Subgroups — #16016413

Order 3328.

506 conjugacy classes of subgroups.

The subgroup lattice has 194 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#13123770	C₂	1	1	yes	maximal, characteristic
2	#13116744	C₂	1	4	yes	maximal, complemented
3	C₁₆×Dic₂₆	C₂	1	4	yes	maximal, complemented
4	#13116594	C₂	1	2	yes	maximal
5	#13116592	C₂	1	1	yes	maximal, characteristic
6	#13116432	C₂	1	2	yes	maximal
7	C₂×C₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
8	#11719175	C₄	1	1	yes	characteristic
9	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	#11718763	C₄	1	2	yes	—
13	#11718762	C₂²	1	2	yes	—
14	C₂×#88052	C₂²	1	2	yes	—
15	#11718585	C₄	1	1	yes	characteristic
16	#11718340	C₄	1	4	yes	—
17	C₈×Dic₂₆	C₂²	1	4	yes	—
18	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
19	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
20	#11718094	C₂²	1	4	yes	—
21	#11718093	C₂²	1	8	yes	—
22	C₁₆×Dic₁₃	C₂²	1	4	yes	—
23	C₄⋊#6090	C₂²	1	4	yes	—
24	C₂²×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
27	#88180	C₂×C₄	1	2	yes	—
28	#88178	C₈	1	4	yes	—
29	C₂×#6099	C₂×C₄	1	2	yes	—
30	#88124	C₈	1	2	yes	—
31	#88123	C₈	1	2	yes	—
32	C₄×Dic₂₆	C₂×C₄	1	4	yes	—
33	C₂×C₁₃×C₁₆	—	2	2	no	—
34	C₂×C₁₃×C₁₆	Q₈	1	4	yes	complemented
35	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
36	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
37	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
38	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₁₃	C₂³	1	2	yes	—
40	#88055	C₂×C₄	1	4	yes	—
41	#88054	C₂×C₄	1	8	yes	—
42	C₈×Dic₁₃	C₂³	1	4	yes	—
43	#88052	—	2	4	no	—
44	C₂×#6090	C₂³	1	4	yes	—
45	C₄⋊#1055	C₂×C₄	1	4	yes	—
46	#33474	—	13	1	no	maximal, Sylow
47	C₂²×C₄×C₁₃	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₂₆	C₂×C₈	1	4	yes	—
50	C₂×C₈×C₁₃	—	2	2	no	—
51	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
52	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
54	C₂×C₈×C₁₃	#41	1	2	yes	—
55	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
56	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
57	C₄²×C₁₃	C₂×C₈	1	1	yes	characteristic
58	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
59	C₂²⋊Dic₁₃	C₂×C₈	1	4	yes	—
60	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
61	C₄⋊Dic₁₃	C₂×C₈	1	4	yes	—
62	#6101	C₂×C₈	1	8	yes	—
63	C₄×Dic₁₃	C₂²×C₄	1	4	yes	—
64	#6099	—	2	4	no	—
65	C₂×#1055	C₂²×C₄	1	4	yes	—
66	C₁₃×C₁₆	—	2	2	no	—
67	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
68	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
69	#6090	—	2	4	no	—
70	#2960	—	13	1	no	—
71	#2183	—	13	4	no	—
72	Q₈×C₁₆	—	13	4	no	—
73	#2151	—	13	2	no	—
74	#2150	—	13	1	no	—
75	#2108	—	13	2	no	—
76	C₂×C₄×C₁₆	D₁₃	1	1	yes	characteristic
77	C₂³×C₁₃	C₂²×C₈	1	1	yes	characteristic
78	C₂×C₄×C₁₃	—	2	4	no	—
79	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
80	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
81	C₂×C₄×C₁₃	C₂²×C₈	1	1	yes	characteristic
82	C₂×C₄×C₁₃	C₂²×C₈	1	2	yes	—
83	C₂×C₄×C₁₃	#131	1	2	yes	—
84	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
85	C₂×Dic₁₃	C₂²×C₈	1	8	yes	—
86	Dic₂₆	—	2	8	no	—
87	C₈×C₁₃	—	2	2	no	—
88	C₈×C₁₃	C₄×Q₈	1	2	yes	—
89	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
90	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
91	#1055	—	2	4	no	—
92	#533	—	13	1	no	—
93	C₂×M₃₂	—	13	2	no	—
94	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
95	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
96	#446	—	13	4	no	—
97	C₈×Q₈	—	13	4	no	—
98	#424	—	13	2	no	—
