Subgroups — C₂×#88324⋊C₁₆ #28307763

Order 13568.

350 conjugacy classes of subgroups.

The subgroup lattice has 182 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811509	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×#88324	C₂	1	1	yes	maximal, characteristic
3	#88324⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719910	C₄	1	2	yes	—
6	C₂×C₄×#88324	C₄	1	1	yes	characteristic
7	C₂×C₄×#88324	C₄	1	1	yes	characteristic
8	#16811509	C₄	1	2	yes	—
9	#16811509	C₂²	1	2	yes	—
10	C₈×#88324	C₄	1	2	yes	—
11	C₈×#88324	C₂²	1	2	yes	—
12	C₂²×#11719908	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₁₆×Dic₅₃	C₄	1	2	yes	—
15	C₁₆×Dic₅₃	C₂²	1	2	yes	—
16	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
17	C₂²×#88324	C₈	1	2	yes	—
18	C₂²×#88324	C₈	1	2	yes	—
19	C₂×#11719910	—	2	4	no	—
20	C₂×#11719910	C₂×C₄	1	4	yes	—
21	C₄×#88324	C₈	1	2	yes	—
22	C₄×#88324	C₈	1	2	yes	—
23	C₄×#88324	C₂×C₄	1	2	yes	—
24	C₄×#88324	C₂×C₄	1	2	yes	—
25	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#11719908	—	2	2	no	—
28	C₂×#11719908	C₂×C₄	1	2	yes	—
29	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
33	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₁₆×C₅₃	—	2	2	no	—
35	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
37	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
38	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
39	C₂×#88324	—	2	4	no	—
40	C₂×#88324	C₁₆	1	8	yes	complemented
41	C₂×#88324	C₂×C₈	1	4	yes	—
42	C₂×#88324	C₂×C₈	1	4	yes	—
43	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
46	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
48	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
49	C₄×Dic₅₃	C₂×C₈	1	1	yes	characteristic
50	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₂×#88322	C₄²	1	1	yes	characteristic
52	C₂×#88322	M₁₆	1	4	yes	—
53	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
54	#11719910	—	2	4	no	—
55	C₁₆×C₅₃	—	2	2	no	—
56	#11719908	—	2	2	no	—
57	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	#97	1	2	yes	—
60	C₂×C₄×C₅₃	#97	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
62	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
63	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
64	C₂×Dic₅₃	C₂×C₁₆	1	4	yes	—
65	C₂×Dic₅₃	M₃₂	1	4	yes	—
66	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
67	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
68	#88324	—	2	4	no	—
69	#88324	C₂×C₁₆	1	8	yes	complemented
70	C₈×C₅₃	#97	1	2	yes	—
71	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
72	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
73	#88322	#97	1	2	yes	—
74	#88322	C₂×M₁₆	1	2	yes	—
75	C₂×#1312	—	53	1	no	maximal, Sylow
76	C₂²×C₅₃	#322	1	2	yes	—
77	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
78	C₂²×C₅₃	#346	1	2	yes	—
79	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
80	C₄×C₅₃	#322	1	2	yes	—
81	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
82	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
83	Dic₅₃	C₄×C₁₆	1	2	yes	—
84	Dic₅₃	#346	1	2	yes	—
85	Dic₅₃	C₂²×C₁₆	1	2	yes	—
86	Dic₅₃	C₂×M₃₂	1	2	yes	—
87	C₂×C₄×C₁₆	—	53	1	no	—
88	C₂×C₄×C₁₆	—	53	1	no	—
89	C₂×C₈²	—	53	1	no	—
90	#1312	—	53	4	no	—
91	C₂×C₅₃	#1312	1	4	yes	complemented
