Subgroups — #15778230

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#6090	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718094	C₂	1	2	yes	maximal
3	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#13112913	C₂	1	2	yes	maximal
5	C₂×C₁₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
8	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
9	C₂×C₈×Dic₁₃	C₄	1	1	yes	characteristic
10	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₁₃	C₂²	1	1	yes	characteristic
12	C₂²×#6090	C₂²	1	2	yes	—
13	C₄²×Dic₁₃	C₄	1	1	yes	characteristic
14	C₂×C₄×#1055	C₄	1	1	yes	characteristic
15	C₁₃×C₄⋊C₁₆	C₄	1	4	yes	complemented
16	#11718094	—	2	4	no	—
17	C₄⋊#6090	C₄	1	4	yes	complemented
18	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
22	C₂×C₄×Dic₁₃	C₈	1	2	yes	—
23	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×Dic₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
26	C₂²×#1055	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₃×C₁₆	—	2	2	no	—
28	C₂×C₁₃×C₁₆	—	2	2	no	—
29	C₂×C₁₃×C₁₆	C₂×C₄	1	4	yes	—
30	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
32	C₈×Dic₁₃	—	2	1	no	—
33	C₈×Dic₁₃	—	2	1	no	—
34	C₈×Dic₁₃	—	2	2	no	—
35	C₈×Dic₁₃	—	2	2	no	—
36	C₈×Dic₁₃	D₄	1	2	yes	—
37	C₈×Dic₁₃	Q₈	1	2	yes	—
38	C₂×#6090	—	2	2	no	—
39	C₂×#6090	—	2	2	no	—
40	C₂×#6090	C₂×C₄	1	4	yes	—
41	C₄×#1055	C₈	1	4	yes	—
42	#12653	—	13	1	no	maximal, Sylow
43	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×C₄×C₁₃	C₂×C₈	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
46	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
47	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃	—	2	1	no	—
49	C₂×C₈×C₁₃	—	2	1	no	—
50	C₂×C₈×C₁₃	C₄²	1	2	yes	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
56	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
57	C₄²×C₁₃	C₄²	1	2	yes	—
58	C₄²×C₁₃	C₂×C₈	1	2	yes	—
59	C₄×Dic₁₃	—	2	4	no	—
60	C₄×Dic₁₃	—	2	1	no	—
61	C₄×Dic₁₃	—	2	1	no	—
62	C₄×Dic₁₃	—	2	4	no	—
63	C₄×Dic₁₃	C₄⋊C₄	1	2	yes	—
64	C₄×Dic₁₃	M₁₆	1	2	yes	—
65	C₂×#1055	—	2	1	no	—
66	C₂×#1055	—	2	1	no	—
67	C₂×#1055	C₄⋊C₄	1	2	yes	—
68	C₂×#1055	C₂×C₈	1	2	yes	—
69	C₂×#1055	C₂×C₈	1	2	yes	—
70	C₂×#1055	M₁₆	1	2	yes	—
71	C₁₃×C₁₆	—	2	4	no	—
72	#6090	—	2	4	no	—
73	C₂×C₄⋊C₁₆	—	13	1	no	—
74	C₂×C₄⋊C₁₆	D₁₃	1	1	yes	characteristic
75	C₂×#346	—	13	2	no	—
76	C₄²×C₈	—	13	1	no	—
77	#1374	—	13	2	no	—
78	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₄×C₈	1	2	yes	—
82	C₂×C₄×C₁₃	M₃₂	1	4	yes	—
83	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×Dic₁₃	—	2	2	no	—
90	C₂×Dic₁₃	—	2	2	no	—
91	C₂×Dic₁₃	—	2	2	no	—
92	C₂×Dic₁₃	—	2	2	no	—
93	C₂×Dic₁₃	C₄⋊C₈	1	4	yes	—
94	C₂×Dic₁₃	M₃₂	1	4	yes	—
95	C₈×C₁₃	—	2	2	no	—
96	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
97	C₈×C₁₃	C₄×D₄	1	1	yes	characteristic
98	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
99	C₈×C₁₃	C₄×Q₈	1	1	yes	characteristic
100	#1055	—	2	2	no	—
101	#1055	C₄⋊C₈	1	4	yes	—
102	C₂²×C₁₆	—	13	2	no	—
103	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
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	—	13	1	no	—
108	C₂×C₄×C₈	—	13	1	no	—
109	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
110	C₄³	—	13	1	no	—
111	C₄⋊C₁₆	—	13	4	no	—
112	C₄⋊C₁₆	Dic₁₃	1	4	yes	complemented
113	#346	—	26	4	no	—
114	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
115	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
116	C₂²×C₁₃	#431	1	1	yes	characteristic
117	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
118	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	#346	1	4	yes	complemented
121	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
122	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
123	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
124	C₄×C₁₃	#431	1	1	yes	characteristic
125	Dic₁₃	—	2	4	no	—
