Subgroups — #15833477

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13125422	C₂	1	1	yes	maximal, characteristic
2	#13123766	C₂	1	2	yes	maximal
3	#13113744	C₂	1	4	yes	maximal, complemented
4	#13113654	C₂	1	4	yes	maximal, complemented
5	#13113290	C₂	1	2	yes	maximal
6	C₂×#11718047	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1566	C₂	1	1	yes	maximal, characteristic
8	#11719175	C₂²	1	2	yes	—
9	C₁₃×#423	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	#11718783	—	2	2	no	—
12	#11718780	—	2	2	no	—
13	#11718777	C₄	1	4	yes	—
14	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
15	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
16	#11718492	—	2	2	no	—
17	#11718492	C₂²	1	4	yes	—
18	#11718491	—	2	2	no	—
19	#11718491	C₂²	1	4	yes	—
20	#11718360	C₄	1	1	yes	characteristic
21	C₄×Dic₅₂	C₂²	1	4	yes	—
22	#11718343	C₄	1	1	yes	characteristic
23	#11718342	C₄	1	2	yes	—
24	C₁₃×#334	C₂²	1	4	yes	—
25	#11718076	C₂²	1	8	yes	—
26	#11718047	C₂²	1	2	yes	—
27	#11718047	C₂²	1	2	yes	—
28	C₂×C₁₃×M₁₆	—	2	2	no	—
29	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
30	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
31	#88180	—	2	2	no	—
32	#88178	—	2	2	no	—
33	#88178	C₂×C₄	1	4	yes	—
34	C₂²×#1055	—	2	1	no	—
35	C₂²×#1055	D₄	1	2	yes	—
36	C₂×Dic₅₂	C₂×C₄	1	4	yes	—
37	#88124	—	2	2	no	—
38	#88123	C₂×C₄	1	2	yes	—
39	#88123	C₂×C₄	1	2	yes	—
40	C₄×Dic₂₆	—	2	2	no	—
41	C₄×Dic₂₆	C₂³	1	4	yes	—
42	C₁₃×C₄⋊C₈	—	2	2	no	—
43	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
44	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₁₃	C₂³	1	2	yes	—
47	#88058	C₂×C₄	1	4	yes	—
48	#88056	C₂×C₄	1	8	yes	—
49	#88055	—	2	4	no	—
50	#88054	—	2	4	no	—
51	#88054	—	2	4	no	—
52	C₈×Dic₁₃	—	2	4	no	—
53	C₄×#1055	—	2	1	no	—
54	C₄×#1055	D₄	1	2	yes	—
55	C₄×#1055	C₂³	1	2	yes	—
56	C₄×#1055	C₂³	1	1	yes	characteristic
57	C₄×#1055	C₂³	1	1	yes	characteristic
58	#17961	—	13	1	no	maximal, Sylow
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
62	C₂×Dic₂₆	—	2	2	no	—
63	C₂×Dic₂₆	C₂²×C₄	1	4	yes	—
64	C₁₃×M₁₆	—	4	4	no	—
65	C₂×C₈×C₁₃	—	2	4	no	—
66	C₂×C₈×C₁₃	—	2	4	no	—
67	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
68	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
69	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
70	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
71	C₄²×C₁₃	#41	1	1	yes	characteristic
72	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₂²⋊Dic₁₃	—	2	4	no	—
74	C₄⋊Dic₁₃	—	2	2	no	—
75	C₄⋊Dic₁₃	—	2	2	no	—
76	C₄⋊Dic₁₃	—	2	2	no	—
