Subgroups — #15863160

Order 3328.

818 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#11718585	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	4	yes	maximal
5	C₂×C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
8	C₂²×#6099	C₂²	1	4	yes	—
9	C₄×C₁₃×M₁₆	C₄	1	4	yes	—
10	C₄×C₁₃×M₁₆	C₄	1	4	yes	—
11	C₂×C₁₃×#97	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
13	#11718585	—	2	4	no	—
14	#11718585	C₂²	1	4	yes	—
15	#11718585	C₂²	1	4	yes	—
16	C₂×C₄⋊#1055	—	2	4	no	—
17	#11718144	C₂²	1	16	yes	—
18	C₂³×C₄×C₁₃	Q₈	1	2	yes	—
19	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
20	C₂×C₁₃×M₁₆	—	2	8	no	—
21	C₂×C₁₃×M₁₆	C₂×C₄	1	8	yes	—
22	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
23	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₁₃	D₄	1	4	yes	—
26	C₂×C₄²×C₁₃	Q₈	1	2	yes	—
27	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
29	C₂×#6099	—	2	4	no	—
30	C₂×#6099	—	2	16	no	—
31	C₂×#6099	C₂³	1	8	yes	—
32	C₂²×#1055	—	2	4	no	—
33	C₁₃×#97	C₂×C₄	1	8	yes	—
34	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
35	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
36	C₄⋊#1055	—	2	8	no	—
37	C₄⋊#1055	—	2	8	no	—
38	#20769	—	13	1	no	maximal, Sylow
39	C₂⁴×C₁₃	C₂×Q₈	1	1	yes	characteristic
40	C₂²×C₄×C₁₃	—	4	4	no	—
41	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
42	C₂²×C₄×C₁₃	C₄⋊C₄	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
45	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
46	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
47	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	#41	1	8	yes	—
49	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
50	C₁₃×M₁₆	—	2	16	no	—
51	C₂×C₈×C₁₃	—	2	8	no	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	8	yes	—
53	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
54	C₄²×C₁₃	C₄⋊C₄	1	4	yes	—
55	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
56	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
57	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
58	C₄²×C₁₃	C₂×D₄	1	2	yes	—
59	C₄²×C₁₃	C₂×D₄	1	2	yes	—
60	#6099	—	2	16	no	—
61	#6099	—	4	8	no	—
62	C₂×#1055	—	2	8	no	—
63	C₂×#1055	—	4	8	no	—
64	C₂×#423	—	13	2	no	—
65	C₂×C₄×M₁₆	D₁₃	1	1	yes	characteristic
66	#1838	—	13	4	no	—
67	#1837	—	13	4	no	—
68	#1738	—	13	4	no	—
69	C₂³×C₁₃	—	4	2	no	—
70	C₂³×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
71	C₂³×C₁₃	#125	1	4	yes	—
72	C₂³×C₁₃	C₂²×Q₈	1	1	yes	characteristic
73	C₂×C₄×C₁₃	—	2	8	no	—
74	C₂×C₄×C₁₃	—	2	4	no	—
75	C₂×C₄×C₁₃	—	4	4	no	—
76	C₂×C₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
79	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
81	C₂×C₄×C₁₃	#117	1	8	yes	—
82	C₂×C₄×C₁₃	#124	1	4	yes	—
83	C₂×C₄×C₁₃	#124	1	4	yes	—
84	C₂×C₄×C₁₃	#125	1	4	yes	—
85	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×#41	1	4	yes	—
88	C₈×C₁₃	—	2	8	no	—
89	#1055	—	4	8	no	—
90	C₂²×M₁₆	—	13	4	no	—
91	C₂²×M₁₆	C₂×D₁₃	1	2	yes	—
92	C₂²×C₄²	C₂×D₁₃	1	1	yes	characteristic
93	#423	—	13	4	no	—
94	#423	—	13	4	no	—
95	#423	—	26	4	no	—
96	C₂×C₄⋊C₈	—	26	4	no	—
97	C₄×M₁₆	Dic₁₃	1	4	yes	—
98	C₄×M₁₆	Dic₁₃	1	4	yes	—
99	C₂×#97	C₂×D₁₃	1	2	yes	—
100	C₂×C₄×C₈	C₂×D₁₃	1	2	yes	—
101	#341	—	13	16	no	—
102	C₂²×C₁₃	—	2	4	no	—
103	C₂²×C₁₃	—	2	2	no	—
104	C₂²×C₁₃	—	4	4	no	—
105	C₂²×C₁₃	#382	1	4	yes	—
106	C₂²×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²×C₁₃	C₂×#125	1	2	yes	—
108	C₄×C₁₃	—	2	4	no	—
109	C₄×C₁₃	—	2	2	no	—
110	C₄×C₁₃	#382	1	4	yes	—
