Subgroups — #16790203

Order 3328.

366 conjugacy classes of subgroups.

The subgroup lattice has 242 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718260	C₂	1	2	yes	maximal
2	#13129875	C₂	1	2	yes	maximal
3	#13129874	C₂	1	2	yes	maximal
4	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#6092	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
12	C₂×C₄×#1055	C₂²	1	1	yes	characteristic
13	#11718260	—	2	4	no	—
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
18	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
19	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
20	C₂×C₄×Dic₁₃	C₈	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
24	C₂²×#1055	C₂³	1	1	yes	characteristic
25	C₂×#6092	—	2	4	no	—
26	C₂×#6092	—	2	4	no	—
27	C₂×#6092	—	2	4	no	—
28	C₄×C₈×C₁₃	—	2	2	no	—
29	C₈×Dic₁₃	—	2	2	no	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₈×Dic₁₃	—	2	2	no	—
32	C₈×Dic₁₃	D₄	1	2	yes	—
33	C₈×Dic₁₃	Q₈	1	2	yes	—
34	C₄×#1055	—	2	2	no	—
35	#12879	—	13	1	no	maximal, Sylow
36	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
37	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
39	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
42	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	—	2	1	no	—
46	C₂×C₈×C₁₃	—	2	1	no	—
47	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
49	C₄²×C₁₃	—	2	1	no	—
50	C₄²×C₁₃	—	2	1	no	—
51	C₄×Dic₁₃	—	2	2	no	—
52	C₄×Dic₁₃	—	2	1	no	—
53	C₄×Dic₁₃	—	2	1	no	—
54	C₄×Dic₁₃	—	2	2	no	—
55	C₄×Dic₁₃	—	2	2	no	—
56	C₄×Dic₁₃	C₁₆	1	4	yes	complemented
57	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
58	C₄×Dic₁₃	M₁₆	1	2	yes	—
59	C₂×#1055	—	2	1	no	—
60	C₂×#1055	—	2	1	no	—
61	C₂×#1055	M₁₆	1	2	yes	—
62	C₂×#1055	C₂×D₄	1	1	yes	characteristic
63	C₂×#1055	C₂×Q₈	1	1	yes	characteristic
64	C₂×#1055	#41	1	2	yes	—
65	C₂×#1055	#41	1	2	yes	—
66	#6092	—	4	4	no	—
67	C₂×C₄⋊C₁₆	—	13	2	no	—
68	C₄²×C₈	—	13	1	no	—
69	#1374	—	13	2	no	—
70	#1374	—	13	2	no	—
71	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	—	2	1	no	—
74	C₂×C₄×C₁₃	—	2	1	no	—
75	C₂×C₄×C₁₃	—	2	1	no	—
76	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
77	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×Dic₁₃	—	2	1	no	—
85	C₂×Dic₁₃	—	2	1	no	—
86	C₂×Dic₁₃	—	2	1	no	—
87	C₂×Dic₁₃	—	2	1	no	—
88	C₂×Dic₁₃	—	2	1	no	—
89	C₂×Dic₁₃	—	2	1	no	—
90	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
91	C₂×Dic₁₃	C₄⋊C₈	1	2	yes	—
92	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
93	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
94	C₂×Dic₁₃	M₃₂	1	2	yes	—
95	C₂×Dic₁₃	M₃₂	1	2	yes	—
96	C₈×C₁₃	—	2	2	no	—
97	C₈×C₁₃	—	2	2	no	—
98	#1055	—	2	2	no	—
99	#1055	#124	1	2	yes	—
100	#1055	#125	1	2	yes	—
101	C₂²×C₁₆	—	13	4	no	—
102	C₂×C₄×C₈	—	13	1	no	—
103	C₂×C₄×C₈	—	13	1	no	—
104	C₂×C₄×C₈	—	13	1	no	—
105	C₂×C₄×C₈	—	13	1	no	—
106	C₂×C₄×C₈	—	13	1	no	—
107	C₂×C₄×C₈	#261	1	1	yes	characteristic
108	C₄³	—	13	1	no	—
109	C₄⋊C₁₆	—	26	4	no	—
110	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
111	C₂²×C₁₃	#431	1	1	yes	characteristic
112	C₂²×C₁₃	#431	1	1	yes	characteristic
113	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
114	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
116	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
117	C₄×C₁₃	—	2	2	no	—
118	C₄×C₁₃	—	2	2	no	—
119	C₄×C₁₃	#382	1	1	yes	characteristic
120	C₄×C₁₃	#423	1	1	yes	characteristic
121	C₄×C₁₃	#432	1	1	yes	characteristic
122	C₄×C₁₃	#432	1	1	yes	characteristic
123	Dic₁₃	—	2	2	no	—
124	Dic₁₃	—	2	2	no	—
125	Dic₁₃	—	2	2	no	—
126	Dic₁₃	C₄⋊C₁₆	1	4	yes	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	C₂×#261	1	1	yes	characteristic
