C₂×C₄	—	8	4	no	—
217	C₂×C₄	—	8	4	no	—
218	C₂×C₄	—	8	4	no	—
219	C₂×C₄	#88072	1	1	yes	characteristic
220	C₂×C₄	#88072	1	1	yes	characteristic
221	C₂×C₄	#88183	1	2	yes	—
222	C₂×C₄	C₂×#6104	1	1	yes	characteristic
223	C₂×C₄	#88196	1	2	yes	—
224	C₂×C₄	C₂×C₂²⋊Dic₁₃	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	—	2	1	no	—
227	C₈	—	104	2	no	—
228	C₈	#88180	1	2	yes	—
229	C₈	#88183	1	2	yes	—
230	C₂²	—	2	8	no	—
231	C₂²	—	2	1	no	—
232	C₂²	—	2	1	no	—
233	C₂²	—	2	1	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	1	no	—
237	C₂²	#11718155	1	2	yes	—
238	C₂²	#11718157	1	2	yes	—
239	C₂²	#11718632	1	2	yes	—
240	C₂²	C₂×#88072	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	2	1	no	—
243	C₄	—	8	4	no	—
244	C₄	#11718628	1	2	yes	—
245	C₄	#11718872	1	1	yes	characteristic
246	C₄	#11718886	1	1	yes	characteristic
247	C₂	—	2	2	no	—
248	C₂	—	2	2	no	—
249	C₂	—	2	2	no	—
250	C₂	#13114922	1	1	yes	characteristic
251	C₂	C₂×#11718155	1	1	yes	characteristic
252	C₂	C₂×#11718157	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#88072
commutator	C₂×C₈×C₁₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₁₃
Fitting	C₂²×C₁₃×#107	C₂
socle	C₂²×C₁₃	C₂×#102
prime radical	C₂⁴×C₁₃	SD₁₆
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂²×#107	D₁₃
hypocenter	C₁₃	#12871
LCS	G ▷ C₂×C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
3. #13112944	C₂	semidirect
201. C₁₃	#12871	semidirect

Maximal subgroups

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

subgroup	index
#13116447	2
#13114874	2
#13112944	2
C₂²×C₁₃×#107	2
#12871	13