92	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
93	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
94	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
95	C₂²×C₁₆	—	53	2	no	—
96	C₂²×C₁₆	—	53	1	no	—
97	C₂²×C₁₆	#6157	1	1	yes	characteristic
98	C₂×C₄×C₈	—	53	1	no	—
99	C₂×C₄×C₈	—	53	1	no	—
100	C₂×C₄×C₈	—	53	1	no	—
101	C₄×C₁₆	—	53	2	no	—
102	C₄×C₁₆	—	53	2	no	—
103	C₄×C₁₆	—	53	2	no	—
104	C₄×C₁₆	—	53	2	no	—
105	C₈²	—	53	2	no	—
106	C₈²	—	53	2	no	—
107	C₅₃	C₂×#1312	1	1	yes	characteristic, complemented, Sylow
108	C₂²×C₈	—	53	2	no	—
109	C₂²×C₈	—	53	1	no	—
110	C₂²×C₈	—	53	2	no	—
111	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
112	C₂×C₄²	—	53	1	no	—
113	C₂×C₁₆	—	2	2	no	—
114	C₂×C₁₆	—	53	4	no	—
115	C₂×C₁₆	—	53	2	no	—
116	C₂×C₁₆	—	106	4	no	—
117	C₂×C₁₆	—	106	2	no	—
118	C₂×C₁₆	C₂×#6157	1	2	yes	—
119	C₄×C₈	—	53	2	no	—
120	C₄×C₈	—	53	2	no	—
121	C₄×C₈	—	53	2	no	—
122	C₄×C₈	—	53	2	no	—
123	C₄×C₈	—	53	1	no	—
124	C₄×C₈	—	53	1	no	—
125	C₄×C₈	—	53	1	no	—
126	C₄×C₈	—	53	1	no	—
127	C₂²×C₄	—	53	1	no	—
128	C₂²×C₄	—	53	1	no	—
129	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
130	C₂×C₈	—	53	8	no	—
131	C₂×C₈	—	53	4	no	—
132	C₂×C₈	—	53	4	no	—
133	C₂×C₈	—	53	1	no	—
134	C₂×C₈	—	53	1	no	—
135	C₂×C₈	—	53	4	no	—
136	C₂×C₈	—	106	4	no	—
137	C₂×C₈	#11719936	1	2	yes	—
138	C₂×C₈	C₄×#6157	1	1	yes	characteristic
139	C₂×C₈	#11719940	1	2	yes	—
140	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
141	C₄²	—	53	1	no	—
142	C₄²	—	53	1	no	—
143	C₄²	—	53	1	no	—
144	C₄²	—	53	1	no	—
145	C₁₆	—	2	2	no	—
146	C₁₆	—	106	4	no	—
147	C₁₆	—	106	2	no	—
148	C₂³	C₈×#6157	1	1	yes	characteristic
149	C₂×C₄	—	53	1	no	—
150	C₂×C₄	—	53	1	no	—
151	C₂×C₄	—	53	1	no	—
152	C₂×C₄	—	53	1	no	—
153	C₂×C₄	—	53	4	no	—
154	C₂×C₄	—	53	4	no	—
155	C₂×C₄	C₈×#6157	1	1	yes	characteristic
156	C₂×C₄	#13133271	1	2	yes	—
157	C₂×C₄	#13133272	1	2	yes	—
158	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
159	C₈	—	53	8	no	—
160	C₈	—	53	2	no	—
161	C₈	—	53	2	no	—
162	C₈	—	106	4	no	—
163	C₈	#13133272	1	2	yes	—
164	C₈	C₂×#11719940	1	1	yes	characteristic
165	C₈	C₂×#11719936	1	1	yes	characteristic
166	C₂²	#88324⋊C₈	1	2	yes	—
167	C₂²	C₁₆×#6157	1	2	yes	—
168	C₂²	#16811510	1	2	yes	—
169	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
170	C₄	—	53	2	no	—
171	C₄	—	53	2	no	—
172	C₄	—	53	2	no	—
173	C₄	—	53	2	no	—
174	C₄	#88324⋊C₈	1	2	yes	—
175	C₄	C₂×#13133271	1	1	yes	characteristic
176	C₄	C₂×#13133272	1	1	yes	characteristic
177	C₂	#88324⋊C₁₆	1	4	yes	direct product factor
178	C₂	C₂×#88324⋊C₈	1	1	yes	characteristic
179	C₂	C₂×C₁₆×#6157	1	1	yes	characteristic
180	C₂	C₂×#16811510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle, prime radical	C₂³×C₅₃	C₄×C₈
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#1312
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. #88324⋊C₁₆	C₂	direct (N × H)
40. C₂×#88324	C₁₆	semidirect
69. #88324	C₂×C₁₆	semidirect
91. C₂×C₅₃	#1312	semidirect
107. C₅₃	C₂×#1312	semidirect
177. C₂	#88324⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16811509	2
C₂×C₈×#88324	2
#88324⋊C₁₆	2
C₂×C₁₆×Dic₅₃	2
C₂×#1312	53

Subgroups — C2×#88324⋊C16 #28307763

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#88324⋊C₁₆ #28307763