Subgroups — #15778229

Order 3328.

414 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718093	C₂	1	2	yes	maximal
2	C₂×#11718053	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₁₃	C₂	1	1	yes	maximal, characteristic
4	#13112913	C₂	1	2	yes	maximal
5	C₂×C₁₃×#346	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×#346	—	2	2	no	—
16	#11718093	C₄	1	8	yes	complemented
17	#11718053	—	2	2	no	—
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	D₄	1	2	yes	—
31	C₄×C₈×C₁₃	Q₈	1	2	yes	—
32	C₈×Dic₁₃	—	2	2	no	—
33	C₈×Dic₁₃	—	2	2	no	—
34	C₈×Dic₁₃	C₈	1	4	yes	—
35	C₈×Dic₁₃	C₂×C₄	1	4	yes	—
36	C₂×#6090	—	2	2	no	—
37	C₂×#6090	—	2	2	no	—
38	C₂×#6090	C₂×C₄	1	4	yes	—
39	C₄×#1055	—	2	2	no	—
40	#12653	—	13	1	no	maximal, Sylow
41	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
42	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
44	C₂²×Dic₁₃	C₂×C₈	1	2	yes	—
45	C₂²×Dic₁₃	C₂×C₈	1	1	yes	characteristic
46	C₂×C₈×C₁₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₁₃	C₂×C₈	1	2	yes	—
48	C₂×C₈×C₁₃	M₁₆	1	2	yes	—
49	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
50	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
52	C₂×C₈×C₁₃	#41	1	2	yes	—
53	C₄²×C₁₃	C₄⋊C₄	1	2	yes	—
54	C₄²×C₁₃	M₁₆	1	2	yes	—
55	C₄×Dic₁₃	—	2	1	no	—
56	C₄×Dic₁₃	—	2	1	no	—
57	C₄×Dic₁₃	—	2	4	no	—
58	C₄×Dic₁₃	—	2	4	no	—
59	C₄×Dic₁₃	C₄²	1	2	yes	—
60	C₄×Dic₁₃	C₂×C₈	1	2	yes	—
61	C₂×#1055	—	2	1	no	—
62	C₂×#1055	—	2	1	no	—
63	C₂×#1055	—	2	1	no	—
64	C₂×#1055	—	2	1	no	—
65	C₂×#1055	C₄²	1	2	yes	—
66	C₂×#1055	C₂×C₈	1	2	yes	—
67	C₁₃×C₁₆	—	2	4	no	—
68	#6090	—	2	4	no	—
69	C₂×C₄⋊C₁₆	—	13	2	no	—
70	C₂×#346	—	13	1	no	—
71	C₂×#346	D₁₃	1	1	yes	characteristic
72	C₄²×C₈	—	13	1	no	—
73	#1374	—	13	2	no	—
74	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
75	C₂×C₄×C₁₃	—	2	2	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	C₄⋊C₈	1	4	yes	—
78	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂²×C₈	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	2	no	—
85	C₂×Dic₁₃	—	2	2	no	—
86	C₂×Dic₁₃	—	2	2	no	—
87	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
88	C₂×Dic₁₃	C₄×C₈	1	2	yes	—
89	C₂×Dic₁₃	M₃₂	1	4	yes	—
90	C₂×Dic₁₃	M₃₂	1	4	yes	—
91	C₈×C₁₃	C₄⋊C₈	1	4	yes	—
92	C₈×C₁₃	C₄×D₄	1	2	yes	—
93	C₈×C₁₃	C₄×Q₈	1	2	yes	—
94	#1055	—	2	2	no	—
95	#1055	—	2	2	no	—
96	C₂²×C₁₆	—	13	2	no	—
97	C₂²×C₁₆	C₂×D₁₃	1	2	yes	—
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	C₂×D₁₃	1	1	yes	characteristic
104	C₄³	—	13	1	no	—
105	C₄⋊C₁₆	—	13	8	no	—
106	#346	—	2	2	no	—
107	#346	—	26	2	no	—
108	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
109	C₂²×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
110	C₂²×C₁₃	#431	1	1	yes	characteristic
111	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
112	C₂²×C₁₃	C₂×M₃₂	1	2	yes	—
113	C₄×C₁₃	—	2	4	no	—
114	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
115	C₄×C₁₃	C₄×M₁₆	1	1	yes	characteristic
116	C₄×C₁₃	C₂×C₄⋊C₈	1	1	yes	characteristic
117	C₄×C₁₃	#431	1	1	yes	characteristic
118	Dic₁₃	—	2	2	no	—
119	Dic₁₃	—	2	4	no	—
120	Dic₁₃	#346	1	4	yes	complemented
121	C₂²×C₈	—	13	1	no	—
122	C₂²×C₈	—	13	1	no	—
123	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
