Subgroups — #15863155

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 200 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123400	C₂	1	2	yes	maximal
2	#13114977	C₂	1	4	yes	maximal, complemented
3	#13114974	C₂	1	4	yes	maximal, complemented
4	#13114563	C₂	1	2	yes	maximal
5	C₂×#11718144	C₂	1	2	yes	maximal
6	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
10	C₂²×#6099	C₂²	1	4	yes	—
11	C₁₃×#405	C₄	1	8	yes	—
12	C₂×C₁₃×#97	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
14	#11718660	—	2	8	no	—
15	#11718585	C₂²	1	8	yes	—
16	#11718144	C₂²	1	4	yes	—
17	#11718144	C₂²	1	8	yes	—
18	#11718144	C₂²	1	4	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	4	no	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
22	C₂²×C₈×C₁₃	—	2	4	no	—
23	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
24	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
26	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
27	C₁₃×#117	C₂×C₄	1	4	yes	—
28	C₁₃×#117	D₄	1	4	yes	—
29	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	Q₈	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
32	C₂×#6099	—	2	4	no	—
33	C₂×#6099	—	2	8	no	—
34	C₂×#6099	—	2	8	no	—
35	C₂×#6099	C₂³	1	8	yes	—
36	C₂²×#1055	—	2	4	no	—
37	C₁₃×#97	C₂×C₄	1	8	yes	—
38	C₄×C₈×C₁₃	C₂×C₄	1	8	yes	—
39	C₄⋊#1055	—	2	16	no	—
40	#20768	—	13	1	no	maximal, Sylow
41	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
43	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×Q₈	1	4	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	#41	1	4	yes	—
47	C₂²×C₄×C₁₃	#41	1	4	yes	—
48	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
49	C₁₃×M₁₆	—	2	8	no	—
50	C₂×C₈×C₁₃	—	2	8	no	—
51	C₂×C₈×C₁₃	—	2	2	no	—
52	C₂×C₈×C₁₃	—	2	2	no	—
53	C₂×C₈×C₁₃	—	2	4	no	—
54	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
55	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
56	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
57	C₁₃×C₄⋊C₄	C₄⋊C₄	1	8	yes	—
58	C₁₃×C₂²⋊C₄	C₄⋊C₄	1	8	yes	—
59	C₄²×C₁₃	C₂²×C₄	1	4	yes	—
60	C₄²×C₁₃	C₂×D₄	1	4	yes	—
61	#6099	—	2	8	no	—
62	#6099	—	2	8	no	—
63	#6099	—	4	8	no	—
64	C₂×#1055	—	2	8	no	—
65	C₂×#1055	—	4	8	no	—
66	#2905	—	13	2	no	—
67	C₂×#405	D₁₃	1	1	yes	characteristic
68	#1838	—	13	4	no	—
69	#1837	—	13	4	no	—
70	#1738	—	13	2	no	—
71	C₂×#341	—	13	2	no	—
72	C₂³×C₁₃	—	4	2	no	—
73	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
74	C₂³×C₁₃	#125	1	4	yes	—
75	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₁₃	—	4	4	no	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	8	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
79	C₂×C₄×C₁₃	#117	1	4	yes	—
80	C₂×C₄×C₁₃	#117	1	4	yes	—
81	C₂×C₄×C₁₃	#124	1	8	yes	—
82	C₂×C₄×C₁₃	#125	1	4	yes	—
83	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
86	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	—	2	2	no	—
89	C₈×C₁₃	—	2	4	no	—
90	#1055	—	4	8	no	—
91	C₂²×M₁₆	—	13	4	no	—
92	C₂²×M₁₆	C₂×D₁₃	1	1	yes	characteristic
93	C₂³×C₈	C₂×D₁₃	1	1	yes	characteristic
94	C₂×#117	C₂×D₁₃	1	1	yes	characteristic
95	#424	—	26	8	no	—
96	#423	—	13	8	no	—
97	#405	Dic₁₃	1	8	yes	—
98	C₂×#97	C₂×D₁₃	1	2	yes	—
99	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
100	#341	—	13	4	no	—
101	#341	—	13	8	no	—
102	#341	—	13	4	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	2	4	no	—
105	C₂²×C₁₃	—	4	4	no	—
106	C₂²×C₁₃	#382	1	4	yes	—
107	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
108	C₂²×C₁₃	C₂×#125	1	2	yes	—
109	C₄×C₁₃	—	2	8	no	—
