Subgroups — #16790238

Order 3328.

342 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13129875	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	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×Dic₁₃	C₈	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	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
23	C₂²×#1055	C₂³	1	1	yes	characteristic
24	C₂×#6092	—	2	4	no	—
25	C₂×#6092	—	2	4	no	—
26	C₂×#6092	—	2	4	no	—
27	C₄×C₈×C₁₃	—	2	2	no	—
28	C₈×Dic₁₃	—	2	2	no	—
29	C₈×Dic₁₃	—	2	2	no	—
30	C₈×Dic₁₃	—	2	2	no	—
31	C₈×Dic₁₃	—	2	2	no	—
32	C₄×#1055	—	2	2	no	—
33	#12894	—	13	1	no	maximal, Sylow
34	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
35	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
36	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
37	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
40	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
41	C₂×C₈×C₁₃	—	2	1	no	—
42	C₂×C₈×C₁₃	—	2	1	no	—
43	C₂×C₈×C₁₃	—	2	1	no	—
44	C₂×C₈×C₁₃	—	2	1	no	—
45	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
47	C₄²×C₁₃	—	2	1	no	—
48	C₄²×C₁₃	—	2	1	no	—
49	C₄×Dic₁₃	—	2	2	no	—
50	C₄×Dic₁₃	—	2	1	no	—
51	C₄×Dic₁₃	—	2	1	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	C₂×#1055	—	2	1	no	—
58	C₂×#1055	—	2	1	no	—
59	C₂×#1055	M₁₆	1	2	yes	—
60	C₂×#1055	#41	1	2	yes	—
61	C₂×#1055	#41	1	2	yes	—
62	C₂×#1055	#41	1	2	yes	—
63	#6092	—	4	4	no	—
64	C₄²×C₈	—	13	1	no	—
65	#1374	—	13	2	no	—
66	#1374	—	13	2	no	—
67	#1374	—	13	2	no	—
68	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
69	C₂×C₄×C₁₃	—	2	1	no	—
70	C₂×C₄×C₁₃	—	2	1	no	—
71	C₂×C₄×C₁₃	—	2	1	no	—
72	C₂×C₄×C₁₃	—	2	1	no	—
73	C₂×C₄×C₁₃	C₂×C₁₆	1	2	yes	—
74	C₂×C₄×C₁₃	M₃₂	1	2	yes	—
75	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
76	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
77	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
78	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
80	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
81	C₂×Dic₁₃	—	2	1	no	—
82	C₂×Dic₁₃	—	2	1	no	—
83	C₂×Dic₁₃	—	2	1	no	—
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	C₂×C₁₆	1	2	yes	—
90	C₂×Dic₁₃	C₂×C₁₆	1	2	yes	—
91	C₂×Dic₁₃	M₃₂	1	2	yes	—
92	C₂×Dic₁₃	M₃₂	1	2	yes	—
93	C₈×C₁₃	—	2	2	no	—
94	C₈×C₁₃	—	2	2	no	—
95	#1055	—	2	2	no	—
96	#1055	#126	1	4	yes	—
97	C₂²×C₁₆	—	13	4	no	—
98	C₂×C₄×C₈	—	13	1	no	—
99	C₂×C₄×C₈	—	13	1	no	—
100	C₂×C₄×C₈	—	13	1	no	—
101	C₂×C₄×C₈	—	13	1	no	—
102	C₂×C₄×C₈	—	13	1	no	—
103	C₂×C₄×C₈	#261	1	1	yes	characteristic
104	C₄³	—	13	1	no	—
105	C₂²×C₁₃	#431	1	1	yes	characteristic
106	C₂²×C₁₃	#431	1	1	yes	characteristic
107	C₂²×C₁₃	#431	1	1	yes	characteristic
108	C₂²×C₁₃	C₂²×C₁₆	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
110	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×M₃₂	1	1	yes	characteristic
112	C₄×C₁₃	—	2	2	no	—
113	C₄×C₁₃	—	2	2	no	—
114	C₄×C₁₃	#383	1	1	yes	characteristic
115	C₄×C₁₃	#432	1	1	yes	characteristic
116	C₄×C₁₃	#432	1	1	yes	characteristic
117	C₄×C₁₃	#432	1	1	yes	characteristic
118	Dic₁₃	—	2	2	no	—
119	Dic₁₃	—	2	2	no	—
120	Dic₁₃	—	2	2	no	—
121	Dic₁₃	—	2	2	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	—	13	1	no	—
124	C₂²×C₈	C₂×#261	1	1	yes	characteristic
125	C₂²×C₈	C₂×#261	1	1	yes	characteristic
126	C₂×C₄²	—	13	1	no	—
127	C₂×C₄²	—	13	1	no	—
128	C₂×C₄²	—	13	1	no	—