125	C₂×C₄²	—	13	1	no	—
126	C₂×C₄²	—	13	2	no	—
127	C₂×C₄²	—	13	2	no	—
128	C₂×C₄²	—	13	1	no	—
129	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
130	C₂×C₁₆	—	2	2	no	—
131	C₂×C₁₆	—	2	2	no	—
132	C₂×C₁₆	—	13	4	no	—
133	C₂×C₁₆	—	26	2	no	—
134	C₂×C₁₆	—	26	2	no	—
135	C₂×C₁₆	C₄×D₁₃	1	4	yes	—
136	C₄×C₈	—	13	4	no	—
137	C₄×C₈	—	13	4	no	—
138	C₄×C₈	—	26	2	no	—
139	C₄×C₈	—	26	2	no	—
140	C₄×C₈	—	26	2	no	—
141	C₄×C₈	Dic₂₆	1	2	yes	—
142	C₄×C₈	C₂²⋊D₁₃	1	2	yes	—
143	C₂×C₁₃	C₄×C₄⋊C₈	1	1	yes	characteristic
144	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
145	C₂×C₁₃	C₄×M₃₂	1	1	yes	characteristic
146	C₂×C₁₃	#2150	1	2	yes	—
147	C₂×C₁₃	#2163	1	2	yes	—
148	C₂²×C₄	—	13	1	no	—
149	C₂²×C₄	—	13	2	no	—
150	C₂²×C₄	—	13	1	no	—
151	C₂²×C₄	C₈×D₁₃	1	2	yes	—
152	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
153	C₂×C₈	—	13	2	no	—
154	C₂×C₈	—	13	2	no	—
155	C₂×C₈	—	26	1	no	—
156	C₂×C₈	—	26	1	no	—
157	C₂×C₈	—	26	1	no	—
158	C₂×C₈	—	26	1	no	—
159	C₂×C₈	C₈×D₁₃	1	2	yes	—
160	C₂×C₈	#6094	1	2	yes	—
161	C₂×C₈	#6101	1	2	yes	—
162	C₂×C₈	C₂×Dic₂₆	1	1	yes	characteristic
163	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
164	C₂×C₈	#6127	1	2	yes	—
165	C₂×C₈	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
166	C₄²	—	13	2	no	—
167	C₄²	—	13	2	no	—
168	C₄²	—	26	4	no	—
169	C₄²	—	26	4	no	—
170	C₄²	—	26	1	no	—
171	C₄²	—	26	1	no	—
172	C₄²	#6094	1	2	yes	—
173	C₄²	#6101	1	2	yes	—
174	C₁₆	—	2	4	no	—
175	C₁₆	—	26	4	no	—
176	C₁₃	#12653	1	1	yes	characteristic, complemented, Sylow
177	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	13	4	no	—
181	C₂×C₄	—	13	2	no	—
182	C₂×C₄	—	13	4	no	—
183	C₂×C₄	—	13	2	no	—
184	C₂×C₄	—	26	2	no	—
185	C₂×C₄	—	26	2	no	—
186	C₂×C₄	—	26	2	no	—
187	C₂×C₄	#88054	1	4	yes	—
188	C₂×C₄	C₄²×D₁₃	1	1	yes	characteristic
189	C₂×C₄	#88126	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
191	C₂×C₄	C₂×#6094	1	1	yes	characteristic
192	C₂×C₄	C₂×#6094	1	1	yes	characteristic
193	C₂×C₄	C₂×#6101	1	1	yes	characteristic
194	C₈	—	26	2	no	—
195	C₈	—	26	2	no	—
196	C₈	#88054	1	4	yes	—
197	C₈	C₄×Dic₂₆	1	2	yes	—
198	C₈	C₄×C₂²⋊D₁₃	1	2	yes	—
199	C₂²	C₄×C₈×D₁₃	1	1	yes	characteristic
200	C₂²	#11718346	1	1	yes	characteristic
201	C₂²	D₁₃×M₃₂	1	2	yes	—
202	C₂²	D₁₃×M₃₂	1	2	yes	—
203	C₂²	C₂×#88054	1	1	yes	characteristic
204	C₄	—	2	4	no	—
205	C₄	—	13	4	no	—
206	C₄	—	26	2	no	—
207	C₄	—	26	4	no	—
208	C₄	#11718346	1	1	yes	characteristic
209	C₄	C₄×#6094	1	1	yes	characteristic
210	C₄	C₄×#6101	1	1	yes	characteristic
211	C₄	C₂×#88054	1	1	yes	characteristic
212	C₂	C₄×#88054	1	1	yes	characteristic
213	C₂	D₁₃×#346	1	1	yes	characteristic
214	C₂	#13116429	1	1	yes	characteristic
215	C₂	#13116591	1	2	yes	—
216	C₂	#13116637	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₁₃
commutator	C₂²×C₁₃	C₂×C₄×C₈
Fitting	C₂×C₁₃×#346	C₂
socle, prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂²×C₈×C₁₃	C₂³
hypercenter	C₂×#346	D₁₃
hypocenter	C₁₃	#12653
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
16. #11718093	C₄	semidirect
120. Dic₁₃	#346	semidirect
176. C₁₃	#12653	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718093	2
C₂×#11718053	2
C₄×C₈×Dic₁₃	2
#13112913	2
C₂×C₁₃×#346	2
#12653	13