130	C₂²×C₈	C₂×#261	1	1	yes	characteristic
131	C₂×C₄²	—	13	1	no	—
132	C₂×C₄²	—	13	1	no	—
133	C₂×C₄²	—	13	1	no	—
134	C₂×C₄²	—	13	1	no	—
135	C₂×C₄²	—	13	1	no	—
136	C₂×C₄²	—	13	1	no	—
137	C₂×C₄²	C₂×#261	1	1	yes	characteristic
138	C₂×C₁₆	—	26	4	no	—
139	C₂×C₁₆	—	26	4	no	—
140	C₂×C₁₆	—	26	4	no	—
141	C₄×C₈	—	2	2	no	—
142	C₄×C₈	—	13	2	no	—
143	C₄×C₈	—	13	2	no	—
144	C₄×C₈	—	26	2	no	—
145	C₄×C₈	—	26	2	no	—
146	C₄×C₈	—	26	2	no	—
147	C₄×C₈	—	26	2	no	—
148	C₂×C₁₃	#1831	1	1	yes	characteristic
149	C₂×C₁₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
150	C₂×C₁₃	#2150	1	1	yes	characteristic
151	C₂×C₁₃	#2162	1	1	yes	characteristic
152	C₂×C₁₃	#2162	1	1	yes	characteristic
153	C₂×C₁₃	#2163	1	1	yes	characteristic
154	C₂×C₁₃	#2163	1	1	yes	characteristic
155	C₂²×C₄	—	13	1	no	—
156	C₂²×C₄	—	13	1	no	—
157	C₂²×C₄	—	13	1	no	—
158	C₂²×C₄	—	13	1	no	—
159	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
160	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
161	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	2	1	no	—
164	C₂×C₈	—	2	1	no	—
165	C₂×C₈	—	2	1	no	—
166	C₂×C₈	—	13	1	no	—
167	C₂×C₈	—	13	1	no	—
168	C₂×C₈	—	13	2	no	—
169	C₂×C₈	—	13	2	no	—
170	C₂×C₈	—	13	2	no	—
171	C₂×C₈	—	26	1	no	—
172	C₂×C₈	—	26	1	no	—
173	C₂×C₈	C₄⋊#261	1	2	yes	—
174	C₂×C₈	#6122	1	2	yes	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₄²	—	13	4	no	—
178	C₄²	—	13	2	no	—
179	C₄²	—	13	2	no	—
180	C₄²	—	26	2	no	—
181	C₄²	—	26	2	no	—
182	C₄²	—	26	2	no	—
183	C₄²	—	26	1	no	—
184	C₄²	—	26	1	no	—
185	C₁₆	—	52	4	no	—
186	C₁₃	#12879	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	13	2	no	—
193	C₂×C₄	—	13	2	no	—
194	C₂×C₄	—	13	2	no	—
195	C₂×C₄	—	13	2	no	—
196	C₂×C₄	—	13	2	no	—
197	C₂×C₄	—	13	2	no	—
198	C₂×C₄	—	26	1	no	—
199	C₂×C₄	—	26	1	no	—
200	C₂×C₄	—	26	1	no	—
201	C₂×C₄	—	26	1	no	—
202	C₂×C₄	—	26	1	no	—
203	C₂×C₄	—	26	1	no	—
204	C₂×C₄	D₁₃⋊C₁₆	1	2	yes	—
205	C₂×C₄	#88098	1	2	yes	—
206	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
207	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
208	C₂×C₄	C₂×C₄⋊#261	1	1	yes	characteristic
209	C₂×C₄	#88237	1	1	yes	characteristic
210	C₂×C₄	#88237	1	1	yes	characteristic
211	C₂×C₄	C₂×#6122	1	1	yes	characteristic
212	C₈	—	2	2	no	—
213	C₈	—	2	2	no	—
214	C₈	—	13	2	no	—
215	C₈	—	13	2	no	—
216	C₈	—	26	2	no	—
217	C₂²	#11719049	1	1	yes	characteristic
218	C₂²	D₁₃⋊(C₄⋊C₈)	1	1	yes	characteristic
219	C₂²	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
220	C₂²	C₂×#88098	1	1	yes	characteristic
221	C₂²	D₁₃⋊M₃₂	1	1	yes	characteristic
222	C₂²	D₁₃⋊M₃₂	1	1	yes	characteristic
223	C₂²	#11719125	1	1	yes	characteristic
224	C₄	—	2	2	no	—
225	C₄	—	2	2	no	—
226	C₄	—	13	4	no	—
227	C₄	—	26	2	no	—
228	C₄	—	26	2	no	—
229	C₄	—	26	2	no	—
230	C₄	#11719055	1	1	yes	characteristic
231	C₄	#11719060	1	1	yes	characteristic
232	C₄	#11719128	1	1	yes	characteristic
233	C₄	#11719129	1	1	yes	characteristic
234	C₂	#13130752	1	1	yes	characteristic
235	C₂	D₁₃⋊(C₄⋊C₁₆)	1	1	yes	characteristic
236	C₂	#13131169	1	1	yes	characteristic
237	C₂	#13131194	1	1	yes	characteristic
238	C₂	#13131195	1	1	yes	characteristic
239	C₂	#13131198	1	1	yes	characteristic
240	C₂	#13131199	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×#261
commutator	C₂²×C₁₃	C₂²×C₁₆
Fitting	C₂×C₄×C₈×C₁₃	C₄
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×#1055	C₂³
hypercenter	C₂×C₄×C₈	#261
hypocenter	C₁₃	#12879
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
56. C₄×Dic₁₃	C₁₆	semidirect
186. C₁₃	#12879	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718260	2
#13129875	2
#13129874	2
C₄×C₈×Dic₁₃	2
#12879	13