126	Dic₁₃	—	2	4	no	—
127	C₂²×C₈	—	13	1	no	—
128	C₂²×C₈	—	13	1	no	—
129	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
130	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
131	C₂×C₄²	—	13	1	no	—
132	C₂×C₄²	—	13	2	no	—
133	C₂×C₄²	—	13	2	no	—
134	C₂×C₄²	—	13	1	no	—
135	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
136	C₂×C₁₆	—	2	2	no	—
137	C₂×C₁₆	—	2	2	no	—
138	C₂×C₁₆	—	13	4	no	—
139	C₂×C₁₆	—	26	2	no	—
140	C₂×C₁₆	—	26	2	no	—
141	C₂×C₁₆	C₂×Dic₁₃	1	4	yes	—
142	C₄×C₈	—	13	2	no	—
143	C₄×C₈	—	13	2	no	—
144	C₄×C₈	—	13	4	no	—
145	C₄×C₈	—	26	2	no	—
146	C₄×C₈	—	26	2	no	—
147	C₄×C₈	—	26	1	no	—
148	C₄×C₈	—	26	1	no	—
149	C₄×C₈	C₄×D₁₃	1	2	yes	—
150	C₄×C₈	C₂×Dic₁₃	1	2	yes	—
151	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
152	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
153	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
154	C₂×C₁₃	#2150	1	2	yes	—
155	C₂×C₁₃	#2163	1	2	yes	—
156	C₂²×C₄	—	13	2	no	—
157	C₂²×C₄	—	13	2	no	—
158	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
159	C₂²×C₄	C₈×D₁₃	1	1	yes	characteristic
160	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
161	C₂×C₈	—	2	1	no	—
162	C₂×C₈	—	2	1	no	—
163	C₂×C₈	—	13	2	no	—
164	C₂×C₈	—	13	2	no	—
165	C₂×C₈	—	13	2	no	—
166	C₂×C₈	—	13	2	no	—
167	C₂×C₈	—	26	1	no	—
168	C₂×C₈	—	26	1	no	—
169	C₂×C₈	C₄×Dic₁₃	1	2	yes	—
170	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
171	C₂×C₈	D₄×D₁₃	1	1	yes	characteristic
172	C₂×C₈	#6129	1	1	yes	characteristic
173	C₂×C₈	Q₈×D₁₃	1	1	yes	characteristic
174	C₂×C₈	#6131	1	1	yes	characteristic
175	C₂×C₈	C₂²×Dic₁₃	1	1	yes	characteristic
176	C₄²	—	13	2	no	—
177	C₄²	—	13	2	no	—
178	C₄²	—	26	4	no	—
179	C₄²	—	26	1	no	—
180	C₄²	—	26	1	no	—
181	C₄²	—	26	4	no	—
182	C₄²	C₈×D₁₃	1	2	yes	—
183	C₄²	C₄×Dic₁₃	1	2	yes	—
184	C₁₆	—	2	4	no	—
185	C₁₆	—	26	4	no	—
186	C₁₃	#12653	1	1	yes	characteristic, complemented, Sylow
187	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	13	4	no	—
190	C₂×C₄	—	13	4	no	—
191	C₂×C₄	—	26	2	no	—
192	C₂×C₄	—	26	2	no	—
193	C₂×C₄	—	26	2	no	—
194	C₂×C₄	—	26	2	no	—
195	C₂×C₄	#88038	1	4	yes	—
196	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
197	C₂×C₄	C₈×Dic₁₃	1	2	yes	—
198	C₂×C₄	C₄⋊C₄×D₁₃	1	1	yes	characteristic
199	C₂×C₄	#88146	1	1	yes	characteristic
200	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
201	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
202	C₂×C₄	M₁₆×D₁₃	1	1	yes	characteristic
203	C₂×C₄	C₂×C₄×Dic₁₃	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	13	4	no	—
206	C₈	—	26	2	no	—
207	C₈	#88141	1	1	yes	characteristic
208	C₈	#88147	1	1	yes	characteristic
209	C₈	D₄×Dic₁₃	1	1	yes	characteristic
210	C₈	Q₈×Dic₁₃	1	1	yes	characteristic
211	C₂²	D₁₃×C₄⋊C₈	1	1	yes	characteristic
212	C₂²	#11718494	1	1	yes	characteristic
213	C₂²	C₂×#88038	1	2	yes	—
214	C₂²	D₁₃×M₃₂	1	2	yes	—
215	C₂²	C₂×C₈×Dic₁₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	26	4	no	—
218	C₄	—	26	4	no	—
219	C₄	#11718094	1	4	yes	complemented
220	C₄	D₁₃×C₄⋊C₈	1	1	yes	characteristic
221	C₄	#11718494	1	1	yes	characteristic
222	C₄	C₄⋊C₄×Dic₁₃	1	1	yes	characteristic
223	C₄	M₁₆×Dic₁₃	1	1	yes	characteristic
224	C₂	C₄⋊C₈×Dic₁₃	1	1	yes	characteristic
225	C₂	C₂×#11718094	1	1	yes	characteristic
226	C₂	M₃₂×Dic₁₃	1	1	yes	characteristic
227	C₂	#13116590	1	2	yes	—
228	C₂	#13116638	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×C₄⋊C₁₆	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₁₃
hypocenter	C₁₃	#12653
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₃×C₄⋊C₁₆	C₄	semidirect
17. C₄⋊#6090	C₄	semidirect
112. C₄⋊C₁₆	Dic₁₃	semidirect
120. C₄×C₁₃	#346	semidirect
186. C₁₃	#12653	semidirect
219. C₄	#11718094	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#6090	2
C₂×#11718094	2
C₄×C₈×Dic₁₃	2
#13112913	2
C₂×C₁₃×C₄⋊C₁₆	2
#12653	13