99	#423	—	13	1	no	—
100	#405	—	13	2	no	—
101	C₂×C₄×C₈	C₂×D₁₃	1	1	yes	characteristic
102	C₄⋊C₁₆	—	13	8	no	—
103	C₄⋊C₁₆	—	13	4	no	—
104	#346	—	13	4	no	—
105	C₄×C₁₆	—	13	4	no	—
106	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
107	C₄×C₁₆	C₂×D₁₃	1	2	yes	—
108	C₂²×C₁₃	—	2	2	no	—
109	C₂²×C₁₃	C₈×Q₈	1	2	yes	—
110	C₂²×C₁₃	C₂³×C₈	1	1	yes	characteristic
111	C₄×C₁₃	—	2	2	no	—
112	C₄×C₁₃	C₈×Q₈	1	2	yes	—
113	C₄×C₁₃	#504	1	4	yes	—
114	C₄×C₁₃	C₂×C₄×Q₈	1	1	yes	characteristic
115	C₄×C₁₃	C₂×#131	1	1	yes	characteristic
116	Dic₁₃	—	2	8	no	—
117	C₂×M₁₆	—	13	2	no	—
118	C₂²×C₈	C₄×D₁₃	1	1	yes	characteristic
119	C₂²×C₈	C₂²×D₁₃	1	1	yes	characteristic
120	C₄⋊Q₈	—	13	2	no	—
121	#125	—	13	2	no	—
122	C₂²⋊Q₈	—	13	4	no	—
123	C₄×Q₈	—	13	4	no	—
124	#117	—	13	2	no	—
125	C₂×C₄²	C₄×D₁₃	1	1	yes	characteristic
126	M₃₂	—	26	4	no	—
127	C₂×C₁₆	—	2	2	no	—
128	C₂×C₁₆	—	13	4	no	—
129	C₂×C₁₆	Dic₂₆	1	4	yes	complemented
130	C₂×C₁₆	C₂²×D₁₃	1	2	yes	—
131	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
132	C₂×C₁₆	C₂²×D₁₃	1	1	yes	characteristic
133	C₄⋊C₈	—	13	8	no	—
134	C₄⋊C₈	—	13	4	no	—
135	#97	—	13	4	no	—
136	C₄×C₈	—	13	4	no	—
137	C₄×C₈	C₄×D₁₃	1	2	yes	—
138	C₄×C₈	C₂²×D₁₃	1	2	yes	—
139	C₂×C₁₃	—	2	2	no	—
140	C₂×C₁₃	C₂×C₈×Q₈	1	1	yes	characteristic
141	C₂×C₁₃	C₂×#504	1	2	yes	—
142	C₂×Q₈	—	13	4	no	—
143	C₂²×C₄	C₈×D₁₃	1	2	yes	—
144	C₂²×C₄	C₂×C₄×D₁₃	1	1	yes	characteristic
145	M₁₆	—	26	4	no	—
146	C₂×C₈	—	2	2	no	—
147	C₂×C₈	—	13	4	no	—
148	C₂×C₈	C₂×Dic₂₆	1	2	yes	—
149	C₂×C₈	C₂×C₄×D₁₃	1	2	yes	—
150	C₂×C₈	C₂×C₄×D₁₃	1	1	yes	characteristic
151	C₂×C₈	#6127	1	2	yes	—
152	C₂×C₈	C₂³×D₁₃	1	1	yes	characteristic
153	C₄⋊C₄	—	13	4	no	—
154	C₄⋊C₄	—	13	8	no	—
155	C₄⋊C₄	—	13	4	no	—
156	C₂²⋊C₄	—	13	4	no	—
157	C₄²	—	13	4	no	—
158	C₄²	C₈×D₁₃	1	1	yes	characteristic
159	C₄²	C₈×D₁₃	1	1	yes	characteristic
160	C₄²	C₂×C₄×D₁₃	1	2	yes	—
161	C₁₆	—	2	2	no	—
162	C₁₆	—	26	4	no	—
163	C₁₆	C₂×Dic₂₆	1	2	yes	—
164	C₁₆	C₂×Dic₂₆	1	2	yes	—
165	C₁₃	#33474	1	1	yes	characteristic, complemented, Sylow
166	C₂³	C₂×C₈×D₁₃	1	1	yes	characteristic
167	Q₈	—	26	8	no	—
168	C₂×C₄	—	2	4	no	—
169	C₂×C₄	—	13	8	no	—
170	C₂×C₄	C₄×Dic₂₆	1	2	yes	—
171	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
172	C₂×C₄	C₂×C₈×D₁₃	1	2	yes	—
173	C₂×C₄	C₂×C₈×D₁₃	1	1	yes	characteristic
174	C₂×C₄	#88155	1	2	yes	—
175	C₂×C₄	C₂²×C₄×D₁₃	1	1	yes	characteristic
176	C₈	—	2	2	no	—
177	C₈	—	26	4	no	—
178	C₈	C₄×Dic₂₆	1	2	yes	—
179	C₈	C₂²×Dic₂₆	1	1	yes	characteristic
180	C₈	C₂×#6127	1	1	yes	characteristic
181	C₂²	—	2	2	no	—
182	C₂²	C₈×Dic₂₆	1	2	yes	—
183	C₂²	C₂²×C₈×D₁₃	1	1	yes	characteristic
184	C₄	—	2	2	no	—
185	C₄	—	26	8	no	—
186	C₄	C₈×Dic₂₆	1	2	yes	—
187	C₄	#11718562	1	4	yes	—
188	C₄	C₂×C₄×Dic₂₆	1	1	yes	characteristic
189	C₄	C₂×#88155	1	1	yes	characteristic
190	C₂	—	2	2	no	—
191	C₂	C₂×C₈×Dic₂₆	1	1	yes	characteristic
192	C₂	C₂×#11718562	1	2	yes	—

	subgroup	G/H
center	C₂×C₁₆	C₂²×D₁₃
commutator, socle	C₂²×C₁₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₁₃
Fitting	C₂×C₄×C₁₃×C₁₆	C₂
prime radical	C₂³×C₁₃	C₂²×C₈
cosocle	C₂×C₈×C₁₃	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₁₃
hypocenter	C₁₃	#33474
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #13116744	C₂	semidirect
3. C₁₆×Dic₂₆	C₂	semidirect
34. C₂×C₁₃×C₁₆	Q₈	semidirect
129. C₂×C₁₆	Dic₂₆	semidirect
165. C₁₃	#33474	semidirect

Maximal subgroups

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

subgroup	index
#13123770	2
#13116744	2
C₁₆×Dic₂₆	2
#13116594	2
#13116592	2
#13116432	2
C₂×C₄×C₁₃×C₁₆	2
#33474	13