77	C₄⋊Dic₁₃	C₂²×C₄	1	4	yes	—
78	#6101	—	2	4	no	—
79	#6101	—	2	4	no	—
80	C₄×Dic₁₃	—	2	4	no	—
81	C₂×#1055	—	2	4	no	—
82	C₂×#1055	—	2	2	no	—
83	C₂×#1055	—	4	2	no	—
84	C₂×#1055	C₂×D₄	1	2	yes	—
85	C₂×#1055	C₂×D₄	1	2	yes	—
86	Dic₅₂	—	2	8	no	—
87	#3170	—	13	1	no	—
88	#2960	—	13	2	no	—
89	#1593	—	13	4	no	—
90	#1579	—	13	4	no	—
91	#1566	D₁₃	1	1	yes	characteristic
92	#1493	—	13	2	no	—
93	C₂×#322	—	13	1	no	—
94	C₂³×C₁₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₁₃	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	#137	1	4	yes	—
101	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
104	C₂×Dic₁₃	—	2	4	no	—
105	C₂×Dic₁₃	—	2	4	no	—
106	Dic₂₆	—	2	8	no	—
107	Dic₂₆	—	4	4	no	—
108	C₈×C₁₃	—	2	2	no	—
109	C₈×C₁₃	—	4	4	no	—
110	C₈×C₁₃	#131	1	4	yes	—
111	#1055	—	2	2	no	—
112	#1055	—	2	2	no	—
113	#1055	—	4	2	no	—
114	#533	—	13	2	no	—
115	#500	—	13	1	no	—
116	#494	—	13	1	no	—
117	#487	—	13	2	no	—
118	#467	—	13	4	no	—
119	#446	—	13	4	no	—
120	#446	—	26	2	no	—
121	C₈×Q₈	—	13	4	no	—
122	C₈×Q₈	—	26	2	no	—
123	C₄×Q₁₆	—	13	4	no	—
124	#424	—	26	2	no	—
125	#423	C₂×D₁₃	1	2	yes	—
126	#405	—	26	2	no	—
127	C₂×C₄×C₈	—	13	1	no	—
128	C₂×C₄×C₈	—	13	1	no	—
129	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
130	#334	C₂×D₁₃	1	4	yes	—
131	Q₈⋊C₈	—	13	8	no	—
132	#322	—	13	2	no	—
133	#322	—	13	2	no	—
134	C₂²×C₁₃	—	2	1	no	—
135	C₂²×C₁₃	—	2	1	no	—
136	C₂²×C₁₃	#441	1	2	yes	—
137	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
138	C₄×C₁₃	—	2	1	no	—
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	#441	1	2	yes	—
141	C₄×C₁₃	#444	1	2	yes	—
142	C₄×C₁₃	C₂×#131	1	2	yes	—
143	C₄×C₁₃	C₂×#137	1	2	yes	—
144	Dic₁₃	—	4	4	no	—
145	Dic₁₃	—	4	4	no	—
146	C₂×Q₁₆	—	13	4	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	13	2	no	—
149	C₂²×C₈	—	26	1	no	—
150	C₂²×C₈	C₄×D₁₃	1	2	yes	—
151	C₄⋊Q₈	—	13	2	no	—
152	C₄⋊Q₈	—	13	2	no	—
153	#125	—	26	2	no	—
154	C₂²⋊Q₈	—	13	4	no	—
155	C₂²⋊Q₈	—	26	2	no	—
156	C₄×Q₈	—	13	4	no	—
157	C₄×Q₈	—	26	2	no	—
158	#117	—	26	2	no	—
159	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
160	#107	—	13	4	no	—
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	26	4	no	—
163	C₄⋊C₈	—	26	4	no	—
164	C₄⋊C₈	C₂²×D₁₃	1	4	yes	—
165	#103	—	13	8	no	—
166	#97	—	26	4	no	—
167	C₄×C₈	—	13	2	no	—
168	C₄×C₈	—	13	1	no	—
169	C₄×C₈	—	13	1	no	—
170	C₄×C₈	—	13	2	no	—
171	C₄×C₈	—	26	4	no	—
172	C₄×C₈	—	26	1	no	—
173	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
174	C₄×C₈	C₄×D₁₃	1	1	yes	characteristic
175	C₄×C₈	C₂²×D₁₃	1	2	yes	—
176	C₂×C₁₃	—	2	2	no	—
177	C₂×C₁₃	#2931	1	1	yes	characteristic
