Subgroups — #16786591

Order 3328.

350 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×D₁₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₈×D₁₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#1057⋊C₁₆	C₂	1	4	yes	maximal, complemented
4	C₄×C₁₆×D₁₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₁₃⋊C₁₆	C₄	1	2	yes	—
6	C₄×D₁₃⋊C₈	C₄	1	1	yes	characteristic
7	C₄×D₁₃⋊C₈	C₄	1	1	yes	characteristic
8	C₂×C₁₆×D₁₃	C₄	1	2	yes	—
9	C₄×C₈×D₁₃	C₂²	1	1	yes	characteristic
10	#11718257	C₂²	1	2	yes	—
11	C₈×#1057	C₄	1	2	yes	—
12	C₈×#1057	C₂²	1	2	yes	—
13	C₄×#6092	C₄	1	2	yes	—
14	C₄×C₁₃×C₁₆	C₄	1	1	yes	characteristic
15	C₁₆×Dic₁₃	C₂²	1	2	yes	—
16	C₄×#6090	C₄	1	1	yes	characteristic
17	C₂×D₁₃⋊C₈	C₈	1	2	yes	—
18	C₂×D₁₃⋊C₈	C₈	1	2	yes	—
19	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×D₁₃	C₂×C₄	1	1	yes	characteristic
21	C₄²×D₁₃	C₂×C₄	1	1	yes	characteristic
22	C₄×#1057	C₈	1	2	yes	—
23	C₄×#1057	C₈	1	2	yes	—
24	C₄×#1057	C₂×C₄	1	2	yes	—
25	C₄×#1057	C₂×C₄	1	2	yes	—
26	C₂×#6092	C₂×C₄	1	4	yes	—
27	D₁₃⋊C₁₆	—	2	4	no	—
28	C₂×C₁₃×C₁₆	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₁₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₁₃	C₂³	1	1	yes	characteristic
32	C₂×#6090	C₂×C₄	1	2	yes	—
33	C₄×#1055	C₂×C₄	1	1	yes	characteristic
34	C₁₆×D₁₃	—	2	4	no	—
35	C₂×#1312	—	13	1	no	maximal, Sylow
36	C₂×C₄×D₁₃	C₄²	1	1	yes	characteristic
37	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
38	C₂×C₄×D₁₃	C₂×C₈	1	1	yes	characteristic
39	C₂×#1057	C₂×C₈	1	4	yes	—
40	C₂×#1057	C₂×C₈	1	4	yes	—
41	D₁₃⋊C₈	—	2	4	no	—
42	D₁₃⋊C₈	C₁₆	1	8	yes	complemented
43	C₂×C₈×C₁₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₁₃	C₄²	1	1	yes	characteristic
46	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
47	C₄×Dic₁₃	C₂×C₈	1	1	yes	characteristic
48	C₄×Dic₁₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×#1055	C₄²	1	1	yes	characteristic
50	C₂×#1055	C₂²×C₄	1	1	yes	characteristic
51	C₈×D₁₃	M₁₆	1	4	yes	—
52	C₈×D₁₃	M₁₆	1	4	yes	—
53	#6092	—	2	4	no	—
54	C₁₃×C₁₆	—	2	2	no	—
55	#6090	—	2	2	no	—
56	C₂×C₄×C₁₆	—	13	1	no	—
57	C₂×C₄×C₁₆	—	13	1	no	—
58	C₂×C₈²	—	13	1	no	—
59	#1312	—	13	4	no	—
60	C₂²×D₁₃	C₄×C₈	1	1	yes	characteristic
61	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
62	C₂×C₄×C₁₃	C₄×C₈	1	1	yes	characteristic
63	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
64	C₂×Dic₁₃	C₄×C₈	1	1	yes	characteristic
65	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
66	C₂×Dic₁₃	C₂²×C₈	1	1	yes	characteristic
67	C₄×D₁₃	#97	1	2	yes	—
68	C₄×D₁₃	#97	1	2	yes	—
69	C₄×D₁₃	C₂×C₁₆	1	4	yes	—
70	C₄×D₁₃	M₃₂	1	4	yes	—
71	#1057	—	2	4	no	—
72	#1057	C₂×C₁₆	1	8	yes	complemented
73	C₈×C₁₃	#97	1	2	yes	—
74	C₈×C₁₃	C₂×M₁₆	1	2	yes	—
75	#1055	#97	1	2	yes	—
76	#1055	C₂×M₁₆	1	2	yes	—
77	C₂²×C₁₆	—	13	2	no	—
78	C₂²×C₁₆	—	13	2	no	—
79	C₂×C₄×C₈	—	13	1	no	—
80	C₂×C₄×C₈	—	13	1	no	—
81	C₂×C₄×C₈	—	13	1	no	—
82	C₄×C₁₆	—	13	2	no	—
83	C₄×C₁₆	—	13	2	no	—
84	C₄×C₁₆	—	13	2	no	—
85	C₄×C₁₆	—	13	1	no	—
86	C₄×C₁₆	#261	1	1	yes	characteristic
87	C₈²	—	13	2	no	—
88	C₈²	—	13	2	no	—