129	C₂×C₄²	—	13	1	no	—
130	C₂×C₄²	—	13	1	no	—
131	C₂×C₄²	—	13	1	no	—
132	C₂×C₄²	C₂×#261	1	1	yes	characteristic
133	C₂×C₁₆	—	26	4	no	—
134	C₂×C₁₆	—	26	4	no	—
135	C₂×C₁₆	—	26	4	no	—
136	C₄×C₈	—	2	2	no	—
137	C₄×C₈	—	26	2	no	—
138	C₄×C₈	—	26	2	no	—
139	C₄×C₈	—	26	2	no	—
140	C₄×C₈	—	26	2	no	—
141	C₄×C₈	—	26	2	no	—
142	C₂×C₁₃	#1839	1	1	yes	characteristic
143	C₂×C₁₃	#2162	1	1	yes	characteristic
144	C₂×C₁₃	#2162	1	1	yes	characteristic
145	C₂×C₁₃	#2162	1	1	yes	characteristic
146	C₂×C₁₃	#2163	1	1	yes	characteristic
147	C₂×C₁₃	#2163	1	1	yes	characteristic
148	C₂×C₁₃	#2163	1	1	yes	characteristic
149	C₂²×C₄	—	13	1	no	—
150	C₂²×C₄	—	13	1	no	—
151	C₂²×C₄	—	13	1	no	—
152	C₂²×C₄	—	13	1	no	—
153	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
154	C₂²×C₄	D₁₃⋊C₈	1	1	yes	characteristic
155	C₂²×C₄	C₂²×#261	1	1	yes	characteristic
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	1	no	—
158	C₂×C₈	—	2	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	13	2	no	—
161	C₂×C₈	—	13	2	no	—
162	C₂×C₈	—	13	2	no	—
163	C₂×C₈	—	13	2	no	—
164	C₂×C₈	—	26	1	no	—
165	C₂×C₈	—	26	1	no	—
166	C₂×C₈	#6118	1	2	yes	—
167	C₂×C₈	#6122	1	2	yes	—
168	C₄²	—	2	1	no	—
169	C₄²	—	2	1	no	—
170	C₄²	—	13	4	no	—
171	C₄²	—	26	2	no	—
172	C₄²	—	26	2	no	—
173	C₄²	—	26	2	no	—
174	C₄²	—	26	1	no	—
175	C₄²	—	26	1	no	—
176	C₄²	—	26	1	no	—
177	C₄²	—	26	1	no	—
178	C₁₆	—	52	4	no	—
179	C₁₃	#12894	1	1	yes	characteristic, complemented, Sylow
180	C₂³	C₂×D₁₃⋊C₈	1	1	yes	characteristic
181	C₂×C₄	—	2	1	no	—
182	C₂×C₄	—	2	1	no	—
183	C₂×C₄	—	2	1	no	—
184	C₂×C₄	—	2	1	no	—
185	C₂×C₄	—	13	2	no	—
186	C₂×C₄	—	13	2	no	—
187	C₂×C₄	—	13	2	no	—
188	C₂×C₄	—	13	2	no	—
189	C₂×C₄	—	26	1	no	—
190	C₂×C₄	—	26	1	no	—
191	C₂×C₄	—	26	1	no	—
192	C₂×C₄	—	26	1	no	—
193	C₂×C₄	—	26	1	no	—
194	C₂×C₄	—	26	1	no	—
195	C₂×C₄	—	26	1	no	—
196	C₂×C₄	—	26	1	no	—
197	C₂×C₄	D₁₃⋊C₁₆	1	2	yes	—
198	C₂×C₄	#88098	1	2	yes	—
199	C₂×C₄	C₂×#6118	1	1	yes	characteristic
200	C₂×C₄	D₁₃⋊M₁₆	1	1	yes	characteristic
201	C₂×C₄	#88237	1	1	yes	characteristic
202	C₂×C₄	#88237	1	1	yes	characteristic
203	C₂×C₄	#88237	1	1	yes	characteristic
204	C₂×C₄	C₂×#6122	1	1	yes	characteristic
205	C₈	—	2	2	no	—
206	C₈	—	2	2	no	—
207	C₈	—	13	4	no	—
208	C₈	—	26	2	no	—
209	C₂²	#11719049	1	1	yes	characteristic
210	C₂²	#11719051	1	1	yes	characteristic
211	C₂²	C₂×D₁₃⋊C₁₆	1	1	yes	characteristic
212	C₂²	C₂×#88098	1	1	yes	characteristic
213	C₂²	D₁₃⋊M₃₂	1	1	yes	characteristic
214	C₂²	D₁₃⋊M₃₂	1	1	yes	characteristic
215	C₂²	#11719125	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	26	2	no	—
219	C₄	—	26	2	no	—
220	C₄	—	26	2	no	—
221	C₄	—	26	2	no	—
222	C₄	#11719054	1	1	yes	characteristic
223	C₄	#11719061	1	1	yes	characteristic
224	C₄	#11719128	1	1	yes	characteristic
225	C₄	#11719130	1	1	yes	characteristic
226	C₂	#13130764	1	1	yes	characteristic
227	C₂	#13131194	1	1	yes	characteristic
228	C₂	#13131195	1	1	yes	characteristic
229	C₂	#13131196	1	1	yes	characteristic
230	C₂	#13131198	1	1	yes	characteristic
231	C₂	#13131199	1	1	yes	characteristic
232	C₂	#13131200	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	#12894
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
179. C₁₃	#12894	semidirect

Maximal subgroups

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

subgroup	index
#13129875	2
#13129874	2
C₄×C₈×Dic₁₃	2
#12894	13