178	C₂×C₁₃	C₂×#441	1	1	yes	characteristic
179	C₂×C₁₃	C₂×#444	1	1	yes	characteristic
180	C₂×Q₈	—	13	4	no	—
181	C₂×Q₈	—	26	2	no	—
182	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₁₃	1	1	yes	characteristic
184	C₂²×C₄	#6131	1	1	yes	characteristic
185	Q₁₆	—	26	8	no	—
186	M₁₆	—	4	4	no	—
187	C₂×C₈	—	2	4	no	—
188	C₂×C₈	—	2	4	no	—
189	C₂×C₈	—	13	2	no	—
190	C₂×C₈	—	13	2	no	—
191	C₂×C₈	—	26	4	no	—
192	C₂×C₈	—	26	2	no	—
193	C₂×C₈	—	52	2	no	—
194	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
195	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
196	C₄⋊C₄	—	13	4	no	—
197	C₄⋊C₄	—	26	2	no	—
198	C₄⋊C₄	—	26	2	no	—
199	C₄⋊C₄	—	26	4	no	—
200	C₄⋊C₄	—	26	2	no	—
201	C₄⋊C₄	—	26	4	no	—
202	C₂²⋊C₄	—	26	4	no	—
203	C₄²	—	26	4	no	—
204	C₄²	C₂×C₄×D₁₃	1	1	yes	characteristic
205	C₄²	D₄×D₁₃	1	1	yes	characteristic
206	C₄²	#6131	1	1	yes	characteristic
207	C₄²	C₂³×D₁₃	1	1	yes	characteristic
208	C₁₃	#17961	1	1	yes	characteristic, complemented, Sylow
209	C₂³	#88147	1	1	yes	characteristic
210	Q₈	—	26	8	no	—
211	Q₈	—	52	4	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	26	4	no	—
215	C₂×C₄	—	26	4	no	—
216	C₂×C₄	#88147	1	1	yes	characteristic
217	C₂×C₄	#88147	1	1	yes	characteristic
218	C₂×C₄	#88147	1	1	yes	characteristic
219	C₂×C₄	#88169	1	4	yes	—
220	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
222	C₂×C₄	C₂×#6131	1	1	yes	characteristic
223	C₈	—	2	2	no	—
224	C₈	—	4	4	no	—
225	C₈	—	26	2	no	—
226	C₈	—	26	2	no	—
227	C₈	—	52	2	no	—
228	C₈	#88161	1	4	yes	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#11718514	1	2	yes	—
232	C₂²	C₂×#88147	1	1	yes	characteristic
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	52	4	no	—
236	C₄	—	52	4	no	—
237	C₄	#11718514	1	2	yes	—
238	C₄	#11718555	1	2	yes	—
239	C₄	C₂×#88161	1	2	yes	—
240	C₄	C₂×#88169	1	2	yes	—
241	C₂	—	2	2	no	—
242	C₂	#13123547	1	1	yes	characteristic
243	C₂	C₂×#11718514	1	1	yes	characteristic
244	C₂	C₂×#11718555	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₁₃
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₄²	C₂³×D₁₃
Fitting	C₁₃×#1566	C₂
socle	C₂²×C₁₃	C₂×C₄×D₄
prime radical	C₂³×C₁₃	C₄×D₄
cosocle	C₄²×C₁₃	C₂⁴
hypercenter	#1566	D₁₃
hypocenter	C₁₃	#17961
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
3. #13113744	C₂	semidirect
4. #13113654	C₂	semidirect
208. C₁₃	#17961	semidirect

Maximal subgroups

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

subgroup	index
#13125422	2
#13123766	2
#13113744	2
#13113654	2
#13113290	2
C₂×#11718047	2
C₁₃×#1566	2
#17961	13