89	C₂²×C₁₃	C₂×C₄×C₈	1	1	yes	characteristic
90	C₂×D₁₃	#322	1	2	yes	—
91	C₂×D₁₃	C₄×C₁₆	1	2	yes	—
92	C₂×D₁₃	#346	1	2	yes	—
93	C₄×C₁₃	C₄×C₁₆	1	2	yes	—
94	C₄×C₁₃	#346	1	2	yes	—
95	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
96	C₄×C₁₃	C₂×#97	1	1	yes	characteristic
97	Dic₁₃	#322	1	2	yes	—
98	Dic₁₃	C₂²×C₁₆	1	2	yes	—
99	Dic₁₃	C₂×M₃₂	1	2	yes	—
100	C₂²×C₈	—	13	2	no	—
101	C₂²×C₈	—	13	1	no	—
102	C₂²×C₈	—	13	1	no	—
103	C₂²×C₈	—	13	2	no	—
104	C₂×C₄²	—	13	1	no	—
105	C₂×C₁₆	—	13	4	no	—
106	C₂×C₁₆	—	13	2	no	—
107	C₂×C₁₆	—	26	4	no	—
108	C₂×C₁₆	—	26	4	no	—
109	C₂×C₁₆	C₂×#261	1	2	yes	—
110	C₄×C₈	—	13	2	no	—
111	C₄×C₈	—	13	2	no	—
112	C₄×C₈	—	13	2	no	—
113	C₄×C₈	—	13	2	no	—
114	C₄×C₈	—	13	1	no	—
115	C₄×C₈	—	13	1	no	—
116	C₄×C₈	—	13	1	no	—
117	C₄×C₈	C₂×#261	1	1	yes	characteristic
118	C₂×C₁₃	C₂×#322	1	1	yes	characteristic
119	C₂×C₁₃	C₂×C₄×C₁₆	1	1	yes	characteristic
120	C₂×C₁₃	C₂×#346	1	1	yes	characteristic
121	D₁₃	#1312	1	4	yes	complemented
122	C₂²×C₄	—	13	1	no	—
123	C₂²×C₄	—	13	1	no	—
124	C₂²×C₄	—	13	1	no	—
125	C₂×C₈	—	13	4	no	—
126	C₂×C₈	—	13	8	no	—
127	C₂×C₈	—	13	1	no	—
128	C₂×C₈	—	13	1	no	—
129	C₂×C₈	—	13	4	no	—
130	C₂×C₈	—	13	4	no	—
131	C₂×C₈	—	13	4	no	—
132	C₂×C₈	—	26	4	no	—
133	C₂×C₈	C₄×#261	1	1	yes	characteristic
134	C₂×C₈	C₂²×#261	1	1	yes	characteristic
135	C₄²	—	13	1	no	—
136	C₄²	—	13	1	no	—
137	C₄²	—	13	1	no	—
138	C₄²	C₄×#261	1	1	yes	characteristic
139	C₁₆	—	2	2	no	—
140	C₁₆	—	26	4	no	—
141	C₁₆	—	26	2	no	—
142	C₁₃	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
143	C₂³	—	13	1	no	—
144	C₂×C₄	—	13	1	no	—
145	C₂×C₄	—	13	1	no	—
146	C₂×C₄	—	13	1	no	—
147	C₂×C₄	—	13	4	no	—
148	C₂×C₄	—	13	4	no	—
149	C₂×C₄	—	13	2	no	—
150	C₂×C₄	—	13	2	no	—
151	C₂×C₄	C₈×#261	1	1	yes	characteristic
152	C₂×C₄	C₈×#261	1	1	yes	characteristic
153	C₂×C₄	C₂×C₄×#261	1	1	yes	characteristic
154	C₈	—	13	8	no	—
155	C₈	—	13	2	no	—
156	C₈	—	13	2	no	—
157	C₈	—	26	4	no	—
158	C₈	#88100	1	2	yes	—
159	C₈	D₁₃⋊M₁₆	1	2	yes	—
160	C₂²	—	13	2	no	—
161	C₂²	—	13	2	no	—
162	C₂²	—	13	2	no	—
163	C₂²	C₂×C₈×#261	1	1	yes	characteristic
164	C₄	—	13	2	no	—
165	C₄	—	13	2	no	—
166	C₄	—	13	2	no	—
167	C₄	C₁₆×#261	1	2	yes	—
168	C₄	#11718216	1	2	yes	—
169	C₄	#11719047	1	1	yes	characteristic
170	C₄	C₂×#88100	1	1	yes	characteristic
171	C₂	—	13	4	no	—
172	C₂	#13130032	1	1	yes	characteristic
173	C₂	C₂×C₁₆×#261	1	1	yes	characteristic
174	C₂	C₂×#11718216	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈	C₂×#261
commutator	C₂×C₁₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#261
Fitting	C₄×C₁₃×C₁₆	C₄
socle	C₂²×C₁₃	C₂×C₄×C₈
prime radical	C₂²×D₁₃	C₄×C₈
cosocle	C₈×Dic₁₃	C₂³
hypercenter	C₄×C₁₆	#261
hypocenter	C₁₃	C₂×#1312
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
3. #1057⋊C₁₆	C₂	semidirect
42. D₁₃⋊C₈	C₁₆	semidirect
72. #1057	C₂×C₁₆	semidirect
121. D₁₃	#1312	semidirect
142. C₁₃	C₂×#1312	semidirect

Maximal subgroups

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

subgroup	index
C₄×D₁₃⋊C₁₆	2
C₈×D₁₃⋊C₈	2
#1057⋊C₁₆	2
C₄×C₁₆×D₁₃	2
C₂×#1312	13