110	C₄×C₁₃	#382	1	4	yes	—
111	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
112	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
113	C₄×C₁₃	C₂×#124	1	2	yes	—
114	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
115	C₂×M₁₆	—	2	4	no	—
116	C₂×M₁₆	—	13	8	no	—
117	C₂×M₁₆	—	26	4	no	—
118	C₂×M₁₆	—	26	8	no	—
119	C₂×M₁₆	—	26	8	no	—
120	C₂×M₁₆	C₂×Dic₁₃	1	4	yes	—
121	C₂²×C₈	—	2	4	no	—
122	C₂²×C₈	—	26	4	no	—
123	C₂²×C₈	C₂×Dic₁₃	1	4	yes	—
124	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
125	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₁₃	1	2	yes	—
127	#117	D₅₂	1	4	yes	—
128	#117	C₂×Dic₁₃	1	4	yes	—
129	C₂×C₄⋊C₄	Dic₂₆	1	2	yes	—
130	C₂×C₂²⋊C₄	Dic₂₆	1	2	yes	—
131	C₂×C₄²	C₂²×D₁₃	1	2	yes	—
132	C₄⋊C₈	—	26	16	no	—
133	#97	C₂×Dic₁₃	1	8	yes	—
134	C₄×C₈	C₂×Dic₁₃	1	8	yes	—
135	C₂×C₁₃	—	2	2	no	—
136	C₂×C₁₃	—	2	4	no	—
137	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
138	C₂×C₁₃	#2916	1	2	yes	—
139	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
140	C₂²×C₄	C₂×Dic₂₆	1	4	yes	—
141	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
142	C₂²×C₄	C₂×D₅₂	1	2	yes	—
143	C₂²×C₄	#6127	1	4	yes	—
144	C₂²×C₄	#6127	1	4	yes	—
145	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
146	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
147	M₁₆	—	2	8	no	—
148	M₁₆	—	26	8	no	—
149	M₁₆	—	26	8	no	—
150	M₁₆	—	52	8	no	—
151	C₂×C₈	—	2	8	no	—
152	C₂×C₈	—	2	2	no	—
153	C₂×C₈	—	2	2	no	—
154	C₂×C₈	—	2	4	no	—
155	C₂×C₈	—	26	8	no	—
156	C₂×C₈	—	52	8	no	—
157	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
158	C₂×C₈	C₂²×Dic₁₃	1	4	yes	—
159	C₂×C₈	C₂²×Dic₁₃	1	2	yes	—
160	C₄⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
161	C₂²⋊C₄	C₄⋊Dic₁₃	1	8	yes	—
162	C₄²	C₂×D₅₂	1	4	yes	—
163	C₄²	C₂²×Dic₁₃	1	4	yes	—
164	C₁₃	#20768	1	1	yes	characteristic, complemented, Sylow
165	C₂³	—	4	2	no	—
166	C₂³	#88124	1	4	yes	—
167	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
168	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
169	C₂×C₄	—	4	4	no	—
170	C₂×C₄	#88124	1	4	yes	—
171	C₂×C₄	#88129	1	8	yes	—
172	C₂×C₄	C₂×C₄⋊Dic₁₃	1	8	yes	—
173	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
174	C₂×C₄	#88180	1	4	yes	—
175	C₂×C₄	#88180	1	4	yes	—
176	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
177	C₂×C₄	C₂×#6127	1	2	yes	—
178	C₂×C₄	C₂×#6127	1	2	yes	—
179	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
180	C₈	—	2	2	no	—
181	C₈	—	2	2	no	—
182	C₈	—	2	4	no	—
183	C₈	—	52	8	no	—
184	C₂²	—	2	2	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	4	4	no	—
187	C₂²	#11718597	1	4	yes	—
188	C₂²	C₂×#88124	1	2	yes	—
189	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
190	C₄	—	2	8	no	—
191	C₄	#11718597	1	4	yes	—
192	C₄	C₂×#88129	1	2	yes	—
193	C₄	C₂×#88180	1	1	yes	characteristic
194	C₄	C₂×#88180	1	1	yes	characteristic
195	C₂	—	2	2	no	—
196	C₂	—	2	4	no	—
197	C₂	C₂×#11718597	1	1	yes	characteristic
198	C₂	#13123447	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₁₃×#405	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×#405	D₁₃
hypocenter	C₁₃	#20768
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405

Complements

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

N	H ≅ G/N	type
2. #13114977	C₂	semidirect
3. #13114974	C₂	semidirect
164. C₁₃	#20768	semidirect

Maximal subgroups

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

subgroup	index
#13123400	2
#13114977	2
#13114974	2
#13114563	2
C₂×#11718144	2
C₂×C₁₃×#405	2
#20768	13