111	C₄×C₁₃	#430	1	4	yes	—
112	C₄×C₁₃	C₂×#117	1	2	yes	—
113	C₄×C₁₃	C₂×#124	1	2	yes	—
114	C₂³×C₄	Dic₂₆	1	2	yes	—
115	C₂³×C₄	C₂²×D₁₃	1	1	yes	characteristic
116	C₂×M₁₆	—	2	8	no	—
117	C₂×M₁₆	—	13	8	no	—
118	C₂×M₁₆	—	26	4	no	—
119	C₂×M₁₆	—	26	16	no	—
120	C₂×M₁₆	C₂×Dic₁₃	1	8	yes	—
121	C₂²×C₈	—	26	4	no	—
122	C₂²×C₈	C₂²×D₁₃	1	4	yes	—
123	C₂×C₄²	Dic₂₆	1	2	yes	—
124	C₂×C₄²	D₅₂	1	4	yes	—
125	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
126	C₂×C₄²	C₂×Dic₁₃	1	2	yes	—
127	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
128	C₂×C₄²	C₂²×D₁₃	1	1	yes	characteristic
129	C₄⋊C₈	—	26	8	no	—
130	C₄⋊C₈	—	26	8	no	—
131	#97	C₂×Dic₁₃	1	8	yes	—
132	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
133	C₄×C₈	C₂×Dic₁₃	1	4	yes	—
134	C₂×C₁₃	—	2	2	no	—
135	C₂×C₁₃	—	2	4	no	—
136	C₂×C₁₃	C₂×#382	1	1	yes	characteristic
137	C₂×C₁₃	C₂×#430	1	2	yes	—
138	C₂⁴	C₂×Dic₂₆	1	1	yes	characteristic
139	C₂²×C₄	—	4	4	no	—
140	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
141	C₂²×C₄	C₄⋊Dic₁₃	1	4	yes	—
142	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
143	C₂²×C₄	C₂×Dic₂₆	1	2	yes	—
144	C₂²×C₄	C₂×Dic₂₆	1	1	yes	characteristic
145	C₂²×C₄	C₂×D₅₂	1	2	yes	—
146	C₂²×C₄	#6127	1	8	yes	—
147	C₂²×C₄	C₂²×Dic₁₃	1	2	yes	—
148	C₂²×C₄	C₂³×D₁₃	1	1	yes	characteristic
149	M₁₆	—	2	16	no	—
150	M₁₆	—	26	16	no	—
151	M₁₆	—	52	8	no	—
152	C₂×C₈	—	2	8	no	—
153	C₂×C₈	—	26	8	no	—
154	C₂×C₈	—	52	8	no	—
155	C₂×C₈	C₂²×Dic₁₃	1	8	yes	—
156	C₄²	C₄⋊Dic₁₃	1	4	yes	—
157	C₄²	C₄⋊Dic₁₃	1	4	yes	—
158	C₄²	C₂×D₅₂	1	2	yes	—
159	C₄²	C₂×D₅₂	1	2	yes	—
160	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
161	C₄²	C₂²×Dic₁₃	1	1	yes	characteristic
162	C₄²	C₂²×Dic₁₃	1	2	yes	—
163	C₁₃	#20769	1	1	yes	characteristic, complemented, Sylow
164	C₂³	—	4	2	no	—
165	C₂³	#88124	1	4	yes	—
166	C₂³	C₂×C₄⋊Dic₁₃	1	2	yes	—
167	C₂³	C₂²×Dic₂₆	1	1	yes	characteristic
168	C₂×C₄	—	2	8	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	4	4	no	—
171	C₂×C₄	—	4	2	no	—
172	C₂×C₄	#88124	1	4	yes	—
173	C₂×C₄	#88129	1	4	yes	—
174	C₂×C₄	#88129	1	4	yes	—
175	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
176	C₂×C₄	C₂×C₄⋊Dic₁₃	1	4	yes	—
177	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
178	C₂×C₄	C₂×C₄⋊Dic₁₃	1	2	yes	—
179	C₂×C₄	#88180	1	8	yes	—
180	C₂×C₄	C₂²×D₅₂	1	1	yes	characteristic
181	C₂×C₄	C₂×#6127	1	4	yes	—
182	C₂×C₄	C₂³×Dic₁₃	1	1	yes	characteristic
183	C₈	—	2	8	no	—
184	C₈	—	52	8	no	—
185	C₂²	—	2	4	no	—
186	C₂²	—	2	2	no	—
187	C₂²	—	4	4	no	—
188	C₂²	#11718597	1	4	yes	—
189	C₂²	C₂×#88124	1	2	yes	—
190	C₂²	C₂²×C₄⋊Dic₁₃	1	1	yes	characteristic
191	C₄	—	2	4	no	—
192	C₄	—	2	2	no	—
193	C₄	#11718597	1	4	yes	—
194	C₄	#11718786	1	4	yes	—
195	C₄	C₂×#88129	1	2	yes	—
196	C₄	C₂×#88180	1	2	yes	—
197	C₂	—	2	2	no	—
198	C₂	—	2	4	no	—
199	C₂	C₂×#11718597	1	1	yes	characteristic
200	C₂	C₂×#11718786	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₅₂
commutator	C₂×C₄×C₁₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×M₁₆	C₂
socle	C₂²×C₁₃	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₁₃	C₂×Q₈
cosocle	C₂²×C₄×C₁₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₁₃
hypocenter	C₁₃	#20769
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
2. #13114977	C₂	semidirect
3. #13114974	C₂	semidirect
163. C₁₃	#20769	semidirect

Maximal subgroups

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

subgroup	index
C₂×#11718585	2
#13114977	2
#13114974	2
#13114563	2
C₂×C₄×C₁₃×M₁₆	2
#20769	13